F(psi,varphi)-Contraction in terms of measure of noncompactness with application for nonlinear integral equations

Farzaneh Nikbakhtsarvestani,Mehdi Asadi,S Mansour Vaezpour

doi:10.1186/s13660-017-1545-2

Abstract

In this paper, some new generalization of Darbo’s fixed point theorem is proved by using a F(psi,varphi)-contraction in terms of a measure of noncompactness. Our result extends to obtaining a common fixed point for a pair of compatible mappings. The paper contains an application for nonlinear integral equations as well.

Highlights

1 Introduction and preliminaries A contractive condition in terms of a measure of noncompactness, which was first used by Darbo, is one of the fruitful tools to obtain fixed point and common fixed point theorems
This paper mainly aims at employing the F(ψ, φ)-contraction and its property in terms of a measure of noncompactness to investigate a fixed point and a common fixed point for a pair of compatible mappings
Let C be a nonempty, bounded, closed, and convex subset of a Banach space E and let T : C → C be a continuous mapping

Summary

And also

S(Cn) ⊂ S(Cn– ) ⊂ co S(Cn– ) = Cn. If μ(CN ) = , for some N ∈ N, T has a fixed point in C, because Schauder’s fixed point theorem guarantees this. S : C∞ → C∞, Schauder’s fixed point theorem ensures S has a fixed point and the set A = {x ∈ C : S(x) = x} is nonempty and closed. To S; T has a fixed point and by continuity of T, B = {x ∈ C : T(x) = x} is nonempty and closed. There exists z ∈ E and it is unique, since g is one to one, such that g(Sz) = Sz, which implies Tz = Sz. T and S have a unique coincidence point from Proposition .

By taking

Proof Put

Also t

Conclusion