Generalized Caristi's Fixed Point Theorems
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Research Article Generalized Caristi’s Fixed Point Theorems Abdul Latif Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Correspondence should be addressed to Abdul Latif, [email protected] Received 27 December 2008; Accepted 9 February 2009 Recommended by Mohamed A. Khamsi We present generalized versions of Caristi’s fixed point theorem for multivalued maps. Our results either improve or generalize the corresponding generalized Caristi’s fixed point theorems due to Bae 2003, Suzuki 2005, Khamsi 2008, and others. Copyright q 2009 Abdul Latif. 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.
1. Introduction A number of extensions of the Banach contraction principle have appeared in literature. One of its most important extensions is known as Caristi’s fixed point theorem. It is well known that Caristi’s fixed point theorem is equivalent to Ekland variational principle 1, which is nowadays an important tool in nonlinear analysis. Many authors have studied and generalized Caristi’s fixed point theorem to various directions. For example, see 2– 8. Kada et al. 9 and Suzuki 10 introduced the concepts of w-distance and τ-distance on metric spaces, respectively. Using these generalized distances, they improved Caristi’s fixed point theorem and Ekland variational principle for single-valued maps. In this paper, using the concepts of w-distance and τ-distance, we present some generalizations of the Caristi’s fixed point theorem for multivalued maps. Our results either improve or generalize the corresponding results due to Bae 4, 11, Kada et al. 9, Suzuki 8, 10, Khamsi 5, and many of others. Let X be a metric space with metric d. We use 2X to denote the collection of all nonempty subsets of X. A point x ∈ X is called a fixed point of a map f : X → X T : X → 2X if x fx x ∈ T x. In 1976, Caristi 12 obtained the following fixed point theorem on complete metric spaces, known as Caristi’s fixed point theorem. Theorem 1.1. Let X be a complete metric space with metric d. Let ψ : X → 0, ∞ be a lower semicontinuous function, and let f : X → X be a single-valued map such that for any
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Fixed Point Theory and Applications
x ∈ X, dx, fx ≤ ψx − ψfx.
1.1
Then f has a fixed point. To generalize Theorem 1.1, one may consider the weakening of one or more of the following hypotheses: i the metric d; ii the lower semicontinuity of the real-valued function ψ; iii the inequality 1.1; iv the function f. In 9, Kada et al. introduced a concept of w-distance on a metric space as follows. A function ω : X × X → 0, ∞ is a w-distance on X if it satisfies the following conditions for any x, y, z ∈ X: w1 ωx, z ≤ ωx, y ωy, z; w2 the map ωx, · : X → 0, ∞ is lower semicontinuous; w3 for any > 0, there exists δ > 0 such that ωz, x ≤ δ and ωz, y ≤ δ imp
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