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Some fixed point theorems for nonexpansive mappings in Hausdorff locally convex spaces Tan, Kok Keong 1970

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SOME FIXED POINT THEOREMS FOR NONEXPANSIVE MAPPINGS IN HAUSDORFF LOCALLY CONVEX SPACES by KOK-KEONG TAN B.S c , Nanyang Un i v e r s i t y , 1966 . A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE"REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of MATHEMATICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1970 I n p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r ag ree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l no t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment o f The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada Supervisor: Professor L. P. Belluce 11 ABSTRACT Let X be a Hausdorff locally convex space, U be a base for closed absolutely convex O-neighborhoods in X , K C X be nonempty. For each U e U , we denote by P^  the gauge of . U . Then T : K -*• K is said to be nonexpansive w.r.t. U i f and only i f for each U e U , Pu(T(x) - T(y)) <_ Py(x - y) for a l l x, y e K ; T : K -»• K is said to be strictly contractive w.r.t. U i f and. only i f for each U e U , there is a constant X with 0 < X < 1 such that u — u Pu(T(x) - T(y)) _< X uP u(x - y) for a l l x, y e K . The concept of nonexpansive (respectively strictly contractive) mappings is originally defined on a metric space. The above definitions are generalizations i f the topology on X is induced by a translation invariant metric, and in particular i f X is a normed space. An analogue of the Banach contraction mapping principle is proved and some examples together with an implicit function theorem are shown as applications. Moreover several fixed point theorems for various kinds of nonexpansive mappings are obtained. The convergence of nets of nonexpansive mappings and that of fixed points are also studied. Finally on sets with 'complete normal structure', a common fixed point theorem is obtained for an arbitrary family of 'commuting' nonexpansive mappings while on sets with 'normal structure', a common fixed point, theorem for an arbitrary family of (not necessarily commuting) 'weakly periodic' nonexpansive mappings is obtained. i i i TABLE OF CONTENTS Page INTRODUCTION , : . 1 CHAPTER I : FIXED POINT THEOREMS FOR CONTRACTIVE MAPPINGS 1-1 : D e f i n i t i o n s and notations 4 1-2 : The Banach contraction mapping p r i n c i p l e . . 6 1-3 : Ite r a t i o n s of contractive mappings. 10 1-4 : Some examples and app l i c a t i o n s 17 1-5 : Nets of contractive mappings. . 27 CHAPTER II : NORMAL STRUCTURE I I - l : Center of a set. 36 II-2 : Normal structure 39 II-3 : Some fi x e d point theorems 45 II-4 : Complete normal structure. 51 CHAPTER I I I : FIXED POINT THEOREMS FOR NONEXPANSIVE MAPPINGS I I I - l : Relative nonexpansive mappings 60 III-2 : A f f i n e mappings and convex mappings 64 III-3 : Mappings with diminishing o r b i t a l diameters 71 III-4 : Bounded mappings 83 III-5 : Weakly p e r i o d i c mappings 87 BIBLIOGRAPHY 92 i v ACKNOWLEDGEMENTS I am greatly indebted to Professor L. P. Belluce for h i s encouragement, guidance and invaluable suggestions during the preparation of this t h e s i s . I also wish to thank Professors C. W. Clark, J. J . F. Fournier, D. T, C. Bures and W. E. Meyers f o r th e i r c a r e f u l reading of th i s t h e s i s . The f i n a n c i a l support of the National Research Council of Canada and the Univ e r s i t y of B r i t i s h Columbia i s g r a t e f u l l y acknowledged. 1. INTRODUCTION Nonexpansive mappings are mappings T : K -> K , where K i s a nonempty subset of a metric space X with metric d , such that d(T(x), T(y)) <_ d(x, y) , for a l l x,y E K . Let H be a H i l b e r t space, (A(t) : T >_ 0} be a family of closed l i n e a r operators on H , f be a mapping of H + x H into H and A(t) , f ( t , u(t)) be p e r i o d i c i n t with a common period E, > 0 . Suppose + A(t)u(t) = f ( t , u(t)) ( t > 0 ) (1) i s a time-dependent non-linear equation of evolution i n H s a t i s f y i n g some conditions. If R i s a s u i t a b l y chosen nonnegative number, f o r each v e B = {v e H : ||v|| <_ R} , define T(v) = u(£) , where u(t) i s the "mild s o l u t i o n " of equation (1) such that u(0) = v . Then F. E. Browder proved i n [5] that T i s a nonexpansive mapping from B into B ,'and moreover, each f i x e d point of T corresponds to a p e r i o d i c s o l u t i o n of equation (1) with period E, . Since nonexpansive mappings originated with and as noted above, have applications to d i f f e r e n t i a l and f u n c t i o n a l equations, the existence of f i x e d points for such mappings has been widely studied. However most of the research so far has been done i n the Banach and metric space s e t t i n g . W. A. K i r k proved i n [12] that i f T : K -> K i s nonexpansive, where K i s a nonempty weakly compact convex subset of a Banach space X and K has "normal structure", then T has a fixed point i n K . If we renorm the space X by an equivalent norm, then T needs not be nonexpansive arid K needs not have "normal structure" with respect to the new norm yet T s t i l l has a f i x e d point i n K . This observation leads us to consider the r e l a t i o n s h i p of a given mapping T and the topology of the space (independent of the norm structure) and to work on neighborhoods instead of norms. This i s the main purpose of t h i s work. In the f i r s t chapter, the notion of a nonexpansive (respectively contractive, s t r i c t l y contractive) mapping i s defined i n a Hausdorff l o c a l l y convex space such that i t w i l l be a ge n e r a l i z a t i o n i f the topology of the space i s induced by a t r a n s l a t i o n i n v a r i a n t metric. Next an analogue of the Banach contraction mapping p r i n c i p l e i s proved and some examples together with an i m p l i c i t function theorem are shown as ap p l i c a t i o n s . Furthermore, convergence of (subsequence of) i t e r a t i o n s of contractive mappings are studied and a r e s u l t of M. E d e l s t e i n i n [9] i s generalized. F i n a l l y we discuss the convergence of a net of contractive mappings and the convergence of a net of fixed points of nonexpansive mappings In the second chapter, we f i r s t study the "center" of a set and then we generalize the notion of "normal str u c t u r e " introduced by M. S. B r o d s k i i and D. P. Milman i n [4] and the notion of complete normal structure introduced by L. P. Belluce and W. A. Kirk i n [ 2 ] . I t i s proved that each compact convex subset of a Hausdorff l o c a l l y convex space has complete normal structure (and so has normal structure) i n our sense. Some r e s u l t s of R. DeMarr i n [6], of L. P. Belluce and W. A. K i r k i n [2] and of L. P. Belluce, W. A. Kirk and E. F. Steiner i n [3] are extended i n our sense. In the t h i r d and f i n a l chapter, we study the fixed point theorems for various kinds of nonexpansive mappings. The notions of nonexpansive mappings with diminishing o r b i t a l diameters and bounded mappings were introduced by L. P. Belluce and W. A. K i r k i n [1] and by W. A. K i r k i n [13] r e s p e c t i v e l y . These notions have been generalized into Hausdorff l o c a l l y convex spaces and much of t h e i r r e s u l t s have been extended i n our general s e t t i n g . The notion of convex mappings i s also generalized and some fi x e d point theorems f o r convex nonexpansive mappings are obtained. F i n a l l y we prove a common fi x e d point theorem f o r an a r b i t r a r y family of (not nec e s s a r i l y "commuting") "weakly p e r i o d i c " (and also periodic) nonexpansive mappings. 4. CHAPTER I FIXED POINT THEOREMS FOR CONTRACTIVE MAPPINGS 1-1. Definitions and notations. In this chapter, X denotes a real or complex Hausdorff locally convex space (T^-l.c.s.) and U denotes a base for the closed absolutely convex neighberhoods of zero (0-nbhds). If U is any closed absolutely convex 0-nbhd, we denote by P^  the gauge of U defined by Pu(x) = inf{X >0 : x e XU} for a l l x e X . If K C X , we denote by K the closure of K , Co(K) the convex hull of K , K1 the interior of K . We shall refer to Robertson [14] and Kelley and Naimioka [11] for properties of gauge function and further notations used thereafter. Definition 1.1. If K C X is non-empty, then a mapping T : K -»• K is said to be nonexpansive with respect to (w.r.t.) U i f and only i f for each U e U, p (T(x) - T(y)) <_? (x - y) for a l l x,y e K . u u ^ Definition 1.2. If K C X is non-empty, then a mapping T : K -*• K is said to be contractive w.r.t. i f and only i f T is nonexpansive w.r.t. U and for each U e U and for any x,y e K , i f P^(x - y) > 0 , then Pu(T(x) - T(y)) < P u(x - y) . 5. l Definition 1.3. If K C X is non-empty, then a mapping T : K -»• K is said to be strictly contractive w.r.t. U i f and only i f for any U e U , there is a constant X such that 0 < X < 1 and u — u P (T(x) - T(y)) < X P (x - y) for a l l x,y e K . U — U U I If K C X is non-empty and T : K -*• K , i t is clear that T is strictly contractive w.r.t. ^ implies T is contractive w.r.t. U which in turn implies T is nonexpansive w.r.t. U and which in turn implies T is continuous. Remark 1.4. If the topology on X is induced by a translation invariant metric d and KC X is non-empty, let = {x e X : d(x,0) <_ r} and U = {B^  : r runs through a net of positive numbers tending to 0} , then a mapping T : K -> K is nonexpansive (respectively contractive and strictly contractive) w.r.t. U i f and only i f d(T(x), T(y)) <_d(x, y) (respectively x ^  y implies d(T(x), T(y)) < d(x, y) and there is a constant X such that 0 <_ X < 1 and d(T(x), T(y)) <_ Xd(x, y) for a l l x,y e K). Remark 1.5. Suppose K C X is nonempty and T : K •> K is nonexpansive w.r.t. ^ . If ^ is the set of a l l closed absolutely convex 0-nbhds in X , i t is clear that T is weakly continuous. 6. Remark 1.6. Let K C X be nonempty and T : K ->• K be nonexpansive w.r.t. U . I f X i s normable, i t i s not known whether there e x i s t s a norm [[ || on X inducing the same topology on X such that ||T(x) - T(y)|| < I  x - y|| , f o r a l l x,y e K . 1—2. The Banach contraction mapping p r i n c i p l e . D e f i n i t i o n 2.1. ! Let K C X be nonempty, T : K •> K and x e K . Then (i ) x i s said to be a fi x e d point of T_ i f and only i f T(x) = x , and ' ( i i ) x i s said to be a per i o d i c point of T_ i f and only i f there i s a N N N-l p o s i t i v e integer N such that T (x) = x , where T (x) = T(T (x)) and ' T° = I , the i d e n t i t y mapping. \ 1 Proposition 2.2. / I f K C X i s nonempty and T : K -+ K i s contractive w.r.t. U , then a f i x e d point of T , whenever i t e x i s t s , i s unique. Proof: Suppose there were £ and n i n K such that £ ^ n and T(£) = £ and T ( n ) = n . Since X i s Hausdorff, there i s a U £ U with ^ - n i U , and so ^ u(5 - n) > 1 > 0 . Since T i s contractive w.r.t. ^ t and P (£ - n) > 0 , we see that P u ( 5 - n) = Pu(T(?) - T(n)) < P (£ - n) , which i s impossible. Hence a fixed point of T , whenever i t e x i s t s , i s unique. Proposition 2 . 3 . Let K C X be nonempty and T K K be contractive w.r.t. U • Then for any x e K , x i s a f i x e d point of T i f and only i f x i s a p e r i o d i c point of T . Proof: Suppose, x e K i s a pe r i o d i c point of T . Let N be N a p o s i t i v e integer such that T (x) = x . Suppose x ^ T(x) . Then there i s a U e U such that p u ( x - T(x)) > 0 . Since T i s contractive w.r.t. U , we have P u(x - T(x)) > P u(T(x) - T 2 ( x ) ) >_ ... >_P u(T N(x) - T N + 1 ( x ) ) = P (x - T(x)), which i s impossible. Hence we must have x - T(x) . The converse i s obvious. The following theorem generalizes the Banach contraction mapping p r i n c i p l e to T 2 _ l . c . s . Theorem 2 . 4 . I f K C X i s nonempty and sequentially complete and T : K K i s s t r i c t l y contractive w.r.t. U , then T has a unique f i x e d . point, say E, , In K . Moreover l i m T n(x) = 5 for a l l x £ K . Proof: Suppose U £ U . Then there i s a constant A such that 0 <_ A u < 1 and P ( T ( x ) - T(y)) <_ A uP u(x - y) for a l l x,y e K . I f x e K , then P u ( T n + 1 ( x ) - T n ( x ) ) < A u P u ( T n ( x ) - T n X ( x ) ) < X nP (T(x) - x) , — u u for a l l n = 1, 2, . .., and so P ( T n + k ( x ) - T n ( x ) ) < P ( T n + k ( x ) - T n + k _ 1 ( x ) ) + ... + P ( T n + 1 ( x ) - T n ( x ) ) u — u u < X n + k _ 1 P (T(x) - x) + ... + X n P (T(x) - x) — u u u u = P (T(x) - x) {X n + k _ 1 + ... + X n} u u u X n < - r ^ — P (T(x) - x) , — 1-X u u for a l l n,k = 1, 2, ... . Choose a p o s i t i v e integer N such that, N n " P (T(x) - x) < 1 , then for m > n > N , say m = n+k , we have -L-~A U U l P u ( T m ( x ) - T n ( x ) ) = P u ( T n + k ( x ) - T n ( x ) ) x n ^ i r r V T ( x ) - x ) U < 1 , so that T m(x) - T n(x) e U . Thus {T n(x) : n = 0 , 1, 2, . .. } i s a Cauchy sequence i n K and so there i s an £ e K such that l i m T n(x) = £ nnw° Since T i s continuous, we see that T(£) = T'(lim T n ( x ) ) = l i m T n + 1 ( x ) = n-x=° n->*=° X u Hence E, i s a fixed point of T , and so must be unique, by Proposition 2.1. I t i s clear that i f K C X i s nonempty, T : K K i s nonexpansive w.r.t. U and i f x e K and there i s a subsequence n. n. i co IT 0 0 X {T (x)}. , of {T (x)} such that E, = l i m T (x) e x i s t s and i s a 1=1 n=l . i-x» fixed point of T , then l i m T n(x) ex i s t s and i s = E, n-*» Corol l a r y 2.5. Let K C X be nonempty and sequentially complete, and N T : K -* K be continuous. I f there i s a p o s i t i v e integer N such that T i s s t r i c t l y contractive w.r.t. Li , then T has a unique f i x e d point i n K . Moreover, l i m T n(x) = £ , for each x e K . n-*» Proof: Since : K •+ K i s s t r i c t l y contractive w.r.t. Li and N K i s se q u e n t i a l l y complete, T has a unique fixed point £ i n K , and N n lim (T ) (x) = E, for a l l x e K , by Theorem 2.4. Thus T i s continuous n-x» implies T(g) = T ( l i m ( T N ) n ( x ) ) n-x=° = l i m ( T N ) n ( T x ) n-*» = 5 and so E, i s a f i x e d point of T . Since a f i x e d point of T i s also a W fix e d point of T , we see that E, must be the unique f i x e d point of T . Thus i t follows that E, = l i m T n(x) , for each x e K . 10. 1-3• Iterations of contractive mappings. The purpose of this section i s to study the i t e r a t i o n s of nonexpansive mappings and contractive mappings and to discuss when the i t e r a t i o n s w i l l converge to a f i x e d point. D e f i n i t i o n 3.1. Suppose K C X i s nonempty. Then T : K -> K i s an  isometry w.r.t. II i f and only i f for any U e U , P (T(x) - T(y)) = P (x - y) for a l l x,y e K . Proposition 3.2. Let K C X be nonempty, T : K ->- K be nonexpansive T w.r.t. U and K = {x e K : there i s an x i n K such that x i s a o l i m i t point of {T n(x Q) : n = 0 , 1, 2, ...}} . Then f o r any x e K , x i s i n i f and only i f x i s a l i m i t point of {T n(x) : n = 0,1,2,...}. Also T(K T) C K T . Proof: Suppose x e K . I f x i s a l i m i t point of n T T {T (x) : n = 0, 1, 2, ...} , then x e K , by d e f i n i t i o n of K T Conversely l e t us assume that x e K . Then there i s an x e K such o that x i s a l i m i t point of {T n(x Q) : n = 0, 1, 2, ...} . Thus for a r b i t r a r i l y f i x e d U e Li and for any p o s i t i v e integer N , there are p o s i t i v e integers m and n such that n > N , m > N + n , x - T n ( x ) e U "• and x - T m ( x ) e U , and so there e x i s t s a p o s i t i v e o I o l integer k = m - n > N such that 11. P (x - T%)) < P (x - T m(x )) + P (Tm(x ) - T m _ n(x)) u — u o u o < i + P„(T n(x) - x) — I u o 1 1 i l + 2 = 1 so that x - T (x) e U . Hence x is a limit point of (Tn(x) : n = 0, 1, 2, ...} . \ Next since T is nonexpansive w.r.t. U , i t is clear that T(KT) C KT Proposition 3.3. Let K C X be nonempty and T : K ->• K be nonexpansive w.r.t. U . Suppose 0 < m^  < m^  < ... , where nw's are positive integers, m. and S = {x e K : lim T """(x) = x} . Then T(S) CS and T is an i-x» isometry on S w.r.t. U • \ m. Proof: If x e S , then lim' T (x) = x. implies m. Tfx)= lim T 1(T(x)) so that T(x) e S . Thus T(S) CS . 1-H» Next suppose there is a U e U and there are x^,x^ e S such that P (T(x n) - T(x„)) ^ P (x, - x_) . Since T is nonexpansive u 1 2 u 1 2 w.r.t. U , we must have P U(T(x ]-) - T(x2>) < Pu(x - x 2) . Let 6 = P (x, - x„) - P ( T ( X T ) - T(x_)) . Choose positive integer i such u 1 2 u 1 2 o m i 1 that i > i Q implies P U ( T ( Xj) ~ x j ) < ^ f° r e a c n 3 = 1> 2 . Since 12. m. m. 1 / . \ m 1 P (T "(x_) - T (x.)) < P (T(x.) - T(x_)) , we have for i > i u l z — u 1 2 c 6 = P (x - x_) - P (T(x_) - T ( x J ) u i I u 1 I m. m. < P u ( x 1 - x 2) - P u(T 1(x ]_) - T 1 ( x 2 ) ) m. m. < P u ( ( x 1 - x 2) - (T 1 ( x 1 ) - T 1 ( x 2 ) ) ) m. m.. < _ P u ( X l - T 1 ( x 1 ) ) + P u ( x 2 - T 1 ( x 2 ) ) < \ 6 + \ 6 = 6 which i s a cont r a d i c t i o n . Therefore T i s an isometry on S w.r.t. U Theorem 3.4. Let K C. X be nonempty and T : K ->• K be nonexpansive T w.r.t. U . If x e K , then T i s an isometry on (T^x) : n = 0, 1, 2, ...} w.r.t. U . T Proof: Let x e K . Suppose there i s a U £ U and there e x i s t nonnegative integers m and n such that P u ( T m + 1 ( x ) - T n + 1 ( x ) ) ± P u ( T m ( x ) - T n ( x ) ) . Then T i s nonexpansive w.r.t. U implies 6 = P u ( T m ( x ) - T n ( x ) ) - P u ( T m + 1 ( x ) - T n + 1 ( x ) ) > 0 . By. Proposition 3.2, x i s a l i m i t point of {T^x): n = 0, .1, 2, ...} . k 1 Choose a p o s i t i v e integer k such that P u ( x - T (x)) < — 6 . Then 13. P u ( T m ( x ) - T n ( x ) ) < P u(T m(x)-T T [ r i" k(x)) + P u ( T m f k ( x ) - T n + k ( x ) ) + P u ( T n + k ( x ) - T n ( x ) ) £ P u ( x - T k ( x ) ) + P ( T ^ C x ) - T n + 1 ( x ) ) + P u ( T k ( x ) - x) < \ 6 + P u ( T m + 1 ( x ) - T n + 1 ( x ) ) + \ 6 = 6 + P u ( T m f l ( x ) - T n + 1 ( x ) ) = P u ( T m ( x ) - T n ( x ) ) , . which i s impossible. Hence we must have for each U e U and for any nonnegative integers m and n , P u ( T m f l ( x ) - T n + 1 ( x ) ) =.P u(T m(x) - T n ( x ) ) , i . e . T i s an isometry on {T n(x) : n = 0, 1, 2, ...} w.r.t. U . Theorem 3.5. Let K C. X be nonempty and T : K •+ K be contractive T T T w.r.t. U . Then Card(K ) <_1 . In case Card(K ) = 1 , K contains only the unique f i x e d point of T . T T Proof: Suppose K ^ 0 and l e t x e K . I f T(x) ^ x , then there e x i s t s U e U with P (x - T(x)) > 0 . Thus T i s cont r a c t i v e 2 w.r.t. U implies P u(T (x) - T(x)) < P u(T(x) - x) . By Theorem 3.4., T i s an isometry on (T n(x) : n = 0, 1, 2, ...} w.r.t. U , and so we must 2 have P (T (x) - T(x)) = P u(T(x) - x) , which i s a c o n t r a d i c t i o n . Thus T T(x) = x . By Proposition 2.2, K = {x} . Coro l l a r y 3.6. Let K C X be nonempty and T : K -> K be contractive 14. w.r.t. U . Suppose there i s an x e K ' and a sequence of s t r i c t l y n. increasing p o s i t i v e integers 1 < n^ < < ... such that £ - l i m T (x ) i-x» exists i n K , then E, i s the unique f i x e d point of T . Moreover, lim T n ( x ) ex i s t s i n K and E, = lim T n ( x ) o o i T Proof: Since E, = l i m T (x ) ex i s t s i n K , E, e K . Hence i-x» by Theorem 3.5., E, must be the unique fixed point of T i n K .. Next suppose U e U . Then there i s a p o s i t i v e integer n. N such that T x ( x )'e E, + U for a l l i >_ N . Thus, for Si = 0, 1, 2, n +Z ' n M + 5 ' P (5 - T N (x )) = P (T*(£) - T (x )) u o u o n < P (£ - T (x )) — u o <_ 1 nN + J l and so T (X Q ) e E, + U for a l l £ = 0, 1, 2, ... and thus T m ( x Q ) e £ + U for a l l m •> n^ . Hence 5 = l i m T n ( x ) n-x° The above c o r o l l a r y generalizes Theorems 1 of M. E d e l s t e i n [9] to Hausdorff l o c a l l y convex spaces. Co r o l l a r y 3.7. Let K C X be nonempty sequentially compact and T : K ->• K be contractive w.r.t. U . . Then T has a unique f i x e d point, say E, , i n K and • E, = l i m T n(x) for any x £ K . n-x» 15. Cor o l l a r y 3.7 can be s l i g h t l y generalized to a kind of mappings, c a l l e d "asymptotically regular mappings". D e f i n i t i o n 3.8. Let K C X be nonempty. Then T : K -> K i s said to be asymptotically regular i f and only i f l i m (T n(x) - ^"'""'"(x)) = 0 , for each n-x>° x e K . I t i s clear from C o r o l l a r y 3.7. that i f K C X i s nonempty sequentially compact and T : K -> K i s contractive w.r.t. ii , then T i s asymptotically regular. Next we have the following Remark 3.9. Let K C X be nonempty and T : K -> K be s t r i c t l y c o ntractive w.r.t. il . Then for any U £ ii and any x £ K , we have X "1 l i m sup {P (T (x) - T J(x))} = 0 . In p a r t i c u l a r , T i s asymptotically n-*=° i,j>n regular. Proof: I f U £ ii , l e t A be a constant such that 0 < X < 1 . u — u and P (T(x) - T(y)) < A P (x - y) f o r a l l x,y £ K . Suppose x e K and u — u u n i s a p o s i t i v e integer, then sup {P u(T 1(x) - T j ( x ) ) } <_ [(A u n)/(1-A )]P (T(x) - x) , so that i,j>n U u u u l i m sup {P (T X(x) - T j(x)} = 0 . n-*°° i,j>n . J • 16. Proposition 3.10. Let K C X be nonempty seq u e n t i a l l y compact and T : K ->• K be continuous and asymptotically regular. Then there i s an x e K such that T(x) = x . Proof: Suppose x e K . Then there are a subsequence n. n. {T 1 ( x ) } K > . of {Tn(x)}°° n and an z e K such that l i m T 1 ( x ) = z . 1=1 n=± Since T i s asymptotically regular, -lim (T n(x) - T n + ^ ( x ) ) = 0 and so , n-x» n. n.+l X X l i m (T (x) - T (x)) = 0 . Since T i s continuous, we see that i-x» n. n.+l n. T(z) = T(lim T """(x)) =lim T 1 (x) = l i m T 1 ( x ) . = z , and so z i s a f i x e d i - x o i-x» i - x o point of T . Proposition 3.11. Let K C X be nonempty sequentially compact and T : K -»- K be nonexpansive w.r.t. U . I f T i s asymptotically regular, then for each x e K , l i m T n(x) e x i s t s i n K and i s a f i x e d point of T n-*>° n. i oo Proof: For each x £ K , there i s a subsequence {T ( x ) o f h. CO 2_ {T (x)} _ such that l i m T (x) e x i s t s i n K and i s a f i x e d point of T n=l Since T i s nonexpansive w.r.t. U , l i m T n(x) e x i s t s and n. l i m T 1 ( x ) = l i m T n(x) . Thus lim T n(x) i s a fixed point of T . 17. 1-4. Some examples and a p p l i c a t i o n s . In t h i s section, some examples of contractive mappings and s t r i c t l y contractive mappings are shown and some app l i c a t i o n s of the Banach contraction mapping p r i n c i p l e are obtained. Example 4.1. Suppose T : X -> X i s nonexpansive w.r.t. U , X q e X , n i s a p o s i t i v e integer and X i s any s c a l a r . Define S : X -> X by A S,(x) = AT n(x) + x , for' a l l x e X . Then for any U £ U , and for a l l A. O x,y £ X , P (S,(x) - S.(y)) = P (XT n(x) - XT n(y)) < |x|P (x - y) , so that u A A u u S i s nonexpansive w.r.t. U i f |x| ^_ 1 and i s s t r i c t l y contractive w.r.t. U i f ' IXI < 1 . Example 4.2. Let K C X be nonempty convex and T : K -»• K . Suppose n a , a n , .... and a > 0 , n > 1 , and a ^ 0 and / a . = 1 . Define o 1 n — , — n . -, i i = l n S : K K by S (x) = I a.T^x) for a l l x £ K where T° = I , the i=0 1 i d e n t i t y mapping on K . Then T i s nonexpansive (re s p e c t i v e l y contractive, s t r i c t l y contractive) w.r.t. U implies S i s nonexpansive (re s p e c t i v e l y contractive, s t r i c t l y contractive) w.r.t. U D e f i n i t i o n 4.3. If U i s any closed absolutely convex 0-nbhd i n X , and K C X i s nonempty and bounded, we define 18. S (K) = i n f {r > 0 : K - K C r U } u <$u(K) i s c a l l e d the diameter of K w.r.t. U . Remark 4.4. I f U i s any closed absolutely convex 0-nbhd i n X , and K C X i s nonempty and bounded, we see that 6 (K) = sup {P u(x - y) : x,y e K} 6 (K) u = 6 ( C o < K ) ) u = 6 (Co(K)) u I t follows that i f K C H C C o ( K ) , then 6 (K) = 6 (H) Proposition 4.5. Let K C X be nonempty convex, and T : K ->• K be nonexpansive w.r.t. U . Suppose a , a^, >_ 0 , n >_ 1 , a^ > 0 n n and I a. = 1 . Define S : K + K by S(x) = £ a.T 1(x) for a l l x e K i=0 1 i=0 1 Then for any x e K , S(x) = x i f and only i f T(x) = x . Proof: I f x e K and T(x) = x , then 19. S(x) = (a 1 + a nT + ... + a T n ) ( x ) o 1 n = a x + a,T(x) + ... + a T n(x) o 1 n = a x + a n x + . . . + a x o 1 n = x Conversely, suppose S(x) = x . Then x = a x + a nT(x) + ... + a T n(x) implies o 1 n r a 1 a (1 - a )x = a,T(x) + ... + a T n(x) , and so x = -r-±-T(x) + ... + - r - ^ T ^ x ) o 1 n 1-a 1-a o . o Let = a ^ / ( l - a Q) . If 3^ = 1 , then x = T(x) . Thus we may assume 3 1 ^ 1 , and so 0 < &1 < 1 . I t follows that x = g T(x) + (1 - S>±)z , W h e r e Z = (1 - « ) ( 1 - a ) T (x) + ... + _ * _ a T n(x) . Since 1 o 1 o n a. 2 £ / i o \ /i \" = 1 » w e see that z e Co({T (x) , T (x)}) . i=z 1 o Suppose x ^ T(x) , then x £ Co ({T(x), T n(x)}) implies {T(x), T n(x)} contains more than one point, and so there i s a U e U • such that d = <$u(Co({T(x) , T n(x)}) = 6 u({T(x), T n(x)}) > 0 . Since P u ( T m ( x ) - T £ ( x ) ) £ P u ( T m _ 1 ( x ) - T A - 1 ( x ) ) whenever m,£ £ {1, 2, n} , we see that there must be a p e {1, 2, ..., n} such that d = P u(T^(x) - x) . Let p Q be the smallest p o s i t i v e integer p such that d = P u ( T P ( x ) - x) . Then 20. d = P u(x - T °(x)) = P u(6 1T(x) - (1 - g 1 ) z - T P°(x)) P P < (3.P (T(x) - T °(x)) + (1 - 8,)P (z - T °(x)) — 1 u 1 u P-1 < 3 xP u(x - T , (x)) + (1 - 3 1)-d and i t follows that < 3 1-d + (1 - 3jW P o _ 1 P u ( x - T 0 (x)) = d , which contradicts our choice of p i f p > 1 ; but for p = 1 , o 1 o o p o _ 1 d = P^(x - T (x)) = P^(x - x) = 0 which i s again a c o n t r a d i c t i o n . Therefore we must have x = T(x) . Example 4.6. Let S be a nonempty Hausdorff l o c a l l y compact to p o l o g i -c a l space and l e t C(S) be the set of a l l complex-(resp. r e a l - ) valued continuous functions on S . For each nonempty compact subset C of S , we define q on C(S) by q (f) = sup | f ( x ) | , f or a l l f e C ( S ) . Then C C xeC q^ i s a semi-norm on C ( S ) for each nonempty compact subset C of S . Let F be the c o l l e c t i o n of a l l such semi-norms, and l e t C(S) have the topology generated by F (Robertson [14]). Then C ( S ) i s a Hausdorff l o c a l l y convex space and a base for closed absolutely convex 0-nbhds i n C(S) i s given by 21. U = {{feC(S) : q 1 ( f ) V - • -Vq n(f)= max q ±(f)<e} :£> 0, q ^ . - . ^ s F , n=l,2,...}. l<i<n F i r s t l y we observe that i f C^, are nonempty compact subsets of S and U = {f e C(S) ; q (f) V . . . v q (f) £ 1} , 1 n then the gauge function P of U i s j u s t q v ••• V I • Thus for 1 • n K C C(S) , T : K K i s nonexpansive w.r.t. U i f and only i f q(T(f) - T(g)) <_ q(f - q) for a l l f ,g e K and a l l q e F; T : K -s- K i s contractive w.r.t. U i f and only i f T i s nonexpansive w.r.t. (J and for any q e F and f,g E K ', q(f - g) > 0 implies q ( t ( f ) - T(g)) < q(f - g); T : K -> K i s s t r i c t l y c ontractive w.r.t. U i f and only i f for any q E F, there i s a constant X with 0 <_ X <1 such that q(T(f) - T(g)) <_ Xq(f - g) for a l l f,g E K . Secondly we observe that i f S i s not compact, then C(S) i s not normable. Indeed, i f C(S) were normable, say by || || , then there i s e > 0 and there are nonempty compact subsets C^, ..., of S such that " {f e C(S) : q v • • • V q (f) £ e ^ ^ f £ C(s) : IIf II 1 ^ -I n n Since L i C. i s compact and S i s not compact, there i s an x £ S such i = l n ' that x £ L J C . . Since S i s completely regular, there i s a continuous 0 i = l 1 n function f on S s a t i s f y i n g f(y) = 0 f o r a l l y e ( J C. , f ( x ) = 1 i = l 1 ° and 0 <_ f(x) <_ 1 for a l l x £ S . I t follows that q (f) = 0 for a l l i i = 1, '2, . . . , n~•• and so q . (Xf) = 0 for a l l i = 1, 2, . . . , n and a l l i r e a l number X and so ||Xf|| < 1 for a l l r e a l numbers X which implies 22. f = 0 on S . This i s a con t r a d i c t i o n . Hence C(S) i s not normable. T h i r d l y i t i s clear that C(S) i s complete. Now l e t K = {f e C(S) : || f ^  = sup | f ( x ) | <_ y} , then X E S i t i s clear that K i s nonempty closed and convex. (1) For each s c a l a r A such that |A| =1 , and each g £ C(S) such that llgll < T , we define T, : K -> K by T, (f) II & iioc _ 4 x,g J X,g = Af +g , Vf e K .• Then we s h a l l show that T i s contractive w.r.t. U with a unique fixed point i n K but i s not s t r i c t l y contractive w.r.t. U Indeed, i f f ^ , e K and C i s any nonempty compact subset of S , then q (T. (f.) - T. (f )) = q (f 2 - f 2) ^c A,g 1 A,g 2 ^c 1 2 2 2 = sup I f (x) - f (x) I xeC = sup|f (x) - f 2 ( x ) | | f x ( x ) + f 2 ( x ) X E C <_ sup |f (x) - f (x) X£C ^ ( f l " f 2 > > and so T^ i s nonexpansive w.r.t. U . Next suppose q^(f^ - f^) > Then there e x i s t s an x £ C such that o f 2 ( x ) - f 2 ( x )| = sup j f , 2 ( x ) - f„ 2(x)| 1 ° 2 ° xeC 1 2 = q (T (f ) - T (f )) c A,g 1 A,g 2 23. I f |f_(x ) + f_(x )| < 1 , then 1 o I o ^ X . g ^ l * " TX,g(f2>> " l f l \ ) - f 2 2 W l < | f l ( x o ) - f 2 ( x o ) | £ q c ( f ! " f 2> ; • i f | f. (x ) + f _ (x ) | =. 1 , then since |f.(x )| <_"kr , we see that J- O £m O IL O " f (x )-=.f (x ) , and so q (T (f ) - T (f )) = 0 < q (f - f ) . Thus l o z o c A,g 1 . A,g 2 c 1 2 X i s contractive w.r.t. U • Now i f . p i s any constant with A>g 0 <_ y < 1 , choose any constant a > 0 with y - < a < . Define h^ = and h 2 = a , then h^,h 2 e ^ ' Since V ~ < a < , we have ( i ) 2 - a 2 > y(-| - a) and so yq (h.. - h,) < q (T. (h.) - T. (h„)) , and I I c l I c A,g 1 A,g 2 hence T^ i s not s t r i c t l y c ontractive w.r.t. U . F i n a l l y i t can be 4 4 63 shown that for each U e U , P (T, ( f n ) - T, (f„) < P ( f . - f_) f o r u A,g 1 A,g 2 — 64 u 1 I 4 a l l f,,f„ e K so that T, i s s t r i c t l y contractive w.r.t. U and so 1 2 A,g, / T^ has a unique f i x e d point i n K , by C o r o l l a r y 2.5. (2) Suppose T : K -* K i s nonexpansive w.r.t. U . For each s c a l a r A with |A| < 1 and each g e C(S) with || g || < — , and oo 4. each p o s i t i v e integer n >_ 3 , we define T^ ^ : K -> K by . T (f) = A ( T f ) n + g , V f £ K . Suppose C i s any nonempty compact X,n,g subset of S and f]_'^2 £ ^ • Then 24. «c<TX.n,g<fl> - T X , n , g < f 2 » = q c a ( T f 1 ) n - A ( T f 2 ) n ) = sup | ( ( T f ) ( x ) ) n - ((Tf ) ( x ) ) n | xeC . = sup | ( T f 1 ) ( x ) - ( T f 2 ) ( x ) | | ( ( T f 1 ) ( x ) ) n _ 1 + . . . + ( ( T f 2 ) ( x ) ) n _ 1 | (n-terms) xeC £ s u p | ( T f 1 ) ( x ) - (Tf )(x)|' { ( | ) n _ 1 + ... + (|) n - 1} (n-terms) xeC = < c< T< fi> - T (V> ^ • ^ I q c ( f l " f2> ' since 0 < — — r < 1 , we see that T, i s s t r i c t l y contractive w.r.t. 2 n - l X,n,g U . Hence by Theorem 2.4., T^ ^ has a unique f i x e d point i n K . (3) Suppose T : K -> K i s nonexpansive w.r.t. U . For each scalar X with 0 < X < 1 , for each g e C(S) with Ijgll^  £. ^  and for each p o s i t i v e integer n > 2 , we define V, : K -* K by - . , A,n,g V, (f) = X ( T f ) n + (1 - X)g , f o r a l l f e K . Then for any nonempty X,n,g • ° compact subset C of S , we have q (V, (f,) - V, ( f . ) ) < X — ^ T J - q (f. - f.) for a l l f. , f . e K , and n c X,n,g 1 X,n,g 2 — 2 n 1 2 1' 2 so V, i s also s t r i c t l y contractive w.r.t. U and hence by Theorem 2.4. X,n,g / V, has a unique f i x e d point i n K . A,n,g Remark 4.7. Suppose X i s sequentially complete and T : X -»- X be I 25. strictly contractive w.r.t. U . Then for arbitrarily fixed y e X , the equation x - Tx = y has a unique solution in X . Proof: Define S : X -+ X by S(x) = T(x) + y . Then S is strictly contractive w.r.t. U and so S has a unique fixed point in X , by Theorem 2.4., and so the equation x|- Tx = y has a unique solution in X . Corresponding to Theorem 2.4., we have the following implicit function theorem which is analogous to a result of E. Dubinsky in [7]: Theorem 4.8. Let K C X be nonempty bounded and sequentially complete, S be a topological space and f : K x S -*• K be continuous. Suppose for each U e (J , there is a constant X with 0 < X < 1 such that u — u • \ P (f(x, s) - f(y, s)) < X P (x - y) for a l l x, y e K and a l l s e S . U — U U ' J Then there is a unique continuous mapping T : S K such that / f(T(s), s) = T(s) , for a l l s e S . Proof: For each s e S , define g g : K -»• K by g g( x) = f(x» s) for a l l x e K . Then g is strictly contractive w.r.t. U . Thus s by Theorem 2.4., there is a unique T(s) e K such that gs(T(s)) = T(s) . Hence there is a unique mapping T : S -»• K such that f(T(s), s) = T(s) , for a l l s e S . It remains to show that T is continuous. 26. Fix any x e K 'we define T : S -> K f o r a l l p o s i t i v e o n c integers n as follows: T n ( s ) = f ( x , s) and T ,, (s) = f ( T (s ) , s) for l o n+1 n a l l s e S and a l l n = 1, 2 Suppose s s i n S , then ( X q , s^) ->• (X q, s) and so ^ ( s ^ ) = f ( x Q > s y ) ~> f ( x Q > s ) = ^ ( s ) , and hence T, i s continuous. . Suppose T, i s continuous and s ->• s i n S , then 1 k y » T (s ) -*• T (s) implies (T. (s ) , s ) •+ (T. ( s ) , s) and so \ + 1 ( s ^ ) = f ( T k ( s ), s ) f ( T k ( s ) , s) = T k + 1 ( s ) , and therefore T k + 1 i s also continuous. Thus by induction, T : S -»- K i s continuous f o r a l l n = 1,2,... . n ' ' Next we want to show that f o r any U e U , 3 a p o s i t i v e integer N(U) such that P u ( T n ( s ) - T(s)) <_ , f o r a l l n > N and a l l s e S. Indeed, since K i s bounded, 6 (K) = sup P (x - y) < » , and so we may choose u 7 r u x,yeK N 1 a p o s i t i v e integer N(U) with A 6 (K) < -r . Thus f or n > N , u u — 3 P u ( T n ( s ) - T(s)) = P u ( f ( T n _ 1 ( s ) , s) - f ( T ( s ) , s » ± A u P u ( T n - l ( s ) - T ( S ) ) < A ^'^C^Cs/- T(s)) = A u n _ 1 P u ( f ( x o , s) - f ( T ( s ) , s)) < A "P (x - T(s)) — u u o < A n6 (K) — u < X N 6 (K) — u u 27. Finally suppose s^ -> s in S . Let U e U , then there is a positive integer N(U) with P^CT^Cs) - T(s)) <_ -j , for a l l n > N and a l l s e S . Since T, , is continuous, there is a u with N~rl O VWV - T N + I ( s ) ) ±i f o r a 1 1 ^ > ^  • T h u s P u(T(s y) - T(s)) -^u^V - W 8 U » + VWV " W s ) ) + P u ( T N + l ( s ) - T ( S ) ) < I + I + I - 3 3 3 = 1 , for a l l u > y and so T(s ) - T(s) e U , for a l l U e U . Therefore o y T(s ) T(s) and hence T is continuous. This completes the proof. 1-5. Nets of contractive mappings. The purpose of this section is to discuss the convergence of nets of contractive mapping and nets of fixed points. Definition 5.1. Let K C X be nonempty. Then T : K -> K is said to be uniformly continuous w.r.t. U i f and only i f for each e > 0 and for each U e U , there is a 6(e, U) > 0 such that for a l l x, y e K , i f P u(x - y) < S , then Pu(T(x) - T(y)) < £ . If K C X is nonempty and T : K -*• K is uniformly . continuous w.r.t. U , then i t is clear that T is continuous. Also i f 28. T : K -> K i s nonexpansive w.r.t. U then T i s uniformly continuous w.r.t. U . D e f i n i t i o n 5.2. Let K C X be nonempty and B C K be nonempty. Let F be a family of mappings from K into i t s e l f . Then F i s said to be equicontinuous w. r. t. U_ on B_ i f and only i f for each e > 0 and each U e U , there i s a 6(e, U) > 0 such that for a l l x,y e B , i f P (x - y) < <5 , then P (T(x) - T(y)) < e for each T e F . u u ' I t i s c l e a r that i f F consists of nonexpansive mappings w.r.t. U, then F i s equicontinuous w.r.t. U D e f i n i t i o n 5.3. Let B,K C X be nonempty with B C K , T : K K and {T } „ be a net of mappings on K (into i t s e l f ) , then (i) T^ T pointwise on B i f and only i f T a ( x ) T(x) f o r each x e B, ( i i ) T^ -> T uniformly w.r.t. _U on B_ If and only i f f o r each e > 0 and each U e U , there i s a a e r such that i f a e T and a > a , ' o — o then P (T (x) - T(x)) < e , for a l l x £ B . •u a Proposition 5.4. Let B C K C X be nonempty and B i s compact. Suppose T : • K -> K and {T } _ i s a net. of mappings on K such that ( i ) T -> T pointwise on B and ( i i ) {T^ : a e r} K_i {T} i s equicontinuous w.r.t. U on B . Then T -> T uniformly w.r.t. U on B . 29. Proof: Suppose £ > 0 and U e u . Then there is a &(z, u ) > 0 such that P (T (x) - T (y)) < -| for each a £ T and u a a 3 P (T(x) - T(y)) < -| for a l l x,y e K with P (x - y) < 6 . Since u 3 u B C U x + 6U1 and B is compact, there exists {x , ..., x } O with X E B 1 N n B C [I x. + (SU1 . Since T -»- T pointwise on B , there are 3 . £ V such that T a ( x j ) - T ^ x j ) e f 1 ) 1 f o r a 1 1 a > g j a n d a 1 1 1 = 1» 2> Take 3 e T such that B_. <_ 3 for a l l j = 1, 2, n . Then for any x £ B , x e x_. + S U 1 for some j e {1, 2, . . ., n} , so that P^(x-x ) < 6 and i t follows that P (T (x) - T (x.))< -f- for a l l a z Y and u a a j 3 Pu(T(x) - T ( x j ) ) < •§ • Hence for a > 3 , P (T (x) - T(x)) u a < P (T (x) - T (x.)) + P (T (x.) - T(x.)) + P (T(x.) - T(x)) — u a a j u a j j u j e , £ , £ < 3 + 3 + 3 = £ Hence P (T (x) - T(x)) < e for a l l x e B , for a l l a > 3 . Therefore u a T -> T uniformly w.r.t. U on B . a Proposition 5.5. Let K C X be nonempty, T : K K be strictly contractive w.r.t... U and .'fx'a^ (xej< ^ e a n e t °^ mappings on K . Suppose (i) there are a,a e K such that T(a) = a and T (a ) = a for a l l ' a a a a a £ T , and (ii) T^ -> T uniformly w.r.t. U on K . Then a^ -*• a . 30. Proof: Suppose 0 < z < 1 and U e Li . Then there i s a constant 0 < X < 1 such that P (T(x) - T(y)) < X P (x - y) f o r a l l u u u u x,y e K . Since T •> T uniformily w.r.t. U on K , there i s an 3 e T a such that i f a > 8 , then P u(T (x) - T(x)) < e ( l - X^) , for a l l x e K. Thus f o r a > B ' P (a - a) = P (T (a ) - T(a)) u c t u c t c t < P (T (a ) - T(a )) + P (T(a ) - T(a)) — u a a a u a < e ( l - X ) + A P (a - a) , ^ u u u a so that P (a - a) < e for a l l a > 8 , and hence a - a e U , for a l l u a a a > B . Therefore a a a Theorem 5.6. Let K C X be nonempty, T : K -* K be s t r i c t l y c ontractive w.r.t. Li and {T } be a net of contractive mappings a azT w.r.t. U on K . Suppose there are a, a e K such that T(a) = a and • a T (a ) = a„ for a l l a e T . If T -»• T pointwise on K and a has a a a a a . v compact nbhd i n K , then a . -/ Proof: Let U e Li such that B = a + U f~\K i s compact. Since {T^ : d z r}0'(T} i s equicontinuous w.r.t. Li on K and hence on B and T^ -»- T pointwise oh K , and hence on B and B i s compact, we see that T^ -> T uniformly.w.r.t. U on B" , by Proposition 5.4. Let X^ be a constant such that 0 < X < 1 and P (T(x) - T(y)) < X P (x - y) f o r a l l u u — u u 31. x, y e K . Choose 3 e V such that P (T (x) - T(x)) < 1 - X f o r a l l J u a u x e B and a l l a > 3 . I f x e B , then x - a e U so that P (x-a) <_ 1, and so for a > 3 , we have P (T (x) - a) < P (T (x) - T(x)) + P (T(x) - a) u a — u a u < (1 - X ) + X P ( x - a ) u u u < (1 - X ) + X -1 — u u = 1 and so T (x) - a £ U for a l l a > 3 and a l l x e B . Thus a T (x) £ a + U fYK = B for a l l x £ B and a l l a > 3 . Thus T : B •> B a a i s contractive w.r.t. U for a l l a > 3 and B i s compact and so by Corol l a r y 3.7. T^ has a unique f i x e d point i n B for each a > 3 . But T has a fixed point a £ K which must also be unique, we see that a £ B for a l l a > 3 • Also f o r x £ B , we see that a P (T(x) - a) = P (T(x) - T(a)) < X P ( x - a ) < X - l < l , s o that u u \ — u u — u T(x) - a £ U for a l l x £ B and so T(x) £ a + U O K = B for a l l x e B. Thus T : B B . Hence by Proposition'5.5., a^ -»• a . Proposition 5.7. Let K C X be nonempty, T : K -> K and {T } _ be a — a aeT net of nonexpansive mappings w.r.t. U on 1 K . Suppose (i) T -»• T point-wise on K , ( i i ) there are, a £ K such that T (a ) = a f o r a l l ' - a a a a a £ T and ( i i i ) there i s a subnet {a .}„, of {a } „ and there i s an a' T a a£T x £ K such that a , -> x . Then T(x ) = x . o a o o o 32. Proof: Suppose U E U . Then there i s an 3' such that 1 P (a , - x ) < -Tr for a l l a' > 3' . Since T (x ) -> T(x ) and so u a o I a o o T , (x ) ->• T(x ) , there i s an v' such that P (T , (x ) - T(x )) < \ a o o u • a o o . 2 for a l l a' > y' . Fix any 6' > 3' and y' J then for a' > 6' , P (a , - T(x )).= P (T ,(a ,) - T(x )) u a o u a a o < P (T ,(a T) - T ,(x )) + P (T ,(x ) - T(x )) — u a a a o u a o o < P (a , - x ) + i u a o 2 1 , 1 < 2 + 2 so that a , - T(x ) e U for a l l a' > 6' . Hence a . -> T(x ). Since a o a o X i s Hausdorff, a , -> T(x ) and a , -> x imply T(x ) = x a o a o o o Cor o l l a r y 5.8. Let K C X be nonempty sequentially compact, T : K K and T^ : K K be nonexpansive w.r.t. U for each n = 1, 2, ... . Suppose T T pointwise on K and there are a e K with T (a ) = a ^ n ^ n n n r for a l l n = 1, 2, ... . Then T has a f i x e d point i n K . D e f i n i t i o n 5.9. Let X. be a T - l . c . s . , i = 1, 2 and U-i be a base for x 2 x closed absolutely convex 0-nbhds i n X^ . Let Tr : X^ x X^ X^ be the natural p r o j e c t i o n for each i = 1, 2 , K C X^ x x^ be nonempty and 33. T : K K . Then T i s said to be s t r i c t l y contractive i n the 1st  va r i a b l e w.r.t. i f and only i f for each y e ^ ( K ) ' ^ o r e a c n u e ^> there i s a constant X such that 0 < X < 1 and u u P u 0 1 ( T ( x 1 , y ) ) - T T 1(T(x 2,y))) U ^ f X j ^ - x 2) ', for a l l x , x 2 e X± with (x , y) , ( x 2, y) e K . Lemma 5.10. Let be T 2-l..c.s. and U i s a closed absolutely convex 0-nbhd i n X. , i =1, 2 . Then l P v ( ( x 1 } x.)) = max {P (x.) : i = 1, 2} , for a l l x. e X. , i = 1, 2. u, xu„ 1 2 u. l x i 1 2 x Proof: P ( ( x n , x,)) = i n f {X >0 : (x., x 0) e X(lL x n ) } u ^ U £ 1 Z X Z 1 z = inf{X > 0 : x. e X U. for a l l i = 1,2} x x <_ max {P (x.) : i = 1, 2} , -i since x. e P (x.)U. f o r a l l i = 1, 2 . But i f x u. x x x P ( ( x n , x 0 ) ) < max{P (x.) : i = 1, 2} = P ( x j , say, then U n *U„ X Z U . X ' U, ± 1 2 , i 1 X l ^ P u 1 x u ( X 1 ' X 2 ) U 1 E n d S ° ( x l ' X2^ ^ P u x u ( ( x l ' X 2 ) ) U 1 X U 2 w h i c h i s impossible. Hence P ((x,, x„)) = max {P (x.) : i = 1, 2} . u,x u„ 1 2 u. x 1 z 1 Theorem 5.11.-Let X. be T . - l . c . s . , U. be a base for closed x 2 I absolutely convex 0-nbhds i n X_^  f o r a l l i = 1, 2 . Suppose M C X^ i s 3 4 . nonempty such that every s t r i c t l y contractive mapping w.r.t. on M has a fi x e d point (e.g. M i s sequentially complete) and N C i s nonempty such that every continuous mapping on N has a fi x e d point. (e.g. N i s compact convex). I f ( i ) T : M x N - > M x N i s uniformly continuous w.r.t. x and ( i i ) T i s s t r i c t l y contractive i n the'1st v a r i a b l e r . t . U , then T has a fi x e d point i n M x N . w Proof: For any y e N , we define : M -> M by T y(x) = TT^ ° T(x, y) , for a l l x e M . Then by ( i i ) , T^ i s s t r i c t l y c ontractive w.r.t. and so by assumption, T^ has a (unique) f i x e d point i n M for each y e N . Define F : N -»- M by F(y) = the unique fi x e d point of T^ i n M , for a l l y e N . Thus T (F(y)) = F(y) f o r a l l y e N . We s h a l l show that F i s continuous. Indeed, suppose y^ ->- y i n N . I f e and ^2 E ^2 ' t l c i e n s i n c e T 1 S uniformly continuous w.r.t. x , for any e > 0 , there i s a• 6 > 0 such that p v u 2 ( ( x i ' - < v y 2 } ) \ 6 i m P l i e s ^ x u ^ v ^ - ^ v ^ < e for a l l (x^, y ^ ) , (x^, y 2 ) £ M x N . Since y^ -> y , there i s an a Q such that a > a implies P (y - y) < 6 . Hence for x., , x„ e M such o r a J 1 2 that P^ (x^ - x 2) < 6 , we have ^ x u ^ ^ W " ( x 2 ' y ) } = m a X ~ X2>> P u 2 ( ^ a - y » < « . for a l l a > a , by Lemma 5.10., and i t follows that o J P (T(x , y ) - T(x_, y)) < s for a l l a > a . Hence for any x e M u ^ u ^ 1 ct z o 35. and a > a Q , we have P ^ V ^ l ' V ~ ^ i ^ ^ ^ ^ u ^ u ^ ^ l 7 ^ ' 1 ^ ^ ^ ' Thus P (T (x) - T (x)) < e , for a l l x E M and for a l l a > a . I t u, y y o 1 J a. follows that T ->• T uniformly w.r.t. on M . Thus by Proposition 5.6., ^Cy^) F(y) , and so F i s continuous. Define G : N N by G(y) = T r 2°T(F(y), y) for a l l y £ N . Then c l e a r l y G i s a continuous mapping on N and so by assumption, there e x i s t s an p £ N such that G(p) = p . Since p = G(p) = TT2 ° T(F(y), y) and F(p) = T (F(p)) = ° T(F(p) , p) , we see that T(F(p), p) = (it1 o T(F(p), p), TT2 ° T(F(p), p) = (F(p), p) , so that (F(p), p) i s a f i x e d point of T i n M x N . 36. CHAPTER II NORMAL STRUCTURE I I - l . Centre of a set. Suppose X i s a T ^ - l . c . s . , H and K are subsets of X such that H i s bounded. I f U i s any closed absolutely convex 0-nbhd, we denote r (U; H) = i n f { r > 0 : x - H C rU} , for any x e X ; r(U; H, K) = i n f { r x ( U ; H) : x e K} ; C(U; H, K) = {x e K : r (U; H) = r(U; H, K)} C(U; H, H) i s c a l l e d the centre of H r e l a t i v e to U (or w.r.t. U) . If H ^ 0 , we see that r (U; H) = sup{P (x - y) : y e H} . The following two lemmas are obvious. Lemma 1.1. If X i s a T ^ - l . c . s . , H C X i s a nonempty bounded subset, U i s a closed absolutely convex 0-nbhd, define f : X ->• H by f(x) = r (U; H) for a l l x e X . Then f i s continuous and f (Xx + (1 - X)y). <_ Xf (x) + (1 - X)f (y) , for a l l x, y e X and 0 £ X <_ 1. Lemma 1.2. Let X be a T 0 - l . c . s . , K C X be nonempty, and H C X 37. be nonempty bounded. I f U i s any closed absolutely convex 0-nbhd, then for any subsets H^, of X such that H C H ^ C Co(H) and K C ^ C C o © , we have- ( i ) r(U; H, K) = r(U; H^, K ) , and ( i i ) C(U; H, K) = C(U; H , K) C C ( U ; H , K ) . Proposition 1.3. Let X be a T2~l.cs., H C K be nonempty bounded and convex, and U be any closed absolutely convex 0-nbhd. I f x e H i s such that r (U; H) < 6 (H) , then there i s an x e H with x u o 0 < r (U; H) < <5 (H) . x u o Proof: We may assume r x ( U ; H) = 0 . Then < U^(H) > 0 implies r^(U; H) > 0 f o r some y £ H . Since Ax + (1 - A)y -> y as A -> 0 + , we have r, , , N (U; H) -> r (U: H) as A -»- 0 + , by Lemma 1.1. Thus there Ax+(1-A)y y i s 0 < A < 1 with Ir (U; H) - r. (U; H) I < ^  r (U; H) , and so — o y A 0 x + \ i ~ A 0 ) y z y r A o X + ( l - A o ) y ( U ; H ) * r y ( U ' H ) > " I r y ( U ' H ) = 1 r y ( U ; H ) > 0 " T a k e x = A x + ( l - A ) y , then x £ H since H i s convex, and r (U; H) > 0. o o o o x o Also by Lemma 1.1., r x ( U ' H ) " rA x + ( l - A ) y ( U ' H ) o < A r (U; H) + (1 - A ) r (U; H) — o x o y = (1 - A Q ) r y ( U ; H) < r (U; H) < 6 (H) y — u 38. Proposition 1.4. Let X be a T ^ - l . c . s . , K C X be nonempty weakly compact convex, H C K be nonempty convex. I f U i s any closed absolutely convex 0-nbhd i n X , then C(U; H, K) i s nonempty closed and convex. Proof: For each p o s i t i v e integer n. , we denote F(y; n) = {x e K : x - y e (r(U; H, K) + ^-)U} , for each y £ H and C n(U) = f*\ F(y; n) = {x e K : x - H C (r(U; H, K) + i)U> . Then i t i s y£H clear that C(U; H, K) = (~\ C (U) . For each p o s i t i v e integer n , since n=l n r(U; H, K) + — > r(U: H, K) , there i s an x £ IC such that n r (U; H) < r(U; H, K) + - , and so x - H C (r(U; H, K) + -)U and so x n — n x £ C^CU) 5 thus (U) 5^  0 • I t i s also c l e a r that F(y, n) i s closed and convex for each p o s i t i v e integer n and therefore C n(U) i s nonempty closed and convex. Moreover, since K i s weakly compact and 0 0 K D C n(U) D C n + 1(D) ^ 0 , we see that (U) 4 0 . Hence C(U; H, K) n=l i s nonempty closed and convex. 39. II-2. Normal Structure. The concept of normal structure was introduced by M. S. Br o d s k i i and D. P. Milman i n [4]. The following d e f i n i t i o n i s i t s gener a l i z a t i o n . D e f i n i t i o n 2.1. Let X be a T - l . c . s . , U be a base for closed absolutely convex 0-nbhds, and K C X . Then K i s said to have normal structure w.r.t. Li i f and'only i f for any bounded convex subset H of K containing more than one point, there i s a U e Li and X q £ H such that r (U; H) < 6 (H) . In th i s case x i s c a l l e d a non-diametral point of x u o o H w.r.t. U . Theorem 2.2. Let X be a T ^ - l . c . s . , U be a base f o r closed absolute l y convex 0-nbhds and K C X be nonempty weakly compact convex. Then K has normal structure w.r.t. Li ±f and only i f for any convex subset H of K containing more than one point there i s a U £ U such that 6 (C(u; H, H)) < <5 (H) . u u Proof: Suppose K has normal structure w.r.t. Li I f H i s nonempty subset of K containing more than one point, then there i s a U £ U and an , x e H such that r (U; H) = r (U; H) < 6 (H) = 6 (H) . I f x x u u z,w £ C(U; H, H) , then z - W E r (U; H)U = r(U; H, H)U implies 40. 6 u(C(U; H, H) <_r(U; H, H) <_ r ^ U ; H) and so <5u(C(TJ; H, H) < 6 (H) . But C(U; "H, H) C C ( U ; H, H) , we see that 6 u(C(U; H, H)) <_ 6 (C(U; H, H) < 5(H) . Conversely, suppose H i s any convex subset of K containing more than one point and U e U i s such that 6 u(C(U; H, H)) < <Su(H) . By Proposition 1.4. C(U; H, H) t 0 . Choose x e C(U; H, H) . Then r (U; H) = r (U; H) = r(U; H, H) . Since there i s a y e H such that r^(U; H) > r(U; H, H) , we see that r(U; H, H) < <S (H) = 6 (H) . By Lemma 1.1., there i s an x £ H such that u u o |r (U; H) - r (U; H) | < -|(6 (H) - r(U; H,. H)) . o Thus r (U; H) < -k<S (H) + r(U; H, H)) < 6 (H) . Hence K has normal x 2 u u o structure w.r.t. U The above theorem generalizes a r e s u l t due to W. A. K i r k i n [13]. Theorem 2.3. Let X be a T ^ - l . c . s . , K Q be compact convex. I f H i s any convex subset of K containing more than one point, then for any closed absolutely convex 0-nbhd U with ^ U ( H ) > 0 > there e x i s t s an x £• H such that x i s a non-diametral point of H w.r.t. U . In o o p a r t i c u l a r K has normal structure w.r.t. any base for closed absolutely convex 0-nbhds. 41. Proof: Choose x , y £ H such that P (x - y ) / 6 (H) = r . n n u n n u CO Since H i s compact, there are x, y £ H and subsequences {x^ *±-]_ °^ {x } : } ., and {y }. , of {y } , with x -> x and y -> y . It n n=l n. i = l n n=l n. n. follows that P (x - y) = P (lim (x - y )) u J u . n. n. X-X>o ! ! = lim P (x - y ) u n. n. i-*» 1 I Let F = { H C H : {x, y} CZ M and P (a - b) = r for — — u a l l a, b e M with a- £ b} . P a r t i a l l y order F by C . Then F ^ 0 since {x, y} £ F . • I f {M. }. „ i s any chain i n F , then c l e a r l y A A£l U M. e F. Therefore by Zorn's Lemma, .F has a maximal element, say M x e r x ° I f M were i n f i n i t e , l e t a, , a„, ... be countably i n f i n i t e d i s t i n c t o 1 2 . CO CO elements of M . Then there i s a subsequence '{a }. , of {a } , and o n. i = l n n=l l a £ H with a a . Thus there i s an i with a e a + rU for a l l n. o n. j l l i > i . I t follows that a , a £ a + rU and so o n. ' n. 0 3 l + l i + 2 o o 2 2 a - a £ -— rU , and hence P (a - a ) < -rr , which i s n. , - n . ,„ 3 u n. n. j 0 — 3 l +1 i +2 I +1 i +2 o o • o o impossible. Thus M must be f i n i t e , say M = {x, , x } where o o I n n 1 — — x. 4 x. i f i 4 i . Denote w = ) — x. . Then w £ H since H i s 1 J , i = l n 1 convex. We s h a l l show that w i s a non-diametral point of H w.r.t. U . • Indeed-we may assume s^ = r w ( U ; H) > 0 . Since H i s compact, there i s an z £ H such that P (w - z) = s by the same proof as above. If u o s = r , then P (z - x.) = r for a l l i = 1, 2, . ... n , for i f o u 1 t = P (z - x. ) < r for some i e {1, 2, ..., n} , then o u 1 o o n t t z - w = I i ( z - x . ) e 7 - rU + — U = ( — r + — ) U , . -, n x .h. n n n n x=l x^x o n-1 t so that P (z - w) < r H < r , which i s impossible. Thus u — n n M {z} £ P and M i s maximal i n F imply M = M U {z} and hence o ' o o o z = x^ . for some j e{ l , 2, n} . But then r = P (x. - z ) =P (x. - x . ) = 0 , again contradicts r > 0 . Hence u J u 3 j s < r and so w i s a non-diametral point of H w.r.t. U . o r - s F i n a l l y choose w £ H with w - w £ — r U . Then o o 2 r - s r + s for any h e H . w - h = w - w + w- h £ — r U + s U = — ^ U . I t o o 2 o 2 r + s follows that r (U; H) < — r . Therefore w i s a non-diametral w — 2 o o point of H w.r.t. U . x The above theorem generalizes a r e s u l t due to R. DeMarr i n [6]. Theorem 2.4. Let X. be T„-l.c.s. and U be a base f o r closed x 2 absolutely convex 0-nbhds i n X. where i = 1, 2 . I f X. has normal x x structure w.r.t. ii. for a l l i = 1, 2 , then X = Xn x X„ has normal x 1 2 structure w.r.t. U = U x U = { u x u . : U. e ii. , 1 = 1,2} 1 2 1 2 x x 4 3 . Proof: Let K be a bounded subset of X containing more than one point. Let ir be the natural projection of X onto X_^  and K. = IT . (K) for i = 1, 2 . Then K. i s bounded and convex for i = 1, 2. i l l I f either or , say K^, contains only one point, say x^ , then contains more than one point; since has normal structure w.r.t. there i s an x„ e K 9 and a U. e U with S (K ) > r (U„; K ) . Thus £ z. ^2 ^2 for any e 5 we have S (K) = i n f { r > 0 : K - K C r (U. x U„) } u^xu^ — 1 1 = i n f { r > 0 : ( x ^ y ^ - ( x ^ y ^ e r(U 1xU 2) for a l l y l 9 y 2 e = i n f { r > 0 : y - y 2 e ru"2 for a l l y , y 2 e K^} = i n f { r > 0 : K 2 - K 2 C r l ^ } • vv > r x 2 ( U 2 ; K 2) = i n f { r > 0 : x 2 - K 2 C rU2> • = i n f { r > 0 : (x. , x) - K C r ( U 1 x U 2)} r ( x x , x 2 ) ( U l X U 2 ^ K> > so that (x^, x 2) i s a non-diametral point of K w.r.t. x U 2 for any e U^ . ; Thus we may assume that for each i = 1, 2 , contains more than one point. I t follows that there are U. e U. and x. £ K. • such 1 1 1 1 that r. = 6 (K.) > r ( U . : K . ) = s . , for a l l i = 1, 2 . Choose any 1 u. 1 x. 1 * 1 1 ' J l •  l Ui £ ^ i 1 ^ ^ } ) P\K , i = 1, 2 . Then u 1 = (x^, v) and u 2 = (w, x 2) for 4 4 . some v £ and w £ . Denote m = + u^) , then m e K since K i s convex. If z £ K , then z = (z^, z^) where £ K\ , i = 1, 2 . Thus m - z = yCu-^  + u,,) - z = - 2 1((x 1, v) + (w, x 2 ) ) - (z , z 2) = (^"(^ + w) ~ zi> + x2^ " z2^ = - z ^ x 2 - z 2) + |(w - z ^ V - z 2) £ I ( S 1 U 1 X 8 2 U 2 ) + 7 ( r l U l ' X r 2 V C | sC^ x u2) + \ r C ^ x u2) = | ( r + s) ( 1 ^ x u2) , where s = max {s^, s 2 ) and r = max {r-^ > r 2 } . Thus r (U, x TJ • K) < -i-(r + s) . Suppose r = r 1 , then since ^-(r + s) < r , m l z * z \ l z K l ~ K l ^ l f ^ r + S ^ U a n c i s o t n e r e a r e vi» 2^ £ K l W l t n y l ~ y2 ^ ^ r + S ^ U l " Choose any z^, z 2 e K with (v^> e K f ° r 1 = x> 2 > then ( Y l, z ±) - (y 2, z 2) ij: | ( r + s ) ^ x u2) implies ^ ( K ) > j ( r + s) . Hence m i s a non-diametral point of K w.r.t. x TJ 2 . Therefore x X 2 has normal structure w.r.t. x 11^ . The above theorem generalizes Theorem 2.1. proved by L. P. Belluce, W. A. K i r k and E. F. Steiner i n [3]. 45. II-3. Some fixed point theorems. Throughout the remaining of t h i s chapter, X w i l l denote a T ^ - l . c . s . , and Li w i l l denote a base for closed absolutely-convex 0-nbhds i n X . D e f i n i t i o n 3.1. Suppose K C X i s nonempty and T : K ->- K . For each nonnegative integer n , we denote 0(T, n, x) = { T n + 1 ( x ) : i = 0, 1, 2, ... for a l l x e K . Theorem 3.2. Let K C X be nonempty bounded closed convex, T : K -> K be nonexpansive w.r.t. Li and M C K be nonempty weakly compact such that (i ) Co(0(T, o, x)) r\ M 4 0 , for a l l x e K and ( i i ) Co(0(T, o, x)) has normal structure w.r.t. Li for a l l x e K . Then there i s an x e M with T(x) = x . Proof: Let G = (AC K : A i s nonempty closed convex and T(A) <C A} . P a r t i a l l y order G by D . Then by weak compactness of M and Zorn's Lemma, G has a minimal element, say . Suppose there were an x e K with T(x) ^ x . Then 0(T, o, x) and thus Co(0(T, o, x)) contains more than one point, and so by ( i i ) , there i s a U £ Li and y £ Co(0(T, o, x)) such that r = r^(U; Co(0(T, o, x))) < 6 u(Co(0(T, o, x)) By Proposition 1.3., we may assume that r > 0 . Define A = {z £ K : z - 0(T, n, x) C rU for some nonnegative integer n } , then A 0 , since y e A . C l e a r l y A i s convex and T(A) C A . Thus T(A) C A since T i s continuous. Since A i s also closed and convex A e G and so A = by the minimality of i n G Define S =' {z e : z - K C rU} . Then c l e a r l y S i s closed and convex. Now i f z £ A , then z - 0(T, n, x) C rD for some nonnegative integer n , so that 0(T, n, x) C z + rU implies C O Co(0(T, n, x)) C z + rU , and so (~\ Co(0(T, m, x)) C Co(0(T, n, x ) ) C z+rU m=0 for a l l z E A . By ( i ) Co(0(T, m, x))/°\M 0 for a l l m = 0, 1, 2,... . CO Hence ^ | (Co(0(T, m, x ) ) / ^ M =/ 0 by weak compactness of M ; and so m=0 CO CO (n Co~(0(T, m, x)))riM $ 0 implies (~\ Co(0(T, m, x)) 4 0 . I f m=0 m=0 CO t £ (~\ Co(0(T, m, x)) , then t £ z + rU for a l l z £ A implies • m=0 z £ t + rU for a l l z £ A and thus = A C t + rU , and hence. oo t - K C rU implies t £ S for a l l t £ f~\ Co(0(t, m, x)) . Thus m=0 S + 0 . Moreover, i f ' T(S) S , take any z e S with T(z) £ S . Let H = (T(z) + rV)r\K • I f y E H , then y £ K C z + rU implies y - z E rU and so T(y) - T(z) £ rU since T i s nonexpansive w.r.t. il I t follows that T(y) £ T(z) + rU ; but T(y) £ K , and so T(y) £ (T(z) + rU)P»K = H . Thus T(H) C H . Since T(z) £ H so that H ^ <J> and c l e a r l y H i s closed and convex, H e G . Hence H = by the minimality of K i n G . But T(z) <J: S implies T(z) + rU and so there exists an • y £ K with y £ (T(z) + rUJO^ = H , so that 47. H C K , which i s a co n t r a d i c t i o n . Hence T(S) C S , and so S £ G Therefore S = again by the minimality of i n G . But 6 (K ) = 6 (S) < r , while 6 (K.) > 6 (Co(0(T, n, x)) > r , which i s u l u — u l — u impossible. This leads to the•conclusion that T(x) = x for a l l x e Since 4 0 , there i s an x e K C K with T(x) = x . By (i ) , x e M . Corol l a r y 3.3. Let K C X be nonempty bounded closed convex with normal structure w.r.t. Li , M C K be weakly compact and T : K K be nonexpansive w.r.t. Li such that Co(0(T, o, x))/^\M 4- 0 for a l l x e K . Then there i s an x e M with T(x) = x . Corol l a r y 3.4. Let K C X be nonempty weakly compact convex with normal structure w.r.t. Li and T : K -> K be nonexpansive w.r.t. Li Then there i s an x e K with T(x) = x . Next w e s h a l l see that the above c o r o l l a r y holds f or a f i n i t e number of "commuting" nonexpansive mappings. D e f i n i t i o n 3.5. Let K C X be nonempty and F be a family of mappings on K . Then (1) for T , £ F , T commutes with T_2 i f and only i f = T T , i . e . T (T 2(x)) = T 2(T (x)) for a l l x £ K , ( i i ) F i s a commuting family i f and only i f . T commutes with T for a l l 48. T r T 2 e F . Theorem 3.6. Let K C X be nonempty weakly compact convex with normal structure w.r.t. U and F be a f i n i t e commuting family of nonexpansive mappings w.r.t. U on K . Then there i s an x e K such that T(x) = x for a l l T e F Proof: Let F = {Tn , .... T } and we may assume that n > 2 . I n - . — weak compactness of K and by Zorn's Lemma, l e t K C K be minimal w.r.t. being nonempty closed convex inv a r i a n t under T_^  for a l l i = l,2,...,n. Since T^ ... T^ : -> i s nonexpansive w.r.t. U and i s weakly compact convex with normal structure w.r.t. U , we see that by C o r o l l a r y 3.4., there i s an x e K. such that T, ... T (x ) = x . Thus the set o 1 1 n o o M = {x s : T^ ... T^(x) = x} i s nonempty. If i e { l , 2, ..., n} and x £ M , then T. ... T (x) = x so that T.(x) =1.(1, ... T (x)) = T, ... T (T.(x)) and I n l l l n l n i so T (x) £ M . Thus T ± ( M ) C M for each i = 1, 2, . . . , n . Also f or each i £ {1, 2, n} and each x £ M , l e t y^ = T^ ... T_^  . . . T n ( x ) > where T. ... T. ... T = T„ ... T i f i = 1, 1 l n . 2 n T, ... T. ... T = T n ... T . i f i = n and T n ... T. ... T 1 l n 1 n-1 1 I n = T 1 . . . T . , T . , 1 . . . T i f 1 < i < n . ' Since T. ( M ) C M for a l l 1 i - l i + l n j — j = 1, 2, . . . , n , we see that y_^  e M . But 4 9 . T.(y.) = T.(T. ... T. ... T (x)) = T n ... T (x) = x . Hence T.(M) = M 1 yi i 1 l n 1 n l for each i e {1, 2, . . ., n} . . I f M contains more than one point, then Co(M) contains more than one point;, since K has normal structure w.r.t. U there e x i s t s a U e U and an x e Co(M) such that 0 < r = r (U; Co(M)) < 6 (Co(M)) . Define C = {y e : r^(U; Co(M)) <_ r} , then C i s nonempty closed and • convex. For each i e {1, 2, n} , i f y c C , then y - Co(M) C rU and so y - M C rU ; since 1\ i s nonexpansive w.r.t. U , i t follows that T (y) - T (M) C rU , and so l^Cy) - M C rU since T ±(M) = M , and hence r^(U; Co(M)) = r y ( U ; M) <_r , so that T ±(y) e C . Thus T (C) C C for each i = 1, 2, . . . , n . Thus C = , by the minimality of K . But then K^^ - Co(M) = C - Co(M) C rU and so 6 (Co(M)) <_r = r (U; Co(M)) < 6 (M) , which i s impossible. Hence M XX X u contains exactly one point, say x , then T_^(x) = x for a l l i = .1, 2, ..., n since T (M) = M for a l l i = l , 2 , . . . , n . One w i l l ask whether the above theorem holds for an i n f i n i t e commuting family F of nonexpansive mappings. This needs a concept so c a l l e d "complete normal structure", and we s h a l l answer t h i s question i n the next section. F i n a l l y we conclude t h i s s ection with some structure of the set of fixed points of a nonexpansive mapping. Theorem 3.7. Let K C X be nonempty weakly compact convex with normal 50. structure w.r.t. ii , T : K -> K be nonexpansive w.r.t. ii and M = {x e K : T(x) = x} . Then f o r any x, y E M with x =f y and 0 < X < 1 , there i s an z E M with P (x - z) = (1 - X)P (x - y) and u u P uCy - z) = XP u(x - y) , for a l l U z ii . I n p a r t i c u l a r , Card M >_ 2 implies Card M > TL0 • Proof: Define A = {z E K : P (x - z) = (1 - X)P (x - y) and P (y - z) = XP (x - y) X u u u u for a l l U £ U} . If U £ ii , then P (x-(Xx+(l-X)y)) = P (( l - X ) x - ( l - X ) y ) = (l-X)P (x-y) u u u and P u ( y - (Xx + (1 - X)y)) = P u(Xy - Xx) = XP y(x - y) , so that Xx + (1 - X)y £ A^ and hence A ^ 0 . Now i f z , z £ A and • A A 1 2 0 <_ y <_ 1 , then P u(x - (Vz1-+ (1 - y ) z 2 ) ) <yP u(x - z ±) + (1 - y)P u(x - z 2) = y ( l - X)P u(x - y) + (1 - y ) ( l - X)P u(x - y) = (1 - X)P u(x - y) , P u ( y - ( y Z ; L + (1 - y ) z 2 ) ) < yP u(y - + (1 - y)P u<y - z 2) = yXP (x - y) + (1 - y)XP (x - y) • u u = XP u(x - y) , and P u(x - y) £ P u ( x - ( y z x + (1 - y)z 2>) + P u ( y - ( y z x + (1 - y ) z 2 ) ) 1 P„Cx - y) , 51. for a l l U e U imply P u(x - (vz± + (1 - y)z 2>) = ( l - A ) P u ( x - y) and P (y - (yz 1 + (1 - y)z„)) = AP (x - y) for a l l U e U . Thus u 1 I u yz + (1 - y)z £ A and so A i s convex. C l e a r l y A i s closed. 1 Z A A A Next i f z E A , then A • P (x - T(z)) = P (T(x) - T(z)) < P (x - z) = (1 - A)P (x - y) , u u — u u P u ( y - T(z)) = P u(T(y) - T(z)) <_ P ^ y - z) = AP^x - y) , P u(x - y) £ P u ( x - T(z)) + P u ( y - T(z)) <_ P ^ x - y) for a l l U £ U , and so P u(x - T(z)) = (1 - X)V^(x - y) and P (y - T(z)) = XP (x - y) for a l l U £ U . Thus T(z) £ A^ and so u u A T(A ) C A . Since A i s weakly compact convex with normal structure A A A w.r.t. U , there i s an z £ A with T(z ) = z , by C o r o l l a r y 3.4. A A A A Thus z £ M . F i n a l l y we s h a l l show that z^ ^ z A i f A., ^ A 0 . A A ^ 1 Z Indeed, choose any U £ U with P (x - y) > 0 . I f z = z^ , then . X]_ A 2 (1 - A )P (x - y) = P (x - z, ) = P (x - z, ) = (1 - A„)P (x - y) so that l u u A ^ . u A 2 2 u A = A . Thus Card M >_ 2 implies Card M >>£ II-4. Complete normal structure. The concept of complete normal structure was f i r s t introduced by L. P. Belluce and W. A. K i r k i n [2]. We s h a l l now generalize t h i s notion to T -1.c.s. 52. D e f i n i t i o n 4.1. Let K C X . Then K i s said to have complete normal  structure w.r.t. _U i f and only i f for each bounded closed convex subset H of K , i f H contains more than one point, then H s a t i s f i e s the following property: (*) For any descending net _ ^ a ^ a e - °^ nonempty subsets of H and for any U e U , i f r(U; W , H) = r(U; H, H) > 0 , for a l l a e r , then C(U; W , H) i s a nonempty proper subset of H. r et Lemma 4.2. • Let K.C X be bounded. Then for each U e U , <S (K) >_ r(U; K, K) • _> -^ -S^ CK) . In- p a r t i c u l a r , i f Card K > 1 , then there i s a U e U such that r(U; K, K) > 0 . Proof: Let U e U . I f r(U; K, K) < "|<5 0 0 , l e t x e K be such that r (U; K) < 4$ (K) . Suppose r (U; K) < r < 4<5 (K) . Since x z u x. 2 u 2r < 6 (K) , there are k.. k, E K such that P (k. - k.) > 2r . But u 1 2 u 1 2 x - K C rU , so that x - k e rU for each i = 1, 2 . Hence P (x - k.) < r , for a l l i = 1,' 2, and so P (k, - k„) < P (k, - x) u i — u 1 z — u 1 + ? u ( x - l ^ ) j< r + r = 2r , which contradicts our choices of k^ and i n K . Thus r(U; K, K) >. j(S (K) . Also i t i s c l e a r that r(U; H,' H) £ 6 U ( K ) • Therefore i f Card K > 1 ,. l e t k , k 2 e K . be such that k^ 4- k 2 , then there i s a U £ U with k^ - k 2 \ U , so that <5 (K) > P (k - k.) > 1 and so r(U; K, K) > 0 . u — u 1 2 53. Proposition 4.3. Let K C X be closed. I f K has complete normal structure, w.r.t. ii , then K has normal structure w.r.t. U Proof: Let H be a bounded convex subset of K containing more than one point, then H i s a bounded closed convex subset of K containing more than one point. Thus there i s a U e U with 6 (H) > 0 u By Lemma 4.2., r(U; H, H) = r(U; H, H) > 0 . Since K has complete , normal structure w.r.t. U . , C(U; H, H) = C(U; H, H) i s a nonempty proper subset of H . Take h e C(U; H, H) , then r h ( U ; H) = r(U; H, H) . Choose any x i n H but not i n C(U; H, H) , then r x ( U ; H) > r(U; H, H). By Lemma 1.1., there i s an x e H such that , o . . Ir, (U; H) - r (U; H) I < -kr (U; H) - r, (U; H)} . I t follows that n x z x h o r (U; H) = r (U; H) < |{r (U; H) + r, (U; H)} < r (U; H) < 6 (H) = <S (H) , x x z x n x u u o o so that x i s a non-diametral point of H w.r.t. U . Hence K has o . normal structure w.r.t. ii Theorem 4.4. Let K C X be compact convex. If H i s any closed convex subset of K containing more than one point, U i s any closed absolutely convex 0-nbhd, and {W } i s a descending net of subsets of H with r(U; W H) = r(U; H, H) > 0 for a l l a z V , then C o C l ^ C(U; W , H)) i s a nonempty proper subset of H . In p a r t i c u l a r , K aer a • ' has complete normal structure w.r.t. ii 5 4 . Proof: P a r t i a l l y order r as follows: i f a, 8 e V , then a < 6 i f and only i f W H> W„ . — a — 3 Case l : Assume that W i s closed and convex for each a a E r . F i r s t we s h a l l show that f o r each E > 0 . there i s an a E T o such that for a l l B e T with g >_ a , sup {inf{P (y-x) : X E ^ W }} < E . 0 y e W g u a e r a Indeed, since each W i s compact, /^ \W i s nonempty. Then for any 1 i ae^ 1 i E > 0 , f~\ W + -Tr eU i s an open set and O W C /*\ W + ^ T E U imply r a 2. a — ' a 2 azT azT. azT there i s an aQ z T such that for - a l l g E r , with g > a , we have C 0 W a + \ z X i ± • T h u s f o r 3 e T with g >_ a Q , i f y s W , then azT y e W + EU and so there i s an x z /~\W such that P (x - y) < he , 8 * a z o " ! , a u o — 2 aeT azT so that inf{P (x - y) : x e } < J e- Hence azY a sup {inf{P (y - x) : x e "^\W }} < e < e , for a l l g e V with 3 > a . yeWg azY Next we s h a l l show that f o r each g E r , r(U; r\ W , H) = r(U;.W , H) = r(U; H, H) . Indeed, l e t r = r(U; ,H) . r, a 8 o a azT azT Since H i s compact convex and W i s nonempty convex, C(U; f\V! ,H) asT a E T i s nonempty closed and convex. Choose any x E C(U; f~\ W , H) . Then asT a r = r(U; f^W , H) = r x ( U ; f\W ) . For each e > 0 , azT azT 1 1 r (U; f~\ W ) < r + -r e , and 3 0 W C x + (r + TTE)U ; also there i s an x 1 1 a o z ' ' a — o z asT aeT W + — E U , and so t u „ a - z E a s r W C x + (r + ^ E ) U + 4 r U = x + (r + e ) U , so that r (U; W ) < r + e • a — o i l o x a _ £ E 55. Hence for each e > 0 , r(U; H, H) = r(U; W , H) <_ r (U; W ) <_ r + e , Oi 2C Ot O e e and thus r(U; H, H) <_ r . For each 8 E r , since W CW , we-see " aeT that 'r = r(U; W , H) <_ r(U; W , H) = r(U; H, H) <_ r Q , so that aeT ' r = r(U; AW , H) = r(U; W H) = r(U; H, H) . ' o . ' 1 a B aeT Third we shall show that 0 f Co(M C(U; W , H)) C C(U; A W , H) . Note that by Proposition 1.4., a — a aeT aeT each C(U; Wa, H) is nonempty. If x e [J C(U; W , H) , then aeT U •,x e C(U; W , H) for some aeT , then r (U; W ) = r(U; W , H) = r , a o x a a o o o o so that r (U; A W ) < r (U; W ) < r . But x ' ' a — x • a — o aeT o r (U; r\ W ) > r(U; A W , H) = r , and so r (U; AW ) = r . Thus x ' 1 ' a — ' ' ' a o ' x ' ' ' a o aeT aeT aeT x E C(U; AW , H) for each x E VJC(U; W , H) so that U C(U; W , H) 1 ' a _ a a a£T asT aET C C(U; H w » R) a n d h e n c e Co"(U C(U; W , H) C. C(U; A W , H) . — I, CL 1 a — r, a asT aeT aeT Finally i t suffices to show that C(U; W , H) C! H . aeT a f Indeed, since & ( f \ W ) > r(U; AW , A W ) > r(U; AW 1, H) = r(U; H, H) aeT aeT aeT aeT > 0 , and W is a compact convex set containing more than one point, aeT a by Theorem 2.3., there is an x E W such that o - 1 ' a aeT r (U; A W ) < S (/~\W ) . But then x ' L a u ' ' a o asT • aeT 56. 6 U ( C ( U ; ft W A , H ) < r Q = r ( U ; ft I T , H ) < r x ( U ; ft < ^ ( ^ V 1 ^ ( H ) aeT aeT o aeT asT so that C ( U ; /^W , H ) C H . r O!r 7^  -Case 2: In general, by Case 1, Proposition 1.4., and Lemma 1.2. , 0 i Co (\J C(U; W , H ) = Co ( U C(U; Co~(W ), H)) r Ct _ Ci, C C(U;/^Co(W ), H ) H . This completes the proof, aer a ^ We are now ready to answer the question raised i n Section II-3. Theorem 4.5. Let K C X be nonempty weakly compact convex with complete normal structure w.r.t. U I f F i s any commuting family of nonexpansive mappings w.r.t. U on K , then there i s an x e K with T(x) = x for a l l T e F . Proof: By weak compactness of K and by Zorn's Lemma, let. K^ be minimal w.r.t. being a nonempty closed convex subset of K inv a r i a n t under each T i n F Suppose K^ contains more than one point, then there i s a U £ U with ^ ( K ) > 0 . By Lemma 4.2., r(U; K.^ K ) > 0 . Let Qt be the family of a l l nonempty f i n i t e subsets of F . For each A e , Let .= ' {x e K : T(x) = x for a l l T £ A} . By Theorem 3.6. , M ^ 0 for each A e Oi . Let A e ($ri be a r b i t r a r i l y A O fixed, and let r = r(U; M , K ) . For each A s (% , let O A X o H = {x e K : r (U;M.) < r } . Since for each x e K , A _L X A o X r (U; M ) <_ r(U; M , K.) i f and only i f r (U; M ) = r(U; M , K) , we X A A J. X A A o o o o see that = C(U; MA , K^ ) , so that is nonempty closed convex, by o o o Lemma 1.1 and Proposition 1.4. It is clear that H„ is closed and convex . A for each A e &l . Denote H = H . Since for any A, B tQl i f 0 AeSt A . ADA — o AC B , then D and so C Hg , i t follows that H q is also convex. Thus H q is convex. Note that for each T e F and each A eQi , T(HA) C H A u { T } . Indeed, i f x e , then x e H A^ T }, and so r x ( U ; MAU{T}J- ro ' I f 7 £ MAU{T} ' t h e n T ( y ) = y s o t h a t P u(T(x)-y) < P u(x - y ) £ r Q implies r T ( x ) ( U ; M A u { T }) < r Q . Thus T(x) e H A ^ T j for each x £ H A so that T(HA) C ^AU^TT. • Hence for each T e- F , T(H ) = T ( U H ) = ( J T(H ) C \ J H. • C \ J H. = H , and so ° A £ ( ^ A Ae&i A ~ Azdi A U { T } ~ Ae(?vA 0 .. ADA ADA • / ADA ADA — o — o ' — o — o T O O C H q ; since T is continuous, T(H) C H . Hence H is nonempty closed convex invariant under each T E F . Thus H = K^, by the minimality of . Suppose E > 0. If X E K 1 = H , then there is an y E H with y - x E — e u • B u t then y E H for some A, E QI with i . 3 A , and so Z A^ • 1 1 — o r (U; M. ) < r < r + ^  s o that y A ' — o o 2 MA C y + (r + 4)UC (x + ^eU) + (r + -k)TJ = x + (r + e)U . • Thus A^ — o I — I o I o Co(MA ) C x + (r + e)U , and hence A Co(M A) C Co(MA ) <Z x + (r + e)U. A — o r. A — A n — o 1 Ae0i 1 A;>A — o I t follows that C\ Co(M ) C A (x + (r + £)U) . But Co(M.) C C o ( M ) , A A . . " T r O A D Aef/i xeK A^ sA — o i f A 3 B and each Co(M ) i s a nonempty weakly closed subset of K , J\ X and so A Co(M^) i s nonempty, since i s weakly compact. Choose any Ae&/ ADA — o , then z £ x + (r + e)U for a l l x e Kn so that Aeoi A 1 ADA — o K C z + (r + e)U . Thus r (U; K_) < r + e so that 1 — z I — o r(U; K , K ) < r (U; K n) < r + e . Thus r(U; L , L ) < r + e for a l l ± 1 — z ± — o ± 1 — o e > 0 and so r(U; K^, K^) _< VQ . On the other hand, r = r(U; M , K n) < r(U; EL , K n) < r . Hence r = r(U; K n, K,) > 0 . o A ± — 1 1 — o O 1 1 O Since f or each A e # t , r(U; M , K ) = r = r(U; K , K ) > 0 , O A X O X X o {M^ : A e&l} forms a decreasing net of nonempty subsets of R ^ C K and K has complete normal structure w.r.t. U , \t C(U; M , K ) i s a A e f o A 1 no nempty proper subset of K . Since H = C(U; M A , ) for each A £&l, X A A X i s a nonempty proper subset of K . Since i s - a net of convex sets, \_) H and so {J H i s also convex. Thus for each Ae&l A £0l T e F , we have 59. T ( U H ) C T ( U H ) = ( J T(H ) C [J H , , C ( J . This contradicts the Ae©i Aefo As&t- Aefh ' AeSc minimality of . Hence contains exactly one point, say X q . Then since T(K.) C l for each T e F , T(x ) = x for each T e F . 1 — 1 o o Corol l a r y 4.6. Let K C X be nonempty compact convex. I f F ' i s any commuting family of nonexpansive mappings w.r.t. U on K , then there i s an x e K with T(x) = x for each T £ F 60. CHAPTER III FIXED POINT THEOREMS FOR NONEXPANSIVE MAPPINGS I I I - l . Relative nonexpansive mappings. In this chapter we s h a l l investigate some f i x e d point theorems for various kinds of nonexpansive mappings. Lemma 1.1. Let X be a to p o l o g i c a l vector space and {x^, x^} be a f i n i t e subset of X . Then Co({x^, . .., x n ^ ) 1 S compact. Proof: Define T : [0, 1] x . . . x [0, 1] x {x±} x . . . x {xn> -»• Co({x 1, x n>) by n T(X, , . . . , A , x, , . . . , x ) = y X.x. . Then c l e a r l y T i s continuous 1 n 1 n . L n l l J 1=1 and onto. Since [0, 1] x . . . x [0, 1] x {x.^ } x . . . x {x^} i s compact, by Tichonov, we see that Co({x^, x }) i s also compact. Next we s h a l l show that Mazur's theorem [6, p. 416-417] holds i n T ^ - l . c . s . Theorem 1.2. Let X be a complete T ^ - l . c . s . I f K C X i s compact, then Co(K) i s also compact. 61. Proof: Since Co(K) i s closed and X i s complete, Co(K) i s complete. Thus by Hausdorff's Theorem on t o t a l boundedness i n [9, p. 61], i t remains to show that Co(K) i s also t o t a l l y bounded. Let U be any closed absolutely convex 0-nbhd i n X . Since K i s compact, there are d i s t i n c t e K with n 1 — 1 K C LJx. + T-U . If y e Co(K) , then y + T U n c ° ( K ) ^ 0 • Choose any i-1 1 1 1 z s y + -^L7^Co(K) , then y - z e -yU and so y = z - (z - y) £ Co(K) - -|u = Co(K) + -|u , for a l l y e Co(K) . Thus — 1 n 1 C o ( K ) C C o ( K ) +-rU . Since K C ( J x . +-rU , define v : K {1,2,.. . ,n} 1=1 1 such that y - x , N E -yU for a l l y e K . If y E Co (K) , then v(y) 4 m m y = 1 ^ i y i ^ o r s o m e • • • » Q K a n d 0 _< ^ <_ 1 with £ ^ i = -» i = l - ^ ^ i-1 thus m m m y - y X.x , N = Y X.y. - y X.x , \ i = l 1 v ( y i } i = l 1 1 i = l 1 v ( y i } m = V X . ( y . - x • , . ) m E IX.(iu) i = l m i I and so y e y X.x , , + T U C Co({x. , . . . , x }) + -r U . L\ l v (y.) 4 — 1 n 4 i = l I Hence Co(K) C Co({x 1, . . . , x n>) + -|- U and so 62. Co(K) C Co(K) + ^ U C Co({x 1, . . . , x ^ ) + -| U . By Lemma 1.1. , Co({x^, x n } ) 1 S compact. Thus there are y^, . .. , y^ i n «- i Co({x , . .., x }) C. Co(K) such that Co({x , . . ., x }) C U Y • + J U • x t follows & £ Co(K) C Co({x x }) + y U C ( \ J ( y . + y U)) + ± U = | J y . + U . j =1 j = l J Hence Co(K) i s also t o t a l l y bounded. Therefore Co(K) i s compact. D e f i n i t i o n 1.3. Let X be a T ^ - l . c . s . , U be a base for closed absolutely convex 0-nbhds i n X . and M, K Cj- X be nonempty with M C K . Then T : K -* K i s said to be nonexpansive r e l a t i v e to M w.r.t. U i f and only i f for any U e U, P (T(x) - T(y)) <_ P u ( x - y) , for a l l x e M and. a l l y e K . Theorem 1.4. Let X be a complete T ^ - l . c . s . , U be a base for closed absolutely convex 0-nbhds i n X , K C X be nonempty bounded closed convex, M C K be nonempty compact and T : K -> K be nonexpansive r e l a t i v e to M w.r.t. U such that ( i ) Co"(0(T, o, x ) ) f \ M ^ 0 for a l l x e K and ( i i ) T(M) C M . Then there i s an x e M with T(x) = x . Proof: Let F = (A C K : A i s nonempty closed convex and T(A) C A} . P a r t i a l l y order F by I> .. Then by ( i ) , compactness of M and Zorn's Lemma, F has a minimal element, say . Let = M. Note that i s nonempty compact and T(M ) C . Let G = (B C M : B i s nonempty compact and T(B) C B}. Then G <fc <j> , for e G . P a r t i a l l y order G by ID . Then by compact-ness of and Zorn's Lemma, G has a minimal element, say . Since T i s continuous on M and so on M 2 and i s compact, T(M 2) i s compact. Thus T(M 2) = M 2 by minimality of M 2 i n G .Suppose M2 contains more than one point, then there i s a U e U with & > ^ ' Since X i s complete and M 2 i s compact, i t follows that Co(M 2) i s also compact, by Theorem 1.2. Since & (Co(M„)) = 5 (M„) > 0 , by Proposition I I - l . 3 . u 2 u 2 and by Theorem II-2.3., there i s an X q e Co(M 2) such that 0 < r = r ^ (U; Co(M 2)) <<5u(Co(M2)) . o Let S = {x e K : r (U; M.) < r = r (U; Co(M n))} . 1 x 2 — x 2 o Then S # (j) since x^ £ S . By Lemma I I - l . l . S i s closed and convex. We s h a l l show that T(S) C S . Indeed, suppose x £ S and z e M 2 = T(M 2), then there i s an w e M 2 C M with T(w) = z . Thus P (T(x) - z) = P (T(x) - T(w)) < P (x - w) < r , and so x , . (U; M ) <_ x XX XX XX J- \2Cj implies T(x) e S for a l l x e S . Hence T(s) C S . Thus S £ F and so S = by the minimality of i n F But S - C rD , and so M2 ^ K i = S implies M 2 ~ M2 — r U ' T t f o l l o w s t h a t <$ (M ) <_ x = r (U; M ) = r (U; Co~(M ) ) < 6 (Co(M )) = 6 (M„) , which i s XX X —^ X JL XX XX Z-o o 64. impossible. Hence must contain only a s i n g l e point, say X q , and so T(x ) = x since T(M„) = M~ . Moreover x e M„ C M . o o 2 2 o 2 — III-2. A f f i n e mappings and convex mappings. Throughout the remaining of t h i s chapter, X w i l l denote a T ^ - l . c . s . and U a base f o r closed absolutely convex 0-nbhds i n X . If K Q i s nonempty, T : K -> K i s said to be a f f i n e i f and only i f f o r any x, y e K and 0 <_ X <_ 1 , T(Xx + (1 - X)y) = XT(x) + (l-X)T(y) Note that i f T i s a f f i n e , i t can be shown by induction that i f n x,, ..., x s K and 0 < A. < 1 , 7 A. = 1 , then 1 n — • i — i i = l n n T( y A.x.) = £ A.T(x.) . A mapping T : K -> K i s said to be convex i = l i = l w.r.t. _U i f and only i f for any U e U , we have P (T(Ax + (1 - A)y)) < AP (T(x)) + (1 - A)P (T(y)) f o r a l l x, y e K and u — u u . 0 _< X <_ 1 . In t h i s case, one can show by induction that for n x 1, • • • , x n E K , 0 <_ X ± <_ 1 , I X± = 1 , i = l n n P (T( Y X.x.)) < Y X.P (T(x.)) f o r a l l U £ il . Note that T i s a f f i n e u . L\ i i — . u\ l u 1 i = l i = l . • . implies T i s convex w.r.t. U 65. Theorem 2.1. Let K C X be nonempty weakly compact convex, T : K -> K be continuous such that I-T i s convex w.r.t. Li on K . Suppose for any U e Li , inf{P (T(x) - x) : x e K} = 0 . Then there i s an x e K with u T(x) = x . Proof: For each U E Li and each r > 0 , l e t H (U) = {z E K : P (T(z) - z) < r} . Since inf{P (T(x) - x) : x e K} = 0 r u — u we see that H (U) =/ 0 for any r > 0 . I f z E H (U) and z z , r A r A then T(z ) - z -> T(z) - z for T i s continuous, and so P (T(z) - z) <_ A A U since P (T(z.) - z.) < r for a l l A . Thus z £ H (U) and so H (U) u A A r r i s closed. Next suppose z^, £ H (U) and 0 <_ A _ < 1 . Then P u ( T ( A z 1 + (1 - X)z 2) - (Az 1 + (1 - A)z 2 ) ) = P u ( ( I - T ) ( A Z ; L + (1 - X)z 2 ) ) < XP ((I - T)(z.)) + (1 - X)P ((I - T ) ( z 9 ) ) — u 1 u 2 <_ Xr + (1 - X)r = r , _ • since I - T i s convex w.r.t. Li . I t follows that Xz^ + (1 - X ) z 2 E H^CU) , and consequently H^CU) i s convex. Note that H^CU) i s a weakly closed nonempty .subset of K , for a l l r > 0 . Now i f r n , . . . , r > 0 , l e t r = min{r n , . . ., r } , then H (TJ) C H (U) for 1 n 1 n r — r . 66. n a l l i = 1, 2, n so that (~\ H (U) D H (U) ^ 0 . Since . , r . — r i = l I {H^(U) : r > 0} has f i n i t e i n t e r s e c t i o n property and K i s weakly compact, H(U) = f~\ H (U) ± 0 . Thus for each U e U , H(U) i s a r>0 r nonempty closed convex and so weakly closed subset of K . If U^, e and D. C U , then P (x) _> P (x) , for a l l x e K , and so U l U2 H (U ) C H (U ) for a l l r > 0 and so H(U ) = /°\ H (U ) C (*\ H (U ) r r i ' r>0 r r>0 r m = H(U„) . If X L , U e U , then there i s a U e U with D C Au, 2 1 m — 1 ' I i = l since U C U. f o r each i = 1, 2, . . . , m , i t follows H(U) CH(!L) for m each d = 1, 2, . . . , m ., and so ft H(U ) D H ( U ) ^ 0 . Therefore i=l 1 ^ {H(U) : U E U} has f i n i t e i n t e r s e c t i o n property. Thus H = O.H(U) i= 0 UEU by weak compactness of K . F i n a l l y we s h a l l show that x e H implies T(x) = x . Indeed i f x e H , then x E H(U) for a l l U e U and so x E H^CU) for a l l r _> 0 and a l l U E U and so x E H (U) for a l l U E U . Thus P (T(x) - x) <_ 1 for a l l U e U implies T(x) - x e U for a l l U E U and so T(x) - x e f~\U = {0}. Consequently T(x) = x . UeU. Theorem 2.2. Let K C X be nonempty closed convex, T : K ->- K and M C K be nonempty weakly compact with MnCo(0(T, o, x)) ^ 0 for a l l x e K . Suppose ( i ) T i s nonexpansive w.r.t. U on 0(T, 1, x) f o r a l l x e K i ( i i ) x =f Tx implies T i s not an isometry w.r.t. (J on 0(T, 1, x) and ( i i i ) I - T n i s convex w.r.t. (J for each n = 1,2,... Then there e x i s t s an x e M with T(x) = x . Proof: By weak compactness of M and by Zorn's Lemma, l e t be minimal w.r.t. being a nonempty closed convex subset of K inv a r i a n t under T . Suppose there were an x e with T(x) f x . Then by ( i i ) , there i s a U e il and there are p o s i t i v e integers k and j with P u ( T k + 1 ( x ) - T j + 1 ( x ) ) i P u ( T k ( x ) - T j ( x ) ) . By ( i ) , i t follows that P u ( T k + 1 ( x ) - T j + 1 ( x ) ) < P u ( T k ( x ) - T j ( x ) ) . Assume k > j and k = n + j , then for w = T^^Cx) and y = T k(x) , we see that P u(w - TnfVj) < P u ( y - Tn(y)) . Define H = {z e K : ? u ( z - Tn(z)) <_ P^(w - T%))}. Then H i s nonempty since w e H . By ( i i i ) , H i s convex. C l e a r l y H i s also closed. Hence H = by the minimality of . But y e K while y £ H which i s a con t r a d i c t i o n . Hence T(x) = x for a l l x z K^. Since ^ 0 , there i s an x. z with T(x) = x . But then x must be i n M since MP\Co(0(T, o, x)) =/ 0 . Theorem 2.3. Let K C X be nonempty bounded closed convex' and sequentially complete, MC K be nonempty compact and T : K -> K be nonexpansive w.r.t. U • Suppose ( i ) M/~\Co(0(T, 1, x)) ^ 0 for a l l x e K and ( i i ) for each x £ K- and U e il , P ( y - T ( y ) ) <_P ( x - T ( x ) ) for a l l y £ Co(0(T, 1, x)) . Then there i s an x e M with T(x ) = x . o o o 68. Proof: Suppose T(x) r x for each x e M . Define A = {T(x) - x : x e M} . Suppose x e M and T(x ) - x ->• y . Then M A A A i s compact implies there i s a subnet {x, } with x, -> x e M . Thus T(x, ) T(x ) , and so T(x, ) - x, -> T(x ) - x . Since X i s T- , X g o X g . X e o o _ 2 y = T(x ) - X q e A . Hence A i s closed. Since o £ A , there i s a U £ U with A A D = 0 , and i t follows that: (*) P (T(m) - m) > 1 for a l l m £ M . Since K i s bounded, there i s 6 > 0 with K S= 6U , and so K - K C 26U and consequently ? u ( x - y) <_ 26 for a l l x, y e K . Choose 0 < q < 1 with (1 - q)•26 < -j . F i x an x^ E K . Define f : K K by f (x) = qT(x) + (1 - q)x Q , for a l l x £ K . By Example 1-4.2.; f i s s t r i c t l y contractive w.r.t. U . Since K i s se q u e n t i a l l y complete, there i s a unique y^ £ K with f ( Y Q ) - YQ , by Theorem 1-2.4. Thus < (1 - q)-26 and so (**) P u ( y Q - T ( y Q ) ) < -| . Since Mr\Co(0(T, 1, y)) 4- 0 , choose any m E M/~\Co(0(T, 1, y )) . Then there exists an z £ Co(0(T, 1, y)) with (***) P (m - z) < \ . Hence u 3 P u(m - T(m)') = P u(m - z + z .- T(z) + T(z) - T(m)) < P (m - z) + P (z - T(z)) + P (T(z) - T(m)) — u u u < P (m - z) + P (y - T(y )) + P (z - m) — u u Jo Jo u 3. 3 3 = 1 , by ( i i ) , (**) and (***) . This contradicts (*) . Hence there i s an x e M such that T(x ) = x o o o Theorem 2.4. Let K C X be nonempty bounded closed convex and seq u e n t i a l l y complete, T : K ->- K be a f f i n e and nonexpansive w.r.t. U Then either (i) K contains only a s i n g l e points or ( i i ) some proper nonempty closed convex subset of K i s inv a r i a n t under T . Proof: Suppose no proper nonempty closed convex subsets of K i s i n v a r i a n t under T , then since T(Co(0(T, 1, x))) C T ( C o ( 0 ( T , 1, x))TC Co(0(T, 1, x)) for each x e K , K = Co(0(T, 1, x)) , for each x e K . F i x any X q e K and l e t M =' { X Q } . Then X q e K = Co(0(T, 1, x)) for each x e K implies MACo(0(T, 1, x)) ^0 for each x e K so that the hypothesis ( i ) of Theorem 2.3. i s s a t i s f i e d . Moreover, for any x e K , i f y e Co(0(T,l,x)), m n. m say y = £ X.T (x) , where 0 X. <_ 1 and £ X. = 1 , then i = l 1 1 i = l 1 70. m n.+l T(y) = I X ±T 1 (x) since T i s a f f i n e , and so for U e U , i = l m n. m n.+l P (y " T(y)) = P ( I X.T ^(x) - I X.T 1 (x)) i = l i = l m n. n.+l < I X.P (T X ( x ) - T 1 (x))-i = l m •<• I X.P (x - T(x)) — . -. l u i = l = P u ( x - T(x)) , so that the hypothesis ( i i ) of Theorem 2.3. i s also s a t i s f i e d . Since M = {x } i s compact, Theorem 2.3. implies T(x ) = x . Thus o o o K = Co(0(T, 1, x^)) = a n d so K contains only a s i n g l e point. Co r o l l a r y 2.5. Let K <C X be nonempty bounded closed convex se q u e n t i a l l y complete, MC K be nonempty weakly compact and T : K -> K be a f f i n e and nonexpansive w.r.t. U . I f MACo(0(T, 1, x)) 0 for each x e K , then there i s an x e M with T(x) = x . Proof: By weak compactness of M and Zorn's Lemma, l e t be minimal w.r.t. being a nonempty closed convex subset of K i n v a r i a n t under T . Since , being a subset of K , i s also bounded and s e q u e n t i a l l y complete, the minimality of and Theorem 2.4. imply contains only a s i n g l e point, say x . Thus T(K^) C implies T(x) = x . Since 71. M P v C o ( 0 ( i y 1, x)) + 0 and {x} = Co(0(T, 1, x)) , we see that x e M . III-3. Mappings with diminishing o r b i t a l diameters. The concept of diminishing o r b i t a l diameters has been introduced by W. A. Kirk i n [1]. We s h a l l generalize t h i s concept into T 2 ~ l . c . s . D e f i n i t i o n 3.1. I f K C X i s nonempty, T : K K and x £ K , then T i s said to have diminishing o r b i t a l diameters w.r.t. _U at_ x i f and only i f for any U e U , $^(0(.T, o, x)) > 0 implies li m 6 (0(T, n, x)) < <5 (0(T, o, x)) . T i s said to have diminishing u u n-*=° o r b i t a l diameters (d.o.d.) w.r.t. _U i f and only i f T has diminishing o r b i t a l diameters w.r.t. U at y for each y i n K . Since 0(T, n+1, x) C 0(T, n, x) f o r each n = 0,1,2,... and each x e K , we see that 6 u(0(T, n+1, x)) <_ 8^(0(1, n, x)) for any U e U so that l i m <5 (0(T, n, x)) e x i s t s . Thus the above notion i s n-*» well-defined. By Remark 1-3.9., T i s s t r i c t l y contractive w.r.t. U implies T has d.o.d. w.r.t. U 72. Theorem 3.2. Let KC! X be nonempty compact and T : K -> K be continuous with diminishing o r b i t a l diameters w.r.t. U . Then there i s an x e K such that T(x) = x . Proof: By compactness of K and Zorn's Lemma, l e t K^ be minimal w.r.t. being a nonempty closed subset of K i n v a r i a n t under T . Suppose there i s an x e with T(x) ^  x , then there i s a U e U with 6 (0(T, o, x)) > 0 , so that lim 5 (0(T, n, x)) < 6 (0(T, o, x)) , since h-*» u T has diminishing o r b i t a l diameters w.r.t. U . Thus there i s a p o s i t i v e integer N with r = &^(0(T, N, x)) < 6^(0(1, o, x)) . I t follows that r = 6 u(0(T, N, x)) < <Su(0(T, o, x) )<_ ^ 0 0 , and so 0 i 0(T, N, x) <^  K . But 0(T, N, x) i s invar i a n t under T , and thus 0(T, N, x) i s invar i a n t under T as T i s continuous, which contradicts the minimality of . Thus T(x) = x for each x e . Since ^ 0 , there i s an x e C K with T(x) = x . Theorem 3.3. . . . Let K C X be nonempty weakly compact, T : K K be nonexpansive w.r.t. U . Suppose there does not e x i s t nonempty proper weakly compact subset of K which i s in v a r i a n t under T . If for some x e K , T has d.o.d. w.r.t. U at x , then T(x) = x . 73. Proof: F i r s t we note that every weakly compact subset of X i s bounded, by Theorem 17.5, p. 155 i n K e l l e y and Namioka [11]. Suppose T(x) i x , then there i s a U e U with 6^(0(T, o, x)) > 0 . Since T has d.o.d. w.r.t. U at x , there i s a p o s i t i v e integer N with r = 6 u(0(T, N, x)) < <5u(0(T, o, x)) . Take X q = T N(x) , then P^Cx^ - T 1 ( x ) ) <_ r for each p o s i t i v e integer i _> N . For each e > 0 , l e t S = {y e X : P ( y - T ^ x ) ) <_r+e, for a l l but a f i n i t e number of p o s i t i v e integers i} . Define S = S £ > 0 C l e a r l y each S i s convex and hence S i s convex. If {y } i s a £ X XET net i n S and y -> y i n X , then for any £ > 0 , there i s an X X o with P (y, - y) < , for a l l X > X Since y, £ S C S ,„ , there u X — 2 — o A — £/2 o i s a p o s i t i v e integer N with P (y - T" L(x)) < r + (E/2) , f o r a l l o u A o i > N . Thus for a l l i > N , — o — o P u ( y - T 1 ( x ) ) < P u ( y - y x ) + P U ( Y X - T X ( x ) ) £ f + ( r + f ) = r + e ; o o thus y £ S , for a l l E > 0 so that y e S = S . Therefore S i s £ £>0 £ also closed. Since X Q E S f l K , we see that M = S A K i s a nonempty weakly compact subset of K . I t i s clear that T(S r\ K) C S P» K , for £ — e each E > 0 and so T(S f~\ K) C S f\K and hence T(M) C M . Thus M = K , by the hypothesis on K . For each £ > 0 , w E K , l e t B £(w) = {y E X : P u(w - y) <_ r + E} . I f {w i s . . . , w } C K = M = S A K , 74. then f o r each e > o , {w , ..., w } C S , and so there are p o s i t i v e 1 n — e integers N n, .... N with P (w. - T 1 ( x ) ) < r + e for a l l i > N. and I n u j v / - j for each j = 1, 2, . .., n . Thus f or N = max{N., , . .., N } and i > N , 1 n P (w. - T 1 ( x ) ) <_ r + e for a l l j = 1, 2, ... , n , so that T X(x) £ B (w.) n 3 _ , £ 3 n for a l l j = 1, 2, . .., n , and hence T (x) E (w . )/OK . I t i s cl e a r 3=1 G 2 that B (w) i s closed and convex f o r each e > 0 and each w E K . Thus £ for E > 0 , {B^(w)/^\K : w E K} i s a family of weakly closed subsets of K with f i n i t e i n t e r s e c t i o n property, and i t follows that C = ( C\ B (w)) r\ K 4- 0 , by weak compactness of K . I t i s also c l e a r £ WEK £ that C C C £ i f 0 < £ < _ £ „ . Thus {C : e > 0} i s a family of e l 2 £ weakly closed subsets of K with f i n i t e i n t e r s e c t i o n property, and hence C = C 4 0 , again by the weak compactness of K . E>0 E I f v £ C , then P^(w - v) <_ r + £ , for each w E K and each E > 0 , so that R V ( U ; K) <_ r + E for a l l E > 0 and thus r V ( U ; K) <_ r . Define R = {y E X : r Y ( U ; K ) £ r} , then R 4 0 since 0 4 C C R H K . C l e a r l y R i s closed and convex, and so R C\ K i s a nonempty weakly closed subset of K . If y e R r\ K , then r^(H; K) <_ r . Fix any £ > 0 , then y - K C ( r + E ) U implies T(y) - T(K) C (r + s ) U since T i s nonexpansive w.r.t. U . Thus T(K) C T(y) +,(r + E ) U , and i t follows T(K A T ( y ) + (r + e ) D ) C T ( K ) C l ( y ) + (r + E ) U . Since T(K) C K , we see that T(K f l T ( y ) + (r + e)U) C K A T ( y ) + (r + s ) U . 75. Hence K/^\T(y) + (r + e)U i s a nonempty weakly compact subset of K inva r i a n t under T , and therefore K A T ( y ) + (r + e)U = K . I t follows that K C T(y) + (r + E ) U and so T(y) - K C (r + e)U . Hence r T ^ ( U ; K) <_ r + e for a l l e > 0 , and so r T ^ ( U ; K) <_ r . Thus T(y) e R A K for each y e R A. K , so that T(R A K) C R A K . By hypothesis on K , RAK = K . But then <$u(K) = <S u(KAR) <_. r < S u(0(T, o, x) ) <_ 6 (K) , which i s impossible. Therefore we must have T(x) = x . Coroll a r y 3.4. Let K C X be nonempty weakly compact and T : K K be nonexpansive w.r.t. U . I f T has d.o.d. w.r.t. U , then there i s an x e K with T(x) = x . Proof: By weak compactness of K and by Zorn's Lemma, l e t be minimal w.r.t. being a nonempty weakly compact subset of K in v a r i a n t under T . Since T has d.o.d. w.r.t. U on K and hence on , T(x) = x , for a l l x e , by Theorem 3.3. Since K ^ 0 , there i s an x e K C K with T(x) = x . Coroll a r y 3.5. Let K C X be nonempty weakly compact convex and T : K ->- K be nonexpansive w.r.t. U . Suppose there i s a p o s i t i v e integer N N such that T : K -> K has d.o.d.w.r.t. U . Then there i s an x e K with T(x) = x . 76. Proof: By weak compactness of K and Zorn's Lemma, l e t be minimal w.r.t. being a nonempty closed convex subset of K in v a r i a n t under N T . Since i s also weakly compact, T : -> K i s nonexpansive-w.r.t. U and has d.o.d. w.r.t. U , there i s an x e with T^(x) = x , by Coro l l a r y 3.4. Suppose T(x) 4 x , then M = {x, T(x), T N - 1 ( x ) } contains more than one point, and so Co(M) contains more than one point. By Lemma 1.1., Co(M) i s compact, and so by Theorem II-2.3., there are a U e U and an x e Co(M) with 0 < r = r (U; Co(M)) < <5 (Co(M)) . o o x u o Let R = {y e K^ : r^(U; M) _< r Q } . By Lemma I I - l . l . , R i s closed and convex. Also• R 4 0 since x e R . If y e R , then r (U: M) < r , o y — o so that P u(T(y)-- x) =-Pu(T(y) - T N ( x ) ) <_ P^Cy - T N _ 1 ( x ) ) _< r Q and for 1 £ m < N , P u(T(y) - T m ( x ) ) _< P ^ y - T m _ 1 ( x ) ) 1 r •' Thus T(y) e R for each y e R . Hence T(R)C. R , and so R = , by the minimality of . But 6 (K ) = 6 (R) < r < 6 (Co(M)) < 6 (K) , which i s impossible. 1 u 1 u — o- u — u Thus T(x) = x . Proposition 3.6. Let K C X be nonempty weakly compact. Then ( i ) for any U e U , there i s an x e K with P (x ) = inf{P (k) : k e K} and u u u u ( i i ) o i K implies there i s a U e U with inf{P (k) : k e K} > 0 . 1 n 77. Proof: (i) Suppose U e U . Then f or any X > 0 such that XU C\ K 4 0 , XU r\ K i s a nonempty weakly closed subset of K . Thus the i n t e r -section of a l l such nonempty sets XUA K i s nonempty, by the weak compactness of K . Take any x . i n that i n t e r s e c t i o n . Then i t i s u c l e a r that P (x )= inf{P (k) : k e K} . u u u ( i i ) I f o £ K , then since K i s weakly closed there i s a closed absolutely convex 0-nbhd U i n the weak o topology such that U' K = 0 . But then U i s also an 0-nbhd i n o o X so that there i s a U z U with U C U . Hence — o P (k) > P (k) > 1 , for a l l k E K . Thus u • — u o inf{P u(k) : k £ K} _> 1 > 0 . I t i s clear that i f K^, K C X are weakly compact, then K^ - i s weakly compact. Thus we have the following: C o r o l l a r y 3.7. If K^, K ^ C X are d i s j o i n t nonempty weakly compact sets, then ( i ) f or each U £ U , there are x^ s and x^ £ such that P u ( x 1 - x 2) = i n f { P u ( k 1 - k 2) : e K^, k 2 e K 2> ; ( i i ) there i s a U £ U with inf{P (k - k2) : k E K , k 2 e K} > 0 . Theorem 3.8. Let K C X be nonempty .weakly compact and T : K -> K be 78. contractive w.r.t. U . Suppose f o r each x e K , there i s a p o s i t i v e integer N(x) such that T N has d.o.d. w.r.t. U at x . Then there i s an x e K with T(x) = x . Proof: By weak compactness of K and by Zorn's Lemma, l e t be minimal w.r.t. being a nonempty weakly closed subset of K inva r i a n t under T . For each n = 2, 3, ... , l e t be minimal w.r.t. being a nonempty weakly closed subset of inva r i a n t under T n . C O Case 1. Suppose ' f~\ K = 0 . Since each K i s weakly n=l m compact, there i s a p o s i t i v e integer m > 2 such that f~\ K =0 while n=l n m-1 m-1 f~\ K 4 0 • Since K and f\ K are nonempty d i s j o i n t weakly•compact n=l n=l m-1 sets, by C o r o l l a r y 3.7., there are U e U , x e K and y e A K such m ' \ n n=l m-1 that P (x - y) = i n f {P ( w - v ) : w e K , v e /"\ K } > 0 . Since u u m ' % n n=l m' . A} J T "(K ) C_ K for each n = 1, 2, . .., n and T " i s contractive n — n x rnt Tn t w.r.t. U , we see that P^T (x) - T (y)) < P^(x - y) . But T m-(x) e K and T m" (y) e r\K imply P (x - y) < P (T m" (x) - T m - ( y ) ) , m * ' ' n u — u n=l which i s a co n t r a d i c t i o n . Hence we must have: Case 2. A K 4 0 • Choose any z E A K . Then tnere , n ' \ n n=l n=l i s a p o s i t i v e integer N(z) such that T^ has d.o.d. w.r.t. U at z . Since f = T^ : -> i s nonexpansive w.r.t. U and f has d.o.d.w.r.t. 79. N U at z e , we must have T ( 2 ) = f(z) = 2 , by Theorem 3.3. Since N T i s contractive and T (z) = z , we must have T(z) = z , . by Proposition 1-2.3. Theorem 3.9. Let K C X be nonempty bounded closed convex, T : K K be nonexpansive w.r.t. U such that no nonempty proper closed convex subset of K i s inva r i a n t under T . Suppose (i) M C K i s nonempty weakly compact such that Co(0(T, o, x))O.M 4 0 for each x e K , and ( i i ) f o r some z e K , T has d.o.d. w..r.t. U at z . Then T(z) = z . Proof: Suppose T(z) =/ z , then 0(T, o, z) contains more than one point. Let U E II with 6^(0(T, o, z)) > 0 , and so lim 8 (0(T, n, z)) < 8 (0(T, o, z)) . Thus there i s a p o s i t i v e integer N n-x» with r = 6 u(0(T, N, z)) < 6 (0(T, o, z)) . Define B = {x E K : r (U; 0(T, n, x) _< r , for some p o s i t i v e integer n} . Since N T (z) E B , B i 0 . C l e a r l y B i s convex and T(B) C B , and so T(B) C 1(B) C B . Hence B = K , by hypothesis on K . Since Co(0(T, i r ^ , z))DCo(0(I, m2, z)) f o r ir^ <_ m2 Co(0(T, m, z)) i s closed convex for each m = o, 1, 2, ... and Co(0(T, m, z))P\M i 0 for each m = 0, 1, 2, ... , i t follows that CO /^\Co(0(T, m, z)) C\¥L f 0 , by weak compactness of M . Suppose m=0 OO t E /^Co(0(T, m, x ) ) A M . If q E K = B" , then for any E > 0 , there i s m=0 80. an e B with ? u ( q - q ) Ji e • But then there i s a p o s i t i v e integer N with r (U; 0(T, N , z)) < r . Thus f o r n > N , £ q £ — — £ £ P (q - T n ( z ) ) < P (q - q ) + P (q - T n ( z ) ) < r + £ , U — U £ U £ — so that r (U; Co(0(T, N , z))) = r (U; 0(T, N , z)) < r + £ . Since q e q e — t E Co(0(T, N , z)) , P (t - q) < r + £ , for any E > 0 so that £ U — P^(t - q) <_ r which i s true f or any q g K . Thus r (U; K) <_ r , C O V t £ n"Co"(0(T, m, z ) ) A M . m=0 Define S =' {x e K : r^CU; K) <_ r} , then S 4 0 since C O 0 4 f~\ Co~(0(T, m, z ) ) A M C S . By Lemma I I - l . l . , S i s closed and convex. m=0 Suppose there i s an x e S with T(x) S . Then there i s an e > 0 such that KCj: T(x) + (r + e ) U . Let H = K A T ( x ) + (r + e ) U . C l e a r l y , H i s a nonempty closed convex subset of K . I f y e H , then y £ K C x + (r + e)U implies P (T(y) - T(x)) <_ P^(y - x) <_ r + e , so that T(y) E K H T(x) + (r + e ) U = H . Thus T(H) C H . Hence by hypothesis on K , K = H = T(x) + (r + e J D H K , so that K C T ( x ) + (r + £)U , which i s a cont r a d i c t i o n . Hence T(S) C S . But then S = K , again by the hypothesis on K . Thus <5 (K) = S (S) < r < $ (0(T, o, z)) < 6 (K) , which i s impossible. Therefore u u — u — u we must have T(z) = z . 81. Co r o l l a r y 3.10. Let K C X be nonempty bounded closed convex, T : K -> K be nonexpansive w.r.t. (J and M C K be nonempty weakly compact. Suppose (i) Co(0(T, o, x)) A M 4 0 for each x e K and ( i i ) T has d.o.d. w.r.t. U . Then there i s an x e M with T(x) = x . Proof: By weak compactness of M and by Zorn's Lemma, l e t K^ be minimal w.r.t. being a nonempty closed convex subset of K inv a r i a n t under T . Thus by Theorem 3.9., T(x) = x for a l l x i n K^ . Since K 1 i 0 , there i s an x e K C K with T(x) = x . By ( i ) , x e M . We s h a l l now discuss the fi x e d point sets of a f i n i t e commuting family of nonexpansive mappings with d.o.d. w.r.t. (J Lemma 3.11. Let K £ x be nonempty bounded, • f = {T^, T n> be a f i n i t e commuting family of nonexpansive mappings w.r.t. U o n K . If for each i = 1, 2, n , T_^  has d.o.d. w.r.t. (J , then f o r any x e K with T^ ... I n ( x ) = x , we have for each U e (J , 5 (OCljL* m> x ) ) = 5 u ( ° ( T i ' °' x ) ) > f o r e a c h i = 1, 2, n , and m = l , 2, ... . Proof: Suppose U-e.-U and m i s any p o s i t i v e integer. I f we denote T T ... T = T ... T , T n ... T n T = T ... T , , and for 1 2 • n 2 n 1 n-1 n. 1 n-1 1 < i < n , T ... T. ... T =• T n ... T T . , n ....T , then f o r each 1 l n 1 l - l l + l n 82. i e (1, 2, n} , 6 U ^ T 1 T i ^ ^ ^ i ' m> 1 S u ^ ° ^ T i ' m ' x ^ since T i s nonexpansive w.r.t. U for each j = 1, 2, n . But (T, ... T. ... T )0(T., m, x) = 0(T., m-1, x) since T n ... T (x) = x , 1 l n l . x 1 n and so 6 (0(1^, m-1, x)) <_ 6 (0(T , m, x)) . Also 0(T ±, m, x) C 0(T., m-1, x) implies 5 ( 0 0 ^ , m, x)) <_ 6 (0 0^, m-1, x)) Hence 6^(0(1 , m, x)) = 5 ( 0 ( ^ 9 m - 1 > x ) ) • Therefore 6 u ( 0 ( T ± , m, x)) = 6 (0(T , o, x)) for each m = 1, 2, ... , and each i = 1, 2, ..., n and each U e U . . Theorem 3.12. Let K C X be nonempty bounded and f = (T ,. . . , T^} be a f i n i t e commuting family of nonexpansive mappings w.r.t. U on K . If T_^  has d.o.d. w.r.t. U on K for each i e {1, 2, n} , then for each x e K , T, ... T (x) = x i f and only i f T.(x) = x for each I n x i = 1, 2, ..., n . In p a r t i c u l a r , {x e K : T, ... T (x) = x} = {x.e K : T.(x) = x for a l l 1 n x i = 1, 2, ..., n} . Proof: • Suppose T^ ... T 00 = x . I f there were some i E {1, 2, . . . , n} with T_^(x) 4 x , then 0(T^, o, x) contains more than one point, and so there w i l l be a U e U with (5^(0(1^, o, x)) > 0 . Thus lim 6 (0(T., m, x)) < S (0(T., o, x)) . By Lemma 3.11., m-*» 83. S (0(T±, m, x)) = S (0(T , o, x)) for a l l m = 1, 2, ... , which i s a contradiction. Hence T\(x) = x for each i = 1, 2, . . . , n . The converse i s obvious. Corollary 3.13. Let K C X be nonempty weakly compact and F = {T^5 T^} be a f i n i t e commuting family of nonexpansive mappings w.r.t. U on K . Suppose T. , . . . , T , T, ... T : K -> K have d.o.d. 1 n 1 n w.r.t. U . Then ( i ) there i s an x e K with T_^(x) = x for each i = 1, 2 n , and ( i i ) {x e K : T 1 ... T n(x) = x} = {x e K : T (x)=x, for a l l i = 1, 2, . . . , n} . Proof: By Corollary 3.4., there i s an x e K with T 1 ... T n(x) = x . By Theorem 3.12., for any y e K , T ... T (y) = y i f and only i f T.(y) = y for each i = 1, 2, . . . , n . III-4. Bounded mappings. In this section we s h a l l discuss further properties of mappings with d.o.d. and their i t e r a t i o n s . I f K C X i s nonempty, we say T : K •> K i s bounded w.r.t. U_ i f and only i f for each U e U there i s a re a l number c > 0 with P (T k(x) - T k(y)) < c P (x - y) , for a l l u u — u u x, y E K , and a l l k = 1, 2, ... . I t i s clear that T i s bounded 84. w.r.t. U implies T i s continuous. Proposition 4.1. Let K C X be nonempty bounded and T : K -> K be bounded w.r.t. U . For each. U e U ,define P u ( x ) = $ (0(1, o, x)) , for each x e K , then p i s continuous, from K into IR u Proof: Suppose x , x e K with x ->- x . Then for any E > 0 , A A £ £ there i s an X such that X > X implies P (x, - x) < min o o u X 2c 2 u Thus for X > X , P (x - x) < ~ and P ( T k ( x J - T k ( x ) ) < c P (x - x) o ' u A z u A — u u A < for a l l k = 1, 2, ... , and so |P (T j(x) - T k ( x ) ) - P (T j(x.) - T k ( x . ) ) | ' U U A A 1 < P u(T j(x) - T k(x) - T j(x x) + T k ( x x ) ) < P (T j(x) - T j(x.)) + P (T k(x) - T k(x.)) U. A U. A < 2 + 2 = e for a l l j , k = 0, 1, 2, ... . Thus for X >X , 85. |p (x) - p (x ) | = |fi ( 0 ( T , o, x)) - 5 ( 0 ( T , o, x , ) ) | U ^ A U U A = | sup P ( T j ( x ) - T k ( x ) ) - sup P ( T j ( x . ) - T k ( x J ) 1,k>0 j,k>0 U X V < sup |P <T j(x)-T k(x)) - P ( T j ( x , ) - T k ( x , ) ) | . i n u U A A J,k>0 < £ Hence p (x ) -> p (x) , and so p i s continuous, u u u Since every nonexpansive mapping T w.r.t. U i s bounded w.r.t. U , we have the following: C o r o l l a r y 4.2. If K C X i s nonempty bounded and T : K K i s nonexpansive w.r.t. U , then f or any U z Li , the mapping p ^ : K E. defined by p (x) = 6 (0(T, o, x)) for each x z K , i s continuous, u u Theorem 4.3. Let K C X be nonempty bounded, T : K -> K be bounded w.r.t. Li' . Suppose ( i ) T has d.o.d. w.r.t. Li and ( i i ) there i s an n 0 0 x £ K such that a subsequence of {T (x ) } n _ ^ converges to z £ K . Then lim T n(x) = z and T(z) = z . Proof: Suppose l i m T (x) = z . I f U E U , then by Proposition 4.1., p : K -> TR. define by p (x) = 5 ( 0 ( T , o, x)) ' for each n k x E K , i s continuous, and so p (z) = lim p ( T (x)) . But k-*» 86. lim p (T . (x)) = lim 6 (0(T, IL , x)) = l i m <5 (0(T, n, x)) , so that . U . U K . U lim 8 (0(.T, n, x)) = p (z) . If p (z) > 0 , then n-x» u 6 u(0(T, o, z)) = P u(z) > 0 and T has d.o.d. w.r.t. U imply li m 8 (0(T, n, z)) < 8 (0(T, o, z)) . Thus there i s a p o s i t i v e integer n-*>= with 6 u(0(T, N, z)) < <$u(0(T, o, z)) . On the other hand, since n. , I L N + n v lim T (x) = z , we have 1 (z) = T (lim T (x)) = l i m T (x) and so k-*» k->co k-^00 •6u(0(T,-N, z)) = P u ( T N ( z ) ) = l i m p u(T k ( x ) ) k-x» = lim 8 (0(T, N+n. , x)) . U K . k-x» = l i m 5 (0(T, n, x)) = 6u(0(T,- o, z)) , which i s a c o n t r a d i c t i o n . Hence lim 8 (0(T, n, x)) = p (z) = 8 (0(T,o, = 0 , for each U e U , and i t follows that T(z) = z and {T (x) } , n=l n k i s a Cauchy sequence; since also l i m T * (x) = z , and X i s Hausdorff, k-*» n n n k we see that l i m T (x) exists and lim T (x) = l i m T (x) = z . C o r o l l a r y 4.4. .Let' K C X be nonempty bounded and T : K -> K be 87. nonexpansive w.r.t. U . I f T has d.o.d. w.r.t. U and there i s an n x e K such that l i m T (x) = z e K , then lim T n(x) = z and T(z) = z . k-*» n-** Since K C X i s sequentially compact implies K i s bounded, we have the following: C o r o l l a r y 4.5. Let K C X be nonempty seq u e n t i a l l y compact-and T : K -> K be bounded w.r.t. 0 . I f T has d.o.d. w.r.t. ti , then f or each x e K , l i m T n(x) e x i s t s i n K and i s a f i x e d point of T i n K . Moreover, T i s asymptotically regular. C o r o l l a r y 4.6. Let K C X be nonempty sequentially compact and T : K -> K be nonexpansive w.r.t. ti . I f T had d.o.d. w.r.t. ti , then for any x e K , lim T n(x) e x i s t s and i s a f i x e d point f o r T i n K . n-x» > III-5. Weakly p e r i o d i c mappings. In t h i s s e c t i o n , the existence of a common fi x e d point for a family of weakly pe r i o d i c nonexpansive mapping i s proved. Theorem 5.1. x Let K C X be nonempty weakly compact convex and F be a (not necessarily f i n i t e and not necessarily commuting) family of nonexpansive mappings w.r.t. (j on K . Suppose for each x z K , x z r\ Co(0(T, 1, x)) and i f M =' {x} U {T. ... T (x) :' {T. , . . . , T } C F), T e F I n 1 n -then Co(M) has normal structure w.r.t. U . Then there i s an x z K with T(x) = x for each T e F • Proof: By weak compactness of K and by Zorn's Lemma, l e t be minimal w.r.t. being a nonempty closed convex subset of K invariant under each T e F . Suppose there•exists an x e K and a T q e F such that T q ( X ) 4 x . Define M= {x} U ... T n(x) : {T±, T } i s any f i n i t e subset of F) , then M , and so Co(M) , contains more than one noint. Thus there are a U e U and an x e Co(M) with o 0 < r = r (U; Co(M)) < 6 (Co(M)) . Define R = {y e 1^ : r y ( U ; M) £ r Q } , then R ± 0 since X q e R . Clearly R i s closed and convex. Suppose T e F . I f y e R , then r^(U; M) £ r^ . I f z e M , then z e Co(0(T, 1, z)) and so for any e > 0 , there i s an z^ e Co(0(T, 1, z)) with P u ( z ~ zj) < £ • But m n. m then z = V X.T 1 ( z ) for some 0 < A. < 1 , y A. = 1 and n. > 1 . l . ^ i — l — . i 1 l i — x=l x=l n . - l Since T (z) e M for each i = 1, 2, ..., m , we see that n. - l P (y - T (2)) < r for each i = 1, 2, ...,m, and thus P u(T(y) - z) = P u(T(y) - + P^z^^ - z) . m n. < P (T(y) - I XT 1 ( z ) ) + e 1=1 m < I X ±P u(T(y) - T 1 ( z ) ) + e i = l m n. -1 1 ( I ^ p u(y - T 1 (z))) + e i = l m 1 o' <( I X.r ) + e i = l so that P (T(y) - z) < r + e for a l l e > 0 implies P (T(y) - z) < u — o u — for each z e M. Thus r T ( y ) ( U ; M) <_ r Q and so T(y) e R . Hence R i s also invariant under each T e F , and thus R = by minimality of But 6 ( X ) = <5 (R) < r < 6 (Co (M)) < 6 ( O , which i s impossible, u l u — o u — u l Therefore we must have T(x) = x for each T e F and each x e . Since 4 0 , F has a common fixed point. Corollary 5.2. Let K C X be nonempty weakly compact convex with normal structure w.r.t. U and F be a (not necessarily f i n i t e and not necessarily commuting) family of nonexpansive mappings w.r.t. U on K . Suppose for each x e K , x e f~\ Co(0(T, 1, x)) . Then there i s an TeF x e K such that T(x) = x , for a l l T e F . 90. D e f i n i t i o n 5.3. Let K C X be nonempty. Then ( i ) T : K -> K i s said to be weakly periodic i f and only i f for each x e K , there i s a subse-n. n, T CO r*l CO n quence {T (x)} of {T (x)} such that T (x) -> x weakly; ( i i ) T : K -> K i s said to be pointwise periodic i f and- only i f for each x e K, there i s a positive integer N(x) such that T^(x) = x ; ( i i i ) T : K -> K i s said to be periodic i f and only i f there i s a positive integer N such that T N(x) = x for a l l x e K . If K C X i s nonempty and T : K -> K , then i t i s clear that T i s periodic implies T i s pointwise periodic; T i s pointwise periodic implies T i s weakly periodic which i n turn implies x E Co(0(T, 1, x)) for each x e K . Hence we have Corollary 5.4. Let K C X be nonempty weakly compact convex with normal structure w.r.t. U and F be a (not necessarily f i n i t e nor commuting) family of weakly periodic (respectively pointwise periodic and periodic) nonexpansive mappings w.r.t. U on K . Then there i s an x E K with T(x) = x for a l l T e F Corollary 5.5. Let K C X be nonempty weakly compact convex and F be a f i n i t e commuting family of pointwise periodic (respectively periodic) nonexpansive mappings w.r.t. U on K . Then there i s an x e K with T(x) = x for each T E F . 91. Proof: Suppose F = {T_, ..., T } . Since T.T. = T.T. for 1 n i j j i a l l i , j = 1,'2, n and 1\ . i s pointwise periodic (respectively periodic) f o r each i = 1, 2, . . . , n , we see that for each x e K , the set m l m n M = {T. ... T (x) : mn, .... m are nonnegative integers} I n 1 n & & i s f i n i t e . Thus Co(M) = Co(M) i s compact, by Lemma 1.1., and so Co(M) has normal structure w.r.t. ii , by Theorem II-2.3. Hence by Theorem 5.1. there i s an x e K with T.(x) = x for each i = 1, 2, . . . , n . BIBLIOGRAPHY L. P. Belluce and W. A. K i r k : Fixed point theorems for c e r t a i n classes of nonexpansive mappings, Proc. Amer. Soc. Vol. 20 (1969), pp. 141-146. L. P. Belluce and W. A. K i r k : Nonexpansive mappings and fi x e d points i n Banach spaces, I l l i n o i s J. Math. 11 (1967)-, pp. 474-479. L. P. Belluce, W. A. K i r k and E. F. Steiner: Normal structures i n Banach spaces, P a c i f i c J. Math. Vol. 26 (1968), pp. 433-440. M. S. B r o d s k i i and D. P. Milman: On the center of a convex set, Dokl. Akad. Nauk SSSR (N.S.) Vol. 59 (1948), pp. 837-840. F. E. Browder: Existence of pe r i o d i c solutions f o r nonlinear equations of evolution, Proc. Nat. Acad. S c i . U. S. A. 53 (1965), pp. 1100-1103. R. DeMarr: Common fi x e d points f o r commuting contraction mappings, P a c i f i c J . Math. Vol. 13 (1963), pp. 1139-1141. E. Dubinsky: Fixed points i n non-normed spaces, Ann. Acad. S c i . Fen. Series A, I. Math. 331 (1963). N. Dunford and J . T. Schwarz: Linear operators, Part I: General Theory, Interscience Publishers Inc., New York (1958). M. E d e l s t e i n : On f i x e d and p e r i o d i c points under contractive mappings, J. London Math. Soc. Vol. 37 (1962), pp. 74-79. J . L. K e l l e y : General topology, D. Van Nostrond, Inc., New York (1955). J. L. Ke l l e y and Isaac Namioka: Linear t o p o l o g i c a l spaces, D. Van Nostrond, Inc., New York (1963). W. A. Ki r k : A f i x e d point, theorem for mappings.which do not increase distances, Amer. Math. Monthly, 72 (1965), pp. 1004-1006. W. A. Ki r k : On mappings with diminishing o r b i t a l diameters, J. London Math. Soc. Vol. 44 (1969), pp. 107-111. A. P. Robertson and Wendy Robertson: Topological vector spaces, Cambridge Uni v e r s i t y Press (1964). 

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