Fixed point solutions of variational inequalities for a semigroup of asymptotically nonexpansive mappings in Banach spaces

Sunthrayuth and Kumam Fixed Point Theory and Applications 2012, 2012:177http://www.fixedpointtheoryandapplications.com/content/2012/1/177

RESEARCH Open Access

Fixed point solutions of variationalinequalities for a semigroup of asymptoticallynonexpansive mappings in Banach spacesPongsakorn Sunthrayuth and Poom Kumam*

*Correspondence:[email protected] of Mathematics,Faculty of Science, King Mongkut’sUniversity of Technology Thonburi(KMUTT), Bangmod, Bangkok,10140, Thailand

AbstractThe purpose of this article is to introduce two iterative algorithms for finding acommon fixed point of a semigroup of asymptotically nonexpansive mappings whichis a unique solution of some variational inequality. We provide two algorithms, oneimplicit and another explicit, from which strong convergence theorems are obtainedin a uniformly convex Banach space, which admits a weakly continuous dualitymapping. The results in this article improve and extend the recent ones announcedby Li et al. (Nonlinear Anal. 70:3065-3071, 2009), Zegeye et al. (Math. Comput. Model.54:2077-2086, 2011) and many others.MSC: 47H05; 47H09; 47H20; 47J25

Keywords: iterative approximation method; common fixed point; semigroup ofasymptotically nonexpansive mapping; strong convergence theorem; uniformlyconvex Banach space

1 IntroductionThroughout this paper, we denote by N and R

+ the set of all positive integers and all pos-itive real numbers, respectively. Let X be a real Banach space. A mapping T : X –→ X issaid to be nonexpansive if

‖Tx – Ty‖ ≤ ‖x – y‖, ∀x, y ∈ X,

and T is asymptotically nonexpansive (see []) if there exists a sequence {kn} of positivereal numbers with limn–→∞ kn = such that

∥∥Tnx – Tny∥∥ ≤ kn‖x – y‖, ∀n≥ and ∀x, y ∈ X.

We denote by Fix(T) the set of fixed points of T , i.e., Fix(T) = {x ∈ X : x = Tx}.Recall that a self-mapping f : X –→ X is a contraction if there exists a constant α ∈ (, )

such that

∥∥f (x) – f (y)∥∥ ≤ α‖x – y‖, ∀x, y ∈ X.

A one-parameter familyS = {T(t) : t ∈R+} ofX into itself is said to be a strongly continuous

semigroup of Lipschitzian mappings if the following conditions are satisfied:

© 2012 Sunthrayuth and Kumam; licensee Springer. This is an Open Access article distributed under the terms of the CreativeCommons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and re-production in any medium, provided the original work is properly cited.

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(i) T()x = x for all x ∈ X ;(ii) T(s + t) = T(s) ◦ T(t) for all s, t ∈R

+;(iii) for each x ∈ X the mapping T(·)x from R

+ into X is continuous;(iv) for each t > , there exists a bounded measurable function Lt : (,∞) –→ [,∞)

such that

∥∥T(t)x – T(t)y∥∥ ≤ Lt‖x – y‖, ∀x, y ∈ X.

A strongly continuous semigroup of Lipschitzianmappings S is called strongly continuoussemigroup of nonexpansive mappings if Lt ≡ for all t > and strongly continuous semi-group of asymptotically nonexpansivemappings if lim supt–→∞ Lt ≤ . Note that for asymp-totically nonexpansive semigroup S , we can always assume that the Lipschitzian constant{Lt}t> is such that Lt ≥ for each t > , Lt is nonincreasing in t and limt–→∞ Lt = ; oth-erwise, we replace Lt for each t > with L̄t := max{sups≥t Ls, }. S is said to have a fixedpoint if there exists x ∈ X such that T(t)x = x for all t ≥ . We denote by Fix(S) the setof fixed points of S , i.e., Fix(S) = ⋂

t≥ Fix(T(t)) (for more details, see [–]).A continuous operator of the semigroup S = {T(t) : t ∈ R

+} is said to be uniform-ly asymptotically regular on X if for all h ≥ and any bounded subset C of X,limt–→∞ supx∈C ‖T(h)T(t)tx–T(t)x‖ = (see [] for examples of uniformly asymptoticallyregular semigroups).Recently, convergence theorems for commonfixed points of a strongly continuous semi-

group of nonexpansivemappings and their generalizations have been studied by numerousauthors (see, e.g., [–]). Construction of fixed points of nonexpansive mappings (and ofcommon fixed points of nonexpansive semigroups) is an important subject in the theoryof nonexpansive mappings and finds application in a number of applied areas, in partic-ular, in image recovery and signal processing (see, e.g., [–]). In the last ten years, theiterative methods for nonexpansive mappings have been applied to solve convex mini-mization problems; see, e.g., [–]. Let H be a real Hilbert space, whose inner productand norm are denoted by 〈·, ·〉 and ‖ · ‖, respectively. Let A be a strongly positive boundedlinear operator on H ; that is, there is a constant γ̄ > with the property

〈Ax,x〉 ≥ γ̄ ‖x‖ for all x ∈H . (.)

A typical problem is to minimize a quadratic function over the set of the fixed points of anonexpansive mapping on a real Hilbert space H :

minx∈F

〈Ax,x〉 – 〈x,b〉, (.)

where C is the fixed point set of a nonexpansive mapping T on H and b is a given pointin H .In , Xu [] proved that the sequence {xn} defined by the iterative method below,

with the initial guess x ∈H chosen arbitrarily,

xn+ = (I – αnA)Txn + αnu, ∀n≥ , (.)

converges strongly to the unique solution of the minimization problem (.) providedthe sequence {αn} satisfies certain conditions. Using the viscosity approximation method,

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Moudafi [] introduced the following iterative process for nonexpansive mappings (see[] for further developments in both Hilbert and Banach spaces). Let f be a contractionon H . Starting with an arbitrary initial x ∈H , we define the sequence {xn} recursively by

xn+ = σnf (xn) + ( – σn)Txn, ∀n≥ , (.)

where {σn} is a sequence in (, ). It is proved in [, ] that under certain appropriateconditions imposed on {σn}, the sequence {xn} generated by (.) strongly converges to aunique solution x* of the variational inequality

⟨(f – I)x*,x – x*

⟩ ≤ , ∀x ∈ F(T). (.)

In , Marino and Xu [] combined the iterative method (.) with the viscosity ap-proximation method (.) considering the following general iterative process:

xn+ = αnγ f (xn) + (I – αnA)Txn, ∀n≥ , (.)

where < γ < γ̄

α. They proved that the sequence {xn} generated by (.) converges strongly

to a unique solution x* of the variational inequality

⟨(γ f –A)x*,x – x*

⟩ ≤ , ∀x ∈ F(T), (.)

which is the optimality condition for the minimization problem

minx∈C

〈Ax,x〉 – h(x),

where h is a potential function for γ f (i.e., h′(x) = γ f (x) for x ∈H).On the other hand, Li et al. [] considered the implicit and explicit viscosity iteration

processes for a nonexpansive semigroup S = {T(t) : t ∈ R+} in a Hilbert space as follows:

xn = αnγ f (xn) + (I – αnA)tn

∫ tn

T(s)xn ds, ∀n ∈ N, (.)

xn+ = αnγ f (xn) + (I – αnA)tn

∫ tn

T(s)xn ds, ∀n ∈ N, (.)

where {αn} and {tn} are two sequences satisfying certain conditions. They proved the se-quence {xn} defined by (.) and (.) converges strongly to x* ∈ Fix(S), which solves thevariational inequality (.). Under the framework of a uniformly convex Banach spacewitha uniformly Gâteaux differentiable norm, Chen and Song [] studied the strong conver-gence of the implicit and explicit viscosity iteration processes for a nonexpansive semi-group S = {T(t) : t ∈R

+} with Fix(S) �= ∅ as follows:

xn = αnf (xn) + ( – αn)tn

∫ tn

T(s)xn ds, ∀n ∈N, (.)

xn+ = αnf (xn) + ( – αn)tn

∫ tn

T(s)xn ds, ∀n ∈N. (.)

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Very recently, Zegeye et al. [] introduced the implicit and explicit iterative processes fora strongly continuous semigroup of asymptotically nonexpansive mappings S = {T(t) : t ∈R

+} in a reflexive and strictly convex Banach spaces with a uniformly Gâteaux differen-tiable norm as follows:

xn = αnu + ( – αn)tn

∫ tn

T(s)xn ds, ∀n ∈N, (.)

xn+ = αnu + ( – αn)tn

∫ tn

T(s)xn ds, ∀n ∈N. (.)

They proved that {xn} defined by (.) and (.) converges strongly to a common fixedpoint of Fix(S) provided certain conditions are satisfied.In this paper, motivated by the above results, we introduce two iterative algorithms for

finding a common fixed point of a semigroup of asymptotically nonexpansive mappingswhich is a unique solution of some variational inequality. We establish the strong con-vergence results in a uniformly convex Banach space which admits a weakly continuousdualitymapping. The results in this article improve and extend the recent ones announcedby Li et al. [], Zegeye et al. [] and many others.

2 PreliminariesThroughout this paper, we write xn ⇀ x (respectively xn ⇀* x) to indicate that the se-quence {xn} weakly (respectively weak*) converges to x; as usual xn –→ x will symbolizestrong convergence; also, a mapping I will denote the identity mapping. Let X be a realBanach space, X* be its dual space. Let U = {x ∈ X : ‖x‖ = }. A Banach space X is said tobe uniformly convex if, for each ε ∈ (, ], there exists a δ > such that for each x, y ∈ U ,‖x – y‖ ≥ ε implies ‖x+y‖

≤ – δ. It is know that a uniformly convex Banach space is re-flexive and strictly convex (see also []). A Banach space is said to be smooth if the limitlimt–→

‖x+ty‖–‖x‖t exists for each x, y ∈ U . It is also said to be uniformly smooth if the limit

is attained uniformly for x, y ∈U .Let ϕ : [,∞) –→ [,∞) be a continuous strictly increasing function such that ϕ() =

and ϕ(t) –→ ∞ as t –→ ∞. This function ϕ is called a gauge function. The dualitymappingJϕ : X –→ X* associated with a gauge function ϕ is defined by

Jϕ(x) ={f * ∈ X* :

⟨x, f *

⟩= ‖x‖ϕ(‖x‖),∥∥f *∥∥ = ϕ

(‖x‖),∀x ∈ X},

where 〈·, ·〉 denotes the generalized duality paring. In particular, the duality mapping withthe gauge function ϕ(t) = t, denoted by J is referred to as the normalized duality mapping.Clearly, the relation Jϕ(x) = ϕ(‖x‖)

‖x‖ J(x) holds for each x �= (see []).Browder [] initiated the study of certain classes of nonlinear operators by means of

the duality mapping Jϕ . Following Browder [], we say a Banach space X has a weaklycontinuous duality mapping if there exits a gauge function ϕ for which the duality map-ping Jϕ(x) is single-valued and continuous from the weak topology to the weak* topology;that is, for each {xn} with xn ⇀ x, the sequence {J(xn)} converges weakly* to Jϕ(x). It isknown that lp has a weakly continuous duality mapping with a gauge function ϕ(t) = tp–

for all < p < ∞. Set (t) =∫ t ϕ(τ )dτ , ∀t ≥ , then Jϕ(x) = ∂(‖x‖), where ∂ denotes the

subdifferential in the sense of convex analysis (recall that the subdifferential of the convexfunction φ : X –→R at x ∈ X is the set ∂φ(x) = {x* ∈ X;φ(y) ≥ φ(x) + 〈x*, y – x〉,∀y ∈ X}).

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In a Banach space X which admits a duality mapping Jϕ with a gauge function ϕ, we saythat an operator A is strongly positive (see []) if there exists a constant γ̄ > with theproperty

⟨Ax, Jϕ(x)

⟩ ≥ γ̄ ‖x‖ϕ(‖x‖) (.)

and

‖aI – bA‖ = sup‖x‖≤

∣∣⟨(aI – bA)x, Jϕ(x)⟩∣∣, a ∈ [, ],b ∈ [–, ]. (.)

As special cases of (.), we have the following results.() If X is a smooth Banach space and ϕ(t) = t for all t ∈ X (see []), then the inequality

(.) reduces to

⟨Ax, J(x)

⟩ ≥ γ̄ ‖x‖. (.)

() If X :=H is a real Hilbert space, then the inequality (.) reduces to (.).The first part of the next lemma is an immediate consequence of the subdifferential

inequality and the proof of the second part can be found in [].

Lemma . ([]) Assume that a Banach space X has a weakly continuous duality map-ping Jϕ with a gauge ϕ.

(i) For all x, y ∈ X , the following inequality holds:

(‖x + y‖) ≤

(‖x‖) + ⟨y, Jϕ(x + y)

⟩.

(ii) Assume that a sequence {xn} in X converges weakly to a point x ∈ X . Then thefollowing identity holds:

lim supn–→∞

(‖xn – y‖) = lim sup

n–→∞

(‖xn – x‖) +(‖y – x‖), ∀x, y ∈ X.

Lemma . ([]) Assume that a Banach space X admits a duality mapping Jϕ with agauge ϕ. Let A be a strongly positive linear bounded operator on X with a coefficient γ̄ > and < ρ ≤ ϕ()‖A‖–. Then ‖I – ρA‖ ≤ ϕ()( – ργ̄ ).

Definition. LetC be a closed convex subset of a real Banach spaceX. LetS = {T(t) : t ∈R

+} be a strongly continuous semigroup of asymptotically nonexpansive mappings fromC into itself such that Fix(S) �= ∅. Then S is said to be almost uniformly asymptoticallyregular (in short a.u.a.r.) on C, if for all h≥ ,

limt–→∞ sup

x∈C

∥∥∥∥t∫ t

T(s)xds – T(h)

(t

∫ t

T(s)xds

)∥∥∥∥ = .

Lemma . ([]) Let C be a closed convex subset of a uniformly convex Banach space Xand S = {T(t) : t ∈R

+} be a strongly continuous semigroup of asymptotically nonexpansive

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mappings from C into itself with a sequence {Lt} ⊂ [,∞) such that Fix(S) �= ∅. Then foreach r > and h ≥ ,

limt–→∞ sup

x∈C∩Br

∥∥∥∥t∫ t

T(s)xds – T(h)

(t

∫ t

T(s)xds

)∥∥∥∥ = .

Lemma . ([]) Assume that {an} is a sequence of nonnegative real numbers such that

an+ ≤ ( – σn)an + δn,

where {σn} is a sequence in (, ) and {δn} is a sequence in R such that(i)

∑∞n= σn = ∞;

(ii) lim supn–→∞δnσn

≤ or∑∞

n= |δn| <∞.Then limn–→∞ an = .

3 Implicit iteration schemeTheorem . Let X be a uniformly convex Banach space which admits a weakly contin-uous duality mapping Jϕ with a gauge ϕ such that ϕ is invariant on [, ]. Let S = {T(t) :t ∈R

+} be a strongly continuous semigroup of asymptotically nonexpansive mappings fromX into itself with a sequence {Lt} ⊂ [,∞) such that Fix(S) �= ∅. Let f : X –→ X be a con-traction mapping with a constant α ∈ (, ) and A : X –→ X be a strongly positive linearbounded operator with a constant γ̄ ∈ (, ) such that < γ < γ̄ ϕ()

α. Let {xn} be a sequence

defined by

xn = αnγ f (xn) + (I – αnA)tn

∫ tn

T(s)xn ds, ∀n≥ , (.)

where {αn} is a sequence in (, ) and {tn} is a positive real divergent sequence which satisfythe following conditions:(C) limn–→∞ αn = ;(C) limn–→∞

( tn

∫ tn Ls ds)–

αn= .

Then the sequence {xn} defined by (.) converges strongly to x* ∈ Fix(S), where x* is theunique solution of the variational inequality

⟨γ f

(x*

)–Ax*, Jϕ

(v – x*

)⟩ ≤ , ∀v ∈ Fix(S). (.)

Proof First, we show that {xn} defined by (.) is well defined. For all n ∈ N, let us definethe mapping

Tfn := αnγ f + (I – αnA)

tn

∫ tn

T(s)ds.

Indeed, for all x, y ∈ X, we have

∥∥Tfnx – Tf

ny∥∥ =

∥∥∥∥αnγ(f (x) – f (y)

)+ (I – αnA)

(tn

∫ tn

(T(s)x – T(s)y

)ds

)∥∥∥∥≤ αnγ

∥∥f (x) – f (y)∥∥ + ‖I – αnA‖

(tn

∫ tn

∥∥T(s)x – T(s)y∥∥ds

)

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≤ αnγα‖x – y‖ + ϕ()( – αnγ̄ )(tn

∫ tn

Ls ds

)‖x – y‖

≤[tn

∫ tn

Ls ds – ϕ()γ̄

(tn

∫ tn

Ls ds

)αn + γααn

]‖x – y‖.

Since limn–→∞( tn

∫ tn Ls ds)–

αn= implies

( tn

∫ tn Ls ds) –

αn< ϕ()γ̄ – γα ≤ ϕ()γ̄

(tn

∫ tn

Ls ds

)– γα,

for sufficiently large n≥ , that is,

tn

∫ tn

Ls ds – ϕ()γ̄

(tn

∫ tn

Ls ds

)αn + γααn < .

Thus, by the Banach contraction mapping principle, there exits a unique fixed point xn ∈X, that is, {xn} defined by (.) is well defined.Next, we show the uniqueness of a solution of the variational inequality (.). Suppose

that x̃,x* ∈ F(S) are solutions of (.), then⟨γ f

(x*

)–Ax*, Jϕ

(x̃ – x*

)⟩ ≤ (.)

and

⟨γ f (x̃) –Ax̃, Jϕ

(x* – x̃

)⟩ ≤ . (.)

Adding up (.) and (.), we obtain

≥ ⟨(γ f

(x*

)–Ax*

)–

(γ f (x̃) –Ax̃

), Jϕ

(x̃ – x*

)⟩=

⟨A

(x̃ – x*

), Jϕ

(x̃ – x*

)⟩– γ

⟨f (x̃) – f

(x*

), Jϕ

(x̃ – x*

)⟩≥ γ̄

∥∥x̃ – x*∥∥ϕ

(∥∥x̃ – x*∥∥)

– γ∥∥f (x̃) – f

(x*

)∥∥∥∥Jϕ(x̃ – x*

)∥∥≥ γ̄

(∥∥x̃ – x*∥∥)

– γα(∥∥x̃ – x*

∥∥)= (γ̄ – γα)

(∥∥x̃ – x*∥∥)

≥ (ϕ()γ̄ – γα

)

(∥∥x̃ – x*∥∥),

which is a contradiction.Wemust have x̃ = x* and the uniqueness is proved. Below, we usex̃ to denote the unique solution of the variational inequality (.).Next, we show that {xn} is bounded. Take p ∈ Fix(S). Then from (.), we get that

‖xn – p‖ =∥∥∥∥αn

(γ f (xn) –Ap

)+ (I – αnA)

(tn

∫ tn

T(s)xn ds – p

)∥∥∥∥≤ αn

∥∥γ f (xn) –Ap∥∥ + ϕ()( – αnγ̄ )

∥∥∥∥ tn

∫ tn

T(s)xn ds – p

∥∥∥∥≤ αnγ

∥∥f (xn) – f (p)∥∥ + αn

∥∥γ f (p) –Ap∥∥

+ ϕ()( – αnγ̄ )(tn

∫ tn

Ls ds

)‖xn – p‖

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≤[tn

∫ tn

Ls ds –

(ϕ()γ̄

(tn

∫ tn

Ls ds

)– γα

)αn

]‖xn – p‖

+ αn∥∥γ f (p) –Ap

∥∥.It follows that

‖xn – p‖ ≤

ϕ()γ̄(

tn

∫ tn Ls ds

)– γα – dn

∥∥γ f (p) –Ap∥∥,

where dn =( tn

∫ tn Ls ds)–

αn. Thus, there exists n≥ such that

‖xn – p‖ ≤ ϕ()γ̄ – γα

∥∥γ f (p) –Ap∥∥.

Hence, {xn} is bounded, so are {f (xn)} and {A( tn

∫ tn T(s)xn ds)}.

Next, we show that ‖xn – T(h)xn‖ –→ as n –→ ∞. From (.), we note that

∥∥∥∥xn – tn

∫ tn

T(s)xn ds

∥∥∥∥ = αn

∥∥∥∥γ f (xn) –A(tn

∫ tn

T(s)xn ds

)∥∥∥∥.

By the condition (C), we obtain

limn–→∞

∥∥∥∥xn – tn

∫ tn

T(s)xn ds

∥∥∥∥ = . (.)

For all h≥ , we note that

∥∥xn – T(h)xn∥∥ ≤

∥∥∥∥xn – tn

∫ tn

T(s)xn ds

∥∥∥∥+

∥∥∥∥ tn

∫ tn

T(s)xn ds – T(h)

(tn

∫ tn

T(s)xn ds

)∥∥∥∥+

∥∥∥∥T(h)(tn

∫ tn

T(s)xn ds

)– T(h)xn

∥∥∥∥≤

∥∥∥∥xn – tn

∫ tn

T(s)xn ds

∥∥∥∥+

∥∥∥∥ tn

∫ tn

T(s)xn ds – T(h)

(tn

∫ tn

T(s)xn ds

)∥∥∥∥+ Lh

∥∥∥∥xn – tn

∫ tn

T(s)xn ds

∥∥∥∥.

By Lemma . and (.), we obtain

limn–→∞

∥∥xn – T(h)xn∥∥ = for all h≥ . (.)

Next, we show that x̃ ∈ Fix(S). By reflexivity of X and boundedness of {xn}, there exists aweakly convergent subsequence {xnj} of {xn} such that xnj ⇀ x̃ ∈ X as j –→ ∞. Since Jϕ is

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weakly continuous, we have by Lemma . that

lim supj–→∞

(‖xnj – x‖) = lim sup

j–→∞

(‖xnj – x̃‖) +(‖x – x̃‖) for all x ∈ X.

Let H(x) = lim supj–→∞ (‖xnj – x‖) for all x ∈ X. It follows that

H(x) =H(x̃) +(‖x – x̃‖) for all x ∈ X.

Since is continuous and limh–→∞ Lh = , it follows from (.) that

H(

limh–→∞

T(h)x̃)= lim

h–→∞H

(T(h)x̃

)

= limh–→∞

lim supj–→∞

(∥∥xnj – T(h)x̃

∥∥)

= limh–→∞

lim supj–→∞

(∥∥T(h)xnj – T(h)x̃

∥∥)

≤ limh–→∞

lim supj–→∞

(Lh‖xnj – x̃‖)

= lim supj–→∞

(‖xnj – x̃‖)

=H(x̃). (.)

On the other hand, we note that

H(

limh–→∞

T(h)x̃)= lim

h–→∞lim supj–→∞

(‖xnj – x̃‖) + lim

h–→∞

(∥∥T(h)x̃ – x̃∥∥)

= lim supj–→∞

(‖xnj – x̃‖) +

(lim

h–→∞∥∥T(h)x̃ – x̃

∥∥). (.)

Combining (.) and (.), we obtain (limh–→∞ ‖T(h)x̃ – x̃‖) ≤ . The property of

implies that limh–→∞ T(h)x̃ = x̃. In fact, since T(t + h)x = T(t)T(h)x for all x ∈ X and t ≥ ,then we have

x̃ = limh–→∞

T(h)x̃ = limh–→∞

T(h + t)x̃ = limh–→∞

T(h)T(t)x̃

= T(t) limh–→∞

T(h)x̃ = T(t)x̃,

for all t ≥ . Hence, x̃ ∈ Fix(S).Next, we show that {xn} is sequentially compact. Since (t) =

∫ t ϕ(τ )dτ , ∀t ≥ and

ϕ : [,∞) –→ [,∞) is the gauge function, then for ≥ k ≥ , ϕ(ky) ≤ ϕ(y) and

(kt) =∫ kt

ϕ(τ )dτ = k

∫ t

ϕ(ky)dy≤ k

∫ t

ϕ(y)dy = k(t).

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By Lemma ., we have

(‖xn – x̃‖) =

(∥∥∥∥αn(γ f (xn) –Ax̃

)+ (I – αnA)

(tn

∫ tn

T(s)xn ds – x̃

)∥∥∥∥)

≤

(∥∥∥∥(I – αnA)(tn

∫ tn

T(s)xn ds – x̃

)∥∥∥∥)+ αn

⟨γ f (xn) –Ax̃, Jϕ(xn – x̃)

⟩

=

(∥∥∥∥(I – αnA)(tn

∫ tn

T(s)xn ds – x̃

)∥∥∥∥)+ αnγ

⟨f (xn) – f (x̃), Jϕ(xn – x̃)

⟩

+ αn⟨γ f (x̃) –Ax̃, Jϕ(xn – x̃)

⟩

≤[tn

∫ tn

Ls ds –

(ϕ()γ̄ – γα

)αn

]

(‖xn – x̃‖)

+ αn⟨γ f (x̃) –Ax̃, Jϕ(xn – x̃)

⟩,

which implies that

(‖xn – x̃‖) ≤

ϕ()γ̄ – γα – dn⟨γ f (x̃) –Ax̃, Jϕ(xn – x̃)

⟩,

where dn =( tn

∫ tn Ls ds)–

αn. Thus, there exists n≥ such that

(‖xn – x̃‖) ≤

ϕ()γ̄ – γα

⟨γ f (x̃) –Ax̃, Jϕ(xn – x̃)

⟩.

In particular, we have

(‖xnj – x̃‖) ≤

ϕ()γ̄ – γα

⟨γ f (x̃) –Ax̃, Jϕ(xnj – x̃)

⟩. (.)

Since Jϕ is single-valued and weakly continuous, it follows that (‖xnj – x̃‖) –→ as j –→∞. The property of implies that xnj –→ x̃ as j –→ ∞.Next, we show that x̃ solves the variational inequality (.). From (.), we derive that

A(tn

∫ tn

T(s)xn ds

)– γ f (xn) =

αn

(tn

∫ tn

T(s)xn ds – xn

). (.)

For all v ∈ Fix(S), it follows from (.) that⟨A

(tn

∫ tn

T(s)xn ds

)– γ f (xn), Jϕ(xn – v)

⟩

=αn

⟨tn

∫ tn

T(s)xn ds – xn, Jϕ(xn – v)

⟩

=αn

[⟨tn

∫ tn

T(s)xn ds – v, Jϕ(xn – v)

⟩–

⟨xn – v, Jϕ(xn – v)

⟩]

≤ αn

[(tn

∫ tn

Ls ds

)

(‖xn – v‖) –(‖xn – v‖)

]

=( tn

∫ tn Ls ds) –

αn

(‖xn – v‖). (.)

Sunthrayuth and Kumam Fixed Point Theory and Applications 2012, 2012:177 Page 11 of 18http://www.fixedpointtheoryandapplications.com/content/2012/1/177

Now, replacing n by nj in (.) and letting j –→ ∞, we notice that

xnj –tnj

∫ tnj

T(s)xnj ds –→ .

By the condition (C), we obtain that

⟨γ f (x̃) –Ax̃, Jϕ(v – x̃)

⟩ ≤ , ∀v ∈ Fix(S).

That is, x̃ is a solution of the variational inequality (.).Finally, we show that {xn} converges strongly to x̃ ∈ Fix(S). Suppose that there exists

another subsequence xni –→ x̂ as j –→ ∞. We note that x̂ ∈ Fix(S) is the solution of thevariational inequality (.). Hence, x̃ = x̂ = x* by uniqueness. In summary, we have shownthat {xn} is sequentially compact and each cluster point of the sequence {xn} is equal to x*.Therefore, we conclude that xn –→ x* as n –→ ∞. This proof is complete. �

Remark . Theorem . extends and generalizes Theorem . of Zegeye et al. [], The-orem . of Chen and Song [] and Theorem . of [], in the following respects:() Theorem . generalizes Theorem . of Zegeye et al. [] to the viscosity iterative

method in a different Banach space which admits a weakly continuous duality mapping.() Theorem . improves Theorem . of Zegeye et al. [] in the sense that our theorem

is applicable in a uniformly convex Banach space without the requirement that S = {T(t) :t ∈R

+} is almost uniformly asymptotically regular.() Theorem . extends Theorem . of Chen and Song [] from a class of strongly

continuous semigroups of nonexpansivemappings to amore general class of strongly con-tinuous semigroups of asymptotically nonexpansive mappings.() Theorem . includes Theorem . of Li et al. [] as a special case.

If S = {T(t) : t ∈R+} is a strongly continuous semigroup of nonexpansive mappings, we

have Lt ≡ and then Theorem . is reduced to the following results.

Corollary . Let X be a uniformly convex Banach space which admits a weakly continu-ous duality mapping Jϕ with a gauge ϕ such that ϕ is invariant on [, ]. Let S = {T(t) : t ∈R

+} be a strongly continuous semigroup of nonexpansive mappings from X into itself suchthat Fix(S) �= ∅. Let f : X –→ X be a contraction mapping with a constant α ∈ (, ) andA : X –→ X be a strongly positive linear bounded operator with a constant γ̄ ∈ (, ) suchthat < γ < γ̄ ϕ()

α. Let {xn} be a sequence defined by

xn = αnγ f (xn) + (I – αnA)tn

∫ tn

T(s)xn ds, ∀n≥ , (.)

where {αn} is a sequence in (, ) such that limn–→∞ αn = and {tn} is a positive real diver-gent sequence. Then the sequence {xn} defined by (.) converges strongly to x* ∈ Fix(S),where x* is the unique solution of the variational inequality

⟨γ f

(x*

)–Ax*, Jϕ

(v – x*

)⟩ ≤ , ∀v ∈ Fix(S). (.)

Sunthrayuth and Kumam Fixed Point Theory and Applications 2012, 2012:177 Page 12 of 18http://www.fixedpointtheoryandapplications.com/content/2012/1/177

Corollary . (Li et al. [, Theorem .]) Let H be a real Hilbert space and C be anonempty closed convex subset of X such that C ± C ⊂ C. Let S = {T(t) : t ∈ R

+} bea strongly continuous semigroup of nonexpansive mappings from C into itself such thatFix(S) �= ∅. Let f : C –→ C be a contraction mapping with a constant α ∈ (, ) andA : C –→ C be a strongly positive linear bounded operator with a constant γ̄ ∈ (, ) suchthat < γ < γ̄

α. Let {xn} be a sequence defined by

xn = αnγ f (xn) + (I – αnA)tn

∫ tn

T(s)xn ds, ∀n≥ , (.)

where {αn} is a sequence in (, ) such that limn–→∞ αn = and {tn} is a positive real diver-gent sequence. Then the sequence {xn} defined by (.) converges strongly to x* ∈ Fix(S),where x* is the unique solution of the variational inequality

⟨γ f

(x*

)–Ax*, v – x*

⟩ ≤ , ∀v ∈ Fix(S). (.)

4 Explicit iteration schemeTheorem . Let X be a uniformly convex Banach space which admits a weakly contin-uous duality mapping Jϕ with a gauge ϕ such that ϕ is invariant on [, ]. Let S = {T(t) :t ∈R

+} be a strongly continuous semigroup of asymptotically nonexpansive mappings fromX into itself with a sequence {Lt} ⊂ [,∞) such that Fix(S) �= ∅. Let f : X –→ X be a con-traction mapping with a constant α ∈ (, ) and A : X –→ X be a strongly positive linearbounded operator with a constant γ̄ ∈ (, ) such that < γ < γ̄ ϕ()

α. For given x ∈ X, let

{xn} be a sequence defined by

xn+ = αnγ f (xn) + (I – αnA)tn

∫ tn

T(s)xn ds, ∀n≥ , (.)

where {αn} is a sequence in (, ) and {tn} is a positive real divergent sequence which satisfythe following conditions:(C) limn–→∞ αn = and

∑∞n= αn = ∞;

(C) limn–→∞( tn

∫ tn Ls ds)–

αn= .

Then the sequence {xn} defined by (.) converges strongly to x* ∈ Fix(S), where x* is theunique solution of the variational inequality

⟨γ f

(x*

)–Ax*, Jϕ

(v – x*

)⟩ ≤ , ∀v ∈ Fix(S). (.)

Proof By the condition limn–→∞ αn = , we may assume, with no loss of generality, thatαn ≤ ϕ()‖A‖– for all n ∈ N. By Lemma ., we have ‖I – αnA‖ ≤ ϕ()( – αnγ̄ ). First, weshow that {xn} is bounded. Take p ∈ Fix(S) and < ε < ϕ()γ̄ – γα.Since limn–→∞

( tn

∫ tn Ls ds)–

αn= implies ϕ()( – αnγ̄ )[(

tn

∫ tn Ls ds) – ] ≤ εαn for suffi-

ciently large n≥ . Then from (.), we get that

‖xn+ – p‖ =∥∥∥∥αn

(γ f (xn) –Ap

)+ (I – αnA)

(tn

∫ tn

T(s)xn ds – p

)∥∥∥∥≤ αn

∥∥γ f (xn) –Ap∥∥ + ϕ()( – αnγ̄ )

∥∥∥∥ tn

∫ tn

T(s)xn ds – p

∥∥∥∥

Sunthrayuth and Kumam Fixed Point Theory and Applications 2012, 2012:177 Page 13 of 18http://www.fixedpointtheoryandapplications.com/content/2012/1/177

≤ αnγ∥∥f (xn) – f (p)

∥∥ + αn∥∥γ f (p) –Ap

∥∥+ ϕ()( – αnγ̄ )

(tn

∫ tn

Ls ds

)‖xn – p‖

≤[ –

(ϕ()γ̄ – γα

)αn + ϕ()( – αnγ̄ )

[(tn

∫ tn

Ls ds

)–

]]‖xn – p‖

+ αn∥∥γ f (p) –Ap

∥∥≤ (

–(ϕ()γ̄ – γα – ε

)αn

)‖xn – p‖ + αn∥∥γ f (p) –Ap

∥∥

=( –

(ϕ()γ̄ – γα – ε

)αn

)‖xn – p‖ + (ϕ()γ̄ – γα – ε

)αn

‖γ f (p) –Ap‖ϕ()γ̄ – γα – ε

.

By induction, we have

‖xn – p‖ ≤ max

{‖x – p‖, ‖γ f (p) –Ap‖

ϕ()γ̄ – γα – ε

}, ∀n≥ .

Hence, {xn} is bounded, so are {f (xn)} and {A( tn

∫ tn T(s)xn ds)}.

Next, we show that ‖xn – T(h)xn‖ –→ as n –→ ∞. From (.), we note that

∥∥∥∥xn+ – tn

∫ tn

T(s)xn ds

∥∥∥∥ = αn

∥∥∥∥γ f (xn) –A(tn

∫ tn

T(s)xn ds

)∥∥∥∥.

By the condition (C), we obtain

limn–→∞

∥∥∥∥xn+ – tn

∫ tn

T(s)xn ds

∥∥∥∥ = . (.)

For all h≥ , we note that

∥∥xn+ – T(h)xn+∥∥ ≤

∥∥∥∥xn+ – tn

∫ tn

T(s)xn ds

∥∥∥∥+

∥∥∥∥ tn

∫ tn

T(s)xn ds – T(h)

(tn

∫ tn

T(s)xn ds

)∥∥∥∥+

∥∥∥∥T(h)(tn

∫ tn

T(s)xn ds

)– T(h)xn+

∥∥∥∥≤

∥∥∥∥xn+ – tn

∫ tn

T(s)xn ds

∥∥∥∥+

∥∥∥∥ tn

∫ tn

T(s)xn ds – T(h)

(tn

∫ tn

T(s)xn ds

)∥∥∥∥+ Lh

∥∥∥∥xn+ – tn

∫ tn

T(s)xn ds

∥∥∥∥.

By Lemma . and (.), we obtain limn–→∞ ‖xn+ – T(h)xn+‖ = and hence

limn–→∞

∥∥xn – T(h)xn∥∥ = for all h≥ . (.)

Sunthrayuth and Kumam Fixed Point Theory and Applications 2012, 2012:177 Page 14 of 18http://www.fixedpointtheoryandapplications.com/content/2012/1/177

Next, we show that

lim supn–→∞

⟨γ f

(x*

)–Ax*, Jϕ

(xn – x*

)⟩ ≤ .

Let {xnj} be a subsequence of {xn} such that

limj–→∞

⟨γ f

(x*

)–Ax*, Jϕ

(xnj – x*

)⟩= lim sup

n–→∞

⟨γ f

(x*

)–Ax*, Jϕ

(xn – x*

)⟩.

By reflexivity of X and boundedness of {xn}, there exists a weakly convergent subsequence{xnj} of {xn} such that xnj ⇀ v ∈ X as j –→ ∞. Since Jϕ is weakly continuous, we have byLemma . that

lim supj–→∞

(‖xnj – x‖) = lim sup

j–→∞

(‖xnj – v‖) +(‖x – v‖) for all x ∈ X.

Let H(x) = lim supj–→∞ (‖xnj – x‖) for all x ∈ X. It follows that

H(x) =H(v) +(‖x – v‖) for all x ∈ X.

Since is continuous and limh–→∞ Lh = , it follows from (.) that

H(

limh–→∞

T(h)v)= lim

h–→∞H

(T(h)v

)

= limh–→∞

lim supj–→∞

(∥∥xnj – T(h)v

∥∥)

= limh–→∞

lim supj–→∞

(∥∥T(h)xnj – T(h)v

∥∥)

≤ limh–→∞

lim supj–→∞

(Lh‖xnj – v‖)

= lim supj–→∞

(‖xnj – v‖)

=H(x̃). (.)

On the other hand, we note that

H(

limh–→∞

T(h)v)= lim

h–→∞lim supj–→∞

(‖xnj – v‖) + lim

h–→∞

(∥∥T(h)v – v∥∥)

= lim supj–→∞

(‖xnj – v‖) +

(lim

h–→∞∥∥T(h)v – v

∥∥). (.)

Combining (.) and (.), we obtain (limh–→∞ ‖T(h)v – v‖) ≤ . The property of

implies that limh–→∞ T(h)v = v. In fact, since T(t + h)x = T(t)T(h)x for all x ∈ X and t ≥ ,then we have

v = limh–→∞

T(h)v = limh–→∞

T(h + t)v = limh–→∞

T(h)T(t)v = T(t) limh–→∞

T(h)v = T(t)v,

Sunthrayuth and Kumam Fixed Point Theory and Applications 2012, 2012:177 Page 15 of 18http://www.fixedpointtheoryandapplications.com/content/2012/1/177

for all t ≥ . Hence, v ∈ Fix(S). Since Jϕ is single-valued and weakly continuous, we obtainthat

lim supn–→∞

⟨γ f

(x*

)–Ax*, Jϕ

(xn – x*

)⟩= lim

j–→∞⟨γ f

(x*

)–Ax*, Jϕ

(xnj – x*

)⟩

=⟨γ f

(x*

)–Ax*, Jϕ

(v – x*

)⟩ ≤ . (.)

Finally, we show that xn –→ x* as n –→ ∞. Now, from Lemma ., we have

(∥∥xn+ – x*

∥∥)

=

(∥∥∥∥αn(γ f (xn) –Ax*

)+ (I – αnA)

(tn

∫ tn

T(s)xn ds – x*

)∥∥∥∥)

=

(∥∥∥∥αnγ(f (xn) – f

(x*

))+ αn

(γ f

(x*

)–Ax*

)

+ (I – αnA)(tn

∫ tn

T(s)xn ds – x*

)∥∥∥∥)

≤

(∥∥∥∥αnγ(f (xn) – f

(x*

))+ (I – αnA)

(tn

∫ tn

T(s)xn ds – x*

)∥∥∥∥)

+ αn⟨γ f

(x*

)–Ax*, Jϕ

(xn+ – x*

)⟩

≤

({ –

(ϕ()γ̄ – γα

)αn + ϕ()( – αnγ̄ )

[(tn

∫ tn

Ls ds

)–

]}∥∥xn – x*∥∥)

+ αn⟨γ f

(x*

)–Ax*, Jϕ

(xn+ – x*

)⟩≤ (

–(ϕ()γ̄ – γα

)αn

)

(∥∥xn – x*∥∥)

+ ϕ()( – αnγ̄ )[(

tn

∫ tn

Ls ds

)–

]

(∥∥xn – x*∥∥)

+ αn⟨γ f

(x*

)–Ax*, Jϕ

(xn+ – x*

)⟩≤ (

–(ϕ()γ̄ – γα

)αn

)

(∥∥xn – x*∥∥)

+ ϕ()( – αnγ̄ )[(

tn

∫ tn

Ls ds

)–

]M + αn

⟨γ f

(x*

)–Ax*, Jϕ

(xn+ – x*

)⟩, (.)

whereM = supn≥{(‖xn – x*‖)}. Put σn := (ϕ()γ̄ – γα)αn and

δn := ϕ()( – αnγ̄ )[(

tn

∫ tn

Ls ds

)–

]M + αn

⟨γ f

(x*

)–Ax*, Jϕ

(xn+ – x*

)⟩.

Then (.) reduces to formula

(∥∥xn+ – x*

∥∥) ≤ ( – σn)(∥∥xn – x*

∥∥)+ δn.

It follows from the conditions (C), (C) and (.) that∑∞

n= σn = ∞ and

lim supn–→∞

δn

σn= lim sup

n–→∞

ϕ()γ̄ – γα

[ϕ()( – αnγ̄ )[(

tn

∫ tn Ls ds) – ]

αnM

+⟨γ f

(x*

)–Ax*, Jϕ

(xn+ – x*

)⟩] ≤ .

Sunthrayuth and Kumam Fixed Point Theory and Applications 2012, 2012:177 Page 16 of 18http://www.fixedpointtheoryandapplications.com/content/2012/1/177

Hence, by Lemma ., we obtain that (‖xn+ – x*‖) –→ as n –→ ∞. The property of implies that xn –→ x* as n –→ ∞. This proof is complete. �

Applications . Let X be a uniformly convex Banach space which admits a weakly con-tinuous duality mapping. Let L(X) be the space of all bounded linear operators on X. For� ∈ L(X), define S := {T(t) : t ∈ R

+} of bounded linear operators by using the followingexponential expression:

T(t) = e–t� :=∞∑k=

(–)k

k!tk�k .

Then, clearly, the family S := {T(t) : t ∈ R+} satisfies the semigroup properties. More-

over, this family forms a one-parameter semigroup of self-mappings of X because et� =[e–t� ]– : X –→ X exists for each t ∈R

+.

Next, the following example shows that all conditions of Theorem . are satisfied.

Example . For instance, let αn =n, tn = n and Lt = +

t+ . Then, clearly, the sequences{αn}, {tn} and {Lt} satisfy our assumptions and the condition (C) in Theorem .. Weshow that the condition (C) is achieved. Indeed, we have

tn

∫ tn Ls ds –

αn=

n

∫ n ( +

s+ )ds – /n

= n{ n

(s + ln(s + )|n

)–

}

= n{ n

(n + ln

(n +

))–

}

=ln

(n +

)n

–→ , as n –→ ∞.

Furthermore, if we take � ∈ L(X) such that ‖T(t)‖ ≤ + t+ and Fix(S) �= ∅ (see, e.g.,

p. of []) then the sequence {xn} defined by (.) converges strongly to x* ∈ Fix(S).

Remark . Theorem . extends and generalizes Theorem . of Zegeye et al. [], The-orem . of Chen and Song [] and Theorem . of Li et al. [] in the following respects:() Theorem . generalizes Theorem . of Zegeye et al. [] to the viscosity iterative

method in a different Banach space which admits a weakly continuous duality mapping.() Theorem . improves Theorem . of Zegeye et al. [] in the sense that our theorem

is applicable in a uniformly convex Banach space without the requirement that S = {T(t) :t ∈R

+} is almost uniformly asymptotically regular.() Theorem . extends Theorem . of Chen and Song [] from a class of strongly

continuous semigroups of nonexpansivemappings to amore general class of strongly con-tinuous semigroups of asymptotically nonexpansive mappings.() Theorem . includes Theorem . of Li et al. [] as a special case.

If S = {T(t) : t ∈R+} is a strongly continuous semigroup of nonexpansive mappings, we

have Lt ≡ and then Theorem . is reduced to the following result.

Sunthrayuth and Kumam Fixed Point Theory and Applications 2012, 2012:177 Page 17 of 18http://www.fixedpointtheoryandapplications.com/content/2012/1/177

Corollary . Let X be a uniformly convex Banach space which admits a weakly continu-ous duality mapping Jϕ with a gauge ϕ such that ϕ is invariant on [, ]. Let S = {T(t) : t ∈R

+} be a strongly continuous semigroup of nonexpansive mappings from X into itself suchthat Fix(S) �= ∅. Let f : X –→ X be a contraction mapping with a constant α ∈ (, ) andA : X –→ X be a strongly positive linear bounded operator with a constant γ̄ ∈ (, ) suchthat < γ < γ̄ ϕ()

α. For given x ∈ C, let {xn} be a sequence defined by

xn+ = αnγ f (xn) + (I – αnA)tn

∫ tn

T(s)xn ds, ∀n≥ , (.)

where {αn} is a sequence in (, ) such that limn–→∞ αn = and∑∞

n= αn = ∞, and {tn} is apositive real divergent sequence. Then the sequence {xn} defined by (.) converges stronglyto x* ∈ Fix(S), where x* is the unique solution of the variational inequality

⟨γ f

(x*

)–Ax*, Jϕ

(v – x*

)⟩ ≤ , ∀v ∈ Fix(S). (.)

Corollary . (Li et al. [, Theorem .]) Let H be a real Hilbert space and C be anonempty closed convex subset of X such that C ± C ⊂ C. Let S = {T(t) : t ∈ R

+} bea strongly continuous semigroup of nonexpansive mappings from C into itself such thatFix(S) �= ∅. Let f : C –→ C be a contraction mapping with a constant α ∈ (, ) andA : C –→ C be a strongly positive linear bounded operator with a constant γ̄ ∈ (, ) suchthat < γ < γ̄

α. For given x ∈ C, let {xn} be a sequence defined by

xn+ = αnγ f (xn) + (I – αnA)tn

∫ tn

T(s)xn ds, ∀n≥ , (.)

where {αn} is a sequence in (, ) such that limn–→∞ αn = and∑∞

n= αn = ∞, and {tn} is apositive real divergent sequence. Then the sequence {xn} defined by (.) converges stronglyto x* ∈ Fix(S), where x* is the unique solution of the variational inequality

⟨γ f

(x*

)–Ax*, v – x*

⟩ ≤ , ∀v ∈ Fix(S). (.)

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsAll authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

AcknowledgementsThe authors were supported by the Higher Education Research Promotion and National Research University Project ofThailand, Office of the Higher Education Commission (NRU-CSEC No. 55000613).

Received: 28 July 2012 Accepted: 2 October 2012 Published: 17 October 2012

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doi:10.1186/1687-1812-2012-177Cite this article as: Sunthrayuth and Kumam: Fixed point solutions of variational inequalities for a semigroup ofasymptotically nonexpansive mappings in Banach spaces. Fixed Point Theory and Applications 2012 2012:177.