Method of semidiscretization in time for quasilinearintegrodifferential equations

D Bahuguna,Reeta Shukla

doi:10.1155/s0161171204203441

Abstract

We consider a class of quasilinear integrodifferential equations in a reflexive Banach space. We apply the method of semidiscretization in time to establish the existence, uniqueness, and continuous dependence on the initial data of strong solutions.

Highlights

Let X and Y be two real reflexive Banach spaces such that the embedding Y X is dense and continuous
Where 0 < T < ∞, A(u) is a linear operator in X for each u in an open subset W of Y, G is a nonlinear Volterra operator defined from C([0, T ]; X) into itself, and the nonlinear map f is defined from [0, T ] × Y × Y into Y
The crucial assumption in these works is that there exists an open subset W of Y such that for each w ∈ W, A(w) generates a C0-semigroup in X, A(·) is locally Lipschitz continuous on W from X into itself, f, defined from W into Y, is bounded and globally Lipschitz continuous from Y into itself, and there exists an isometric isomorphism S : Y → X such that

Summary

Introduction

Let X and Y be two real reflexive Banach spaces such that the embedding Y X is dense and continuous. We use the ideas and techniques of Zeidler [10] and the method of semidiscretization in time to establish existence, uniqueness, and continuous dependence on initial data of strong solutions to (1.1) on [0, T ] for some 0 < T ≤ T . After proving a priori estimates for the approximate solution, the convergence of the approximate solution to the unique solution of the evolution equation is established.

Results

Conclusion