Some fixed point theorems for Meir-Keeler condensing operators and application to a system of integral equations. (English) Zbl 07094826

Belhadj, Maha; Ben Amar, Afif; Boumaiza, Mohamed

Bull. Belg. Math. Soc. - Simon Stevin 26, No. 2, 223-239 (2019).

Summary: We introduce the concept of Meir-Keeler condensing operator in a Banach space via an arbitrary measure of weak noncompactness. We prove some generalizations of Darbo’s fixed point theorem by considering a measure of weak noncompactness which not necessary has the maximum property. We prove some coupled fixed point theorems and we apply them in order to establish the existence of weak solutions for a system of functional integral equations of Volterra type.

MSC:

47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10	Fixed-point theorems
47H30	Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
45B05	Fredholm integral equations

Keywords:

measure of weak noncompactness; condensing map; Darbo fixed point theorem; non-linear integral equations