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UBC Theses and Dissertations

On non-linear time-lag evolution equations Lam, Che-Bor

Abstract

The purpose of this thesis is to obtain existence theorems for perturbated evolution equations in Hilbert spaces and Banach spaces. A practical example of the perturbations under consideration is an operator with a time-lag or delayed argument. Throughout the thesis, the Galerkin approximation method will be used to establish the existence theorems. In chapter one, we study the problem in Hilbert spaces. The key is to obtain a priori estimates on the time fractional derivatives of the approximate solutions. We shall prove that there exists a solution with time fractional derivatives of order less than 1/2. In chapter two, we consider the problem in Banach spaces. Again, we apply the Galerkin approximation method, but with a special basis. In chapter three, we study periodic solutions of evolution equations. Here we use the Schauder-Tychonov fixed point theorem to prove the existence of periodic solution to the approximating equations. In the last chapter, we give examples of appliactions of the various theorems proved in the first three chapters.

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