The 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(R + × R + , E) using functions that are defined, continuous and bounded on R + × 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 R + × R + with values in the space ℓ ∞ consisting of real bounded sequences endowed with the standard supremum norm. An example exemplifies our investigations.