Approximation of Common Solutions of Fixed Point Problem for Hemicontractive-type Mapping , Split Equilibrium and Variational Inequality Problems

In this paper, we introduce and study an iterative approximation method for finding a common element of the set of fixed points of multi-valued hemicontractive-type mapping, the set of solutions of split equilibrium problem and the set of solutions of variational inequality problem for continuous monotone mapping in real Hilbert spaces. Furthermore, we proved strong convergence theorem for the sequences generated by the proposed iterative algorithm under some conditions imposed on parameters. Our results improve and generalize most of the results that have been previously proved by some authors recently.


Introduction
Let H be a real Hilbert space with inner product ., .and induced norm . .Let C be a nonempty subset of H.A mapping S : C −→ H is called Lipschitzian if there exists L ≥ 0 such that Sx − Sy ≤ L x − y , for all x, y ∈ C. (1.1) If L = 1 in (1.1), then S is called nonexpansive mapping and if 0 ≤ L < 1, then S is called a contraction.
A mapping S : C −→ H is said to be k−strictly pseudocontractive if there exists k ∈ [0, 1) such that Sx − Sy 2 ≤ x − y 2 + k x − Sx − (y − Sy) 2 , for all x, y ∈ C.
And S is called pseudocontractive mapping if 2 , for all x, y ∈ C.
Observe that the class of pseudocontractive mappings properly contains the classes of k−strictly pseudocontractive mappings and nonexpansive mappings (see [6,15]).
A mapping S : C −→ H with nonempty set of fixed points, F(S) = {x ∈ C : Sx = x} = ∅, is said to be quasi-nonexpansive if Sx − p ≤ x − p holds for all p ∈ F(S) and x ∈ C. The mapping S is called demicontractive if there exists a constant k ∈ [0, 1) such that Sx − p 2 ≤ x − p 2 + k x − Sx 2 , for all p ∈ F(S) and x ∈ C.And S is called hemicontractive if Sx − p 2 ≤ x − p 2 + x − Sx 2 holds for all p ∈ F(S) and x ∈ C.
We note that the class of hemicontractive mappings properly contains the class of pseudocontractive mappings S with F(S) = ∅ and the classes of quasi-nonexpansive and demicontractive mappings (see, for example, [16,27]).If L = 1, then S is called nonexpansive; ii) k−strictly pseudocontractive if there exists k ∈ [0, 1) such that for all x, y ∈ C, u ∈ Sx and v ∈ Sy; iii) pseudocontractive if for all x, y ∈ C, u ∈ Sx and v ∈ Sy.
We observe from the definitions that every multi-valued nonexpansive mapping is k−strictly pseudocontractive mapping and every multi-valued k−strictly pseudocontractive mapping is pseudocontractive mapping, however, the inclusions are strict (see [6,27]).
An element x ∈ C is called a fixed point of a multi-valued mapping S : C −→ CB(C) if x ∈ Sx.We denote the set of fixed points of a mapping S by F(S).We write x n x to indicate that the sequence {x n } converges weakly to x and x n → x to indicate that the sequence {x n } converges strongly to x.
Given a multi-valued mapping S : ) is a multi-valued nonexpansive mapping, then (I − S) is demiclosed at zero, where C is a closed and convex subset of a Hilbert space H.For more details as regards the demiclosedness principle for nonexpansive mappings (see [1]).
Definition 1.2 Let S : C −→ CB(C) be a multi-valued mapping.Then, S is said to be 1.quasi-nonexpansive if F(S) = ∅ and for all p ∈ F(S), x ∈ C, we have 2. demicontractive-type if F(S) = ∅, there exists a constant k ∈ [0, 1) and for all p ∈ F(S), x ∈ C, we have 3. hemicontractive-type if F(S) = ∅ and for all p ∈ F(S), x ∈ C, We observe that every nonexpansive mapping with nonempty set of fixed points is quasi-nonexpansive mapping, every k−strictly pseudocontractive mapping S with F(S) = ∅ and S(p) = {p}, ∀p ∈ F(S) is demicontractivetype mapping, and every pseudocontractive mapping S with F(S) = ∅ and S(p) = {p}, ∀p ∈ F(S) is hemicontractivetype mapping.It is also easy to see that every quasi-nonexpansive mapping is demicontractive-type mapping, and every demicontractive-type mapping is hemicontractive-type mapping.However, the inclusions are proper (see, for example, [15,27]).
Many authors have shown their interest in studying the existence and approximation of fixed points of nonlinear mappings (including hemicontractive-type mapping) (see, for example, [6,12,20,21,27] and the references therein).
Recall that a mapping A : C −→ H is called α-inverse strongly monotone if there exists a positive real number α Clearly, the class of monotone mappings includes the class of α-inverse strongly monotone mappings but the inclusion is proper (see [15]).
Note that every α-inverse strongly monotone mapping A is 1 α −Lipschitz mapping and for each λ ∈ (0, 2α], I − λA is a nonexpansive mapping from C into H (see, for example, [15,25]).It is known that if S : C −→ H is nonexpansive, then A := I − S is 1  2 −inverse strongly monotone mapping (for more details, see [23]).Let C be a nonempty, closed and convex subset of a real Hilbert space H and let A : C −→ H be a nonlinear mapping.The classical variational inequality problem, which was developed as a tool for solving partial differential equations by Stampacchia [22], is the problem of finding u ∈ C such that v − u, Au ≥ 0, for all v ∈ C. (1. 2) The set of solutions of the variational inequality problem (1.2) is denoted by V I(C, A).Variational inequality problem encompasses many mathematical problems, among others, including nonlinear equations, complementarity problems and fixed point problems.Variational inequality problems have been extensively studied by several authors (see, for example, [2,10,14,26,31,32] and the references cited therein).
Let C be a nonempty, closed and convex subset of a real Hilbert space H and let F : C × C −→ R be a bifunction, where R is the set of real numbers.The equilibrium problem, which was initially introduced by Blum and Oettli [3] in 1994, is the problem of finding a point x ∈ C such that F(x, y) ≥ 0, for all y ∈ C. ( The set of solutions of problem (1.3) is denoted by EP(F).Various problems arising in physics, optimization, economics, engineering and transportation can be reduced to finding solutions of equilibrium problem.Many authors have considered an iterative algorithm for approximating solutions of equilibrium problems (1.3) (see, for example, [7,15,16,24,26] and the references cited therein).The equilibrium problem (1.3) includes variational inequality problems, optimization problems, Nash equilibrium problems and fixed point problems as special cases.
As a generalization of the equilibrium problem (1.3),He [9] introduced the following class of split equilibrium problems that enable us to find a solution of one equilibrium problem such that its image under a given bounded linear operator is a solution of another equilibrium problem in different subsets of spaces.
Throughout the rest of this paper unless otherwise stated, let H 1 and H 2 be real Hilbert spaces and let C and Q be nonempty, closed and convex subsets of H 1 and H 2 , respectively. Let and such that In this work we denote the set of solutions of split equilibrium problem (1.4) and (1.5) by Ω.That is, and Bp ∈ EP(F 2 )}.

Given two nonlinear mappings
which was studied by Censor et al. [4].
fixed point problem (SFPP), which is formulated as finding a point x * ∈ C with the property: which was studied by Censor and Segal [5], and Eslamian [8].
On the other hand, if we let H 1 = H 2 , B = I, Q = C and F 2 ≡ 0, then the split equilibrium problem reduces to the equilibrium problem (1.3).Hence, split equilibrium problem includes split variational inequality, split fixed point problems and the classical equilibrium problems as a special case.Thus, the split equilibrium problem is quite general.
For finding a solution of split equilibrium problems, He [9] also considered the following iterative algorithm: where B * is the adjoint of the bounded linear operator B. Then, under mild conditions on parameters {r n } and µ, the author proved that the sequences {x n } and {u n } generated by (1.6) converge weakly to a point p ∈ Ω.
Motivated by the works of He [9], Kazmi and Rizvi [11] studied the problem of finding a common point of the set of fixed points of nonexpansive single-valued self-mapping S : C −→ C, the sets of solutions of split equilibrium and variational inequality problems by considering the following iterative algorithm: where A : C −→ H 1 is an α−inverse strongly monotone mapping, F 2 is upper semi-continuous in the first argument and the sequences {α n }, {β n }, {γ n }, {λ n } and {r n } satisfy some fitting conditions.Then, they proved that the sequence {x n } generated by (1.7) converges strongly to p = P Θ v, where Θ = F(S) ∩ Ω ∩ V I(C, A).
We remark that the result of Kazmi and Rizvi [11] improves and extends the result of He [9], however, it is restricted to single-valued nonexpansive and α−inverse monotone mappings; and the assumption F 2 is upper semi-continuous in the first argument is strong.However, recently, Meche et al. [17] observed that the condition F 2 is upper semi-continuous assumed in [11] can be dispensed with, and they investigated the following iterative algorithm for approximating a common solution of split equilibrium problem, variational inequality problem for Lipschitz monotone mapping A : C −→ H 1 and fixed point problem for nonexpansive multi-valued mapping S: where B : H 1 −→ H 2 is a bounded linear operator and B * is the adjoint of B, v n ∈ Sy n , f : C −→ C is a contraction mapping , and {γ n } ⊂ (0, 1 L ), {b n }, {a n } ⊂ (0, 1) satisfying some appropriate restrictions.They proved that the sequence {x n } generated by (1.8) converges strongly to a point p ∈ Θ = F(S) ∩ Ω ∩ V I(C, A), where p = P Θ f (p).

Motivated and inspired by the above results, we have raised the following natural question:
Question: Is it possible to introduce an iterative algorithm which converges strongly to a common element of the set of fixed points of class of multi-valued mappings more general than nonexpansive mappings, the set of solutions of variational inequality problem and the set of solutions of split equilibrium problem?
It is our purpose in this paper to construct an iterative algorithm and prove that the algorithm converges strongly to a common element of fixed point set of a Lipschitz hemicontractive-type multi-valued mapping, solution sets of split equilibrium and variational inequality problems in the framework of real Hilbert spaces.The results presented in this paper extend and improve the corresponding results announced by Censor et al. [4], Eslamian [8], He [9], Kazmi and Rizvi [11], Meche et al. [14,17] and some other results in this research field.

Preliminaries
In this section, we collect some concepts and results that play a crucial role in the sequel.
Recall that for every point x in a real Hilbert space H, there exists a unique nearest point in C, denoted by P C x, such that The mapping P C : H −→ 2 C is called the metric projection of H onto C and it is characterized by In what follows, we shall make use of the following assumption.Let F : C × C → R be a bifunction satisfying the following conditions: (A2) F is monotone, i.e., F(x, y) + F(y, x) ≤ 0, ∀ x, y ∈ C; (A3) lim t↓0 F(tz + (1 − t)x, y) ≤ F(x, y), ∀ x, y, z ∈ C; (A4) for each x ∈ C, y → F(x, y) is convex and lower semicontinuous.
In the proof of our main result, we make use the following lemmas.
Lemma 2.1 [28] Let {b n } be a sequence of nonnegative real numbers such that where {α n } ⊂ (0, 1) and δ n ⊂ R satisfying the following restrictions: Then, Lemma 2.2 [30] Let H be a real Hilbert space.Then, for all x i ∈ H and 1 the following equality holds: Lemma 2.3 Let H be a real Hilbert space.Then, for every x, y ∈ H, we have the following: Lemma 2.4 [18] Let H be a Hilbert space.Let A, B ∈ CB(H) and a ∈ A. Then, for ε > 0, there exists a point b ∈ B such that ||a − b|| ≤ D(A, B) + ε.In particular, for every a ∈ A there exists an element b ∈ B such that ||a − b|| ≤ 2D(A, B).
Lemma 2.5 [3,7] Let F 1 be a bifunction from C × C into R satisfying Assumption 2. For s > 0 and for all x ∈ H 1 , define a mapping T F 1 s : H 1 −→ C as follows: Then, we have the following: 1. T F 1 s is nonempty and single valued; Furthermore, assume that F 2 : Q × Q −→ R satisfies Assumption 2. For r > 0 and for all u ∈ H 2 , define a mapping T F 2 r : H 2 −→ Q as follows: Then, the following hold: i. T F 2 r is nonempty and single valued; ii. iii.
Lemma 2.6 [29] Let A be a continuous monotone mapping from C into H 1 .Then, for any t > 0 and x ∈ H 1 , there exists Moreover, the mapping J t : satisfies the following: 1. J t is single-valued; 2. J t is firmly nonexpansive, i.e., J t x − J t y 2 ≤ J t x − J t y, x − y , ∀ x, y ∈ H 1 ; 3. F(J t ) = V I(C, A);

V I(C, A) is closed and convex.
Lemma 2.7 [13] Let {a n } be a sequence of real numbers such that there exist a subsequence {n i } of {n} such that a n i < a n i +1 , for all i ∈ N.Then, there exists a nondecreasing sequence {m k } ⊂ N such that m k → ∞ and the following properties are satisfied by all (sufficiently large) numbers k ∈ N: In fact, m k = max{j ≤ k : a j ≤ a j+1 }.

Main Results
In this section, we prove strong convergence theorems for a split equilibrium problem, variational inequality problem and a fixed point problem for a Lipschitz hemicontractive-type multi-valued mapping.
First, we show that B * (I − T F 2 r )B is a 1 2d −inverse strongly monotone mapping.Since T F 2 r is nonexpansive, we have I − T F 2 r is 1 2 −inverse strongly monotone mapping.Thus, for all x, y ∈ H 1 , from Cauchy Schwartz inequality, we get that r )B is nonexpansive.Hence, from the nonexpansiveness of T F 1 s , we obtain that Now, Let p ∈ Θ.Then, we have Sp = p, J t p = p and p ∈ Ω, hence p = T F 1 s p and Bp = T F 2 r Bp which implies that T F 1 s (I − λB * (I − T F 2 r )B)p = p.Thus, using (3.2), we get that Since J t is nonexpansive, from (3.3) we have that Since S is hemicontractive-type mapping and w n ∈ Su n , we get that On the other hand, Since S is hemicontractive-type mapping and v n ∈ Sy n , from (3.1), (3.4) and Lemma 2.2, we obtain that Substituting (3.6) into (3.5),we obtain that From the assumption that v n − w n ≤ 2D(Ty n , Tu n ) and S is L−Lipschitzian mapping, and Lemma 2.2, we find that Thus, by substituting (3.8) into (3.7),we obtain that (3.9) Hence, from Lemma 2.2, (3.4), (3.9) and condition (i), we have that and from condition (ii), we see that for all n ≥ 0. Thus, combining (3.10) and (3.11), we obtain that Then, by induction, we have that Therefore, the sequence {x n } is bounded.This completes the proof.
Theorem 3.2 Let H 1 and H 2 be real Hilbert spaces.Let C and Q be nonempty, closed and convex subsets of H 1 and H 2 , respectively.Let A : C −→ H 1 be a continuous monotone mapping and B : H 1 −→ H 2 be a bounded linear operator.Let ) for all n ≥ 0, where v n ∈ Sy n and w n ∈ Su n are such that v n − w n ≤ 2D(Sy n , Su n ), s, r, t > 0, λ ∈ (0, 1 d ), d = B * B , where B * is the adjoint of B, {a n }, {α n } ⊂ (0, 1), and {b n },{c n } ⊂ [a, b] for some a, b ∈ (0, 1) satisfying the following control conditions: Then, the sequence {x n } converges strongly to q ∈ Θ, where q = P Θ (u).
First, we note that P Θ is well defined because Θ is nonempty, closed and convex subset of C and from Theorem 3.1 it follows that the sequence {x n } is bounded and so are the sequences {y n }, {z n } and {u n }.Now, let p ∈ Θ.
Then, since T F 1 s is nonexpansive, we have that Hence, the 1 2d −inverse strong monotonicity of B * (I − T F 2 r )B and the fact that Bp = T F 2 r Bp give that On the other hand, from (3.12), (3.4), Lemma 2.2 and Lemma 2.3 (ii), we get Therefore, using (3.9), (3.13) and (3.14), we get the following: Then, we complete the proof by the next two cases.
Case 1: Suppose that there exists a positive integer n 0 such that { x n − p } is decreasing for all n ≥ n 0 .Then, the sequence { x n − p } is convergent and from (3.11) and (3.15), we have that Hence, assumption of {c n }, convergence of {||x n − p||} and the fact that a n → 0 as n → ∞ imply that and hence But this Thus, substituting (3.9) and (3.18) into (3.14),we obtain that  On the other hand, since J t is firmly nonexpansive, from Lemma 2.3 (i), we get that This together with (3.3) give that Then, by substituting (3.9) and (3.21) into (3.14),we obtain that Hence, the assumption that α n → 0 as n → ∞ and (3.In addition, from (3.11) and (3.22), we also have And from the Lipschitz condition of S, (3.12) and (3.23), we get that Therefore, from (3.20), (3.25), (3.26), definition of {x n+1 } and the assumption that α n → 0 as n → ∞, we find that Moreover, from (3.11) and (3.22), we infer that Now, let q = P Θ (u).Then, we show that lim sup n→∞ u − q, x n+1 − q ≤ 0.
Since the sequence {x n+1 } is bounded in a real Hilbert space H 1 , we can choose a subsequence {x Since C is closed and convex, C is weakly closed.So, we have w ∈ C and from (3.27), we find that x n i w as i → ∞ and thus it follows from (3.20) and (3.25) that z n i w and y n i w as i → ∞.
Next, we claim that w ∈ Θ.From (3.24) and the hypothesis that (I − S) is demiclosed at zero, we obtain that w ∈ F(S).
Since (I − λB * (I − T F 2 r )B) is nonexpansive, from (3.17This with the fact that Bw = T F 2 r Bw gives that w = T F 1 s w, and hence w ∈ EP(F 1 ).Therefore, w ∈ Ω.
On the other hand, from (3.1) and (3.25), we have lim n→∞ z n i − J t z n i = lim n→∞ z n i − y n i = 0.
Since z n i w and J t is nonexpansive, then (I − J t ) is demiclosed at zero and so, we get that w = J t w and hence w ∈ V I(C, A).
Therefore, w ∈ Θ.Thus, since q = P Θ (u) and x n i +1 w, from the property of metric projection P C given in (2.2), we have lim sup n→∞ u − q, x n+1 − q = lim i→∞ u − q, x n i +1 − q = u − q, w − q ≤ 0.
Case 2. Suppose that there exists a subsequence {n j } of {n} such that x n j − p < x n j +1 − p , for all j ∈ N.Then, by Lemma 2.7, there exists a nondecreasing sequence {m k } ⊂ N such that m k → ∞, and Then, since q = P Θ (u), following the processes in Case 1, we get that lim sup k→∞ u − q, x m k +1 − q ≤ 0. (3.31) Now, since q ∈ Θ, from (3.22), we have that and hence (3.30) and (3.32) imply that Hence, in view of the fact that a m k > 0, we have that x m k − q 2 ≤ 2 u − q, x m k +1 − q .
Thus, using (3.31) we obtain that x m k − q → 0 as k → ∞.This together with (3.32) implies that x m k +1 − q → 0 as k → ∞.Because x k − q ≤ x m k +1 − q for all k ∈ N, we get that x k → q.Therefore, from the above two cases, we deduce that the sequence {x n } converges strongly to q = P Θ (u).This completes the proof.
If, in Theorem 3.2, we assume that S is a single-valued Lipschitz hemicontractive mapping, then we obtain the following result:

LetDefinition 1 . 1
CB(C) denote the family of nonempty, closed and bounded subsets of C, and K(C) denote the family of nonempty and compact subsets of C. The Hausdorff metric on CB(C) is defined by D(A, B) = max sup x∈A d(x, B), sup y∈B d(y, A) , for all A, B ∈ CB(C), where d(x, B) = inf{ x − b : b ∈ B}.Let S : C −→ CB(C) be a multi-valued mapping.Then, S is said to be i) Lipschitzian if there exists a constant L ≥ 0 such that D(Sx, Sy) ≤ L x − y , for all x, y ∈ C.

F 1 :
C × C −→ R and F 2 : Q × Q −→ R be bifunctions satisfying Assumption 2. Let S : C −→ CB(C) be a L−Lipschitz hemicontractive-type multi-valued mapping.Assume that Θ = F(S) ∩ Ω ∩ V I(C, A) is nonempty and Sp = {p} for all p ∈ Θ.Let x 0 , u ∈ C be arbitrary and let {x n } be a sequence in C generated by be bifunctions satisfying Assumption 2. Let S : C −→ CB(C) be a L−Lipschitz hemicontractive-type multi-valued mapping.Assume that Θ = F(S) ∩ Ω ∩ V I(C, A) is nonempty, closed and convex, Sp = {p} for all p ∈ Θ and (I − S) is demiclosed at zero.Let x 0 , u ∈ C be arbitrary and let {x n } be a sequence in C generated by

Hence, since {x
n } and {z n } are bounded and a n → 0 as n → ∞, from (3.16) and assumption of {c n }, we obtain that lim n→∞ z n − x n = 0. (3.20)
be bifunctions and let B : H 1 −→ H 2 be a bounded linear operator, then the split equilibrium problem (SEP) is the problem of finding a point x * ∈ C such that Theorem 3.1 Let H 1 and H 2 be real Hilbert spaces.Let C and Q be nonempty, closed and convex subsets of H 1 and H 2 , respectively.Let A : C −→ H 1 be a continuous monotone mapping and B : H 1 −→ H 2 be a bounded linear operator.Let .17) And Since T F 1 s is firmly nonexpansive and (I − λB * (I − T F 2 r )B) is nonexpansive, from (3.1) and Lemma 2.3 (i), we have that