Abstract:
In this paper, we prove a common fixed point theorem for a pair of weakly compatible mappings in fuzzy metric space using the joint common limit in the range property of mappings called (JCLR) property. An example is also furnished which demonstrates the validity of main result. We also extend our main result to two finite families of self mappings. Our results improve and generalize results of Cho et al. [Y. J. Cho, S. Sedghi and N. Shobe, “Generalized fixed point theorems for compatible mappings with some types in fuzzy metric spaces,” Chaos, Solitons & Fractals, Vol. 39, No. 5, 2009, pp. 2233-2244.] and several known results existing in the literature.

Abstract:
This paper discusses the monotone variational inequality over the solution set of the variationalinequality problem and the fixed point set of a nonexpansive mapping. The strong convergencetheorem for the proposed algorithm to the solution is guaranteed under some suitable assumptions.

Abstract:
We introduce a general implicit iterative scheme base on viscosity approximation method with a ϕ-strongly pseudocontractive mapping for finding a common element of the set of solutions for a system of mixed equilibrium problems, the set of common fixed point for a nonexpansive semigroup, and the set of solutions of system of variational inclusions with set-valued maximal monotone mapping and Lipschitzian relaxed cocoercive mappings in Hilbert spaces. Furthermore, we prove that the proposed iterative algorithm converges strongly to a common element of the above three sets, which is a solution of the optimization problem related to a strongly positive bounded linear operator.

Abstract:
We introduce a modified block hybrid projection algorithm for solving the convex feasibility problems for an infinite family of closed and uniformly quasi- -asymptotically nonexpansive mappings and the set of solutions of the generalized equilibrium problems. We obtain a strong convergence theorem for the sequences generated by this process in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in this paper improve and extend some recent results. 1. Introduction and Preliminaries The convex feasibility problem (CFP) is the problem of computing points laying in the intersection of a finite family of closed convex subsets , of a Banach space This problem appears in various fields of applied mathematics. The theory of optimization [1], Image Reconstruction from projections [2], and Game Theory [3] are some examples. There is a considerable investigation on (CFP) in the framework of Hilbert spaces which captures applications in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning [4]. The advantage of a Hilbert space is that the projection onto a closed convex subset of is nonexpansive. So projection methods have dominated in the iterative approaches to (CFP) in Hilbert spaces. In 1993, Kitahara and Takahashi [5] deal with the convex feasibility problem by convex combinations of sunny nonexpansive retractions in a uniformly convex Banach space. It is known that if is a nonempty closed convex subset of a smooth, reflexive, and strictly convex Banach space, then the generalized projection (see, Alber [6] or Kamimura and Takahashi [7]) from onto is relatively nonexpansive, whereas the metric projection from onto is not generally nonexpansive. We note that the block iterative method is a method which is often used by many authors to solve the convex feasibility problem (CFP) (see, [8, 9], etc.). In 2008, Plubtieng and Ungchittrakool [10] established strong convergence theorems of block iterative methods for a finite family of relatively nonexpansive mappings in a Banach space by using the hybrid method in mathematical programming. Let be a nonempty closed convex subset of a real Banach space with and being the dual space of . Let be a bifunction of into and a monotone mapping. The generalized equilibrium problem, denoted by , is to find such that The set of solutions for the problem (1.1) is denoted by , that is If , the problem (1.1) reducing into the equilibrium problem for , denoted by , is to find such that If , the problem (1.1) reducing into the

Abstract:
The purpose of this paper is to introduce and study a modified Halpern’s iterative scheme for solving the split feasibility problem (SFP) in the setting of infinite-dimensional Hilbert spaces. Under suitable conditions a strong convergence theorem is established. The main result presented in this paper improves and extends some recent results done by Xu (Iterative methods for the split feasibility problem in infinite-dimensional Hilbert space, Inverse Problem 26 (2010) 105018) and some others.

Abstract:
We introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of two countable families of relatively quasi-nonexpansive mappings, the set of the variational inequality for an -inverse-strongly monotone operator, the set of solutions of the generalized mixed equilibrium problem and zeros of a maximal monotone operator in the framework of a real Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2 uniformly convex and uniformly smooth Banach space. The results presented in this paper improve and extend some recent results. 1. Introduction Let be a Banach space with norm , a nonempty closed convex subset of , and let denote the dual of . Let be a bifunction, be a real-valued function, and a mapping. The generalized mixed equilibrium problem, is to find such that The set of solutions to (1.1) is denoted by , that is, If , the problem (1.1) reduces into the mixed equilibrium problem for , denoted by , which is to find such that If , the problem (1.1) reduces into the mixed variational inequality of Browder type, denoted by , which is to find such that If and the problem (1.1) reduces into the equilibrium problem for , denoted by , which is to find such that If , the problem (1.3) reduces into the minimize problem, denoted by , is to find such that The above formulation (1.4) was shown in [1] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, variational inequality problems, vector equilibrium problems, Nash equilibria in noncooperative games. In addition, there are several other problems, for example, the complementarity problem, fixed point problem and optimization problem, which can also be written in the form of an . In other words, the is an unifying model for several problems arising in physics, engineering, science, optimization, economics, and so forth. In the last two decades, many papers have appeared in the literature on the existence of solutions of ; see, for example, [1, 2] and references therein. Some solution methods have been proposed to solve the ; see, for example, [1, 3–11] and references therein. A Banach space is said to be strictly convex if for all with and . Let be the unit sphere of . Then a Banach space is said to be smooth if the limit exists for each It is also said to be uniformly smooth if the limit exists uniformly for . Let be a Banach space. The modulus of convexity of is the function defined by A Banach space is uniformly convex if and only if

Abstract:
We introduce a new general system of generalized nonlinear mixed composite-type equilibria and propose a new iterative scheme for finding a common element of the set of solutions of a generalized equilibrium problem, the set of solutions of a general system of generalized nonlinear mixed composite-type equilibria, and the set of fixed points of a countable family of strict pseudocontraction mappings. Furthermore, we prove the strong convergence theorem of the purposed iterative scheme in a real Hilbert space. As applications, we apply our results to solve a certain minimization problem related to a strongly positive bounded linear operator. Finally, we also give a numerical example which supports our results. The results obtained in this paper extend the recent ones announced by many others.

Abstract:
The main purpose of this paper is considering the lacunary sequence spaces defined by Karakaya (2007), by proving the property ( ) and Uniform Opial property. 1. Introduction Let be a real Banach space and let (resp., ) be a closed unit ball (resp., the unit sphere) of . For any subset of , we denote by the convex hull of . The Banach space is uniformly convex , if for each there exists such that for the inequality implies (see [1]). A Banach space has the property if for each there exists such that implies , where denotes the Kuratowski measure noncompactness of a subset of defined as the infimum of all such that can be covered by a finite union of sets of diameter less than . The following characterization of the property is very useful (see [2]): A Banach space has the property if and only if for each there exists such that for each element and each sequence in with there is an index for which where . A Banach space is nearly uniformly convex if for each and every sequence in with , there exists such that . Define for any the drop determined by by . A Banach space has the drop property (write ) if for every closed set disjoint with there exists an element such that . A point is an of if for any sequence in such that as , the week convergence of to implies that as . If every point in is an -point of , then is said to have the property . A Banach space is said to have the uniform Kadec-Klee property (abbreviated as (UKK)) if for every there exists such that for every sequence in with and as , we have . Every (UKK) Banach space has -property (see [3]). The following implications are true in any Banach spaces,(1.1) where denotes the property of reflexivity (see [3–6]). A Banach space is said to have the Opial property (see [7]) if every sequence weakly convergent to satisfies for every . Opial proved in [7] that the sequence space have this property but do not have it. A Banach space is said to have the uniform Opial property (see [8]), if for each there exists such that for any weakly null sequence in and with there holds For example, the space in [9, 10] has the uniform Opial property. The Opial property is important because Banach spaces with this property have the weak fixed point property (see [11]) and the geometric property involving fixed point theory can be found, for example, in [9, 12–14]. For a bounded subset , the set measure of noncompactness was defined in [15] by The ball measure of noncompactness was defined in [16, 17] by The functions and are called the Kuratowski measure of noncompactness and the Hausdorff measure of noncompactness in

Abstract:
We introduce a new iterative algorithm for finding a common element of the set of solutions of a system of generalized mixed equilibrium problems, zero set of the sum of a maximal monotone operators and inverse-strongly monotone mappings, and the set of common fixed points of an infinite family of nonexpansive mappings with infinite real number. Furthermore, we prove under some mild conditions that the proposed iterative algorithm converges strongly to a common element of the above four sets, which is a solution of the optimization problem related to a strongly positive bounded linear operator. The results presented in the paper improve and extend the recent ones announced by many others.

Abstract:
The purpose of this paper is to present a new hybrid block iterative scheme by the generalized - projection method for finding a common element of the fixed point set for a countable family of uniformly quasi--asymptotically nonexpansive mappings and the set of solutions of the system of generalized mixed equilibrium problems in a strictly convex and uniformly smooth Banach space with the Kadec-Klee property. Furthermore, we prove that our new iterative scheme converges strongly to a common element of the aforementioned sets. The results presented in this paper improve and extend important recent results in the literature.