### 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.

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