Let E be a real Banach space and let J denotes the normalized duality mapping from E into 2 E∗given by

STRONG CONVERGENCE THEOREMS FOR NONEXPANSIVE MAPPINGS BY VISCOSITY APPROXIMATION METHODS IN BANACH SPACES

Xiaolong QIN, Yongfu SU and Changqun WU

Abstract. In this paper, we introduce a modified Ishikawa iterative process for a pair of nonexpansive mappings and obtain a strong con- vergence theorem in the framework of uniformly Banach spaces. Our results improve and extend the recent ones announced by Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51-60], Xu [H.K. Xu, Viscosity approxima- tion methods for nonexpansive mappings. J. Math. Anal. Appl. 298 (2004) 279-291] and some others.

1. Introduction and Preliminaries

Let E be a real Banach space and let J denotes the normalized duality mapping from E into 2 ^E

given by

J (x) = {f ∈ E ^∗ : hx, f i = kxk ² = kf k ² }, x ∈ E,

where E ^∗ denotes the dual space of E and h·, ·i denotes the generalized duality pairing. Recall that a self mapping f : C → C is a contraction on C if there esists a constant α ∈ (0, 1) such that

kf (x) − f (y)k ≤ αkx − yk, x, y ∈ C.

We use Π ^C to denote the collection of all contractions on C. That is, Π ^C = {f |f : C → C a contraction}. Note that each f ∈ Π ^C has a unique fixed point in C. Also, recall that T is nonexpansive if

kT x − T yk ≤ kx − yk for all x, y ∈ C.

A point x ∈ C is a fixed point of T provided T x = x. Denote by F (T ) the set of fixed points of T ; that is, F (T ) = {x ∈ C : T x = x}. Given a real number t ∈ (0, 1) and a contraction f ∈ Π C . We define a mapping T t x = tf (x) + (1 − t)T x, x ∈ C. It is obviously that T ^t is a contraction on

Mathematics Subject Classification. Primary 47H09; Secondary 65J15.

Key words and phrases. Nonexpansive map; Iteration scheme; Sunny and nonexpansive retraction; viscosity method.

113

C . In fact, for x, y ∈ C, we obtain

kT t x − T t yk ≤ kt(f (x) − f (y)) + (1 − t)(T x − T y)k

≤ αtkx − yk + (1 − t)kT x − T yk

≤ αtkx − yk + (1 − t)kx − yk

= (1 − t(1 − α))kx − yk.

Let x ^t be the unique fixed point of T ^t . That is, x ^t is the unique solution of the fixed point equation

(1.1) x t = tf (x ^t ) + (1 − t)T x ^t .

A special case has been considered by Browder [1] in a Hilbert space as follows. Fix u ∈ C and define a contraction S t on C by

S t x = tu + (1 − t)T x, x ∈ C.

If we use z t to denote the unique fixed point of S t , which yields that z t = tu + (1 − t)T z ^t .

In 1967, Browder [1] proved the following theorem.

Theorem 1.1 In a Hilbert space, as t → 0, z ^t converges strongly to a fixed point of T that is closet to u, that is, the nearest point projection of u onto F (T ).

Also, In 1967, Halpern [5] firstly introduced this iteration scheme (1.2)

( x ₀ = x ∈ C chosen arbitrarily, x n +1 = α ⁿ u + (1 − α ⁿ )T x ⁿ , which is the special cases of

(1.3)

( x ₀ = x ∈ C chosen arbitrarily, x n +1 = α ⁿ f (x ⁿ ) + (1 − α ⁿ )T x ⁿ .

In [9], Moudafi proposed a viscosity approximation method of selecting a particular fixed point of a given nonexpansive mapping in Hilbert spaces.

If H is a Hilbert space, T : C → C is a nonexpansive self-mapping on a nonempty closed convex C of H and f : C → C is a contraction, he proved the following theorems.

Theorem 1.2 (Moudafi [9]). The sequence {x ⁿ } generated by the scheme x n = 1

1 +  ⁿ T x n +  n

1 +  ⁿ f (x ⁿ )

converges strongly to the unique solution of the variational inequality:

¯

x ∈ F (T ), such that h(I − f )¯ x, x ¯ − xi ≤ 0, ∀x ∈ F (T ),

where { ⁿ } is a sequence of positive numbers tending to zero.

Theorem 1.3 (Moudafi [9]). With and initial z 0 ∈ C defined the sequence {z ⁿ } by

z n +1 = 1

1 +  ⁿ T z n +  n

1 +  ⁿ f (z ⁿ ).

Supposed that lim ^n→∞  n = 0, and P ∞

n =1  = ∞ and lim ^n→∞ | _ ¹

n+1

− ¹ _ | = 0.

Then {z ⁿ } converges strongly to the unique solution of the unique solutions of the variational inequality:

¯

x ∈ F (T ) such that h(I − f )¯ x, x ¯ − xi ≤ 0, ∀x ∈ F (T ).

Recently Xu [14] studied the viscosity approximation methods proposed by Moudafi [9] for nonexpansive mappings in a uniformly smooth Banach space. More precisely, he proved following theorems.

Theorem 1.4 (Xu [14]). Let E be a uniformly smooth Banach space, C a closed convex subset of E and T : C → C a nonexpansive mapping with F (T ) 6= ∅, and f ∈ Π ^C . Then the path {x ^t } defined by x t = tf (x ^t ) + (1 − t)T x ^t , t ∈ (0, 1), converges strongly to a point in F (T ). If we define Q : Π C → F (T ) by Q(f ) = lim t→ 0 x t , the Q(f ) solves the variational inequality

h(I − f )Q(f ), j(Q(f ) − x)i, f ∈ Π ^C , x ∈ F (T ).

Theorem 1.5 (Xu [14]). Let E be a uniformly smooth Banach space, C a closed convex subset of E and T : C → C a nonexpansive mapping with F (T ) 6= ∅ and f ∈ Π C . Assume that α n ∈ (0, 1) satisfies the following conditions

(i) lim ^n→∞ α n = 0;

(ii) P ∞

n =0 α n = ∞;

(iii) either lim ^{n→∞ α} _α

n

= 1 or P ∞

n =0 |α ⁿ +1 − α ⁿ | ≤ ∞. Then the sequence {x ⁿ } generated by

x ₀ ∈ C, x n +1 = α ⁿ f (x ⁿ ) + (1 − α ⁿ )T x ⁿ , n = 0, 1, 2, . . . converges strongly to a fixed point of T.

Two classical iteration processes are often used to approximate a fixed point of a nonexpansive mapping. The first one is introduced by Mann [8]

and is defined as

(1.4) x n +1 = α ⁿ x n + (1 − α ⁿ )T x ⁿ , n ≥ 0,

where the initial guess x ₀ is taken in C arbitrarily and the sequence {α n } ^∞ n =0

is in the interval [0, 1].

The second iteration process is referred to as Ishikawa’s iteration process [6] which is defined recursively by

(1.5)

( y n = β n x n + (1 − β n )T x n , x n +1 = α ⁿ x n + (1 − α ⁿ )T y ⁿ ,

where the initial guess x 0 is taken in C arbitrarily, {α ⁿ } and {β n } are sequences in the interval [0, 1]. But both (1.4) and (1.5) have only weak convergence, in general (see [4] for an example). For example, Reich [11], shows that if E is a uniformly convex and has a F r´ ehet differentiable norm and if the sequence {α n } is such that α n (1 − α n ) = ∞, then the sequence {x n } generated by processes (1.4) converges weakly to a point in F (T ). (An extension of this result to processes (1.5) can be found in [13].) Therefore, many authors attempt to modify (1.4) and (1.5) to have strong convergence.

Recently, Kim and Xu [7] introduced the following iteration process in the framework of Banach spaces.

(1.6)



 

 

x ₀ ∈ C chosen arbitrarily, y n = β ⁿ x n + (1 − β ⁿ )T x ⁿ , x n +1 = α ⁿ u + (1 − α ⁿ )y ⁿ . More precisely, they proved the following theorem:

Theorem 1.6 (Kim and Xu [7]). Let C be a closed convex subset of a uniformly smooth Banach space E and let T : C → C be a nonexpansive mapping such that F (T ) 6= ∅. Give a point u ∈ C and given sequences {α ⁿ } and {β ⁿ } in (0, 1), the following conditions are satisfied:

(i) α ⁿ → 0, β n → 0, P ∞

n =0 α n = ∞ and P ∞

n =0 β n = ∞, (ii) P ∞

n =0 |α ⁿ +1 − α ⁿ | < ∞, P ∞

n =0 |β ⁿ +1 − β ⁿ | < ∞.

Define a sequence {x n } in C by (1.6). Then {x n } strongly to converges to a fixed point of T .

In this paper, we use viscosity approximation methods to study strong convergence of a pair of nonexpansive mappings in the framework of uni- formly smooth Banach spaces. We introduce the composite iteration process as follows:

(1.7)



 

 

z n = γ ⁿ x n + (1 − γ ⁿ )T 2 x n , y n = β n x n + (1 − β n )T ₁ z n , x n +1 = α n f (x n ) + (1 − α n )y n ,

where the sequence {α ⁿ } in (0,1) and {β n }, {γ n } are sequences in [0,1]. We

prove, under certain appropriate assumptions on the sequences {α ⁿ }, {β ⁿ }

and {γ n }, that {x n } defined by (1.7) converges to a common fixed point of

T ₁ and T 2 , which solves some variational inequality.

If {γ ⁿ } = 1 in (1.7) this can be viewed as a modified Mann iteration process

(1.8)

( y n = β ⁿ x n + (1 − β ⁿ )T 1 x n , x n +1 = α ⁿ f (x ⁿ ) + (1 − α ⁿ )y ⁿ .

If {γ n } = 1 and {β n } = 0 in (1.7), then (1.7) reduces to (1.3) which consid- ered by Xu [14].

It is our purpose in this paper is to introduce this composite iteration scheme for approximating a common fixed point of two nonexpansive map- pings by using viscosity methods in the framework of uniformly smooth Banach spaces. we establish the strong convergence of the sequence {x n } defined by (1.7). Our results improve and extend the ones announced by Kim and Xu [7], Xu [14] and some others.

We need the following definitions and lemmas for the proof of our main results.

The norm of E is said to be Gˆateaux differentiable (and E is said to be smooth) if

(1.9) lim

t→ 0

kx + tyk − kxk t

exists for each x, y in its unit sphere U = {x ∈ E : kxk = 1}. It is said to be uniformly Fr´echet differentiable (and E is said to be uniformly smooth ) if the limit in (1.9) is attained uniformly for (x, y) ∈ U × U .

Lemma 1.1 A Banach space E is uniformly smooth if and only if the duality map J is single-valued and norm-to-norm uniformly continuous on bounded sets of E.

Lemma 1.2 In a Banach space E, there holds the inequality kx + yk ² ≤ kxk ² + 2hy, j(x + y)i, x, y ∈ E where j(x + y) ∈ J(x + y).

Lemma 1.3 ( Xu [15], [16]). Let {α n } be a sequence of nonnegative real numbers satisfying the property

α n +1 ≤ (1 − γ n )α n + γ n σ n , n ≥ 0, where {γ ⁿ } ^∞ n =0 ⊂ (0, 1) and {σ n } ^∞ n =0 such that

(i) lim ^n→∞ γ n = 0 and P ∞

n =0 γ n = ∞, (ii) either lim sup n→∞ σ n ≤ 0 or P ^∞

n =0 |γ n σ n | < ∞.

Then {α ⁿ } ^∞ _{n =0} converges to zero.

Recall that if C and D are nonempty subsets of a Banach space E such that C is nonempty closed convex and D ⊂ C, then a map Q : C → D is sunny ([2], [12]) provided Q(x + t(x − Q(x))) = Q(x) for all x ∈ C and t ≥ 0 whenever x + t(x − Q(x)) ∈ C. A sunny nonexpansive retraction is a sunny retraction,which is also nonexpansive. Sunny nonexpansive retractions play an important role in our argument. They are characterized as follows [2, 3, 12]: if E is a smooth Banach space, then Q : C → D is a sunny nonexpansive retraction if and only if there holds the inequality

hx − Qx, J(y − Qx)i ≤ 0 for all x ∈ C and y ∈ D.

Reich [10] showed that if E is uniformly smooth and if D is the fixed point set of a nonexpansive mapping from C into itself, then there is a sunny nonexpansive retraction from C onto D and it can be constructed as follows.

Lemma 1.4 (Reich [10]). Let E be a uniformly smooth Banach space and let T : C → C be a nonexpansive mapping with a fixed point x ^t ∈ C of the contraction C 3 x 7→ tu + (1 − t)T x converging strongly as t → 0 to a fixed point of T . Define Q : C → F (T ) by Qu = s − lim

t→ 0 x t . Then Q is the unique sunny nonexpansive retract from C onto F (T ); that is, Q satisfies the property

hu − Qu, J(z − Qu)i ≤ 0, u ∈ C, z ∈ F (T ).

Lemma 1.5 (Xu [14]). Let E be a uniformly smooth Banach space and let T : C → C be a nonexpansive mapping with a fixed point x ^t ∈ C of the contraction C 3 x 7→ tu + (1 − t)T x converges strongly as t → 0 to a fixed point of T . Define Q : Π C → F (T ) by

(1.10) Qf = s − lim

t→ 0 x t , f ∈ Π C . Then Q(f ) solves the variational inequality

(1.11) h(I − f )Q(f ), J(Q(f ) − p)i ≤ 0, f ∈ Π C , p ∈ F (T ).

In particular, if f = u is a constant, then (1.10) is reduced to the sunny nonexpansive retract from C onto F (T ):

(1.12) hu − Qu, J(p − Qu)i ≤ 0, u ∈ C, p ∈ F (T ).

2. Main Results

Theorem 2.1 Let C be a closed convex subset of a uniformly smooth Ba-

nach space E and let T ₁ , T ₂ : C → C be a pair of nonexpansive mappings

such that F (T 1 T ₂ ) = F (T 1 ) T F (T 2 ) 6= ∅. The initial guess x 0 ∈ C is chosen

arbitrarily and given sequences {α ⁿ } ^∞ n =0 in (0,1) and {β ⁿ } ^∞ n =0 and {γ ⁿ } ^∞ n =0

in [0,1], the following conditions are satisfied (i) P ^∞

n =0 α n = ∞, α n → 0;

(ii) β ⁿ → 0, γ n → 0;

(iii) P ^∞

n =0 |α n +1 − α n | < ∞, P ^∞

n =0 |β n +1 − β n | < ∞ and P ^∞

n =0 |γ n +1 − γ n | <

∞.

Let {x ⁿ } ^∞ n =1 be the composite process defined by



 

 

z n = γ n x n + (1 − γ n )T 2 x n , y n = β ⁿ x n + (1 − β ⁿ )T 1 z n , x n +1 = α ⁿ f (x ⁿ ) + (1 − α ⁿ )y ⁿ .

Then {x n } ^∞ n =1 converges strongly to some common fixed point p ∈ F (T ₁ ) ∩ F (T 2 ) which solves the variational inequality

(2.1) h(I − f )Q(f ), J(Q(f ) − p)i ≤ 0, f ∈ Π C , p ∈ F (T ₁ ) ∩ F (T 2 ).

Proof. First we observe that {x ⁿ } ^∞ n =0 is bounded. Indeed, taking a fixed point p of F (T ₁ ) ∩ F (T ₂ ), we note that

(2.2) kz n − pk ≤ γ n kx n − pk + (1 − γ n )kT ₂ x n − pk ≤ kx n − pk.

It follows that (2.3)

ky n − pk ≤ β n kx n − pk + (1 − β n )kT 1 z n − pk

≤ β n kx n − pk + (1 − β n )kz ⁿ − pk

≤ kx n − pk.

It follows from (2.3) that

kx ⁿ +1 − pk ≤ α ⁿ kf (x ⁿ ) − pk + (1 − α ⁿ )ky ⁿ − pk

≤ α ⁿ kf (x ⁿ ) − f (p)k + α ⁿ kf (p) − pk + (1 − α ⁿ )kx ⁿ − pk

≤ max{ 1

1 − α kf (p) − pk, kx n − pk}.

Now, an induction yields

(2.4) kx n − pk ≤ max{ 1

1 − α kf (p) − pk, kx ₀ − pk}. n ≥ 0,

which implies that {x n } is bounded, so are {T ₂ x n }, {f (x n )} {y n }, {z n } and {T ₁ z n }.

Since condition (i), we obtain

(2.6) kx n +1 − y n k = α n kf (x n ) − y n k → 0, as n → ∞.

Next, we claim that

(2.6) kx n +1 − x n k → 0.

In order to prove (2.6) from

( x n +1 = α ⁿ f (x ⁿ ) + (1 − α ⁿ )y ⁿ , x n = α ⁿ⁻ 1 f (x ⁿ ) + (1 − α ⁿ⁻ 1 )y ⁿ . We have

x n +1 − x ⁿ =(1 − α ⁿ )(y ⁿ − y ⁿ⁻ 1 )

+ (α ⁿ⁻ 1 − α ⁿ )(y ⁿ⁻ 1 − f (x ⁿ⁻ 1 )) + α ⁿ (f (x ⁿ ) − f (x ⁿ⁻ 1 )).

It follows that

kx ⁿ +1 − x ⁿ k ≤ (1 − α ⁿ )ky ⁿ − y ⁿ⁻ 1 k (2.7)

+ |α n− 1 − α n |ky n− 1 − f (x n− 1 )k + αα n kx n − x n− 1 k.

Similarly, Since

( y n = β ⁿ x n + (1 − β ⁿ )T 1 z n ,

y n− 1 = β ⁿ⁻ 1 x n− 1 + (1 − β ⁿ⁻ 1 )T 1 z n− 1 . We obtain

y n − y ⁿ⁻ 1 = (1 − β ⁿ )(T 1 z n − T 1 z n− 1 ) + β ⁿ (x ⁿ − x ⁿ⁻ 1 ) + (T 1 z n− 1 − x ⁿ⁻ 1 )(β ⁿ⁻ 1 − β ⁿ ).

It follow that

(2.8)

ky n − y n− 1 k ≤ (1 − β n )kT 1 z n − T ₁ z n− 1 k + β n kx n − x n− 1 k + kT 1 z n− 1 − x n− 1 k|β n− 1 − β n |

≤ (1 − β n )kz ⁿ − z n− 1 k + β n kx n − x n− 1 k + kT 1 z n− 1 − x ⁿ⁻ 1 k|β ⁿ⁻ 1 − β ⁿ |.

On the other hand, from

( z n = γ ⁿ x n + (1 − γ ⁿ )T 2 x n ,

z n− 1 = γ n− 1 x n− 1 + (1 − γ n− 1 )T 2 z n− 1 , we also can obtain

z n − z n− 1 =(1 − γ ⁿ )(T 2 x n − T ₂ x n− 1 ) + γ ⁿ (x ⁿ − x n− 1 ) + (γ ⁿ⁻ 1 − γ n )(T 2 x n− 1 − x n− 1 ),

which yields that

(2.9) kz ⁿ − z ⁿ⁻ 1 k ≤ kx ⁿ − x ⁿ⁻ 1 k + |γ ⁿ⁻ 1 − γ ⁿ |kT 2 x n− 1 − x ⁿ⁻ 1 k.

Substituting (2.9) into (2.8), we get (2.10)

ky n − y n− 1 k ≤ (1 − β n )(kx n − x n− 1 k + |γ n− 1 − γ n |kT 2 x n− 1 − x n− 1 k)

+ β n kx n − x n− 1 k + kT 1 z n− 1 − x n− 1 k|β n− 1 − β n |.

That is,

(2.11) ky n − y n− 1 k ≤ kx n − x n− 1 k + |γ n− 1 − γ n |kT ₂ x n− 1 − x n− 1 k + kT ₁ z n− 1 − x n− 1 k|β n− 1 − β n |.

Similarly, substitute (2.11) into (2.7) yields that (2.12)

kx ⁿ +1 − x ⁿ k ≤ (1 − α ⁿ )(kx ⁿ − x ⁿ⁻ 1 k + |γ ⁿ⁻ 1 − γ ⁿ |kT 2 x n− 1 − x ⁿ⁻ 1 k + kT 1 z n− 1 − x ⁿ⁻ 1 k|β ⁿ⁻ 1 − β ⁿ |)

+ |α ⁿ⁻ 1 − α ⁿ |ky ⁿ⁻ 1 − f (x ⁿ⁻ 1 )k + αα ⁿ kx ⁿ − x ⁿ⁻ 1 k

≤ (1 − (1 − α)α ⁿ )kx ⁿ − x ⁿ⁻ 1 k

+ M 1 (|α n− 1 − α n | + |β n− 1 − β n | + |γ n− 1 − γ n |), where M 1 is a constant such that

M ₁ ≥ max{ky n− 1 − f (x n− 1 )k, kx n− 1 − T ₂ x n− 1 k, kx n− 1 − T ₁ z n− 1 k}

for all n. By assumptions (i)-(iii), we have that

n→∞ lim α n = 0,

∞

X

n =1

(1 − α)α ⁿ = ∞, and

∞

X

n =1

(|β ⁿ − β n− 1 | + |α n − α n− 1 | + |γ n − γ n− 1 |) < ∞.

Hence, Lemma 1.3 is applicable to (2.12) and we obtain (2.6) holds. Observe that

(2.13)

kT ₁ T ₂ x n − x n k

≤ kx n − x n +1 k + kx n +1 − y n k + ky n − T ₁ z n k + kT ₁ z n − T ₁ T ₂ x n k

≤ kx n − x n +1 k + kx n +1 − y n k + β n kx n − T ₁ z n k + kz n − T ₂ x n k

≤ kx ⁿ − x ⁿ +1 k + kx ⁿ +1 − y ⁿ k + β ⁿ kx ⁿ − T 1 z n k + γ ⁿ kx ⁿ − T 2 x n k.

Since assumption lim ^n→∞ β n = lim ^n→∞ γ n = 0, (2.5) and (2.6), we know (2.14) kT 1 T ₂ x n − x n k → 0.

Put T = T 1 T ₂ . Since T 1 and T 2 are nonexpansive, we have T is also nonex- pansive. Next, we claim that

(2.15) lim sup

n→∞

hf (q) − q, J(x n − q)i ≤ 0,

where q = Qf = s−lim t→ 0 x t with x t being the fixed point of the contraction

x 7→ tf (x) + (1 − t)T x, where T = T ₁ T ₂ . From x ^t solves the fixed point

equation

x t = tf (x ^t ) + (1 − t)T x ^t . Thus we have

kx t − x n k = k(1 − t)(T x t − x n ) + t(f (x ^t ) − x ⁿ )k.

It follows from Lemma 1.2 that

(2.16)

kx t − x n k ² ≤ (1 − t) ² kT x t − x n k ² + 2thf (x t ) − x n , J(x t − x n )i

≤ (1 − 2t + t ² )kx ^t − x n k ² + f ⁿ (t)

+ 2thf (x t ) − x t , J(x t − x n )i + 2tkx t − x n k ² , where

(2.17) f n (t) = (2kx ^t − x ⁿ k + kx ⁿ − T x ⁿ k)kx ⁿ − T x ⁿ k → 0, as n → 0.

It follows that

(2.18) hx ^t − f (x ^t ), J(x ^t − x ⁿ )i ≤ t

2 kx ^t − x ⁿ k ² + 1

2t f n (t).

Let n → ∞ in (2.18) and note (2.17) yields

(2.19) lim sup

n→∞

hx t − f (x t ), J(x t − x n )i ≤ t 2 M ₂ ,

where M 2 > 0 is a constant such that M 2 ≥ kx t − x n k ² for all t ∈ (0, 1) and n ≥ 1. Taking t → 0 from (2.19), we have

lim sup

t→ 0

lim sup

n→∞

hx t − f (x t ), J(x t − x n )i ≤ 0.

So, for any  > 0, there exists a positive number δ 1 such that, for t ∈ (0, δ 1 ), we get

(2.20) lim sup

n→∞

hx t − f (x t ), J(x t − x n )i ≤  2 .

On the other hand, since x ^t → q as t → 0, from Lemma 1.1, there exists δ ₂ > 0 such that, for t ∈ (0, δ 2 ) we have

|hf (q) − q, J(x n − q)i − hx t − f (x t ), J(x t − x n )i|

≤ |hf (q) − q, J(x n − q)i − hf (q) − q, J(x n − x t )i|

+|hf (q) − q, J(x n − x t )i − hx t − f (x t ), J(x t − x n )i|

≤ |hf (q) − q, J(x n − q) − J(x n − x t )i|

+|hf (q) − f (x ^t ) − q + x ^t , J (x ⁿ − q)i|

≤ kf (q) − qkkJ(x n − q) − J(x n − x t )k +kf (q) − f (x ^t ) − q + x ^t kkx n − qk

< ₂ ^ .

Picking δ = min{δ 1 , δ ₂ }, ∀t ∈ (0, δ), we have

hf (q) − q, J(x n − q)i ≤ hx t − f (x t ), J(x ^t − x n )i + 

2 .

That is, lim sup

n→∞

hf (q) − q, J(x n − q)i ≤ lim sup

n→∞

hx t − f (x t ), J(x ^t − x n )i +  2 . It follows from (2.21) that

lim sup

n→∞

hf (q) − q, J(x n − q)i ≤ .

Since  is chosen arbitrarily, we have

(2.21). lim sup

n→∞

hf (q) − q, J(x n − q)i ≤ 0

Finally, we show that x n → q strongly and this concludes the proof.

Indeed, using Lemma 1.2 again we obtain

kx n +1 − qk ² = k(1 − α ⁿ )(y ⁿ − q) + α n (f (x ⁿ ) − q)k ²

≤ (1 − α n ) ² ky n − qk ² + 2α n hf (x n ) − q, J(x n +1 − q)i

≤ (1 − α ⁿ ) ² kx ⁿ − qk ²

+ 2α ⁿ hf (x ⁿ ) − f (q), J(x ⁿ +1 − q)i + 2α ⁿ hf (q) − q, J(x ⁿ +1 − q)i

≤ (1 − α n ) ² kx n − qk ² + 2α ⁿ αkx n − qkkx n +1 − qk + 2α ⁿ hf (q) − q, J(x n +1 − q)i

≤ (1 − α n ) ² kx n − qk ² + α n α(kx n − qk ² + kx n +1 − qk ² ) + 2α n hf (q) − q, J(x n +1 − q)i.

Therefore, we obtain kx n +1 − qk ²

≤ 1 − (2 − α)α ⁿ + α ² n

1 − αα n

kx n − qk ² − 2α ⁿ 1 − αα n

hf (q) − q, J(x n +1 − q)i

≤ 1 − (2 − α)α ⁿ 1 − αα n

kx n − qk ² − 2α ⁿ 1 − αα n

hf (q) − q, J(x n +1 − q)i + M ₂ α ² _n

= (1 − 2(1 − α)α ⁿ 1 − αα n

)kx n − qk ² + 2(1 − α)α ⁿ

1 − αα n

( M ₂ (1 − αα ⁿ )α ⁿ

2(1 − α) + 1

1 − α hf (q) − q), J(x n +1 − q)i.

Now we apply Lemma 1.3 and use (2.21) to see that kx ⁿ − qk → 0. This completes the proof.

As corollaries of Theorem 2.1, we have the following.

Corollary 2.2 Let C be a closed convex subset of a uniformly smooth Ba-

nach space E and let T 1 : C → C be a nonexpansive mapping such that

F (T 1 ) 6= ∅. The initial guess x 0 ∈ C is chosen arbitrarily and given se- quences {α ⁿ } ^∞ n =0 in (0,1) and {β ⁿ } ^∞ n =0 in [0,1], the following conditions are satisfied

(i) P ∞

n =0 α n = ∞, α ⁿ → 0, ; (ii) β n < a, for some a ∈ [0, 1);

(iii) P ∞

n =0 |α n +1 − α n | < ∞, P ∞

n =0 |β n +1 − β n | < ∞.

Let {x ⁿ } ^∞ n =1 be the composite process defined by (1.8), then {x ⁿ } ^∞ n =1 con- verges strongly to some fixed point p ∈ F (T 1 ) which Q(f ) solves the varia- tional inequality

h(I − f )Q(f ), J(Q(f ) − p)i ≤ 0, f ∈ Π ^C , p ∈ F (T 1 ).

Proof. By taking {γ ⁿ } = 1, we can obtain the desired conclusion. This completes the proof.

Corollary 2.3 (Xu [14]). Let E be a uniformly smooth Banach space, C a closed convex subset of E and T : C → C a nonexpansive mapping with F (T ) 6= ∅, and f ∈ Π ^C . Assume that α ⁿ ∈ (0, 1) satisfies the following conditions

(i) lim ^n→∞ α n = 0;

(ii) P ^∞

n =0 α n = ∞;

(iii) P ∞

n =0 |α n +1 − α n | ≤ ∞. Then the sequence {x n } generated by x ₀ ∈ C, x n +1 = α ⁿ f (x ⁿ ) + (1 − α ⁿ )T x ⁿ , n = 0, 1, 2, . . . converges strongly to Q(f ), which solves the variational inequality

h(I − f )Q(f ), J(Q(f ) − p)i ≤ 0, f ∈ Π ^C , p ∈ F (T ).

Proof. By taking {γ ⁿ } = 1 and {β ⁿ }=0, we can obtain the desired conclu- sion. This completes the proof.

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Xiaolong Qin

Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, PR China

e-mail address : [email protected] Yongfu Su

Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, PR China

e-mail address: [email protected] Changqun Wu

School of Business and Administration, Henan University, Kaifeng 475001, PR China

e-mail address : [email protected]

(Received December 7, 2006 )