Summary: In this paper, based on some geometrical properties of CATp(0) spaces, for p≥2, we obtain two fixed point results for monotone multivalued nonexpansive mappings and proximally monotone nonexpansive mappings. Which under some assumptions, reduce to coincide and generalize a fixed point result for monotone nonexpansive mappings. This work is a continuity of the previous work of Ran and Reurings, Nieto and Rodríguez-López done for monotone contraction mappings.

Summary: The extended tanh method with a computerized symbolic computation is used for constructing the traveling wave solutions of coupled nonlinear equations arising in physics. The obtained solutions include solitons, kinks and plane periodic solutions. The applied method will be used to solve the generalized coupled Hirota Satsuma KdV equation.

Summary: Best proximity points in the Hilbert ball (JNCA). Volume 17. Number 6. pp. 1083-1094. Best proximity points in the Hilbert ball. Abdul Rahim Khan and Sami Atif Shukri, Key words, Mathematices Subject Classification. Hilbert ball, Best proximity point, coupled best proximity point, nonexpansive mapping, firmly nonexpansive mapping, Primary 47H10, 54H25; Secondary 47H09, 46C20. ONLINE SUBSCRIPTION (Library Only) PDF, PDF. Open Access: until 31 OCT. (Free) Flash, Flash. Copyright© 2016 Yokohama Publishers, For Editor, For Authors.

Summary: In this paper I seek soliton solutions of two-component generalizations of the Kaup-Kupershmidt and Sawada-Kotera equations, for this purpose I will apply the extended tanh method. The extended tanh method with a computerized symbolic computation, is used for constructing the travelling wave solutions of coupled nonlinear equations arising in physics. The obtained solutions include soliton, kink and plane periodic solutions. KeyWords: Soliton Solutions; Kaup-Kupershmidt Equation; Sawada-Kotera Equation

Summary: The existence of best proximity point is an important aspect of optimization theory. We define the concept of proximally monotone Lipschitzian mappings on a partially ordered metric space. Then we obtain sufficient conditions for the existence and uniqueness of best proximity points for these mappings in partially ordered CAT (0) spaces. This work is a continuation of the work of Ran and Reurings [Proc. Amer. Math. Soc. 132 (2004), 1435–1443] and Nieto and Rodr ıguez-López [Order, 22 (2005), 223–239] for the new class of mappings introduced herein.

Summary: In this work, we study existence and convergence of best proximity points of a cyclic contraction mapping in a complete CATp(0) metric space, with p≥2. The case of coupled best proximity points of a pair of cyclic contraction mappings is also discussed. As an application, we provide sufficient conditions to obtain an extension of the Banach Contraction Principle for coupled fixed points.

Summary: Some fixed point theorems for discontinuous mappings in Banach spaces by Berinde and Pacurar [Fixed point theorems for non-self single-valued almost contractions, Fixed Point Theory 14 (2013), 301-311] and Kirk [Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type, Israel J. Math. 17 (1974), 339-346] are extended to uniformly convex metric spaces.

Summary: We extend the Gromov geometric definition of CAT (0) spaces to the case where the comparison triangles are not in the Euclidean plane but belong to a general Banach space. In particular, we study the case where the Banach space is , for .

Summary: A general viscosity iterative method for a finite family of generalized asymptotically quasi-nonexpansive mappings in a convex metric space is introduced. Special cases of the new iterative method are the viscosity iterative method of Chang et al. (Appl. Math. Comput. 212:51-59, 2009), an analogue of the viscosity iterative method of Fukhar-ud-din et al. (J. Nonlinear Convex Anal. 16:47-58, 2015) and an extension of the multistep iterative method of Yildirim and Özdemir (Arab. J. Sci. Eng. 36:393-403, 2011). Our results generalize and extend the corresponding known results in uniformly convex Banach spaces and spaces simultaneously.

Summary: In this paper, we prove the analog to Browder and Göhde fixed point theorem for G-nonexpansive mappings in complete hyperbolic metric spaces uniformly convex. In the linear case, this result is refined. Indeed, we prove that if X is a Banach space uniformly convex in every direction endowed with a graph G, then every G-nonexpansive mapping T: A→ A, where A is a nonempty weakly compact convex subset of X, has a fixed point provided that there exists u0∈ A such that T (u0) and u0 are G-connected. c 2016 All rights reserved.

Summary: In this work, we study existence and convergence of best prox-imity points of a cyclic contraction mapping in a complete CATp(0)metric space, with p ≥ 2. The case of coupled best proximity points of apair of cyclic contraction mappings is also discussed. As an applicati on,we provide sufficient conditions to obtain an extension of the BanachContraction Principle for coupled fixed points.

Summary: Strong convergence and△-convergence of an implicit Ishikawa type algorithm associated with two nonexpansive mappings on a hyperbolic metric space is established.

Summary: In this work, some geometrical properties of Hilbert spaces are investigated in lp spaces, for p ≥ 2. As an application, we obtain an extension of the Banach Contraction Principle for best proximity points. The case of nonexpansive mappings is also discussed.