A New Hybrid Projection Algorithm for System of Equilibrium Problems and Variational Inequality Problems and Two Finite Families of Quasi-φ-Nonexpansive Mappings

and Applied Analysis 3 problems for two countable families of quasi-φ-nonexpansive mappings and the common solution of variational inequality problems for a finite family of monotone mappings in a uniformly smooth and strictly convex real Banach space. Then, we prove a strong convergence theorem of the iterative procedure generated by the conditions. The results obtained in this paper extend and improve several recent results in this area.


Introduction
Throughout this paper, we denote by R and N the set of all real numbers and the set of all positive numbers, respectively.We also assume that  is a real Banach space and  * is the dual space of .Let  be a nonempty, closed and convex subset of a real Banach space  with the dual  * .We recall the following definitions.
Clearly, the class of monotone mappings include the class of -inverse strongly monotone mappings.
Let  :  →  * be a monotone mapping.The variational inequality problem is to find a point  ∈  such that ⟨ − , ⟩ ≥ 0, ∀ ∈ . ( The set of the solution of the variational inequality problem is denoted by VI(, ).
Let  be a nonempty, closed and convex subset of a smooth, strictly convex and reflexive Banach space , let  :  →  be a mapping, and () be the set of fixed points of .
A point  ∈  is said to be a fixed point of  if  = .The set of the solution of the fixed point of  is denoted by () := { ∈  :  = }.
A point  ∈  is said to be an asymptotic fixed point of  if there exists a sequence {  } ⊂  such that   ⇀  and ‖  −   ‖ → 0. We denoted the set of all asymptotic fixed points of  by F().
A point  ∈  is said to be a strong asymptotic fixed point of  if there exists a sequence {  } ⊂  such that   →  and ‖  −   ‖ → 0. We denoted the set of all strong asymptotic fixed points of  by F().
Let  :  ×  → R be a bifunction.The equilibrium problem is to find a point  ∈  such that  (, ) ≥ 0, ∀ ∈ .
The set of the solution of equilibrium problem is denoted by EP().Numerous problems in sciences, mathematics, optimizations, and economics reduced to find a solution of equilibrium problems.The equilibrium problems include variational inequality problems and fixed point problem, and optimization problems as special cases (see, e.g., [1][2][3]).Recently, many authors have considered the problem for finding the common solution of fixed point problems, the common solution of equilibrium problems, and the common solution of variational inequality problems.In 1953, Mann [4] introduced the iterative sequence {  } ∈N which is defined by where the initial element  0 ∈  is arbitrary,  is a nonexpansive mapping, and {  } is the sequence in [0, 1] such that lim  → ∞   = 0 and ∑ ∞ =1   = ∞.The sequence of ( 7) is generally referred to as the Mann iteration.
In 2009, Takahashi and Zembayashi [5] introduced the following iterative scheme by the shrinking projection method, and they proved that {  } ∈N converges strongly to  = Π EP()∩() , under appropriate conditions.Theorem TZ.Let  be a uniformly smooth and uniformly convex real Banach space, and let  be a nonempty, closed and convex subset of .Let  be a bifunction from  ×  to R satisfying (A1)-(A4) and let  be a relatively nonexpansive mapping from  into itself such that () ∩ EP() ̸ = 0. Let {  } be a sequence generated by for every  ∈ N ∪ {0}, where  is the duality mapping on , the sequence Then, the sequence {  } converges strongly to Π ()∩EP()  0 , where Π ()∩EP()  0 is the generalized projection of  onto () ∩ EP().
In 2009, Qin et al. [6] extended the iterative process (8) from a single relatively nonexpansive mapping to two relatively quasi-nonexpansive mappings.In 2011, Zegeye and Shahzad [7] introduced an iterative process for finding an element in the common fixed point set of finite family of closed relatively quasi-nonexpansive mappings, common solutions of finite family of equilibrium problems, and common solutions of the finite family of variational inequality problems for monotone mappings in Banach spaces.
In this paper, motivated and inspired by the previously mentioned above results, we introduce a new iterative procedure for solving the common solution of system of equilibrium problems for a finite family of bifunctions satisfying certain conditions and the common solution of fixed point problems for two countable families of quasi--nonexpansive mappings and the common solution of variational inequality problems for a finite family of monotone mappings in a uniformly smooth and strictly convex real Banach space.Then, we prove a strong convergence theorem of the iterative procedure generated by the conditions.The results obtained in this paper extend and improve several recent results in this area.

Preliminaries
The space  is said to be smooth if   () > 0, ∀ > 0 and is said to be uniformly smooth if and only if lim  → 0 +   ()/ = 0.
The modulus of convexity of  is the function A Banach space  is said to be uniformly convex if and only if   () > 0 for all  ∈ (0, 2].
We recall the following definitions.
Definition 1.Let  be a nonempty set.
Remark 2. We here the following basic properties.
(2) The class of quasi--nonexpansive mappings contains properly the class of weak relatively nonexpansive mappings as a subclass, but the converse may be not true.
(3) The class of weak relatively nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true.
(4) The class of quasi--nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true.
(5) If  is a real uniformly smooth Banach space, then  is uniformly continuous on each bounded subset of .
(6) If  is a strictly convex reflexive Banach space, then Let  be a real Banach space and {  } be a sequence in .We denote by   →  and   ⇀  the strong convergence and weak convergence of {  }, respectively.The normalized duality mapping  from  to 2  * is defined by where ⟨⋅, ⋅⟩ denotes the duality pairing.It is well known that if  is smooth, then  is single-valued and demicontinuous, and if  is uniformly smooth, then  is uniformly continuous on bounded subset of .Moreover, if  is reflexive and strictly convex Banach space with a strictly convex dual, then  −1 is single-valued, one-to-one, surjective, and it is the duality mapping from  * to  and so  −1 =   * and  −1  =   (see [11,12]).We note that in a Hilbert space , the mapping  is the identity operator.Now, let  be a smooth and strictly convex reflexive Banach space.As Alber (see [13]) and Kamimura and Takahashi (see [14]) did, the Lyapunov functional  :  ×  → R + is defined by It follows from Kohsaka and Takahashi (see [15]) that (, ) = 0 if and only if  =  and that Further suppose that  is nonempty, closed and convex subset of .The generalized, projection (Alber see [13]) Π  :  →  is defined by for each  ∈ , Remark 3. If  is a real Hilbert space , then (, ) = ‖ − ‖ 2 and Π  =   (the metric projection of  onto ).
Lemma 4 (Alber [13]).Let  be a nonempty, closed and convex subset of a smooth and strictly convex reflexive Banach space , and let  ∈ .Then Lemma 5 (Kamimura and Takahashi [14]).Let  be a nonempty, closed and convex subset of a smooth and strictly convex reflexive Banach space , and let  ∈  and p ∈ . Then, Lemma 6 (Qin et al. [6] and Kohsaka and Takahashi [16]).
Lemma 7 (Kamimura and Takahashi [14]).Let  be a uniformly convex and smooth real Banach space and let {  } and {  } be two sequences of .If (  ,   ) → 0 and either Lemma 8. Let  be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property, let {  } and {  } be two sequences of , and  ∈ .If   →  and (  ,   ) → 0, then   → .

Main Results
In this section, we prove a strong convergence theorem which solves the problem for finding a common solution of the system of equilibrium problems and variational inequality problems and fixed point problems in Banach spaces.
Proof.We will complete this proof by seven steps below.
Step 1.We will show that   is closed and convex for each  ≥ 0.
From the definition of   , it is obvious that   is closed.Moreover, since It follows that   is convex for each  ≥ 0. Therefore,   is closed and convex for each  ≥ 0.
Step 2. We will show that Ω ⊂   for each  ≥ 0.
Hence, the sequence {  } is well defined.
Step 3. We will show that the sequence{  } is bounded.
Let  := ⋂ ∞ =0   .From Ω ⊂   for each  ≥ 0 and {  } is well defined.From the assumption of , we see that  is closed and convex subset of .
Step 4. We will show that there exists p ∈  such that   → p, as  → ∞.
Without a loss of generalization, we can assume that   ⇀  0 ∈ .Since {  } is bounded and  is reflexive.Since  := ⋂ ∞ =0   is closed and convex, it follows that  0 ∈ , ∀ ≥ 0.Moreover, by using the weak lower semicontinuous of the norm on  and (35), we obtain (39) By using Lemma 5, we have ⟨ p −  0 ,  p −  0 ⟩ = 0 and hence, p =  0 .By the definition of , we get lim inf (60) Step 6.We will show that p ∈ Ω.
Substep 1.We will show that p ∈ ⋂  =1 VI(,   ).From the definition of    of algorithm (32), we have Let {  } ∈N ⊂ N be such that    =  1 , for all  ∈ N. Then from (61), we obtain and that is Now, we set   =  + (1 − ) p, for all  ∈ (0, 1] and  ∈ .Therefore, we get   ∈ .From (63), it follows that From the continuity of  and ( 41) and (54), we have    → p,    → p, as  → ∞, we obtain Since  1 is a monotone mapping, we also have Thus, it follows that Abstract and Applied Analysis and hence If  → 0, we obtain This implies that p ∈ VI(,  1 ).
Similarly, let {  } ∈N ⊂ N be such that    =  2 , for all  ∈ N.
Similarly, let {  } ∈N ⊂ N be such that    =  2 , for all  ∈ N.Then, we have again that p ∈ EP( 2 ).
From algorithm (32) and Lemma Moreover, the demicontinuity of  −1 implies that     ⇀ p as  → ∞.Thus, the Kadec-Klee property of , we obtain Let {  } ∈N ⊂ N be such that    =  1 , for all  ∈ N.
Similarly, let {  } ∈N ⊂ N be such that    =  2 , for all  ∈ N.Then, we have again that p ∈ ( 2 ).
Similarly, let {  } ∈N ⊂ N be such that    =  2 , for all  ∈ N.Then, we have again that p ∈ ( 2 ).
From   = Π   ( 0 ), we have Taking  → ∞ in (91), one has Now, we have p ∈ Ω and by Lemma 5, we get This completes the proof of Theorem 13.
If we set  =  =  = 1 in Theorem 13, then we obtain the following result.

Application to Relatively Nonexpansive Mappings.
If we change the condition (2) in Theorem 13 as follows: {  }  =1 and {  }  =1 are finite families of relatively nonexpansive mappings.From Remark 2(3) and (2) every relatively nonexpansive mappings is weak relatively nonexpansive mappings and every weak relatively nonexpansive mappings is quasi-nonexpansive mappings.Then, we obtain the following result.

Application to Hilbert Spaces.
If  = , a real Hilbert space, then  is uniformly smooth and strictly convex real Banach space.In this case,  =  and Π  =   .Then, we obtain the following result.
Then, the sequence {  } ∞ =0 converges strongly to an element of Ω.
Remark 22.Our theorem extends and improves the corresponding results in [5][6][7] in the following aspect.
(a) For the mapping, we extend the mappings from nonexpansive mappings, relatively nonexpansive mappings, and weak relatively nonexpansive mappings to more general than quasi--nonexpansive mappings.
(b) For the common solution, we extend the common solution of a single finite family of quasi--nonexpansive mappings to two finite families of quasi--nonexpansive mappings.