http://www.hindawi.comThe latest articles from Hindawi Publishing Corporation© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2010/708984.htmlFixed point and coincidence results are presented for multivalued generalized φ-weak contractive mappings on complete metric spaces, where φ:[0,+∞)→[0,+∞) is a lower semicontinuous function with φ(0)=0 and φ(t)>0 for all t>0. Our results extend previous results by Zhang and Song (2009), as well as by Rhoades (2001), Nadler (1969), and Daffer and Kaneko (1995).Behzad Djafari Rouhani and Sirous Moradi© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2010/518243.htmlTaking into account possibly inexact data, we study iterative schemes for approximating fixed points and attractors of contractive and nonexpansive set-valued mappings, respectively. More precisely, we are concerned with the existence of convergent trajectories of nonstationary dynamical systems induced by approximations of a given set-valued mapping.Simeon Reich and Alexander J. Zaslavski© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2010/864536.htmlIn cone uniform and uniform spaces, we introduce the three kinds of dissipative set-valued dynamic systems with generalized pseudodistances and not necessarily lower semicontinuous entropies, we study the convergence of dynamic processes and generalized sequences of iterations of these dissipative dynamic systems, and we establish conditions guaranteeing the existence of periodic points and endpoints of these dissipative dynamic systems and the convergence to these periodic points and endpoints of dynamic processes and generalized sequences of iterations of these dissipative dynamic systems. The paper includes examples.Kazimierz Włodarczyk and Robert Plebaniak© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2009/412898.htmlSome new weakly contractive type multimaps in the setting of metric spaces are introduced, and we prove some results on the existence of fixed points for such maps under certain conditions. Our results extend and improve several known results including the corresponding recent fixed point results of Pathak and Shahzad (2009), Latif and Abdou (2009), Latif and Albar (2008), Cirić (2008), Feng and Liu (2006), and Klim and Wardowski (2007).Abdul Latif and Afrah A. N. Abdou© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2010/546761.htmlFixed Point Theory for multivalued mappings has many useful applications in Applied Sciences, in particular, in Game Theory and Mathematical Economics. Thus, it is natural to try of extending the known fixed point results for single-valued mappings to the setting of multivalued mappings. Some theorems of existence of fixed points of single-valued mappings have already been extended to the multivalued case. However, many other questions remain still open, for instance, the possibility of extending the well-known Kirk's Theorem, that is: do Banach spaces with weak normal structure have the fixed point property (FPP) for multivalued nonexpansive mappings? There are many properties of Banach spaces which imply weak normal structure and consequently the FPP for single-valued mappings (for example, uniform convexity, nearly uniform convexity, uniform smoothness,…). Thus, it is natural to consider the following problem: do these properties also imply the FPP for multivalued mappings? In this way, some partial answers to the problem of extending Kirk's Theorem have appeared, proving that those properties imply the existence of fixed point for multivalued nonexpansive mappings. Here we present the main known results and current research directions in this subject. This paper can be considered as a survey, but some new results are also shown.T. Domínguez Benavides and B. Gavira© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2010/734181.htmlThe purpose of this paper is to study the robustness of Mann type algorithm in the sense that approximately perturbed mapping does not alter the convergence of Mann type algorithm. It is proven that Mann type algorithm with perturbed mapping xn+1=λnxn+(1−λn)(Txn+en)−λnμnF(xn) remains convergent in a Banach space setting where λn,μn∈[0,1], T a nonexpansive mapping, en, n=0,1,…, errors and F a strongly accretive and strictly pseudocontractive mapping.L. C. Ceng, Y. C. Liou, and J. C. Yao© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2010/805326.htmlIn the paper by Hu in 2008, the author proved a strong convergence result for nonexpansive mappings using a modified Halpern's iteration algorithm. Unfortunately, the case limn→∞βn=1 does not guarantee the strong convergence of the sequence {xn}. In this note, we provide a counter-example to the theorem.Shuang Wang© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2009/319804.htmlWe show some sufficient conditions on a Banach space X concerning the generalized James constant, the generalized Jordan-von Neumann constant, the generalized Zbagănu constant, the coefficient ε˜0(X), the weakly convergent sequence coefficient WCS(X), and the coefficient of weak orthogonality, which imply the existence of fixed points for multivalued nonexpansive mappings. These fixed point theorems improve some previous results in the recent papers.Zhanfei Zuo and Yunan Cui© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2010/281381.htmlThe purpose of this work is to present some (local and global) fixed point results for singlevalued and multivalued generalized contractions on spaces endowed with vector-valued metrics. The results are extensions of some theorems given by Perov (1964), Bucur et al. (2009), M. Berinde and V. Berinde (2007), O'Regan et al. (2007), and so forth.Alexandru-Darius Filip and Adrian Petruşel© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2009/520976.htmlThis paper provides a few convergence results of the Ishikawa iteration sequence with errors for nonexpansive and quasi-nonexpansive mappings in hyperspaces. The results presented in this paper improve and generalize some results in the literature.Zeqing Liu, Jeong Sheok Ume, and Shin Min Kang© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2010/321594.htmlThe first continuation method for contractive maps in the setting of a metric space was given by Granas. Later, Frigon extended Granas theorem to the class of weakly contractive maps, and recently Agarwal and O'Regan have given the corresponding result for a certain type of quasicontractions which includes maps of Kannan type. In this note we introduce the concept of weakly Kannan maps and give a fixed point theorem, and then a continuation method, for this class of maps.David Ariza-Ruiz and Antonio Jiménez-Melado© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2010/278973.htmlWe introduce an iterative algorithm for finding a common element of the set of solutions of quasivariational inclusion problems and of the set of fixed points of strict pseudocontractions in the framework Hilbert spaces. The results presented in this paper improve and extend the corresponding results announced by many others.Yu Li and Changqun Wu© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2009/257089.htmlWe introduce a new approximation scheme combining the viscosity method with extragradient method for finding a common element of the set of solutions of a mixed equilibrium problem and the set of fixed points of a finite family of nonexpansive mappings and the set of the variational inequality for a monotone, Lipschitz continuous mapping. We obtain a strong convergence theorem for the sequences generated by these processes in Hilbert spaces. Based on this result, we also get some new and interesting results. The results in this paper generalize, extend, and improve some well-known results in the literature.Jian-Wen Peng and Soon-Yi Wu© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2009/609281.htmlIn this manuscript, a class of self-mappings on cone Banach spaces which have at least one fixed point is considered. More precisely, for a closed and convex subset C of a cone Banach space with the norm ∥x∥P=d(x,0), if there exist a, b, s and T:C→C satisfies the conditions 0≤s+|a|−2b<2(a+b) and 4ad(Tx,Ty)+b(d(x,Tx)+d(y,Ty))≤sd(x,y) for all x,y∈C , then T has at least one Fixed point.Erdal Karapınar© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2010/108343.htmlWe establish two fixed point theorems for nonlinear operators on Banach spaces partially ordered by a cone. The first fixed point theorem is concerned with a class of mixed monotone operators. In the second fixed point theorem, the nonlinear operators are neither monotone nor mixed monotone. We also provide an illustrative example for our second result.Hui-Sheng Ding, Jin Liang, and Ti-Jun Xiao© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2010/240450.htmlFor a monotone operator T, we shall show weak convergence of Rockafellar's proximal point algorithm to some zero of T and strong convergence of the perturbed version of Rockafellar's to PZu under some relaxed conditions, where PZ is the metric projection from H onto Z=T−10. Moreover, our proof techniques are simpler than some existed results.Xinkuan Chai, Bo Li, and Yisheng Song© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2010/281890.htmlLet X be a generalized convex metric space, and let S, T be a pair of asymptotically nonexpansive mappings. In this paper, we will consider an Ishikawa type iteration process with errors to approximate the unique common fixed point of S and T.Chao Wang, Jin Li, and Daoli Zhu© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2010/634109.htmlWe use chain methods to prove fixed point results for maximalizing mappings in posets. Concrete examples are also presented.S. Heikkilä© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2010/471781.htmlMotivated by Halpern's result, we prove strong convergence theorem of an iterative sequence in CAT(0) spaces. We apply our result to find a common fixed point of a family of nonexpansive mappings. A convergence theorem for nonself mappings is also discussed.Satit Saejung© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2010/407651.htmlWe introduce two general iterative schemes for finding a common fixed point of a countable family of relatively nonexpansive mappings in a Banach space. Under suitable setting, we not only obtain several convergence theorems announced by many authors but also prove them under weaker assumptions. Applications to the problem of finding a common element of the fixed point set of a relatively nonexpansive mapping and the solution set of an equilibrium problem are also discussed.Daruni Boonchari and Satit Saejung© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2010/241908.htmlA real-valued continuous function f(t) on an interval (α,β) gives rise to a map X↦f(X) via functional calculus from the convex set of n×n Hermitian matrices all of whose eigenvalues belong to the interval. Since the subpace of Hermitian matrices is provided with the order structure induced by the cone of positive semidefinite matrices, one can consider convexity of this map. We will characterize its convexity by the following trace-inequalities: Tr(f(B)−f(A))(C−B)≤Tr(f(C)−f(B))(B−A) for A≤B≤C. A related topic will be also discussed.Tsuyoshi Ando© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2010/765206.htmlWe provide a general regularization method for monotone variational inequalities, where the regularizer is a Lipschitz continuous and strongly monotone operator. We also introduce an iterative method as discretization of the regularization method. We prove that both regularization and iterative methods converge in norm.Xiubin Xu and Hong-Kun Xu© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2010/926209.htmlWe obtain some new existence theorems of the maximal and minimal fixed points for discontinuous monotone operator on an order interval in an ordered normed space. Moreover, the maximal and minimal fixed points can be achieved by monotone iterative method under some conditions. As an example of the application of our results, we show the existence of extremal solutions to a class of discontinuous initial value problems.Yujun Cui and Xingqiu Zhang© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2010/970579.htmlWith a recent result of Suzuki (2001) we extend Caristi-Kirk's fixed point theorem, Ekeland's variational principle, and Takahashi's minimization theorem in a complete metric space by replacing the distance with a τ-distance. In addition, these extensions are shown to be equivalent. When the τ-distance is l.s.c. in its second variable, they are applicable to establish more equivalent results about the generalized weak sharp minima and error bounds, which are in turn useful for extending some existing results such as the petal theorem.Zili Wu© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2010/962628.htmlThe purpose of this paper is to introduce hybrid projection algorithms for finding a common element of the set of common fixed points of a countable family of relatively nonexpansive mappings and the set of solutions of an equilibrium problem in the framework of Banach spaces. Moreover, we apply our result to the problem of finding a common element of an equilibrium problem and the problem of finding a zero of a maximal monotone operator. Our result improve and extend the corresponding results announced by Takahashi and Zembayashi (2008 and 2009), and many others.Somyot Plubtieng and Wanna Sriprad© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2010/719631.htmlWe prove the weak and strong convergence of the implicit iterative process to a common fixed point of an asymptotically quasi-I-nonexpansive mapping T and an asymptotically quasi-nonexpansive mapping I, defined on a nonempty closed convex subset of a Banach space.Farrukh Mukhamedov and Mansoor Saburov© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2010/493298.htmlWe extend the celebrated result of W. A. Kirk that a metric space X is complete if and only if every Caristi self-mapping for X has a fixed point, to partial metric spaces.Salvador Romaguera© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2010/754320.htmlWe consider a modified Halpern type iterative algorithm for a family of quasi-ϕ-nonexpansive mappings in the framework of Banach spaces. Strong convergence theorems of the purposed iterative algorithms are established.Xiaolong Qin, Yeol Je Cho, Sun Young Cho, and Shin Min Kang© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2009/574387.htmlCone-valued lower semicontinuous maps are used to generalize Cristi-Kirik's fixed point theorem to Cone metric spaces. The cone under consideration is assumed to be strongly minihedral and normal. First we prove such a type of fixed point theorem in compact cone metric spaces and then generalize to complete cone metric spaces. Some more general results are also obtained in quasicone metric spaces.Thabet Abdeljawad and Erdal Karapinar© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/fpta/2010/291851.htmlA mixed monotone variational inequality (MMVI) problem in a Hilbert space H is formulated to find a point u∗∈H such that 〈Tu∗,v−u∗〉+φ(v)−φ(u∗)≥0 for all v∈H, where T is a monotone operator and φ is a proper, convex, and lower semicontinuous function on H. Iterative algorithms are usually applied to find a solution of an MMVI problem. We show that the iterative algorithm introduced in the work of Wang et al., (2001) has in general weak convergence in an infinite-dimensional space, and the algorithm introduced in the paper of Noor (2001) fails in general to converge to a solution.Xiwen Lu, Hong-Kun Xu, and Ximing Yin© 2010, Hindawi Publishing Corporation. All rights reserved.