Regularity for Solutions of Second-Order Nonlinear Integrodifferential Functional Equations

Han-Geul Kim,Jin-Mun Jeong,Dong-Gun Park

doi:10.1155/2010/280463

Abstract

We deal with the well-posedness for solutions of nonlinear integrodifferential equations of second-order in Hilbert spaces by converting the problem into the contraction mapping principle with more general conditions on the principal operators and the nonlinear terms and obtain a variation of constant formula of solutions of the given nonlinear equations.

Highlights

Let H and V be two complex Hilbert spaces
We deal with the well-posedness for solutions of nonlinear integrodifferential equations of secondorder in Hilbert spaces by converting the problem into the contraction mapping principle with more general conditions on the principal operators and the nonlinear terms and obtain a variation of constant formula of solutions of the given nonlinear equations
The subset of this paper is to consider the initial value problem of the following second-order nonlinear integrodifferential control system on Hilbert spaces: x t Ax t f t, x t h t, 0 < t ≤ T, 1.1 x 0 x0, x 0 x1, where the nonlinear term is given by t f t, x k t − s g s, x s ds

Summary

Introduction

Let H and V be two complex Hilbert spaces. Assume that V is a dense subspace in H and the injection of V into H is continuous. Under the assumption that h ∈ L2 0, T ; V ∗ and the local Lipschitz continuity of the nonlinear mapping from V into H not from H into itself , we deal with regularity for the solution of the given equation 1.1 which will enable us to obtain a global existence theorem for the strict solution x belonging to C1 0, T ; V ∩ C 0, T ; H ∩ W2,2 0, T ; V ∗ ; namely, assuming the Lipschitz continuity of nonlinear terms, we show that the well-posedness and stability properties for a class with nonlinear perturbation of secondorder are similar to those of its corresponding linear system. We will develop and apply the existence theory for first-order differential equations see [16, 17] to study certain second-order differential equations associated with nonlinear maximal monotone operators in Hilbert spaces

Linear Hyperbolic Equations

Hyperbolic Equations with Nonlinear Perturbations