Weighted pseudo-almost periodic solutions to a neutral delay integral equation of advanced type

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Abstract

This paper studies suitable sufficient conditions to ensure the existence and uniqueness of weighted pseudo-almost periodic solutions to a neutral delay integral equation of advanced type introduced by T.A. Burton in the literature. The abstract results are then utilized to characterize weighted pseudo-almost periodic solutions to the well-known logistic equation.

Introduction

In Diagana [7], a new generalization of Bohr almost periodic functions was introduced. This new concept is called weighted pseudo-almost periodicity and implements in a natural fashion the notion of pseudo-almost periodicity introduced in the literature in the early nineties by Zhang [9], [10], [11]. To construct those new spaces, the main idea consists of enlarging the so-called ergodic component, utilized in Zhang’s definition of pseudo-almost periodicity, with the help of a weighted measure dμ(x)=ρ(x)dx, where ρ:R(0,) is a locally integrable function over R, which is commonly called weight.

In this paper we investigate suitable sufficient conditions for the existence and uniqueness of weighted pseudo-almost periodic solutions to the abstract integral equation of the form u(t)=f(u(h1(t)))+tQ(s,u(s),u(h2(s)))C(ts)ds+g(t) for each tR, where f,g,h1,h2,C:RR are continuous functions with hi(R)=R for i=1,2, and Q:R×R×RR is jointly continuous.

Setting h1(t)=h2(t)=tp, where p>0 is a constant, in Eq. (1.1), one obtains the so-called neutral delay integral equation of advanced type u(t)=f(u(tp))+tQ(s,u(s),u(sp))C(ts)ds+g(t), which was introduced in the literature by Burton [4] as an intermediate step while studying the existence and uniqueness of (periodic) bounded solutions to the logistic differential equation given by u(t)=au(t)+αu(tp)q(t,u(t),u(tp)) where a>0, 0|α|<1, and p>0 are respectively constants.

Thus under some suitable assumptions, the existence and uniqueness of a weighted pseudo-almost periodic solution to Eq. (1.1) are obtained (Theorem 3.2). Next we make use of the previous result to prove the existence and uniqueness of a weighted pseudo-almost periodic solution to the logistic equation (Theorem 3.4).

The existence of almost periodic, asymptotically almost periodic, pseudo-almost periodic, and weighted pseudo-almost periodic solutions is among the most attractive topics in the qualitative theory of differential equations due to their applications, especially in biology, economics, and physics. Some contributions related to pseudo-almost periodic solutions to abstract differential and partial differential equations have recently been made; among them are [1], [2], [3], [5], [6], [8]. However, the existence of weighted pseudo-almost periodic solutions to integral equations, especially those of the form Eq. (1.1), is an untreated topic and this provides the main motivation of the present paper. In particular, we will make use of our result related to Eq. (1.1) to discuss the existence and uniqueness of weighted pseudo-almost periodic solutions to the logistic differential equation, that is, Eq. (1.3).

Section snippets

Weighted pseudo-almost periodic functions

Let (X,), (Y,Y) be two Banach spaces. The collection of all bounded linear operators from X into Y with be denoted as B(X,Y). This is simply denoted as B(X) when X=Y.

Let U denote the collection of all functions (weights) ρ:R(0,) which are locally integrable over R such that ρ(x)>0 for almost each xR. From now on, if ρU and for r>0, we then set m(r,ρ)rrρ(x)dx.

As in the particular case when ρ(x)=1 for each xR, we are exclusively interested in those weights, ρ, for which limrm(r,ρ)=

The main result

Throughout the rest of the paper, we suppose that X=Y=R, equipped with the classical absolute value. Note however that when dealing with the pseudo-almost periodicity of Q it would be more convenient to choose X=R×R; see (H.3).

Throughout the rest of the paper, we let ρU and assume supsR[ρ(s+τ)ρ(s)]<andsupr>0[m(r+τ,ρ)m(r,ρ)]< for each τR (the space PAP(X,ρ) is then translation invariant).

We now require the following assumptions:

  • (H.1)

    The functions f,g:RR belong to PAP(R,ρ). Moreover, f

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