Hybrid Iterations for the Fixed Point Problem and Variational Inequalities

A hybrid iterative algorithm with Meir-Keeler contraction is presented for solving the fixed point problem of the pseudocontractive mappings and the variational inequalities. Strong convergence analysis is given as lim 𝑛→∞ 𝑑(𝑆𝑇𝑥 𝑛 ,𝑇𝑆𝑥 𝑛 ) .


Introduction
Throughout, we assume that H is a real Hilbert space with the inner ⟨⋅, ⋅⟩ and the norm ‖ ⋅ ‖ and C ⊂ H is a nonempty closed convex set.
We use Fix(T) to denote the set of fixed points of T.
Definition 3. A mapping T : C → C is said to be -Lipschitzian if for all , † ∈ C, where > 0 is a constant.
If = 1, T is said to be nonexpansive. One of our purposes of this paper is to find the fixed points of the pseudocontractive mappings in Hilbert spaces. In the literature, there are a large number references associated with the fixed point algorithms for the pseudocontractive mappings. See, for instance, [1][2][3][4][5][6][7][8][9]. The first interesting algorithm for finding the fixed points of the Lipschitz pseudocontractive mappings in Hilbert spaces was presented by Ishikawa [4] in 1974.
Ishikawa proved that the sequence { } generated by (4) converges strongly to a fixed point of T provided C is a compact set.
Zhou's Algorithm. For any 0 ∈ C, define the sequence { } iteratively by where { } and { } are two real sequences in (0, 1) satisfying the following conditions: Zhou proved that the sequence { } generated by (5) converges strongly to proj Fix(T) ( 0 ) without the compactness assumption.
Definition 4. A mapping A : C → H is said to be inverse strongly monotone if there exists > 0 such that for all , V ∈ C.
The variational inequality problem is to find ∈ C such that The set of solutions of the variational inequality problem is denoted by VI(C, A). It is well known that variational inequality theory has emerged as an important tool in studying a wide class of obstacles, unilateral and equilibrium problems, which arise in several branches of pure and applied sciences in a unified and general framework. For related work, please refer to [10][11][12][13][14][15][16][17][18] and the references therein. Motivated and inspired by the related work on the fixed point problem and the variational inequality problem in the literature, the purpose of this paper is continuous to study algorithmic approach to the fixed point problem of the pseudocontractive mappings and the variational inequality problem in Hilbert spaces. We suggest a hybrid algorithm with Meir-Keeler contraction and consequently we prove the strong convergence of the presented algorithm.

Preliminaries
Recall that the metric projection proj C : The metric projection proj C is a typical firmly nonexpansive mapping, that is, for all , † ∈ H. It is well known that, in a real Hilbert space H, the following equality holds: for all , † ∈ H and ∈ [0, 1].
Lemma 5 (see [9]). Let H be a real Hilbert space and let C be a closed convex subset of H. Let T : C → C be a continuous pseudocontractive mapping. Then, Let {C } ⊂ H be a sequence of nonempty closed convex sets. We define the symbols -C and -C as follows.
If C 0 satisfies the following: then we say that {C } converges to C 0 in the sense of Mosco [19] and we write C 0 = -lim → ∞ C . It is easy to show that if {C } is nonincreasing with respect to inclusion, then {C } converges to ⋂ ∞ =1 C in the sense of Mosco. Tsukada [20] proved the following theorem for the metric projection.
Lemma 6 (see [20]). Let H be a Hilbert space. Let {C } be a sequence of nonempty closed convex subsets of H. If C 0 = -lim → ∞ C exists and is nonempty, then, for each ∈ H, {proj C ( )} converges strongly to proj C 0 ( ), where proj C and proj C 0 are the metric projections of H onto C and C 0 , respectively.
Let (E, ) be a complete metric space. A mapping : E → E is called a Meir-Keeler contraction [21] if, for any > 0, there exists > 0 such that for all , † ∈ E. It is well known that the Meir-Keeler contraction is a generalization of the contraction.
Lemma 7 (see [21]). A Meir-Keeler contraction defined on a complete metric space has a unique fixed point.
Lemma 8 (see [22]). Let be a Meir-Keeler contraction on a convex subset C of a Banach space E. Then, for any > 0, there exists ∈ (0, 1) such that for all , † ∈ C.
Lemma 9 (see [22]). Let C be a convex subset of a Banach space E. Let T be a nonexpansive mapping on C and let be a Meir-Keeler contraction on C. Then the following holds.
(i) T is a Meir-Keeler contraction on C.

Main Results
In this section, we firstly introduce a hybrid iterative algorithm for finding the common element of the fixed point problem and the variational inequality problem.
Next, we show the strong convergence of (14).

Remark 12.
Note that Λ is a closed convex subset of C. Thus proj Λ is well defined. Since is a Meir-Keeler contraction of C, it follows that proj Λ is a Meir-Keeler contraction of C by Lemma 9. According to Lemma 7, there exists a unique fixed Proof. The outline of our proof is as follows.
Proof of Step 2. In fact, it is obvious from the assumption that C 0 = C is closed convex. Suppose that C is closed and convex for some ∈ N. For any ∈ C , we know that ‖ − ‖ ≤ ‖ − ‖ is equivalent to So C +1 is closed and convex. By induction, we deduce that C is closed and convex for all ∈ N.

Proof of Step 3. Firstly, from
Step 2, we note that { } is well defined. Since ⋂ ∞ =1 C is closed convex, we also have that proj ⋂ ∞ =1 C is well defined and so proj ⋂ ∞ =1 C is a Meir-Keeler contraction on C. By Lemma 7, there exists a unique Since C is a nonincreasing sequence of nonempty closed convex subsets of H with respect to inclusion, it follows that Setting := proj C (]) and applying Lemma 6, we can conclude that Now, we show that lim → ∞ ‖ − ]‖ = 0. Assume that = lim → ∞ ‖ − ]‖ > 0. Then, for any with 0 < < , we can choose 1 > 0 such that Since is a Meir-Keeler contraction, for the positive , there exists another 2 > 0 such that for all , ∈ C.
In fact, we can choose a common > 0 such that (28) and (29) hold. If 1 > 2 , then If 1 ≤ 2 , then, from (29), it follows that for all , ∈ C. Thus, we have for all , ∈ C. Since → ], there exists 0 ∈ N such that for all ≥ 0 . Now, we consider two possible cases.
Case 2 (‖ − ]‖ > + for all ≥ 0 ). Now, we prove that Case 2 is impossible. Suppose that Case 2 is true. By Lemma 8, there exists ∈ (0, 1) such that for all ≥ 0 . Thus we have for all ≥ 0 . It follows that which gives a contradiction. Hence we obtain Proof of Step 4. By Step 3, we deduce immediately that { } is bounded. Observe that Therefore, we have Since +1 ∈ C +1 , we have This together with (44) implies that From (15) and (24), we have Then we have By (46) and (48), we obtain Since proj C is firmly nonexpansive, we have It follows that and so This together with (46) and (49) implies that Note that It follows that Since → ], we have → ] by (54). So, from (56) and Lemma 5, we deduce that ] ∈ Fix(T).