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The modified general iterative methods for asymptotically nonexpansive semigroups in Banach spaces

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Abstract

In this paper, we introduce the modified general iterative methods for finding a common fixed point of asymptotically nonexpansive semigroups, which is a unique solution of some variational inequality. We prove the strong convergence theorems of such iterative scheme in a real Banach space which has a uniformly Gâteaux differentiable norm and admits the duality mapping j φ and uniform normal structure. The main result extends various results existing in the current literature.

1 Introduction

Let E be a normed linear space, K be a nonempty, closed and convex subset of E. Let T be a self-mapping on K. Then T is said to be asymptotically nonexpansive if there exists a sequence { k n }[1,) with lim n k n =1 such that

T n x T n y k n xy,for all x,yK and n1.

The class of asymptotically nonexpansive maps was introduced by Goebel and Kirk [1] as an important generalization of the class of nonexpansive maps (i.e., mapping T:KK such that TxTyxy, x,yK).

A mapping T is said to be uniformly L-Lipschitzian, if there exists a constant L>0 such that

T n x T n y Lxy,for all x,yK and n1.

It is clear that every asymptotically nonexpansive is uniformly L-Lipschitzian with a constant L= sup n 1 k n 1. We use F(T) to denote the set of fixed points of T, that is, F(T)={xK:Tx=x}.

A self-mapping f:KK is a contraction on K if there exists a constant α(0,1) such that

f ( x ) f ( y ) αxy, for all x,yK.
(1.1)

We use Π K to denote the collection of all contractions on K. That is,

Π K ={f:f is a contraction on K}.

A family S={T(s):0s<} of mappings of K into itself is called an asymptotically nonexpansive semigroup on K if it satisfies the following conditions:

  1. (i)

    T(0)x=x for all xK;

  2. (ii)

    T(s+t)=T(s)T(t) for all s,t0;

  3. (iii)

    there exists a sequence { k n }[1,) with lim n k n =1 such that for all x,yK T n (t)x T n (t)y k n xy, t0, n1;

  4. (iv)

    for all xK, the mapping tT(t)x is continuous.

An asymptotically nonexpansive semigroup S is called nonexpansive semigroup if k n =1 for all n1. We denote by F(S) the set of all common fixed points of S, that is,

F(S):= { x K : T ( t ) x = x , 0 t < } = t 0 F ( T ( t ) ) .

A gauge function φ is a continuous strictly increasing function φ:[0,)[0,) such that φ(0)=0 and φ(t) as t. Let E be the dual space of E. The duality mapping J φ :E 2 E associated to a gauge function φ is defined by

J φ (x)= { f E : x , f = x φ ( x ) , f = φ ( x ) } ,xE.

In particular, the duality mapping with the gauge function φ(t)=t, denoted by J, is referred to as the normalized duality mapping. Clearly, there holds the relation J φ (x)= φ ( x ) x J(x) for all x0 (see [2]). Set

Φ(t)= 0 t φ(τ)dτ,t0,

then

J φ (x)=Φ ( x ) ,xE,

where denotes the subdifferential in the sense of convex analysis. Furthermore, Φ is a continuous convex and strictly increasing function on [0,) (see [3]).

In a Banach space E having duality mapping J φ with a gauge function φ, an operator A is said to be strongly positive [4] if there exists a constant γ ¯ >0 with the property

A x , J φ ( x ) γ ¯ xφ ( x )
(1.2)

and

αIβA= sup x 1 | ( α I β A ) x , J φ ( x ) | ,α[0,1],β[1,1],
(1.3)

where I is the identity mapping. If E:=H is a real Hilbert space, then the inequality (1.2) reduces to

Ax,x γ ¯ x 2 for all xH.
(1.4)

Let uC. Then, for each t(0,1) and for a nonexpansive map T, there exists a unique point x t C satisfying the following condition:

x t =(1t)T x t +tu,

since the mapping G t (x)=(1t)Tx+tu is a contraction. When H is a Hilbert space and T is a self-map, Browder [5] showed that { x t } converges strongly to an element of F(T), which is nearest to u as t 0 + . This result was extended to various more general Banach space by Morales and Jung [6], Takahashi and Ueda [7], Reich [8] and a host of other authors. Many authors (see, e.g. [9, 10]) have also shown convergence of the path

x n =(1 α n ) T n x n + α n u

in Banach spaces for asymptotically nonexpansive mapping self-map T under some conditions on α n . In 2009, motivated and inspired by Moudafi [11], Shahzad and Udomene [12] introduced and studied the iterative procedures for the approximation of common fixed points of asymptotically nonexpansive mappings in a real Banach space with uniformly Gâteaux differentiable norm and uniform normal structure.

Let S be a nonexpasive semigroup on K. In 2002, Suzuki [13] introduced, in Hilbert space, the implicit iteration

u n =(1 α n )T( t n ) u n + α n u,uK,n1,
(1.5)

where { α n } is a sequence in (0,1), { t n } is a sequence of positive real numbers. Under certain restrictions to the sequence { α n } and { t n }, Suzuki proved strong convergence of (1.5) to a member of F(S) nearest to u. In 2005, Xu [14] extended Suzuki [13]’s result from Hilbert space to a uniformly convex Banach space having a weakly continuous duality map j φ with gauge function φ. In 2009, Chang et al. [15] introduced the following implicit and explicit schemes for an asymptotically nonexpansive semigroup:

y n =(1 α n ) T n ( t n ) y n + α n u,uK,n1,
(1.6)

and

x n + 1 =(1 β n ) T n ( t n ) x n + β n u,uK,n0,
(1.7)

where α n , β n (0,1) and t n R + in a real Banach space with uniformly Gâteaux differentiable norm and uniform normal structure. Suppose, in addition, that lim n x n T(t) x n =0 and lim n y n T(t) y n =0 uniformly in t[0,). Then the { x n } and { y n } converge strongly to a point of F(S).

Very recently, motivated and inspired by Moudafi [11], Cholumjiak and Suantai [16] studied the following implicit and explicit viscosity methods:

y n = α n f( y n )+(1 α n )T( t n ) y n ,n1,
(1.8)

and

x n + 1 = α n f( x n )+(1 α n )T( t n ) x n ,n0.
(1.9)

They obtained the strong convergence theorems in the frame work of a real reflexive strictly convex Banach space which has a uniformly Gâteaux differentiable norm and admits the duality mapping j φ . For more related results, see [1719].

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,xx,b,
(1.10)

where C is the fixed point set of a nonexpansive mapping T on H and b is a given point in H. In 2009, motivated and inspired by Marino and Xu [20], Li et al. [21] introduced the following general iterative procedures for the approximation of common fixed points of a nonexpansive semigroup {T(s):s0} on a nonempty, closed and convex subset K in a Hilbert space:

y n = α n γf( y n )+(I α n A) 1 t n 0 t n T(s) y n ds,n1,
(1.11)

and

x n + 1 = α n γf( x n )+(I α n A) 1 t n 0 t n T(s) x n ds,n0,
(1.12)

where { α n } and { t n } are sequences in [0,1] and (0,), respectively, A is a strongly positive bounded linear operator on C and f is a contraction on C. And their convergence theorems can be proved under some appropriate control conditions on parameter { α n } and { t n }. Furthermore, by using these results, they obtained two mean ergodic theorems for nonexpansive mappings in a Hilbert space. Many authors extended the Li et al. [21]’s results in direction of algorithms and spaces (see [2227]).

In this paper, inspired and motivated by Chang et al. [15], Cholamjiak and Suantai [16], Li, Li and Su [21], Wangkeeree and Wangkeeree [24] and Wangkeeree et al. [4], we introduce the following iterative approximation methods (1.13) and (1.14) for the class of strongly continuous semigroup of asymptotically nonexpansive mappings S={T(s):0s<}:

y n = α n γf( y n )+(I α n A) T n ( t n ) y n ,n1,
(1.13)

and

x n + 1 = β n γf( x n )+(I β n A) T n ( t n ) x n ,n0,
(1.14)

where A is a strongly positive bounded linear operator on K and f is a contraction on K. The strong convergence theorems of the iterative approximation methods (1.13) and (1.14) in a real Banach space which has a uniformly Gâteaux differentiable norm and admits the duality mapping j φ are studied. Moreover, we study the strong convergence results of the following two iterative approximation methods (1.15) and (1.16):

w n + 1 = β n γf ( T n ( t n ) w n ) +(I β n A) T n ( t n ) w n ,n0,
(1.15)

and

z n + 1 = T n ( t n ) ( β n γ f ( z n ) + ( I β n A ) T n ( t n ) z n ) ,n0.
(1.16)

2 Preliminaries

Throughout this paper, let E be a real Banach space and E be its dual space. 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. A Banach space E is said to uniformly convex if, for any ϵ(0,2], there exists δ>0 such that, for any x,yU, xyϵ implies x + y 2 1δ. It is known that a uniformly convex Banach space is reflexive and strictly convex (see also [28]). Let U={xE:x=1}. A Banach space E is said to be smooth if the limit

lim t 0 x + t y x t

exists for all x,yU. In this case, the norm of E is said to be Gâteaux differentiable. The space E is said to have a uniformly Gâteaux differentiable if for each yU, the limit attained uniformly for xU. The space E is said to have a Fréchet differentiable if for each xU, the limit attained uniformly for yU and uniformly Fréchet differentiable if, the limit attained uniformly for x,yU. It is well known that (uniformly) Gâteaux differentiable of the norm of E implies (uniformly) Fréchet differentiable.

The following Lemma can be found in [16].

Lemma 2.1 [[16], Lemma 2.6]

Let E be a Banach space which has a uniformly Gâteaux differentiable norm and admits the duality mapping J φ , then J φ is uniformly continuous from the norm topology of E to the weak topology of E on each bounded subset of E.

The next lemma is an immediate consequence of the subdifferential inequality can be found in [3].

Lemma 2.2 [3]

Assume that a Banach space E which admits a duality mapping J φ with gauge φ. For all x,yE, the following inequality holds:

Φ ( x + y ) Φ ( x ) + y , j φ ( x + y ) , j φ (x+y) J φ (x+y).

Let K be a nonempty, bounded, closed and convex subset of a Banach space E. The diameter of K be defined by d(K):=sup{xy:x,yK}. For each xK, denote r(x,K)=sup{xy:x,yK} and denote by r(K):=inf{r(x,K):xK} the chebyshev radius of K relative to itself. The normal structure coefficient N(E) of E is defined by

N(E):=inf { d ( K ) r ( K ) : K  is a bounded, closed and convex of  E  with  d ( K ) > 0 } .

A Banach space E is said to have uniform normal structure if N(E)>1. It is known that every Banach space with a uniform normal structure is reflexive. Every uniformly convex and uniformly smooth Banach spaces have uniform normal structure.

The following existence theorem of an asymptotically nonexpansive mapping is useful tools for our proof.

Theorem 2.3 [[9], Theorem 1]

Suppose E is a Banach space with uniformly normal structure, K is a nonempty bounded subset of E, and T:KK is a uniformly k-Lipschitzian mapping with k<N ( X ) 1 / 2 . Suppose also there exists a nonempty, bounded, closed and convex subset K of K with the following property (P):

x K implies ω w (x) K ,

where ω w (x) is the weak ω-limit set if T at x, i.e., the set

{ y E : y = weak lim j T n j x for some n j } .

Then T has a fixed point in K .

In order to prove our main result, we need the following lemmas and definitions.

Let l be the Banach space of all bounded real-valued sequences. Let LIM be a continuous linear functional on l satisfying LIM=1=LIM(1). Then we know that LIM is mean on if and only if

inf{ a n :nN}LIM(a)sup{ a n :nN}

for every a=( a 1 , a 2 ,) l . Occasionally, we shall use LIM n ( a n ) instead of LIM(a). A mean LIM on is called a Banach limit if

LIM n ( a n )= LIM n ( a n + 1 )

for every a=( a 1 , a 2 ,) l . Using the Hahn-Banach theorem, or the Tychonoff fixed-point theorem, we can prove the existence of a Banach limit. We know that if μ is a Banach limit, then

lim inf n a n LIM n ( a n ) lim sup n a n

for every a=( a 1 , a 2 ,) l .

Subsequently, the following result was showed in [16].

Proposition 2.4 [[16], Proposition 3.2]

Let K be a nonempty, closed and convex subset of a real Banach space E which has a uniformly Gâteaux differentiable norm and admits the duality mapping J φ . Suppose that { x n } is a bounded sequence of K and let LIM n be a Banach limit and zE. Then

LIM n Φ ( x n z ) = inf y K LIM n Φ ( x n y ) ,

if and only if

LIM n y z , j φ ( x n z ) 0,yK.

The next valuable lemma is proved for applying our main results.

Lemma 2.5 [[4], Lemma 3.1]

Assume that a Banach space E which admits a duality mapping J φ with gauge φ. Let A be a strong positive linear bounded operator on E with coefficient γ ¯ >0 and 0<ρφ(1) A 1 . Then IρAφ(1)(1ρ γ ¯ ).

In the following, we also need the following lemma that can be found in the existing literature [29].

Lemma 2.6 [[29], Lemma 2.1]

Let { a n } be a sequence of nonnegative real number satisfying the property

a n + 1 (1 γ n ) a n + γ n β n ,n0,

where { γ n }(0,1) and { β n }R such that n = 0 γ n = and lim sup n β n 0. Then { a n } converges to zero, as n.

Lemma 2.7 [3]

Let C be a nonempty, closed and convex subset of a reflexive Banach space E and f:C(,] a proper lower semicontinuous convex function such that f( x n ) as x n . Then there exists x 0 D(f) such that f( x 0 )= inf x C f(x).

3 Main theorem

Theorem 3.1 Let E be a real Banach space with uniform normal structure which has a uniformly Gateaux differentiable norm and admits the duality mapping J φ , K be a nonempty, bounded, closed and convex subset of E such that K±KK. Let S={T(s):s0} be an asymptotically nonexpansive semigroup on K with a sequence { k n }[1,+), lim n k n =1 and sup n 1 k n <N ( E ) 1 / 2 such that F(S). Let f Π K with coefficient α(0,1), A a strongly positive bounded linear operator on K with coefficient γ ¯ >0 and 0<γ< φ ( 1 ) γ ¯ α and let { α n } and { t n } be sequences of real numbers such that 0< α n <1, t n >0. Then the following hold:

  1. (i)

    If k n 1 α n <φ(1) γ ¯ γα, n1, then there exists a sequence { y n }E defined by

    y n = α n γf( y n )+(I α n A) T n ( t n ) y n ,n1.
    (3.1)
  2. (ii)

    Suppose, in addition, lim n y n T(t) y n =0 uniformly in t[0,) and the real sequence { α n } satisfies lim n α n =0 and lim n k n 1 α n =0.

Then { y n } converges strongly as n to a common fixed point x ˜ in F(S) which solves the variational inequality:

( A γ f ) x ˜ , J φ ( x ˜ z ) 0,zF(S).
(3.2)

Proof We first show that the uniqueness of a solution of the variational inequality (3.2). Suppose both x ˜ F(S) and x F(S) are solutions to (3.2), then

( A γ f ) x ˜ , J φ ( x ˜ x ) 0
(3.3)

and

( A γ f ) x , J φ ( x x ˜ ) 0.
(3.4)

Adding (3.3) and (3.4), we obtain

( A γ f ) x ˜ ( A γ f ) x , J φ ( x ˜ x ) 0.
(3.5)

Noticing that for any x,yK,

(3.6)

Therefore, x ˜ = x and the uniqueness is proved. Below, we use x ˜ to denote the unique solution of (3.2). Since lim n α n =0, we may assume, without loss of generality, that α n <φ(1) A 1 . For each integer n1, define a mapping G n :KK by

G n (y)= α n γf(y)+(I α n A) T n ( t n )y,yK.

We shall show that G n is a contraction mapping. For any x,yK,

G n ( x ) G n ( y ) = α n γ f ( x ) + ( I α n A ) T n ( t n ) x α n γ f ( y ) ( I α n A ) T n ( t n ) y d s α n γ ( f ( x ) f ( y ) ) + ( I α n A ) ( T n ( t n ) x T n ( t n ) y ) α n γ α x y + φ ( 1 ) ( 1 α n γ ¯ ) k n x y = ( α n γ α + φ ( 1 ) ( 1 α n γ ¯ ) k n ) x y ( k n α n γ α + φ ( 1 ) α n γ ¯ k n ) x y ( k n α n ( φ ( 1 ) γ ¯ k n γ α ) ) x y .

Since 0< k n 1 α n <φ(1) γ ¯ γα, we have

0< k n 1 α n <φ(1) γ ¯ γαφ(1) γ ¯ k n γα.

It then follows that 0<( k n α n (φ(1) γ ¯ k n γα))<1. We have G n is a contraction map with coefficient ( k n α n (φ(1) γ ¯ k n γα)). Then, for each n1, there exists a unique y n K such that G n ( y n )= y n , that is,

y n = α n γf( y n )+(I α n A) T n ( t n ) y n ,n1.

Hence, (i) is proved.

  1. (ii)

    Define μ:KR by

    μ(y)= LIM n Φ ( y n y ) ,yK,

where LIM n is a Banach limit on l . Since μ is continuous and convex and g(z) as z, and E is reflexive, by Lemma 2.7, g attains its infimum over E. Let uK be such that

LIM n Φ ( y n u ) = inf y E LIM n Φ ( y n y ) .
(3.7)

Let

K := { z E : μ ( z ) = inf y K μ ( y ) } .

We have that K is a nonempty, bounded, closed and convex subset of K and also has the property (P), indeed, if x K and w ω w (x), i.e. w=weak lim j T m j x as j. Notice that, lim n y n T(t) y n =0 uniformly in t[0,), by induction we can prove that for all m1

lim n y n T m ( t ) y n =0 uniformly in t[0,).
(3.8)

From (3.8) and weakly lower semicontinuous of μ, and for each h0, we have that

μ ( w ) lim inf j μ ( T m j ( h ) x ) lim sup m μ ( T m ( h ) x ) = lim sup m LIM n Φ ( y n T m ( h ) x ) lim sup m [ LIM n Φ ( y n T m ( h ) y n + T m ( h ) y n T m ( h ) x ) ] = lim sup m LIM n Φ ( T m ( h ) y n T m ( h ) x ) lim sup m LIM n Φ ( k m y n x ) = LIM n Φ ( y n x ) = μ ( x ) = inf y K μ ( y ) ,

which implies that K satisfies the property (P). By Theorem 2.3, there exists a element zK such that zF(S) K .

Since K±KK, we have z+γf(z)AzK. By Proposition 2.4,

LIM n z + γ f ( z ) A z z , J φ ( y n z ) 0,

it implies that

LIM n γ f ( z ) A z , J φ ( y n z ) 0.
(3.9)

In fact, since Φ(t)= 0 t φ(τ)dτ, t0, and φ:[0,)[0,) is a gauge function, then for 1k0, φ(kx)φ(x) and

Φ(kt)= 0 k t φ(τ)dτ=k 0 t φ(kx)dxk 0 t φ(x)dx=kΦ(t).

It follows from Lemma 2.2 that

Φ ( y n z ) = Φ ( ( I α n A ) T n ( t n ) y n ( I α n A ) z + α n ( γ f ( y n ) γ f ( z ) + γ f ( z ) A z ) ) Φ ( ( I α n A ) T n ( t n ) y n ( I α n A ) z + α n γ ( f ( y n ) f ( z ) ) ) + α n γ f ( z ) A z , J φ ( y n z ) Φ ( φ ( 1 ) ( 1 α n γ ¯ ) T n ( t n ) y n T n ( t n ) z + α n γ α y n z ) + α n γ f ( z ) f ( z ) , J φ ( y n z ) Φ ( φ ( 1 ) ( 1 α n γ ¯ ) ( k n ) y n z + α n γ α y n z ) + α n γ f ( z ) A z , J φ ( y n z ) Φ ( [ φ ( 1 ) ( 1 α n γ ¯ ) ( k n ) + α n γ α ] y n z ) + α n γ f ( z ) A z , J φ ( y n z ) [ φ ( 1 ) ( 1 α n γ ¯ ) k n + α n γ α ] Φ ( y n z ) + α n γ f ( z ) A z , J φ ( y n z ) .

This implies that

Φ ( y n z ) 1 1 φ ( 1 ) ( 1 α n γ ¯ ) k n + α n γ α α n γ f ( z ) A z , J φ ( y n z ) ,

also

Φ ( y n z ) 1 ( φ ( 1 ) γ ¯ α γ ) ( 1 α n γ ¯ ) d n γ f ( z ) A z , J φ ( y n z ) ,

where d n = k n 1 α n . Thus

LIM n Φ ( y n z ) LIM n ( 1 ( φ ( 1 ) γ ¯ α γ ) ( 1 α n γ ¯ ) d n γ f ( z ) A z , J φ ( y n z ) ) 0 ,

and hence

( φ ( 1 ) γ ¯ γ α ) LIM n Φ ( y n z ) 0.

Since φ(1) γ ¯ >γα, LIM n Φ( y n z)=0, and then there exists a subsequence { y n j } of { y n } such that y n j z as j, we shall denoted by { y j }.

Next, we prove that z solves the variational inequality (3.2). From (3.1), we have

(Aγf) y n = 1 α n (I α n A) ( I T n ( t n ) ) y n .

On the other hand, note for all x,yK,

For pF(S), we have

( A γ f ) y n , J φ ( y n p ) = 1 α n ( I α n A ) ( I T n ( t n ) y n , J φ ( y n p ) = 1 α n ( I T n ( t n ) ) y n ( I T n ( t n ) ) p , J φ ( y n p ) + A ( I T n ( t n ) ) y n , J φ ( y n p ) k n 1 α n Φ ( x y ) + A ( I T n ( t n ) ) y n , J φ ( y n p ) k n 1 α n Φ ( x y ) + A y n T n ( t n ) y n M ,

where M sup n 1 φ( y n p). Replacing y n with y n j and letting j, note that y n T n ( t n ) y n 0 and lim n k n 1 α n =0, we have that

( A γ f ) z , J φ ( z p ) 0,pF(T).

That is, zF(S) is a solution of (3.2). Then z= x ˜ . In summary, we have that each cluster point of { y n } converges strongly to x ˜ as n. This completes the proof. □

If AI, the identity mapping on K, and γ=1, then Theorem 3.1 reduces to the following corollary.

Corollary 3.2 Let E be a real Banach space with uniform normal structure which has a uniformly Gateaux differentiable norm and admits the duality mapping J φ , K be a nonempty, bounded, closed and convex subset of E. Let S={T(s):s0} be an asymptotically nonexpansive semigroup on K with a sequence { k n }[1,+), lim n k n =1 and sup n 1 k n <N ( E ) 1 / 2 such that F(S). Let f Π K with coefficient α(0,1) and let { α n } and { t n } be sequences of real numbers such that 0< α n <1, t n >0. Then the following hold:

  1. (i)

    If k n 1 α n <1α, n1, then there exists a sequence { y n }K

defined by

y n = α n f( y n )+(1 α n ) T n ( t n ) y n ,n1.
(3.10)
  1. (ii)

    Suppose, in addition, lim n y n T(t) y n =0 uniformly in t[0,) and the real sequences { α n } satisfies lim n α n =0 and lim n k n 1 α n =0.

Then { y n } converges strongly as n to a common fixed point x ˜ in F(S) which solves the variational inequality:

( I f ) x ˜ , J φ ( x ˜ z ) 0,zF(S).
(3.11)

If fu, the constant mapping on K, then Corollary 3.2 reduces to the following corollary.

Corollary 3.3 Let E be a real Banach space with uniform normal structure which has a uniformly Gateaux differentiable norm and admits the duality mapping J φ , K be a nonempty, bounded, closed and convex subset of E. Let S={T(s):s0} be an asymptotically nonexpansive semigroup on K with a sequence { k n }[1,+), lim n k n =1 and sup n 1 k n <N ( E ) 1 / 2 such that F(S). Let { α n } and { t n } be sequences of real numbers such that 0< α n <1, t n >0. Then the following hold:

  1. (i)

    If k n 1 α n <1, n1, then there exists a sequence { y n }K defined by

    y n = α n u+(1 α n ) T n ( t n ) y n ,n1.
    (3.12)
  2. (ii)

    Suppose, in addition, lim n y n T(t) y n =0 uniformly in t[0,) and the real sequences { α n } satisfies lim n α n =0 and lim n k n 1 α n =0.

Then { y n } converges strongly as n to a common fixed point x ˜ in F(S), which solves the variational inequality:

x ˜ u , J φ ( x ˜ z ) 0,zF(S).
(3.13)

Next, we present the convergence theorem for the explicit scheme.

Theorem 3.4 Let E be a real Banach space with uniform normal structure, which has a uniformly Gateaux differentiable norm and admits the duality mapping J φ , K be a nonempty, bounded, closed and convex subset of E such that K±KK. Let S={T(s):s0} be an asymptotically nonexpansive semigroup on K with a sequence { k n }[1,+), lim n k n =1 and sup n 1 k n <N ( E ) 1 / 2 such that F(S). Let f Π K with coefficient α(0,1), A a strongly positive bounded linear operator on K with coefficient γ ¯ >0 and 0<γ< φ ( 1 ) γ ¯ α . Let { β n } and { t n } be sequences of real numbers such that 0< β n <1, t n 0,

(C1) lim n β n =0;

(C2) lim n k n 1 β n =0;

(C3) n = 0 β n =.

For any x 0 K, let the sequences { x n } be defined by

x n + 1 = β n γf( x n )+(I β n A) T n ( t n ) x n ds,n0.
(3.14)

Suppose, in addition, lim n x n T(t) x n =0 uniformly in t[0,). Then { x n } converge strongly as n to the same point x ˜ in F(S), which solves the variational inequality (3.2).

Proof By Theorem 3.1, there exists a unique solution x ˜ in F(S) which solves the variational inequality (3.2) and y m x ˜ as m. Next, we prove that

lim sup n γ f ( x ˜ ) A x ˜ , J φ ( x n x ˜ ) 0.
(3.15)

For all m1, n1, we have

y m x n = α m γ f ( y m ) + ( I α m A ) T m ( t m ) y m x n = α m ( γ f ( y m ) A y m ) + ( T m ( t m ) y m T m ( t m ) x n ) + ( T m ( t m ) x n x n ) + α m ( A y m A T m ( t m ) y m ) .

It follows from Lemma 2.2 that

Φ ( y m x n ) = Φ ( ( I α m A ) T m ( t m ) y m ( I α m A ) x n + α m ( γ f ( y m ) γ f ( x ˜ ) + γ f ( x ˜ ) A x ˜ + A x ˜ A x n ) ) Φ ( ( I α m A ) T m ( t m ) x n ( I α m A ) x n + α m γ ( f ( x m ) f ( x ˜ ) ) + A x ˜ A x n ) + α n γ f ( x ˜ ) A ( x ˜ ) , J φ ( y n x ˜ ) Φ ( φ ( 1 ) ( 1 α n γ ¯ ) T n ( t n ) x n T n ( t n ) x ˜ + α n γ α x n x ˜ ) + α n γ f ( x ˜ ) f ( x ˜ ) , J φ ( y n x ˜ ) Φ ( φ ( 1 ) ( 1 α n γ ¯ ) k n x n x ˜ + α n γ α x n x ˜ ) + α n γ f ( x ˜ ) A ( x ˜ ) , J φ ( x n x ˜ ) Φ ( [ φ ( 1 ) ( 1 α n γ ¯ ) k n + α n γ α ] x n x ˜ ) + α n γ f ( x ˜ ) A ( x ˜ ) , J φ ( x n x ˜ ) [ φ ( 1 ) ( 1 α n γ ¯ ) k n + α n γ α ] Φ ( x n x ˜ ) + α n γ f ( x ˜ ) A ( x ˜ ) , J φ ( x n x ˜ ) [ ( 1 φ ( 1 ) α n γ ¯ ) ( k n 1 ) + 1 α n ( φ ( 1 ) γ ¯ γ α ) ] Φ ( x n x ˜ ) + α n γ f ( x ˜ ) A ( x ˜ ) , J φ ( x n x ˜ ) ( 1 α n ( φ ( 1 ) γ ¯ γ α ) ) Φ ( x n x ˜ ) + ( 1 φ ( 1 ) α n γ ¯ ) ( k n 1 ) M + α n γ f ( x ˜ ) A ( x ˜ ) , J φ ( x n x ˜ ) .
(3.16)

It follows that

Since K is bounded, so that { x n } and { y m } are all bounded, and hence

(3.17)

where M is a constant satisfying M sup n , m N φ( x n y m ). By our hypothesis, lim n x n T(t) x n =0, uniformly in t[0,). By induction, we can prove that for all m1

lim n x n T m ( t ) x n =0,uniformly in t[0,).

Hence for all m1, we have

lim n x n T m ( t m ) x n =0,as n.
(3.18)

Therefore, taking upper limit as n in (3.17), we have

(3.19)

Since K is bounded, it follows from (C1) that

y m T m ( t m ) y n = α n γ f ( y m ) + A T m ( t m ) y m 0as m.
(3.20)

And then, taking upper limit as m in (3.19), by (C3) and (3.20), we get

lim sup m lim sup n γ f ( y m ) A y m , J φ ( x n y m ) 0.
(3.21)

On the other hand, since lim m y m = x ˜ due to the fact the duality mapping J φ is norm-to-weak uniformly continuous on bounded subset of E, it implies that

Therefore, for any given ε>0, there exists a positive number N such that for all mN

γ f ( x ˜ ) A x ˜ , J φ ( x n x ˜ ) γ f ( y m ) A y m , J φ ( x n y m ) +ε.

It follows from (3.21) that

Since ε is arbitrary, we have

lim sup n γ f ( x ˜ ) A x ˜ , J φ ( x n x ˜ ) 0.
(3.22)

Finally, we show that x n x ˜ as n.

Φ ( x n + 1 x ˜ ) = Φ ( ( I β n A ) T n ( t n ) x n ( I β n A ) x ˜ + β n ( γ f ( x n ) γ f ( x ˜ ) + γ f ( x ˜ ) A ( x ˜ ) ) ) Φ ( ( I β n A ) T n ( t n ) x n ( I β n A ) x ˜ + β n γ ( f ( x n ) f ( x ˜ ) ) ) + β n γ f ( x ˜ ) A ( x ˜ ) , J φ ( y n x ˜ ) Φ ( φ ( 1 ) ( 1 β n γ ¯ ) T n ( t n ) x n T n ( t n ) x ˜ + β n γ α x n x ˜ ) + β n γ f ( x ˜ ) f ( x ˜ ) , J φ ( y n x ˜ ) Φ ( φ ( 1 ) ( 1 β n γ ¯ ) k n x n x ˜ + β n γ α x n x ˜ ) + β n γ f ( x ˜ ) A ( x ˜ ) , J φ ( x n x ˜ ) Φ ( [ φ ( 1 ) ( 1 β n γ ¯ ) k n + β n γ α ] x n x ˜ ) + β n γ f ( x ˜ ) A ( x ˜ ) , J φ ( x n x ˜ ) [ φ ( 1 ) ( 1 β n γ ¯ ) k n + β n γ α ] Φ ( x n x ˜ ) + β n γ f ( x ˜ ) A ( x ˜ ) , J φ ( x n x ˜ ) [ ( 1 φ ( 1 ) β n γ ¯ ) ( k n 1 ) + 1 β n ( φ ( 1 ) γ ¯ γ α ) ] Φ ( x n x ˜ ) + β n γ f ( x ˜ ) A ( x ˜ ) , J φ ( x n x ˜ ) ( 1 β n ( φ ( 1 ) γ ¯ γ α ) ) Φ ( x n x ˜ ) + ( 1 φ ( 1 ) β n γ ¯ ) ( k n 1 ) M + β n γ f ( x ˜ ) A ( x ˜ ) , J φ ( x n x ˜ ) ,
(3.23)

where M >0 such that sup n 1 Φ( x n x ˜ ) M . Put

s n = β n ( φ ( 1 ) γ ¯ γ α )

and

σ n = ( 1 φ ( 1 ) β n γ ¯ φ ( 1 ) γ ¯ γ α ) ( k n 1 β n ) M + 1 ( φ ( 1 ) γ ¯ γ α ) γ f ( x ˜ ) A ( x ˜ ) , J φ ( x n x ˜ ) .

Then (3.23) is reduced to

Φ ( x n + 1 x ˜ ) (1 s n )Φ ( x n x ˜ ) + s n σ n .
(3.24)

Applying Lemma 2.6 to (3.24), we conclude that Φ( x n + 1 x ˜ )0 as n; that is, x n x ˜ as n. This completes the proof. □

Using Theorem 3.4, we obtain the following two strong convergence theorems of new iterative approximation methods for an asymptotically nonexpansive semigroup S.

Theorem 3.5 Let E be a real Banach space with uniform normal structure, which has a uniformly Gateaux differentiable norm and admits the duality mapping J φ , K be a nonempty, bounded, closed and convex subset of E such that K±KK. Let S={T(s):s0} be an asymptotically nonexpansive semigroup on K with a sequence { k n }[1,+), lim n k n =1 and sup n 1 k n <N ( E ) 1 / 2 such that F(S). Let f Π K with coefficient α(0,1), A a strongly positive bounded linear operator on K with coefficient γ ¯ >0 and 0<γ< φ ( 1 ) γ ¯ α . Let { β n } and { t n } be sequences of real numbers such that 0< β n <1, t n 0,

(C1) lim n β n =0;

(C2) lim n k n 1 β n =0;

(C3) n = 0 β n =.

For any w 0 K, let the sequence { w n } be defined by

w n + 1 = β n γf ( T n ( t n ) w n ) +(I β n A) T n ( t n ) w n ,n0.
(3.25)

Then { w n } converges strongly as n to a point x ˜ in F(S) which solves the variational inequality (3.2).

Proof Let { x n } be the sequence given by x 0 = w 0 and

x n + 1 = β n γf( x n )+(I β n A) T n ( t n ) x n ,n0.

By Theorem 3.4, x n x ˜ . We claim that w n x ˜ . We calculate the following:

x n + 1 w n + 1 = β n γ f ( x n ) f ( T n ( t n ) w n ) + I β n A T n ( t n ) x n T n ( t n ) w n β n γ α x n T n ( t n ) w n + φ ( 1 ) ( 1 β n γ ¯ ) ( k n ) x n w n β n γ α x n T n ( t n ) x ˜ + β n γ α T n ( t n ) x ˜ T n ( t n ) w n + φ ( 1 ) ( 1 β n γ ¯ ) ( k n ) x n w n β n γ α x n x ˜ + β n γ α k n x ˜ w n + φ ( 1 ) ( 1 β n γ ¯ ) ( k n ) x n w n β n γ α x n x ˜ + β n γ α w n x n + β n γ α k n x ˜ x n + φ ( 1 ) ( 1 β n γ ¯ ) ( k n ) x n w n = φ ( 1 ) ( 1 β n γ ¯ ) ( k n ) x n w n + β n γ α w n x n + β n ( γ α + k n ) x n x ˜ = ( φ ( 1 ) ( 1 β n γ ¯ ) ( k n ) + β n γ α ) x n w n + β n ( γ α + k n ) x n x ˜ [ ( 1 φ ( 1 ) β n γ ¯ ) ( k n 1 ) + 1 β n ( φ ( 1 ) γ ¯ γ α ) ] x n w n + β n ( γ α + k n ) x n x ˜ ( 1 β n ( φ ( 1 ) γ ¯ γ α ) ) x n w n + ( 1 φ ( 1 ) β n γ ¯ ) ( k n 1 ) M + β n ( γ α + k n ) x n x ˜ ,

where M>0 such that sup n 1 x n w n M. Put

s n = β n ( φ ( 1 ) γ ¯ γ α )

and

σ n = ( 1 φ ( 1 ) β n γ ¯ φ ( 1 ) γ ¯ γ α ) ( k n 1 β n ) M+ ( γ α + k n ) ( φ ( 1 ) γ ¯ γ α ) x n x ˜ .

Then we have that

x n + 1 w n + 1 (1 s n ) x n w n + s n σ n .
(3.26)

It follows from (C3), lim n x n x ˜ =0 and Lemma 2.6 that x n w n 0 as n; that is, w n x ˜ as n. □

Theorem 3.6 Let E be a real Banach space with uniform normal structure which has a uniformly Gâteaux differentiable norm and admits the duality mapping J φ , K be a nonempty, bounded, closed and convex subset of E such that K±KK. Let S={T(s):s0} be an asymptotically nonexpansive semigroup on K with a sequence { k n }[1,+), lim n k n =1 and sup n 1 k n <N ( E ) 1 / 2 such that F(S). Let f Π K with coefficient α(0,1), A a strongly positive bounded linear operator on K with coefficient γ ¯ >0 and 0<γ< φ ( 1 ) γ ¯ α . Let { β n } and { t n } be sequences of real numbers such that 0< β n <1, t n 0,

(C1) lim n β n =0;

(C2) lim n k n 1 β n =0;

(C3) n = 0 β n =.

For any z 0 K, let the sequence { z n } be defined by

z n + 1 = T n ( t n ) ( β n γ f ( z n ) + ( I β n A ) z n ) ,n0.
(3.27)

Then { z n } converges strongly as n to a point x ˜ in F(S) which solves the variational inequality (3.2).

Proof Define the sequences { w n } and { σ n } by

w n = β n γf( z n )+(I β n A) z n and σ n = β n + 1 ,n0.

We have that

w n + 1 = β n + 1 γf( z n + 1 )+(I β n + 1 A) z n + 1 = σ n γf ( T n ( t n ) w n ) +(I σ n A) T n ( t n ) w n .

It follows from Theorem 3.5 that { w n } converges strongly to x ˜ . Thus, we have

z n x ˜ z n w n + w n x ˜ = β n γ f ( z n ) A z n + w n x ˜ β n M + w n x ˜ 0 as  n ,

where M>0 such that M sup n 1 γf( z n )A z n . Hence, { z n } converges strongly to x ˜ . □

If AI, the identity mapping on E, and γ=1, then Theorem 3.4 reduces to the following corollary.

Corollary 3.7 Let E be a real Banach space with uniform normal structure which has a uniformly Gâteaux differentiable norm and admits the duality mapping J φ , K be a nonempty, bounded, closed and convex subset of E. Let S={T(s):s0} be an asymptotically nonexpansive semigroup on K with a sequence { k n }[1,+), lim n k n =1 and sup n 1 k n <N ( E ) 1 / 2 such that F(S). Let f Π K with coefficient α(0,1). Let { β n } and { t n } be sequences of real numbers such that 0< β n <1, t n 0,

(C1) lim n β n =0;

(C2) lim n k n 1 β n =0;

(C3) n = 0 β n =.

For any x 0 K, let the sequence { x n } be defined by

x n + 1 = β n f( x n )+(1 β n ) T n ( t n ) x n ds,n0.
(3.28)

Suppose, in addition, lim n x n T(t) x n =0 uniformly in t[0,). Then { x n } converges strongly as n to a point x ˜ in F(S), which solves the variational inequality (3.11).

If fu, then Corollary 3.7 reduces to the following corollary.

Corollary 3.8 Let E be a real Banach space with uniform normal structure which has a uniformly Gateaux differentiable norm and admits the duality mapping J φ , K be a nonempty, bounded, closed and convex subset of E such that K±KK. Let S={T(s):s0} be an asymptotically nonexpansive semigroup on K with a sequence { k n }[1,+), lim n k n =1 and sup n 1 k n <N ( E ) 1 / 2 such that F(S). Let f Π K with coefficient α(0,1). Let { β n } and { t n } be sequences of real numbers such that 0< β n <1, t n 0,

(C1) lim n β n =0;

(C2) lim n k n 1 β n =0;

(C3) n = 0 β n =.

For any x 0 K, let the sequence { x n } be defined by

x n + 1 = β n u+(1 β n ) T n ( t n ) x n ds,n0.
(3.29)

Suppose, in addition, lim n x n T(t) x n =0 uniformly in t[0,). Then { x n } converges strongly as n to a point x ˜ in F(S) which solves the variational inequality (3.13).

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Acknowledgements

The first author is supported by the ‘Centre of Excellence in Mathematics’ under the Commission on Higher Education, Ministry of Education, Thailand.

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Correspondence to Rabian Wangkeeree.

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Keywords

  • Banach Space
  • Variational Inequality
  • Convex Subset
  • Nonexpansive Mapping
  • Duality Mapping