Multivalued generalized nonlinear contractive maps and fixed points

doi:10.1016/j.na.2010.10.017

Nonlinear Analysis: Theory, Methods & Applications

Volume 74, Issue 4, 15 February 2011, Pages 1436-1444

https://doi.org/10.1016/j.na.2010.10.017 Get rights and content

Abstract

We introduce some notions of generalized nonlinear contractive maps and prove some fixed point results for such maps. Consequently, several known fixed point results are either improved or generalized including the corresponding recent fixed point results of Ciric [L.B. Ciric, Multivalued nonlinear contraction mappings, Nonlinear Anal. 71 (2009) 2716–2723], Klim and Wardowski [D. Klim, D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (2007) 132–139], Feng and Liu [Y. Feng, S. Liu, Fixed point theorems for multivalued contractive mappings and multivalued Caristi type mappings, J. Math. Anal. Appl. 317 (2006) 103–112] and Mizoguchi and Takahashi [N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989) 177–188].

Section snippets

Introduction and preliminaries

Let $(X, d)$ be a metric space, $2^{X}$ a collection of nonempty subsets of $X$ , and $C B (X)$ a collection of nonempty closed bounded subsets of $X, C l (X)$ a collection of nonempty closed subsets of $X, K (X)$ a collection of nonempty compact subsets of $X$ .

For any $A, B \in C B (X)$ , let $H (A, B) = max {sup_{x \in A} d (x, B), sup_{y \in B} d (y, A)},$ where $d (x, B) = {inf}_{y \in B} d (x, y)$ . $H$ is called the Hausdorff metric induced by $d$ .

An element $x \in X$ is called a fixed point of a multivalued map $T : X \to 2^{X}$ if $x \in T (x)$ . We denote $Fix (T) = {x \in X : x \in T (x)}$ . A sequence ${x_{n}}$ in $X$

The results

First we prove a fixed point result which improves the recent fixed point result [5, Theorem 2.1] of Ciric.

Theorem 2.1

Let $(X, d)$ be a complete metric space with a $w$ -distance $ω$ . Let $T : X \to C l (X)$ be a multivalued map. Assume that the following conditions hold:

(I)
there exist a constant $b \in (0, 1)$ and a function $φ : [0, \infty) \to [b, 1)$ such that for each $t \in [0, \infty)$ $\underset{r \to t^{+}}{lim sup} φ (r) < 1$
(II)
the map $f : X \to R$ , defined by $f (x) = ω (x, T (x))$ is lower semicontinuous
(III)
for any $x \in X$ , there exists $y \in T (x)$ satisfying $\sqrt{φ (f (x))} ω (x, y) \leq f (x),$ and $f (y) \leq φ (f (x)) ω (x, y) .$

Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions which led to an improved presentation of the manuscript.

References (8)

N. Mizoguchi et al.
Fixed point theorems for multivalued mappings on complete metric spaces
J. Math. Anal. Appl.
(1989)
Y. Feng et al.
Fixed point theorems for multivalued contractive mappings and multivalued Caristi type mappings
J. Math. Anal. Appl.
(2006)
D. Klim et al.
Fixed point theorems for set-valued contractions in complete metric spaces
J. Math. Anal. Appl.
(2007)
L.B. Ciric
Multivalued nonlinear contraction mappings
Nonlinear Anal.
(2009)

There are more references available in the full text version of this article.

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