Strong Convergence Theorems for Two Total Quasi-φ-Asymptotically Nonexpansive Mappings in Banach Spaces

The purpose of this paper is to establish some strong convergence theorems for a common fixed point of two total quasi-φasymptotically nonexpansivemappings in Banach space bymeans of the hybridmethod inmathematical programming.The results presented in this paper extend and improve on the corresponding ones announced by Martinez-Yanes and Xu (2006), Plubtieng and Ungchittrakool (2007), Qin et al. (2009), and many others.


Introduction
Let  be a smooth Banach space and let  * be the topological dual of .Let the function  :  ×  → R + be defined by  (, ) = ‖‖ 2 − 2⟨, ⟩ +          2 , ∀, ∈ , where  is the normalized duality mapping from  to  * .Let  be a nonempty subset of  and  :  →  a nonlinear mapping.Recall that a mapping  is called nonexpansive if ‖ − ‖ ≤ ‖ − ‖, ∀,  ∈ .A point  ∈  is said to be a fixed point of  [1] if  = .Let () := { ∈  :  = } be the set of fixed points of .A point  ∈  is said to be an asymptotic fixed point of  [1] if  contains a sequence {  } which converges weakly to  such that the strong lim  → ∞ (  −   ) = 0.The set of asymptotic fixed points of  will be denoted by F().
Iterative approximation of fixed points for nonexpansive mappings has been considered by many papers for either the Mann iteration [8] or the Ishkawa iteration [9]; see, for example, [9][10][11][12] and the references therein.
(i) One of the methods is to combine Halpern iteration with Mann iteration; see, for example, C.E. Chidume and C.O. Chidume [10] and Hu [12].
Motivated and inspired by [2,4,10,[12][13][14][15][16][17], the purpose of this paper is to modify Halpern iteration (2) by means of both methods above for two total quasi--asymptotically nonexpansive mappings and then to prove the strong convergence in the framework of Banach spaces.The results presented in this paper extend and improve the corresponding results of Martinez-Yanes and Xu [15], Plubtieng and Ungchittrakool [16], Qin et al. [17], and others.
Throughout the paper, we denote ⇀ for weak convergence and → for strong convergence.

Preliminaries
Let  be a real Banach space endowed with the norm ‖ ⋅ ‖ and let  * be the topological dual of .For all  ∈  and  * ∈  * , the value of  at  is denoted by ⟨, ⟩ and is called the duality pairing.Then, the normalized duality mapping  :  → 2  * is defined by It is well known that the operator  is well defined and  is the identity mapping if and only if  is a Hilbert space.But in general,  is nonlinear and multiple-valued.
The following basic properties for a Banach space  can be found in [18].
(i) If  is uniformly smooth, then  is unformly continuous on each bounded subset of .
(ii) If  is reflexive and strictly convex, then  −1 is normweak-continuous.
(iii) If  is a smooth, strictly convex, and reflexive Banach space, then  is single-valued, one-to-one, and onto.
(iv) A Banach space  is uniformly smooth if and only if  * is uniformly convex.
(vi) Every uniformly smooth Banach space is reflexive.
Moreover, we have the Lyapunov functional  defined by (1).It is obvious from the definition of the function  that we have the following property.
Lemma 1 (see [20]).Let  be a smooth, strictly convex, and reflexive Banach space and let  be a nonempty closed convex subset of .Then the following conclusions hold: If  is a real Hilbert space, then (, ) = ‖ − ‖ 2 and Π  =   (the metric projection of  onto a closed convex subset ).
Lemma 4 (see [4]).Let  be a smooth, strictly convex, and reflexive Banach space with Kadec-Klee property, and let  be a nonempty closed convex subset of .Let  :  →  be a closed and totally quasi--asymptotically nonexpansive mapping with nonnegative real sequences {  }, {]  } and a strictly increasing continuous function  : then the fixed point set () of  is a closed and convex subset of .
The main results of this paper are stated as follows.
Proof.We divide the proof of Theorem 6 into six steps.
(I) We first show that F is a closed and convex subset in .
By Lemma 4, it is trivial to show that () and () are two closed and convex subsets of .Therefore F is closed and convex in .
(II) Next we prove that   is a closed and convex subset in  for all  ≥ 1.
As a matter of fact, by hypothesis,  1 =  is closed and convex.Suppose that   is closed and convex for some  ≥ 1.
From the definition of , we may know that and thus  +1 is closed and convex.Therefore, by induction principle,   are closed and convex for all  ≥ 1.This also shows that Π F  0 is well defined.
Moreover, it follows from Property 1(ii) that we have Combining ( 7) with ( 8), we have This shows that  ∈  +1 .Therefore, by induction principle, we have F ⊂   for all  ≥ 1.
(IV) Next we prove that {  } converges strongly to some point  * ∈ .
Proof.The proof is similar to that of Theorem 6 and hence we omit it.This completes the proof.