Iterative Algorithms for a Finite Family of Multivalued Quasi-Nonexpansive Mappings

LetK be a nonempty closed and convex subset of a uniformly convex real Banach space E and let T 1 , . . . , T m : K → 2 K bemmultivalued quasi-nonexpansive mappings. A new iterative algorithm is constructed and the corresponding sequence {x n } is proved to be an approximating fixed point sequence of each T i ; that is, limd (x n ; Tx n ) = 0. Then, convergence theorems are proved under appropriate additional conditions. Our results extend and improve some important recent results (e.g., Abbas et al. (2011)).


Introduction
Let (, ) be a metric space,  a nonempty subset of , and  :  → 2  a multivalued mapping.An element  ∈  is called a fixed point of  if  ∈ .For single valued mapping, this reduces to  = .The fixed point set of  is denoted by () := { ∈  : }.
Interest in the study of fixed point theory for multivalued nonlinear mappings stems, perhaps, mainly from its usefulness in real-world applications such as Game Theory and Nonsmooth Differential Equations.Game Theory.Nash showed the existence of equilibria for noncooperative static games as a direct consequence of multivalued Brouwer or Kakutani fixed point theorem.More precisely, under some regularity conditions, given a game, there always exists a multivalued mapping whose fixed points coincide with the equilibrium points of the game.This, among other things, made Nash a recipient of Nobel Prize in Economic 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.This is part of the problem that is being addressed by iterative methods for fixed point of multivalued mappings.Nonsmooth Differential Equations.A large number of problems from mechanics and electrical engineering lead to differential inclusions and differential equations with discontinuous right-hand sides, for example, a dry friction force of some electronic devices.Below are two models: where  and  0 are fixed in R.These types of differential equations do not have solutions in the classical sense.A generalized notion of solution is what is called a solution in the sense of Fillipov.Consider the following multivalued initial value problem: 2

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Under some conditions, the solutions set of ( 1) and ( 2) coincides with the fixed point set of some multivalued mappings.
On the other hand, Abbas et al. [11] introduced a new one-step iterative process for approximating a common fixed point of two multivalued nonexpansive mappings in a real uniformly convex Banach space and established weak and strong convergence theorems for the proposed process under some basic boundary conditions.Let ,  :  → () be two multivalued nonexpansive mappings.They introduced the following iterative scheme: where and {  }, {  }, and {  } are real sequences in (0, 1) satisfying The following lemma is a consequence of the definition of Hausdorff metric, as remarked by Nadler Jr. [6].
Following the work of Abbas et al. [11], Rashwan and Altwqi [16] introduced a new scheme for approximation of a common fixed point of three multivalued nonexpansive mappings in uniformly convex Banach space.Let , ,  :  → () be three multivalued nonexpansive mappings.They employed the following iterative process: where   ∈   ,   ∈   , and and {  }, {  }, and {  } are real sequences in (0, 1) satisfying Remark 2. Note that if   ,   , and   are known, then the existence of  +1 ,  +1 , and  +1 satisfying ( 10) is guaranteed by Lemma 1.
Before we state the result of Rashwan and Altwqi [16], we need the following definition.
Let  = () ∩ () ∩ () be the set of all common fixed points of the mappings , , and .
Then, they proved the following theorem.
Theorem BS (Bunyawat and Suantai [17]).Let  be a real Banach space and  a closed convex subset of .
It is our purpose in this paper to construct a new iterative algorithm and prove strong convergence theorems for approximating a common fixed point of a finite family of multivalued quasi-nonexpansive mappings in uniformly convex real Banach spaces.The class of mappings used in our theorems is much more larger than that of multivalued nonexpansive mappings.Our theorems generalize and extend those of Abbas et al. [11], Rashwan and Altwqi [16], and Bunyawat and Suantai [17] and many other important results.

Main Results
In this paper, we propose the following iterative algorithm.
Proof.From Theorem 5, we have lim  → ∞ (  ,     ) = 0 for all  = 1, 2, . . ., .Using the fact that { 1 , . . .,   } satisfies condition ( * ), it follows that there exists some  0 such that lim  → ∞ ((  , (  0 ))) = 0. Thus there exist a subsequence {   } of {  } and a sequence {  } ⊂ (  0 ) such that By setting   in place of  and following the same arguments as in the proof of Lemma 4, we obtain from inequality (15 We now show that {  } is a Cauchy sequence in .Notice that This shows that {  } is a Cauchy sequence in  and thus converges strongly to some  ∈ .Using the fact that   is quasi-nonexpansive and   → , we have  14) converges strongly to a common fixed point of {  ,  = 1, . . ., }.
Remark 13.The recursion formula (14) used in our theorems is easier to use than the recursion formula (6) of Abbas et al., the one of Rashwan and Altwqi (9), and the one of Bunyawat and Suantai (12) in the following sense: in our algorithm,    ∈     for  = 1, . . .,  and do not have to satisfy the restrictive conditions: (7) in the recursion formula (6), (10) in the recursion formulas (9), and similar restrictive conditions in the recursion formula (12).