Research Article Iterative Algorithms for a Finite Family of Multivalued Quasi-Nonexpansive Mappings C. Diop, M. Sene, and N. Djitté Department of Mathematics, Gaston Berger University, P.O. Box 234, Saint Louis, Senegal Correspondence should be addressed to N. Djitt´ e; ngalla.djitte@ugb.edu.sn Received 21 July 2014; Revised 13 October 2014; Accepted 16 October 2014; Published 13 November 2014 Academic Editor: Ting-Zhu Huang Copyright © 2014 C. Diop et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let  be a nonempty closed and convex subset of a uniformly convex real Banach space  and let  1 ,...,  :→2  be  multi- valued quasi-nonexpansive mappings. A new iterative algorithm is constructed and the corresponding sequence {  } is proved to be an approximating fxed point sequence of each   ; that is, lim (  ;  )=0. Ten, convergence theorems are proved under appropriate additional conditions. Our results extend and improve some important recent results (e.g., Abbas et al. (2011)). 1. Introduction Let (,) be a metric space,  a nonempty subset of , and :→2  a multivalued mapping. An element ∈ is called a fxed point of  if ∈. For single valued mapping, this reduces to =. Te fxed point set of  is denoted by ():={∈:}. For several years, the study of fxed point theory for mul- tivalued nonlinear mappings has attracted, and continues to attract, the interest of several well known mathematicians (see, e.g., Brouwer [1], Kakutani [2], Nash [3, 4], Geanakoplos [5], Nadler Jr. [6], and Downing and Kirk [7]). Interest in the study of fxed point theory for multivalued nonlinear mappings stems, perhaps, mainly from its useful- ness in real-world applications such as Game Teory and Nonsmooth Diferential Equations. Game Teory. Nash showed the existence of equilibria for noncooperative static games as a direct consequence of mul- tivalued Brouwer or Kakutani fxed point theorem. More pre- cisely, under some regularity conditions, given a game, there always exists a multivalued mapping whose fxed points coin- cide with the equilibrium points of the game. Tis, among other things, made Nash a recipient of Nobel Prize in Eco- nomic Sciences in 1994. However, it has been remarked that the applications of this theory to equilibrium are mostly static: they enhance understanding conditions under which equilibrium may be achieved but do not indicate how to construct a process starting from a nonequilibrium point and convergent to equilibrium solution. Tis is part of the prob- lem that is being addressed by iterative methods for fxed point of multivalued mappings. Nonsmooth Diferential Equations. A large number of prob- lems from mechanics and electrical engineering lead to difer- ential inclusions and diferential equations with discontinu- ous right-hand sides, for example, a dry friction force of some electronic devices. Below are two models:   =(,), a.e.∈:=[−,],(0)= 0 , (1) where  and  0 are fxed in R. Tese types of diferential equations do not have solutions in the classical sense. A gen- eralized notion of solution is what is called a solution in the sense of Fillipov. Consider the following multivalued initial value problem: −  2   2 ∈− 1 4 − 1 4 sign (−1) on Ω=(0,); (0)=0; ()=0. (2) Hindawi Publishing Corporation Advances in Numerical Analysis Volume 2014, Article ID 181049, 6 pages http://dx.doi.org/10.1155/2014/181049