Abstract:
This survey covers in our opinion the most important results in the theory of continuous selections of multivalued mappings (approximately) from 2002 through 2012. It extends and continues our previous such survey which appeared in Recent Progress in General Topology, II, which was published in 2002. In comparison, our present survey considers more restricted and specific areas of mathematics. Note that we do not consider the theory of selectors (i.e. continuous choices of elements from subsets of topological spaces) since this topics is covered by another survey in this volume.

Abstract:
The purpose of this paper is to prove end-point theorems for multivalued mappings satisfying comparatively a more general contractive condition in ordered complete metric spaces. Afterwards, we extend the results of previous sections and prove common end-point results for a pair of -weakly isotone increasing multivalued mappings in the underlying spaces. Finally, we present common end point for a pair of -weakly isotone increasing multivalued mappings satisfying weakly contractive condition.

Abstract:
In recent years, the definition of relatively nonexpansive multivalued mapping and the definition of weak relatively nonexpansive multivalued mapping have been presented and studied by many authors. In this paper, we give some results about weak relatively nonexpansive multivalued mappings and give two examples which are weak relatively nonexpansive multivalued mappings but not relatively nonexpansive multivalued mappings in Banach space 2 and [0,1](1<<

Abstract:
Necessary and sufficient conditions for the existence of common stationarypoints of two multivalued mappings and common stationary point theorems formultivalued mappings on bounded metric spaces are given. Our results extendthe theorems due to Fisher in 1979, 1980, and 1983 and Ohta and Nikaido in 1994.

Abstract:
Based on the notion of -accretive mappings and the resolvent operators associated with -accretive mappings due to Lan et al., we study a new class of multivalued nonlinear variational inclusion problems with -accretive mappings in Banach spaces and construct some new iterative algorithms to approximate the solutions of the nonlinear variational inclusion problems involving -accretive mappings. We also prove the existence of solutions and the convergence of the sequences generated by the algorithms in -uniformly smooth Banach spaces.

Abstract:
The purpose of this paper is to ensure the existence of fixed points for multivalued nonexpansive weakly inward nonself-mappings in uniformly convex metric spaces. This extends a result of Lim (1980) in Banach spaces. All results of Dhompongsa et al. (2005) and Chaoha and Phon-on (2006) are also extended.

Abstract:
We give some initial properties of a subset of modular metric spaces and introduce some fixed-point theorems for multivalued mappings under the setting of contraction type. An appropriate example is as well provided. The stability of fixed points in our main theorems is also studied.

Abstract:
This paper deals with the study of parameter dependence of extensions of Lipschitz mappings from the point of view of continuity. We show that if assuming appropriate curvature bounds for the spaces, the multivalued extension operators that assign to every nonexpansive (resp. Lipschitz) mapping all its nonexpansive extensions (resp. Lipschitz extensions with the same Lipschitz constant) are lower semi-continuous and admit continuous selections. Moreover, we prove that Lipschitz mappings can be extended continuously even when imposing the condition that the image of the extension belongs to the closure of the convex hull of the image of the original mapping. When the target space is hyperconvex one can obtain in fact nonexpansivity.

Abstract:
In this paper, we introduce the notion of multivalued contractive mappings in complex valued metric space and prove common fixed point theorems for two multivalued contractive mappings in complex valued metric spaces without using the notion of continuity. Our results improve and extend the results of Azam et al. (2011). 1. Introduction The Banach fixed point theorem was used to establish the existence of a unique solution for a nonlinear integral equation [1]. Moreover, this theorem plays an important role in several branches of mathematics. For instance, it has been used to show the existence of solutions of nonlinear Volterra integral equations and nonlinear integrodifferential equations in the Banach spaces and to show the convergence of algorithms in computational mathematics. Because of its importance and usefulness for mathematical theory, it has become a very popular tool of mathematical analysis in many directions. Nadler [2] introduced the concept of multivalued contraction mappings and obtained the fixed points results for multivalued mappings. Huang and Zhang [3] introduced the notion of cone metric space which is a generalization of metric spaces. They extended Banach contraction principle to cone metric spaces. Since then, Arshad et al. [4], Azam and Arshad [5], Latif and Shaddad [6], Karap？nar [7], and many others obtained fixed point theorems in cone metric spaces (see [8]). The fixed point results regarding rational contractive conditions cannot be extended in cone metric spaces. Azam et al. [9] introduced the concept of complex valued metric spaces and obtained sufficient conditions for the existence of common fixed points of a pair of mappings satisfying contractive type condition involving rational inequalities. In the same way, Rouzkard and Imdad [10] established some common fixed point theorems satisfying certain rational expressions in complex valued metric spaces which generalize, unify, and complement the results of Azam et al. [9]. Recently, Sintunavarat and Kumam [11] obtained common fixed point results by replacing constant of contractive condition with control functions. For more details in the subject, we refer to [12–19]. The aim of this paper is to extend the results of Azam et al. [9] to multivalued mappings in complex valued metric spaces. 2. Preliminaries Let be the set of complex numbers and . Define a partial order on as follows: It follows that if one of the following conditions is satisfied: In particular, we will write if and one of (i), (ii), and (iii) is satisfied and we will write if only (iii) is satisfied.

Abstract:
Berinde and Borcut (2011), introduced the concept of tripled fixed point for single mappings in partially ordered metric spaces. Samet and Vetro (2011) established some coupled fixed point theorems for multivalued nonlinear contraction mappings in partially ordered metric spaces. In this paper, we obtain existence of tripled fixed point of multivalued nonlinear contraction mappings in the framework of partially ordered metric spaces. Also, we give an example.