Rhoades type fixed point theorems for two hybrid pairs of mappings in metrically convex spaces

A result on common fixed points for two hybrid pairs of mappings is proved without continuity and commutativity requirements which either partially or completely generalize several earlier results due to Ahmed and Rhoades (2001) [6], Rhoades (1997) [5] and some others. An illustrative example is furnished besides deriving some related results.     

Some generalizations of fixed point theorems and common fixed point theorems

We present some fixed point theorems and common fixed point theorems which generalize and unify previous known results.     

Remarks on some recent metrical common fixed point theorems

In an interesting article, Bouhadjera and Godet-Thobie (2009) [6] introduced notions of subcompatibility and subsequential continuity, and utilized them to prove several common fixed point theorems. The results of the aforementioned article contain flaws, and they are not correct in their present form. But these results can be recovered in two ways by strengthening either of the newly introduced definitions. The results, in their corrected form, still bring about noted improvements over a multitude of relevant fixed point theorems of the existing literature, substantiating the utility of these (two) new notions.     

Comments on some fixed point theorems in hyperconvex metric spaces

In this paper, we show that a number of known fixed point theorems for the Fan-Browder type maps or acyclic maps defined on (subsets of) hyperconvex metric spaces are simple consequences of the previously known theorems for corresponding maps defined on generalized convex spaces.     

Minimax and fixed point theorems

Mathematische Annalen -     

Common fixed point theorems for some new generalized contractive type mappings

In this paper, some new generalized contractive type conditions for a pair of mappings in metric space are defined. Some common fixed point results for these mappings are presented.     

KKM mappings in cone metric spaces and some fixed point theorems

In this paper, we define KKM mappings in cone metric spaces and define Nℜ-cone metric spaces to obtain some fixed point theorems and hence generalize the results obtained in [A. Amini, M. Fakhar, J. Zafarani, KKM mapping in metric spaces, Nonlinear Anal. 60 (2005) 1045-1052].     

Edelstein's method and fixed point theorems for some generalized nonexpansive mappings

A new condition for mappings, called condition (C), which is more general than nonexpansiveness, was recently introduced by Suzuki [T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008) 1088-1095]. Following the idea of Kirk and Massa Theorem in [W.A. Kirk, S. Massa, Remarks on asymptotic and Chebyshev centers, Houston J. Math. 16 (1990) 364-375], we prove a fixed point theorem for mappings with condition (C) on a Banach space such that its asymptotic center in a bounded closed and convex subset of each bounded sequence is nonempty and compact. This covers a result obtained by Suzuki [T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008) 1088-1095]. We also present fixed point theorems for this class of mappings defined on weakly compact convex subsets of Banach spaces satisfying property (D). Consequently, we extend the results in [T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008) 1088-1095] to many other Banach spaces.     

Generalized KKM theorems and common fixed point theorems

In this paper, we first prove a generalized KKM theorem, and then use this generalized KKM theorem to establish the generalized equi-KKM theorem, common fixed point theorems for a family of multivalued maps, and the Kakutani-Fan-Glicksberg fixed point theorem. We also show that an existence theorem of the common fixed point theorem is equivalent to the Kakutani-Fan-Glicksberg fixed point theorem.     

Fuzzy mappings and fixed point theorems

We extend Heilpern's fixed point theorem for fuzzy contraction mappings to a pair of generalized fuzzy contraction mappings. Also we prove a fixed point theorem for nonexpansive fuzzy mappings on a compact star-shaped subset of a Banach space.     

On fixed point theorems and nonsensitivity

Sensitivity is a prominent aspect of chaotic behavior of a dynamical system. We study the relevance of nonsensitivity to fixed point theory in affine dynamical systems. We prove a fixed point theorem which extends Ryll-Nardzewski??s theorem and some of its generalizations. Using the theory of hereditarily nonsensitive dynamical systems we establish left amenability of Asp(G), the algebra of Asplund functions on a topological group G (which contains the algebra WAP(G) of weakly almost periodic functions). We note that, in contrast to WAP(G) where the invariant mean is unique, for some groups (including the integers) there are uncountably many invariant means on Asp(G). Finally, we observe that dynamical systems in the larger class of tame G-systems need not admit an invariant probability measure, and the algebra Tame(G) is not left amenable.