Almost periodic solutions of neutral functional differential equations

Abstract

In this paper we study a non-autonomous neutral functional differential equation in a Banach space. Applying the theory of semigroups of operators to evolution equations and Krasnoselskii's fixed point theorem we establish the existence and uniqueness of a mild almost periodic solution of the problem under consideration.The theory of almost periodic functions was first treated and created by Bohr during 1924Bohr during -1926. Later on Bochner extended the Bohr theory to general abstract spaces. The theory has been widely treated by Favard, Levitan [1], Besicovich [2] in monographs. Amerio [3] has extended certain results of Favard and Bochner to differential equations in abstract spaces. Functional differential equations arise as models in several physical phenomena, for example, reaction-diffusion equations, climate models, coupled oscillators, population ecology, neural networks, the propagation of waves etc. A class of functional differential equations called neutral differential equations arises in many phenomena such as in the study of oscillatory systems and also in the modeling of several physical problems [4]. There exists an extensive literature for ordinary neutral functional differential equations; as a reference one can see Hale and Lunel's book [5] and references therein. A partial neutral differential equation with finite delay case arises, for instance, from the transmission line theory. And unbounded delay arises, for instance, from the description of the heat conduction in materials with fading memory (Gurtin and Pipkin [6]).The purpose of this paper is to deal with the existence and uniqueness of an almost periodic solution of the following functional differential equation in a complex Banach space X :where A P(X ) is the set of all almost periodic functions from R to X (cf. Definition 2.1) and the family {A(t) : t ∈ R} of operators in X generates an exponentially stable evolution system {U (t, s), t ≥ s}. We also discuss the differential

Almost periodic solutions of nonlinear functional differential equations

“…The authors have shown the existence of an almost periodic solution of a neutral functional differential equation using the Krasnoselskiis fixed point theorem and the theory of semigroups in their previous work [1]. A similar result has also been established by the authors [2] using the theory of monotone operators.…”

Square-mean almost automorphic mild solutions to some stochastic differential equations in a Hilbert space

“…They were introduced by Bochner [1,2]; for more details about this topic, we refer the reader to [3,4]. In recent years, the existence of almost periodic and almost automorphic solutions on different kinds of deterministic differential equations have been considerably investigated in lots of publications [5][6][7][8][9][10][11][12][13][14][15] because of its significance and applications in physics, mechanics, and mathematical biology.…”

Almost periodic solutions of neutral functional differential equations

“…For example, these equations arise in the study of two or more simple oscillatory systems with some interconnections between them [1,2], and in modeling physical problems such as vibration of masses attached to an elastic bar [2]. Qualitative analysis such as periodicity, almost periodicity and stability of functional equations have been studied by many researchers (see [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]20] and the references cited therein). More recently researchers have given special attention to the study of neutral differential equations, see [3,4,8,18,12,13,15,16] and references cited therein.…”

“…Qualitative analysis such as periodicity, almost periodicity and stability of functional equations have been studied by many researchers (see [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]20] and the references cited therein). More recently researchers have given special attention to the study of neutral differential equations, see [3,4,8,18,12,13,15,16] and references cited therein. Recently, Islam and Raffoul [12] have studied the periodic solution of a nonlinear neutral system of the form dx(t) dt = A(t)x(t) + d dt Q (t, x(t − g(t))) + G(t, x(t), x(t − g(t))), (1.1) where A(t) is a nonsingular n × n matrix with continuous-real-valued functions as its elements.…”