VISCOSITY APPROXIMATION METHODS OF NONEXPANSIVE MAPPINGS IN HILBERT SPACES AND APPLICATIONS

In this paper, we introduce a new viscosity approximation method of nonexpan- sive mappings in Hilbert spaces. The strong convergence theorems of the rules are proved under certain assumptions imposed on the sequence of parameters. Moreover, we give appli- cations of the purposed viscosity method.


Introduction
In this paper, we shall take H as a real Hilbert space, •, • as the inner product, • as the induced norm, and C as a nonempty closed subset of H.In order to verify the weak convergence of an algorithm to a fixed point of a nonexpansive mapping we need the demiclosedness principle: Theorem 1.4.[2] (The demiclosedness principle) Let C be a nonempty closed convex subset of a real Hilbert space H and T : C → C such that x n ⇀ x * ∈ C and (I − T )x n → 0. Then x * = T x * .Here → and ⇀ denote strong and weak convergence, respectively.
Moreover, the following result gives the conditions for the convergence of a nonnegative real sequence.
The following strong convergence theorem, which is also called the viscosity approximation method for nonexpansive mappings in real Hilbert spaces is given by Moudafi [6] in 2000.
Theorem 1.6.Let C be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonexpansive mapping of C into itself such that F (T ) := {x ∈ H : T (x) = x} is nonempty.Let f be a contraction of C into itself.Consider the sequence where the sequence {ǫ n } ∈ (0, 1) satisfies: (1) Then {x n } converges strongly to a fixed point x * of the mapping T , which is also the unique solution of the variational inequality In 2015, Xu et al. [9] applied the viscosity method on the midpoint rule for nonexpansive mappings and they give the generalized viscosity implicit rule: This, using contraction, regularizes the implicit midpoint rule for nonexpansive mappings.They also proved that the sequence generated by the generalized viscosity implicit rule converges strongly to a fixed point of T .Ke and Ma [5], motivated and inspired by the idea of Xu et al. [9], proposed two generalized viscosity implicit rules: Our contribution in this direction is the following new viscosity method: (1.1)

The Main Result
Theorem 2.1.Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be a nonexpansive mapping with F (T ) = ∅ and f : C → C be a contraction with coefficient θ ∈ [0, 1).Pick any x 0 ∈ C, let {x n } be a sequence generated by the viscosity method (1.1), where {α n }, {β n } and {γ n } are sequences in (0, 1) satisfying the following conditions: (i) Then {x n } converges strongly to a fixed point x * of the mapping T, which is also the unique solution of the variational inequality In other words, x * is the unique fixed point of the contraction Proof.The proof is divided into four steps: Step 1. ({x n } is bounded) Take p ∈ F (T ) arbitrarily, we have (2.1) Since γ n ∈ (0, 1), 1 − γn 2 ≥ 0.Moreover, by equation (2.1) and α n + β n + γ n = 1, we get Thus, we have Now by induction, we obtain Hence, we concluded that where M > 0 is a constant such that It gives that is, Note that for γ n ∈ (0, 1), 1 − γn 2 ≤ 1 and βn(1−θ) Step 3.
Indeed, we take a subsequence {x n i } of {x n } which converges weakly to a fixed point p of T .Without loss of generality, we may assume that {x n i } ⇀ p. From lim n→∞ x n − T x n = 0 and Theorem 1.4, we have p = T p.This together with the property of the metric projection implies that lim sup Step 4. (x n → x * as n → ∞) Take x * ∈ F (T ) the unique fixed point of the contraction P F (T ) f. Consider where It becomes Solving this quadratic inequality for x n+1 +xn 2 − x * , we get which reduces to . Let .

A More General System of Variational Inequalities
Let C be a nonempty closed convex subset of a real Hilbert space H and {A i } N i=1 : C → H be a family of mappings.In [1], Cai and Bu considered the problem of finding The equation (3.1) can be written as which is a more general system of variational inequalities in Hilbert spaces, where λ i > 0 for all i ∈ {1, 2, 3, ..., N }.We also have following lemmas.
Lemma 3.1.[1] Let C be a nonempty closed convex subject of a real Hilbert space H.For i ∈ {1, 2, 3, ..., N }, let A i : C → H be δ i -inverse-strongly monotone for some positive real number δ i , namely, Let C be a nonempty closed convex subject of a real Hilbert spaces H. Let A i : C → H be a nonlinear mapping, where i ∈ {1, 2, 3, ..., N }.For given ) is a solution of the problem (3.1) if and only if that is, From Lemma 3.2, we know that x * 1 = G(x * 1 ), that is, x * 1 is a fixed point of the mapping G, where G is defined by (3.2).Moreover, if we find the fixed point x * 1 , it is easy to get the other points by (3.3).Applying Theorem 2.1 we get the result Theorem 3.3.Let C be a nonempty closed convex subject of a real Hilbert space H.For i ∈ {1, 2, 3, ..., N }, let A i : C → H be δ i -inverse-strongly monotone for some positive real number δ i with F (G) = ∅, where G : C → C is defined by Let f : C → C be a contraction with coefficient θ ∈ [0, 1).Pick any x 0 ∈ C, let {x n } be a sequence generated by where {α n }, {β n } and {γ n } are sequences in (0, 1) satisfying the conditions (i)-(iv) and (vi) lim n→∞ x n − G(x n ) = 0. Then {x n } converges strongly to a fixed point x * of the nonexpansive mapping G which is also the unique solution of the variational inequality In other words, x * is the unique fixed point of the contraction P F (G) f , that is, P F (G) f (x * ) = x * .

The Constrained Convex Minimization Problem
Now, we consider the following constrained convex minimization problem: where φ : C → R is a real valued convex function and assumes that the problem (3.4) is consistent.Let Ω denote its solution set.For the minimization problem (3.4), if φ is (Fréchet) differentiable, then we have the following lemma.
Lemma 3.4.(Optimality Condition) [7] A necessary condition of optimality for a point x * ∈ C to be a solution of the minimization problem (3.4) is that x * solves the variational inequality ∇φ(x * ), x − x * ≥ 0, ∀x ∈ C. (3.5) Equivalently, x * ∈ C solves the fixed point equation for every constant λ > 0. If, in a addition φ is convex, then the optimality condition (3.5) is also sufficient.
It is well known that the mapping P C (I − λA) is nonexpansive when the mapping A is δ-inverse-strongly monotone and 0 < λ < 2δ.We therefore have the following result.Theorem 3.5.Let C be a nonempty closed convex subset of a real Hilbert space H.For the minimization problem (3.4), assume that φ is (Fréchet) differentiable and the gradient ∇φ is a δ-inverse-strongly monotone mapping for some positive real number δ.Let f : C → C be a contraction with coefficient θ ∈ [0, 1).Pick any x 0 ∈ C. Let {x n } be a sequence generated by where {α n }, {β n } and {γ n } are sequences in (0, 1) satisfying the conditions (i)-(iv) and (vii) lim n→∞ x n − P C (I − λ∇φ)(x n ) = 0. Then {x n } converges strongly to a solution x * of the minimization problem (3.4), which is also the unique solution of the variational inequality In other words, x * is the unique fixed point of the contraction P Ω f , that is, P Ω f (x * ) = x * .
On the bases of above lemma, we have the following result.Theorem 3.8.Let C be a nonempty closed convex subset of the real Hilbert space H.For i = 1, 2, 3, ..., N , let {T i } N i=1 be a finite family of K i -strictly pseudo-contractive mappings of C into itself with K i ≤ ω i and N i=1 F (T i ) = ∅.Let λ 1 , λ 2 , λ 3 , ..., λ N be real numbers with 0 < λ i < ω 2 , ∀i = 1, 2, 3, ..., N and ω 1 + ω 2 < 1.Let K be the K-mapping generated by T 1 , T 2 , T 3 , ..., T N and λ 1 , λ 2 , λ 3 , ..., λ N .Let f : C → C be a contraction with coefficient θ ∈ [0, 1).Pick any x 0 ∈ C, let {x n } be sequence generated by where {α n }, {β n } and {γ n } are sequences in (0, 1) satisfying the conditions (i)-(iv) and (viii) lim n→∞ x n − K(x n ) = 0. Then {x n } converges strongly to a fixed point x * of the mappings {T i } N i=1 , which is also the unique solution of the variational inequality In other words, x * is the unique fixed point of the contraction P N i=1 F (T i ) f , that is, P N i=1 F (T i ) f (x * ) = x * .

Definition 1 . 1 .
Let T : H → H be a mapping.Then T is called nonexpansive ifT (x) − T (y) ≤ x − y , ∀x, y ∈ H. Definition 1.2.A mapping f : H → H is called a contraction if for all x, y ∈ H and θ ∈ [0, 1) f (x) − f (y) ≤ θ x − y .Definition 1.3.P c : H → C is called a metric projection if for every x ∈ H there exists a unique nearest point in C, denoted by P c x, such that x − P c x ≤ x − y , ∀y ∈ C.