Fixed point results for multivalued contractive maps

B A Bin Dehaish,Abdul Latif

doi:10.1186/1687-1812-2012-61

B A Bin Dehaish, Abdul Latif

Open Access

https://doi.org/10.1186/1687-1812-2012-61

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Abstract

Abstract Using the concept of u-distance, we prove a fixed point theorem for multivalued contractive maps. We also establish a multivalued version of the Caristi's fixed point theorem and common fixed point result. Consequently, several known fixed point results are either improved or generalized including the corresponding fixed point results of Caristi, Mizoguchi-Takahashi, Kada et al., Suzuki-Takahashi, Suzuki, and Ume. Mathematics Subject Classification (2000): 47H10; 54H25.

Highlights

Let X be a complete metric space with metric d
Kada et al [7] introduced the notion of w-distance on a metric space as follows: A function ω : X × X ® R+ is called w-distance on X if it satisfies the following for x, y, z Î X: (w1) ω(x, z) ≤ ω(x, y) + ω(y, z); (w2) the map ω(x,.) : X ® R+ is lower semicontinuous; i.e., for {yn} in X with yn ® y
A single-valued map g : X ® X is p-contractive if there exist a u-distance p on X and a constant r Î (0, 1) such that for each x, y Î X

Summary

Introduction

Let X be a complete metric space with metric d. Investigations on the existence of fixed points for multivalued maps in the setting of metric spaces was initiated by Nadler [2]. Using the concept of Hausdorff metric, he generalized Banach contraction principle which states that each multivalued contraction map T : X ® CB(X) has fixed point provided X is complete.

Results

Conclusion