On existence of solutions of a quadratic Urysohn integral equation on an unbounded interval

Abstract

We show that $\omega_0 (X) = \lim_{T\to\infty} \lim_{\varepsilon\to 0} \omega^T (X, \varepsilon)$ is a measure of noncompactness defined on some subsets of the space $C(\mathbb{R}^+) = \{x\colon \mathbb{R}^+ \to \mathbb{R},\ x\ \text{continuous}\}$ furnished with the distance defined by the family of seminorms $|x|_n$. Moreover, using a technique associated with the measures of noncompactness, we prove the existence of solutions of a quadratic Urysohn integral equation on an unbounded interval. This measure allows to obtain theorems on the existence of solutions of a integral equations on an unbounded interval under a weaker assumptions then the assumptions of theorems obtained by applying two-component measures of noncompactness