On measures of noncompactness in the space of functions defined on the half-axis with values in a Banach space

In the paper we construct measures of noncompactness in the space of functions defined, continuous and bounded on the real half-axis and taking values in an arbitrary Banach space E. We assume that in the space E is given a measure of noncompactness which is utilized in the construction of measures in question. One of the constructed measures of noncompactness is applied in the proof of a theorem on the existence of solutions of an infinite system of quadratic integral equations in the space of functions defined, continuous and bounded on the real half-axis and with values in the space of bounded real sequences. An example illustrating the mentioned existence result is included.