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

Leszek Olszowy

Commentationes Mathematicae, Tome 48 (2008) no. 1, pp. 103-112.

Résumé

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.

Détail

DOI : 10.14708/cm.v48i1.5264

Mots-clés : Quadratic Urysohn integral, measure of noncompactness, Tichonov fixed point theorem