Acta Universitatis Apulensis ISSN: 1582-5329 No. 28/2011 pp. 179-188 ON MULTIVALUED CARISTI TYPE FIXED POINT THEOREMS Abdul Latif And Marwan A. Kutbi Abstract In this paper, we prove some multivalued Caristi type fixed point theorems. These results generalize the corresponding generalized Caristi’s fixed point theorems due to Kada-Suzuki-Takahashi (1996), Bae (2003), Suzuki (2005), Khamsi (2008) and others. 2000 Mathematics Subject Classification : 47H09, 54H25. 1. Introduction In 1976, Caristi [3] proved very interesting fixed point theorem on complete met- ric spaces, which is a genalizatrion of the well-known Banach contraction principle. The Caristi’s fixed point theorem, equivalent to Ekland variational principle [4], is an important tool in nonlinear analysis and has extensive applications in the fields of variational inequalities, optimization, control theory and differential equations. Many authors have studied and generalized Caristi’s fixed point theorem to various directions. Kada et al. [7] introduced the concept of w-distance on metric space and improved single-valued Caristi’s fixed point theorem. Recently, generalizing the concept of w-distance, Suzuki [10] introduced the concept of τ -distance on metric spaces and proved Caristi’s fixed point theorem for singlevalued maps with respect to τ -distance. In this note, we prove some multivalued Caristi type fixed point theorem with respect to τ -distance which are mentioned without proof in [9]. We present these here with all details since the results are not well known. In fact, these results generalize the corresponding fixed point theorems due to Kada-Suzuki-Takahashi (1996), Bae (2003), Suzuki (2005), Khamsi (2008) and others. 2. Preliminaries Let X be a metric space with metric d. We use 2 X 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 → 2 X ) if x = f (x)(x ∈ T (x)). 179