This article is part of the series S. Park's Contribution to the Development of Fixed Point Theory and KKM Theory.

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Strong convergence theorems for variational inequalities and fixed points of a countable family of nonexpansive mappings

Aunyarat Bunyawat1 and Suthep Suantai2*

Author Affiliations

1 Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

2 Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

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Fixed Point Theory and Applications 2011, 2011:47  doi:10.1186/1687-1812-2011-47


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2011/1/47


Received: 25 January 2011
Accepted: 5 September 2011
Published: 5 September 2011

© 2011 Bunyawat and Suantai; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A new general iterative method for finding a common element of the set of solutions of variational inequality and the set of common fixed points of a countable family of nonexpansive mappings is introduced and studied. A strong convergence theorem of the proposed iterative scheme to a common fixed point of a countable family of nonexpansive mappings and a solution of variational inequality of an inverse strongly monotone mapping are established. Moreover, we apply our main result to obtain strong convergence theorems for a countable family of nonexpansive mappings and a strictly pseudocontractive mapping, and a countable family of uniformly k-strictly pseudocontractive mappings and an inverse strongly monotone mapping. Our main results improve and extend the corresponding result obtained by Klin-eam and Suantai (J Inequal Appl 520301, 16 pp, 2009).

Mathematics Subject Classification (2000): 47H09, 47H10

Keywords:
countable family of nonexpansive mappings; variational inequality; inverse strongly monotone mapping; strictly pseudocontractive mapping; countable family of uniformly k-strictly pseudocontractive mappings

1 Introduction

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. In this paper, we always assume that a bounded linear operator A on H is strongly positive, that is, there is a constant γ ̄ > 0 such that A x , x γ ̄ | | x | | 2 for all x H. Recall that a mapping T of H into itself is called nonexpansive if ||Tx - Ty|| ≤ ||x - y|| for all x, y H. The set of all fixed points of T is denoted by F(T), that is, F(T) = {x C : x = Tx}. A self-mapping f : H H is a contraction on H if there is a constant α ∈ [0, 1) such that ||f(x) - f(y) || ≤ α ||x - y|| for all x, y H.

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on H:

min x F 1 2 A x , x - x , b , (1.1)

where F is the fixed point set of a nonexpansive mapping T on H and b is a given point in H. A mapping B of C into H is called monotone if 〈Bx - By, x - y〉 ≥ 0 for all x, y C. The variational inequality problem is to find x C such that 〈Bx, y - x〉 ≥ 0 for all y C. The set of solutions of the variational inequality is denoted by VI(C, B). A mapping B of C to H is called inverse strongly monotone if there exists a positive real number β such that 〈x - y, Bx - By〉 ≥ β ||Bx - By||2 for all x, y C.

Starting with an arbitrary initial x0 H, define a sequence {xn} recursively by

x n + 1 = ( I - α n A ) T x n + α n b n 0 . (1.2)

It is proved by Xu [1] that the sequence {xn} generated by (1.2) converges strongly to the unique solution of the minimization problem (1.1) provided the sequence {αn} satisfies certain conditions.

On the other hand, Moudafi [2] introduced the viscosity approximation method for nonexpansive mappings. Let f be a contraction on H. Starting with an arbitrary initial x0 H, define a sequence {xn} recursively by

x n + 1 = ( 1 - σ n ) T x n + σ n f ( x n ) n 0 , (1.3)

where {σn} is a sequence in (0, 1). It is proved by Moudafi [2] and Xu [3] that under certain appropriate conditions imposed on {σn}, the sequence {xn} generated by (1.3) strongly converges to the unique solution x* in C of the variational inequality

( I - f ) x * , x - x * 0 x C .

Recently, Marino and Xu [4] combined the iterative method (1.2) with the viscosity approximation method (1.3) and considered the following general iteration process:

x n + 1 = ( I - α n A ) T x n + α n γ f ( x n ) n 0 (1.4)

and proved that if the sequence {αn} satisfies appropriate conditions, the sequence {xn} generated by (1.4) converges strongly to the unique solution of the variational inequality

( A - γ f ) x * , x - x * 0 x C

which is the optimality condition for the minimization problem

min x C 1 2 A x , x - h ( x ) ,

where h is a potential function for γ f (i.e., h'(x) = γ f(x) for x H).

Chen, Zhang and Fan [5] introduced the following iterative process: x0 C,

x n + 1 = α n f ( x n ) + ( 1 - α n ) T P C ( x n - λ n B x n ) , n 0 , (1.5)

where {αn} ⊂ (0, 1) and {λn} ⊂ [a, b] for some a, b with 0 < a < b < 2β.

They proved that under certain appropriate conditions imposed on {αn} and {λn}, the sequence {xn} generated by (1.5) converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inverse strongly monotone mapping (say x ̄ C ), which solves the variational inequality

( I - f ) x ̄ , x - x ̄ 0 x F ( T ) V I ( C , B ) .

Klin-eam and Suantai [6] modify the iterative methods (1.4) and (1.5) by proposing the following general iterative method: x0 C,

x n + 1 = P C ( α n γ f ( x n ) + ( I - α n A ) T P C ( x n - λ n B x n ) ) , n 0 , (1.6)

where PC is the projection of H onto C, f is a contraction, A is a strongly positive linear bounded operator, B is a β-inverse strongly monotone mapping, {αn} ⊂ (0, 1) and {λn} ⊂ [a, b] for some a, b with 0 < a < b < 2β. They noted that when A = I and γ = 1, the iterative scheme (1.6) reduced to the iterative scheme (1.5).

Wangkeeree, Petrot and Wangkeeree [7] introduced the following iterative process:

x 0 = x H , y n = β n x n + ( 1 - β n ) T n x n , x n + 1 = α n γ f ( x n ) + ( I - α n A ) y n , n 0 (1.7)

where {αn}and{βn} ⊂ (0, 1) and Tn is a countable family of nonexpansive mappings, f is a contraction, and A is a strongly positive linear bounded operator. They proved that under certain appropriate conditions imposed on {αn}, {βn} and {Tn}, the sequence {xn} converges strongly to x ̃ , which solves the variational inequality:

( A - γ f ) x ̃ , x ̃ - z 0 z F ( T ) .

In this paper, motivated and inspired by Klin-eam and Suantai [6], we introduced the following iteration to find some solutions of variational inequality and fixed points of countable family of nonexpansive mappings in a Hilbert spaces H: x0 C,

x n + 1 = P C ( α n γ f ( x n ) + ( I - α n A ) T n P C ( x n - λ n B x n ) ) , n 0 , (1.8)

where PC is the projection of H onto C, f is a contraction, A is a strongly positive linear bounded operator, Tn is a countable family of nonexpansive mappings of C into itself, B is a β-inverse strongly monotone mapping, {αn} ⊂ (0, 1), and {λn} ⊂ [a, b] for some a, b with 0 < a < b < 2β.

2 Preliminaries

Let H be a real Hilbert space with inner product 〈·,·〉 and norm || · ||, and let C be a closed convex subset of H. We write xn x to indicate that the sequence {xn} converges weakly to x, and xn x implies that {xn} converges strongly to x. For every point x H, there exists a unique nearest point in C, denoted by PCx, such that ||x - PCx|| ≤ ||x - y|| for all y C and PCx is called the metric projection of H onto C. We know that PC is a nonexpansive mapping of H onto C. It is also known that PC satisfies 〈x - y, PCx - PCy〉 ≥ ||PCx - PCy||2 for every x, y H. Moreover, PCx is characterized by the properties: PCx C and 〈x - PCx, PCx - y〉 ≥ 0 for all y C. In the context of the variational inequality problem, this implies that

u V I ( C , A ) u = P C ( u - λ A u ) , λ > 0 .

A set-valued mapping T : H → 2H is called monotone if for all x, y H, f Tx and g Ty imply 〈x - y, f - g〉 ≥ 0. A monotone mapping T : H → 2H is maximal if the graph G(T) of T is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if for (x, f ) ∈ H × H, 〈x - y, f - g〉 ≥ 0 for every (y, g) ∈ G(T) implies f Tx. Let A be an inverse strongly monotone mapping of C into H, and let NCv be the normal cone to C at v C, i.e., NCv = {w H : 〈v - u, w〉 ≥ 0, ∀u C}, and define

T v = A v + N C v , v C , , v C .

Then, T is maximal monotone and 0 ∈ Tv if and only if v V I(C, A).

Lemma 2.1 Let C be a closed convex subset of a real Hilbert space H. Given x H and y C, then

(i) y = PCx if and only if the inequality x - y, y - z〉 ≥ 0 for all z C,

(ii) PC is nonexpansive,

(iii) x - y, PCx - PCy〉 ≥ ||PCx - PCy||2 for all x, y H,

(iv) x - PCx, PCx - y〉 ≥ 0 for all x H and y C.

Lemma 2.2 [4]Assume A is a strongly positive linear bounded operator on a Hilbert space H with coefficient γ ̄ > 0 and 0 < ρ ≤ ||A|| -1, then | | I - ρ A | | 1 - ρ γ ̄ .

Lemma 2.3 [8]Assume {an} is a sequence of nonnegative real numbers such that

a n + 1 ( 1 - γ n ) a n + δ n , n 0

where {γn} ⊂ (0, 1) and {δn} is a sequence in such that

(i) n = 1 γ n = ,

(ii) lim supn→∞ δn n ≤ 0 or n = 1 | δ n | < .

Then, lim n→∞ an = 0.

Lemma 2.4 [9]Let C be a closed convex subset of a real Hilbert space H, and let T : C C be a nonexpansive mapping such that F(T) ∅. If a sequence {xn} in C such that xn z and xn - Txn → 0, then z = Tz.

To deal with a family of mappings, the following conditions are introduced: Let C be a subset of a real Banach space E, and let { T n } n = 1 be a family of mappings of C such that n = 1 F ( T n ) . Then, {Tn} is said to satisfy the AKTT-condition [10] if for each bounded subset B of C,

n = 1 s u p { | | T n + 1 z - T n z | | : z B } < .

Lemma 2.5 [10]Let C be a nonempty and closed subset of a Banach space E and let {Tn} be a family of mappings of C into itself which satisfies the AKTT-condition. Then, for each x C, {Tnx} converges strongly to a point in C. Moreover, let the mapping T be defined by

T x = lim n T n x x C .

Then, for each bounded subset B of C,

lim sup n { | | T z T n z | | : z B } = 0.

In the sequel, we will write ({Tn}, T ) satisfies the AKTT-condition if {Tn} satisfies the AKTT-condition, and T is defined by Lemma 2.5 with F ( T ) = n = 1 F ( T n ) .

3 Main results

In this section, we prove a strong convergence theorem for a countable family of nonexpansive mappings.

Theorem 3.1 Let C be a closed convex subset of a real Hilbert space H, and let B : C H be a β-inverse strongly monotone mapping, also let A be a strongly positive linear bounded operator of H into itself with coefficient γ ̄ > 0 such that ||A|| = 1 and let f : C C be a contraction with coefficient α(0 < α < 1). Assume that 0 < γ < γ ̄ α . Let {Tn} be a countable family of nonexpansive mappings from a subset C into itself with F = n = 1 F ( T n ) V I ( C , B ) . Suppose {xn} is the sequence generated by the following algorithm: x0 C,

x n + 1 = P C ( α n γ f ( x n ) + ( I - α n A ) T n P C ( x n - λ n B x n ) )

for all n = 0, 1, 2, ..., where {αn} ⊂ (0, 1) and {λn} ⊂ (0, 2β ). If {αn} and {λn} are chosen so that λn ∈ [a, b] for some a, b with 0 < a < b < 2β ,

( C 1 ) lim n 0 α n = 0 ; ( C 2 ) n = 1 α n = ; ( C 3 ) n = 1 α n + 1 - α n < ; ( C 4 ) n = 1 λ n + 1 - λ n < .

Suppose that ({Tn}, T ) satisfies the AKTT-condition. Then, {xn} converges strongly to q F, where q = PF (γ f + I - A)(q) which solves the following variational inequality:

( γ f - A ) q , p - q 0 p F .

Proof. First, we show that the sequence {xn} is bounded. Consider the mapping I -λnB. Since B is a β-inverse strongly monotone mapping, we have that for all x, y C,

| | ( I - λ n B ) x - ( I - λ n B ) y | | 2 = | | ( x - y ) - λ n ( B x - B y ) | | 2 (1) = | | x - y | | 2 - 2 λ n x - y , B x - B y + λ n 2 | | B x - B y | | 2 (2) | | x - y | | 2 + λ n ( λ n - 2 β ) | | B x - B y | | 2 . (3) (4) 

For 0 < λn < 2β, implies that ||k(I - λnB)x - (I- λnB)y||2 ≤ ||x - y||2.

So, the mapping I - λnB is nonexpansive.

Put yn = PC(xn - λnBxn) for all n ≥ 0. Let u F. Then u = PC(u - λnBu).

From PC is nonexpansive implies that

| | y n - u | | = | | P C ( x n - λ n B x n ) - P C ( u - λ n B u ) | | (1) | | ( x n - λ n B x n ) - ( u - λ n B u ) | | (2) = | | ( I - λ n B ) x n - ( I - λ n B ) u | | . (3) (4)

Since I - λnB is nonexpansive, we have that ||yn - u|| ≤ ||xn - u||. Then

| | x n + 1 - u | | = | | P C ( α n γ f ( x n ) + ( I - α n A ) T n y n ) - u | | (1) | | α n γ f ( x n ) + ( I - α n A ) T n y n - u | | (2) = | | α n ( γ f ( x n ) - A u ) + ( I - α n A ) ( T n y n - u ) | | . (3) (4)

Since A is strongly positive linear bounded operator, we have

| | x n + 1 - u | | α n | | γ f ( x n ) - A u | | + ( 1 - α n γ ̄ ) | | T n y n - u | | (1) α n | | γ f ( x n ) - γ f ( u ) | | + α n | | γ f ( u ) - A u | | + ( 1 - α n γ ̄ ) | | T n y n - u | | . (2) (3) 

By contraction of f, we have

| | x n + 1 - u | | α γ α n | | x n - u | | + α n | | γ f ( u ) - A u | | + ( 1 - α n γ ̄ ) | | T n y n - u | | (1)  = α γ α n | | x n - u | | + α n | | γ f ( u ) - A u | | + ( 1 - α n γ ̄ ) | | T n y n - T n u | | (2)  α γ α n | | x n - u | | + α n | | γ f ( u ) - A u | | + ( 1 - α n γ ̄ ) | | y n - u | | (3)  α γ α n | | x n - u | | + α n | | γ f ( u ) - A u | | + ( 1 - α n γ ̄ ) | | x n - u | | (4)  ( α γ α n + 1 - α n γ ̄ ) | | x n - u | | + α n | | γ f ( u ) - A u | | (5)  ( 1 - α n ( γ ̄ - α γ ) ) | | x n - u | | + α n ( γ ̄ - α γ ) | | γ f ( u ) - A u | | γ ̄ - α γ (6)  m a x | | x n - u | | , | | γ f ( u ) - A u | | γ ̄ - α γ . (7)  (8) 

It follows from induction that | | x n - u | | m a x | | x 0 - u | | , | | γ f ( u ) - A u | | γ ̄ - α γ , n ≥ 0.

Therefore, {xn} is bounded, so are {yn}, {Tnyn}, {Bxn}, and {f (xn)}.

Next, we show that ||xn+1 - xn|| → 0 and ||yn - Tnyn|| → 0 as n → ∞.

Since PC is nonexpansive, we also have

| | y n + 1 - y n | | = | | P C ( x n + 1 - λ n + 1 B x n + 1 ) - P C ( x n - λ n B x n ) | | (1) | | x n + 1 - λ n + 1 B x n + 1 - ( x n - λ n B x n ) | | (2) | | x n + 1 - λ n + 1 B x n + 1 - ( x n - λ n + 1 B x n ) | | + | λ n - λ n + 1 | | | B x n | | (3) = | | ( I - λ n + 1 B ) x n + 1 - ( I - λ n + 1 B ) x n | | + | λ n - λ n + 1 | | | B x n | | . (4)  (5) 

Since I - λnB is nonexpansive, we have

| | y n + 1 - y n | | | | x n + 1 - x n | | + | λ n - λ n + 1 | | | B x n | | .

So we obtain

| | x n + 1 x n | | = | | P C ( α n γ f ( x n ) + ( I α n A ) T n y n ) P C ( α n 1 γ f ( x n 1 ) + ( I α n 1 A ) T n 1 y n 1 ) | | | | α n γ ( f ( x n ) f ( x n 1 ) ) + γ ( α n α n 1 ) f ( x n 1 ) + ( I α n A ) ( T n y n T n 1 y n 1 ) + ( α n α n 1 ) A T n 1 y n 1 | | α n α γ | | x n x n 1 | | + γ | α n α n 1 | | | f ( x n 1 ) | | + ( 1 α n γ ¯ ) | | T n y n T n 1 y n 1 | | + | α n α n 1 | | | A T n 1 y n 1 | | α n α γ | | x n x n 1 | | + γ | α n α n 1 | | | f ( x n 1 ) | | + ( 1 α n γ ¯ ) ( | | T n y n T n y n 1 | | + | | T n y n 1 T n 1 y n 1 | | ) + | α n α n 1 | | | A T n 1 y n 1 | | α n α γ | | x n x n 1 | | + γ | α n α n 1 | | | f ( x n 1 ) | | + ( 1 α n γ ¯ ) ( | | y n y n 1 | | + | | T n y n 1 T n 1 y n 1 | | ) + | α n α n 1 | | | A T n 1 y n 1 | | = α n α γ | | x n x n 1 | | + γ | α n α n 1 | | | f ( x n 1 ) | | + ( 1 α n γ ¯ ) | | y n y n 1 | | + ( 1 α n γ ¯ ) | | T n y n 1 T n 1 y n 1 | | + | α n α n 1 | | | A T n 1 y n 1 | | α n α γ | | x n x n 1 | | + γ | α n α n 1 | | | f ( x n 1 ) | | + ( 1 α n γ ¯ ) | | x n x n 1 | | + ( 1 α n γ ¯ ) | λ n 1 λ n | | | B x n 1 | | + ( 1 α n γ ¯ ) | | T n y n 1 T n 1 y n 1 | | + | α n α n 1 | | | A T n 1 y n 1 | | = ( 1 ( γ ¯ α γ ) α n ) | | x n x n 1 | | + γ | α n α n 1 | | | f ( x n 1 ) | | + ( 1 α n γ ¯ ) | λ n 1 λ n | | | B x n 1 | | + ( 1 α n γ ¯ ) | | T n y n 1 T n 1 y n 1 | | + | α n α n 1 | | | A T n 1 y n 1 | | ( 1 ( γ ¯ α γ ) α n ) | | x n x n 1 | | + 2 L | α n α n 1 | + M | λ n 1 λ n | + s u p y { y n } | | T n y T n 1 y | | ,

where L = max{supn∈ℕ ||ATn-1yn - 1||, supn∈ℕ γ ||f (xn - 1)||} and M = sup{||Bxn-1|| : n∈ℕ}.

Since {Tn} satisfies the AKTT-condition, we get that

n = 1 sup y { y n } | | T n y - T n - 1 y | | < .

From condition (C3), (C4) and by Lemma 2.3, we have ||xn+1 - xn|| → 0.

For u F and u = PC(u - λnBu), we have

| | x n + 1 u | | 2 = | | P C ( α n γ f ( x n ) + ( I α n A ) T n y n ) P C ( u ) | | 2 | | α n ( γ f ( x n ) A u ) + ( I α n A ) ( T n y n u ) | | 2 ( α n | | γ f ( x n ) A u | | + | | I α n A | | | | T n y n u | | ) 2 ( α n | | γ f ( x n ) A u | | + ( 1 α n γ ¯ ) | | y n u | | ) 2 α n | | γ f ( x n ) A u | | 2 + ( 1 α n γ ¯ ) | | y n u | | 2 + 2 α n ( 1 α n γ ¯ ) | | γ f ( x n ) A u | | | | y n u | | α n | | γ f ( x n ) A u | | 2 + ( 1 α n γ ¯ ) | | ( I λ n B ) x n ( I λ n B ) u | | 2 + 2 α n ( 1 α n γ ¯ ) | | γ f ( x n ) A u | | | | y n u | | α n | | γ f ( x n ) A u | | 2 + ( 1 α n γ ¯ ) ( | | x n u | | 2 2 λ n x n u , B x n B u + λ n 2 | | B x n B u | | 2 ) + 2 α n ( 1 α n γ ¯ ) | | γ f ( x n ) A u | | | | y n u | | α n | | γ f ( x n ) A u | | 2 + ( 1 α n γ ¯ ) ( | | x n u | | 2 2 λ n β | | B x n B u | | 2 + λ n 2 | | B x n B u | | 2 ) + 2 α n ( 1 α n γ ¯ ) | | γ f ( x n ) A u | | | | y n u | | = α n | | γ f ( x n ) A u | | 2 + ( 1 α n γ ¯ ) ( | | x n u | | 2 + λ n ( λ n 2 β ) | | B x n B u | | 2 ) + 2 α n ( 1 α n γ ¯ ) | | γ f ( x n ) A u | | | | y n u | | α n | | γ f ( x n ) A u | | 2 + | | x n u | | 2 + ( 1 α n γ ¯ ) b ( b 2 β ) | | B x n B u | | 2 + 2 α n ( 1 α n γ ¯ ) | | γ f ( x n ) A u | | | | y n u | | .

So, we obtain

- ( 1 - α n γ ̄ ) b ( b - 2 β ) | | B x n - B u | | 2 α n | | γ f ( x n ) - A u | | 2 + ( | | x n - u | | + | | x n + 1 - u | | ) ( | | x n - u | | - | | x n + 1 - u | | ) + ε n α n | | γ f ( x n ) - A u | | 2 + ε n + | | x n - x n + 1 | | ( | | x n - u | | + | | x n + 1 - u | | ) ,

where ε n = 2 α n ( 1 - α n γ ̄ ) | | γ f ( x n ) - A u | | | | y n - u | | .

Since αn → 0 and ||xn+1 - xn|| → 0, we obtain ||Bxn - Bu|| → 0 as n → ∞.

Further, by Lemma 2.1, we have

| | y n u | | 2 = | | P C ( x n λ n B x n ) P C ( u λ n B u ) | | 2 ( x n λ n B x n ) ( u λ n B u ) , y n u = 1 2 ( | | ( x n λ n B x n ) ( u λ n B u ) | | 2 + | | y n u | | 2 | | ( x n λ n B x n ) ( u λ n B u ) ( y n u ) | | 2 ) 1 2 ( | | x n u | | 2 + | | y n u | | 2 | | ( x n y n ) λ n ( B x n B u ) | | 2 ) | | x n u | | 2 | | x n y n | | 2 + 2 λ n x n y n , B x n B u λ n 2 | | B x n B u | | 2 .

So, we have

| | x n + 1 - u | | 2 = | | P C ( α n γ f ( x n ) + ( I - α n A ) T n y n ) - P C ( u ) | | 2 (1) | | α n ( γ f ( x n ) - A u ) + ( I - α n A ) ( T n y n - u ) | | 2 (2) ( α n | | γ f ( x n ) - A u | | + | | I - α n A | | | | T n y n - u | | ) 2 (3) ( α n | | γ f ( x n ) - A u | | + ( 1 - α n γ ̄ ) | | y n - u | | ) 2 (4) α n | | γ f ( x n ) - A u | | 2 + ( 1 - α n γ ̄ ) | | y n - u | | 2 (5) + 2 α n ( 1 - α n γ ̄ ) | | γ f ( x n ) - A u | | | | y n - u | | (6) α n | | γ f ( x n ) - A u | | 2 + ( 1 - α n γ ¯ ) | | x n - u | | 2 - ( 1 - α n γ ¯ ) | | x n - y n | | 2 (7) + 2 ( 1 - α n γ ̄ ) λ n x n - y n , B x n - B u - ( 1 - α n γ ̄ ) λ n 2 | | B x n - B u | | 2 (8) + 2 α n ( 1 - α n γ ̄ ) | | γ f ( x n ) - A u | | | | y n - u | | , (9) (10) 

which implies

( 1 - α n γ ̄ ) | | x n - y n | | 2 α n | | γ f ( x n ) - A u | | 2 + ( | | x n - u | | + | | x n + 1 - u | | ) | | x n - x n + 1 | | (1) + 2 ( 1 - α n γ ̄ ) λ n x n - y n , B x n - B u - ( 1 - α n γ ̄ ) λ n 2 | | B x n - B u | | 2 (2) + 2 α n ( 1 - α n γ ̄ ) | | γ f ( x n ) - A u | | | | y n - u | | . (3) (4) 

Since αn → 0, ||xn+1 - xn|| → 0, and ||Bxn - Bu|| → 0, we obtain ||xn - yn|| → 0 as n → ∞.

Next, we have

| | x n + 1 - T n y n | | = | | P C ( α n γ f ( x n ) + ( I - α n A ) T n y n ) - P C ( T n y n ) | | (1) | | α n γ f ( x n ) + ( I - α n A ) T n y n - T n y n | | (2) = α n | | γ f ( x n ) - A T n y n | | . (3) (4)

Since αn → 0 and {f (xn)}, {ATnyn} are bounded, we have ||xn+1 - Tnyn|| → 0 as n → ∞. Since

| | x n - T n y n | | | | x n - x n + 1 | | + | | x n + 1 - T n y n | | ,

it implies that ||xn - Tnyn|| → 0 as n → ∞. Since

| | x n - T n x n | | | | x n - T n y n | | + | | T n y n - T n x n | | (1) | | x n - T n y n | | + | | y n - x n | | , (2) (3)

we obtain ||xn - Tnxn|| → 0 as n → ∞. Moreover, from

| | y n - T n y n | | | | y n - x n | | + | | x n - T n y n | | ,

it follows that ||yn - Tnyn|| → 0 as n → ∞.

By ||yn - xn|| → 0, ||Tnyn - xn|| → 0 and Lemma 2.5, we have

| | T x n - x n | | | | T x n - T y n | | + | | T y n - T n y n | | + | | T n y n - x n | | (1) | | x n - y n | | + s u p { | | T n z - T z | | : z { y n } } + | | T n y n - x n | | . (2) (3)

Hence, limn→∞ ||Txn - xn|| = 0. Observe that PF (γ f +I - A) is a contraction.

By Lemma 2.2, we have that | | I - A | | 1 - γ ̄ , and since 0 < γ < γ ̄ α , we get

| | P F ( γ f + I - A ) x - P F ( γ f + I - A ) y | | | | ( γ f + I - A ) x - ( γ f + I - A ) y | | (1)  γ | | f ( x ) - f ( y ) | | + | | I - A | | | | x - y | | (2)  γ α | | x - y | | + ( 1 - γ ̄ ) | | x - y | | (3)  = ( 1 - ( γ ̄ - γ α ) ) | | x - y | | . (4)  (5) 

Then, Banach's contraction mapping principle guarantees that PF (γ f +I - A) has a unique fixed point, say q H. That is, q = PF (γ f + I - A)q. By Lemma 2.1, we obtain

( γ f - A ) q , x - q 0 for all x F . (3.1)

Choose a subsequence { y n k } of {yn} such that

lim sup n ( γ f A ) q , T n y n q = lim k ( γ f A ) q , T n k y n k q .

As { y n k } is bounded, there exists a subsequence { y n k i } of { y n k } which converges weakly to p. Without loss of generality, we may assume that y n k p .

Since ||yn - Tnyn|| → 0, we obtain T n k y n k p . Since ||xn - Txn|| → 0, ||xn - yn|| → 0 and by Lemma 2.4-2.5, we have p n = 1 F ( T n ) . Let

S v = B v + N C v , v C , , v C .

where NCv is normal cone to C at v C, that is NCv = {w H : 〈v - u, w〉 ≥ 0, ∀u C}. Then S is a maximal monotone. Let (v, w) ∈ G(S). Since w - Bv NCv and yn C, we have 〈v - yn, w - Bv〉 ≥ 0. On the other hand, by Lemma 2.1 and from yn = PC(xn - λnBxn), we have

v - y n , y n - ( x n - λ n B x n ) 0 (1) v - y n , ( y n - x n ) λ n + B x n 0 . (2) (3)

Hence,

v - y n k , w v - y n k , B v (1) v - y n k , B v - v - y n k , y n k - x n k λ n + B x n k (2) = v - y n k , B v - B x n k - y n k - x n k λ n (3) = v - y n k , B v - B y n k + v - y n k , B y n k - B x n k - v - y n k , y n k - x n k λ n (4) v - y n k , B y n k - B x n k - v - y n k , y n k - x n k λ n . (5) (6) 

This implies 〈v - p, w〉 ≥ 0. Since S is maximal monotone, we have p S -10 and hence p V I(C, B). We obtain that p F. By (3.1), we have 〈(γ f - A)q, p - q〉 ≤ 0. It follows that

limsup n ( γ f - A ) q , T n y n - q = lim k ( γ f - A ) q , T n k y n k - q = ( γ f - A ) q , p - q 0 .

Finally, we prove xn q. By ||yn - u|| ≤ ||xn - u|| and Schwarz inequality, we have

| | x n + 1 q | | 2 = | | P C ( α n γ f ( x n ) + ( I α n A ) T n y n ) P C ( q ) | | 2 | | α n ( γ f ( x n ) A q ) + ( I α n A ) ( T n y n q ) | | 2 | | ( I α n A ) ( T n y n q ) | | 2 + α n 2 | | γ f ( x n ) A q | | 2 + 2 α n ( I α n A ) ( T n y n q ) , γ f ( x n ) A q ( 1 α n γ ¯ ) 2 | | y n q | | 2 + α n 2 | | γ f ( x n ) A q | | 2 + 2 α n T n y n q , γ f ( x n ) A q 2 α n 2 A ( T n y n q ) , γ f ( x n ) A q ( 1 α n γ ¯ ) 2 | | x n q | | 2 + α n 2 | | γ f ( x n ) A q | | 2 + 2 α n T n y n q , γ f ( x n ) γ f ( q ) + 2 α n T n y n q , γ f ( q ) A q 2 α n 2 A ( T n y n q ) , γ f ( x n ) A q ( 1 α n γ ¯ ) 2 | | x n q | | 2 + α n 2 | | γ f ( x n ) A q | | 2 + 2 α n | | T n y n q | | | | γ f ( x n ) γ f ( q ) | | + 2 α n T n y n q , γ f ( q ) A q 2 α n 2 A ( T n y n q ) , γ f ( x n ) A q ( 1 α n γ ¯ ) 2 | | x n q | | 2 + α n 2 | | γ f ( x n ) A q | | 2 + 2 γ α α n | | y n q | | | | x n q | | + 2 α n T n y n q , γ f ( q ) A q 2 α n 2 A ( T n y n q ) , γ f ( x n ) A q ( 1 α n γ ¯ ) 2 | | x n q | | 2 + α n 2 | | γ f ( x n ) A q | | 2 + 2 γ α α n | | x n q | | 2 + 2 α n T n y n q , γ f ( q ) A q 2 α n 2 A ( T n y n q ) , γ f ( x n ) A q ( ( 1 α n γ ¯ ) 2 + 2 γ α α n ) | | x n q | | 2 + α n ( 2 T n y n q , γ f ( q ) A q + α n | | γ f ( x n ) A q | | 2 + 2 α n | | A ( T n y n q ) | | | | γ f ( x n ) A q | | ) = ( 1 2 ( γ ¯ γ α ) α n ) | | x n q | | 2 + α n ( 2 T n y n q , γ f ( q ) A q + α n | | γ f ( x n ) A q | | 2 + 2 α n | | A ( T n y n q ) | | | | γ f ( x n ) A q | | + α n γ ¯ 2 | | x n q | | 2 ) .

Since {xn}, {f (xn)} and {Tnyn} are bounded, we can take a constant η > 0 such that

η | | γ f ( x n ) - A q | | 2 + 2 | | A ( T n y n - q ) | | | | γ f ( x n ) - A q | | + γ ̄ 2 | | x n - q | | 2

for all n ≥ 0. It follows that

| | x n + 1 - q | | 2 ( 1 - 2 ( γ ̄ - γ α ) α n ) | | x n - q | | 2 + α n β n , (3.2)

where βn = 2〈Tnyn - q, γ f(q) - Aq〉 +ηαn. By lim supn→∞〈(γ f - A)q, Tnyn - q〉 ≤ 0, we get lim sup n→∞ βn ≤ 0. By Lemma 2.3 and (3.2), we can conclude that xn q. This completes the proof. ■

Corollary 3.2 Let C be a closed convex subset of a real Hilbert space H, and let B : C H be a β-inverse strongly monotone mapping, also let f : C C be a contraction with coefficient α(0 < α < 1). Let {Tn} be a countable family of nonexpansive mappings from a subset C into itself with F = n = 1 F ( T n ) V I ( C , B ) . Suppose {xn} is the sequence generated by the following algorithm: x0 C,

x n + 1 = α n f ( x n ) + ( 1 - α n ) T n P C ( x n - λ n B x n )

for all n = 0, 1, 2, ..., where {αn} ⊂ (0, 1) and {λn} ⊂ (0, 2β ). If {αn} and {λn} are chosen so that λn ∈ [a, b] for some a, b with 0 < a < b < 2β,

( C 1 ) lim n 0 α n = 0 ; ( C 2 ) n = 1 α n = ; ( C 3 ) n = 1 α n + 1 - α n < ; ( C 4 ) n = 1 λ n + 1 - λ n < .

Suppose that ({Tn}, T ) satisfies the AKTT-condition. Then {xn} converges strongly to q F, where q = PF (γ f + I - A)(q) which solves the following variational inequality:

( γ f - I ) q , p - q 0 p F .

Proof. Taking A = I and γ = 1 in Theorem 3.1, we get the results. ■

4 Applications

In this section, we apply the iterative scheme (1.8) and Theorem 3.1 for finding a common fixed point of countable family of nonexpansive mappings and strictly pseudocontractive mapping and inverse strongly monotone mapping.

A mapping T : C C is called strictly pseudocontractive if there exists k with 0 ≤ k < 1 such that

| | T x - T y | | 2 | | x - y | | 2 + k | | ( I - T ) x - ( I - T ) y | | 2 x , y C .

If k = 0, then T is nonexpansive. Put B = I - T, where T : C C is a strictly pseudocontractive mapping with k. Then, B is ((1 - k)/2)-inverse strongly monotone and B -1 (0) = F(T). Hence, for all x, y C,

| | ( I - B ) x - ( I - B ) y | | 2 | | x - y | | 2 + k | | B x - B y | | 2 .

Conversely, since H is a real Hilbert space, we have

| | ( I - B ) x - ( I - B ) y | | 2 | | x - y | | 2 + | | B x - B y | | 2 - 2 x - y , B x - B y .

Thus, we have

x - y , B x - B y 1 - k 2 | | B x - B y | | 2 .

Theorem 4.1 Let C be a closed convex subset of a real Hilbert space H, and let A be a strongly positive linear bounded operator of H into itself with coefficient γ ̄ > 0 such that ||A|| = 1 and let f : C C be a contraction with coefficient α(0 < α < 1). Assume that 0 < γ < γ ̄ α . Let {Tn} be a family of nonexpansive mappings of C into itself and let S be a strictly pseudocontractive mapping of C into itself with β such that F = n = 1 F ( T n ) F ( S ) . Suppose {xn} is a sequence generated by the following algorithm: x0 C,

x n + 1 = P C ( α n γ f ( x n ) + ( I - α n A ) T n ( ( 1 - λ n ) x n - λ n S x n ) )

for all n = 0, 1, 2, ..., where {αn} ⊂ [0, 1) and {λn} ⊂ [0, 1 - β). If {αn} and {λn} are chosen so that λn ∈ [a, b] for some a, b with 0 < a < b < 1 - β,

( C 1 ) lim n 0 α n = 0 ; ( C 2 ) n = 1 α n = ; ( C 3 ) n = 1 α n + 1 - α n < ; ( C 4 ) n = 1 λ n + 1 - λ n < .

Suppose that ({Tn}, T) satisfies the AKTT-condition. Then, {xn} converges strongly to q F, such that

( γ f - A ) q , p - q 0 p F .

Proof. Put B = I - S, then B is ((1 - k)/2)-inverse strongly monotone and F(S) = V I(C, B) and PC(xn - λnBxn) = (1 - λn)xn +λnSxn. Therefore, by Theorem 3.1, the conclusion follows. ■

Lemma 4.2 [9]Let T : C H be a k-strictly pseudocontractive, then

(i) the fixed point set F(T) of T is closed convex so that the projection PF(T) is well defined;

(ii) define a mapping S : C H by

S x = μ x + ( 1 - μ ) T x , x C . (4.1)

If μ ∈ [k, 1), then S is a nonexpansive mapping such that F(T) = F(S).

A family of mappings { T n : C H } n = 1 is called a family of uniformly k-strictly pseudocontractions, if there exists a constant k ∈ [0, 1) such that | | T n x - T n y | | 2 | | x - y | | 2 + k | | ( I - T n ) x - ( I - T n ) y | | 2 x , y C , n 1 .

Let {Tn : C C} be a countable family of uniformly k-strictly pseudocontractions. Let { S n : C C } n = 1 be the sequence of mappings defined by (4.1), i.e.,

S n x = μ x + ( 1 - μ ) T n x , x C , n 1 with μ [ k , 1 ) .

Corollary 4.3 Let C be a closed convex subset of a real Hilbert space H, and let B : C H be a β-inverse strongly monotone mapping, also let A be a strongly positive linear bounded operator of H into itself with coefficient γ ̄ > 0 such that ||A|| = 1 and let f : C C be a contraction with coefficient α(0 < α < 1). Assume that 0 < γ < γ ̄ α . Let {Tn} be a countable family of uniformly k-strictly pseudocontractions from a subset C into itself with F = n = 1 F ( T n ) V I ( C , B ) . Suppose {xn} is the sequence generated by the following algorithm: x0 C,

x n + 1 = P C ( α n γ f ( x n ) + ( I - α n A ) S n P C ( x n - λ n B x n ) )

for all n = 0, 1, 2, ..., where {αn} ⊂ (0, 1) and {λn} ⊂ (0, 2β). If {αn} and {λn} are chosen so that λn ∈ [a, b] for some a, b with 0 < a < b < 2β,

( C 1 ) lim n 0 α n = 0 ; ( C 2 ) n = 1 α n = ; ( C 3 ) n = 1 α n + 1 - α n < ; ( C 4 ) n = 1 λ n + 1 - λ n < .

Then, {xn} converges strongly to q F, where q = PF (γ f + I - A)(q) which solves the following variational inequality:

( γ f - A ) q , p - q 0 p F .

Proof. Let {Tn} be a countable family of uniformly k-strictly pseudo-contractions from a subset C into itself. Set Sn = μI + (1 - μ)Tn where μ ∈ [k, 1). By Lemma 4.2, we have Sn is nonexpansive and F (Sn) = F (Tn). Therefore, by Theorem 3.1, the conclusion follows. ■

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

AB study and researched nonlinear analysis and also wrote this article. SS participated in the process of the study and helped to draft the manuscript. All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank the Centre of Excellence in Mathematics for financial support under the project RG-1-53-02-2. The first author is also supported by the Graduate School, Chiang Mai University, Thailand.

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