Weighted pseudo-almost periodic solutions to a neutral delay integral equation of advanced type
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 , where is a locally integrable function over , 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 for each , where are continuous functions with for , and is jointly continuous.
Setting , where is a constant, in Eq. (1.1), one obtains the so-called neutral delay integral equation of advanced type 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 where , , and 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 , be two Banach spaces. The collection of all bounded linear operators from into with be denoted as . This is simply denoted as when .
Let denote the collection of all functions (weights) which are locally integrable over such that for almost each . From now on, if and for , we then set
As in the particular case when for each , we are exclusively interested in those weights, , for which
The main result
Throughout the rest of the paper, we suppose that , equipped with the classical absolute value. Note however that when dealing with the pseudo-almost periodicity of it would be more convenient to choose ; see (H.3).
Throughout the rest of the paper, we let and assume for each (the space is then translation invariant).
We now require the following assumptions:
- (H.1)
The functions belong to . Moreover,
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