Abstract:
We introduce a class of η-accretive mappings in a real Banach space and show that the η-proximal point mapping for η-m-accretive mapping is Lipschitz continuous. Further, we develop an iterative algorithm for a class of general variational-like inclusions involving η-accretive mappings in real Banach space, and discuss its convergence criteria. The class of η-accretive mappings includes several important classes of operators that have been studied by various authors.

Abstract:
Metric projection operators can be defined in similar wayin Hilbert and Banach spaces. At the same time, they differ signifitiantly in their properties. Metric projection operator in Hilbert space is a monotone and nonexpansive operator. It provides an absolutely best approximation for arbitrary elements from Hilbert space by the elements of convex closed sets . This leads to a variety of applications of this operator for investigating theoretical questions in analysis and for approximation methods. Metric projection operators in Banach space do not have properties mentioned above and their applications are not straightforward. Two of the most important applications of the method of metric projection operators are as follows: 1. Solve a variational inequality by the iterative-projection method, 2. Find common point of convex sets by the iterative-projection method. In Banach space these problems can not be solved in the framework of metric projection operators. Therefore, in the present paper we introduce new generalized projection operators in Banach space as a natural generalization of metric projection operators in Hilbert space. In Sections 2 and 3 we introduce notations and recall some results from the theory of variational inequalities and theory of approximation. Then in Sections 4 and 5 we describe the properties of metric projection operators $P_\Omega$ in Hilbert and Banach spaces and also formulate equivalence theorems between variational inequalities and direct projection equations with these

Abstract:
In 2007 H. Long-Guang and Z. Xian, [H. Long-Guang and Z. Xian, Cone Metric Spaces and Fixed Point Theorems of Contractive Mapping, J. Math. Anal. Appl., 322(2007), 1468-1476], generalized the concept of a metric space, by introducing cone metric spaces, and obtained some fixed point theorems for mappings satisfying certain contractive conditions. The main question was "Are cone metric spaces a real generalization of metric spaces?" Throughout this paper we answer the question in the negative, proving that every cone metric space is metrizable and the equivalent metric satisfies the same contractive conditions as the cone metric. So most of the fixed point theorems which have been proved are straightforward results from the metric case.

Abstract:
In this paper the notion of modular cone metric space is introduced and some properties of such spaces are investigated. Also we define convex modular cone metric which takes values in CR(Y) where Y is a compact Hausdorff space. Then a fixed point theorem is proved for contractions in these spaces. Furthermore, we make a remark on paper [9] and it will be proved that their fixed point result in modular metric spaces is not true.

Abstract:
In this paper the concept of a partial cone metric space is investigated, some continuity type theorems, and fixed point theorems of contractive mappings in this generalized setting are proved as well as some theorems related to topological properties.

Abstract:
We prove that `volume cone implies metric cone' in the setting of RCD spaces, thus generalising to this class of spaces a well known result of Cheeger-Colding valid in Ricci-limit 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 use a Mann-type iteration scheme and the metric projection operator (the nearest-point projection operator) to approximate the solutions of variational inequalities in uniformly convex and uniformly smooth Banach spaces.

Abstract:
We use a Mann-type iteration scheme and the metric projection operator (the nearest-point projection operator) to approximate the solutions of variational inequalities in uniformly convex and uniformly smooth Banach spaces.