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

We show that ${\omega }_0(X)=\underset{T\to \mathrm{\infty }}{lim}\underset{\epsilon \to 0}{lim}{\omega }^T(X,\epsilon )$ is a measure of noncompactness defined on some subsets of the space $C({\mathbb{R}}^{+})=\{x:{\mathbb{R}}^{+}\to \mathbb{R},x\text{continuous}\}$ 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.