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Strong convergence theorem for amenable semigroups of nonexpansive mappings and variational inequalities

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Abstract

In this paper, using strongly monotone and lipschitzian operator, we introduce a general iterative process for finding a common fixed point of a semigroup of nonexpansive mappings, with respect to strongly left regular sequence of means defined on an appropriate space of bounded real-valued functions of the semigroups and the set of solutions of variational inequality for β-inverse strongly monotone mapping in a real Hilbert space. Under suitable conditions, we prove the strong convergence theorem for approximating a common element of the above two sets.

Mathematics Subject Classification 2000: 47H09, 47H10, 43A07, 47J25

1 Introduction

Throughout this paper, we assume that H is a real Hilbert space with inner product and norm are denoted by 〈. , .〉 and || . ||, respectively, and let C be a nonempty closed convex subset of H. A mapping T of C into itself is called nonexpansive if || Tx - Ty ||≤|| x - y ||, for all x, y H. By Fix(T), we denote the set of fixed points of T (i.e., Fix(T) = {x H : Tx = x}), it is well known that Fix(T) is closed and convex. Recall that a self-mapping f : CC is a contraction on C if there exists a constant α [0, 1) such that || f(x) - f(y) ||≤ α || x - y || for all x, y C.

Let B : CH be a mapping. The variational inequality problem, denoted by VI(C, B), is to fined x C such that

B x , y - x 0 ,
(1)

for all y C. The variational inequality problem has been extensively studied in literature. See, for example, [1, 2] and the references therein.

Definition 1.1 Let B : CH be a mapping. Then B

(1) is called η-strongly monotone if there exists a positive constant η such that

B x - B y , x - y η x - y 2 , x , y C ,

(2) is called k-Lipschitzian if there exist a positive constant k such that

B x - - B y k x - y , x , y C ,

(3) is called β-inverse strongly monotone if there exists a positive real number β > 0 such that

B x - B y , x - y β B x - B y 2 , x , y C .

It is obvious that any β-inverse strongly monotone mapping B is 1 β -Lipschitzian.

Moudafi [3] introduced the viscosity approximation method for fixed point of nonexpansive mappings (see [4] for further developments in both Hilbert and Banach spaces). Starting with an arbitrary initial x0 H, define a sequence {x n } recursively by

x n + 1 = ( 1 - α n ) T x n + α n f ( x n ) , n 0 ,
(2)

where α n is sequence in (0, 1). Xu [4, 5] proved that under certain appropriate conditions on {α n }, the sequences {x n } generated by (2) strongly converges to the unique solution x* in Fix(T) of the variational inequality:

( f - I ) x * , x - x * 0 , x F i x ( T ) .

Let A is strongly positive operator on C. That is, there is a constant γ ̄ >0 with the property that

A x , x γ ̄ x 2 , x C .

In [5], it is proved that the sequence {x n } generated by the iterative method bellow with initial guess x0 H chosen arbitrarily,

x n + 1 = ( I - α n A ) T x n + α n u , n 0 ,
(3)

converges strongly to the unique solution of the minimization problem

min x F i x ( T ) 1 2 A x , x - x , b ,

where b is a given point in H.

Combining the iterative method (2) and (3), Marino and Xu [6] consider the following iterative method:

x n + 1 = ( I - α n A ) T x n + α n γ f ( x n ) , n 0 ,
(4)

it is proved that if the sequence {α n } of parameters satisfies the following conditions:

(C1) α n → 0,

(C2) n = 0 α n =,

C3) either n = 0 α n + 1 - α n < or lim n α n + 1 α n =1.

then, the sequence {x n } generated by (4) converges strongly, as n → ∞, to the unique solution of the variational inequality:

( γ f - A ) x * , x - x * 0 , x F i x ( T ) ,

which is the optimality condition for minimization problem

min x F i x ( T ) 1 2 A x , x - h ( x ) ,

where h is a potential function for γf (i.e., h'(x) = γf(x), for all x H). Some people also study the application of the iterative method (4) [7, 8].

Yamada [9] introduce the following hybrid iterative method for solving the variational inequality:

x n + 1 = T x n - μ α n F ( T x n ) , n ,
(5)

where F is k-Lipschitzian and η-strongly monotone operator with k > 0, η > 0, 0<μ< 2 η k 2 , then he proved that if {α n } satisfying appropriate conditions, then {x n } generated by (5) converges strongly to the unique solution of the variational inequality:

F x * , x - x * 0 , x F i x ( T ) .

In 2010, Tian [10] combined the iterative (4) with the iterative method (5) and considered the iterative methods:

x n + 1 = ( I - μ α n F ) T x n + α n γ f ( x n ) , n 0 ,
(6)

and he prove that if the sequence {α n } of parameters satisfies the conditions (C1), (C2), and (C3), then the sequences {x n } generated by (6) converges strongly to the unique solution x* Fix(T) of the variational inequality:

( μ F - γ f ) x * , x - x * 0 , x F i x ( T ) .

In this paper motivated and inspired by Atsushiba and Takahashi [11], Ceng and Yao [12], Kim [13], Lau et al. [14], Lau et al [15], Marino and Xu [6], Piri and Vaezi [16], Tian [10], Xu [5] and Yamada [9], we introduce the following general iterative algorithm: Let x0 C and

y n = β n x n + ( 1 - β n ) P C ( x n - δ n B x n ) , x n + 1 = α n γ f ( x n ) + ( I - α n μ F ) T μ n y n , n 0 .
(7)

where P C is a metric projection of H onto C, B is β-inverse strongly monotone, φ = {T t : t S} is a nonexpansive semigroup on H such that the set F=Fix ( φ ) VI ( C , B ) ,, X is a subspace of B(S) such that 1 X and the mapping t → 〈T t x, y〉 is an element of X for each x, y H, and {μ n } is a sequence of means on X. Our purpose in this paper is to introduce this general iterative algorithm for approximating a common element of the set of common fixed point of a semigroup of nonexpansive mappings and the set of solutions of variational inequality for β-inverse strongly monotone mapping which solves some variational inequality. We will prove that if {μ n } is left regular and the sequences {α n }, {β n }, and {δ n } of parameters satisfies appropriate conditions, then the sequences {x n } and {y n } generated by (7) converges strongly to the unique solution x * F of the variational inequalities:

( μ F - γ f ) x * , x - x * 0 , x F , B x * , y - x * 0 y C .

2 Preliminaries

Let S be a semigroup and let B(S) be the space of all bounded real-valued functions defined on S with supremum norm. For s S and f B(S), we define elements l s f and r s f in B(S) by

( l s f ) ( t ) = f ( s t ) , ( r s f ) ( t ) = f ( t s ) , t S .

Let X be a subspace of B(S) containing 1 and let X* be its topological dual. An element μ of X* is said to be a mean on X if || μ || = μ(1) = 1. We often write μ t (f(t)) instead of μ(f) for μ X* and f X. Let X be left invariant (resp. right invariant), i.e., l s (X) X (resp. r s (X) X) for each s S. A mean μ on X is said to be left invariant (resp. right invariant) if μ(l s f) = μ(f) (resp. μ(r s f) = μ(f)) for each s S and f X. X is said to be left (resp. right) amenable if X has a left (resp. right) invariant mean. X is amenable if X is both left and right amenable. As is well known, B(S) is amenable when S is a commutative semigroup, see [15]. A net {μ α } of means on X is said to be strongly left regular if

lim α l s * μ α - μ α = 0 ,

for each s S, where l s * is the adjoint operator of l s .

Let S be a semigroup and let C be a nonempty closed and convex subset of a reflexive Banach space E. A family φ = {T t : t S} of mapping from C into itself is said to be a nonexpansive semigroup on C if T t is nonexpansive and T ts = T t T s for each t, s S. By Fix(φ), we denote the set of common fixed points of φ, i.e.,

F i x ( φ ) = t S { x C : T t x = x } .

Lemma 2.1[15]Let S be a semigroup and C be a nonempty closed convex subset of a reflexive Banach space E. Let φ = {T t : t S} be a nonexpansive semigroup on H such that {T t x : t S} is bounded for some x C, let X be a subspace of B(S) such that 1 X and the mapping t → 〈T t x, y*is an element of X for each x C and y* E*, and μ is a mean on X. If we write T μ x instead of T t xdμ ( t ) , then the followings hold.

(i) T μ is non-expansive mapping from C into C.

(ii) T μ x = x for each x Fix(φ).

(iii) T μ x c o { T t x : t S } for each x C.

Let C be a nonempty subset of a Hilbert space H and T : CH a mapping. Then T is said to be demiclosed at v H if, for any sequence {x n } in C, the following implication holds:

x n u C , T x n v i m p l y T u = v ,

where → (resp. ) denotes strong (resp. weak) convergence.

Lemma 2.2[17]Let C be a nonempty closed convex subset of a Hilbert space H and suppose that T : CH is nonexpansive. Then, the mapping I - T is demiclosed at zero.

Lemma 2.3[18]For a given x H, y C,

y = P C x y - x , z - y 0 , z C .

It is well known that P C is a firmly nonexpansive mapping of H onto C and satisfies

P C x - P C y 2 P C x - P C y , x - y , x , y H
(8)

Moreover, P C is characterized by the following properties: P C x C and for all x H, y C,

x - P C x , y - P C x 0 .
(9)

It is easy to see that (9) is equivalent to the following inequality

x - y 2 x - P C x 2 + y - P C x 2 .
(10)

Using Lemma 2.3, one can see that the variational inequality (24) is equivalent to a fixed point problem.

It is easy to see that the following is true:

u V I ( C , B ) u = P C ( u - λ B u ) , λ > 0 .
(11)

A set-valued mapping U : H → 2 H is called monotone if for all x, y H, f Ux and g Uy imply 〈x - y, f - g〉 ≥ 0. A monotone mapping U : H → 2 H is maximal if the graph of G(U) of U is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping U is maximal if and only if for (x, f) H × H, 〈x - y, f - g〉 ≥ 0 for every (y, g) G(U) implies that f Ux. Let B be a monotone mapping of C into H and let N C x be the normal cone to C at x C, that is, N C x = {y H : 〈z - x, y〉 ≤ 0, z C} and define

U x = B x + N C x , x C , x C .
(12)

Then U is the maximal monotone and 0 Ux if and only if x VI(C, B); see [19].

The following lemma is well known.

Lemma 2.4 Let H be a real Hilbert space. Then, for all x, y H

x - y 2 x 2 + 2 y , x + y , .

Lemma 2.5[5]Let {a n } be a sequence of nonnegative real numbers such that

a n + 1 ( 1 - b n ) a n + b n c n , n 0 ,

where {b n } and {c n } are sequences of real numbers satisfying the following conditions:

(i) {b n } (0, 1), n = 0 b n =,

(ii) either limsup n c n 0or n = 0 b n c n <.

Then, lim n a n =0.

As far as we know, the following lemma has been used implicitly in some papers; for the sake of completeness, we include its proof.

Lemma 2.6 Let H be a real Hilbert space and F be a k-Lipschitzian and η-strongly monotone operator with k > 0, η > 0. Let0<μ< 2 η k 2 andτ=μ ( η - μ k 2 2 ) . Then fort ( 0 , min { 1 , 1 τ } ) , I - t μ F is contraction with constant 1 - t τ.

Proof. Notice that

( I - t μ F ) x - ( I - t μ F ) y 2 = ( I - t μ F ) x - ( I - t μ F ) y , ( I - t μ F ) x - ( I - t μ F ) y = x - y 2 + t 2 μ 2 F x - F y 2 - 2 t μ x - y , F x - F y x - y 2 + t 2 μ 2 k 2 x - y 2 - 2 t μ η x - y 2 x - y 2 + t μ 2 k 2 x - y 2 - 2 t μ η x - y 2 = 1 - 2 t μ η - μ k 2 2 x - y 2 = ( 1 - 2 t τ ) x - y 2 ( 1 - t τ ) 2 x - y 2 .

It follows that

( I - t μ F ) x - ( I - t μ F ) y ( 1 - t τ ) x - y ,

and hence I - tμF is contractive due to 1 - (0, 1). □

Notation Throughout the rest of this paper, F will denote a k-Lipschitzian and η-strongly monotone operator of C into H with k > 0, η > 0, f is a contraction on C with coefficient 0 < α < 1. We will also always use γ to mean a number in ( 0 , τ α ) , where τ=μ ( η - μ k 2 2 ) and 0<μ< 2 η k 2 . The open ball of radius r centered at 0 is denoted by B r and for a subset D of H, by c o D, we denote the closed convex hull of D. For ε > 0 and a mapping T : DH, we let F ε (T; D) be the set of ε-approximate fixed points of T, i.e., F ε (T; D) = {x D : ||x - T x || ≤ ε }. Weak convergence is denoted by and strong convergence is denoted by →.

3 Main results

Theorem 3.1 Let S be a semigroup, C a nonempty closed convex subset of real Hilbert space H and B : CH be a β-inverse strongly monotone. Let φ = {T t : t S} be a nonexpansive semigroup of C into itself such thatF=Fix ( φ ) VI ( C , B ) ,, X a left invariant subspace of B(S) such that 1 X, and the function t → 〈T t x, yis an element of X for each x C and y H, {μ n } a left regular sequence of means on X such that n = 1 μ n + 1 - μ n <. Let {α n } and {β n } be sequences in (0, 1) and {δ n } be a sequence in [a, b], where 0 < a < b < 2β. Suppose the following conditions are satisfied.

(B1) limn→∞α n = 0, limn→∞β n = 0,

(B2) n = 1 α n =,

(B3) n = 1 α n + 1 - α n <, n = 1 β n + 1 - β n <, n = 1 δ n + 1 - δ n <.

If {x n } and {y n } be generated by x0 C and

y n = β n x n + ( 1 - β n ) P C ( x n - δ n B x n ) , x n + 1 = α n γ f ( x n ) + ( I - α n μ F ) T μ n y n , n 0 .

Then, {x n } and {y n } converge strongly, as n → ∞, to x * F, which is a unique solution of the variational inequalities:

( μ F - γ f ) x * , x - x * 0 , x F , B x * , y - x * 0 y C .

Proof. Since {α n } satisfies in condition (B1), we may assume, with no loss of generality, that α n <min { 1 , 1 τ } . Since B is β-inverse strongly monotone and δ n < 2β, for any x, y C, we have

| | ( I δ n B ) x ( I δ n B ) y | | 2 = | | ( x y ) δ n ( B x B y ) | | 2 = | | x y | | 2 2 δ n x y , B x B y + δ n 2 | | B x B y | | 2 | | x y | | 2 2 δ n β | | B x B y | | 2 + δ n 2 | | B x B y | | 2 = | | x y | | 2 + δ n ( δ n 2 β ) | | B x B y | | 2 | | x y | | 2 .

It follows that

( I - δ n B ) x - ( I - δ n B ) y x - y .
(13)

Let pF, in the context of the variational inequality problem, the characterization of projection (11) implies that p = P C (p - δ n B p ). Using (13), we get

y n - p = β n x n + ( 1 - β n ) P C ( x n - δ n B x n ) - p = β n [ x n - p ] + ( 1 - β n ) [ P C ( x n - δ n B x n ) - P C ( p - δ n B p ) ] β n x n - p + ( 1 - β n ) P C ( x n - δ n B x n ) - P C ( p - δ n B p ) β n x n - p + ( 1 - β n ) x n - p = x n - p .
(14)

We claim that {x n } is bounded. Let pF, using Lemma 2.6 and (14), we have

x n + 1 - p = α n γ f ( x n ) + ( I - α n μ F ) T μ n y n - p = α n γ f ( x n ) + ( I - α n μ F ) T μ n y n - ( I - α n μ F ) p - α n μ F p α n γ f ( x n ) - μ F p + ( I - α n μ F ) T μ n y n - ( I - α n μ F ) p α n γ f ( x n ) - γ f ( p ) + α n γ f ( p ) - μ F p + ( 1 - α n τ ) T μ n y n - p α n γ α x n - p + α n γ f ( p ) - μ F p + ( 1 - α n τ ) y n - p α n γ α x n - p + α n γ f ( p ) - μ F p + ( 1 - α n τ ) x n - p = ( 1 - α n ( τ - γ α ) ) x n - p + α n γ f ( p ) - μ F p max { x n - p , ( τ - γ α ) - 1 γ f ( p ) - μ F p } .

By induction we have,

| | x n p | | max { ( τ γ α ) 1 | | γ f ( p ) μ F p | | , | | x 0 p | | } = M 0 .

Hence, {x n } is bounded and also {y n } and {f(x n )} are bounded. Set D = {y H : ||y - p||≤ = M0}. We remark that D is φ-invariant bounded closed convex set and {x n }, {y n } D. Now we claim that

limsup n sup y D T μ n y - T t T μ n y = 0 , t S .
(15)

Let ε > 0. By [[20], Theorem 1.2], there exists δ > 0 such that

c o F δ ( T t ; D ) + B δ F ε ( T t ; D ) , t S .
(16)

Also by [[20], Corollary 1.1], there exists a natural number N such that

1 N + 1 i = 0 N T t i s y - T t 1 N + 1 i = 0 N T t i s y δ ,
(17)

for all t, s S and y D. Let t S. Since {μ n } is strongly left regular, there exists N0 such that μ n - l t i * μ n δ ( M 0 + p ) for nN0 and i = 0, 1, 2,..., N. Then we have

sup y D T μ n y 1 N + 1 i = 0 N T t i s y d μ n s = sup y D sup z = 1 | T μ n y , z 1 N + 1 i = 0 N T t i s y d μ n s , z | = sup y D sup z = 1 | 1 N + 1 i = 0 N ( μ n ) s T s y , z 1 N + 1 i = 0 N ( μ n ) s T t i s y , z | 1 N + 1 i = 0 N sup y D sup z = 1 | ( μ n ) s T s y , z ( l t i * μ n ) s T s y , z | max i = 1,2, N μ n l t i * μ n ( M 0 + p ) δ , n N 0 .
(18)

By Lemma 2.1, we have

1 N + 1 i = 0 N T t i s y d μ n s c o 1 N + 1 i = 0 N T t i ( T s y ) : s s .
(19)

It follows from (16), (17), (18), and (19) that

T μ n ( y ) c o 1 N + 1 i = 0 N T t i s ( y ) : s S + B δ c o F δ ( T t ; D ) + B δ F ε ( T t ; D ) ,

for all y D and nN0. Therefore,

lim sup n sup y D T t ( T μ n y ) - T μ n y ε .

Since ε > 0 is arbitrary, we get (15). In this stage, we will show

lim n x n - T t x n = 0 , t S .
(20)

Let t S and ε > 0. Then, there exists δ > 0, which satisiies (16). From limn→∞α n = 0 and (15) there exists N1 such that α n δ ( τ + μ k ) M 0 and T μ n y n F δ ( T t ; D ) , for all nN1. By Lemma 2.6 and (14), we have

α n γ f ( x n ) - μ F T μ n y n α n ( γ f ( x n ) - f ( p ) + γ f ( p ) - μ F p + μ F p - μ F T μ n y n ) α n ( γ α x n - p + γ f ( p ) - μ F p + μ k y n - p ) α n ( γ α M 0 + ( τ - γ α ) M 0 + μ k M 0 ) α n ( τ + μ k ) M 0 δ ,

for all nN1. Therefore, we have

x n + 1 = T μ n y n + α n [ γ f ( x n ) + μ F ( T μ n y n ) ] F δ ( T t ; D ) + B δ F ε ( T t ; D ) ,

for all nN1. This shows that

x n - T t x n ε , n N 1 .

Since ε > 0 is arbitrary, we get (20).

Let

Q= P F . Then Q(I - μF + γ f) is a contraction of H into itself. In fact, we see that

Q ( I - μ F + γ f ) x - Q ( I - μ F + γ f ) y ( I - μ F + γ f ) x - ( I - μ F + γ f ) y ( I - μ F ) x - ( I - μ F ) y + γ f ( x ) - f ( y ) = lim n I - 1 - 1 n μ F x - I - 1 - 1 n μ F y + γ f ( x ) - f ( y ) lim n ( 1 - ( 1 - 1 n ) τ ) x - y + γ α x - y = ( 1 - τ ) x - y + γ α x - y ,

and hence Q(I - μF + γ f) is a contraction due to (1 - (τ -γα)) (0, 1).

Therefore, by Banachs contraction principal, P F ( I - μ F + γ f ) has a unique fixed point x*. Then using (9), x* is the unique solution of the variational inequality:

( γ f - μ F ) x * , x - x * 0 , x F .
(21)

We show that

limsup n γ f ( x * ) - μ F x * , x n - x * 0 .
(22)

Indeed, we can choose a subsequence { x n i } of {x n } such that

limsup n γ f ( x * ) - μ F x * , x n - x * = lim i γ f ( x * ) - μ F x * , x n i - x * .
(23)

Because { x n i } is bounded, we may assume that x n i z. In terms of Lemma 2.2 and (20), we conclude that z Fix (φ).

Now, let us show that z VI (C, B). Let w n = P C (x n - δ n Bx n ), it follows from the definition of {y n } that

y n + 1 y n = β n + 1 x n + 1 + ( 1 β n + 1 ) P C ( x n + 1 δ n + 1 B x n + 1 ) β n x n ( 1 β n ) P C ( x n δ n B x n ) = β n + 1 ( x n + 1 x n ) + ( β n + 1 β n ) x n + ( 1 β n + 1 ) P C ( x n + 1 δ n + 1 B x n + 1 ) ( 1 β n + 1 ) P C ( x n δ n + 1 B x n ) + ( 1 β n + 1 ) P C ( x n δ n + 1 B x n ) ( 1 β n ) P C ( x n δ n B x n ) β n + 1 x n + 1 x n + | β n + 1 β n | x n + ( 1 β n + 1 ) P C ( x n + 1 δ n + 1 B x n + 1 ) P C ( x n δ n + 1 B x n ) + P C ( x n δ n + 1 B x n ) P C ( x n δ n B x n ) + β n P C ( x n δ n B x n ) β n + 1 P C ( x n δ n + 1 B x n ) ] β n + 1 x n + 1 x n + | β n + 1 β n | x n + ( 1 β n + 1 ) x n + 1 x n + | δ n + 1 δ n | B x n + β n P C ( x n δ n B x n ) β n P C ( x n δ n + 1 B x n ) + β n P C ( x n δ n + 1 B x n ) β n + 1 P C ( x n δ n + 1 B x n ) β n + 1 x n + 1 x n + | β n + 1 β n | x n + ( 1 β n + 1 ) x n + 1 x n + | δ n + 1 δ n | B x n + β n | δ n + 1 δ n | B x n + | β n + 1 β n | P C ( x n δ n + 1 B x n ) = x n + 1 x n + | β n + 1 β n | x n + ( 1 + β n ) | δ n + 1 δ n | B x n + | β n + 1 β n | | | P C ( x n δ n + 1 B x n ) | | .

Using the last inequality, we get

x n + 1 - x n = α n γ f ( x n ) + ( I - α n μ F ) T μ n y n - α n - 1 γ f ( x n - 1 ) - ( I - α n - 1 μ F ) T μ n - 1 y n - 1 = α n γ f ( x n ) - α n γ f ( x n - 1 ) + ( α n - α n - 1 ) γ f ( x n - 1 ) + ( I - α n μ F ) T μ n y n - ( I - α n μ F ) T μ n - 1 y n - 1 + ( I - α n μ F ) T μ n - 1 y n - 1 - ( I - α n - 1 μ F ) T μ n - 1 y n - 1 α n γ α x n - x n - 1 + α n - α n - 1 γ f ( x n - 1 ) + ( 1 - α n τ ) T μ n y n - T μ n - 1 y n - 1 + α n - α n - 1 μ F T μ n - 1 y n - 1 α n γ α x n - x n - 1 + α n - α n - 1 γ f ( x n - 1 ) + ( 1 - α n τ ) y n - y n - 1 + ( 1 - α n τ ) T μ n y n - 1 - T μ n - 1 y n - 1 + α n - α n - 1 μ F T μ n - 1 y n - 1 α n γ α x n - x n - 1 + α n - α n - 1 γ f ( x n - 1 ) + ( 1 - α n τ ) x n - x n - 1 + ( 1 - α n τ ) β n - β n - 1 x n - 1 + ( 1 - α n τ ) ( 1 + β n - 1 ) δ n - δ n - 1 B x n - 1 + ( 1 - α n τ ) β n - β n - 1 P C ( x n - 1 - δ n B x n - 1 ) + ( 1 - α n τ ) T μ n y n - 1 - T μ n - 1 y n - 1 + α n - α n - 1 μ F T μ n - 1 y n - 1 .

Thus, for some large enough constant M > 0, we have

x n + 1 - x n ( 1 - α n ( τ - γ α ) ) x n - x n - 1 + [ α n - α n - 1 + β n - β n - 1 + δ n - δ n - 1 + μ n - μ n - 1 ] M .

Therefore, using condition B3 and Lemma 2.5, we get

lim n x n + 1 - x n = 0 .
(24)

Let pF, from (11) and deiinition of {y n }, we have

y n p 2 = β n x n + ( 1 β n ) P C ( x n δ n B x n ) p 2 = β n ( x n p ) + ( 1 β n ) ( P C ( x n δ n B x n ) P C ( p δ n B p ) ) 2 β n x n p 2 + ( 1 β n ) ( x n p ) δ n ( B x n B p ) ) 2 = β n x n p 2 + ( 1 β n ) x n p 2 + δ n 2 ( 1 β n ) B x n B p 2 2 δ n ( 1 β n ) x n p , B x n B p x n p 2 + δ n 2 ( 1 β n ) B x n B p 2 2 δ n ( 1 β n ) β B x n B p 2 = x n p 2 + δ n ( 1 β n ) ( δ n 2 β ) B x n B p 2 .
(25)

Using (25), we have

x n + 1 - p 2 = α n γ f ( x n ) + ( I - α n μ F ) T μ n y n - p 2 = α n ( γ f ( x n ) - μ F T μ n y n ) + ( T μ n y n - p ) 2 = α n 2 γ f ( x n ) - μ F T μ n y n 2 + T μ n y n - p + 2 α n γ f ( x n ) - μ F T μ n y n , T μ n y n - p α n 2 γ f ( x n ) - μ F T μ n y n 2 + y n - p 2 + 2 α n γ f ( x n ) - μ F T μ n y n , T μ n y n - p 2 α n 2 γ f ( x n ) - μ F T μ n y n 2 + x n - p 2 + δ n ( 1 - β n ) ( δ n - 2 β ) B x n - B p 2 + 2 α n γ f ( x n ) - μ F T μ n y n , T μ n y n - p = α n 2 γ f ( x n ) - μ F T μ n y n 2 + x n - p 2 + δ n ( δ n - 2 β n ) B x n - B p 2 - δ n β n ( δ n - 2 β n ) B x n - B p 2 + 2 α n γ f ( x n ) - μ F T μ n y n , T μ n y n - p .
(26)

Therefore,

- δ n ( δ n - 2 β n ) B x n - B p 2 α n 2 γ f ( x n ) - μ F T μ n y n 2 + [ x n - p + x n + 1 - p ] x n + 1 - x n - δ n β n ( δ n - 2 β n ) B x n - B p 2 + 2 α n γ f ( x n ) - μ F T μ n y n , T μ n y n - p .

Hence, using condition B1 and (24), we get

lim n B x n - B p = 0 .
(27)

From (8), we have

w n p 2 = P C ( x n δ n B x n ) P C ( p δ n B p ) 2 ( x n δ n B x n ) ( p δ n B p ), w n p = 1 2 [ ( x n δ n B x n ) ( p δ n B p ) 2 + w n p 2 ( x n δ n B x n ) ( p δ n B p ) ( w n p ) 2 ] 1 2 [ x n p 2 + w n p 2 ( x n δ n B x n ) ( p δ n B p ) ( w n p ) 2 ] = 1 2 [ x n p 2 + w n p 2 x n w n 2 + 2 δ n x n w n , B x n B p δ n 2 B x n B p 2 ].

So we obtain

w n - p 2 x n - p 2 - x n - w n 2 + 2 δ n x n - w n , B x n - B p - δ n 2 B x n - B p 2 .
(28)

It follows from (26) and (28) that

x n + - p 2 α n 2 γ f ( x n ) - μ F T μ n y n 2 + y n - p 2 + 2 α n γ f ( x n ) - μ F T μ n y n , T μ n y n - p α n 2 γ f ( x n ) - μ F T μ n y n 2 + β n x n + ( 1 - β n ) P C ( x n - δ n B x n ) - p 2 + 2 α n γ f ( x n ) - μ F T μ n y n , T μ n y n - p α n 2 γ f ( x n ) - μ F T μ n y n 2 + β n x n - p 2 + ( 1 - β n ) w n - p 2 + 2 α n γ f ( x n ) - μ F T μ n y n , T μ n y n - p α n 2 γ f ( x n ) - μ F T μ n y n 2 + β n x n - p 2 + ( 1 - β n ) x n - p 2 - ( 1 - β n ) x n - w n 2 + 2 δ n ( 1 - β n ) x n - w n , B x n - B p - δ n 2 ( 1 - β n ) B x n - B p 2 + 2 α n γ f ( x n ) - μ F T μ n y n , T μ n y n - p .

Which implies that

x n - w n 2 α n 2 γ f ( x n ) - μ F T μ n y n 2 + [ x n - p + x n + 1 - p ] x n + 1 - x n + β n x n - w n 2 + 2 δ n ( 1 - β n ) x n - w n B x n - B p - δ n 2 ( 1 - β n ) B x n - B p 2 + 2 α n γ f ( x n ) - μ F T μ n y n , T μ n y n - p .

Therefore, using condition B1, (24) and (27), we get

lim n x n - w n = 0 .
(29)

Let U : H →2 H be a set-valued mapping is defined by

U x = A x + N C x , x C , , x C .

where N C x is the normal cone to C at x C. Since B is relaxed, β-inverse strongly monotone. Thus, U is maximal monotone see [19]. Let (x, y) G (U), hence y - Bx N C x. Since w n = P C (x n - ζ n Bx n ) therefore, 〈x - w n , y - Bx〉 ≥ 0. On the other hand from w n = P C (x n - ζ n Bx n ), we have

x - w n , w n - ( x n - δ n B x n ) 0 ,

that is

x - w n , w n - x n δ n + B x n 0 .

Therefore, we have

x w n i , y x w n i , B x x w n i , B x x w n i , w n i x n i δ n i + B x n i ) = x w n i , B x w n i x n i δ n i B x n i = x w n i , B x B w n i + x w n i , B w n i B x n i x w n i , w n i x n i δ n i x w n i , B w n i B x n i x w n i , w n i x n i δ n i x w n i , B w n i B x n i x w n i | | w n i x n i δ n i .

Noting that lim i w n i - x n i =0, x n i z, x n i z and B is 1 β -lipschitzian, we obtain

x - z , y 0 .

Since U is maximal monotone, we have z U-10, and hence z VI(C, B). Therefore, zF.

Since x n i z from (21) and (23), we have

limsup n γ f ( x * ) - μ F x * , x n - x * 0 .

Finally, we prove that x n x* as n → ∞. By Lemmas 2.4, 2.6, and (14), we have

x n + 1 - x * 2 = α n γ f ( x n ) + ( I - α n μ F ) T μ n y n - x * 2 = α n γ f ( x n ) - α n μ F x * + ( I - α n μ F ) T μ n y n - ( I - α n μ F ) x * 2 ( I - α n μ F ) T μ n y n - ( I - α n μ F ) x * 2 + 2 α n γ f ( x n ) - μ F x * , x n + 1 - x * ( 1 - α n τ ) 2 y n - x * 2 + 2 α n γ f ( x n ) - μ F x * , x n + 1 - x * ( 1 - α n τ ) 2 y n - x * 2 + 2 α n γ f ( x n ) - γ f ( x * ) , x n + 1 - x * + 2 α n γ f ( x * ) - μ F x * , x n + 1 - x * . ( 1 - α n τ ) 2 y n - x * 2 + α n γ α [ x n - x * 2 + x n + 1 - x * 2 ] + 2 α n γ f ( x * ) - μ F x * , x n + 1 - x * . ( 1 - α n τ ) 2 x n - x * 2 + α n γ α [ x n - x * 2 + x n + 1 - x * 2 ] + 2 α n γ f ( x * ) - μ F x * , x n + 1 - x * .
(30)

So from (30), we reach the following

x n + 1 - x * 2 1 + α 2 τ 2 - 2 α n τ + α n γ α 1 - α n γ α x n - x * 2 + 2 α n 1 - α n γ α γ f ( x * ) - μ F x * , x n + 1 - x * ( 1 - α n 2 ( τ - γ α ) - α n τ 2 1 - α n γ α ) x n - x * 2 + α n 2 ( τ - γ α ) - α n τ 2 1 - α n γ α 2 2 ( τ - γ α ) - α n τ 2 γ f ( x * ) - μ F x * , x n + 1 - x *

It follows that

x n + 1 - x * 2 ( 1 - b n ) x n - x * 2 + b n c n ,
(31)

where

b n = α n 2 ( τ - γ α ) - α n τ 2 1 - α n γ α , c n = 2 2 ( τ - γ α ) - α n τ 2 γ f ( x * ) - μ F x * , x n + 1 - x *

Since α n → o and n = 0 α n =, we have n = 0 b n = and by (22), we get lim supn→∞c n ≤ 0. Consequently, applying Lemma 2.5, to (31), we conclude that xnx*. Since || y n - x* || ≤ || x n - x* ||, we have ynx*. □

Corollary 3.2 Let {α n }, {β n }, {δ n } and B be as in Theorem 3.1. Let T a nonexpansive mapping of C into C such thatF=Fix ( T ) VI ( C , B ) . Suppose x0 H and {x n } and {y n } be generated by the iteration algorithm

y n = β n x n + ( 1 - β n ) P C ( x n - δ n B x n ) , x n + 1 = α n γ f ( x n ) + ( I - α n μ F ) n n - 1 0 n - 1 n T ( t ) y n d t , n 0 .

Then {x n } and {y n } convergence strongly to x* which is the unique solution of the systems of variational inequalities:

( μ F - γ f ) x * , x - x * 0 , x F , B x * , y - x * 0 y C ,

Proof. Take λ n = n - 1 n , for n , we define μ n ( f ) = 1 λ n 0 λ n f ( t ) d t for each f C(+), where C(+) denotes the space of all real-valued bounded continuous functions on R+ with supremum norm. Then, {μ n } is regular sequence of means on C(+) such that

μ n + 1 - μ n 2 1 - λ n λ n + 1

for more details, see [21]. Further, for each y C, we have

T μ n y = 1 λ n 0 λ n T ( t ) y d t .

On the other hand

n = 1 μ n + 1 - μ n 2 n = 1 λ n + 1 - λ n λ n + 1 = 2 n = 1 n n + 1 - n - 1 n n n + 1 = 2 n = 1 1 n 2 <

Now, apply Theorem 3.1 to conclude the result. □

Corollary 3.3 Let S, φ, X, {μ n }, F, {α n }, {β n }, and {δ n } be as in Theorem 3.1. Let A be a strongly positive bounded linear operator with coefficient γ ̄ >0, ζ a number in ( 0 , τ ̄ α ) , where τ ̄ = μ ̄ ( γ ̄ - μ ̄ A 2 2 ) and0< μ ̄ < 2 γ ̄ A 2 . If {x n } and {y n } are generated by x0 C and

y n = β n x n + ( 1 - β n ) P C ( x n - δ n A x n ) , x n + 1 = α n γ f ( x n ) + ( I - α n μ ̄ A ) T μ n y n , n 0 .

Then, {x n } and {y n } converge strongly, as n → ∞, to x * F, which is a unique solution of the variational inequalities:

( μ F - γ f ) x * , x - x * 0 , x F , A x * , y - x * 0 y C .

Proof. Because A is strongly positive bounded linear operator on H with coefficient γ ̄ , we have

A x - A y , x - y γ ̄ x - y 2 .

Therefore, A is γ ̄ -strongly monotone.

On the other hand

A x - A y A x - y .

Therefore,

γ ̄ A 2 A x - A y 2 A x - A y , x - y .

Hence, A is γ ̄ A 2 -inverse strongly monotone. Now apply Theorem 3.1 to conclude the result. □

Corollary 3.4 Let {α n }, {β n } and B be as in Theorem 3.1. Let u, x0 C and {x n } and {y n } be generated by the iterative algorithm

y n = β n x n + ( 1 - β n ) P C ( x n - δ n B x n ) , x n + 1 = α n u + ( I - α n μ F ) y n , n 0 .

Then {x n } and {y n } convergence strongly to x* which is the unique solution of the systems of variational inequalities:

( μ F - γ f ) x * , x - x * 0 , x F , B x * , y - x * 0 y C .

Proof. It is sufficient to take f= 1 γ u and φ = {I} in Theorem 3.1. □

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