On a class of innite system of third-order differential equations in lp via measure of noncompactness

Abstract

In this paper, with the help of measure of noncompactness together with Darbo-type fixed point theorem, we focus on the infinite system of third-order differential equations
$$u_{i}^{\prime\prime\prime}+au_{i}^{\prime\prime}+bu_{i}^{\prime}+cu_{i}
=f_{i}(t,u_{1}(t),u_{2}(t),\ldots)$$
where $ f_{i}\in C(\mathbb{R}\times \mathbb{R}^{\infty}, \mathbb{R}) $ is
$ \omega $-periodic with respect to the first coordinate and $ a,b,c \in \mathbb{R} $ are constants.
The aim of this paper is to obtain the results with respect to the existence of $ \omega $-periodic solutions of the aforementioned system in the Banach sequence space
$ \ell_{p} $ ($ 1\leq p<\infty $) utilizing the respective Green's function.
Furthermore, some examples are provided to support our main results.