Solvability of infinite systems of third-order differential equations in $$c_{0}$$ by Meir–Keeler condensing operators

Abstract

Throughout this work, using the technique of measure of noncompactness together with Meir–Keeler condensing operators, we study the solvability of the following infinite system of third-order differential equations in the Banach sequence space $$c_0 $$ as a closed subspace of $$ \ell ^{\infty } $$ : $$\begin{aligned} u_{i}^{\prime \prime \prime }+au_{i}^{\prime \prime }+bu_{i}^{\prime }+cu_{i} =f_{i}(t,u_{1}(t),u_{2}(t),\ldots ) \end{aligned}$$ 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 constant. Our approach depends on the Green’s function corresponding to the aforesaid system and deduce some conclusions relevant to the existence of $$ \omega $$ -periodic solutions in Banach sequence space $$c_{0} $$ . In addition, some examples are supplied to illustrate the usefulness of the outcome.