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On positive solutions of a volterra equation of the second kind Thompson, Desmond Edwin 1970

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ON POSITIVE SOLUTIONS OF A VOLTERRA EQUATION OF THE SECOND KIND by DESMOND EDWIN THOMPSON B.Sc . (Special Hons); Univers i ty of The West Indies, 1967 A THESIS SUMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department " of MATHEMATICS We accept th is thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1970 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree tha permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver 8, Canada ( i i ) ABSTRACT Vol te r ra Integral Equations of the second kind occur i n many problems i n Physics and Engineering. Here we study the conditions for and behaviour of pos i t ive solut ions of these equations. Examples have been given to point out some of the d i f f i c u l t i e s that occur i n the theory. TABLE. OF CONTENTS Introductions Chapter one Bounds and Monotone Solutions Chapter two Comparison and Approximate Theorems Chapter three Asymptotic Behaviour Bibliography ( iv) ACKNOWLEDGEMENTS I am indebted to my supervisor Dr. G. HUIGE, for h i s generous and valuable assistance i n the research and w r i t i n g of th i s paper. My thanks to Dr. G.W. Bluman, who read th i s thesis and provided excel lent suggestions. I am grateful to the Univers i ty of B .C . and the National Research Council for the i r f i n a n c i a l support. Las t , but not l eas t , I wish to thank Mrs. Y . S . Choo for typing the thes is . INTRODUCTION In th i s paper we w i l l consider a Vol te r ra equation of the second k ind , u(x) = f(x) + K(x , t )u ( t )d t (0.1) o where K(x , t ) i s a VOLTERRA TYPE•KERNEL that i s , ^ K ( x , t ) = 0 for t > x . • In the f i r s t part we w i l l be concerned with comparison and approximation theorems. Some compairson theorems have been given i n [1] and [7]. Here we apply stronger conditions to f(x) and K(x , t ) and obtain a proper ' ; • j ! inequal i ty between so lu t ions . Their ideas, as w e l l as that of an approximate so lu t ion for a d i f f e r e n t i a l equation as given i n [ 5 ] h a v e , been combined to give a theorem re l a t i ng approximate solutions to that of (0.1) . Some discussion on monotone solut ions has been given. [Even thbuj the conditions for such solut ions are very strong, examples have been given to show that weaker conditions may work. On the question of upper bounds most of the mater ia l has been taken from [1], [3], [5] and [6] . A discussion i s given i n the f i r s t sec t ion . In the l a s t sect ion we deal with conditions for pos i t ive solut ions of (0.1) when K ( x , t ) < 0 . In th i s case the equation then Considered i's - 2 -u(x) = f(x) - K(x , t )u ( t )d t where K(x , t ) >_ 0 . A study of the asymptotic behaviour of pos i t ive solut ions of th is equation i s given . - 3 -CHAPTER 1 BOUNDS AND MONOTONE SOLUTIONS For completeness we state here the p r i n c i p a l resul ts we need for Vol te r ra equations. Let I '= { x : 0 <_ x < 00 } . S = { t : t _< x x E I > . and K(x , t ) be a VOLTERRA TYPE kernel defined on I x I . Define K , ( x , t ) = K(x , t ) and for n >^  2 K n ( x , t ) = K(x,s)K , ( s , t )d s n-1 By the resolvent kernel of K(x , t ) we mean the function H(x, t ) given by the series H(x, t ) = £ K (x , t ) . This ser ies converges uniformly on k=l n I x I , provided K i s continuous . i We quote here without proof the fol lowing theorem. ^ Let f(x) be continuous on I and K(x , t ) be a VOLTERRA TYPE v kernel continuous on I x S . Then there i s a unique so lu t ion to the equation - 4 -rx u(x) = f(x) + K(x , t )u ( t )d t rx given by u(x) ^ f(x) + H(x , t ) f ( t )d t (0.2) For the proof of th i s theorem see [2] or [4] . We begin th is sect ion with a simple but very important lemma. Unless otherwise stated we w i l l assume that f e C(I) and K ( x , t ) e C(I x s) . Lemma 1.1 I f K(x , t ) >_ 0 and f (x) >_ 0 then u(x) >_ f (x) >_ 0 . Proof I f K(x , t ) >_ 0 then the resolvent kernel H(x, t ) >_ 0 and hence i t follows from (0.2) that u(x) >_ f (x) >_ 0 . Corol lary 1.1 I f K(x , t ) > 0 and f(x) > 0 then u(x) > f(x) > 0 . Proof Now u(x) = f(x) + K(x , t )u ( t )d t and hence from lemma 1.1 u(x) > 0 and so u(x) > f(x) > 0 . We now give some resul ts on monotone so lu t ions . The f i r s t theorem gives conditions on both f(x) and K(x , t ) . However, i t w i l l be seen that these conditions can be relaxed. Even though we have not been able to prove theorems with ' s o f t ' conditions on K , the point i s made i n examples 1.1 and 1.2 . - 5 -THEOREM 1.1 If f (x) >_ 0 and monotonic increasing and K(x,t). >_ 0 and monotonic increasing i n x, then u(x) i s monotonic increasing . Proof; Let x > s then u(x) - u(s) = f(x) - f ( s ) + K(x,t)u(t)dt - K( s , t ) u ( t ) d t r s (K(x,t) - K ( s , t ) ) u ( t ) d t + x K(x,t)u(t)dt but from lemma 1.1 u(x) > 0 and so u(x) - u(s) > 0 Cor o l l a r y I f we replace f(x) >_ 0 by f(x) > 0 and K(x,t) >0 by \ K(x,t) > 0 i n the above theorem then u(x) i s s t r i c t l y monotonic increasing Proof: From C o r o l l a r y 1.1 u(x) > 0 and so from rs [K(x,t) - K ( s , t ) ] u ( t ) d t + fx u(x) - u(s) = f(x) - • f ( s ) + we have u(x) - u(s) > f(x) - f ( s ) > 0 . K(x,t ) u ( t ) d t i . e u(x) > u(s) . II As we can see from Theorem 1.1 the condition that K(x,t) i s monotonic increasing i n x i s quite strong. However,'the theorem i s not true f o r - 6 -arb i t ra ry K(x , t ) >_ 0 (see example 1.2). On.the other hand i n some cases (see example 1.1) we can s t i l l . o b t a i n monotonic increasing solut ions when K(x , t ) i s monotonic decreasing i n x . I t should be noted that for th is to be true i t appears that K ( x , t ) should not decrease too fast . Before we give the example we. quote a lemma due to TRICOMI . A(x) Lemma 1.2. I f K ( x , t ) = ^ F£J then the so lu t ion of (0.1) can be wr i t t en rx as u(x) = f(x) + e X _ t K ( x , t ) f ( t ) d t . •'o For the proof see [2] pages 17-18 . EXAMPLE 1.1 Consider the case when f(x) e C ' ( I ) with f(x) > 0 and -x f ' ( x ) >_ 0 . Take K(x , t ) = 7 7 ^ - = - — as i n lemma 1.2 A(t) e " t then u(x) = f(x) + x- t -x t , , , , t e e e f ( t )d t rx = f(x) + f ( t )d t u ' (x) = f ' ( x ) + f(x) > 0 EXAMPLE 1.2 Take A(x) = e" From lemma 1.2 n -x and f(x) = :e X-- 7 -x u(x) = e + -t+x -x t t, e e e e dt n n n n I / N X , X _ X X i / - i n - l \ x - x u (x) = e + e -e + (1 - nx )e rx n e dt x n-1 x-x = 2e + (1 - nx )e e dt . f 1 t n u'(1) = 2e + (1 - n) e dt < 0 for n > 7 THEOREM 1.2 I f u(x) i s monotonic increasing with K(x , t ) >_ 0 and x K(x , t )d t < 1 for a l l x e I i f f(x) > 0 then f(x) < u(x) < f(x) K(x , t )d t rx Proof: u(x) = f(x) + K(x , t )u ( t )d t r x < f(x) + u(x) K(x , t )d t rx that i s u(x)(1 - K(x , t )d t ) < f(x) or u(x) < f(x) rx K(x , t )d t - 8 -that u(x) >_ f(x) follows from lemma 1.1 11 As seen from above, the conditions for a ' n i c e ' upper bound i s quite strong. In [7] the case K(x , t ) = u(x)k(x-t) was considered and the fol lowing theorem stated. THEOREM 1.3 I f the l i m i t c = l im f(x) exis t s and K(x , t ) = k(x- t ) i s absolutely integrable i . e k( t ) dt < co then the l i m i t of u(x) given by (0.1) i s l i m u(x) = 3 x-*» provided Re(s) > 0 .00 i f and only i f 1 -e S t k ( t ) d t + 1 , k ( t )d t In general the resolvent kernel cannot be eas i ly calculated and so the need for upper bounds that remain close to the so lu t ion i s extremely important. I f K ( x , t ) = g(x)h(t) then as shown i n [3] H(x, t ) = g(x)h(t) exp( g( t )h( t )d t ) In some cases we can wri te K(x , t ) < _ ( x ) K 2 ( t ) ^ and apply GRONWALL1S INEQUALITY and obtain u(x) £ f(x) + K 1 (x)(exp K 1 ( s ) K 2 ( s ) d s ) ( K 2 ( s ) f ( s )ds ) (1.1)' •'o Jo For proof see [5] or [1] - 9 -T h i s f o r m h o w e v e r a s was p o i n t e d o u t i n [6] d o e s n o t s a y much a b o u t t h e b e h a v i o u r o f u ( x ) a t i n f i n i t y . S o m e t i m e s ( s e e e x a m p l e g i v e n b e l o w ) i t i s b e t t e r t o w r i t e t h e u p p e r b o u n d f o r K ( x , t ) i n t h e f o r m . n K ( x , t ) < I K ( x ) H ( t ) i = l 1 I f t h i s i s d o n e t h e n we c a n a p p l y t h e f o l l o w i n g t h e o r e m d u e t o W I L L E T T . ! The p r o o f i s g i v e n i n [6] . THEOREM 1.4 S u p p o s e t h a t n u ( x ) <_ f ( x ) + I w . ( x ) i = l 1 x v ^ ( s ) u ( s ) d s w h e r e w . v . ( i , i = 1... n ) a n d v . u a r e i n t e g r a b l e o n I . T h e n u < E f ( x ) x j J. x — j n w h e r e E ^ ( i = 0, 1 • • • n ) i s d e f i n e d i n d u c t i v e l y a s a c o m p o s i t i o n o f i + 1 f u n c t i o n a l o p e r a t o r s , t h a t i s , E . = D.D. n ••• D w h e r e x x x-1 o D f ( x ) = f ( x ) o rx D f = f + ( E 1 _ 1 w i ) ( e x p v . E . n w . ) . ( x x-1 x v . f d t ) x T h e f o l l o w i n g e x a m p l e w a s c o n s i d e r e d i n [6] . - 10 -Let u(x) < x + fX ? -At (A 2xe Z + l ) u ( t )d t o Here A. Is a r ea l parameter and the problem i s to determine • the asymptotic behaviour of u as A —> oo , i n pa r t i cu l a r , to prove that u = 0(1) uniformly for x r e s t r i c t ed to compact subintervals of [0, oo) . o -A> , I f we take w ^ x ) = 1 v 1 (x) = 1 and w 2 (x) = x and v 2 (x) = X then the desired resu l t follows by d i rec t app l ica t ion of theorem 1.4. However i f we apply 1.1 d i r e c t l y i . e take K(x , t ) < max(l, A 2 x)e ^ t or K ( x , t ) <_ (1 + A 2 x)max(l , e X t ) then th is does not produce that u must be bounded as A —> oo \ - 11 -CHAPTER 2 COMPARISON AND APPROXIMATION THEOREMS I n t h i s c h a p t e r we w i l l be m a i n l y c o n c e r n e d w i t h t h e e q u a t i o n u. (x) = f . ( x ) + i x K i ( x , t ) u j . ( t ) d t , ( i = 1,2) (2.1) Here we assume t h a t f\ (x) .£ L 2 ( I ) and K ± ( x , t ) e L 2 ( I * I ) , f o r i = ,1,2 THEOREM 2 .1 . L e t K ( x , t ) be a y o l t e r r a t y p e k e r n e l w i t h | K x ( x , t ) | <_ K 2 ( x , t ) and I ^ C x ) | < f ^ x ) ; K ( x , t ) and f ^ ( x ) p o s s i b l y complex.. I f u_^(x) i s t h e u n i q u e s o l u t i o n o f t h e e q u a t i o n u . ( x ) = f . ( x ) + K , ( x , t ) u . ( t ) d t , th e n j u ^ ( x ) | <_ u,,(x) . I n f a c t , u 2(x) - I^Cx) ! > f 2 (x) - | f 1 ( x ) | P r o o f : The p r o o f i s a consequence o f Lemma 1.1. The r e a d e r i s r e f e r r e d t o [1] f o r d e t a i l s . We now c o n s i d e r t h e case when f ^ ( x ) i s r e a l - v a l u e d . - 12 -THEOREM 2.2. Let K (x , t ) i = 1,2 be a Vol te r ra type kernel such that K 2 ( x , t ) _>-K (x , t ) >_ 0 . I f f 2 ( x ) ' _> f ^ x ) with f 2 ( x ) _> 0 and u^(x) i s the unique so lu t ion of (2.1) then u 2 (x) >_ u^(x) . In fact u 2 (x) - u (x) _> f 2 ( x ) - f x ( x ) . Proof: Let • H . (x , t ) be the resolvent kernel for (2 .1) . Then i t i s eas i ly seen that 0 <_ H ^ x . t ) <_ H 2 ( x , t ) . u 2 (x) - u x (x) = f 2 ( x ) - f (x) + H 2 ( x , t ) f 2 ( t ) d t x H 1 ( x , t ) f 1 ( t ) d t f 2 ( x ) - f 1 ( x ) + [H 2 (x , t ) - H 1 ( x , t ) ] f 2 ( t ) d t r x + H ^ x . t ) . [ £ 2 ( t ) - f x ( t ) ] d t and so u 2 (x) - u ^ x ) >_ f 2 ( x ) - f 1 ( x ) This completes the proof. I t should be noted that Theorem 2.2 i s appl icable when Theorem 2.1 i s not. THEOREM 2.3. Let f (x) e L 2 ( I ) ., K(x , t ) >_ 0 and K(x , t ) e L 2 ( l x l ) . I f u(x) i s the unique so lu t ion to (0.1) and v(x) e L 2 ( I ) wi th - 13 -v(x) < f (x) + K( x , t ) v ( t ) d t then u(x) > v(x) r x Proof: Let v(x) = g(x) + K( x , t ) v ( t ) d t , then f(x) > g(x) u(x) - v(x) = f(x) - g(x) + x K(x,.t) [u(t) - v ( t ) ] d t and so from Lemma 1.1 u(x) - v(x) > 0 , i . e . u(x) > v(x) . n n THEOREM 2.4. Let K(x,t) = £ K.(x,t) , F(x) = J f.(x) with i = l 1 ' i = l 1 K ±(x,t) >_ 0 , f ( x) > 0 and K ±(x,t) e L 2 ( I x l ) f (x) ' e L 2 ( I ) f o r i = l , . . . , n . If u ^ ( x ) I s t n e unique s o l u t i o n of u.(x) = f.(x) + K ±(x,t)-u ( t ) d t and u(x) i s the unique s o l u t i o n of u(x) = F(x) + K(x,t)u(t)dt n then ^ u. (x) <_ u(x) i = l x Proof: The proof i s by induction on n . Theorem true for n = 1 n-1 Assume true for n = k-1 . i . e . u.(x) <_ v(x) where i = l 1 - 14 -k-1 rx k-1 v(x) = I f (x) + I K (x , t )v ( t )d t i = l o i = l k-1 rx k-1 u (x) + v(x) = f (x) + I f . (x ) + 7 K . ( x , t ) v ( t ) d 1=1 J o 1=1 r x + K (x , t )u ( t)dt n n since from Lemma 1.1 u (x) > 0 . From Theorem 2.3 we have n — u (x) + v(x) < u(x) n —' i . e . k-1 u (x) + T u. (x) < u(x) n . L , i — i = l or I u. (x) <_ u(x) i = l This completes the proof. We should note here that even i f we deal with d i s t i n c t functions i n L^ ( i . e . f 2 ( x ) > f^(x) say) we cannot ,in general obtain a d i rec t inequal i ty for so lu t ions . I f however, the solutions are con-tinuous then th is can be done. Let us consider the case when f(x) . and K(x , t ) are continuous - 15 -THEOREM 2.5. Let u(x) = f(x) + K(x , t )u ( t )d t , ex v(x) < f(x) + K(x , t )v ( t )d t with f(x) _> 0 and K(x , t ) >_ 0 . I f v(x) i s continuous then u(x) > v(x) for x e I Proof: u(0) = f(0) > v(0) that i s u(0) > v(0) . Suppose the theorem i s f a l se , By cont inui ty of u(x) and v(x) there exis t s x^ > 0 such that v ( x x ) = u(x x ) where v(x) < u(x) for 0 <_ x < x^ . Now r x n u( X ; L ) = f ( X ; L ) + L K ( x l 5 t ) u ( t ) d t >_ f (x 1 ) + K ( x 1 t ) v ( t ) d t > v ( x 1 ) This contradicts u(x^) = v(x^) . Hence the theorem i s true. COROLLARY 2 .1 . I f w(x) > f(x) + K(x, t )w(t )d t u(x) = f(x) + K(x , t )u ( t )d t v(x) < f(x) + K(x , t ) v ( t ) d t where w(x) i s continuous, then - 16 -w(x) > u(x) > v(x) . COROLLARY 2.2. Let u (x) be the unique solut ions of i u .(x) = f ± (x ) + K i ( n , t ) u . ( t ) d t for i = 1,2 . I f f (x) e C(I) and K ± ( x , t ) e C ( M ) wi th f 1 ( x ) > f 2 ( x ) _> 0 , K ^ x . t ) >_ K 2 ( x , t ) >0 then u±(x) > u 2 (x ) for x e I . i Proof: u ^ x ) = f ^ x ) + x K 1 ( x , t ) u 1 ( t ) d t > f 2 ( x ) + K 2 ( x , t ) u 1 ( t ) d t hence from Theorem 2.4 we have u^(x) > u 2 (x ) for- x e I DEFINITION 2 .1 . We say y(x,6) i s a 6-approximate so lu t ion of (0.1) i f |y(x,6) - f(x) - K ( x , t ) y ( t ) d t | < 6(x) . THEOREM 2.6. Let G(x, t ) , K(x , t ) e L ^ T * ! ) with G(x, t ) >_K(x,t) _> I f v(x) i s the so lu t ion of v(x) = 6(x) + G(x , t )v ( t )d t and u(x) i s the unique so lu t ion of (0,1) then |y(x,6) - u (x ) | <^v(x) . Proof: Consider the case when y(x) <^  o"(x) + f (x) + Since u(x) i s a so lu t ion of (0.1) we have x K ( x , t ) y ( t ) d t y ( x ) - u ( x ) < 6 ( x ) + K ( x , t ) [ y ( t ) - u ( t ) ] d t L e t w ( x ) = 6 ( x ) + x K ( x , t ) w ( t ) d t t h e n f r o m T h e o r e m 2 . 3 we h a v e y ( x ) - u ( x ) <. w ( x ) <_ v ( x ) , t h a t i s y ( x ) *~ u ( x ) < v ( x ) . • I f we t a k e y ( x ) >_ - 6 ( x ) + f ( x ) + x K ( x , t ) y ( t ) d t t h e n y ( x ) - u ( x ) >_ -6 ( x ) + x K ( x , t ) y ( t ) - u ( t ) d t a n d s o u ( x ) - y ( x ) <^  6 ( x ) + x K ( x , t ) [ u ( t ) - y ( t ) ] d t H e n c e f r o m T h e o r e m 2 . 3 u ( x ) - y ( x ) _< v ( x ) t h e r e f o r e we h a v e | u ( x ) - y ( x , 6 ) | •_< v ( x ) . I DEPENDENCE ON INITIAL VALUE In the case of d i f f e r e n t i a l equation of the form y ' = f (x ,y ) where f i s continuous and y (x Q ) = y 0 we know that solut ions depend continuously on the i n i t i a l value x Q . With th i s i s mind we can wr i te the so lu t ion of (0.1) as a function of two var iables that i s , the so lu t ion of u(x) = f(x) + fx (K(x , t )u( t )d t can be wr i t t en as u(x,s) , for x >_ s s The next theorem shows that u(x,s) i s monotone decreasing i n the second var iable . THEOREM 2.3 Let u(x,s) be the so lu t ion of u(x,s) = f(x) + rx K ( x , t ) u ( t , s ) d t s where f (x) >_ 0 and K(x , t ) >^  0 ; then for > s 2 , UC X JSJ^) <_ u ( x , s 2 ) rx Proof: u ( x , s 1 ) = f(x) + K ( x , t ) u ( t , s 1 ) d t s l rx _< f(x) + K ( x , t ) u ( t , s 1 ) d t J s 2 hence from theorem 2.1 we have u(x,s. l ) <^  u ( x , s 2 ) - 19 -CHAPTER 3 ASYMPTOTIC BEHAVIOUR From the above discussion we can see that f(x) acts as 'boundary l i n e ' depending on whether K ( x , t ) ^ 0 or K(x , t ) <_ 0 . In the case when f(x) and K ( x , t ) are continuous, i t i s impossible to obtain solut ions u(x) <_ f(x) when K ( x , t ) i s p o s i t i v e . The natural question therefore a r i ses : are there solut ions u(x) such that 0 <^  u(x) <_ f (x)? In the next theorem we g i v e s u f f i c i e n t conditions for such solut ions • . L . Instead of considering equation (0.1) with K ( x , t ) <_ 0 we w i l l consider u(x) = f (x) -with K(x , t ) > 0 . ' ' X K(x , t )u ( t )d t (3.1) o THEOREM 3.1 Let f (x) > 0 and K ( x , t ) > 0 i f < ^ ( y ? t ) f ° r X 5 7 0 < t < x then the so lu t ion to (3.1) i s such that f(x) > u(x) > 0 . rx Proof: u(x) = f(x) K(x , t )u ( t )d t that i s f(x) > u(x) . o I t remains to prove that u(x) cannot be negative anywhere on I . - 20 -u(0) = f(0) > 0 . Suppose the theorem i s f a l s e . Then by con t i n u i t y there e x i s t s x 1 > 0 and 6 > 0 such that u(x) > 0 0 < x <_ Xj u(xj) - 0 u(x) < 0 x 1 < x £ x 1 + 6 i . e f o r we have x^ < x <_ X j + 6 0 > u(x) > f(x) rx. K(x,t)u(t)dt f f x V f ( x l )  r w ( f C x \ _•- 1 f ( X l ) ; , 1 U 1 ; f(x) J K(x, t ) u ( t ) d t } - f ( x , ) 1 t U l ; rx. K ( x 1 , t ) u ( t ) d t } f(x) u(x,) 0 . i . e . u(x) >^  0 f o r x1 < x <_ x1 + 6 contradiction to the f a c t that u(x) < 0 i n that i n t e r v a l Therefore the theorem i s true . - 21 -i . e . f (x) >_ u(x) ^. 0 . Corol lary 3.1 I f K ( x , t ) i s monotonic decreasing i n x and f(x) montonic increas ing, the conditions for the above theorem i s s a t i s f i e d . Corol lary 3.2 I f we take the case K ( x , t ) = k(x - t ) a l l we need i s that k( t ) be monotonic decreasing . We now study the asymptotic behaviour of u(x) given cer ta in . conditions on K(x , t ) and f(x) . THEOREM 3.2 I f u(x) >_ 0 i s a so lu t ion of (3.1) and i f co > K ( x , t ) > 0 K ( x , t ) monotone increasing i n x and f(x) monotone increasing and ! f (x) < B < co then l i m u(x) = 0 . ' — x-*» . , Proof: Suppose that u(x) —f-> 0 then there ex i s t a sequence {x } such n that x < x , T —> oo and n n+1 u(x ) > a > 0 for some a > 0 . n — We claim there ex i s t 3 > 0 such that u(x) >-^a i f x < x < x + 3 . — / n n - 22 -Now u(x) - u(x ) n f(x) - f (x ) -n = f(x) - f ( x n ) . + K(x , t )u ( t )d t + K(x , t )u ( t )d t o • 'o X n K(x , t )u ( t )d t -n n K(x , t )u ( t )d t K(x , t )u ( t )d t n n K(x , t )u ( t )d t n n fx K(x , t )u ( t )d t - K(x , t )u ( t )d t x n rx > - K(x , t )u ( t )d t x n > - B K(x , t )d t : M = / sup u(t) < B < oo . n x <t<x n-i . e . u(x) > u(x ) -• B n x K(x , t ) d t > j n rx +B provided n K(x , t )d t < x n a/2 .B". Since we may take | x n - x n + 1 | ^.1 f o r / a l l n •>_ 0 and 8 < .1 . we have K<x,t)u(t)dt >.~ A I 1 x <x-3 n. A = . i n f K ( x , t ) ( x , t ) e l x l -> oo as x co . i . e . B > f(x) = u(x) + x K(x , t )u ( t )d t - 23 -and so f (x) — > co as x -»• co T h i s c o n t r a d i c t s t h e h y p o t h e s i s t h a t f ( x ) < B < co T h e r e f o r e u ( x ) — > 0 as x oo THEOREM 3.3 L e t u ( x ) be a s o l u t i o n o f ( 3 . 1 ) . I f l j m K ( x , t ) d t = 0 w i t h A < oo , l i m x K ( x , t ) d t = C < oo and f ( x ) < B < oo w i t h l i m sup f ( x ) > 0 t h e n u ( x ) —f-> 0 >. P r o o f : Suppose u ( x ) — > 0 t h e n f o r E > 0 t h e r e e x i s t N > 0 such t h a t u ( x ) < e and and ^hence K ( x , t ) d t < e x > N . f ( x ) = u ( x ) + = u ( x ) + K ( x , t ) u ( t ) d t A K ( x , t ) u ( t ) d t + K ( x , t ) u ( t ) d t JA fA < u ( x ) + B rx • K ( x , t ) d t + e JA K ( x , t ) d t f o r x > N . < e + Be + E C f o r x > N t h a t i s f ( x ) — > 0 - 24 -this contradicts l im sup f(x) > 0 Therefore u(x) —/-> 0 Corol lary I f we consider the case K(x , t ) = k(x - t) and then the above theorem i s s t i l l true . k( t )d t < co THEOREM 3.3 I f u(x) > 0 i s a so lu t ion of (3.1) and u(x) i s monotonic decreasing with l i m K(x , t )d t = oo and f (x) <_ B < co then u(x) —> 0 •'o Proof: u(x) = f(x) - K(x , t )u ( t )d t u(x) + u(x) J K(x , t )d t < f(x) u(x) < f(x> f X 1 + K(x , t )d t -> 0 as x —> oo - 25 -BIBLIOGRAPHY [1] P.R. BEESACK : Comparison theorems and inequa l i t i e s for Vo l t e r r a Integral Equations. B u l l . Amer. Math. S o c , 20, (1969), 61-66 . [2] F .G . TRICOMI Integral Equations, Interscience New York, 1957 . j [3] S.C. CHU and F .T . METCALF. On Gronwall 's Inequal i ty , Proc. Amer. Math. S o c , 18, (1967) 439-440 . I [4] W. POGORZELSKI, Integral Equations and the i r Appl icat ions Vol 1, Pergamon Prees, 1966 . [5] E .A . CODDINGTON and N. LEVINSON, Theory of Ordinary D i f f e r e n t i a l Equation McGRAW-HILL New York, 1955 . [6] D. WILLETT A l i nea r general izat ion of Gronwall 's I n e q u a l i l i t y . Proc, Amer. Math. S o c , 16, (1965) 774-778 . [7] N . DISTEFANO A Vol te r ra In tegral Equation i n the s t a b i l i t y of some l i n e a r hereditary phenomena. J . Math. Anal and App l , 23, (1968)- I 365-383 . ! 

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