Common Fixed Point Theorems for a Finite Family of Discontinuous and Noncommutative Maps
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Research Article Common Fixed Point Theorems for a Finite Family of Discontinuous and Noncommutative Maps Lai-Jiu Lin and Sung-Yu Wang Department of Mathematics, National Changhua University of Education, Changhua 50058, Taiwan Correspondence should be addressed to Lai-Jiu Lin, [email protected] Received 30 December 2010; Accepted 20 February 2011 Academic Editor: Jong Kim Copyright q 2011 L.-J. Lin and S.-Y. Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study common fixed point theorems for a finite family of discontinuous and noncommutative single-valued functions defined in complete metric spaces. We also study a common fixed point theorem for two multivalued self-mappings and a stationary point theorem in complete metric spaces. Throughout this paper, we establish common fixed point theorems without commuting and continuity assumptions. In contrast, commuting or continuity assumptions are often assumed in common fixed point theorems. We also give examples to show our results. Results in this paper except those that generalized Banach contraction principle and those improve and generalize recent results in fixed point theorem are original and different from any existence result in the literature. The results in this paper will have some applications in nonlinear analysis and fixed point theory.
1. Introduction and Preliminaries Let X, d be a metric space and T : X X be a multivalued map. We say that x ∈ X is a stationary point of T if T x {x}. The existence theorem of stationary point was first considered by Dancs et al. 1. If S is a self-mapping multivalued or single valued defined on X, we denote FS the collection of all the fixed points of S. In this paper,we need the following definitions. Definition 1.1. A function f : X → X is called i contraction if there exists r ∈ 0, 1 such that d fx, f y ≤ rd x, y ,
∀x, y ∈ X,
1.1
2
Fixed Point Theory and Applications ii kannan if there exists α ∈ 0, 1/2 such that d fx, f y ≤ αd x, fx αd y, f y ,
∀x, y ∈ X,
1.2
iii quasicontractive if there is a constant r ∈ 0, 1 such that d fx, f y ≤ rM x, y ,
∀x, y ∈ X,
1.3
where Mx, y max{dx, y, dx, fx, dy, fy, dx, fy, dy, fx}. iv weakly contractive if there exists a lower semicontinuous and nondecreasing function ϕ : 0, ∞ → 0, ∞ with ϕt 0 if and only if t 0 such that d fx, f y ≤ d x, y − ϕ d x, y ,
∀x, y ∈ X.
1.4
It is known that every contraction and every Kannan mapping has a unique fixed point in complete metric spaces Banach 2, Kannan 3 and every quasicontractive mapping has a ´ c 4, Rhoades 5. In 2001, Rhoades 6 proved that unique fixed point in Banach spaces Ciri´ every weakly contractive mapping has a unique fixed point in a complete metric space. Let f and g be self-maps defined on X;
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