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Some generalizations of Darbo fixed point theorem and applications. (English) Zbl 1290.47053

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 \[ x(t)=f(t,x(t))+\int_0^t g(t,s,x(s))\,ds \qquad \text{for }\;t\in \mathbb R_+. \]

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
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Full Text: Euclid