Existence and stability for some partial neutral functional differential equations with infinite delay

Abstract

In this paper, we study a class of partial neutral functional differential equations with infinite delay. We suppose that the linear part is not necessarily densely defined but satisfies the resolvent estimates of the Hille-Yosida theorem. We give some sufficient conditions ensuring the existence, uniqueness and regularity of solutions. A principle of linearized stability is also established in the autonomous case. To illustrate our abstract results, we conclude this work by an example.

Existence Results for Semilinear Perturbed Functional Differential Equations of Neutral Type with Infinite Delay

“…For example, if we consider the operator is not dense in C([0, 1], R) with the sup-norm (see [9] for more examples and remarks concerning the nondensely defined operators). Recently, evolution functional differential equations with nondensely defined linear operators have received much attention (see for instance the papers by Adimy et al [1], Adimy and Ezzinbi [2], Ezzinbi and Liu [10]). In this section, we extend the results of the previous section to the case where the operator A is nondensely defined.…”

“…From the definition it follows that [1]). Moreover, y satisfies the following variation of constants formula:…”

Periodic solutions of abstract neutral functional differential equations with infinite delay

“…For autonomous ANFDE, a concept of mild solution is given in Hernández and Henríquez [19]. Subsequently, some concepts of integral and strict solutions have been discussed in Adimy and Ezzinbi [2] and in Adimy, Bouzahir and Ezzinbi [4]. Recently, using properties of the non-autonomous abstract Cauchy problem, a concept of mild solution for non-autonomous ANFDE is considered in Fu and Ezzinbi [11].…”

“…ANFDE's have been studied by several authors last years, including Adimy [3,4,1] and Hernández and Henríquez [18,19] for the unbounded delay case. In addition, equations of type (1.1) have been used by several authors to model control systems.…”

Existence of Solutions for Second Order Partial Neutral Functional Differential Equations

“…First-order abstract partial neutral functional differential equations have also been studied by different authors, including Adimy et al [1,2,3], Hale [11], Wu and Xia [26] and Wu [27] for finite delay, and Hernández & Henríquez [14,15] and Hernández [16] for the unbounded delay case. In addition, some abstract second-order neutral Cauchy problems have been considered recently in the literature, see for instance [5,6,7,22].…”