Measures of noncompactness in the Banach space $BC(\mathbb{R_+}\times\mathbb{R_+},E)$ and its application to infinite system of integral equation in two variables.

Abstract

he purpose of this paper is to study the existence of solutions to an infinite system of Volterra-Hammerstein type nonlinear integral equations in two variables in Banach space $BC(\mathbb{R_+}\times\mathbb{R_+},E)$ using functions that are defined, continuous and bounded on $\mathbb{R_+}\times\mathbb{R_+}$, taking values in a given Banach space $E$. The method used in our research is linked to the creation of a suitable measure of noncompactness in the space of functions defined, continuous and bounded on $\mathbb{R_+}\times\mathbb{R_+}$ with values in the space $\ell_\infty$ consisting of real bounded sequences endowed with the standard supremum norm. An example exemplifies our investigations.