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

Abstract

The purpose of this article is to introduce two iterative algorithms for finding a common fixed point of a semigroup of asymptotically nonexpansive mappings which is a unique solution of some variational inequality. We provide two algorithms, one implicit and another explicit, from which strong convergence theorems are obtained in a uniformly convex Banach space, which admits a weakly continuous duality mapping. The results in this article improve and extend the recent ones announced by 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.

1 Introduction

Throughout this paper, we denote by and R + the set of all positive integers and all positive real numbers, respectively. Let X be a real Banach space. A mapping T:XX is said to be nonexpansive if

TxTyxy,x,yX,

and T is asymptotically nonexpansive (see [1]) if there exists a sequence { k n } of positive real numbers with lim n k n =1 such that

T n x T n y k n xy,n1 and x,yX.

We denote by Fix(T) the set of fixed points of T, i.e., Fix(T)={xX:x=Tx}.

Recall that a self-mapping f:XX is a contraction if there exists a constant α(0,1) such that

f ( x ) f ( y ) αxy,x,yX.

A one-parameter family S={T(t):t R + } of X into itself is said to be a strongly continuous semigroup of Lipschitzian mappings if the following conditions are satisfied:

  1. (i)

    T(0)x=x for all xX;

  2. (ii)

    T(s+t)=T(s)T(t) for all s,t R + ;

  3. (iii)

    for each xX the mapping T()x from R + into X is continuous;

  4. (iv)

    for each t>0, there exists a bounded measurable function L t :(0,)[0,) such that

    T ( t ) x T ( t ) y L t xy,x,yX.

A strongly continuous semigroup of Lipschitzian mappings S is called strongly continuous semigroup of nonexpansive mappings if L t 1 for all t>0 and strongly continuous semigroup of asymptotically nonexpansive mappings if lim sup t L t 1. Note that for asymptotically nonexpansive semigroup S, we can always assume that the Lipschitzian constant { L t } t > 0 is such that L t 1 for each t>0, L t is nonincreasing in t and lim t L t =1; otherwise, we replace L t for each t>0 with L t ¯ :=max{ sup s t L s ,1}. S is said to have a fixed point if there exists x 0 X such that T(t) x 0 = x 0 for all t0. We denote by Fix(S) the set of fixed points of S, i.e., Fix(S)= t 0 Fix(T(t)) (for more details, see [24]).

A continuous operator of the semigroup S={T(t):t R + } is said to be uniformly asymptotically regular on X if for all h0 and any bounded subset C of X, lim t sup x C T(h)T(t)txT(t)x=0 (see [5] for examples of uniformly asymptotically regular semigroups).

Recently, convergence theorems for common fixed points of a strongly continuous semigroup of nonexpansive mappings and their generalizations have been studied by numerous authors (see, e.g., [610]). Construction of fixed points of nonexpansive mappings (and of common fixed points of nonexpansive semigroups) is an important subject in the theory of nonexpansive mappings and finds application in a number of applied areas, in particular, in image recovery and signal processing (see, e.g., [1116]). In the last ten years, the iterative methods for nonexpansive mappings have been applied to solve convex minimization problems; see, e.g., [1719]. 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 is a constant γ ¯ >0 with the property

Ax,x γ ¯ x 2 for all xH.
(1.1)

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 F 1 2 Ax,xx,b,
(1.2)

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

In 2003, Xu [19] proved that the sequence { x n } defined by the iterative method below, with the initial guess x 0 H chosen arbitrarily,

x n + 1 =(I α n A)T x n + α n u,n0,
(1.3)

converges strongly to the unique solution of the minimization problem (1.2) provided the sequence { α n } satisfies certain conditions. Using the viscosity approximation method, Moudafi [20] introduced the following iterative process for nonexpansive mappings (see [19] for further developments in both Hilbert and Banach spaces). Let f be a contraction on H. Starting with an arbitrary initial x 0 H, we define the sequence { x n } recursively by

x n + 1 = σ n f( x n )+(1 σ n )T x n ,n0,
(1.4)

where { σ n } is a sequence in (0,1). It is proved in [19, 20] that under certain appropriate conditions imposed on { σ n }, the sequence { x n } generated by (1.4) strongly converges to a unique solution x of the variational inequality

( f I ) x , x x 0,xF(T).
(1.5)

In 2006, Marino and Xu [21] combined the iterative method (1.3) with the viscosity approximation method (1.4) considering the following general iterative process:

x n + 1 = α n γf( x n )+(I α n A)T x n ,n0,
(1.6)

where 0<γ< γ ¯ α . They proved that the sequence { x n } generated by (1.6) converges strongly to a unique solution x of the variational inequality

( γ f A ) x , x x 0,xF(T),
(1.7)

which is the optimality condition for the minimization problem

min x C 1 2 Ax,xh(x),

where h is a potential function for γf (i.e., h (x)=γf(x) for xH).

On the other hand, Li et al.[22] considered the implicit and explicit viscosity iteration processes for a nonexpansive semigroup S={T(t):t R + } in a Hilbert space as follows:

x n = α n γf( x n )+(I α n A) 1 t n 0 t n T(s) x n ds,nN,
(1.8)
x n + 1 = α n γf( x n )+(I α n A) 1 t n 0 t n T(s) x n ds,nN,
(1.9)

where { α n } and { t n } are two sequences satisfying certain conditions. They proved the sequence { x n } defined by (1.8) and (1.9) converges strongly to x Fix(S), which solves the variational inequality (1.7). Under the framework of a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, Chen and Song [23] studied the strong convergence of the implicit and explicit viscosity iteration processes for a nonexpansive semigroup S={T(t):t R + } with Fix(S) as follows:

x n = α n f( x n )+(1 α n ) 1 t n 0 t n T(s) x n ds,nN,
(1.10)
x n + 1 = α n f( x n )+(1 α n ) 1 t n 0 t n T(s) x n ds,nN.
(1.11)

Very recently, Zegeye et al.[7] introduced the implicit and explicit iterative processes for a strongly continuous semigroup of asymptotically nonexpansive mappings S={T(t):t R + } in a reflexive and strictly convex Banach spaces with a uniformly Gâteaux differentiable norm as follows:

x n = α n u+(1 α n ) 1 t n 0 t n T(s) x n ds,nN,
(1.12)
x n + 1 = α n u+(1 α n ) 1 t n 0 t n T(s) x n ds,nN.
(1.13)

They proved that { x n } defined by (1.12) and (1.13) converges strongly to a common fixed point 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 mappings which is a unique solution of some variational inequality. We establish the strong convergence results in a uniformly convex Banach space which admits a weakly continuous duality mapping. The results in this article improve and extend the recent ones announced by Li et al.[22], Zegeye et al.[7] and many others.

2 Preliminaries

Throughout this paper, we write x n x (respectively x n x) to indicate that the sequence { x n } weakly (respectively weak) converges to x; as usual x n x will symbolize strong convergence; also, a mapping I will denote the identity mapping. Let X be a real Banach space, X be its dual space. Let U={xX:x=1}. A Banach space X is said to be uniformly convex if, for each ϵ(0,2], there exists a δ>0 such that for each x,yU, xyϵ implies x + y 2 1δ. It is know that a uniformly convex Banach space is reflexive and strictly convex (see also [24]). A Banach space is said to be smooth if the limit lim t 0 x + t y x t exists for each x,yU. It is also said to be uniformly smooth if the limit is attained uniformly for x,yU.

Let φ:[0,)[0,) be a continuous strictly increasing function such that φ(0)=0 and φ(t) as t. This function φ is called a gauge function. The duality mapping J φ :X 2 X associated with a gauge function φ is defined by

J φ (x)= { f X : x , f = x φ ( x ) , f = φ ( x ) , x X } ,

where , denotes the generalized duality paring. In particular, the duality mapping with the gauge function φ(t)=t, denoted by J is referred to as the normalized duality mapping. Clearly, the relation J φ (x)= φ ( x ) x J(x) holds for each x0 (see [25]).

Browder [25] initiated the study of certain classes of nonlinear operators by means of the duality mapping J φ . Following Browder [25], we say a Banach space X has a weakly continuous duality mapping if there exits a gauge function φ for which the duality mapping J φ (x) is single-valued and continuous from the weak topology to the weak topology; that is, for each { x n } with x n x, the sequence {J( x n )} converges weakly to J φ (x). It is known that l p has a weakly continuous duality mapping with a gauge function φ(t)= t p 1 for all 1<p<. Set Φ(t)= 0 t φ(τ)dτ, t0, then J φ (x)=Φ(x), where denotes the subdifferential in the sense of convex analysis (recall that the subdifferential of the convex function ϕ:XR at xX is the set ϕ(x)={ x X;ϕ(y)ϕ(x)+ x ,yx,yX}).

In a Banach space X which admits a duality mapping J φ with a gauge function φ, we say that an operator A is strongly positive (see [26]) if there exists a constant γ ¯ >0 with the property

A x , J φ ( x ) γ ¯ xφ ( x )
(2.1)

and

aIbA= sup x 1 | ( a I b A ) x , J φ ( x ) | ,a[0,1],b[1,1].
(2.2)

As special cases of (2.1), we have the following results.

  1. (1)

    If X is a smooth Banach space and φ(t)=t for all tX (see [27]), then the inequality (2.1) reduces to

    A x , J ( x ) γ ¯ x 2 .
    (2.3)
  2. (2)

    If X:=H is a real Hilbert space, then the inequality (2.1) reduces to (1.1).

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 [28].

Lemma 2.1 ([28])

Assume that a Banach space X has a weakly continuous duality mapping J φ with a gauge φ.

  1. (i)

    For all x,yX, the following inequality holds:

    Φ ( x + y ) Φ ( x ) + y , J φ ( x + y ) .
  2. (ii)

    Assume that a sequence { x n } in X converges weakly to a point xX. Then the following identity holds:

    lim sup n Φ ( x n y ) = lim sup n Φ ( x n x ) +Φ ( y x ) ,x,yX.

Lemma 2.2 ([26])

Assume that a Banach space X admits a duality mapping J φ with a gauge φ. Let A be a strongly positive linear bounded operator on X with a coefficient γ ¯ >0and0<ρφ(1) A 1 . ThenIρAφ(1)(1ρ γ ¯ ).

Definition 2.3 Let C be a closed convex subset of a real Banach space X. Let S={T(t):t R + } be a strongly continuous semigroup of asymptotically nonexpansive mappings from C into itself such that Fix(S). Then S is said to be almost uniformly asymptotically regular (in short a.u.a.r.) on C, if for all h0,

Lemma 2.4 ([7])

Let C be a closed convex subset of a uniformly convex Banach space X andS={T(t):t R + }be a strongly continuous semigroup of asymptotically nonexpansive mappings from C into itself with a sequence{ L t }[1,)such thatFix(S). Then for eachr>0andh0,

Lemma 2.5 ([29])

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.

3 Implicit iteration scheme

Theorem 3.1 Let X be a uniformly convex Banach space which admits a weakly continuous duality mapping J φ with a gauge φ such that φ is invariant on[0,1]. LetS={T(t):t R + }be a strongly continuous semigroup of asymptotically nonexpansive mappings from X into itself with a sequence{ L t }[1,)such thatFix(S). Letf:XXbe a contraction mapping with a constantα(0,1)andA:XXbe a strongly positive linear bounded operator with a constant γ ¯ (0,1)such that0<γ< γ ¯ φ ( 1 ) α . Let{ x n }be a sequence defined by

x n = α n γf( x n )+(I α n A) 1 t n 0 t n T(s) x n ds,n1,
(3.1)

where{ α n }is a sequence in(0,1)and{ t n }is a positive real divergent sequence which satisfy the following conditions:

(C1) lim n α n =0;

(C2) lim n ( 1 t n 0 t n L s d s ) 1 α n =0.

Then the sequence{ x n }defined by (3.1) converges strongly to x Fix(S), where x is the unique solution of the variational inequality

γ f ( x ) A x , J φ ( v x ) 0,vFix(S).
(3.2)

Proof First, we show that { x n } defined by (3.1) is well defined. For all nN, let us define the mapping

T n f := α n γf+(I α n A) 1 t n 0 t n T(s)ds.

Indeed, for all x,yX, we have

T n f x T n f y = α n γ ( f ( x ) f ( y ) ) + ( I α n A ) ( 1 t n 0 t n ( T ( s ) x T ( s ) y ) d s ) α n γ f ( x ) f ( y ) + I α n A ( 1 t n 0 t n T ( s ) x T ( s ) y d s ) α n γ α x y + φ ( 1 ) ( 1 α n γ ¯ ) ( 1 t n 0 t n L s d s ) x y [ 1 t n 0 t n L s d s φ ( 1 ) γ ¯ ( 1 t n 0 t n L s d s ) α n + γ α α n ] x y .

Since lim n ( 1 t n 0 t n L s d s ) 1 α n =0 implies

( 1 t n 0 t n L s d s ) 1 α n <φ(1) γ ¯ γαφ(1) γ ¯ ( 1 t n 0 t n L s d s ) γα,

for sufficiently large n1, that is,

1 t n 0 t n L s dsφ(1) γ ¯ ( 1 t n 0 t n L s d s ) α n +γα α n <1.

Thus, by the Banach contraction mapping principle, there exits a unique fixed point x n X, that is, { x n } defined by (3.1) is well defined.

Next, we show the uniqueness of a solution of the variational inequality (3.2). Suppose that x ˜ , x F(S) are solutions of (3.2), then

γ f ( x ) A x , J φ ( x ˜ x ) 0
(3.3)

and

γ f ( x ˜ ) A x ˜ , J φ ( x x ˜ ) 0.
(3.4)

Adding up (3.3) and (3.4), we obtain

0 ( γ f ( x ) A x ) ( γ f ( x ˜ ) A x ˜ ) , J φ ( x ˜ x ) = A ( x ˜ x ) , J φ ( x ˜ x ) γ f ( x ˜ ) f ( x ) , J φ ( x ˜ x ) γ ¯ x ˜ x φ ( x ˜ x ) γ f ( x ˜ ) f ( x ) J φ ( x ˜ x ) γ ¯ Φ ( x ˜ x ) γ α Φ ( x ˜ x ) = ( γ ¯ γ α ) Φ ( x ˜ x ) ( φ ( 1 ) γ ¯ γ α ) Φ ( x ˜ x ) ,

which is a contradiction. We must have x ˜ = x and the uniqueness is proved. Below, we use x ˜ to denote the unique solution of the variational inequality (3.2).

Next, we show that { x n } is bounded. Take pFix(S). Then from (3.1), we get that

x n p = α n ( γ f ( x n ) A p ) + ( I α n A ) ( 1 t n 0 t n T ( s ) x n d s p ) α n γ f ( x n ) A p + φ ( 1 ) ( 1 α n γ ¯ ) 1 t n 0 t n T ( s ) x n d s p α n γ f ( x n ) f ( p ) + α n γ f ( p ) A p + φ ( 1 ) ( 1 α n γ ¯ ) ( 1 t n 0 t n L s d s ) x n p [ 1 t n 0 t n L s d s ( φ ( 1 ) γ ¯ ( 1 t n 0 t n L s d s ) γ α ) α n ] x n p + α n γ f ( p ) A p .

It follows that

x n p 1 φ ( 1 ) γ ¯ ( 1 t n 0 t n L s d s ) γ α d n γ f ( p ) A p ,

where d n = ( 1 t n 0 t n L s d s ) 1 α n . Thus, there exists n1 such that

x n p 1 φ ( 1 ) γ ¯ γ α γ f ( p ) A p .

Hence, { x n } is bounded, so are {f( x n )} and {A( 1 t n 0 t n T(s) x n ds)}.

Next, we show that x n T(h) x n 0 as n. From (3.1), we note that

x n 1 t n 0 t n T ( s ) x n d s = α n γ f ( x n ) A ( 1 t n 0 t n T ( s ) x n d s ) .

By the condition (C1), we obtain

lim n x n 1 t n 0 t n T ( s ) x n d s =0.
(3.5)

For all h0, we note that

x n T ( h ) x n x n 1 t n 0 t n T ( s ) x n d s + 1 t n 0 t n T ( s ) x n d s T ( h ) ( 1 t n 0 t n T ( s ) x n d s ) + T ( h ) ( 1 t n 0 t n T ( s ) x n d s ) T ( h ) x n x n 1 t n 0 t n T ( s ) x n d s + 1 t n 0 t n T ( s ) x n d s T ( h ) ( 1 t n 0 t n T ( s ) x n d s ) + L h x n 1 t n 0 t n T ( s ) x n d s .

By Lemma 2.4 and (3.5), we obtain

lim n x n T ( h ) x n =0for all h0.
(3.6)

Next, we show that x ˜ Fix(S). By reflexivity of X and boundedness of { x n }, there exists a weakly convergent subsequence { x n j } of { x n } such that x n j x ˜ X as j. Since J φ is weakly continuous, we have by Lemma 2.1 that

lim sup j Φ ( x n j x ) = lim sup j Φ ( x n j x ˜ ) +Φ ( x x ˜ ) for all xX.

Let H(x)= lim sup j Φ( x n j x) for all xX. It follows that

H(x)=H( x ˜ )+Φ ( x x ˜ ) for all xX.

Since Φ is continuous and lim h L h =1, it follows from (3.6) that

H ( lim h T ( h ) x ˜ ) = lim h H ( T ( h ) x ˜ ) = lim h lim sup j Φ ( x n j T ( h ) x ˜ ) = lim h lim sup j Φ ( T ( h ) x n j T ( h ) x ˜ ) lim h lim sup j Φ ( L h x n j x ˜ ) = lim sup j Φ ( x n j x ˜ ) = H ( x ˜ ) .
(3.7)

On the other hand, we note that

H ( lim h T ( h ) x ˜ ) = lim h lim sup j Φ ( x n j x ˜ ) + lim h Φ ( T ( h ) x ˜ x ˜ ) = lim sup j Φ ( x n j x ˜ ) + Φ ( lim h T ( h ) x ˜ x ˜ ) .
(3.8)

Combining (3.7) and (3.8), we obtain Φ( lim h T(h) x ˜ x ˜ )0. The property of Φ implies that lim h T(h) x ˜ = x ˜ . In fact, since T(t+h)x=T(t)T(h)x for all xX and t0, then we have

x ˜ = lim h T ( h ) x ˜ = lim h T ( h + t ) x ˜ = lim h T ( h ) T ( t ) x ˜ = T ( t ) lim h T ( h ) x ˜ = T ( t ) x ˜ ,

for all t0. Hence, x ˜ Fix(S).

Next, we show that { x n } is sequentially compact. Since Φ(t)= 0 t φ(τ)dτ, t0 and φ:[0,)[0,) is the gauge function, then for 1k0, φ(ky)φ(y) and

Φ(kt)= 0 k t φ(τ)dτ=k 0 t φ(ky)dyk 0 t φ(y)dy=kΦ(t).

By Lemma 2.1, we have

Φ ( x n x ˜ ) = Φ ( α n ( γ f ( x n ) A x ˜ ) + ( I α n A ) ( 1 t n 0 t n T ( s ) x n d s x ˜ ) ) Φ ( ( I α n A ) ( 1 t n 0 t n T ( s ) x n d s x ˜ ) ) + α n γ f ( x n ) A x ˜ , J φ ( x n x ˜ ) = Φ ( ( I α n A ) ( 1 t n 0 t n T ( s ) x n d s x ˜ ) ) + α n γ f ( x n ) f ( x ˜ ) , J φ ( x n x ˜ ) + α n γ f ( x ˜ ) A x ˜ , J φ ( x n x ˜ ) [ 1 t n 0 t n L s d s ( φ ( 1 ) γ ¯ γ α ) α n ] Φ ( x n x ˜ ) + α n γ f ( x ˜ ) A x ˜ , J φ ( x n x ˜ ) ,

which implies that

Φ ( x n x ˜ ) 1 φ ( 1 ) γ ¯ γ α d n γ f ( x ˜ ) A x ˜ , J φ ( x n x ˜ ) ,

where d n = ( 1 t n 0 t n L s d s ) 1 α n . Thus, there exists n1 such that

Φ ( x n x ˜ ) 1 φ ( 1 ) γ ¯ γ α γ f ( x ˜ ) A x ˜ , J φ ( x n x ˜ ) .

In particular, we have

Φ ( x n j x ˜ ) 1 φ ( 1 ) γ ¯ γ α γ f ( x ˜ ) A x ˜ , J φ ( x n j x ˜ ) .
(3.9)

Since J φ is single-valued and weakly continuous, it follows that Φ( x n j x ˜ )0 as j. The property of Φ implies that x n j x ˜ as j.

Next, we show that x ˜ solves the variational inequality (3.2). From (3.1), we derive that

A ( 1 t n 0 t n T ( s ) x n d s ) γf( x n )= 1 α n ( 1 t n 0 t n T ( s ) x n d s x n ) .
(3.10)

For all vFix(S), it follows from (3.10) that

A ( 1 t n 0 t n T ( s ) x n d s ) γ f ( x n ) , J φ ( x n v ) = 1 α n 1 t n 0 t n T ( s ) x n d s x n , J φ ( x n v ) = 1 α n [ 1 t n 0 t n T ( s ) x n d s v , J φ ( x n v ) x n v , J φ ( x n v ) ] 1 α n [ ( 1 t n 0 t n L s d s ) Φ ( x n v ) Φ ( x n v ) ] = ( 1 t n 0 t n L s d s ) 1 α n Φ ( x n v ) .
(3.11)

Now, replacing n by n j in (3.11) and letting j, we notice that

x n j 1 t n j 0 t n j T(s) x n j ds0.

By the condition (C2), we obtain that

γ f ( x ˜ ) A x ˜ , J φ ( v x ˜ ) 0,vFix(S).

That is, x ˜ is a solution of the variational inequality (3.2).

Finally, we show that { x n } converges strongly to x ˜ Fix(S). Suppose that there exists another subsequence x n i x ˆ as j. We note that x ˆ Fix(S) is the solution of the variational inequality (3.2). Hence, x ˜ = x ˆ = x by uniqueness. In summary, we have shown that { x n } is sequentially compact and each cluster point of the sequence { x n } is equal to x . Therefore, we conclude that x n x as n. This proof is complete. □

Remark 3.2 Theorem 3.1 extends and generalizes Theorem 3.5 of Zegeye et al.[7], Theorem 3.1 of Chen and Song [23] and Theorem 3.1 of [22], in the following respects:

  1. (1)

    Theorem 3.1 generalizes Theorem 3.5 of Zegeye et al. [7] to the viscosity iterative method in a different Banach space which admits a weakly continuous duality mapping.

  2. (2)

    Theorem 3.1 improves Theorem 3.5 of Zegeye et al. [7] 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.

(3) Theorem 3.1 extends Theorem 3.1 of Chen and Song [23] from a class of strongly continuous semigroups of nonexpansive mappings to a more general class of strongly continuous semigroups of asymptotically nonexpansive mappings.

  1. (4)

    Theorem 3.1 includes Theorem 3.1 of Li et al. [22] as a special case.

If S={T(t):t R + } is a strongly continuous semigroup of nonexpansive mappings, we have L t 1 and then Theorem 3.1 is reduced to the following results.

Corollary 3.3 Let X be a uniformly convex Banach space which admits a weakly continuous duality mapping J φ with a gauge φ such that φ is invariant on[0,1]. LetS={T(t):t R + }be a strongly continuous semigroup of nonexpansive mappings from X into itself such thatFix(S). Letf:XXbe a contraction mapping with a constantα(0,1)andA:XXbe a strongly positive linear bounded operator with a constant γ ¯ (0,1)such that0<γ< γ ¯ φ ( 1 ) α . Let{ x n }be a sequence defined by

x n = α n γf( x n )+(I α n A) 1 t n 0 t n T(s) x n ds,n1,
(3.12)

where{ α n }is a sequence in(0,1)such that lim n α n =0and{ t n }is a positive real divergent sequence. Then the sequence{ x n }defined by (3.12) converges strongly to x Fix(S), where x is the unique solution of the variational inequality

γ f ( x ) A x , J φ ( v x ) 0,vFix(S).
(3.13)

Corollary 3.4 (Li et al. [[22], Theorem 3.1])

Let H be a real Hilbert space and C be a nonempty closed convex subset of X such thatC±CC. LetS={T(t):t R + }be a strongly continuous semigroup of nonexpansive mappings from C into itself such thatFix(S). Letf:CCbe a contraction mapping with a constantα(0,1)andA:CCbe a strongly positive linear bounded operator with a constant γ ¯ (0,1)such that0<γ< γ ¯ α . Let{ x n }be a sequence defined by

x n = α n γf( x n )+(I α n A) 1 t n 0 t n T(s) x n ds,n1,
(3.14)

where{ α n }is a sequence in(0,1)such that lim n α n =0and{ t n }is a positive real divergent sequence. Then the sequence{ x n }defined by (3.14) converges strongly to x Fix(S), where x is the unique solution of the variational inequality

γ f ( x ) A x , v x 0,vFix(S).
(3.15)

4 Explicit iteration scheme

Theorem 4.1 Let X be a uniformly convex Banach space which admits a weakly continuous duality mapping J φ with a gauge φ such that φ is invariant on[0,1]. LetS={T(t):t R + }be a strongly continuous semigroup of asymptotically nonexpansive mappings from X into itself with a sequence{ L t }[1,)such thatFix(S). Letf:XXbe a contraction mapping with a constantα(0,1)andA:XXbe a strongly positive linear bounded operator with a constant γ ¯ (0,1)such that0<γ< γ ¯ φ ( 1 ) α . For given x 1 X, let{ x n }be a sequence defined by

x n + 1 = α n γf( x n )+(I α n A) 1 t n 0 t n T(s) x n ds,n1,
(4.1)

where{ α n }is a sequence in(0,1)and{ t n }is a positive real divergent sequence which satisfy the following conditions:

(C1) lim n α n =0and n = 1 α n =;

(C2) lim n ( 1 t n 0 t n L s d s ) 1 α n =0.

Then the sequence{ x n }defined by (4.1) converges strongly to x Fix(S), where x is the unique solution of the variational inequality

γ f ( x ) A x , J φ ( v x ) 0,vFix(S).
(4.2)

Proof By the condition lim n α n =0, we may assume, with no loss of generality, that α n φ(1) A 1 for all nN. By Lemma 2.2, we have I α n Aφ(1)(1 α n γ ¯ ). First, we show that { x n } is bounded. Take pFix(S) and 0<ϵ<φ(1) γ ¯ γα.

Since lim n ( 1 t n 0 t n L s d s ) 1 α n =0 implies φ(1)(1 α n γ ¯ )[( 1 t n 0 t n L s ds)1]ϵ α n for sufficiently large n1. Then from (4.1), we get that

x n + 1 p = α n ( γ f ( x n ) A p ) + ( I α n A ) ( 1 t n 0 t n T ( s ) x n d s p ) α n γ f ( x n ) A p + φ ( 1 ) ( 1 α n γ ¯ ) 1 t n 0 t n T ( s ) x n d s p α n γ f ( x n ) f ( p ) + α n γ f ( p ) A p + φ ( 1 ) ( 1 α n γ ¯ ) ( 1 t n 0 t n L s d s ) x n p [ 1 ( φ ( 1 ) γ ¯ γ α ) α n + φ ( 1 ) ( 1 α n γ ¯ ) [ ( 1 t n 0 t n L s d s ) 1 ] ] x n p + α n γ f ( p ) A p ( 1 ( φ ( 1 ) γ ¯ γ α ϵ ) α n ) x n p + α n γ f ( p ) A p = ( 1 ( φ ( 1 ) γ ¯ γ α ϵ ) α n ) x n p + ( φ ( 1 ) γ ¯ γ α ϵ ) α n γ f ( p ) A p φ ( 1 ) γ ¯ γ α ϵ .

By induction, we have

x n pmax { x 1 p , γ f ( p ) A p φ ( 1 ) γ ¯ γ α ϵ } ,n1.

Hence, { x n } is bounded, so are {f( x n )} and {A( 1 t n 0 t n T(s) x n ds)}.

Next, we show that x n T(h) x n 0 as n. From (4.1), we note that

x n + 1 1 t n 0 t n T ( s ) x n d s = α n γ f ( x n ) A ( 1 t n 0 t n T ( s ) x n d s ) .

By the condition (C1), we obtain

lim n x n + 1 1 t n 0 t n T ( s ) x n d s =0.
(4.3)

For all h0, we note that

x n + 1 T ( h ) x n + 1 x n + 1 1 t n 0 t n T ( s ) x n d s + 1 t n 0 t n T ( s ) x n d s T ( h ) ( 1 t n 0 t n T ( s ) x n d s ) + T ( h ) ( 1 t n 0 t n T ( s ) x n d s ) T ( h ) x n + 1 x n + 1 1 t n 0 t n T ( s ) x n d s + 1 t n 0 t n T ( s ) x n d s T ( h ) ( 1 t n 0 t n T ( s ) x n d s ) + L h x n + 1 1 t n 0 t n T ( s ) x n d s .

By Lemma 2.4 and (4.3), we obtain lim n x n + 1 T(h) x n + 1 =0 and hence

lim n x n T ( h ) x n =0for all h0.
(4.4)

Next, we show that

lim sup n γ f ( x ) A x , J φ ( x n x ) 0.

Let { x n j } be a subsequence of { x n } such that

lim j γ f ( x ) A x , J φ ( x n j x ) = lim sup n γ f ( x ) A x , J φ ( x n x ) .

By reflexivity of X and boundedness of { x n }, there exists a weakly convergent subsequence { x n j } of { x n } such that x n j vX as j. Since J φ is weakly continuous, we have by Lemma 2.1 that

lim sup j Φ ( x n j x ) = lim sup j Φ ( x n j v ) +Φ ( x v ) for all xX.

Let H(x)= lim sup j Φ( x n j x) for all xX. It follows that

H(x)=H(v)+Φ ( x v ) for all xX.

Since Φ is continuous and lim h L h =1, it follows from (4.4) that

H ( lim h T ( h ) v ) = lim h H ( T ( h ) v ) = lim h lim sup j Φ ( x n j T ( h ) v ) = lim h lim sup j Φ ( T ( h ) x n j T ( h ) v ) lim h lim sup j Φ ( L h x n j v ) = lim sup j Φ ( x n j v ) = H ( x ˜ ) .
(4.5)

On the other hand, we note that

H ( lim h T ( h ) v ) = lim h lim sup j Φ ( x n j v ) + lim h Φ ( T ( h ) v v ) = lim sup j Φ ( x n j v ) + Φ ( lim h T ( h ) v v ) .
(4.6)

Combining (4.5) and (4.6), we obtain Φ( lim h T(h)vv)0. The property of Φ implies that lim h T(h)v=v. In fact, since T(t+h)x=T(t)T(h)x for all xX and t0, then we have

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

for all t0. Hence, vFix(S). Since J φ is single-valued and weakly continuous, we obtain that

lim sup n γ f ( x ) A x , J φ ( x n x ) = lim j γ f ( x ) A x , J φ ( x n j x ) = γ f ( x ) A x , J φ ( v x ) 0 .
(4.7)

Finally, we show that x n x as n. Now, from Lemma 2.1, we have

Φ ( x n + 1 x ) = Φ ( α n ( γ f ( x n ) A x ) + ( I α n A ) ( 1 t n 0 t n T ( s ) x n d s x ) ) = Φ ( α n γ ( f ( x n ) f ( x ) ) + α n ( γ f ( x ) A x ) + ( I α n A ) ( 1 t n 0 t n T ( s ) x n d s x ) ) Φ ( α n γ ( f ( x n ) f ( x ) ) + ( I α n A ) ( 1 t n 0 t n T ( s ) x n d s x ) ) + α n γ f ( x ) A x , J φ ( x n + 1 x ) Φ ( { 1 ( φ ( 1 ) γ ¯ γ α ) α n + φ ( 1 ) ( 1 α n γ ¯ ) [ ( 1 t n 0 t n L s d s ) 1 ] } x n x ) + α n γ f ( x ) A x , J φ ( x n + 1 x ) ( 1 ( φ ( 1 ) γ ¯ γ α ) α n ) Φ ( x n x ) + φ ( 1 ) ( 1 α n γ ¯ ) [ ( 1 t n 0 t n L s d s ) 1 ] Φ ( x n x ) + α n γ f ( x ) A x , J φ ( x n + 1 x ) ( 1 ( φ ( 1 ) γ ¯ γ α ) α n ) Φ ( x n x ) + φ ( 1 ) ( 1 α n γ ¯ ) [ ( 1 t n 0 t n L s d s ) 1 ] M + α n γ f ( x ) A x , J φ ( x n + 1 x ) ,
(4.8)

where M= sup n 1 {Φ( x n x )}. Put σ n :=(φ(1) γ ¯ γα) α n and

δ n :=φ(1)(1 α n γ ¯ ) [ ( 1 t n 0 t n L s d s ) 1 ] M+ α n γ f ( x ) A x , J φ ( x n + 1 x ) .

Then (4.8) reduces to formula

Φ ( x n + 1 x ) (1 σ n )Φ ( x n x ) + δ n .

It follows from the conditions (C1), (C2) and (4.7) that n = 1 σ n = and

lim sup n δ n σ n = lim sup n 1 φ ( 1 ) γ ¯ γ α [ φ ( 1 ) ( 1 α n γ ¯ ) [ ( 1 t n 0 t n L s d s ) 1 ] α n M + γ f ( x ) A x , J φ ( x n + 1 x ) ] 0 .

Hence, by Lemma 2.5, we obtain that Φ( x n + 1 x )0 as n. The property of Φ implies that x n x as n. This proof is complete. □

Applications 4.2 Let X be a uniformly convex Banach space which admits a weakly continuous duality mapping. Let L(X) be the space of all bounded linear operators on X. For ΨL(X), define S:={T(t):t R + } of bounded linear operators by using the following exponential expression:

T(t)= e t Ψ := k = 0 ( 1 ) k k ! t k Ψ k .

Then, clearly, the family S:={T(t):t R + } satisfies the semigroup properties. Moreover, this family forms a one-parameter semigroup of self-mappings of X because e t Ψ = [ e t Ψ ] 1 :XX exists for each t R + .

Next, the following example shows that all conditions of Theorem 4.1 are satisfied.

Example 4.3 For instance, let α n = 1 n , t n = n 2 and L t =1+ 1 t + 1 . Then, clearly, the sequences { α n }, { t n } and { L t } satisfy our assumptions and the condition (C1) in Theorem 4.1. We show that the condition (C2) is achieved. Indeed, we have

1 t n 0 t n L s d s 1 α n = 1 n 2 0 n 2 ( 1 + 1 s + 1 ) d s 1 1 / n = n { 1 n 2 ( s + ln ( s + 1 ) | 0 n 2 ) 1 } = n { 1 n 2 ( n 2 + ln ( n 2 + 1 ) ) 1 } = ln ( n 2 + 1 ) n 0 , as  n .

Furthermore, if we take ΨL(X) such that T(t)1+ 1 t + 1 and Fix(S) (see, e.g., p.160 of [30]) then the sequence { x n } defined by (4.1) converges strongly to x Fix(S).

Remark 4.4 Theorem 4.1 extends and generalizes Theorem 3.5 of Zegeye et al.[7], Theorem 3.2 of Chen and Song [23] and Theorem 3.2 of Li et al.[22] in the following respects:

  1. (1)

    Theorem 4.1 generalizes Theorem 3.5 of Zegeye et al. [7] to the viscosity iterative method in a different Banach space which admits a weakly continuous duality mapping.

  2. (2)

    Theorem 4.1 improves Theorem 3.5 of Zegeye et al. [7] 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.

(3) Theorem 4.1 extends Theorem 3.2 of Chen and Song [23] from a class of strongly continuous semigroups of nonexpansive mappings to a more general class of strongly continuous semigroups of asymptotically nonexpansive mappings.

  1. (4)

    Theorem 4.1 includes Theorem 3.2 of Li et al. [22] as a special case.

If S={T(t):t R + } is a strongly continuous semigroup of nonexpansive mappings, we have L t 1 and then Theorem 4.1 is reduced to the following result.

Corollary 4.5 Let X be a uniformly convex Banach space which admits a weakly continuous duality mapping J φ with a gauge φ such that φ is invariant on[0,1]. LetS={T(t):t R + }be a strongly continuous semigroup of nonexpansive mappings from X into itself such thatFix(S). Letf:XXbe a contraction mapping with a constantα(0,1)andA:XXbe a strongly positive linear bounded operator with a constant γ ¯ (0,1)such that0<γ< γ ¯ φ ( 1 ) α . For given x 1 C, let{ x n }be a sequence defined by

x n + 1 = α n γf( x n )+(I α n A) 1 t n 0 t n T(s) x n ds,n1,
(4.9)

where{ α n }is a sequence in(0,1)such that lim n α n =0and n = 1 α n =, and{ t n }is a positive real divergent sequence. Then the sequence{ x n }defined by (4.9) converges strongly to x Fix(S), where x is the unique solution of the variational inequality

γ f ( x ) A x , J φ ( v x ) 0,vFix(S).
(4.10)

Corollary 4.6 (Li et al. [[22], Theorem 3.2])

Let H be a real Hilbert space and C be a nonempty closed convex subset of X such thatC±CC. LetS={T(t):t R + }be a strongly continuous semigroup of nonexpansive mappings from C into itself such thatFix(S). Letf:CCbe a contraction mapping with a constantα(0,1)andA:CCbe a strongly positive linear bounded operator with a constant γ ¯ (0,1)such that0<γ< γ ¯ α . For given x 1 C, let{ x n }be a sequence defined by

x n + 1 = α n γf( x n )+(I α n A) 1 t n 0 t n T(s) x n ds,n1,
(4.11)

where{ α n }is a sequence in(0,1)such that lim n α n =0and n = 1 α n =, and{ t n }is a positive real divergent sequence. Then the sequence{ x n }defined by (4.11) converges strongly to x Fix(S), where x is the unique solution of the variational inequality

γ f ( x ) A x , v x 0,vFix(S).
(4.12)

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Acknowledgements

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

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Sunthrayuth, P., Kumam, P. Fixed point solutions of variational inequalities for a semigroup of asymptotically nonexpansive mappings in Banach spaces. Fixed Point Theory Appl 2012, 177 (2012). https://doi.org/10.1186/1687-1812-2012-177

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Keywords

  • iterative approximation method
  • common fixed point
  • semigroup of asymptotically nonexpansive mapping
  • strong convergence theorem
  • uniformly convex Banach space