Weak and strong convergence theorems for an asymptotically $k$-strict pseudo-contraction and a mixed equilibrium problem

DOI 10.4134/JKMS.2009.46.3.561

WEAK AND STRONG CONVERGENCE THEOREMS FOR AN ASYMPTOTICALLY k-STRICT PSEUDO-CONTRACTION

AND A MIXED EQUILIBRIUM PROBLEM

Yonghong Yao, Haiyun Zhou, and Yeong-Cheng Liou

Abstract. We introduce two iterative algorithms for finding a common element of the set of fixed points of an asymptotically k-strict pseudo- contraction and the set of solutions of a mixed equilibrium problem in a Hilbert space. We obtain some weak and strong convergence theorems by using the proposed iterative algorithms. Our results extend and improve the corresponding results of Tada and Takahashi [16] and Kim and Xu [8, 9].

1. Introduction

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let ϕ : C → R be a real valued function and Θ : C × C → R be an equilibrium bifunction, i.e., Θ(u, u) = 0 for each u ∈ C. Now we concern the following mixed equilibrium problem (MEP) which is to find x ^∗ ∈ C such that (MEP) Θ(x ^∗ , y) + ϕ(y) − ϕ(x ^∗ ) ≥ 0, ∀y ∈ C.

In particular, if ϕ ≡ 0, this problem reduces to the equilibrium problem (EP), which is to find x ^∗ ∈ C such that

(EP) Θ(x ^∗ , y) ≥ 0, ∀y ∈ C.

Denote the set of solutions of (MEP) by Ω and the set of solutions of (EP) by Γ. The mixed equilibrium problems include fixed point problems, optimiza- tion problems, variational inequality problems, Nash equilibrium problems and the equilibrium problems as special cases; see, e.g., [1, 3, 4, 10, 22]. Some methods have been proposed to solve the equilibrium problems and the mixed equilibrium problems, see, e.g., [2, 5, 6, 7, 10, 12, 14, 15, 16, 17, 18, 19, 20, 21].

Received August 25, 2007.

2000 Mathematics Subject Classification. 49J30, 47H10, 47H17, 49M05, 90C25, 90C99.

Key words and phrases. mixed equilibrium problems, fixed point problems, iterative al- gorithm, asymptotically k-strict pseudo-contraction, Hilbert space.

The first two authors were supported by National Natural Science Foundation of China Grant 10771050 and the third author was supported by the grant NSC 97-2221-E-230-017.

c

°2009 The Korean Mathematical Society

561

On the other hand, recently, Kim and Xu [8, 9] introduced some itera- tive methods for solving fixed point problems of asymptotically nonexpansive mappings and asymptotically k-strict pseudo-contractions, respectively. The corresponding iterative algorithms are as follows. The first one introduced in [8] is:

(KX1)

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

C n = {z ∈ C : ky n − zk ² ≤ kx n − zk ² + θ n }, Q n = {z ∈ C : hx n − z, x 0 − x n i ≥ 0}, x n+1 = P C

∩Q

x 0 ,

where T : C → C is an asymptotically nonexpansive mapping and θ n = (1 − α n )(k ² _n − 1)(diam C) ² → 0 as n → ∞. And the second one introduced in [9] is:

(KX2)

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

C n = {z ∈ C : ky n − zk ² ≤ kx n − zk ²

+ [k − α n (1 − α n )]kx n − T ⁿ x n k ² + θ n }, Q n = {z ∈ C : hx n − z, x 0 − x n i ≥ 0},

x n+1 = P C

∩Q

x 0 ,

where T : C → C is an asymptotically k-strict pseudo-contraction and θ n =

∆ ² _n (1 − α n )γ n → 0(n → ∞), ∆ n = sup{kx n − zk : z ∈ F (T )} < ∞. Subse- quently, Kim and Xu proved that the iterative algorithm (KX1) and (KX2) are strongly convergent. For more details, see [8, 9]. However, we note that the (n + 1)th iterate x _n+1 is defined as the projection of the initial guess x ₀ onto the intersection of two closed convex subsets C n and Q n . Therefore, an interesting problem is how to construct appropriately C n and Q n such that the computations become easier.

Motivated by the above works, in this paper we introduce two iterative algo- rithms for finding a common element of the set of fixed points of an asymptoti- cally k-strict pseudo-contraction and the set of solutions of a mixed equilibrium problem in a Hilbert space. We obtain some weak and strong convergence the- orems by using the proposed iterative algorithms. Our results extend and improve the corresponding results of Tada and Takahashi [16], Kim and Xu [8, 9].

2. Preliminaries

Let H be a real Hilbert space with inner product h·, ·i and norm k · k. Let C be a nonempty closed convex subset of H. Then, for any x ∈ H, there exists a unique nearest point in C, denoted by P C (x), such that

kx − P C (x)k ≤ kx − yk

for all y ∈ C. Such a P C is called the metric projection of H onto C. We know that P C is nonexpansive. Further, for x ∈ H and x ^∗ ∈ C,

(1) x ^∗ = P C (x) ⇔ hx − x ^∗ , x ^∗ − yi ≥ 0 for all y ∈ C.

Recall that a mapping T : C → C is said to be an asymptotically k-strict pseudo-contraction if, there exists a constant k ∈ [0, 1) satisfying

(2) kT ⁿ x − T ⁿ yk ² ≤ (1 + γ n )kx − yk ² + kk(I − T ⁿ )x − (I − T ⁿ )yk ² for all x, y ∈ C and all integers n ≥ 1, where γ n ≥ 0 for all n and such that γ n → 0 as n → ∞. Note that if k = 0, then T is an asymptotically nonexpansive mapping, that is, there exists a sequence {γ n } of nonnegative numbers with γ n → 0 such that

kT ⁿ x − T ⁿ yk ≤ (1 + γ n )kx − yk ² for all x, y ∈ C and all integers n ≥ 1.

In the sequel, we will use F (T ) to denote the set of fixed points of T . For given sequence {x n } ⊂ C, let ω w (x n ) = {x : ∃x n

→ x weakly} denote the weak ω-limit set of {x n }.

In this paper, for solving the mixed equilibrium problems for an equilibrium bifunction Θ : C × C → R, we assume that Θ satisfies the following conditions:

(H1) Θ is monotone, i.e., Θ(x, y) + Θ(y, x) ≤ 0 for all x, y ∈ C;

(H2) for each fixed y ∈ C, x 7→ Θ(x, y) is concave and upper semicontinuous;

(H3) for each x ∈ C, y 7→ Θ(x, y) is convex.

A mapping η : C × C → H is called Lipschitz continuous, if there exists a constant λ > 0 such that

kη(x, y)k ≤ λkx − yk, ∀x, y ∈ C.

A differentiable function K : C → R on a convex set C is called:

(i) η-convex if

K(y) − K(x) ≥ hK

(x), η(y, x)i, ∀x, y ∈ C, where K

is the Frechet derivative of K at x;

(ii) η-strongly convex if there exists a constant σ > 0 such that K(y) − K(x) − hK

(x), η(y, x)i ≥ (σ/2)kx − yk ² , ∀x, y ∈ C.

Let C be a nonempty closed convex subset of a real Hilbert space H, ϕ : C → R be real-valued function and Θ : C × C → R be an equilibrium bifunction.

Let r be a positive number. For a given point x ∈ C, the auxiliary problem for (MEP) consists of finding y ∈ C such that

Θ(y, z) + ϕ(z) − ϕ(y) + 1

r hK

(y) − K

(x), η(z, y)i ≥ 0, ∀z ∈ C.

Let S r : C → C be the mapping such that for each x ∈ C, S r (x) is the solution set of the auxiliary problem, i.e., ∀x ∈ C,

S r (x) =

½

y ∈ C : Θ(y, z) + ϕ(y) + 1

r hK

(y) − K

(x), η(z, y)i ≥ 0, ∀z ∈ C

¾ .

We need the following important and interesting result for proving our main results.

Lemma 2.1 ([21]). Let C be a nonempty closed convex subset of a real Hilbert space H and let ϕ : C → R be a lower semicontinuous and convex functional.

Let Θ : C×C → R be an equilibrium bifunction satisfying conditions (H1)-(H3).

Assume that

(i) η : C × C → H is Lipschitz continuous with constant λ > 0 such that (a) η(x, y) + η(y, x) = 0, ∀x, y ∈ C,

(b) η(·, ·) is affine in the first variable,

(c) for each fixed y ∈ C, x 7→ η(y, x) is sequentially continuous from the weak topology to the weak topology;

(ii) K : C → R is η-strongly convex with constant σ > 0 and its deriva- tive K

is sequentially continuous from the weak topology to the strong topology;

(iii) for each x ∈ C, there exist a bounded subset D x ⊂ C and z x ∈ C such that for any y ∈ C \ D x ,

Θ(y, z x ) + ϕ(z x ) − ϕ(y) + 1

r hK

(y) − K

(x), η(z x , y)i < 0.

Then there hold the following:

(I) S r is single-valued;

(II) S r is nonexpansive if K

is Lipschitz continuous with constant ν > 0 such that σ ≥ λν and

hK

(x 1 ) − K

(x 2 ), η(u 1 , u 2 )i ≥ hK

(u 1 ) − K

(u 2 ), η(u 1 , u 2 )i, ∀(x 1 , x 2 ) ∈ C × C where u i = S r (x i ) for i = 1, 2;

(III) F (S r ) = Ω;

(IV) Ω is closed and convex.

We also need the following lemmas.

Lemma 2.2. Let H be a real Hilbert space. There hold the following well- known identities:

(i) kx − yk ² = kxk ² − 2hx, yi + kyk ² ∀x, y ∈ H.

(ii) ktx+(1−t)yk ² = tkxk ² +(1−t)kyk ² −t(1−t)kx−yk ² ∀t ∈ [0, 1], ∀x, y ∈ H.

Lemma 2.3 ([9]). Assume C is a closed convex subset of a real Hilbert space

H and let T : C → C be an asymptotically k-strict pseudo-contraction. Then

the following conclusions hold:

(i) for each n ≥ 1, T ⁿ satisfies the Lipschitz condition:

kT ⁿ x − T ⁿ yk ≤ L n kx − yk ∀x, y ∈ C, where L n = ^k+

√ 1+γ

(1−k)

1−k ;

(ii) the mapping I − T is demiclosed at zero, that is, if {x n } is a sequence in C such that x n → x ^∗ weakly and (I − T )x n → 0 strongly, then (I − T )x ^∗ = 0;

(iii) the fixed point set F (T ) of T is closed and convex so that the projection P F (T ) is well-defined.

Lemma 2.4 ([11]). Let C be a closed convex subset of a real Hilbert space H.

Let {x n } be a sequence in H and u ∈ H. Let q = P C u. If {x n } is such that ω w (x n ) ⊂ C and satisfies the condition

kx n − uk ≤ ku − qk for all n, then x n → q.

3. Main results

In this section, we first introduce the following new iterative algorithm.

Algorithm 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H, ϕ : C → R be a lower semicontinuous and convex real valued function, Θ : C×C → R be an equilibrium bifunction and T : C → C be an asymptotically k-strict pseudo-contraction. Assume that F (T ) ∩ Ω is nonempty and bounded.

Let r be a positive parameter and δ ∈ (k, 1) be a constant. Let {α n } be a sequence in [0, 1]. Define the sequences {x n } and {y n } by the following manner:

(3)

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x 0 ∈ C chosen arbitrarily, Θ(y n , x) + ϕ(x) − ϕ(y n ) + 1

r hK

(y n ) − K

(x n ), η(x, y n )i ≥ 0, ∀x ∈ C, z n = (1 − α n )x n + α n [δy n + (1 − δ)T ⁿ y n ],

C n = {z ∈ C : kz n − zk ² ≤ kx n − zk ² + θ n }, Q n = {z ∈ C : hx n − z, x 0 − x n i ≥ 0}, x n+1 = P C

∩Q

x 0 ,

where θ n = γ n ∆ ² _n , ∆ n = sup{kx n − pk : p ∈ F (T ) ∩ Ω} < ∞.

Now we give a strong convergence result concerning iterative Algorithm 3.1 as follows.

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let ϕ : C → R be a lower semicontinuous and convex functional. Let Θ : C × C → R be an equilibrium bifunction satisfying conditions (H1)-(H3) and let T : C → C be an asymptotically k-strict pseudo-contraction. Assume that F (T ) ∩ Ω is nonempty and bounded. Assume that:

(i) η : C × C → H is Lipschitz continuous with constant λ > 0 such that;

(a) η(x, y) + η(y, x) = 0, ∀x, y ∈ C, (b) η(·, ·) is affine in the first variable,

(c) for each fixed y ∈ C, x 7→ η(y, x) is sequentially continuous from the weak topology to the weak topology;

(ii) K : C → R is η-strongly convex with constant σ > 0 and its derivative K

is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constant ν > 0 such that σ ≥ λν;

(iii) for each x ∈ C; there exist a bounded subset D x ⊂ C and z x ∈ C such that, for any C 3 y / ∈ D x ,

Θ(y, z x ) + ϕ(z x ) − ϕ(y) + 1

r hK

(y) − K

(x), η(z x , y)i < 0;

(iv) α n ∈ [a, 1] for some a ∈ (0, 1).

Then the sequence {x n } generated iteratively by (3) converges strongly to P _{F (T )∩Ω} x 0 provided S r is firmly nonexpansive.

Proof. First, we show that the sequence {x n } is well-defined. It is obvious that C n and Q n are closed and convex. Let p ∈ F (T ) ∩ Ω. From y n = S r x n , we have

(4) ky n − pk = kS r x n − S r pk ≤ kx n − pk.

From Lemma 2.2, (2) and (4), we obtain (5)

kz n − pk ² ≤ (1 − α n )kx n − pk ² + α n kδ(y n − p) + (1 − δ)(T ⁿ y n − p)k ²

= (1 − α n )kx n − pk ² + α n [δky n − pk ² + (1 − δ)kT ⁿ y n − pk ²

− δ(1 − δ)kT ⁿ y n − y n k ² ]

≤ (1 − α n )kx n − pk ² + α n {δky n − pk ² + (1 − δ)[(1 + γ n )ky n − pk ² + kky n − T ⁿ y n k ² ] − δ(1 − δ)ky n − T ⁿ y n k ² }

= (1 − α n )kx n − pk ² + α n {ky n − pk ² + (1 − δ)γ n ky n − pk ² + (1 − δ)(k − δ)ky n − T ⁿ y n k ² }

≤ (1 − α n )kx n − pk ² + α n (1 + γ n )ky n − pk ²

≤ (1 + γ n )kx n − pk ² .

Hence, we have kz n − pk ² ≤ kx n − pk ² + θ n . This implies that p ∈ C n ; thus,

(6) F (T ) ∩ Ω ⊂ C n

for every n ≥ 0. Next we show by induction that F (T ) ∩ Ω ⊂ C n ∩ Q n for each n ≥ 0. Since F (T ) ∩ Ω ⊂ C 0 and Q 0 = C, we get

F (T ) ∩ Ω ⊂ C 0 ∩ Q 0 .

Suppose that F (T )∩Ω ⊂ C k ∩Q k for k ∈ N. Then, there exists x n+1 ∈ C k ∩Q k

such that

x n+1 = P C

∩Q

x 0 . Therefore, for each z ∈ C k ∩ Q k , we have

hx k+1 − z, x 0 − x k+1 i ≥ 0.

Note that F (T ) ∩ Ω ⊂ C k ∩ Q k . Hence, for any z ∈ F (T ) ∩ Ω we have hx k+1 − z, x 0 − x k+1 i ≥ 0,

therefore z ∈ Q _k+1 . So, we get

F (T ) ∩ Ω ⊂ Q k+1 . From this and (6), we have

F (T ) ∩ Ω ⊂ C k+1 ∩ Q k+1 . This denotes that the sequence {x n } is well-defined.

Since F (T ) ∩ Ω is a nonempty closed convex subset of C, there exists a unique z

∈ F (T ) ∩ Ω such that z

= P _{F (T )∩Ω} x 0 . From x n+1 = P C

∩Q

x 0 , we have

kx n+1 − x 0 k ≤ kz − x 0 k

for all z ∈ C n ∩ Q n . Since z

∈ F (T ) ∩ Ω ⊂ C n ∩ Q n , we have (7) kx n+1 − x 0 k ≤ kz

− x 0 k

for every n ≥ 0. Therefore, {x n } is bounded, so are {y n } and {z n }. Since x n = P Q

x 0 and x n+1 ∈ Q n , we get

kx n − x 0 k ≤ kx n+1 − x 0 k.

This together with the boundedness of {kx n −x 0 k} implies that lim n→∞ kx n −x 0 k exists. The fact that x n+1 ∈ Q n implies that hx n+1 −x n , x n −x 0 i ≥ 0. Applying Lemma 2.2, we obtain

(8)

kx n+1 − x n k ² = k(x n+1 − x 0 ) − (x n − x 0 )k ²

= kx n+1 − x 0 k ² − kx n − x 0 k ² − 2hx n+1 − x n , x n − x 0 i

≤ kx n+1 − x 0 k ² − kx n − x 0 k ²

→ 0.

From x n+1 ∈ C n , we have

kx n − z n k ≤ kx n − x n+1 k + kx n+1 − z n k

≤ (2 + γ n )kx n − x n+1 k

→ 0.

For p ∈ F (T ) ∩ Ω, noting that S r is firmly nonexpansive, we have ky n − pk ² = kS r x n − S r pk ²

≤ kS r x n − S r p, x n − pi

= hy n − p, x n − pi

= 1

2 {ky _n − pk ² + kx _n − pk ² − kx _n − y _n k ² }, and hence,

ky n − pk ² ≤ kx n − pk ² − kx n − y n k ² . From (5), we have

kz n − pk ² ≤ (1 − α n )kx n − pk ² + α n (1 + γ n )ky n − pk ²

≤ (1 − α n )kx n − pk ² + α n (1 + γ n ){kx n − pk ² − kx n − y n k ² }

= (1 + α n γ n )kx n − pk ² − α n (1 + γ n )kx n − y n k ² , that is,

(9)

kx n − y n k ² ≤ 1

α n (1 + γ n ) {kx n − pk ² − kz n − pk ² } + γ n kx n − pk ² 1 + γ n

≤ 1

α n (1 + γ n ) kx n − z n k{kx n − pk + kz n − pk} + γ n kx n − pk ² 1 + γ n

→ 0.

Combining (8) and (9), we have

n→∞ lim ky n+1 − y n k = 0.

By the fact α n (1 − δ)(T ⁿ y n − y n ) = z n − (1 − α n )x n − α n y n , we get kα n (1 − δ)(T ⁿ y n − y n )k ≤ kz n − x n k + α n kx n − y n k → 0, which implies that

kT ⁿ y n − y n k → 0.

Next we show that

n→∞ lim ky _n − T y _n k = 0.

As a matter of fact, we have

ky n − T y n k ≤ ky n − T ⁿ y n k + kT ⁿ y n − T ⁿ⁺¹ y n k + kT ⁿ⁺¹ y n − T y n k

≤ (1 + L ₁ )ky _n − T ⁿ y _n k + kT ⁿ y _n − T ⁿ⁺¹ y _n k.

Note that

kT ⁿ y n − T ⁿ⁺¹ y n k ≤ kT ⁿ y n − y n k + ky n − y n+1 k + ky n+1 − T ⁿ⁺¹ y n+1 k + kT ⁿ⁺¹ y n+1 − T ⁿ⁺¹ y n k

≤ kT ⁿ y n − y n k + (1 + L n+1 )ky n − y n+1 k

+ ky n+1 − T ⁿ⁺¹ y n+1 k.

Therefore, we have

(10) ky n − T y n k ≤ (2 + L 1 )ky n − T ⁿ y n k + (1 + L n+1 )ky n − y n+1 k + ky n+1 − T ⁿ⁺¹ y n+1 k → 0.

Since {y n } is bounded, there exists a subsequence {y n

} of {y n } which converges weakly to w. From (7), we also obtain that T y n

→ w weakly. Next we show that w ∈ Ω. Since y _n = S _r x _n , we derive

Θ(y n , x) + ϕ(x) − ϕ(y n ) + 1

r hK

(y n ) − K

(x n ), η(x, y n )i ≥ 0, ∀x ∈ C.

From the monotonicity of Θ, we have 1

r hK

(y n ) − K

(x n ), η(x, y n )i + ϕ(x) − ϕ(y n ) ≥ −Θ(y n , x) ≥ Θ(x, y n ), and hence

*

K

(y n

) − K

(x n

)

r , η(x, y n

) +

+ ϕ(x) − ϕ(y n

) ≥ Θ(x, y n

).

Since ^K

0

(y

)−K

(x

)

r → 0 and y n

→ w weakly, from the weak lower semicon- tinuity of ϕ and Θ(x, y) in the second variable y, we have

Θ(x, w) + ϕ(w) − ϕ(x) ≤ 0

for all x ∈ C. For 0 < t ≤ 1 and x ∈ C, let x t = tx + (1 − t)w. Since x ∈ C and w ∈ C, we have x t ∈ C and hence Θ(x t , w) + ϕ(w) − ϕ(x t ) ≤ 0. From the convexity of equilibrium bifunction Θ(x, y) in the second variable y, we have

0 = Θ(x t , x t ) + ϕ(x t ) − ϕ(x t )

≤ tΘ(x _t , x) + (1 − t)Θ(x _t , w) + tϕ(x) + (1 − t)ϕ(w) − ϕ(x _t )

≤ t[Θ(x t , x) + ϕ(x) − ϕ(x t )],

and hence Θ(x t , x) + ϕ(x) − ϕ(x t ) ≥ 0. Then, we have Θ(w, x) + ϕ(x) − ϕ(w) ≥ 0 for all x ∈ C and hence w ∈ Ω.

We shall prove that w ∈ F (T ). As a matter of fact, Lemma 2.3 and (10) guarantee that every weak limit point of {y n } is a fixed point of T . That is, ω w (y n ) ⊂ F (T ). Hence, w ∈ F (T ).

Therefore, we have

w ∈ F (T ) ∩ Ω.

This fact, the inequality (7) and Lemma 2.4 ensure the strong convergence of {x n } to P F (T )∩Ω x 0 . This completes the proof. ¤

As a direct consequence of Theorem 3.1, we obtain the following.

Corollary 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let ϕ : C → R be a lower semicontinuous and convex functional.

Let Θ : C × C → R be an equilibrium bifunction satisfying conditions (H1)- (H3) and let T : C → C be an asymptotically nonexpansive mapping such that F (T ) ∩ Ω 6= ∅. Assume that:

(i) η : C × C → H is Lipschitz continuous with constant λ > 0 such that;

(a) η(x, y) + η(y, x) = 0, ∀x, y ∈ C, (b) η(·, ·) is affine in the first variable,

(c) for each fixed y ∈ C, x 7→ η(y, x) is sequentially continuous from the weak topology to the weak topology;

(ii) K : C → R is η-strongly convex with constant σ > 0 and its derivative K

is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constant ν > 0 such that σ ≥ λν;

(iii) for each x ∈ C; there exist a bounded subset D x ⊂ C and z x ∈ C such that, for any C 3 y / ∈ D x ,

Θ(y, z x ) + ϕ(z x ) − ϕ(y) + 1

r hK

(y) − K

(x), η(z x , y)i < 0;

(iv) α n ∈ [a, 1] for some a ∈ (0, 1).

Then the sequence {x n } generated iteratively by (3) converges strongly to P F (T )∩Ω x 0 provided S r is firmly nonexpansive.

Corollary 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be an asymptotically k-strict pseudo-contraction such that F (T ) is nonempty and bounded. Let δ ∈ (k, 1) be a constant and {α n } be a sequence in [0, 1]. Assume α n ∈ [a, 1] for some a ∈ (0, 1). For given x 0 ∈ C arbitrarily, let the sequence {x _n } generated iteratively by

(11)

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 

z n = (1 − α n )x n + α n [δx n + (1 − δ)T ⁿ x n ], C n = {z ∈ C : kz n − zk ² ≤ kx n − zk ² + θ n }, Q n = {z ∈ C : hx n − z, x 0 − x n i ≥ 0}, x n+1 = P C

∩Q

x 0 ,

where θ n = γ n ∆ ² _n , ∆ n = sup{kx n − pk : p ∈ F (T )} < ∞. Then the sequence {x n } defined by (11) converges strongly to P _{F (T )} x 0 .

Proof. Set ϕ(x) = 0 and Θ(x, y) = 0 for all x, y ∈ C and put r = 1. Take

K(x) = ^kxk ₂

and η(y, x) = y − x for all x, y ∈ C. Then we have y n = x n .

Hence, by the similar argument as that in the proof of Theorem 3.1, we can

obtain the desired result. This completes the proof. ¤

Corollary 3.3. Let C be a nonempty closed convex subset of a real Hilbert

space H. Let T : C → C be an asymptotically nonexpansive mappings such

that F (T ) is nonempty and bounded. Let δ ∈ (k, 1) be a constant and {α n } be

a sequence in [0, 1]. Assume α n ∈ [a, 1] for some a ∈ (0, 1). Then the sequence {x n } defined by (11) converges strongly to P _{F (T )} x 0 .

Now we give another iterative algorithm as follows.

Algorithm 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H, ϕ : C → R be a lower semicontinuous and convex real valued function, Θ : C×C → R be an equilibrium bifunction and T : C → C be an asymptotically k-strict pseudo-contraction. Let r be a positive parameter and δ ∈ (k, 1) be a constant. Let {α n } be a sequence in [0, 1]. Define the sequences {x n } and {y n } by the following manner:

(12)

 

 

 



x 0 ∈ C chosen arbitrarily, Θ(y n , x) + ϕ(x) − ϕ(y n ) + 1

r hK

(y n ) − K

(x n ), η(x, y n )i ≥ 0, ∀x ∈ C, x n+1 = (1 − α n )x n + α n [δy n + (1 − δ)T ⁿ y n ].

Finally we state and prove a weak convergence theorem concerning Algo- rithm 3.2.

Theorem 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let ϕ : C → R be a lower semicontinuous and convex functional. Let Θ : C × C → R be an equilibrium bifunction satisfying conditions (H1)-(H3) and let T : C → C be an asymptotically k-strict pseudo-contraction such that F (T ) ∩ Ω 6= ∅. Assume that:

(i) η : C × C → H is Lipschitz continuous with constant λ > 0 such that;

(a) η(x, y) + η(y, x) = 0, ∀x, y ∈ C, (b) η(·, ·) is affine in the first variable,

(c) for each fixed y ∈ C, x 7→ η(y, x) is sequentially continuous from the weak topology to the weak topology;

(ii) K : C → R is η-strongly convex with constant σ > 0 and its derivative K

is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constant ν > 0 such that σ ≥ λν;

(iii) for each x ∈ C, there exist a bounded subset D x ⊂ C and z x ∈ C such that, for any C 3 y / ∈ D x ,

Θ(y, z x ) + ϕ(z x ) − ϕ(y) + 1

r hK

(y) − K

(x), η(z x , y)i < 0;

(iv) α _n ∈ [a, b] for some a, b ∈ (0, 1) and P _∞

n=1 γ _n < ∞.

Then the sequence {x n } generated iteratively by (12) converges weakly to w ∈ P _{F (T )∩Ω} provided S _r is firmly nonexpansive, where w = lim _n→∞ P _{F (T )∩Ω} (x _n ).

Proof. By Lemma 2.1, {x n } and {y n } are all well-defined. Let p ∈ F (T ) ∩ Ω, from y n = S r x n , we have

ky n − pk = kS r x n − S r pk ≤ kx n − pk.

Therefore, we have

kx n+1 − pk ² ≤ (1 − α n )kx n − pk ² + α n kδ(y n − p) + (1 − δ)(T ⁿ y n − p)k ²

= (1 − α n )kx n − pk ² + α n [δky n − pk ² + (1 − δ)kT ⁿ y n − pk ²

− δ(1 − δ)ky n − T ⁿ y n k ² ]

≤ (1 − α n )kx n − pk ² + α n {δky n − pk ² + (1 − δ)[(1 + γ n )

× ky n − pk ² + kky n − T ⁿ y n k ² ] − δ(1 − δ)ky n − T ⁿ y n k ² }

= (1 − α n )kx n − pk ² + α n {[1 + (1 − δ)γ n ]ky n − pk ² + (1 − δ)(k − δ)ky n − T y n k ² }

≤ (1 − α n )kx n − pk ² + α n (1 + γ n )ky n − pk ²

≤ (1 + γ n )kx n − pk ² .

This implies that lim n→∞ kx n − pk exists. Hence, {x n }, {y n } are all bounded and

n→∞ lim kx n+1 − x n k = 0.

Next, for p ∈ F (T ) ∩ Ω, noting S r is firmly nonexpansive, we have ky n − pk ² = kS r x n − S r pk ²

≤ kS r x n − S r p, x n − pi

= hy n − p, x n − pi

= 1

2 {ky n − pk ² + kx n − pk ² − kx n − y n k ² }, and hence,

ky n − pk ² ≤ kx n − pk ² − kx n − y n k ² . Therefore, we have

kx n+1 − pk ² ≤ (1 − α n )kx n − pk ² + α n (1 + γ n )ky n − pk ²

≤ (1 − α n )kx n − pk ² + α n (kx n − pk ² − kx n − y n k ² ) + α n γ n ky n − pk ²

≤ kx n − pk ² − α n kx n − y n k ² + α n γ n ky n − pk ² . So, we obtain

kx n − y n k ² ≤ 1

α n {kx n − pk ² − kx n+1 − pk ² } + γ n ky n − pk ² → 0.

As {x n } is bounded, there exists a subsequence {x n

} of {x n } which converges

weakly to w. From kx n − y n k → 0, we also have that y n

→ w weakly. First, as

in the proof of Theorem 3.1, we can show w ∈ Ω. Let us show that w ∈ F (T ).

Let p ∈ F (T ) ∩ Ω. Since α n (1 − δ)T ⁿ y n = x n+1 − (1 − α n )x n − αδy n , we have kα n (1 − δ)(T ⁿ y n − x n )k = kx n+1 − x n + α n δ(x n − y n )k

≤ kx n+1 − x n k + α n kx n − y n k

→ 0, which implies that

kT ⁿ y n − x n k → 0.

Hence

kT ⁿ y n − y n k ≤ kT ⁿ y n − x n k + ky n − x n k → 0.

Repeating the similar argument as Theorem 3.1, we can obtain

n→∞ lim ky n − T y n k = 0.

From this and y n

→ w weakly, we obtain w ∈ F (T ). Then, w ∈ F (T ) ∩ Ω.

Let {x _n

} be another subsequence of {x _n } such that x _n

→ w

weakly. Then, we have

w

∈ F (T ) ∩ Ω.

If w 6= w

, from the opial theorem [20], we get

n→∞ lim kx n − wk = lim inf

i→∞ kx n

− wk

< lim inf

i→∞ kx n

− w

k

= lim

n→∞ kx n − w

k

= lim inf

j→∞ kx n

− w

k

< lim inf

j→∞ kx n

− wk

= lim

n→∞ kx n − wk.

This is a contradiction. So we have w = w

. This implies that x n → F (T ) ∩ Ω weakly.

Let z n = P F (T )∩Ω (x n ). Since w ∈ F (T ) ∩ Ω, we have hx n − z n , z n − wi ≥ 0.

Hence, we have that {z n } converges strongly to some w 0 ∈ F (T ) ∩ Ω. Since {x n } converges weakly to w, we have

hw − w 0 , w 0 − wi ≥ 0.

Therefore, we obtain

w = w 0 = lim

n→∞ P _{F (T )∩Ω} (x n ).

This completes the proof. ¤

Corollary 3.4. Let C be a nonempty closed convex subset of a real Hilbert space H. Let ϕ : C → R be a lower semicontinuous and convex functional.

Let Θ : C × C → R be an equilibrium bifunction satisfying conditions (H1)- (H3) and let T : C → C be an asymptotically nonexpansive mapping such that F (T ) ∩ Ω 6= ∅. Assume that:

(i) η : C × C → H is Lipschitz continuous with constant λ > 0 such that;

(a) η(x, y) + η(y, x) = 0, ∀x, y ∈ C, (b) η(·, ·) is affine in the first variable,

(c) for each fixed y ∈ C, x 7→ η(y, x) is sequentially continuous from the weak topology to the weak topology;

(ii) K : C → R is η-strongly convex with constant σ > 0 and its derivative K

is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constant ν > 0 such that σ ≥ λν;

(iii) for each x ∈ C, there exist a bounded subset D x ⊂ C and z x ∈ C such that, for any C 3 y / ∈ D x ,

Θ(y, z x ) + ϕ(z x ) − ϕ(y) + 1

r hK

(y) − K

(x), η(z x , y)i < 0;

(iv) α n ∈ [a, b] for some a, b ∈ (0, 1) and P _∞

n=0 γ n < ∞.

Then the sequence {x n } generated iteratively by (12) converges weakly to w ∈ P F (T )∩Ω provided S r is firmly nonexpansive, where w = lim n→∞ P F (T )∩Ω (x n ).

Corollary 3.5. Let C be a nonempty closed convex subset of a real Hilbert space H. Let ϕ : C → R be a lower semicontinuous and convex functional. Let Θ : C × C → R be an equilibrium bifunction satisfying conditions (H1)-(H3) such that Ω 6= ∅. Assume that:

(i) η : C × C → H is Lipschitz continuous with constant λ > 0 such that;

(a) η(x, y) + η(y, x) = 0, ∀x, y ∈ C, (b) η(·, ·) is affine in the first variable,

(c) for each fixed y ∈ C, x 7→ η(y, x) is sequentially continuous from the weak topology to the weak topology;

(ii) K : C → R is η-strongly convex with constant σ > 0 and its derivative K

is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constant ν > 0 such that σ ≥ λν;

(iii) for each x ∈ C, there exist a bounded subset D x ⊂ C and z x ∈ C such that, for any C 3 y / ∈ D x ,

Θ(y, z x ) + ϕ(z x ) − ϕ(y) + 1

r hK

(y) − K

(x), η(z x , y)i < 0.

Let the sequences {x n } and {y n } be generated by

 

 

 



x ₀ ∈ C chosen arbitrarily, Θ(y n , x) + ϕ(x) − ϕ(y n ) + 1

r hK

(y n ) − K

(x n ), η(x, y n )i ≥ 0, ∀x ∈ C, x n+1 = (1 − α n )x n + α n y n ,

where r is a positive parameter and {α n } is a sequence ∈ [a, b] for some a, b ∈ (0, 1).

Then the sequence {x n } converges weakly to w ∈ Ω provided S r is firmly non- expansive, where w = lim n→∞ P Ω (x n ).

Proof. Taking T = I in Theorem 3.2, we can obtain our desired result. This

completes the proof. ¤

Corollary 3.6. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be an asymptotically k-strict pseudo-contraction such that F (T ) 6= ∅. Let δ ∈ (k, 1) be a constant and {α n } be a real sequence in [0, 1]. Assume that α _n ∈ [a, b] for some a, b ∈ (0, 1) and P _∞

n=1 γ _n < ∞. For given x 0 ∈ C arbitrarily, let the sequence {x n } generated iteratively by

x n+1 = (1 − α n )x n + α n [δy n + (1 − δ)T ⁿ y n ].

Then {x n } converges weakly to w ∈ P F (T ) , where w = lim n→∞ P F (T ) (x n ).

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Yonghong Yao

Department of Mathematics Tianjin Polytechnic University Tianjin 300160, China

E-mail address: [email protected] Haiyun Zhou

Department of Mathematics

Shijiazhuang Mechanical Engineering College Shijiazhuang 050003, China

E-mail address: [email protected] Yeong-Cheng Liou

Department of Information Management Cheng Shiu University

Kaohsiung 833, Taiwan

E-mail address: simplex [email protected]