Aghajani, Asadollah; Banaś, Józef; Sabzali, Navid Some generalizations of Darbo fixed point theorem and applications. (English) Zbl 1290.47053 Bull. Belg. Math. Soc. - Simon Stevin 20, No. 2, 345-358 (2013). In this paper, the authors establish some extensions of the well-known Darbo fixed point theorem. In particular, they prove the following result.Let \(\Omega\) be a nonempty, bounded, closed and convex subset of a Banach space \(E\) and let \(T: \Omega \to \Omega\) be a continuous mapping satisfying the inequality \[ \mu(TX)\leq\varphi(\mu(X)) \] for any nonempty subset \(X\) of \(\Omega\), where \(\mu\) is an arbitrary measure of noncompactness and \(\varphi: \mathbb R_+\to\mathbb R_+\) is a nondecreasing function such that \(\lim\limits_{n\to\infty}\varphi^n(t)=0\) for each \(t\geq 0\). Then \(T\) has at least one fixed point in the set \(\Omega\). Next, they show an application of that fixed point theorem proving the existence of a continuous bounded solution to the following Volterra integral equation with perturbation \[ x(t)=f(t,x(t))+\int_0^t g(t,s,x(s))\,ds \qquad \text{for }\;t\in \mathbb R_+. \] Reviewer: Dariusz Bugajewski (Poznań) Cited in 3 ReviewsCited in 91 Documents MSC: 47H10 Fixed-point theorems 47H08 Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc. 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations Keywords:measure of noncompactness; modulus of continuity; functional integral equation; Darbo fixed point theorem PDFBibTeX XMLCite \textit{A. Aghajani} et al., Bull. Belg. Math. Soc. - Simon Stevin 20, No. 2, 345--358 (2013; Zbl 1290.47053) Full Text: Euclid