On existence of solution of a class of quadratic-integral equations using contraction defined by simulation functions and measure of noncompactness

Abstract

In this paper we have introduced a new type of contraction condition using a class of simulation functions, in the sequel using the new contraction definition, involving measure of noncompactness; we establish few results on existence of fixed points of continuous functions defined on a subset of Banach space. This result also generalizes other related results obtained in Arab [Arab, R., Some generalizations of Darbo fixed point theorem and its application, Miskolc Math. Notes, 18 (2017), No. 2, 595–610], Banas [Bana ´ s, J. and Goebel, K., ´ Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, Dekker, New York, 60 (1980)]. The obtained results are used in establishing existence theorems for a class of nonlinear quadratic equation (which generalizes several types of fractional-quadratic integral equations such as Abel’s integral equation) defined on a closed and bounded subset of R. The existence of solution is established with the aid of a measure of noncompactness defined on function space C(I) introduced in [Banas, J. and Olszowy, L., ´ Measures of Noncompactness related to monotonicity, Comment. Math., 41 (2001), 13–23].