Fixed point theory for multivalued φ-contractions

Vasile L Lazăr

doi:10.1186/1687-1812-2011-50

Abstract

AbstractThe purpose of this paper is to present a fixed point theory for multivalued φ-contractions using the following concepts: fixed points, strict fixed points, periodic points, strict periodic points, multivalued Picard and weakly Picard operators; data dependence of the fixed point set, sequence of multivalued operators and fixed points, Ulam-Hyers stability of a multivalued fixed point equation, well-posedness of the fixed point problem, limit shadowing property of a multivalued operator, set-to-set operatorial equations and fractal operators. Our results generalize some recent theorems given in Petruşel and Rus (The theory of a metric fixed point theorem for multivalued operators, Proc. Ninth International Conference on Fixed Point Theory and its Applications, Changhua, Taiwan, July 16-22, 2009, 161-175, 2010).2010 Mathematics Subject Classification47H10; 54H25; 47H04; 47H14; 37C50; 37C70

Highlights

The purpose of this paper is to present a fixed point theory for multivalued -contractions using the following concepts: fixed points, strict fixed points, periodic points, strict periodic points, multivalued Picard and weakly Picard operators; data dependence of the fixed point set, sequence of multivalued operators and fixed points, Ulam-Hyers stability of a multivalued fixed point equation, well-posedness of the fixed point problem, limit shadowing property of a multivalued operator, set-toset operatorial equations and fractal operators
Our results generalize some recent theorems given in Petruşel and Rus
The purpose of this paper is to present a fixed point theory for multivalued -contractions in terms of the following:

Summary

Introduction

Ψ(h); (iv) (sequence of operators) Let T, Tn : X ® Pcl(X), n Î N be multivalued -contractions such that Tn(x) →H T(x)as n ® +∞, uniformly with respect to each × Î X. Let (X, d) be a complete metric space and T : X ® Pcl(X) be a multivalued -contraction with (SF)T ≠ ∅.

Objectives

Results

Conclusion