The general iterative scheme for semigroups of nonexpansive mappings and variational inequalities with applications

Mathematical and Computer Modelling 57 (2013) 1289–1297

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The general iterative scheme for semigroups of nonexpansive mappings and variational inequalities with applications Liping Yang ∗ School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510520, China

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Article history: Received 20 May 2011 Received in revised form 8 July 2012 Accepted 26 August 2012 Keywords: Nonexpansive semigroups Strong convergence Variational inequality Reflexive Banach spaces Strictly convex Banach spaces

abstract In this paper, we introduce the implicit and explicit viscosity iteration schemes for nonexpansive semigroups {T (t ) : t ≥ 0}. Additionally, it proves that the proposed iterative schemes converge strongly to a unique common fixed point of {T (t ) : t ≥ 0} in the framework of reflexive and strictly convex Banach space, which solves some variational inequality. The main results of this paper improve and generalize recent known results in the current literature. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction Let E be a real Banach space, K be a nonempty closed convex subset of E. Recall that a mapping f : K → K is a contraction on K if there exists a constant α ∈ (0, 1) such that ∥f (x) − f (y)∥ ≤ α∥x − y∥, x, y ∈ K . We use ΠK to denote the collection of mappings f verifying the above inequality. That is, ΠK = {f : K → K | f is a contraction with constant α}. Note that each f ∈ ΠK has a unique fixed point in K . Recall that a one-parameter family T = {T (t ) : 0 ≤ t < ∞} is said to be a Lipschitzian semigroup on K (see, e.g., [1]) if the following conditions are satisfied. (i) (ii) (iii) (iv)

T (0)x = x, x ∈ K . T (t + s)x = T (t )T (s)x, t , s ≥ 0, x ∈ K . For each x ∈ K , the map t → T (t )x is continuous on [0, ∞). There exists a bounded measurable function Lt : (0, ∞) → (0, ∞) such that, for each t > 0, ∥T (t )x − T (t )y∥ ≤ Lt ∥x − y∥, x, y ∈ K .

A Lipschitzian semigroup T is said to be nonexpansive semigroup if Lt = 1 for all t > 0, and asymptotically nonexpansive if lim supt →∞ Lt ≤ 1, respectively. We use F (T ) to denote the common fixed point set of the semigroup T , that is, F (T ) = {x ∈ K : T (s)x = x, ∀s > 0}. A continuous operator of the semigroup T is said to be uniformly asymptotically regular (u.a.r.) on K if for all h ≥ 0 and any bounded subset C of K , limt →∞ supx∈C ∥T (h)T (t )x − T (t )x∥ = 0 (see [2]). The variational inequality problem was first introduced by Hartman and Stampacchia [3]. Then, the variational inequality has achieved an increasing attention in many research fields, such as mathematical programming, constrained linear and nonlinear optimization, automatic control, manufacturing system design, signal and image processing and the complementarity problem in economics and pattern recognition (see [4–6] and the references therein). Nowadays, the

∗

Tel.: +86 20 37629035. E-mail address: [email protected].

0895-7177/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2012.08.015

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L. Yang / Mathematical and Computer Modelling 57 (2013) 1289–1297

theory of variational inequalities and fixed point theory are two important and dynamic areas in nonlinear analysis and optimization. One promising approach to handle these problems is to develop some kind of iterative schemes to compute the approximate solutions of variational inequalities and to find a common fixed point of a given family of operators. We now have a variety of techniques to suggest and analyze various iterative algorithms for solving variational inequalities and the related optimization problems. The fixed point theory has played an important role in the development of various algorithms for solving variational inequalities. Let H be a real Hilbert space, whose inner product and norm are denoted by ⟨·, ·⟩ and ∥ · ∥, respectively. Assume that A is strongly positive bounded linear operator on H, that is, there is a constant γ¯ > 0 with property

⟨Ax, x⟩ ≥ γ¯ ∥x∥2 ∀x ∈ H . A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space H: min x∈C

1 2

⟨Ax, x⟩ − ⟨x, b⟩,

where C is the fixed point set of a nonexpansive mapping T on H and b is a given point in H. Recently, Marino and Xu [7] have considered the following general iteration process xn+1 = (I − αn A)Txn + αn γ f (xn ),

n ≥ 0,

(1.1)

and proved that if the sequence {αn } satisfies appropriate conditions, the sequence {xn } generated by (1.1) converges strongly to the unique solution of the variational inequality

⟨(A − γ f )x∗ , x − x∗ ⟩ ≥ 0,

x ∈ Fix(T ),

which is the optimality condition for the minimization problem min

x∈Fix(T )

1 2

⟨Ax, x⟩ − h(x),

where h is a potential function for γ f (i.e., h′ (x) = γ f (x), for x ∈ H) and Fix(T ) = {x ∈ H : Tx = x}. Question. Can the iteration sequence (1.1) provide the same result for the more general class of continuous semigroup of nonexpansive mappings in Banach space? Inspired by the above results, the purpose of this paper is to study the convergence problems of the implicit and explicit viscosity iterative processes for nonexpansive semigroup {T (t ) : t ≥ 0} in general Banach spaces. And it provides an affirmative answer to the above Question. We establish the strong convergence results which generalize the corresponding results given by Marino and Xu [7], and Song and Xu [8]. 2. Preliminaries Recall that a Banach space E is said to be strictly convex if ∥x∥ = ∥y∥ = 1, x ̸= y implies ∥x + y∥/2 < 1. In a strictly convex Banach space E, we have if ∥x∥ = ∥y∥ = ∥tx + (1 − t )y∥, for t ∈ (0, 1) and x, y ∈ E, then x = y. Let E be a Banach space with dimension E ≥ 2. The modulus of E is the function δE : (0, 2] → [0, 1] defined by

δE (ε) = inf { 1 − ∥x + y∥/2 : ∥x∥ = ∥y∥ = 1, ∥x − y∥ = ε } . A Banach space E is uniformly convex if and only if δE (ε) > 0 for all ε ∈ (0, 2]. Let S (E ) = { x ∈ E : ∥x∥ = 1 }. The space E is said to be smooth if lim (∥x + ty∥ − ∥x∥)/t

t →0

exists for all x, y ∈ S (E ). For any x, y ∈ E (x ̸= 0), we denote this limit by (x, y). The norm ∥ · ∥ of E is said to be Fréchet differentiable if for all x ∈ S (E ), the limit (x, y) exists uniformly for all y ∈ S (E ). E is said to have a uniformly Gâteaux differentiable norm if for each y ∈ S (E ) the limit (x, y) is attained uniformly for x ∈ S (E ). ∗ Let E ∗ denote the dual space of a Banach space E. The normalized duality mapping J : E → 2E is defined by J (x) = {x∗ ∈ E ∗ : ⟨x, x∗ ⟩ = ∥x∥2 = ∥x∗ ∥2 },

∀x ∈ E ,

where ⟨·, ·⟩ denotes the generalized pairing. If E is a Hilbert space, then J = I (the identity mapping). It is well-known that if E is smooth, then J is single-valued, which is denoted by j. And if E has a uniformly Gâteaux differentiable norm then the duality mapping is norm-to-weak∗ uniformly continuous on bounded sets. Let µ be a continuous linear functional on l∞ and (a0 , a1 , . . .) ∈ l∞ . We write µn (an ) instead of µ((a0 , a1 , . . .)). Recall a Banach limit µ is a bounded function functional on l∞ such that

∥µ∥ = µn (1) = 1,

lim inf an ≤ µn (an ) ≤ lim sup an , n→∞

n→∞

µn (an+r ) = µn (an ) for any fixed positive integer r and for all (a0 , a1 , . . .) ∈ l∞ .

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Recall that an operator A is strongly positive on a smooth Banach space if there exists a constant γ¯ > 0 with the property

⟨Ax, J (x)⟩ ≥ γ¯ ∥x∥2 ,

∥aI − bA∥ = sup |⟨(aI − bA)x, J (x)⟩|,

(2.2)

∥x∥≤1

where a ∈ [0, 1], b ∈ [−1, 1], I is the identity mapping and J is the normalized duality mapping. Lemma 2.1. Assume that A is a strongly positive linear bounded operator on a smooth Banach space E with coefficient γ¯ > 0 and 0 < ρ < ∥A∥−1 . Then ∥I − ρ A∥ ≤ 1 − ρ γ¯ . Proof. The proof follows as in the proof of Lemma 2.5 of [7].



Let f be a given contraction on E with contraction coefficient 0 < α < 1, t ∈ (0, 1) such that t < ∥A∥−1 and 0 < γ

< (γ¯ /α). Consider a mapping Tn on E defined by Tn u := t γ f (u) + (I − tA)T (tn )u,

u ∈ E.

Then, by Lemma 2.1, we have

∥Tn x − Tn y∥ ≤ t γ ∥f (x) − f (y)∥ + ∥(I − tA)(T (tn )x − T (tn )y)∥ ≤ (1 − t (γ¯ − γ α))∥x − y∥, for all x, y ∈ E. Hence Tn has a unique fixed point, denoted u, which uniquely solves the fixed point equation u = t γ f (u) + (I − tA)T (tn )u. Let (X , d) be a metric space. A subset B of X is said to be a Chebyshev set, if for each x ∈ X , there exists a unique element y ∈ B such that d(x, y) = d(x, B), where d(x, B) = infy∈B d(x, y). Lemma 2.2 (See [9, Theorem 5.1.18]). E is a reflexive strictly convex Banach space if and only if every nonempty closed convex subset of E is a Chebyshev set. Lemma 2.3 (See [10]). Let E be a real normed linear space and J the normalized duality map on E. Then for any given x, y ∈ E, the following inequality holds:

∥x + y∥2 ≤ ∥x∥2 + 2⟨y, j(x + y)⟩,

∀j(x + y) ∈ J (x + y).

Lemma 2.4 (See [11]). Let {xn }, {yn } be two bounded sequences in a Banach space E and βn ∈ [0, 1] with 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1. Suppose xn+1 = βn yn +(1 −βn )xn for all integers n ≥ 0 and lim supn→∞ (∥yn+1 − yn ∥−∥xn+1 − xn ∥) ≤ 0. Then limn→∞ ∥xn − yn ∥ = 0. Lemma 2.5 (See [12]). Let {an } be a sequence of nonnegative real numbers satisfying the following relation: an+1 ≤ (1 − ρn )an + ρn σn ,

n ≥ 0,

where {ρn } and {σn } are sequences of real numbers such that (i) 0 < ρn < 1; (ii) ∞ n=1 |ρn σn | is convergent. Then an → ∞ as n → ∞.

∞

n =1

ρn = ∞; (iii) lim supn→∞ σn ≤ 0 or

3. Main results The following lemma summarizes the properties of the implicit iterative sequence {un }. Lemma 3.1. Let K be a nonempty closed convex subset of a reflexive, smooth and strictly convex Banach space E with a uniformly Gâteaux differentiable norm. Let {T (t ) : t ≥ 0} be a uniformly asymptotically regular nonexpansive semigroup on K such that F (T ) ̸= ∅, and f ∈ ΠK . Let A be a strongly positive linear bounded self-adjoint operator with coefficient γ¯ > 0. Assume that 0 < γ < (γ¯ /α). Let {un } be a sequence defined by un = αn γ f (un ) + (I − αn A)T (tn )un for all n ≥ 1 such that limn→∞ tn = ∞, limn→∞ αn = 0. Then (i) the sequence {un } is bounded; (ii) the sequence {un } is relatively sequentially compact.

(3.1)

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Proof. (i) Since αn → 0 as n → ∞, we may assume, with no loss of generality, that αn < ∥A∥−1 for all n ≥ 1. Then

∥ un − p ∥ = ≤ ≤ ≤ =

∥(I − αn A)(T (tn )un − p) + αn (γ f (un ) − Ap)∥ (1 − αn γ¯ )∥T (tn )un − T (tn )p∥ + αn ∥γ f (un ) − Ap∥ (1 − αn γ¯ )∥un − p∥ + αn ∥γ f (un ) − γ f (p) + γ f (p) − Ap∥ (1 − αn γ¯ )∥un − p∥ + αn γ α∥un − p∥ + αn ∥γ f (p) − Ap∥ (1 − αn (γ¯ − γ α))∥un − p∥ + αn ∥γ f (p) − Ap∥.

Therefore ∥un − p∥ ≤ ∥γ f (p) − Ap∥/(γ¯ − γ α). Thus, {un } is bounded, so are {f (un )} and {T (tn )un }. (ii) It follows from (3.1) and limn→∞ αn = 0 that we obtain ∥un − T (tn )un ∥ = αn ∥γ f (un ) − AT (tn )un ∥ ≤ αn (∥γ f (un )∥ + ∥AT (tn )un ∥) → 0 as n → ∞. Define the mapping h : K → R by h(s) = µn ∥un − s∥2 ,

∀s ∈ K .

Since E is reflexive and h is continuous, convex and h(s) → ∞ as ∥s∥ → ∞, we get that h attains its infimum over K (see, e.g. [13]). Therefore C = {y ∈ K : h(y) = minx∈K h(x)} is nonempty. And it is also closed, convex, bounded. Since {T (t )} is u.a.r. nonexpansive semigroup and limn→∞ tn = ∞, then for all t > 0, lim ∥T (t )(T (tn )un ) − T (tn )un ∥ ≤ lim sup ∥T (t )(T (tn )x) − T (tn )x∥ → 0,

n→∞

n→∞ x∈D

where D is any bounded subset of K containing {un }. Hence,

∥un − T (t )un ∥ ≤ ∥un − T (tn )un ∥ + ∥T (tn )un − T (t )(T (tn )un )∥ + ∥T (t )(T (tn )un ) − T (t )un ∥ ≤ 2∥un − T (tn )un ∥ + ∥T (tn )un − T (t )(T (tn )un )∥. Thus, for all t > 0, we have lim ∥un − T (t )un ∥ = 0.

(3.2)

n→∞

Furthermore, we have h(T (t )y) = µn ∥un − T (t )y∥2 = µn ∥T (t )un − T (t )y∥2

≤ µn ∥un − y∥2 . Hence T (t )y ∈ C . It follows from Lemma 2.2 that there exists a unique v ∈ C such that

∥p − v∥ = min ∥p − x∥. x∈C

Since T (t )p = p and T (t )v ∈ C ,

∥p − T (t )v∥ = ∥T (t )p − T (t )v∥ ≤ ∥p − v∥. Hence T (t )v = v by the uniqueness of v ∈ C . Thus v ∈ F (T ). Since C is a closed convex set, there exists a weakly convergent subsequence {uni } ⊂ {un } by reflexivity of E and boundedness of {un } such that uni ⇀ v as i → ∞. From (3.1) we get that

∥un − v∥2 = = ≤ =

∥αn (γ f (un ) − Av) + (I − αn A)(T (tn )un − v)∥2 ⟨αn (γ f (un ) − Av) + (I − αn A)(T (tn )un − v), j(un − v)⟩ (1 − αn γ¯ )∥un − v∥2 + αn ⟨γ f (un ) − Av, j(un − v)⟩ (1 − αn γ¯ )∥un − v∥2 + αn ⟨γ f (un ) − γ f (v), j(un − v)⟩ + αn ⟨γ f (v) − Av, j(un − v)⟩

≤ (1 − αn (γ¯ − γ α))∥un − v∥2 + αn ⟨γ f (v) − Av, j(un − v)⟩.

(3.3)

Thus

∥un − v∥2 ≤ ⟨γ f (v) − Av, j(un − v)⟩/(γ¯ − γ α).

(3.4)

∗

Since j is norm-to-weak uniformly continuous on bounded subset of E, it follows from (3.4) that

µn ∥un − v∥2 ≤ µn ⟨γ f (v) − Av, j(un − v)⟩/(γ¯ − γ α) → 0, as n → ∞. Hence, there exists a subsequence {uni } ⊂ {un } such that uni → v as i → ∞. The proof is completed.



Theorem 3.2. Let K be a nonempty closed convex subset of a reflexive, smooth and strictly convex Banach space E with a uniformly Gâteaux differentiable norm. Let {T (t ) : t ≥ 0} be a uniformly asymptotically regular nonexpansive semigroup on K such that F (T ) ̸= ∅, and f ∈ ΠK . Let A be a strongly positive linear bounded self-adjoint operator with coefficient γ¯ > 0. Assume that

L. Yang / Mathematical and Computer Modelling 57 (2013) 1289–1297

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0 < γ < (γ¯ /α). Let {un } be a sequence defined by (3.1) for all n ≥ 1 such that limn→∞ tn = ∞, limn→∞ αn = 0. Then the sequence {un } converges strongly to a point v of F (T ) which solves the variational inequality:

⟨(A − γ f )v, j(v − z )⟩ ≤ 0,

z ∈ F (T ).

(3.5)

Proof. We first show the uniqueness of solutions of the variational inequality (3.5). Suppose p, q ∈ F (T ) satisfy (3.5), we get that

⟨(A − γ f )p, j(p − q)⟩ ≤ 0.

(3.6)

⟨(A − γ f )q, j(q − p)⟩ ≤ 0.

(3.7)

Adding up (3.6) and (3.7), we have that 0 ≥ ⟨(A − γ f )p − (A − γ f )q, j(p − q)⟩

= ⟨A(p − q), j(p − q)⟩ − γ ⟨f (p) − f (q), j(p − q)⟩ ≥ γ¯ ∥p − q∥2 − γ ∥f (p) − f (q)∥ ∥p − q∥ ≥ (γ¯ − γ α)∥p − q∥2 . Hence p = q and the uniqueness is proved. It follows from Lemma 3.1(ii) that there exists a subsequence {uni } ⊂ {un } such that uni → v as i → ∞ and v ∈ F (T ). Next, we prove that v solves Eq. (3.5). Indeed, it follows from (3.1) that we get

(A − γ f )un = −(I − αn A)(I − T (tn ))un /αn . Notice

⟨(I − T (tn ))un − (I − T (tn ))z , j(un − z )⟩ = ∥un − z ∥2 − ∥T (tn )un − T (tn )z ∥ ∥un − z ∥ ≥ ∥un − z ∥2 − ∥un − z ∥2 = 0. Let z ∈ F (T ). It follows that

⟨(A − γ f )uni , j(uni − z )⟩ = − =−

1

αni 1

αni

⟨(I − αni A)(I − T (tni ))uni , j(uni − z )⟩ ⟨(I − T (tni ))uni − (I − T (tni ))z , j(uni − z )⟩ + ⟨A(I − T (tni ))uni , j(uni − z )⟩

≤ ⟨A(I − T (tni ))uni , j(uni − z )⟩.

(3.8)

Let i → ∞ in (3.8), we have ⟨(A − γ f )v, j(v − z )⟩ ≤ 0. That v ∈ F (T ) is a solution of (3.5). Now, we have proved that {un } is relatively sequentially compact and each cluster point of {un } (as n → ∞) equals v . Thus un → v as n → ∞. The proof is completed.  Remark 3.3. (1) Theorem 3.2 extends Theorem 3.2 of Marino and Xu [7] from a real Hilbert space to reflexive strictly convex space and from nonexpansive mappings to nonexpansive semigroups. (2) Taking A = I and γ = 1 in Theorem 3.2, we get Theorem 3.2 of Song and Xu [8]. Lemma 3.4. Let K be a nonempty closed convex subset of a reflexive, smooth and strictly convex Banach space E with a uniformly Gâteaux differentiable norm. Let {T (t ) : t ≥ 0} be a uniformly asymptotically regular nonexpansive semigroup on K such that F (T ) ̸= ∅, and f ∈ ΠK . Let A be a strongly positive linear bounded self-adjoint operator with coefficient γ¯ > 0. Let {xn } be a sequence generated in the following iterative process: xn+1 = αn γ f (xn ) + δn xn + ((1 − δn )I − αn A)T (tn )xn , where 0 < γ < (γ¯ /α), the given sequences {αn } and {δn } are in (0, 1) satisfying the following conditions. (1) n=1 αn = ∞, limn→∞ αn = 0. (2) 0 < lim infn→∞ δn ≤ lim supn→∞ δn < 1. (3) h, tn ≥ 0 such that tn+1 = h + tn and limn→∞ tn = ∞.

∞

Then (i) the sequence {xn } is bounded; (ii) the sequence limn→∞ ∥xn+1 − xn ∥ = 0.

(3.9)

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Proof. Since αn → 0 as n → ∞, we may assume, with no loss of generality, that αn < (1 − δn )∥A∥−1 for all n. Since A is a linear bounded operator on E, it follows from (2.2) that

∥A∥ = sup{|⟨Au, Ju⟩| : ∥u∥ = 1, u ∈ E }. Notice that

⟨((1 − δn )I − αn A)u, Ju⟩ = 1 − δn − αn ⟨Au, Ju⟩ ≥ 1 − δn − αn ∥A∥ ≥ 0. Therefore

∥(1 − δn )I − αn A∥ = sup{⟨((1 − δn )I − αn A)u, Ju⟩ : ∥u∥ = 1, u ∈ E } = sup{1 − δn − αn ⟨Au, Ju⟩ : ∥u∥ = 1, u ∈ E } ≤ 1 − δn − αn γ¯ . (i) Take a point q ∈ F (T ) to get

∥xn+1 − q∥ = ≤ ≤ ≤ =

∥αn (γ f (xn ) − Aq) + δn (xn − q) + ((1 − δn )I − αn A)(xn − q)∥ αn ∥γ f (xn ) − Aq∥ + δn ∥xn − q∥ + ∥(1 − δn )I − αn A∥∥xn − q∥ αn γ ∥f (xn ) − f (q)∥ + αn ∥γ f (q) − Aq∥ + δn ∥xn − q∥ + (1 − δn − αn γ¯ )∥xn − q∥ αn αγ ∥xn − q∥ + αn ∥γ f (q) − Aq∥ + (1 − αn γ¯ )∥xn − q∥ αn ∥γ f (q) − Aq∥ + (1 − αn (γ¯ − αγ ))∥xn − q∥

≤ max{∥xn − q∥, (γ¯ − αγ )−1 ∥Aq − γ f (q)∥}.

(3.10)

It follows from (3.10) by induction that

∥xn − q∥ ≤ max {∥x0 − q∥, ∥Aq − γ f (q)∥/(γ¯ − γ α)} ,

n ≥ 0.

Hence, {xn } is bounded, so are {f (un )} and {T (tn )un }. (ii) Putting ln = (xn+1 − δn xn )/(1 − δn ), we have xn+1 = δn xn + (1 − δn )ln ,

∀n ≥ 0 .

(3.11)

Observe that from the definition of ln , we get ln+1 − ln =

xn+2 − δn+1 xn+1

−

xn+1 − δn xn

1 − δn+1 1 − δn αn γ f (xn ) + ((1 − δn )I − αn A)T (tn )xn αn+1 γ f (xn+1 ) + ((1 − δn+1 )I − αn+1 A)T (tn+1 )xn+1 − = 1 − δn+1 1 − δn αn+1 αn = γ f (xn+1 ) − γ f (xn ) + T (tn+1 )xn+1 − T (tn )xn 1 − δn+1 1 − δn αn αn+1 + AT (tn )xn − AT (tn+1 )xn+1 1 − δn 1 − δn+1 αn+1 αn = [γ f (xn+1 ) − AT (tn+1 )xn+1 ] + [AT (tn )xn − γ f (xn )] 1 − δn+1 1 − δn + T (tn+1 )xn+1 − T (tn+1 )xn + T (tn+1 )xn − T (tn )xn .

It follows that

∥ln+1 − ln ∥ − ∥xn+1 − xn ∥ ≤

αn+1 1 − δn+1

(∥γ f (xn+1 )∥ + ∥AT (tn+1 )xn+1 ∥) +

αn 1 − δn

(∥AT (tn )xn ∥ + ∥γ f (xn )∥)

+ ∥T (tn+1 )xn+1 − T (tn+1 )xn ∥ + ∥T (tn+1 )xn − T (tn )xn ∥ − ∥xn+1 − xn ∥ αn+1 (∥γ f (xn+1 )∥ + ∥AT (tn+1 )xn+1 ∥) ≤ 1 − δn+1 αn + (∥AT (tn )xn ∥ + ∥γ f (xn )∥) + ∥T (h)T (tn )xn − T (tn )xn ∥. 1 − δn Since {T (t ) : t ≥ 0} is uniformly asymptotically regular and limn→∞ tn = ∞, it follows that lim ∥T (h)T (tn )xn − T (tn )xn ∥ ≤ lim sup ∥T (h)T (tn )x − T (tn )x∥ = 0,

n→∞

n→∞ x∈B

(3.12)

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where B is any bounded set containing {xn }. Moreover, since {xn }, {f (xn )} and {T (tn )xn } are bounded, and αn → 0 as n → ∞, (3.12) implies that lim sup(∥ln+1 − ln ∥ − ∥xn+1 − xn ∥) ≤ 0. n→∞

Hence, by Lemma 2.4 we have limn→∞ ∥ln − xn ∥ = 0. Noting (3.11), we have xn+1 − xn = (1 − δn )(ln − xn ). Consequently limn→∞ ∥xn+1 − xn ∥ = 0. This completes the proof.  Theorem 3.5. Let K be a nonempty closed convex subset of a reflexive, smooth and strictly convex Banach space E with a uniformly Gâteaux differentiable norm. Let {T (t ) : t ≥ 0} be a uniformly asymptotically regular nonexpansive semigroup on K such that F (T ) ̸= ∅, and f ∈ ΠK . Let A be a strongly positive linear bounded self-adjoint operator with coefficient γ¯ > 0. Let {xn } be a sequence generated by (3.9) such that 0 < γ < (γ¯ /α), the given sequences {αn } and {δn } are in (0, 1) satisfying the following conditions. (1) n=1 αn = ∞, limn→∞ αn = 0. (2) 0 < lim infn→∞ δn ≤ lim supn→∞ δn < 1. (3) h, tn ≥ 0 such that tn+1 = h + tn and limn→∞ tn = ∞.

∞

Then {xn } converges strongly to q, as n → ∞, q is the element of F (T ) such that q is the unique solution in F (T ) to the variational inequality (3.5). Proof. We first show that lim supn→∞ ⟨(γ f − A)q, j(xn − q)⟩ ≤ 0. Indeed, it follows from (3.9) that

∥xn+1 − T (tn )xn ∥ = ∥αn (γ f (xn ) − AT (tn )xn ) + δn (xn − T (tn )xn )∥ ≤ αn (γ ∥f (xn )∥ + ∥AT (tn )xn ∥) + δn ∥xn − xn+1 ∥ + δn ∥xn+1 − T (tn )xn ∥ and so

∥xn+1 − T (tn )xn ∥ ≤

αn 1 − δn

(γ ∥f (xn )∥ + ∥AT (tn )xn ∥) +

δn 1 − δn

∥xn − xn+1 ∥.

It follows from Lemma 3.4 and limn→∞ αn = 0 that limn→∞ ∥xn+1 − T (tn )xn ∥ = 0. Therefore

∥xn − T (tn )xn ∥ ≤ ∥xn − xn+1 ∥ + ∥xn+1 − T (tn )xn ∥ → 0 and for any t ≥ 0,

∥xn − T (t )xn ∥ ≤ ∥xn − T (tn )xn ∥ + ∥T (tn )xn − T (t )T (tn )xn ∥ + ∥T (t )T (tn )xn − T (t )xn ∥ ≤ 2∥xn − T (tn )xn ∥ + ∥T (tn )xn − T (t )T (tn )xn ∥ → 0, that is

∥xn − T (t )xn ∥ → 0 as n → ∞.

(3.13)

For each m ≥ 0, let um ∈ K be the unique fixed point of the contraction mapping Gm (tm )x := αm γ f (x) + (I − αm A)S (tm )x, where S (t ) := (1 −δ)I +δ T (t ) for δ ∈ (0, 1). Thus, S := {S (t ) : t ≥ 0} is a strongly continuous semigroup of nonexpansive mapping and F (S ) = F (T ). It follows from (3.13) that

∥xn − S (t )xn ∥ = ∥xn − [(1 − δ)xn + δ T (t )xn ]∥ → 0

(3.14)

for any t ≥ 0. Moreover, we have um − xn = (I − αm A)(S (tm )um − xn ) + αm (γ f (um ) − Axn ). It follows from Lemma 2.3 that

∥um − xn ∥2 = (1 − γ¯ αm )2 ∥S (tm )um − xn ∥2 + 2αm ⟨γ f (um ) − Axn , j(um − xn )⟩ ≤ (1 − γ¯ αm )2 [∥S (tm )um − S (tm )xn ∥ + ∥S (tm )xn − xn ∥]2 + 2αm ⟨(γ f − A)um , j(um − xn )⟩ + 2αm ⟨A(um − xn ), j(um − xn )⟩ ≤ (1 − γ¯ αm )2 [∥um − xn ∥2 + ∥S (tm )xn − xn ∥(2∥um − xn ∥ + ∥S (tm )xn − xn ∥)] + 2αm ⟨(γ f − A)um , j(um − xn )⟩ + 2αm ⟨A(um − xn ), j(um − xn )⟩ = (1 − 2γ¯ αm + (γ¯ αm )2 )∥um − xn ∥2 + fm (tn ) + 2αm ⟨(γ f − A)um , j(um − xn )⟩ + 2αm ⟨A(um − xn ), j(um − xn )⟩,

(3.15)

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L. Yang / Mathematical and Computer Modelling 57 (2013) 1289–1297

where fm (tn ) = (1 − γ¯ αm )2 ∥S (tm )xn − xn ∥(2∥um − xn ∥ + ∥S (tm )xn − xn ∥) → 0 as n → ∞. Since A is a linear strong positive operator, it follows from (2.2) that

⟨A(um − xn ), j(um − xn )⟩ ≥ γ¯ ∥um − xn ∥2 .

(3.16)

It follows from (3.15) and (3.16) that 2αm ⟨(A − γ f )um , j(um − xn )⟩ ≤ ((γ¯ αm )2 − 2γ¯ αm )∥um − xn ∥2 + fm (tn ) + 2αm ⟨A(um − xn ), j(um − xn )⟩

≤ (γ¯ αm2 − 2αm )⟨A(um − xn ), j(um − xn )⟩ + fm (tn ) + 2αm ⟨A(um − xn ), j(um − xn )⟩ = γ¯ αm2 ⟨A(um − xn ), j(um − xn )⟩ + fm (tn ). This implies

⟨(A − γ f )um , j(um − xn )⟩ ≤ ≤

γ¯ αm 2

γ¯ αm 2

⟨A(um − xn ), j(um − xn )⟩ + M+

1 2αm

1 2αm

fm (tn )

fm (tn ),

(3.17)

where M > 0 is a constant such that ⟨A(um − xn ), j(um − xn )⟩ ≤ M. Let n → ∞ first, and then m → ∞ in (3.17), we have lim sup lim sup⟨(A − γ f )um , j(um − xn )⟩ ≤ 0. m→∞

(3.18)

n→∞

By Theorem 3.2, um → q ∈ F (T ), as m → ∞. Moreover, j is norm-to-norm∗ uniformly continuous on the bounded set. And since

⟨(γ f − A)q, j(xn − q)⟩ = ⟨(γ f − A)q, j(xn − q) − j(xn − um )⟩ + ⟨A(um − q), j(xn − um )⟩ + ⟨γ (f (q) − f (um )), j(xn − um )⟩ + ⟨(γ f − A)um , j(xn − um )⟩, we get that lim sup⟨(γ f − A)q, j(xn − q)⟩ ≤ 0. n→∞

Next, we show that xn → q as n → ∞. Indeed, it follows from (3.9) and Lemma 2.3 that

∥xn+1 − q∥2 = ∥((1 − δn )I − αn A)(T (tn )xn − q) + αn (γ f (xn ) − Aq) + δn (xn − q)∥2 ≤ ∥((1 − δn )I − αn A)(T (tn )xn − q) + δn (xn − q)∥2 + 2αn ⟨(γ f (xn ) − Aq), j(xn+1 − q)⟩ ≤ [(1 − δn − αn γ¯ )∥xn − q∥ + δn ∥xn − q∥]2 + 2αn γ ⟨f (xn ) − f (q), j(xn+1 − q)⟩ + 2αn ⟨(γ f − A)q, j(xn+1 − q)⟩ ≤ (1 − αn γ¯ )2 ∥xn − q∥2 + 2αn γ ∥f (xn ) − f (q)∥∥xn+1 − q∥ + 2αn ⟨(γ f − A)q, j(xn+1 − q)⟩ ≤ (1 − αn γ¯ )2 ∥xn − q∥2 + 2αn γ α∥xn − q∥∥xn+1 − q∥ + 2αn ⟨(γ f − A)q, j(xn+1 − q)⟩ ≤ (1 − αn γ¯ )2 ∥xn − q∥2 + αn γ α(∥xn − q∥2 + ∥xn+1 − q∥2 ) + 2αn ⟨(γ f − A)q, j(xn+1 − q)⟩ = ((1 − αn γ¯ )2 + αn γ α)∥xn − q∥2 + αn γ α∥xn+1 − q∥2 + 2αn ⟨(γ f − A)q, j(xn+1 − q)⟩. This implies that

((1 − αn γ¯ )2 + αn γ α) 2αn ∥xn − q∥2 + ⟨(γ f − A)q, j(xn+1 − q)⟩ 1 − αn γ α 1 − αn γ α   2αn (γ − γ α) (αn γ¯ )2 2αn = 1− ∥xn − q∥2 + ∥xn − q∥2 + ⟨(γ f − A)q, j(xn+1 − q)⟩ 1 − αn γ α 1 − αn γ α 1 − αn γ α    2αn (γ − γ α) 2αn (γ − γ α) αn γ 2 ≤ 1− ∥xn − q∥2 + M1 1 − αn γ α 1 − αn γ α 2(γ − γ α)  1 + ⟨(γ f − A)q, j(xn+1 − q)⟩ , γ − γα

∥xn+1 − q∥2 ≤

where M1 is an appropriate constant such that M1 = supn ∥xn − q∥2 . Put

ρn = σn =

2αn (γ − γ α) 1 − αn γ α

,

αn γ 2 1 M1 + ⟨(γ f − A)q, j(xn+1 − q)⟩, 2(γ − γ α) γ − γα

L. Yang / Mathematical and Computer Modelling 57 (2013) 1289–1297

1297

that is

∥xn+1 − q∥2 ≤ (1 − ρn )∥xn − q∥2 + ρn σn . ∞ It is easily seen that limn→∞ ρn = 0, n=1 ρn = ∞ and lim supn→∞ σn ≤ 0. By Lemma 2.5, we conclude that xn → q. The proof is completed.



Remark 3.6. (1) Theorem 3.5 extends Theorem 3.4 of Marino and Xu [7] from a real Hilbert space to reflexive strictly convex space and from nonexpansive mappings to nonexpansive semigroups. (2) Taking A = I , δn = 0 and γ = 1 in Theorem 3.5, we get Theorem 4.2 of Song and Xu [8]. As application of Theorem 3.5, we can obtain the following result. Theorem 3.7. Let K be a nonempty closed convex subset of a Hilbert space E. Let {T (t ) : t ≥ 0} be a uniformly asymptotically regular nonexpansive semigroup on K such that F (T ) ̸= ∅, and f ∈ ΠK . Let A be a strongly positive linear bounded self-adjoint operator with coefficient γ¯ > 0. Let {xn } be a sequence generated by (3.9) such that 0 < γ < (γ¯ /α), the given sequences {αn } and {δn } are in (0, 1) satisfying the following conditions. (1) n=1 αn = ∞, limn→∞ αn = 0. (2) 0 < lim infn→∞ δn ≤ lim supn→∞ δn < 1. (3) h, tn ≥ 0 such that tn+1 = h + tn and limn→∞ tn = ∞.

∞

Then {xn } converges strongly, as n → ∞, to the solution of the variational inequality

⟨(A − γ f )q, q − z ⟩ ≤ 0,

z ∈ F (T ),

which is the optimality condition for the minimization problem min

x∈F (T )

1 2

⟨Ax, x⟩ − h(x),

where h is a potential function for γ f (i.e., h′ (x) = γ f (x), for x ∈ H). Proof. If E is a Hilbert space, then J = I. We can obtain the desired conclusion easily from Theorem 3.5. This completes the proof.  Acknowledgments The author thanks the referees for the comments that helped improve the presentation of this article. This work was supported in part by the National Natural Science Foundation of China (60974143) and the Natural Science Foundation of Guangdong Province (10151009001000039). References [1] H.K. Xu, Strong asymptotic behavior of almost-orbits of nonlinear semigroups, Nonlinear Anal. 46 (2001) 135–151. [2] A. Aleyner, Y. Censor, Best approximation to common fixed points of a semigroup of nonexpansive operators, J. Nonlinear Convex Anal. 6 (2005) 137–151. [3] P. Hartman, G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta Math. 115 (1996) 153–188. [4] P. Harker, J. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Math. Program. 48 (1990) 161–220. [5] A. Cochocki, R. Unbehauen, Neural Networks for Optimization and Signal Processing, Wiley, New York, 1993. [6] F. Facchinei, J. Pang, Finite-Dimensional Variational Inequalities and Nonlinear Complementarity Problems, Springer-Verlag, New York, 2003. [7] G. Marino, H.X. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 318 (2006) 43–52. [8] Y. Song, S. Xu, Strong convergence theorems for nonexpansive semigroups in Banach spaces, J. Math. Anal. Appl. 338 (2008) 152–161. [9] Robert E. Megginson, An Introduction to Banach Space Theory, Springer-Verlag, New York, 1998. [10] C.H. Morales, J.S. Jung, Convergence of paths for pseudocontractive mappings in Banach spaces, Proc. Amer. Math. Soc. 128 (2000) 3411–3419. [11] T. Suzuki, Strong convergence of Krasnoselskii and Mann’s sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl. 305 (2005) 227–239. [12] H.K. Xu, An iterative approach to guadratic optimization, J. Optim. Theory Appl. 116 (2003) 659–678. [13] V. Barbu, Th. Precupanu, Convexity and Optimization in Banach Spaces, Editura Academiei, Bucharest, 1978.