Convergence Theorems for Equilibrium Problems and Fixed-Point

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2012, Article ID 416476, 23 pages doi:10.1155/2012/416476

Research Article Convergence Theorems for Equilibrium Problems and Fixed-Point Problems of an Infinite Family of ki -Strictly Pseudocontractive Mapping in Hilbert Spaces Haitao Che,1, 2 Meixia Li,1 and Xintian Pan1 1 2

School of Mathematics and Information Science, Weifang University, Shandong Weifang 261061, China School of Management Science, Qufu Normal University, Shandong Rizhao 276800, China

Correspondence should be addressed to Haitao Che, [email protected] Received 28 February 2012; Revised 8 June 2012; Accepted 24 June 2012 Academic Editor: Jong Hae Kim Copyright q 2012 Haitao Che et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We first extend the definition of Wn from an infinite family of nonexpansive mappings to an infinite family of strictly pseudocontractive mappings, and then propose an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of an infinite family of ki -strictly pseudocontractive mappings in Hilbert spaces. The results obtained in this paper extend and improve the recent ones announced by many others. Furthermore, a numerical example is presented to illustrate the effectiveness of the proposed scheme.

1. Introduction Let H be a real Hilbert space with inner product ·, · and induced norm · . Let C be a nonempty closed convex subset of H and let F : C × C → R be a bifunction. We consider the following equilibrium problem EP which is to find z ∈ C such that EP : F z, y ≥ 0,

∀y ∈ C.

1.1

Denote the set of solutions of EP by EPF. Given a mapping T : C → H, let Fx, y T x, y − x for all x, y ∈ C. Then, z ∈ EPF if and only if T x, y − x ≥ 0 for all y ∈ C, that is, z is a solution of the variational inequality. Numerous problems in physics, optimization, and

2

Journal of Applied Mathematics

economics reduce to find a solution of 1.1. Some methods have been proposed to solve the equilibrium problem 1–13. A mapping B : C → C is called θ-Lipschitzian if there exists a positive constant θ such that Bx − By ≤ θx − y,

∀x, y ∈ C.

1.2

B is said to be η-strongly monotone if there exists a positive constant η such that 2 Bx − By, x − y ≥ ηx − y ,

∀x, y ∈ C.

1.3

A mapping S : C → C is said to be k-strictly pseudocontractive mapping if there exists a constant 0 ≤ k < 1 such that Sx − Sy2 ≤ x − y2 kI − Sx − I − Sy2 ,

1.4

for all x, y ∈ C and FS denotes the set of fixed point of the mapping S, that is FS {x ∈ C : Sx x}. If k 1, then S is said to a pseudocontractive mapping, that is, Sx − Sy2 ≤ x − y2 I − Sx − I − Sy2 ,

1.5

is equivalent to

I − Sx − I − Sy, x − y ≥ 0,

1.6

for all x, y ∈ C. The class of k-strict pseudo-contractive mappings extends the class of nonexpansive mappings A mapping T is said to be nonexpansive if T x − T y ≤ x − y, for all x, y ∈ C. That is, S is nonexpansive if and only if S is a 0-strict pseudocontractive mapping. Clearly, the class of k-strictly pseudocontractive mappings falls into the one between classes of nonexpansive mappings and pseudo-contractive mapping. In 2006, Marino and Xu 14 introduced the general iterative method and proved that for a given x0 ∈ H, the sequence {xn } generated by the algorithm xn1 αn γfxn I − αn BT xn ,

n ∈ N,

1.7

where T is a self-nonexpansive mapping on H, f is an α-contraction of H into itself i.e., fx − fy ≤ αx − y, for all x, y ∈ H and α ∈ 0, 1, {αn } ⊂ 0, 1 satisfies certain conditions, B is strongly positive bounded linear operator on H, and converges strongly to fixed point x∗ of T which is the unique solution to the following variational inequality: γf − B x∗ , x∗ − x ≤ 0,

∀x ∈ FT .

1.8


3

Tian 15 considered the following iterative method, for a nonexpansive mapping T :H → H with FT / ∅, xn1 αn γfxn I − μαn F T xn ,

n ∈ N,

1.9

where F is k-Lipschitzian and η-strongly monotone operator. The sequence {xn } converges strongly to fixed-point q in FT which is the unique solution to the following variational inequality:

γf − μF q, p − q ≤ 0,

p ∈ FT .

1.10

For finding a common element of EPF ∩ FS, S. Takahashi and W. Takahashi 16 introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solution 1.1 and the set of fixed points of a nonexpansive mapping in a Hilbert space. Let S : C → H be a nonexpansive mapping. Starting with arbitrary initial point x1 ∈ H, define sequences {xn } and {un } recursively by 1 F un , y y − un , un − xn ≥ 0, rn xn1 αn fxn 1 − αn Sun ,

∀y ∈ C,

1.11

∀n ∈ N.

They proved that under certain appropriate conditions imposed on {αn } and {rn }, the sequences {xn } and {un } converge strongly to z ∈ FS ∩ EPF, where z PFS∩EPF fz. Liu 17 introduced the following scheme: x1 ∈ H and 1 F un , y y − un , un − xn ≥ 0, rn yn βn un 1 − βn Sun , xn1 αn γfxn I − αn Byn ,

∀y ∈ C, 1.12 ∀n ∈ N,

where S is a k-strict pseudo-contractive mapping and B is a strongly positive bounded linear operator. They proved that under certain appropriate conditions imposed on {αn }, {βn }, and {rn }, the sequence {xn } converges strongly to z ∈ FS ∩ EPF, where z PFS∩EPF I − B γfz. In 18, the concept of W mapping had been modified for a countable family {Tn }n∈N of nonexpansive mappings by defining the sequence {Wn }n∈N of W-mappings generated by {Tn }n∈N and {λn } ⊂ 0, 1, proceeding backward Un,n1 : I, Un,n : λn Tn Un,n1 1 − λn I, ···

4

Journal of Applied Mathematics Un,k : λk Tk Un,k1 1 − λk I, ··· Un,2 : λ2 T2 Un,3 1 − λ2 I, Wn Un,1 : λ1 T1 Un,2 1 − λ1 I. 1.13

Yao et al. 19 using this concept, introduced the following algorithm: x1 ∈ H and

xn1

1 F un , y y − un , un − xn ≥ 0, ∀y ∈ C, rn αn fxn βn xn 1 − αn − βn Wn un , ∀n ∈ N.

1.14

They proved that under certain appropriate conditions imposed on {αn } and {rn }, the se quences {xn } and {un } converge strongly to z ∈ ∞ i1 FTi ∩ EPF. Colao and Marino 20 considered the following explicit viscosity scheme

xn1

1 F un , y y − un , un − xn ≥ 0, ∀y ∈ C, rn αn rfxn βn xn 1 − βn I − αn A Wn un , ∀n ∈ N,

1.15

where A is a strongly positive operator on H. Under certain appropriate conditions, the sequences {xn } and {un } converge strongly to z ∈ ∞ i1 FTi ∩ EPF. Motivated and inspired by these facts, in this paper, we first extend the definition of Wn from an infinite family of nonexpansive mappings to an infinite family of strictly pseudocontractive mappings, and then propose the iteration scheme 3.2 for finding an element of EPF ∞ i1 FSi , where {Si } is an infinite family of ki -strictly pseudo-contractive mappings of C into itself. Finally, the convergence theorem of the iteration scheme is obtained. Our results include Yao et al. 19, Colao and Marino 20 as some special cases.

2. Preliminaries Throughout this paper, we always assume that C is a nonempty closed convex subset of a Hilbert space H. We write xn x to indicate that the sequence {xn } converges weakly to x. xn → x implies that {xn } converges strongly to x. We denote by N and R the sets of positive integers and real numbers, respectively. For any x ∈ H, there exists a unique nearest point in C, denoted by PC x, such that x − PC x ≤ x − y,

∀y ∈ C.

2.1

Such a PC is called the metric projection of H onto C. It is known that PC is nonexpansive. Furthermore, for x ∈ H and u ∈ C, u PC x ⇐⇒ x − u, u − y ≥ 0,

∀y ∈ C.

2.2


5

It is widely known that H satisfies Opial s condition 21, that is, for any sequence {xn } with xn x, the inequality lim infxn − x < lim infxn − y

n→∞

n→∞

2.3

holds for every y ∈ H with y / x. In order to solve the equilibrium problem for a bifunction F : C × C → R, we assume that F satisfies the following conditions: A1 Fx, x 0, for all x ∈ C. A2 F is monotone, that is, Fx, y Fy, x ≤ 0, for all x, y ∈ C. A3 limt↓0 Ftz 1 − tx, y ≤ Fx, y, for all x, y, z ∈ C. A4 For each x ∈ C, y → Fx, y is convex and lower semicontinuous. Let us recall the following lemmas which will be useful for our paper. Lemma 2.1 see 22. Let F be a bifunction from C ×C into R satisfying (A1), (A2), (A3), and (A4). Then, for any r > 0 and x ∈ H, there exists z ∈ C such that 1 F z, y y − z, z − x ≥ 0, r

∀y ∈ C.

2.4

Furthermore, if Tr x {z ∈ C : Fz, y 1/ry − z, z − x ≥ 0, ∀y ∈ C}, then the following hold: 1 Tr is single-valued. 2 Tr is firmly nonexpansive, that is, Tr x − Tr y2 ≤ Tr x − Tr y, x − y,

∀x, y ∈ H.

2.5

3 FTr EP F. 4 EP F is closed and convex. Lemma 2.2 see 23. Let S : C → H be a k-strictly pseudo-contractive mapping. Define T : C → H by T x λx 1 − λSx for each x ∈ C. Then, as λ ∈ k, 1, T is nonexpansive mapping such that FT FS. Lemma 2.3 see 24. In a Hilbert space H, there holds the inequality x y2 ≤ x2 2y, x y,

∀x, y ∈ H.

2.6

Lemma 2.4 see 25. Let H be a Hilbert space and C be a closed convex subset of H, and T : C → C a nonexpansive mapping with FT / ∅. If {xn } is a sequence in C weakly converging to x and if {I − T xn } converges strongly to y, then I − T x y.

6


Lemma 2.5 see 26. Let {xn } and {zn } be bounded sequences in a Banach space E and {γn } be a sequence in 0, 1 satisfying the following condition 0 < lim inf γn ≤ lim sup γn < 1. n→∞

n→∞

2.7

Suppose that xn1 γn xn 1 − γn zn , n ≥ 0 and limn → ∞ supzn1 − zn − xn1 − xn ≤ 0. Then limn → ∞ zn − xn 0. Lemma 2.6 see 27. Assume that {an } is a sequence of nonnegative real numbers such that an1 ≤ 1 − bn an bn δn ,

n ≥ 0,

2.8

where {bn } is a sequence in 0, 1 and {δn } is a sequence in R, such that i ∞ i1 bi ∞. ii limn → ∞ sup δn ≤ 0 or ∞ i1 |bn δn | < ∞. Then, limn → ∞ an 0. Let {Si } be an infinite family of ki -strictly pseudo-contractive mappings of C into itself, we define a mapping Wn of C into itself as follows, Un,n1 : I, Un,n : τn Sn Un,n1 1 − τn I, ··· Un,k : τk Sk Un,k1 1 − τk I,

2.9

··· Un,2 : τ2 S2 Un,3 1 − τ2 I, Wn Un,1 : τ1 S1 Un,2 1 − τ1 I, where 0 ≤ τi ≤ 1, Si σi I 1 − σi Si and σi ∈ ki , 1 for i ∈ N. We can obtain Si is a nonexpansive mapping and FSi FSi by Lemma 2.2. Furthermore, we obtain that Wn is a nonexpansive mapping. Remark 2.7. If ki 0, and σi 0 for i ∈ N, then the definition of Wn in 2.9 reduces to the definition of Wn in 1.13. To establish our results, we need the following technical lemmas. Lemma 2.8 see 18. Let C be a nonempty closed convex subset of a strictly convex Banach space. Let {Si } be an infinite family of nonexpansive mappings of C into itself and let {τi } be a real sequence such that 0 < τi ≤ b < 1 for every i ∈ N. Then, for every x ∈ C and k ∈ N, the limit limn → ∞ Un,k x exists.


7

In view of the previous lemma, we will define Wx : lim Wn x lim Un,1 x, n→∞

n→∞

x ∈ C.

2.10

Lemma 2.9 see 18. Let C be a nonempty closed convex subset of a strictly convex Banach space. Let {Si } be an infinite family of nonexpansive mappings of C into itself such that ∞ FSi / ∅ and i1 ∞ let {τi } be a real sequence such that 0 < τi ≤ b < 1 for every i ∈ N. Then, FW i1 FSi / ∅. The following lemmas follow from Lemmas 2.2, 2.8, and 2.9. Lemma 2.10. Let C be a nonempty closed convex subset of a strictly convex Banach space. Let {Si } be ∅. an infinite family of ki -strictly pseudo-contractive mappings of C into itself such that ∞ i1 FSi / Define Si σi I 1 − σi Si and σi ∈ ki , 1 and let {τi } be a real sequence such that 0 < τi ≤ b < 1 ∞ ∅. for every i ∈ N. Then, FW ∞ i1 FSi i1 FSi / Lemma 2.11 see 28. Let C be a nonempty closed convex subset of a Hilbert space. Let {Si } be an infinite family of nonexpansive mappings of C into itself such that ∞ / ∅ and let {τi } be a real i1 FSi sequence such that 0 < τi ≤ b < 1 for every i ∈ N. If K is any bounded subset of C, then lim supWx − Wn x 0.

n → ∞ x∈K

2.11

3. Main Results Let H be a real Hilbert space and F be a k-Lipschitzian and η-strongly monotone operator with k > 0, η > 0, 0 < μ < 2η/k2 and 0 < t < 1. Then, for t ∈ min{0, {1, 1/τ}}, S I − tμF : H → H is a contraction with contractive coefficient 1 − tτ and τ 1/2μ2η − μk2 . In fact, from 1.2 and 1.3, we obtain Sx − Sy2 x − y − tμFx − Fy2 2 2 x − y t2 μ2 Fx − Fy − 2tμFx − Fy, x − y 2 2 2 ≤ x − y k2 t2 μ2 x − y − 2tημx − y

3.1

2 ≤ 1 − tμ 2η − μk2 x − y 2 ≤ 1 − tτ2 x − y . Thus, S 1 − tμF is a contraction with contractive coefficient 1 − tτ ∈ 0, 1. Now, we show the strong convergence results for an infinite family ki -strictly pseudocontractive mappings in Hilbert space. Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H and F be a bifunction from C × C → R satisfying (A1)–(A4). Let Si : C → C be a ki -strictly pseudo-contractive ∅ and {τi } be a real sequence such that 0 < τi ≤ b < 1, i ∈ N. Let mapping with ∞ i1 FSi ∩ EP / f be a contraction of H into itself with β ∈ 0, 1 and B be k-Lipschitzian and η-strongly monotone

8


operator on H with coefficients k, η > 0, 0 < μ < 2η/k2 , 0 < r < 1/2μ2η − μk2 /β τ/β and τ < 1. Let {xn } be a sequence generated by 1 F un , y y − un , un − xn ≥ 0, λn

xn1

∀y ∈ C,

yn δn un 1 − δn Wn un , αn rfxn βn xn 1 − βn I − μαn B yn ,

3.2 ∀n ∈ N,

where un Tλn xn and {Wn : C → C} is the sequence defined by 2.9. If {αn }, {βn }, {δn }, and {λn } satisfy the following conditions: i {αn } ⊂ 0, 1, limn → ∞ αn 0,

∞

i1

αn ∞,

ii 0 < limn → ∞ inf βn ≤ limn → ∞ sup βn < 1, iii 0 < limn → ∞ inf δn ≤ limn → ∞ sup δn < 1, limn → ∞ |δn1 − δn | 0, iv {λn } ⊂ 0, ∞, limn → ∞ λn > 0, limn → ∞ |λn1 − λn | 0. Then {xn } converges strongly to z ∈ variational inequality

∞

i1

FSi ∩ EP / ∅, where z is the unique solution of

lim sup rf − μB z, p − z ≤ 0,

n→∞

∀p ∈

∞ FSi ∩ EP / ∅, i1

3.3

that is, z PFW∩EP F I − μB rfz, which is the optimality condition for the minimization problem 1

min z∈ ∞ i1 FSi ∩EP 2

μBz, z − hz,

3.4

where h is a potential function for rf (i.e., h z rfz for z ∈ H). Proof . We divide the proof into five steps. Step 1. We prove that {xn } is bounded. Noting the conditions i and ii, we may assume, without loss of generality, that αn /1 − βn ≤ min{1, 1/τ}. For x, y ∈ C, we obtain 1 − βn I − αn μB x − 1 − βn I − αn μB y

αn αn μB x − I − μB y 1 − βn 1 − βn

αn ≤ 1 − βn 1 − τ x − y 1 − βn 1 − βn − αn τ x − y.

≤ 1 − βn I−

3.5


9

∅. Since un Tλn xn and p Tλn p, then from Lemma 2.1, we know Take p ∈ ∞ i1 FSi ∩ EP / that, for any n ∈ N, un − p Tλ xn − Tλ p ≤ xn − p. n n

3.6

Furthermore, since Wn p p and 3.6, we have yn − p δn un 1 − δn Wn un − p δn un − p 1 − δn Wn un − p ≤ δn un − p 1 − δn Wn un − p ≤ un − p ≤ xn − p.

3.7

Thus, it follows from 3.7 that xn1 − p αn rfxn βn xn 1 − βn I − μαn B yn − p αn r fxn − f p αn rf p − μBp βn xn − p 1 − βn I − μαn B yn − p ≤ αn rβxn − p αn rf p − μBp βn xn − p 1 − βn − ταn yn − p ≤ 1 − αn τ − rβ xn − p αn rf p − μBp

3.8

rf p − μBp ≤ max xn − p , . τ − rβ

By induction, we have rf p − μBp xn − p ≤ max x1 − p, , τ − rβ

n ≥ 1.

3.9

Hence, {xn } is bounded and we also obtain that {un }, {Wn un }, {yn }, {Byn }, and {fxn } are all bounded. Without loss of generality, we can assume that there exists a bounded set K ⊂ C such that {un }, {Wn un }, {yn }, {Byn }, {fxn } ∈ K, for all n ∈ N. Step 2. We show that limn → ∞ xn − xn1 0. Let xn1 1 − βn zn βn xn . We note that xn1 − βn xn αn rfxn zn 1 − βn

1 − βn I − μαn B yn , 1 − βn

3.10

10


and then 1 − βn1 I − μαn1 B yn1 zn1 − zn 1 − βn1 αn rfxn 1 − βn I − μαn B yn − 1 − βn αn1 rfxn1

3.11

αn1 αn rfxn1 − μByn1 − rfxn − μByn yn1 − yn . 1 − βn1 1 − βn

Therefore, zn1 − zn ≤

αn1 rfxn1 μByn1 1 − βn1

αn rfxn μByn yn1 − yn . 1 − βn

3.12

It follows from 3.2 that yn1 − yn δn1 un1 1 − δn1 Wn1 un1 − δn un 1 − δn Wn un ≤ |δn1 − δn |un δn1 un1 − un 1 − δn1 Wn1 un1 − Wn un

3.13

|δn1 − δn |Wn un . We will estimate un1 − un . From un1 Tλn1 xn1 and un Tλn xn , we obtain 1 F un1 , y y − un1 , un1 − yn1 ≥ 0, λn1 1 F un , y y − un , un − yn ≥ 0, λn

∀y ∈ C,

∀y ∈ C.

3.14 3.15

Taking y un in 3.14 and y un1 in 3.15, we have Fun1 , un

1 λn1

un − un1 , un1 − xn1 ≥ 0,

1 Fun , un1 un1 − un , un − xn ≥ 0. λn

3.16

So, from A2, one has

un − xn un1 − xn1 − un1 − un , λn λn1

≥ 0,

3.17


11

furthermore, λn un1 − un , un − un1 − xn − un1 − xn1 ≥ 0. λn1

3.18

Since limn → ∞ λn > 0, we assume that there exists a real number such that λn > a > 0 for all n ∈ N. Thus, we obtain

λn un1 − un , xn1 − xn 1 − un1 − xn1 λn1 λn ≤ un1 − un xn1 − xn 1 − − x , u n1 n1 λn1

un1 − un 2 ≤

3.19

which means λn un1 − xn1 un1 − un ≤ xn1 − xn 1 − λn1 ≤ xn1 − xn

1 |λn1 − λn |un1 − xn1 a

3.20

≤ xn1 − xn L1 |λn1 − λn |, where L1 sup{un1 − xn1 : n ∈ N}. Next, we estimate Wn1 un1 − Wn un . Notice that Wn1 un1 − Wn un Wn1 un1 − Wn1 un Wn1 un − Wn un ≤ un1 − un Wn1 un − Wn un .

3.21

From 2.9, we obtain Wn1 un − Wn un τ1 S1 Un1,2 un − τ1 S1 Un,2 un ≤ τ1 Un1,2 un − Un,2 un τ1 τ2 S2 Un1,3 un − τ2 S2 Un,3 un ≤ τ1 τ2 Un1,3 un − Un,3 un ≤ ··· ≤ τ1 τ2 · · · τn Un1,n1 un − Un,n1 un ≤ L2

n

τi ,

i1

where L2 ≥ 0 is a constant such that Un1,n1 un − Un,n1 un ≤ L2 , for all n ∈ N.

3.22

12

Journal of Applied Mathematics Substituting 3.20 and 3.22 into 3.21, we obtain Wn1 un1 − Wn un ≤ xn1 − xn L1 |λn1 − λn | L2

n

τi .

3.23

i1

Hence, we have yn1 − yn ≤ |δn1 − δn |un Wn un xn1 − xn 1 − δn1 L2

n

τi L1 |λn1 − λn |

3.24

i1

≤ L3 |δn1 − δn | xn1 − xn 1 − δn1 L2

n τi L1 |λn1 − λn |, i1

where L3 sup{un Wn un : n ∈ N}. Furthermore, zn1 − zn ≤

αn1 rfxn1 μByn1 1 − βn1

αn rfxn μByn 1 − βn

xn1 − xn L1 |λn1 − λn | L2 1 − δn1

n

3.25 τi

i1

L3 |δn1 − δn |. It follows from 3.25 that zn1 − zn − xn1 − xn ≤

αn1 rfxn1 μByn1 αn rfxn μByn 1 − βn1 1 − βn L1 |λn1 − λn | L2 1 − δn1

n

3.26

τi L3 |δn1 − δn |.

i1

By the conditions i, iii, and iv, we obtain lim supzn1 − zn − xn1 − xn ≤ 0.

n→∞

3.27

Hence, by Lemma 2.5, one has lim zn − xn 0,

n→∞

3.28


13

which implies lim xn1 − xn lim 1 − βn zn − xn 0.

n→∞

3.29

n→∞

Step 3. We claim that limn → ∞ Wun − un 0. Notice that Wun − un Wun − Wn un Wn un − un ≤ Wun − Wn un Wn un − un

3.30

≤ supWu − Wn u Wn un − un . u∈K

It follows from 3.2 that Wn un − un Wn un − yn yn − un ≤ yn − un Wn un − yn yn − un δn Wn un − un ≤ xn − un yn − xn δn Wn un − un .

3.31

By the condition iii, we obtain Wn un − un ≤

1 xn − un yn − xn . 1 − δn

First, we show limn → ∞ xn −un 0. From 3.2, for all p ∈ Lemma 2.3 and noting that · is convex, we obtain

∞

i1

3.32 FSi ∩EPF, applying

xn1 − p2 αn rfxn βn xn 1 − βn I − μαn B yn − p2 2 αn rfxn μByn βn xn − p 1 − βn yn − p 2 ≤ βn xn − p 1 − βn yn − p 2αn rfxn μByn , xn1 − p 2 2 ≤ βn xn − p 1 − βn yn − p 2αn rfxn μByn xn1 − p 2 2 ≤ βn xn − p 1 − βn un − p 2αn rfxn μByn xn1 − p.

3.33

Since un Tλn xn , p Tλn p, we have un − p2 Tλ xn − Tλ p2 ≤ xn − p, un − p n n

1 xn − p2 un − p2 − xn − un 2 , 2

3.34

14


which implies un − p2 ≤ xn − p2 − xn − un 2 .

3.35

Substituting 3.35 into 3.33, we have xn1 − p2 ≤ xn − p2 − 1 − βn xn − un 2 2αn rfxn μByn xn1 − p,

3.36

which means 2 2 1 − βn xn − un 2 ≤ xn − p − xn1 − p 2αn rfxn μByn xn1 − p ≤ xn1 − xn xn − p xn1 − p 2αn rfxn μByn xn1 − p. 3.37

Noticing limn → ∞ αn 0 and limn → ∞ inf1 − βn > 0, we have lim xn − un 0.

n→∞

3.38

Second, we show limn → ∞ yn − xn 0. It follows from 3.2 that yn − xn ≤ yn − xn1 xn1 − xn αn rfxn βn xn 1 − βn I − μαn B yn − yn xn1 − xn ≤ αn rfxn μByn βn xn − yn xn1 − xn .

3.39

This implies that 1 − βn yn − xn ≤ αn rfxn μByn xn1 − xn .

3.40

Noticing limn → ∞ αn 0, limn → ∞ inf1 − βn > 0 and 3.30, we have lim yn − xn 0.

n→∞

3.41

Thus, substituting 3.41 and 3.38 into 3.32, we obtain lim Wn un − un 0.

n→∞

3.42

Furthermore, 3.42, 3.30, and Lemma 2.11 lead to lim Wun − un 0.

n→∞

3.43


15

Step 4. Letting z PFW∩EPF I − μB rfz, we show lim sup rf − μB z, xn − z ≤ 0.

n→∞

3.44

We know that PFW∩EPF I − μB rf is a contraction. Indeed, for any x, y ∈ H, we have PFW∩EPF I − μB rf x − PFW∩EPF I − μB rf y ≤ I − μB rf x − I − μB rf y ≤ 1 − τ rβ x − y,

3.45

and hence PFW∩EPF I − μB rf is a contraction due to 1 − τ rβ ∈ 0, 1. Thus, Banach’s Contraction Mapping Principle guarantees that PFW∩EPF I − μB rf has a unique fixed point, which implies z PFW∩EPF I − μB rfz. Since {uni } ⊂ {un } is bounded in C, without loss of generality, we can assume that {uni } ω, it follows from 3.43 that Wuni ω. Since C is closed and convex, C is weakly closed. Thus we have ω ∈ C. Let us show ω ∈ FW. For the sake of contradiction, suppose that ω ∈ / FW, that is, } ω, by the Opial condition, we have Wω ω. Since {u / ni lim infuni − ω < lim infuni − Wω

n→∞

n→∞

≤ lim inf{uni − Wuni Wuni − Wω} n→∞

3.46

≤ lim inf{uni − Wuni uni − ω}. n→∞

It follows 3.43 that lim infuni − ω < lim infuni − ω.

n→∞

n→∞

3.47

This is a contradiction, which shows that ω ∈ FW. Next, we prove that ω ∈ EPF. By 3.2, we obtain 1 F un , y y − un , un − xn ≥ 0. λn

3.48

1 y − un , un − xn ≥ F y, un . λn

3.49

1 y − uni , uni − xni ≥ F y, uni . λni

3.50

It follows from A2 that

Replacing n by ni , we have

16


Since 1/λni uni −xni → 0 and {uni } ω, it follows from A4 that Fy, ω ≥ 0 for all y ∈ C. Put zt ty 1 − tω for all t ∈ 0, 1 and y ∈ C. Then, we have zt ∈ C and then Fzt , ω ≥ 0. Hence, from A1 and A4, we have 0 Fzt , zt ≤ tF zt , y 1 − tF zt , y ≤ tF zt , y ,

3.51

which means Fzt , y ≥ 0. From A3, we obtain Fω, y ≥ 0 for y ∈ C and then ω ∈ EPF. Therefore, ω ∈ FW ∩ EPF. Since z PFW∩EPF I − μB rfz, it follows from 3.38, 3.42, and Lemma 2.11 that lim sup rf − μB z, xn − z ≤ lim rf − μB z, xni − z n→∞

i→∞

lim

i→∞

rf − μB z, xni − uni

lim

rf − μB z, uni − Wni uni

i→∞

lim

rf − μB z, Wni uni − Wuni

i→∞

lim

i→∞

3.52

rf − μB z, Wuni − z

rf − μB z, ω − z ≤ 0.

Step 5. Finally we prove that xn → ω as n → ∞. In fact, from 3.2 and 3.7, we obtain 2 xn1 − ω2 αn rfxn βn xn 1 − βn I − μαn B yn − ω αn r fxn − fω αn rfω − μBω 2 βn xn − ω 1 − βn I − μαn B yn − ω αn r fxn − fω, xn1 − ω αn rfω − μBω, xn1 − ω βn xn − ω, xn1 − ω 1 − βn I − μαn B yn − ω , xn1 − ω xn − ω2 xn1 − ω2 αn rfω − μBω, xn1 − ω 2 2 yn − ω xn1 − ω2 xn − ω2 xn1 − ω2 βn 1 − βn − αn τ 2 2

1 − αn τ − rβ ≤ xn − ω2 xn1 − ω2 αn rfω − μBω, xn1 − ω , 2 3.53 ≤ αn rβ


17

which implies 1 − αn τ − rβ xn − ω2 xn1 − ω ≤ 1 αn τ − rβ 2αn τ − rβ rfω − μBω, xn1 − ω 1 αn τ − rβ τ − rβ ≤ 1 − αn τ − rβ xn − ω2 2αn τ − rβ rfω − μBω, xn1 − ω. 1 αn τ − rβ τ − rβ 2

3.54

From condition i and 3.7, we know that ni1 αn τ − rβ ∞ and limi → ∞ sup2/1 αn τ − rβτ − rβrfω − μBω, xn1 − ω ≤ 0. we can conclude from Lemma 2.6 that xn → ω as n → ∞. This completes the proof of Theorem 3.1. Remark 3.2. If r 1, μ 1, B I and δi 0, ki 0, σi 0 for i ∈ N, then Theorem 3.1 reduces to Theorem 3.5 of Yao et al. 19. Furthermore, we extend the corresponding results of Yao et al. 19 from one infinite family of nonexpansive mapping to an infinite family of strictly pseudo-contractive mappings. Remark 3.3. If μ 1 and δi 0, ki 0, σi 0 for i ∈ N, then Theorem 3.1 reduces to Theorem 10 of Colao and Marino 20. Furthermore, we extend the corresponding results of Colao and Marino 20 from one infinite family of nonexpansive mapping to an infinite family of strictly pseudo-contractive mappings, and from a strongly positive bounded linear operator A to a k-Lipschitzian and η-strongly monotone operator B. Theorem 3.4. Let C be a nonempty closed convex subset of a real Hilbert space H and F be a bifunction from C × C → R satisfying (A1)–(A4). Let S : C → C be a nonexpansive mapping with FS ∩ EP / ∅. Let f be a contraction of H into itself with β ∈ 0, 1 and B be k-Lipschitzian and η-strongly monotone operator on H with coefficients k, η > 0, 0 < μ < 2η/k2 , 0 < r < 1/2μ2η − μk2 /β τ/β and τ < 1. Let {xn } be sequence generated by 1 F un , y y − un , un − xn ≥ 0, λn

xn1

∀y ∈ C,

yn δn un 1 − δn Sn un , αn rfxn βn xn 1 − βn I − μαn B yn ,

3.55 ∀n ∈ N,

where un Tλn xn . If {αn }, {βn }, {δn }, and {λn } satisfy the following conditions: i {αn } ⊂ 0, 1, limn → ∞ αn 0,

∞

i1

αn ∞,

ii 0 < limn → ∞ inf βn ≤ limn → ∞ sup βn < 1, iii 0 < limn → ∞ inf δn ≤ limn → ∞ sup δn < 1, limn → ∞ |δn1 − δn | 0, iv {λn } ⊂ 0, ∞, limn → ∞ λn > 0, limn → ∞ |λn1 − λn | 0.

18


∅, where z is the unique solution of variaThen {xn } converges strongly to z ∈ FS ∩ EP / tional inequality lim sup rf − μB z, p − z ≤ 0,

n→∞

∀p ∈ FS ∩ EP / ∅,

3.56

that is, z PFS∩EP F I − μB rfz. Proof . By Theorem 3.1, letting ki 0, σi 0, τi 1 and Si S for i ∈ N, we can obtain Theorem 3.4.

4. Numerical Example Now, we present a numerical example to illustrate our theoretical analysis results obtained in Section 3. Example 4.1. Let H R, C −1, 1, Sn I, τn τ ∈ 0, 1, λn 1, n ∈ N, Fx, y 0, for all x, y ∈ C, B I, r μ 1, fx 1/10x, for all x, with contraction coefficient β 1/5, δn 1/2, αn 1/n, βn 1/4 1/2n for every n ∈ N. Then {xn } is the sequence generated by xn1

1−

9 xn , 10n

4.1

and {xn } → 0, as n → ∞, where 0 is the unique solution of the minimization problem min x∈C

9 2 x c. 20

4.2

Proof. We divide the proof into four steps. Step 1. We show Tλn x PC x,

∀x ∈ H,

4.3

where ⎧x ⎨ , PC x |x| ⎩ x,

x∈ / C, x ∈ C.

4.4

Since Fx, y 0, for all x, y ∈ C, due to the definition of Tλn x, for all x ∈ H, by Lemma 2.1, we obtain Tλn x z ∈ C : y − z, z − x ≥ 0, ∀y ∈ C .

4.5

By the property of PC , for x ∈ C, we have Tλn x PC x Ix. Furthermore, it follows from 3 in Lemma 2.1 that EPF C.

4.6


19

Step 2. We show that Wn I.

4.7

It follows from 2.9 that W1 U1,1 τ1 S1 U1,2 1 − τ1 I τ1 S1 1 − τ1 I, W2 U2,1 τ1 S1 U2,2 1 − τ1 I τ1 S1 τ2 S2 U2,3 1 − τ2 I 1 − τ1 I τ1 τ2 S1 S2 τ1 1 − τ2 S1 1 − τ1 I, W3 U3,1 τ1 S1 U3,2 1 − τ1 I

4.8

τ1 S1 τ2 S2 U3,3 1 − τ2 I 1 − τ1 I τ1 τ2 S1 S2 U3,3 τ1 1 − τ2 S1 1 − τ1 I τ1 τ2 S1 S2 τ3 S3 U3,4 1 − τ3 I τ1 1 − τ2 S1 1 − τ1 I τ1 τ2 τ3 S1 S2 S3 τ1 τ2 1 − τ3 S1 S2 τ1 1 − τ2 S1 1 − τ1 I. Furthermore, we obtain Wn Un,1 τ1 τ2 τ3 · · · τn S1 S2 S3 · · · Sn τ1 τ2 · · · τn−1 1 − τn S1 S2 · · · Sn−1 τ1 τ2 · · · τn−2 1 − τn−1 S1 S2 · · · Sn−2 · · · τ1 1 − τ2 S1 1 − τ1 I.

4.9

Since Si I, τi τ for i ∈ N, one has Wn τ n τ n−1 1 − τ · · · τ1 − τ 1 − τ I I.

4.10

Step 3. We show that xn1

9 1− xn , 10n

4.11

{xn } → 0, as n → ∞, where 0 is the unique solution of the minimization problem min x∈C

9 2 x c. 20

4.12

20

Journal of Applied Mathematics Table 1: This table shows the value of sequence {xn } on each iteration step initial value x1 0.2.

n 1 2 3 4 5 .. . 9 10 .. . 14 15 16

xn 0.2000 0.0200 0.0110 0.0077 0.0060 .. . 0.0032 0.0029 .. . 0.0021 0.0019 0.0018

n 17 18 19 20 21 .. . 26 27 .. . 30 31 32

xn 0.0017 0.0016 0.0016 0.0015 0.0014 .. . 0.0012 0.0011 .. . 0.0010 0.0009 0.0009

In fact, we can see that B I is k-Lipschitzian and η-strongly monotone operator on H with coefficient k 1, η 3/4 such that 0 < μ < 2η/k2 , 0 < r < 1/2μ2η − μk2 /β τ/β, so we take r μ 1. Since Sn I, n ∈ N, we have ∞ FSi H. i1

4.13

Furthermore, we obtain ∞ FSi ∩ EPF C −1, 1. i1

4.14

Next, we need prove {xn } → 0, as n → ∞. Since yn un for all n ∈ N, we have xn1 αn rfxn βn xn 1 − βn I − μαn B yn

4.15 9 1− xn , 10n for all n ∈ N. Thus, we obtain a special sequence {xn } of 3.2 in Theorem 3.1 as follows

9 xn . xn1 1 − 10n

4.16

By Lemma 2.6, it is obviously that xn → 0, 0 is the unique solution of the minimization problem min x∈C

where c is a constant number.

9 2 x c, 20

4.17


21

0.2 0.18

The value of sequence

0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

0

5

10

15

20

25

30

35

Iteration steps

Figure 1: The corresponding graph at x 0.2.

Step 4. Finally, we use software Matlab 7.0 to give the numerical experiment results and then obtain Table 1 which show that the iteration process of the sequence {xn } is a monotonedecreasing sequence and converges to 0. From Table 1 and the corresponding graph Figure 1, we show that the more the iteration steps are, the more slowly the sequence {xn } converges to 0.

Acknowledgments The authors thank the anonymous referees and the editor for their constructive comments and suggestions, which greatly improved this paper. This project is supported by the Natural Science Foundation of China Grants nos. 11171180, 11171193, 11126233, and 10901096 and Shandong Provincial Natural Science Foundation Grants no. ZR2009AL019 and ZR2011AM016 and the Project of Shandong Province Higher Educational Science and Technology Program Grant no. J09LA53.

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