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Iterative algorithms approach to a general system of nonlinear variational inequalities with perturbed mappings and fixed point problems for nonexpansive semigroups

Journal of Inequalities and Applications20122012:133

https://doi.org/10.1186/1029-242X-2012-133

Received: 10 February 2012

Accepted: 11 June 2012

Published: 11 June 2012

Abstract

In this paper, we introduce new iterative algorithms for finding a common element of the set of solutions of a general system of nonlinear variational inequalities with perturbed mappings and the set of common fixed points of a one-parameter nonexpansive semigroup in Banach spaces. Furthermore, we prove the strong convergence theorems of the sequence generated by these iterative algorithms under some suitable conditions. The results obtained in this paper extend the recent ones announced by many others.

Mathematics Subject Classification (2010): 47H09, 47J05, 47J25, 49J40, 65J15

Keywords

one-parameter nonexpansive semigroupperturbed mappingiterative algorithmvariational inequalitystrong convergenceBanach spacecommon fixed points

1 Introduction

Variational inequality theory has been studied widely in several branches of pure and applied sciences. Indeed, applications of variational inequalities span as diverse disciplines as differential equations, time-optimal control, optimization, mathematical programming, mechanics, finance, and so on (see, e.g., [1, 2] for more details). Note that most of the variational problems include minimization or maximization of functions, variational inequality problems, quasivariational inequality problems, decision and management sciences, and engineering sciences problems. For more details, we recommend the reader [38, 2931].

Let X be a real Banach space, and X* be its dual space. The duality mapping J : X 2 X * is defined by
J x = f X * : x , f = | | x | | 2 , | | f | | = | | x | | ,

where 〈·, ·〉 denotes the duality pairing between X and X*. If X : = H is a real Hilbert space, then J = I where I is the identity mapping. It is well known that if X is smooth, then J is single-valued, which is denoted by j (see [9]).

Let C be a nonempty closed and convex subset of X and T be a self-mapping of C. We denote → and by strong and weak convergence, respectively. Recall that a mapping T : C → C is said to be L-Lipschitzian if there exists a constant L > 0 such that
| | T x - T y | | L | | x - y | | , x , y C .

If 0 < L < 1, then T is a contraction and if L = 1, then T is a nonexpansive mapping. We denote by Fix(T) the set of all fixed points set of the mapping T, i.e., Fix(T) = {x C : Tx = x}.

A mapping F : CX is said to be accretive if there exists j(x - y) J(x - y) such that
F x - F y , j x - y 0 , x , y C .
A mapping F : CX is said to be strongly accretive if there exists a constant η > 0 and j(x - y) J(x - y) such that
F x - F y , j x - y η | | x - y | | 2 , x , y C .

Remark 1.1. If X : = H is a real Hilbert space, accretive and strongly accretive mappings coincide with monotone and strongly monotone mappings, respectively.

Let H be a real Hilbert space, whose inner product and norm are denoted by〈·, ·〉 and ||·||, respectively. Let A be a strongly positive bounded linear operator on H, that is, there exists a constant γ ¯ > 0 such that
A x , x γ ¯ | | x | | 2 , x H .
(1.1)

Remark 1.2. From the definition of operator A, we note that a strongly positive bounded linear operator A is a ||A||-Lipschitzian and η-strongly monotone operator.

Let C be a nonempty closed and convex subset of a real Banach space X. Recall that the classical variational inequality is to find x* C such that
Ψ x * , j x - x * 0 , x C ,
(1.2)
where Ψ: CX is a nonlinear mapping and j(x - x*) J(x - x*). The set of solution of variational inequality is denoted by VI(C, Ψ). If X : = H is a real Hilbert space, then (1.2) reduces to find x* C such that
Ψ x * , x - x * 0 , x C .
(1.3)
A typical problem is to minimize a quadratic function over the set of the fixed points of a nonex-pansive mapping on a real Hilbert space H
min x C 1 2 A x , x - x , u ,
(1.4)

where C is the fixed point set of a nonexpansive mapping T on H and u is a given point in H.

In 2001, Yamada [10] introduced a hybrid steepest descent method for a nonexpansive mapping T as follows:
x n + 1 = T x n - μ λ n F T x n , n 0 ,
(1.5)
where F is a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0 and 0 < μ < 2 η κ 2 . He proved that if {λ n } satisfying appropriate conditions, then the sequence {x n }generated by (1.5) converges strongly to the unique solution of variational inequality
F x * , x - x * 0 , x F i x T .
(1.6)
In 2006, Marino and Xu [11] introduced and considered the following general iterative method:
x n + 1 = α n γ f x n + I - α n A T x n , n 0 ,
(1.7)
where A is a strongly positive bounded linear operator on a real Hilbert space H. They, proved that, if the sequence {α n } of parameters satisfies appropriate conditions, then the sequence {x n } generated by (1.7) converges strongly to the unique solution of the variational inequality
γ f - A x * , x - x * 0 , x Fix T ,
(1.8)
which is the optimality condition for the minimization problem
min x C 1 2 A x , x - h x ,
(1.9)

where C is the fixed point set of a nonexpansive mapping T and h is a potential function for γf (i.e., h'(x) = γf(x) for all x H).

Recently, Tian [12] combined the iterative method (1.7) with the Yamada's method (1.5) and considered the general iterative method for a nonexpansive mapping T as follows:
x n + 1 = α n γ f x n + I - α n μ F T x n , n 0 .
(1.10)
Then, he proved that the sequence {x n } generated by (1.10) converges strongly to the unique solution of variational inequality
γ f - μ F x * , x - x * 0 , x Fix T .
(1.11)
Let Ψ1, Ψ2 : CX be two mappings. Yao et al. [7] considered the following problem of finding (x*, y*) C × C such that
ρ 1 Ψ 1 y * + x * - y * , j x - x * 0 , x C , ρ 2 Ψ 2 x * + y * - x * , j x - y * 0 , x C ,
(1.12)
which is called a general system of nonlinear variational inequalities in Banach spaces, where ρ1 > 0 and ρ2 > 0 are two constants. In particular, if ρ1 = 1 and ρ2 = 1 then problem (1.12) reduces to problem of finding (x*, y*) C × C such that
Ψ 1 y * + x * - y * , j x - x * 0 , x C , Ψ 2 x * + y * - x * , j x - y * 0 , x C ,
(1.13)

which is defined by Yao et al. [13].

Very recently, Yao et al. [7] introduced an iterative algorithm for solving the problem (1.12). To be more precise, they proved the following theorem.

Theorem YLKY[7]Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space X and let Q C be a sunny nonexpansive retraction from X onto C. Let the mappings Ψ1, Ψ2 : C → X be α-inverse-strongly accretive and β-inverse-strongly accretive, respectively. Let A : C → X be a strongly positive linear bounded operator with coefficient γ ¯ > 0 . Let Ω: = VI(C, Ψ1) ∩ VI(C, Ψ2). For given x0 C, let the sequence {x n } be generated by
z n = Q C x n - ρ 2 Ψ 2 x n , y n = Q C z n - ρ 1 Ψ 1 z n , x n + 1 = β n x n + 1 - β n Q C I - α n A y n , n 0 .
(1.14)

Suppose that {α n } and {β n } are sequences in [0, 1] satisfying the following conditions:

(C1) lim n→∞ α n = 0 and n = 1 α n = ;

(C2) 0 < lim inf n→∞ β n lim sup n→∞ β n < 1.

Then, {x n } converges strongly to x ^ Ω which solves the variational inequality (1.12).

On the other hand, motivated and inspired by the idea of Tian [12] and Yao et al. [7], we consider and introduce the following system of variational inequalities in Banach spaces: Let C be a nonempty closed and convex subset of a Banach space X. Let Ψ i , Φ i : CX (i = 1, 2) be a mapping. First, we consider the following problem of finding (x*, y*) C × C such that
ρ 1 Ψ 1 + Φ 1 y * + x * - y * , j x - x * 0 , x C , ρ 2 Ψ 2 + Φ 2 x * + y * - x * , j x - y * 0 , x C ,
(1.15)

which is called a general system of nonlinear variational inequalities with perturbed mapping in Banach spaces, where ρ1 > 0 and ρ2 > 0 are two constants. In particular, if Φ1 = Φ2 = 0 then problem (1.15) reduces to problem (1.12). Further, if Φ1 = Φ2 = 0 and ρ1 = ρ2 = 1 then problem (1.15) reduces to problem (1.13). Second, we introduce iterative algorithms (3.15) below for finding a common element of the set of solutions of a general system of nonlinear variational inequalities with perturbed mappings (1.15) and the set of common fixed points of a one-parameter nonexpansive semigroup in Banach spaces. Furthermore, we show that our iterative algorithm converges strongly to a common element of the two aforementioned sets under some suitable conditions. Our results extend the main result of Tian [12] and Yao et al. [7] and the methods of the proof in this paper are also new and different.

2 Preliminaries

Let U = {x X : ||x|| = 1}. A Banach space X is said to be strictly convex if | | x + y | | 2 < 1 for all x, y U with xy. A Banach space X is called uniformly convex if for each ε > 0 there is a δ > 0 such that for x, y X with ||x||, ||y|| ≤ 1 and ||x - y|| ≥ ε, ||x + y|| ≤ 2(1 - δ) holds. The modulus of covexity of X defined by
δ X ϵ = inf 1 - | | 1 2 x + y | | : | | x | | , | | y | | 1 , | | x - y | | ϵ ,
for all ε [0 2]. It is known that every uniformly convex Banach space is strictly convex and reflexive [9]. The norm of X is said to be Gâteaux differentiable if the limit
lim t 0 | | x + t y | | - | | x | | t
(2.1)

exists for each x, y U. In this case X is smooth. The norm of X is said to be Fréchet differentiable if for each x U, the limit (2.1) is attained uniformly for y U. The norm of X is called uniformly Fréchet differentiable if the limit (2.1) is attained uniformly for x, y U. It is well known that (uniform) Fréchet differentiability of the norm of X implies (uniform) Gâteaux differentiability of the norm of X.

Let ρ X : [0, ) → [0, ) be the modulus of smoothness of X defined by
ρ X τ = sup 1 2 | | x + y | | + | | x - y | | - 1 : x U , | | y | | τ .

A Banach space X is said to be uniformly smooth if ρ X t t 0 as t→ 0. Suppose that q > 1, then X is said to be q-uniformly smooth if there exists c > 0 such that ρ X (t) ≤ ct q . It is easy to see that if X is q-uniformly smooth, then q ≤ 2 and X is uniformly smooth. It is well known that X is uniformly smooth if and only if the norm of X is uniformly Fré chet differentiable and hence the norm of X is Fré chet differentiable, in particular, the norm of X is Fré chet differentiable. Typical examples of both uniformly convex and uniformly smooth Banach spaces are L p , where p > 1. More precisely, L p is min{p, 2}-uniformly smooth for every p > 1.

Definition 2.1. A one-parameter family S = T t : t > 0 from C into itself is said to be a nonexpansive semigroup on C if it satisfies the following conditions:
  1. (i)

    T(0)x = x for all x C;

     
  2. (ii)

    T(s + t)x = T(s)T(t)x for all x C and s, t > 0;

     
  3. (iii)

    for each x C the mapping t T(t)x is continuous;

     
  4. (iv)

    ||T(t)x - T(t)y|| ≤ ||x - y|| for all x, y C and t > 0.

     

Remark 2.2. We denote by Fix S the set of all common fixed points of  S , that is Fix S  : =  t > 0 Fix T t = x C : T t x = x . We know that Fix S is nonempty if C is bounded [14].

Now, we present the concept of a uniformly asymptotically regular semigroup [1517].

Definition 2.3. Let C be a nonempty closed and convex subset of a Banach space X , S = T t : t > 0 be a continuous operator semigroup on C. Then  S is said to be uniformly asymptotically regular (in short, u.a.r.) on C if for all h ≥ 0 and any bounded subset B of C such that
lim t sup x B T h T t x - T t x = 0 .

The nonexpansive semigroup {σ t : t > 0} defined by the following lemma is an example of u.a.r. operator semigroup. Other examples of u.a.r. operator semigroup can be found in [15].

Lemma 2.4. [18]Let C be a nonempty closed and convex subset of a uniformly convex Banach space X, B be a bounded closed and convex subset of C. If we denote S = T t : t > 0 is a nonexpansive semi-group on C such that Fix S = t > 0 Fix T t . For all h ≥ 0, the set σ t x = 1 t 0 t T s x d s , then
lim t sup x B σ t x - T h σ t x = 0 .
Example 2.5. The set {σ t : t > 0} defined by Lemma 2.4 is u.a.r. nonexpansive semigroup. In fact, it is obvious that {σ t : t > 0} is a nonexpansive semigroup. For each h > 0, we have
| | σ t x - σ h σ t x | | = | | σ t x - 1 h 0 h T s σ t x d s | | = 1 h 0 h σ t x - T s σ t x d s 1 h 0 h σ t x - T s σ t x d s .
By Lemma 2.4, we obtain that
lim t sup x B σ t x - σ h σ t x 1 h 0 h lim t sup x B σ t x - T s σ t x d s = 0 .
Let D be a nonempty subset of C. A mapping Q : CD is said to be sunny[19] if
Q Q x + t x - Q x = Q x ,

whenever Qx + t(x - Qx) C for x C and t ≥ 0. A mapping Q : CD is said to be retraction if Qx = x for all x D. Furthermore, Q is a sunny nonexpansive retraction from C onto D if Q is a retraction from C onto D which is also sunny and nonexpansive. A subset D of C is called a sunny nonexpansive retraction of C if there exists a sunny nonexpansive retraction from C onto D. It is well known that if X : = H is a real Hilbert space, then a sunny nonexpansive retraction Q C is coincident with the metric projection from X onto C. The following lemmas concern the sunny nonex-pansive retraction.

Lemma 2.6. [19]Let C be a closed and convex subset of a smooth Banach space X. Let D be a nonempty subset of C. Let Q : C → D be a retraction and let J be the normalized duality mapping on X. Then the following are equivalent:
  1. (a)

    Q is sunny and nonexpansive.

     
  2. (b)

    ||Qx - Qy||2 x - y, J(Qx - Qy)〉, x, y C.

     
  3. (c)

    x - Qx, J(y - Qx)〉 0, x C, y D.

     

Lemma 2.7. [20]If X is a strictly convex and uniformly smooth Banach space and if T : C → C is a nonexpansive mapping having a nonempty fixed point set Fix(T), then the set Fix(T) is a sunny nonexpansive retraction of C.

Let be the set of positive integers and let l be the Banach space of bounded valued functions on with supremum norm. Let LIM be a linear continuous functional on l and let x = (a1, a2,...) l . Then sometimes, we denote by LIM n (a n ) the value of LIM(x). We know that there exists a linear continuous functional LIM on l such that LIM = LIM(1) = 1 and LIM(a n ) = LIM(an+1) for each x = (a1, a2,...) l . Such a LIM is called a Banach limit. Let LIM be a Banach limit. Then
lim inf n a n L I M x lim sup n a n ,
Let l be a Banach space of all bounded real-valued sequences. A Banach limit LIM n [9] is a linear continuous functional on l such that
| | LIM | | = 1 , lim inf n a n LI M n a n lim sup n a n ,

for each x = (a1, a2,...) l . Specially, if a n a, then LIM(x) = a[9].

In order to prove our main results, we need the following lemmas.

Lemma 2.8. [21]Let X be a real 2-uniformly smooth Banach space with the best smoothness constant K > 0. Then the following inequality holds:
x + y 2 x 2 + 2 y , J x + 2 K y 2 , x , y X .
Lemma 2.9. [22]In a real Banach space X, the following inequality holds:
| | x + y | | 2 | | x | | 2 + 2 y , j x + y , x , y X ,

where j(x + y) J(x + y).

Lemma 2.10. [23]Let {x n } and {l n } be bounded sequences in a Banach space X and let {β n } be a sequence in [0, 1] with 0 < lim inf n→∞ β n lim sup n→∞ β n < 1. Suppose xn+1= (1 - β n )l n + β n x n for all integers n ≥ 0 and lim sup n→∞ (||ln+1- l n || - ||xn+1- x n ||) 0. Then, lim n→∞ ||l n - x n || = 0.

Lemma 2.11. [24]Let C be a closed and convex subset of a strictly convex Banach space X. Let T1and T2be two nonexpansive mappings from C into itself with Fix(T1) Fix(T2) = Ø. Define a mapping S by
S x = δ T 1 x + 1 - δ T 2 x , x C ,

where δ is a constant in (0, 1). Then S is nonexpansive and Fix(S) = Fix(T1) ∩ Fix(T2).

Lemma 2.12. [9]Let C be a closed and convex subset of a reflexive Banach space X. Let μ be a proper convex lower semicontinuous function of C into (-∞, ∞] and suppose that μ(x n ) → ∞ as ||x n || → ∞. Then, there exists z C such that μ(z) = infxC{μ(x): x C}.

Lemma 2.13. [25]Let C be a nonempty closed convex subset of a smooth Banach space X and let S = T h : h > 0 be a u.a.r. nonexpansive semigroup on C such Fix S  =  h > 0 F i x T h and at least there exists a T(h) which is demicompact. Then, for each x C, there exists a sequence {T(t k ): t k > 0, k } {T(h): h > 0} such that {T (t k )x} converges strongly to some point in Fix S , where lim k→∞ t k = .

Lemma 2.14. [26]Let C be a nonempty closed and convex subset of a real uniformly convex Banach space and T : C → C be a nonexpansive mapping such that Fix(T) ≠ Ø . Then, I - T is demiclosed at zero.

Lemma 2.15. [27]Let X be a real smooth and uniformly convex Banach space and let r > 0. Then there exists a strictly increasing, continuous and convex function g : [0, 2r] → such that g(0) = 0 and
g | | x - y | | | | x | | 2 - 2 x , j y + | | y | | 2 , x , y B r .
Lemma 2.16. [28]Assume that {a n } is a sequence of nonnegative real numbers such that
a n + 1 1 - σ n a n + δ n ,
where {σ n } is a sequence in (0, 1) and {δ n } is a sequence in such that
  1. (i)

    n = 0 σ n = ;

     
  2. (ii)

    lim sup n δ n σ n 0 or n = 0 | δ n | < .

     

Then, lim n→∞ a n = 0.

Lemma 2.17. Let C be a nonempty closed and convex subset of a real 2-uniformly smooth Banach space X. Let F : CX be a κ-Lipschitzian and η-strongly accretive operator with constants κ, η > 0. Let 0 < μ < η κ 2 K 2 and τ = μ(η - μκ2K2). Then, for each t ( 0 , min { 1 , 1 2 τ } ) , the mapping S : CC defined by S : = (I - tμF) is contractive with a constant 1 - tτ.

Proof. Since 0 < μ < η κ 2 K 2 and t ( 0 , min { 1 , 1 2 τ } ) . This implies that 1 - tτ (0, 1). From Lemma 2.8, for all x, y C, we have
| | S x - S y | | 2 = | | I - t μ F x - I - t μ F y | | 2 = | | x - y - t μ F x - F y | | 2 | | x - y | | 2 - 2 t μ F x - F y , j x - y + 2 t 2 μ 2 K 2 | | F x - F y | | 2 | | x - y | | 2 - 2 t μ η | | x - y | | 2 + 2 t 2 μ 2 κ 2 K 2 | | x - y | | 2 1 - 2 t μ η - μ κ 2 K 2 | | x - y | | 2 1 - t μ η - μ κ 2 K 2 2 | | x - y | | 2 = 1 - t τ 2 | | x - y | | 2 .
It follows that
| | S x - S y | | 1 - t τ | | x - y | | .

Hence, we have S : = (I - tμF) is contractive with a constant 1 - . This proof is complete.

Lemma 2.18. Let C be a nonempty closed and convex subset of a real 2-uniformly smooth and uniformly convex Banach space X. Let the mappings Ψ, Φ: C → H be β ̃ -inverse strongly accretive and γ ̃ -inverse strongly accretive, respectively. Then, we have
| | I - ρ Ψ + Φ x - I - ρ Ψ + Φ y | | 2 | | x - y | | 2 + 4 ρ 2 K 2 ρ - β ̃ 2 K 2 | | Ψ x - Ψ y | | 2 + 4 ρ K 2 ρ - γ ̃ 2 K 2 | | Φ x - Φ y | | 2 .

In particular, if 0 < ρ < min β ̃ 2 K 2 , γ ̃ 2 K 2 , then I - ρ(Ψ + Φ) is nonexpansive.

Proof. By the convexity of ||·||2 and Lemma 2.8, for all x, y C, we have
| | I - ρ Ψ + Φ x - I - ρ Ψ + Φ y | | 2 = | | x - y - ρ Ψ + Φ x - Ψ + Φ y | | 2 = | | 1 2 x - y - 2 ρ Ψ x + Φ y + 1 2 x - y - 2 ρ Φ x - Φ y | | 2 1 2 | | x - y - 2 ρ Ψ x - Ψ y | | 2 + 1 2 | | x - y - 2 ρ Φ x - Φ y | | 2 1 2 | | x - y | | 2 - 4 ρ Ψ x - Ψ y , j x - y + 8 ρ 2 K 2 | | Ψ x - Ψ y | | 2 + 1 2 | | x - y | | 2 - 4 ρ Φ x - Φ y , j x - y + 8 ρ 2 K 2 | | Φ x - Φ y | | 2 1 2 | | x - y | | 2 - 4 ρ β ̃ | | Ψ x - Ψ y | | 2 + 8 ρ 2 K 2 | | Ψ x - Ψ y | | 2 + 1 2 | | x - y | | 2 - 4 ρ γ ̃ | | Φ x - Φ y | | 2 + 8 ρ 2 K 2 | | Φ x - Φ y | | 2 = | | x - y | | 2 + 4 ρ K 2 ρ - β ̃ 2 K 2 | | Ψ x - Ψ y | | 2 + 4 ρ K 2 ρ - γ ̃ 2 K 2 | | Φ x - Φ y | | 2 .

It is clear that, if 0 < ρ < min β ̃ 2 K 2 , γ ̃ 2 K 2 , then I - ρ(Ψ + Φ) is nonexpansive. This proof is complete.

Lemma 2.19. Let C be a nonempty closed and convex subset of a real 2-uniformly smooth and uniformly convex Banach space X. Let Q C be a sunny nonexpansive retraction from X onto C. Let Ψ i : C → H (i = 1, 2) be β ̃ i -inverse-strongly accretive and Φ i : C → H (i = 1, 2) be γ ̃ i -inverse-strongly accretive. Let G: C → C be the mapping defined by
G x : = Q C Q C x - ρ 2 Ψ + Φ x - ρ 1 Ψ 1 + Φ 1 Q C x - ρ 2 Ψ 2 + Φ 2 x , x C .

If 0 < ρ 1 < min β ̃ 1 2 K 2 , γ ̃ 1 2 K 2 and 0 < ρ 2 < min β ̃ 2 2 K 2 , γ ̃ 2 2 K 2 , then G : C → C is nonexpansive.

Proof. By Lemma 2.18, for all x, y C, we have
| | G x - G y | | = | | Q C Q C x - ρ 2 Ψ + Φ x - ρ 1 Ψ 1 + Φ 1 Q C x - ρ 2 Ψ 2 + Φ 2 x - Q C Q C y - ρ 2 Ψ + Φ y - ρ 1 Ψ 1 + Φ 1 Q C y - ρ 2 Ψ 2 + Φ 2 y | | | | I - ρ 1 Ψ 1 + Φ 1 Q C I - ρ 2 Ψ 2 + Φ 2 x - I - ρ 1 Ψ 1 + Φ 1 Q C I - ρ 2 Ψ 2 + Φ 2 y | | | | Q C I - ρ 2 Ψ 2 + Φ 2 x - Q C I - ρ 2 Ψ 2 + Φ 2 | | | | I - ρ 2 Ψ 2 + Φ 2 x - I - ρ 2 Ψ 2 + Φ 2 | | | | x - y | | ,

which implies that G : CC is nonexpansive. This proof is complete.

Lemma 2.20. Let C be a nonempty closed and convex subset of a real 2-uniformly smooth and uniformly convex Banach space X. Let Q C be a sunny nonexpansive retraction from X onto C. Let Ψ i : C → H (i = 1, 2) be β ̃ i -inverse-strongly accretive and Φ i : C → H (i = 1, 2) be γ ̃ i -inverse-strongly accretive. For given(x*, y*) C × C is a solution of the problem (1.15) if and only if x* Fix(G) and y* = Q C (x* - ρ22 + Φ2)x*), where G is the mapping defined as in Lemma 2.19.

Proof. Let (x*, y*) C × C be a solution of the problem (1.15). Then, we can rewrite (1.15) as
y * - ρ 1 Ψ 1 + Φ 1 y * - x * , j x - x * 0 , x C , x * - ρ 2 Ψ 2 + Φ 2 x * - y * , j x - y * 0 , x C .
(2.2)
From Lemma 2.6(c), we can deduce that (2.2) is equivalent to
x * = Q C y * - ρ 1 Ψ 1 + Φ 1 y * , y * = Q C x * - ρ 2 Ψ 2 + Φ 2 x * .
(2.3)

This proof is complete.

3 Main results

Let C be a nonempty closed and convex subset of a real 2-uniformly smooth and uniformly convex Banach space X. Let Q C be a sunny nonexpansive retraction from X onto C. Let F : C → X be a κ-Lipschitizian and η-strongly accretive operator with constants κ, η > 0, V : C → C be an L-Lipschitzian mapping with a constant L ≥ 0 and T : C → C be a nonexpansive mapping with Fix(T) = Ø. Let 0 < μ < η κ 2 K 2 and 0 ≤ γL < τ, where τ = μ(η - μκ2K2). For each t ( 0 , min { 1 , 1 2 τ } ) , consider the mapping S t : C → C defined by
S t x = Q C t γ V x + I - t μ F T x , x C .
It is easy to see that S t is contractive. Indeed, from Lemma 2.17, for all x, y C, we have
| | S t x - S t y | | = | | Q C t γ V x + I - t μ F T x - Q C t γ V y + I - t μ F T y | | | | t γ V x + I - t μ F T x - t γ V y + I - t μ F T y | | = | | t γ V x - V y + I - t μ F T x - T y | | t γ | | V x - V y | | + 1 - t τ | | T x - T y | | 1 - τ - γ L t | | x - y | | .
Hence S t is contractive. By the Banach contraction principle, S t has a unique fixed point, denoted by x t , which uniquely solves the fixed point equation
x t = Q C t γ V x t + I - t μ F T x t .
(3.1)
Lemma 3.1. Let C be a nonempty closed and convex subset of a real 2-uniformly smooth and uniformly convex Banach space X. Let Q C be a sunny nonexpansive retraction from X onto C. Let F : C → X be a κ-Lipschitizian and η-strongly accretive operator with constants κ, η > 0, V : C → C be an L-Lipschitzian mapping with a constant L ≥ 0 and T : C → C be a nonexpansive mapping with Fix(T) = Ø. Let 0 < μ < η κ 2 K 2 and 0 ≤ γL < τ, where τ = μ(η - μκ2K2). Then the net {x t } defined by (3.1) converges strongly to x ^ Fix T as t → 0, where x ^ is the unique solution of the variational inequality
μ F - γ V x ^ , j x ^ - v 0 , v F i x T .
(3.2)
Proof. We observe that
μ κ 2 K 2 > 0 η - μ κ 2 K 2 < η μ η - μ κ 2 K 2 < μ η τ < μ η .
It follows that
0 γ L < τ < μ η .
(3.3)
First, we show the uniqueness of a solution of the variational inequality (3.2). Suppose that x ^ , x ̃ Fix T are solution of (3.2), then
μ F - γ V x ^ , j x ^ - x ̃ 0
(3.4)
and
μ F - γ V x ̃ , j x ̃ - x ^ 0 .
(3.5)
Adding up (3.4) and (3.5), we have
0 μ F - γ V x ^ - μ F - γ V x ̃ , j x ^ - x ̃ = μ F x ^ - F x ̃ , j x ^ - x ̃ - γ V x ^ - V x ̃ , j x ^ - x ̃ μ η | | x ^ - x ̃ | | 2 - γ | | V x ^ - V x ̃ | | | | x ^ - x ̃ | | μ η - γ L | | x ^ - x ̃ | | 2 .

Note that (3.3) implies that x ^ = x ̃ and the uniqueness is proved. Below, we use x ^ to denote the unique solution of (3.2).

Next, we show that {x t } is bounded. Without loss of generality, we may assume that t min { 1 , 1 2 τ } . Take p Fix(T). From Lemma 2.17, we have
| | x t - p | | = | | Q C t γ V x t + I - t μ F T x t - Q C p | | | | t γ V x t - μ F p + I - t μ F T x t - p | | t γ | | V x t - V p | | + t | | γ V p - μ F p | | + 1 - t τ | | T x t - p | | 1 - τ - γ L t | | x t - p | | + t | | γ V p - μ F p | | .
It follows that
| | x t - p | | | | γ V p - μ F p | | τ - γ L .
Hence, {x t } is bounded, so are {V x t } and {FTx t }. Assume {t n } (0, 1) is such that t n → 0 as n → ∞. Set x n : = x t n . Define a mapping ϕ : C by
ϕ x : = L I M n | | x n - x | | 2 , x C ,
where LIM n is a Banach limit on l . Note that X is reflexive and ϕ is continuous, convex functional and ϕ (x) → ∞ as ||x|| → ∞. From Lemma 2.12, there exists z C such that ϕ(z) = infxCϕ(x). This implies that the set
K : = z C : ϕ z = inf x C ϕ x .
Observe that
| | x n - T x n | | = | | Q C t n γ V x n + I - t n μ F T x n - Q C T x n | | t n | | γ V x n - μ F T x n | | 0  as  n .
(3.6)
For all z C, we have
ϕ T z = L I M n | | x n - T z | | 2 = L I M n | | T x n - T z | | 2 L I M n | | x n - z | | 2 = ϕ z ,
which implies that T(K) K; that is, K is invariant under T. Since X is a uniformly smooth Banach space, it has the fixed point property for nonexpansive mapping T. Then, there exists x ̃ K such that T x ̃ = x ̃ . Since x ̃ is also minimization of ϕ over C, it follows that x C and t (0, 1),
ϕ x ̃ ϕ x ̃ + t x - μ F x ̃ .
From Lemma 2.9, we have
| | x n x ˜ + t ( μ F x ˜ x ) | | 2 | | x n x ˜ | | 2 + 2 t μ F x ˜ x , j ( x n x ˜ + t ( μ F x ˜ x ) ) .
Taking Banach limit over n ≥ 1, then
LIM n | | x n x ˜ + t ( μ F x ˜ x ) | | 2 LIM n | | x n x ˜ | | 2 + 2 t LIM n μ F x ˜ x , j ( x n x ˜ + t ( μ F x ˜ x ) ) ,
which in turn implies that
2 t LI M n x - μ F x ̃ , j x n - x ̃ + t μ F x ̃ - x LI M n | | x n - x ̃ | | 2 - L I M n | | x n - x ̃ + t μ F x ̃ - x | | 2 0 ,
and hence
LI M n x - μ F x ̃ , j x n - x ̃ + t μ F x ̃ - x 0 .
Again since X is a uniformly smooth Banach space, we have that the duality mapping j is norm-to-norm uniformly continuous on a bounded subset of C (see [9], Lemma 1), letting t → 0, we obtain
LI M n x - μ F x ̃ , j x n - x ̃ 0 , x C .
In particular
LI M n γ V x ̃ - μ F x ̃ , j x n - x ̃ 0 .
(3.7)
Set x t = Q C y t , where y t = tγVx t + (I - tμF)Tx t . Notice that x n : = x t n and y n : = y t n . For x ̃ Fix T , by Lemma 2.6(c), we have
| | x n - x ̃ | | 2 = y n - x ̃ , j x n - x ̃ + Q C y n - y n , j Q C y n - x ̃ y n - x ̃ , j x n - x ̃ = t n γ V x n - μ F x ̃ , j x n - x ̃ + I - t n μ F T x n - x ̃ , j x n - x ̃ t n γ V x n - μ F x ̃ , j x n - x ̃ + 1 - t n τ | | x n - x ̃ | | 2 .
Thus, we have
| | x n - x ̃ | | 2 1 τ γ V x n - μ F x ̃ , j x n - x ̃ = 1 τ γ V x n - V x ̃ , j x n - x ̃ + γ V x ̃ - μ F x ̃ , j x n - x ̃ 1 τ γ L | | x n - x ̃ | | 2 + γ V x ̃ - μ F x ̃ , j x n - x ̃ ,
which implies that
| | x n - x ̃ | | 2 1 τ - γ L γ V x ̃ - μ F x ̃ , j x n - x ̃ .
It follows from (3.7) that
LI M n | | x n - x ̃ | | 2 1 τ - γ L LI M n γ V x ̃ - μ F x ̃ , j x n - x ̃ 0 .
(3.8)

This implies that LI M n | | x n - x ̃ | | = 0 . Hence, there exists a subsequence x n i of {x n } such that x n i x ̃ as i → ∞. From (3.6) and Lemma 2.14, we get x ̃ Fix T .

Next, we show that x ̃ solves the variational inequality (3.2). We note that
x t = Q C y t = Q C y t - y t + t γ V x t + I - t μ F T x t ,
which derives that
μ F - γ V x t = 1 t Q C y t - y t - 1 t I - T x t + μ F x t - F T x t .
(3.9)
Note that I - T is accretive (i.e., 〈(I - T )x - (I - T )y, j(x - y)〉 0, for x, y C). For all v Fix(T), it follows from (3.9) and Lemma 2.6(c) that
μ F - γ V x t , j x t - v = 1 t Q C y t - y t , j Q C y t - v - 1 t I - T x t - I - T v , j x t - v + μ F x t - F T x t , j x t - v μ F x t - F T x t , j x t - v μ | | F x t - F T x t | | | | x t - v | | | | x t - T x t | | M ,
(3.10)

where M = supn ≥ 1{μκ||x t - v||} and t 0 , min 1 , 1 2 τ . Now, replacing t in (3.10) with t n and taking the limit as n → ∞, we noticing that x t n - T x t n x ̃ T x ̃ = 0 for x ̃ Fix T , we obtain μ F - γ V x ̃ , j x ̃ - v 0 . Hence x ̃ Fix T is the solution of the variational inequality (3.2). Consequently, x ^ = x ̃ by uniqueness. Therefore, x t x ^ as t → 0. This completes the proof.

Lemma 3.2. Let C be a nonempty closed and convex subset of a real 2-uniformly smooth and uniformly convex Banach space X. Let Q C be a sunny nonexpansive retraction from X onto C. Let F : C → X be a κ-Lipschitizian and η-strongly accretive operator with constants κ, η > 0, V : C → C be an L-Lipschitzian mapping with a constant L ≥ 0 and T : C → C be a nonexpansive mapping with Fix(T) = Ø. Let 0 < μ < η κ 2 K 2 and 0 ≤ γL < τ, where τ = μ(η - μκ2K2). Assume that the net {x t } defined by (3.1) converges strongly to x ^ Fix T as t → 0. Suppose that {x n } is bounded and lim n→∞ ||x n - Tx n || = 0. Then
lim sup n γ V x ^ - μ F x ^ , j x n - x ^ 0 .
(3.11)
Proof. Notice that x t = Q C y t , where y t = tγVx t + (I - tμF)Tx t . We note that
y t - x n = t γ V x t + I - t μ F T x t - x n = t γ V x t - μ F x t + T x t - x n + t μ F x t - F T x t = t γ V x t - μ F x t + T x t - T x n + T x n - x n + t μ F Q C y t - F Q C T x t .
It follows from Lemma 2.6(c) that
| | x t - x n | | 2 = y t - x n , j x t - x n + Q C y t - y t , j Q C y t - x n y t - x n , j x t - x n = t γ V x t - μ F x t , j x t - x n + T x t - T x n , j x t - x n + T x n - x n , j x t - x n + t μ F Q C y t - F Q C T x t , j x t - x n t γ V x t - μ F x t , j x t - x n + | | x t - x n | | 2 + | | T x n - x n | | | | x t - x n | | + t μ κ | | Q C y t - Q C T x t | | | | x t - x n | | t γ V x t - μ F x t , j x t - x n + | | x t - x n | | 2 + | | T x n - x n | | | | x t - x n | | + t 2 μ κ | | γ V x t - μ F T x t | | | | x t - x n | | ,
which in turn implies that
γ V x t - μ F x t , j x n - x t | | T x n - x n | | t | | x t - x n | | + t μ κ | | γ V x t - μ F T x t | | .
(3.12)
Since x n - Tx n → 0 as n → ∞, taking the upper limit as n → ∞ firstly, and then as t → 0 in (3.12), we have
lim sup t 0 lim sup n γ V x t - μ F x t , j x n - x t 0 .
(3.13)
On the other hand, we note that
γ V x ^ - μ F x ^ , j x n - x ^ = γ V x ^ - μ F x ^ , j x n - x ^ - γ V x ^ - μ F x ^ , j x n - x t + γ V x ^ - μ F x ^ , j x n - x t - γ V x ^ - μ F x t , j x n - x t + γ V x ^ - μ F x t , j x n - x t - γ V x t - μ F x t , j x n - x t + γ V x t - μ F x t , j x n - x t = γ V x ^ - μ F x ^ , j x n - x ^ - j x n - x t + μ F x t - F x ^ , j x n - x t + γ V x ^ - V x t , j x n - x t + γ V x t - μ F x t , j x n - x t .
Taking the upper limit as n → ∞, we have
lim sup n γ V x ^ - μ F x ^ , j x n - x ^ lim sup n γ V x ^ - μ F x ^ , j x n - x ^ - j x n - x t + μ κ + γ L | | x t - x ^ | | lim sup n | | x n - x t | | + lim sup n γ V x t - μ F x t , j x n - x t .
Since X is a uniformly smooth Banach space, we have that the duality mapping j is norm-to-norm uniformly continuous on bounded subset of C (see [9], Lemma 1), then
lim sup t 0 lim sup n γ V x ^ - μ F x ^ , j x n - x ^ - j x n - x t = 0 .
(3.14)
Then, from (3.13) and (3.14), we have
lim sup n γ V x ^ - μ F x ^ , j x n - x ^ = lim sup t 0 lim sup n γ V x ^ - μ F x ^ , j x n - x ^ lim sup t 0 lim sup n γ V x t - μ F x t , j x n - x t 0 .

This completes the proof.

Theorem 3.3. Let C be a nonempty closed and convex subset of a real 2-uniformly smooth and uniformly convex Banach space X. Let Q C be a sunny nonexpansive retraction from X onto C. Let F : C → X be a κ-Lipschitizian and η-strongly accretive operator with constants κ, η > 0, V : C → C be an L-Lipschitzian mapping with a constant L ≥ 0. Let 0 < μ < η κ 2 K 2 and 0 ≤ γL < τ, where τ = μ(η - μκ2K2). Let S = T h : h > 0 be a u.a.r. nonexpansive semigroup from C into itself such that F i x S : = h > 0 Fix T h and least there exists a T(h) which is demicompact. Let Ψ i : C → X (i = 1, 2) be β ̃ i -inverse-strongly accretive and Φ i : CX (i = 1, 2) be γ ̃ i -inverse-strongly accretive. Assume that Ω : = Fix S Fix G , where G is defined as in Lemma 2.19. For given x1 C, let {x n } be a sequence defined by
z n = Q C x n - ρ 2 Ψ 2 + Φ 2 x n , y n = Q C z n - ρ 1 Ψ 1 + Φ 1 z n , x n + 1 = β n x n + 1 - β n Q C α n γ V x n + I - α n μ F T t n y n , n 1 ,
(3.15)

where ρ i 0 , min β ̃ i 2 K 2 , γ ̃ i 2 K 2 for all i= 1, 2. Suppose that {α n } and {β n } are sequences in [0, 1] and {t n } is a sequence in (0, ) satisfying the following conditions:

(C1) lim n→∞ α n = 0 and n = 1 α n = ;

(C2) 0 < lim inf n→∞ β n lim sup n→∞ β n < 1;

(C3) tn+1= h + t n , for all h > 0 and lim n→∞ t n = .

Then, the sequence {x n } defined by (3.15) converges strongly to x ^ Ω as n → ∞, which x ̃ is the unique solution of the variational inequality
μ F - γ V x ̃ , j x ̃ - v 0 , v Ω ,
(3.16)

and x ^ , y ^ is the solution of the problem(1.15), where y ^ = Q C x ^ - ρ 2 Ψ 2 + Φ 2 x ^ .

Proof. Note that from the condition (C 1), we assume without loss generality, that α n min { 1 , 1 2 τ } for all n ≥ 1. First, we show that {x n } is bounded. Take x * Ω . It follows from Lemma 2.19 that
x * = Q C [ Q C ( x 2 ( Ψ 2 + Φ 2 ) x * ) 1 ( Ψ 1 + Φ 1 ) Q C ( x * 2 ( Ψ 2 + Φ 2 ) x * ) ] .
Put y* = Q C (x* - ρ22 + Φ2)x*), then x* = Q C (y* - ρ11 + Φ1)y*). From (3.15), we observe that
| | y n - x * | | = | | G x n - G x * | | | | x n - x * | | .
Set u n = Q C [α n γVx n + (I - α n μF)T(t n )y n ]. From Lemma 2.17, we have
| | u n - x * | | = | | Q C α n γ V x n + I - α n μ F T t n y n - Q C x * | | | | α n γ V x n - μ F x * + I - α n μ F T t n y n - x * | | α n γ | | V x n - V x * | | + α n | | γ V x * - μ F x * | | + 1 - α n τ | | y n - x * | | 1 - τ - γ L α n | | x n - x * | | + α n | | γ V x * - μ F x * | | .
It follows that
| | x n + 1 - x * | | = | | β n x n - x * + 1 - β n u n - x * | | β n | | x n - x * | | + 1 - β n | | u n - x * | | β n | | x n - x * | | + 1 - β n 1 - τ - γ L α n | | x n - x * | | + α n | | γ V x * - μ F x * | | = 1 - τ - γ L α n 1 - β n | | x n - x * | | + α n 1 - β n τ - γ L | | γ V x * - μ F x * | | τ - γ L .
By induction, we have
| | x n - x * | | max | | x 1 - x * | | , | | γ V x * - μ F x * | | τ - γ L , n 1 .

Hence, {x n } is bounded, so are {z n }, {y n } and {u n }.

Next, we show that ||xn+1- x n || → 0 as n → ∞. We observe that
| | y n + 1 - y n | | = | | G x n + 1 - G x n | | | | x n + 1 - x * | | .
(3.17)
Now, we can take a constant M > 0 such that
M = sup n 1 γ | | V x n + 1 | | + μ | | F T t n + 1 y n + 1 | | , γ | | V x n | | + μ | | F T t n y n | | .
Then, we have
| | u n + 1 - u n | | = | | Q C α n + 1 γ V x n + 1 + I - α n + 1 μ F T t n + 1 y n + 1 - Q C α n γ V x n + I - α n μ F T t n y n | | | | α n + 1 γ V x n + 1 + I - α n + 1 μ F T t n + 1 y n + 1 - α n γ V x n - I - α n μ F T t n y n | | α n + 1 | | γ V x n + 1 - μ F T t n + 1 y n + 1 | | + α n | | γ V x n - μ F T t n y n | | + | | T t n + 1 y n + 1 - T t n y n | | α n + 1 + α n M + | | y n + 1 - y n | | + | | T h T t n y n - T t n y n | | .
(3.18)
Combining (3.17) and (3.18), we obtain
| | u n + 1 u n | | | | x n + 1 x n | | ( α n + 1 + α n ) M + | | T ( h ) T ( t n ) y n T ( t n ) y n | | .
(3.19)
Since {T(h): h > 0} is a u.a.r. nonexpansive semigroup and lim n→∞ t n = , then for all h > 0, and for any bounded subset B of C containing {x n } and {y n }, we obtain that
lim n | | T h T t n y n - T t n y n | | lim n sup ω B | | T h T t n ω - T t n ω = 0 .
(3.20)
Consequently, it follows from the conditions (C 1), (C 2) and (3.19) that
lim sup n | | u n + 1 - u n | | - | | x n + 1 - x n | | 0 .
Hence, by Lemma 2.10, we obtain that
lim n | | u n - x n | | = 0 .
(3.21)
Consequently, we have
lim n | | x n + 1 - x n | | = lim n 1 - β n | | u n - x n | | = 0 .
(3.22)
We observe that
| | u n - T t n y n | | = | | Q C α n γ f x n + I - α n μ F T t n y n - Q C T t n y n | | α n | | γ f x n - μ F T t n y n | | 0  as  n .
(3.23)
From (3.21) and (3.23), we have
| | x n - T t n y n | | | | x n - u n | | + | | u n - T t n y n | | 0  as  n .
(3.24)
Next, we show that lim n→∞ ||x n - T(h)x n || = 0, h > 0. For all x*, y* Ω, by the convexity of || · ||2 and Lemma 2.8, we have
| | z n - y * | | 2 = | | Q C x n - ρ 2 Ψ 2 + Φ 2 x n - Q C x * - ρ 2 Ψ 2 + Φ 2 x * | | 2 | | x n - ρ 2 Ψ 2 + Φ 2 x n - x * - ρ 2 Ψ 2 + Φ 2 x * | | 2 = | | x n - x * - ρ 2 Ψ 2 + Φ 2 x n - Ψ 2 + Φ 2 x * | | 2 = | | 1 2 x n - x * - 2 ρ 2 Ψ 2 x n - Ψ 2 x * + 1 2 x n - x * - 2 ρ 2 Φ 2 x n - Φ 2 x * | | 2 1 2 | | x n - x * - 2 ρ 2 Ψ 2 x n - Ψ 2 x * | | 2 + 1 2 | | x n - x * - 2 ρ 2 Φ 2 x n - Φ 2 x * | | 2 1 2 | | x n - x * | | 2 - 4 ρ 2 Ψ 2 x n - Ψ 2 x * , x n - x * + 8 ρ 2 2 K 2 | | Ψ 2 x n - Ψ 2 x * | | 2 + 1 2 | | x n - x * | | 2 - 4 ρ 2 Φ 2 x n - Φ 2 x * , x n - x * + 8 ρ 2 2 K 2 | | Φ 2 x n - Φ 2 x * | | 2 1 2 | | x n - x * | | 2 - 4 ρ 2 β ̃ 2 | | Ψ 2 x n - Ψ 2 x * | | 2 + 8 ρ 2 2 K 2 | | Ψ 2 x n - Ψ 2 x n x * | | 2 + 1 2 | | x n - x * | | - 4 ρ 2 γ ̃ 2 | | Φ 2 x n - Φ 2 x * | | 2 + 8 ρ 2 2 K 2 | | Φ 2 x n - Φ 2 x n x * | | 2 = | | x n - x * | | 2 + 4 ρ 2 K 2 ρ 2 - β ̃ 2 2 K 2 | | Ψ 2 x n - Ψ 2 x * | | 2 + 4 ρ 2 K 2 ρ 2 - γ 2 2 K 2 | | Φ 2 x n - Φ 2 x * | | 2 .
(3.25)
In a similar way, we can get
| | y n - x * | | 2 | | z n - y * | | 2 + 4 ρ 1 K 2 ρ 1 - β ̃ 1 2 K 2 | | Ψ 1 z n - Ψ 1 y * | | 2 + 4 ρ 1 K 2 ρ 1 - γ ̃ 1 2 K 2 | | Φ 1 z n - Φ 1 y * | | 2 .
(3.26)
Substituting (3.25) into (3.26), we have
| | y n - x * | | 2 | | x n - x * | | 2 + 4 ρ 2 K 2 ρ 2 - β ̃ 2 2 K 2 | | Ψ 2 x n - Ψ 2 x * | | 2 + 4 ρ 2 K 2 ρ 2 - γ ̃ 2 2 K 2 | | Φ 2 x n - Φ 2 x * | | 2 + 4 ρ 1 K 2 ρ 1 - β ̃ 1 2 K 2 | | Ψ 1 z n - Ψ 1 y * | | 2 + 4 ρ 1 K 2 ρ 1 - γ ̃ 1 2 K 2 | | Φ 1 z n - Φ 1 y * | | 2 .
(3.27)
Set u n = Q C v n , where v n = α n γVx n + (I - α n μF)T(t n )y n . From Lemma 2.6(c), we have
| | u n - x * | | 2 = v n - x * , j u n - x * + Q C v n - v n , j Q C v n - x * v n - x * , j u n - x * = α n γ V x n - μ F x * , j u n - x * + I - α n μ F T t n y n - x * , j u n - x * 1 - α n τ | | y n - x * | | | | u n - x * | | + α n γ V x n - μ F x * , j u n - x * 1 - α n τ 2 | | y n - x * | | 2 + | | u n - x *