Abstract:
We prove a strong convergence theorem for a common fixed point of two sequences of strictly pseudocontractive mappings in Hilbert spaces. We also provide some applications of the main theorem to find a common element of the set of fixed points of a strict pseudocontraction and the set of solutions of an equilibrium problem in Hilbert spaces. The results extend and improve the recent ones announced by Marino and Xu (2007) and others. 1. Introduction Let be a real Hilbert space and a nonempty closed convex subset of . Let be a self-mapping of . Recall that is said to be a strict pseudocontraction if there exists a constant such that for all . (We also say that is a -strict pseudocontraction if satisfies (1.1)). We use to denote the set of fixed points of (i.e., ). Note that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings which are mappings on such that for all . That is, is nonexpansive if and only if is a 0-strict pseudocontraction. In 1953, Mann [1] introduced the following iterative scheme: where the sequence is chosen in . Mann's iteration process (1.3) has been extensively investigated for nonexpansive mappings. One of the fundamental convergence results was proved by Reich [2]. In an infinite-dimensional Hilbert space, the Mann's iteration (1.3) can conclude only weak convergence [3, 4]. In 1967, Browder and Petryshyn [5] established the first convergence result for a -strict pseudocontraction in a real Hilbert space. They proved weak and strong convergence theorems by using (1.3) with a constant control sequence for all . However, this scheme has only weak convergence even in a Hilbert space. Therefore, many authors try to modify the normal Mann's iteration process to have strong convergence; see, for example, [6–10] and the references therein. Attempts to modify (1.3) so that strong convergence is guaranteed have been made. In 2003, Nakajo and Takahashi [9] proposed the following modification of (1.3) for a single nonexpansive mapping by using the hybrid projection method in a Hilbert space where denotes the metric projection from onto a closed convex subset of . They proved that if the sequence is bounded above from one, then defined by (1.4) converges strongly to . In 2007, Marino and Xu [11] proved the following strong convergence theorem by using the hybrid projection method for a strict pseudocontraction. They defined a sequence as follows: They proved that if , then defined by (1.5) converges strongly to . Motivated and inspired by the above-mentioned results, it is the purpose of this paper to

Abstract:
We provide some new type of mappings associated with pseudocontractions by introducing some actual examples in smooth and strictly convex Banach spaces. Moreover, we also find the significant inequality related to the mappings mentioned in the paper and the mappings defined from generalized mixed equilibrium problems on Banach spaces. We propose an iterative shrinking projection method for finding a common solution of generalized mixed equilibrium problems and fixed point problems of closed and -quasi-strict pseudo-contractions. Our results hold in reflexive, strictly convex, and smooth Banach spaces with the property (). The results of this paper improve and extend the corresponding results of Zhou and Gao (2010) and many others.

Abstract:
We prove a strong convergence theorem by using a hybrid algorithm in order to find a common fixed point of Lipschitz pseudocontraction and κ-strict pseudocontraction in Hilbert spaces. Our results extend the recent ones announced by Yao et al. (2009) and many others.

Abstract:
The convex feasibility problem (CFP) of finding a point in the nonempty intersection is considered, where is an integer and the 's are assumed to be convex closed subsets of a Banach space . By using hybrid iterative methods, we prove theorems on the strong convergence to a common fixed point for a finite family of relatively nonexpansive mappings. Then, we apply our results for solving convex feasibility problems in Banach spaces.

Abstract:
The convex feasibility problem (CFP) of finding a point in the nonempty intersection i=1NCi is considered, where N 1 is an integer and the Ci's are assumed to be convex closed subsets of a Banach space E. By using hybrid iterative methods, we prove theorems on the strong convergence to a common fixed point for a finite family of relatively nonexpansive mappings. Then, we apply our results for solving convex feasibility problems in Banach spaces.

Abstract:
We create some new ideas of mappings called quasi-strict -pseudocontractions. Moreover, we also find the significant inequality related to such mappings and firmly nonexpansive mappings within the framework of Hilbert spaces. By using the ideas of metric -projection, we propose an iterative shrinking metric -projection method for finding a common fixed point of a quasi-strict -pseudocontraction and a countable family of firmly nonexpansive mappings. In addition, we provide some applications of the main theorem to find a common solution of fixed point problems and generalized mixed equilibrium problems as well as other related results. 1. Introduction It is well known that the metric projection operators in Hilbert spaces and Banach spaces play an important role in various fields of mathematics such as functional analysis, optimization theory, fixed point theory, nonlinear programming, game theory, variational inequality, and complementarity problem (see, e.g., [1, 2]). In 1994, Alber [3] introduced and studied the generalized projections from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces. Moreover, Alber [1] presented some applications of the generalized projections to approximately solve variational inequalities and von Neumann intersection problem in Banach spaces. In 2005, Li [2] extended the generalized projection operator from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces and studied some properties of the generalized projection operator with applications to solve the variational inequality in Banach spaces. Later, Wu and Huang [4] introduced a new generalized -projection operator in Banach spaces. They extended the definition of the generalized projection operators introduced by [3] and proved some properties of the generalized -projection operator. Fan et al. [5] presented some basic results for the generalized -projection operator and discussed the existence of solutions and approximation of the solutions for generalized variational inequalities in noncompact subsets of Banach spaces. Let be a real Hilbert space; a mapping with domain and range in is called firmly nonexpansive if nonexpansive if Throughout this paper, stands for an identity mapping. The mapping is said to be a strict pseudocontraction if there exists a constant such that In this case, may be called a -strict pseudocontraction. We use to denote the set of fixed points of (i.e. . is said to be a quasi-strict pseudocontraction if the set of fixed point is nonempty and if there exists a constant such that Construction of fixed

Abstract:
We are concerned with the study of a multistep iterative scheme with errors involving a finite family of nonexpansive nonself-mappings. We approximate the common fixed points of a finite family of nonexpansive nonself-mappings by weak and strong convergence of the scheme in a uniformly convex Banach space. Our results extend and improve some recent results, Shahzad (2005) and many others.

Abstract:
We are concerned with the study of a multistep iterative scheme with errors involving a finite family of nonexpansive nonself-mappings. We approximate the common fixed points of a finite family of nonexpansive nonself-mappings by weak and strong convergence of the scheme in a uniformly convex Banach space. Our results extend and improve some recent results, Shahzad (2005) and many others.

Abstract:
Recently, Basha (2013) addressed a problem that amalgamates approximation and optimization in the setting of a partially ordered set that is endowed with a metric. He assumed that if and are nonvoid subsets of a partially ordered set that is equipped with a metric and is a non-self-mapping from to , then the mapping has an optimal approximate solution, called a best proximity point of the mapping , to the operator equation , when is a continuous, proximally monotone, ordered proximal contraction. In this note, we are going to obtain his results by omitting ordering, proximal monotonicity, and ordered proximal contraction on . 1. Introduction Let be a non-self-mapping from to , where and are nonempty subsets of a metric space . Clearly, the set of fixed points of can be empty. In this case, one often attempts to find an element that is in some sense closest to . Best approximation theory and best proximity point analysis are applicable for solving such problems. The well-known best approximation theorem, due to Fan [1], asserts that if is a nonempty, compact, and convex subset of a normed linear space and is a continuous function from to , then there exists a point in such that the distance of to is equal to the distance of to . Such a point is called a best approximation point of in . A point in is said to be a best proximity point for , if the distance of to is equal to the distance of to . The aim of best proximity point theory is to provide sufficient conditions that assure the existence of best proximity points. Investigation of several variants of contractions for the existence of a best proximity point can be found in [1–15]. In most of the papers on the best proximity, the ordering, proximal monotonicity, and ordered proximal contraction on the mapping play a key role. A natural question arises that it is possible that we can have other ways that may not require the ordering as well as proximal monotonicity and ordered proximal contraction on the mapping . Very recently, Basha [5] addressed a problem that amalgamates approximation and optimization in the setting of a partially ordered set that is endowed with a metric. He assumed that if and are nonvoid subsets of a partially ordered set that is equipped with a metric and is a non-self-mapping from to , then the mapping has an optimal approximate solution, called a best proximity point of the mapping , to the operator equation , when is a continuous, proximally monotone, ordered proximal contraction. In this note, we are going to obtain his results by omitting ordering, proximal monotonicity, and