Existence of Solutions of Nonlinear Neutral Integrodifferential Equations in Banach Spaces

Abstract

In this paper we prove the existence of mild solutions of a nonlinear neutral integrodifferential equation in a Banach space. The results are obtained by using the Schaefer fixed point theorem. As an application the controllability problem for the neutral system is discussed.

Nonlinear neutral integrodifferential equations on unbounded intervals

“…It seems to us that not many existence results have so far been established to solve the general problem (1)-(2). Our main result, in particular, extends the main result (Theorem 3.1, p. 96) of Dauer and Balachandran [3] in twofold: Our operator A(t) is time-dependent and the interval on which equation (1) is considered is infinite. Furthermore, our hypotheses are simpler than those in Dauer and Balachandran [3].…”

“…t 0 h(t, s, x s )ds , t ∈ [0, +∞) (1) with the boundary condition The theory of neutral delay differential equations has extensively been studied in the literature (see [1,3,5,7] and the references therein). In particular, Benchohra, Henderson, and Ntouyas [1] recently obtained existence results for impulsive neutral functional differential equations in Banach spaces on finite intervals.…”

Semilinear Volterra integrodifferential equations with nonlocal initial conditions

“…, u t p = u 0 , (1.3b) where p ∈ N, 0 < t 1 < · · · < t p ≤ T are given, u 0 ∈ X, −A generates a linear C 0 -semigroup on X, while f : [0, T ]×X → X and g : [0, T ] p ×X p → X satisfy Lipschitz conditions. See also [4,5,8,9,13] for related results, including applications to integrodifferential equations. Next, Ntouyas and Tsamatos [17,18] studied nonlocal semilinear problems in the absence of Lipschitz conditions by using compactness arguments.…”

Existence of solutions for abstract neutral integro-differential equations with unbounded delay

“…It is convenient to note that in the papers [4], [3], [6], [7], [13], among several other works, an alternative assumption (which is really a particular case of condition (H g )) has been used to treat neutral systems. In these works it is assumed that the semigroup (T (t)) t 0 is compact and the set of operators {AT (t) : t ∈ (0, b]} is bounded.…”