American Journal of Applied Mathematics and Statistics, 2017, Vol. 5, No. 6, 175-190 Available online at http://pubs.sciepub.com/ajams/5/6/1 ©Science and Education Publishing DOI:10.12691/ajams-5-6-1 Approximation for a Common Fixed Point for Family of Multivalued Nonself Mappings Mollalgn Haile Takele 1,2,* , B. Krishna Reddy 2 1 Department of Mathematics, College of Science, Bahir Dar University, Ethiopia 2 Department of Mathematics, University College of Science, Osmania University, India *Corresponding author: mollalgnh@bdu.edu.et Abstract In this paper, we introduce Mann type iterative method for finite and infinite family of multivalued nonself and non expansive mappings in real uniformly convex Banach spaces. We extend the result to the class of quasi non expansive mappings in real Uniformity convex Banach spaces. We also extend for approximating a common fixed point for the class of multivalued, strictly pseudo contractive and generalized strictly pseudo contractive nonself mappings in real Hilbert spaces. We prove both weak and strong convergence results of the iterative method. Keywords: fixed point, nonself mapping, nonexpansive mapping, strictly pseudo contractive, generalized strictly pseudo contractive mappings, multivalued mapping, Mann type iterative method, uniformly convex Banach space Cite This Article: Mollalgn Haile Takele, and B. Krishna Reddy, “Approximation for a Common Fixed Point for Family of Multivalued Nonself Mappings.” American Journal of Applied Mathematics and Statistics, vol. 5, no. x (2017): 175-190. doi: 10.12691/ajams-5-6-1. 1. Introduction Fixed point theory for multi-valued mappings becomes very interesting for numerous researchers of the field because of its many real world applications in convex optimization, game theory and differential inclusions. Multi-valued mappings are also important in solving critical points in optimal control and other problems (Agarwal et al [2] pp 188). In single valued case, for example in studying the operator equation 0 Au = (when the mapping A is monotone) if K is a subset of a Hilbert space H, then : AK H → is monotone mapping if , 0, , Ax Ay x y xy K − − ≥ ∀ ∈ , Browder [5] introduced a new operator T defined by T I A = − , where I is the identity mapping on the Hilbert space H, the operator is called pseudo contractive operator and the solutions of 0 Au = are the fixed points of the pseudo contractive mapping T and vice versa. Consider a mapping : AK H → and the Variational inequality * * , 0, Ax x x − ≥ x K ∀∈ , in which the problem is to find * x K ∈ satisfying the in equality, this problem is the Variational inequality problem arises in convex optimization, differential inclusions. Let : f K →ℜ be convex, continuously differentiable function. Thus, * * ( ), 0 fx x x ∇ − ≥ x K ∀∈ is Variational inequality for A f =∇ , this inequality is optimality condition for minimization problem min () xK fx ∈ which appears in many areas. An example of a monotone operator in optimization theory is the multi-valued mapping of the sub differential of the functional , f : ( ) 2 H f Df H ∂ ⊆ → and is defined by { } () : , () (), , fx g H x yg fx f y y K ∂ = ∈ − ≤ − ∀∈ (1.1) and 0 () fx ∈∂ satisfies the condition ,0 0 () () . x y fx f y y K − = ≤ − ∀∈ In particular, if : f K →ℜ is convex, continuously differentiable function then A f =∇ , the gradient is a sub differential which is single valued mapping and the condition () 0 fx ∇ = is operator equation and ( ), 0 fx x y ∇ − ≥ is Variational in equality and both conditions are closely related to optimality conditions. Thus, finding fixed point or common fixed point for Multi valued mapping is important in many practical areas. Let K be a non empty subset of a real normed space E , then ( ) CB E denotes the set of non empty, closed and bounded subsets of E . We say K is proximal, if for every , x E ∈ there exists some y K ∈ such that { } inf , . x y x z z K − = − ∀∈ We denote the family of nonempty proximal bounded subsets of K by Prox(K). We observe that, in Hilbert spaces by projection theorem every non empty, closed and convex subset of H is proximal. Also Agarwal et al [2] presented that every nonempty, closed and convex subset of a uniformly convex Banach space is proximal. For , AB in ( ) CB E , we