Fixed Points and Stability for Integral-Type Multivalued Contractive Mappings

Zhefu An,Liangshi Zhao,Mengyao Li,Huseyin Isik

doi:10.1155/2021/5542414

Abstract

The existence and iterative approximations of fixed points concerning two classes of integral-type multivalued contractive mappings in complete metric spaces are proved, and the stability of fixed point sets relative to these multivalued contractive mappings is established. The results obtained in this article generalize and improve some known results in the literature. An illustrative example is given.

Highlights

IntroductionThe famous Banach fixed point theorem has both various extensions and valuable applications in a mass of differential equations, difference equations, functional equations, matrix equations, and integral equations ([1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26])
In 2002, Branciari [3] obtained an interesting integral-type fixed point theorem for the contractive mapping of integral type, which is an integral version of the Banach contraction mapping
In 1969, Nadler [15] gave a multivalued analog of the Banach fixed point theorem by using the Hausdorff metric and introducing the multivalued contraction mapping, that is, he presented a nice fixed point theorem for the multivalued contraction mapping

Summary

Introduction

The famous Banach fixed point theorem has both various extensions and valuable applications in a mass of differential equations, difference equations, functional equations, matrix equations, and integral equations ([1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]). Let ðX, ρÞ be a complete metric space and T : X ⟶ CBðXÞ be a multivalued contraction mapping, that is, there exists a constant r ∈ 1⁄20, 1Þ satisfying. The researchers [4, 11, 13, 22,23,24] gained fixed point theorems for several multivalued contractive mappings and studied the stability of fixed point sets with respect to the multivalued contractive mappings. By combining the ideas of Nadler, Branciari, and Lim, in this article, we study the existence and iterative approximations of fixed points concerning two classes of integral-type multivalued contractive mappings in complete metric spaces and present stability of fixed point sets relative to a sequence of integral-type multivalued contractive mappings. An example is presented to illustrate the efficiency of our results

Preliminaries

Fixed Point Theorems and an Example

On Stability of Fixed Point Sets

Conclusion