Shrinking Projection Method of Common Fixed Point Problems for

International Scholarly Research Network ISRN Mathematical Analysis Volume 2011, Article ID 529407, 17 pages doi:10.5402/2011/529407

Research Article Shrinking Projection Method of Common Fixed Point Problems for Discrete Asymptotically Strictly Pseudocontractive Semigroups and Mixed Equilibrium Problems in Hilbert Spaces Pattanapong Tianchai Faculty of Science, Maejo University, Chiangmai 50290, Thailand Correspondence should be addressed to Pattanapong Tianchai, [email protected] Received 13 July 2011; Accepted 18 August 2011 Academic Editor: A. Levy Copyright q 2011 Pattanapong Tianchai. 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. This paper is concerned with a common element of the set of common fixed point for a discrete asymptotically strictly pseudocontractive semigroup and the set of solutions of the mixed equilibrium problems in Hilbert spaces. The strong convergence theorem for the above two sets is obtained by a general iterative scheme based on the shrinking projection method which extends and improves the corresponding ones due to Kim Proceedings of the Asian Conference on Nonlinear Analysis and Optimization Matsue, Japan, 2008, 139–162.

1. Introduction Throughout this paper, we always assume that C is a nonempty closed convex subset of a real Hilbert space H with inner product and norm denoted by ·, · and · , respectively. The domain of the function ϕ : C → R ∪ {∞} is the set dom ϕ x ∈ C : ϕx < ∞ .

1.1

Let ϕ : C → R ∪ {∞} be a proper extended real-valued function, and let Φ be a bifunction from C × C into R such that C ∩ dom ϕ / ∅, where R is the set of real numbers. The so-called mixed equilibrium problem is to find x ∈ C such that Φ x, y ϕ y − ϕx ≥ 0,

∀y ∈ C.

1.2

2

ISRN Mathematical Analysis

The set of solutions of problem 1.2 is denoted by MEP Φ, ϕ; that is, MEP Φ, ϕ x ∈ C : Φ x, y ϕ y − ϕx ≥, ∀y ∈ C .

1.3

It is obvious that if x is a solution of problem 1.2 then x ∈ dom ϕ. As special cases of problem 1.2, we have the following. i If ϕ 0, then problem 1.2 is reduced to find x ∈ C such that Φ x, y ≥ 0,

∀y ∈ C.

1.4

We denote by EP Φ the set of solutions of equilibrium problem, problem 1.4 which was studied by Blum and Oettli 1. ii If Φx, y Bx, y − x for all x, y ∈ C where a mapping B : C → H, then problem 1.4 is reduced to find x ∈ C such that Bx, y − x ≥ 0,

∀y ∈ C.

1.5

We denote by V IC, B the set of solutions of variational inequality problem, problem 1.5 which was studied by Hartman and Stampacchia 2. iii If Φ 0, then problem 1.2 is reduced to find x ∈ C such that ϕ y − ϕx ≥ 0,

∀y ∈ C.

1.6

We denote by Argminϕ the set of solutions of minimize problem. Recall that PC is the metric projection of H onto C; that is, for each x ∈ H, there exists the unique point in PC x ∈ C such that x − PC x miny∈C x − y. A mapping T : C → C is called nonexpansive if T x − T y ≤ x − y for all x, y ∈ C, and a mapping f : C → C is called a contraction if there exists a constant α ∈ 0, 1 such that fx − fy ≤ αx − y for all x, y ∈ C. A point x ∈ C is a fixed point of T provided T x x. We denote by FT the set of fixed points of T ; that is, FT {x ∈ C : T x x}. If C is a nonempty bounded closed convex subset of H and T is a nonexpansive mapping of C into itself, then FT is nonempty see 3. Iterative methods are often used to solve the fixed point equation T x x. The most well-known method is perhaps the Picard successive iteration method when T is a contraction. Picard’s method generates a sequence {xn } successively as xn1 T xn for all n ∈ N with x1 x chosen arbitrarily, and this sequence converges in norm to the unique fixed point of T . However, if T is not a contraction for instance, if T is a nonexpansive, then Picard’s successive iteration fails, in general, to converge. Instead, Mann’s iteration method for a nonexpansive mapping T see 4 prevails and generates a sequence {xn } recursively by xn1 αn xn 1 − αn T xn ,

∀n ∈ N,

1.7

where x1 x ∈ C chosen arbitrarily and the sequence {αn } lies in the interval 0, 1. Recall that a mapping T : C → C is said to be as follows.


3

i κ-strictly pseudocontractive see 5 if there exists a constant κ ∈ 0, 1 such that T x − T y2 ≤ x − y2 κI − T x − I − T y2 ,

∀x, y ∈ C,

1.8

in brief, we use κ-SPC to denote the κ-strictly pseudocontractive, it is obvious that T is a nonexpansive if and only if T is a 0-SPC. ii Asymptotically κ-SPC see 6 if there exists a constant κ ∈ 0, 1 and a sequence {γn } of nonnegative real numbers with limn → ∞ γn 0 such that n T x − T n y2 ≤ 1 γn x − y2 κI − T n x − I − T n y2 ,

∀x, y ∈ C,

1.9

for all n ∈ N, if κ 0 then T is an asymptotically nonexpansive with kn 1 γn for all n ∈ N; that is, T is an asymptotically nonexpansive see 7 if there exists a sequence {kn } ⊂ 1, ∞ with limn → ∞ kn 1 such that n T x − T n y ≤ kn x − y,

∀x, y ∈ C,

1.10

for all n ∈ N, it is known that the class of κ-SPC mappings, and the classes of asymptotically κ-SPC mappings are independent see 8. The Mann’s algorithm for nonexpansive mappings has been extensively investigated see 5, 9, 10 and the references therein. One of the well-known results is proven by Reich 10 for a nonexpansive mapping T on C, which asserts the weak convergence of the sequence {xn } generated by 1.7 in a uniformly convex Banach space with a Frechet

differentiable norm under the control condition ∞ n1 αn 1 − αn ∞. Recently, Marino and Xu 11 devoloped and extended Reich’s result to SPC mapping in Hilbert space setting. More precisely, they proved the weak convergence of the Mann’s iteration process 1.7 for a κ-SPC mapping T on C, and, subsequently, this result was improved and carried over the class of asymptotically κ-SPC mappings by Kim and Xu 12. It is known that the Mann’s iteration 1.7 is in general not strongly convergent see 13. The strong convergence is guaranteed and has been proposed by Nakajo and Takahashi 14, they modified the Mann’s iteration method 1.7 which is to find a fixed point of a nonexpansive mapping by a hybrid method, which called the shrinking projection method or the CQ method as the following theorem. Theorem NT. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonexpansive mapping of C into itself such that FT / ∅. Suppose that x1 x ∈ C chosen arbitrarily, and let {xn } be the sequence defined by yn αn xn 1 − αn T xn , Cn z ∈ C : yn − z ≤ xn − z , Qn {z ∈ C : xn − z, x1 − xn ≥ 0}, xn1 PCn ∩Qn x1 ,

∀n ∈ N,

where 0 ≤ αn ≤ α < 1. Then {xn } converges strongly to PFT x1 .

1.11

4


Subsequently, Marino and Xu [15] introduced an iterative scheme for finding a fixed point of a κ-SPC mapping as the following theorem. Theorem MX. Let C be a closed convex subset of a Hilbert space H, and, T : C → C be a κ-SPC mapping for some 0 ≤ κ < 1. Assume that FT / ∅. Suppose that x1 x ∈ C chosen arbitrarily, and let {xn } be the sequence defined by yn αn xn 1 − αn T xn , 2 Cn z ∈ C : yn − z ≤ xn − z2 1 − αn κ − αn xn − T xn 2 ,

1.12

Qn {z ∈ C : xn − z, x1 − xn ≥ 0}, xn1 PCn ∩Qn x1 ,

∀n ∈ N,

where 0 ≤ αn < 1. Then the sequence {xn } converges strongly to PFT x1 . Quite recently, Kim and Xu [12] has improved and carried Theorem MX over the more wider class of asymptotically κ-SPC mappings as the following theorem. Theorem KX. Let C be a closed convex subset of a Hilbert space H, and, T : C → C be an asymptotically κ-SPC mapping for some 0 ≤ κ < 1 and a bounded sequence {γn } ⊂ 0, ∞ such that limn → ∞ γn 0. Assume that FT is a nonempty bounded subset of C. Suppose that x1 x ∈ C chosen arbitrarily, and let {xn } be the sequence defined by yn αn xn 1 − αn T n xn , 2 Cn z ∈ C : yn − z ≤ xn − z2 κ − αn 1 − αn xn − T n xn 2 θn ,

1.13


∀n ∈ N,

where θn Δ2n 1 − αn γn → 0 as n → ∞, Δn sup{xn − z : z ∈ FT } < ∞, and 0 ≤ αn < 1 such that lim supn → ∞ αn < 1 − κ. Then the sequence {xn } converges strongly to PFT x1 . Recall that a discrete family S {Tn : n ≥ 0} of self-mappings of C is said to be a Lipschitzian semigroup on C if the following conditions are satisfied. 1 T0 I where I denotes the identity operator on C. 2 Tnm x Tn Tm x, ∀n, m ≥ 0, ∀x ∈ C. 3 There exists a sequence {Ln } of nonnegative real numbers such that Tn x − Tn y ≤ Ln x − y,

∀x, y ∈ C, ∀n ≥ 0.

1.14

A discrete Lipschitzian semigroup S is called nonexpansive semigroup if Ln 1 for all n ≥ 0, contraction semigroup if 0 < Ln < 1 for all n ≥ 0 and, asymptotically nonexpansive semigroup if lim supn → ∞ Ln ≤ 1, respectively. We use FS to denote the common fixed point set of the semigroup S; that is, FS {x ∈ C : Tn x x, ∀n ≥ 0}.


5

Very recently, Kim 16 introduced asymptotically κ-SPC semigroup, a discrete family S {Tn : n ≥ 0} of self-mappings of C which is said to be asymptotically κ-SPC semigroup on C if, in addition to 1, 2 and the following condition 3 are satisfied. 3 There exists a constant κ ∈ 0, 1 and a bounded sequence {Ln } of nonnegative real numbers with lim supn → ∞ Ln ≤ 1 such that Tn x − Tn y2 ≤ Ln x − y2 κI − Tn x − I − Tn y2 , ∀x, y ∈ C, ∀n ≥ 0.

1.15

Note that for both discrete asymptotically nonexpansive semigroups and discrete asymptotically κ-SPC semigroups, we can always assume that the Lipschitzian constants {Ln }n≥0 are such that Ln ≥ 1 for all n ≥ 0 and limn → ∞ Ln 1; otherwise, we replace Ln for all n ≥ 0 with Ln max{supm≥n Lm , 1}. Therefore, for a single asymptotically κ-SPC mapping T : C → C note that 1.15 immediately reduces to 1.9 by taking Tn ≡ T n and γn Ln − 1 such that Ln ≥ 1 for all n ≥ 0 and limn → ∞ Ln 1. To be more precise, Kim also showed in the framework of Hilbert spaces for the asymptotically κ-SPC semigroups that Tn is continuous on C for all n ≥ 0 and that FS is closed and convex see Lemma 3.2 in 16, and the demiclosedness principle see Theorem 3.3 in 16 holds in the sense that if {xn } is a sequence in C such that xn z

and lim supm → ∞ lim supn → ∞ xn − Tm xn 0 then, z ∈ FS ∞ n0 FTn , and he also introduced an iterative scheme to find a common fixed point of a discrete asymptotically κ-SPC semigroup and a bounded sequence {Ln } ⊂ 1, ∞ such that limn → ∞ Ln 1 as follows: x0 x ∈ C chosen arbitrarily, yn αn xn 1 − αn Tn xn , 2 Cn z ∈ C : yn − z ≤ xn − z2 1 − αn θn κ − αn xn − Tn xn 2 ,

1.16


∀n ∈ N ∪ {0},

where θn Ln − 1 · sup{xn − z2 : z ∈ FS} < ∞. He proved that under the parameter 0 ≤ αn < 1 for all n ∈ N ∪ {0}, if FS is a nonempty bounded subset of C, then the sequence {xn } generated by 1.16 converges strongly to PFS x0 . Inspired and motivated by the works mentioned above, in this paper, we introduce a general iterative scheme 3.1 below to find a common element of the set of common fixed point for a discrete asymptotically κ-SPC semigroup and the set of solutions of the mixed equilibrium problems in Hilbert spaces. The strong convergence theorem for the above two sets is obtained based on the shrinking projection method which extend and improve the corresponding ones due to Kim 16.

2. Preliminaries Let C be a nonempty closed convex subset of a real Hilbert space H. For a sequence {xn } in H, we denote the strong convergence and the weak convergence of {xn } to x ∈ H by xn → x and xn x, respectively, and the weak ω-limit set of {xn } by ωw xn {x : ∃xnj x}.

6


For solving the mixed equilibrium problem, let us assume that the bifunction Φ : C × C → R, the function ϕ : C → R ∪ {∞} and the set C satisfy the following conditions. A1 Φx, x 0 for all x ∈ C. A2 Φ is monotone; that is, Φx, y Φy, x ≤ 0 for all x, y ∈ C. A3 For each x, y, z ∈ C, limΦ tz 1 − tx, y ≤ Φ x, y . t↓0

2.1

A4 For each x ∈ C, y → Φx, y is convex and lower semicontinuous. A5 For each y ∈ C, x → Φx, y is weakly upper semicontinuous. B1 For each x ∈ C and r > 0, there exists a bounded subset Dx ⊂ C and yx ∈ C such that for any z ∈ C \ Dx , 1 Φ z, yx ϕ yx − ϕz yx − z, z − x < 0. r

2.2

B2 C is a bounded set. Lemma 2.1 see 17. Let H be a Hilbert space. For any x, y ∈ H and λ ∈ R, one has λx 1 − λy2 λx2 1 − λy2 − λ1 − λx − y2 .

2.3

Lemma 2.2 see 3. Let C be a nonempty closed convex subset of a Hilbert space H. Then the following inequality holds:

x − PC x, PC x − y ≥ 0,

∀x ∈ H, y ∈ C.

2.4

Lemma 2.3 see 18. Let C be a nonempty closed convex subset of a Hilbert space H, Φ : C × C → R satisfying the conditions A1–A5, and let ϕ : C → R ∪ {∞} be a proper lower semicontinuous and convex function. Assume that either B1 or B2 holds. For r > 0, define a mapping Sr : C → C as follows: Sr x

1 z ∈ C : Φ z, y ϕ y − ϕz y − z, z − x ≥ 0, ∀y ∈ C , r

for all x ∈ C. Then, the following statement hold. 1 For each x ∈ C, Sr x / ∅. 2 Sr is single-valued. 3 Sr is firmly nonexpansive; that is, for any x, y ∈ C,

2.5


7 Sr x − Sr y2 ≤ Sr x − Sr y, x − y .

2.6

4 FSr MEP Φ, ϕ. 5 MEP Φ, ϕ is closed and convex. Lemma 2.4 see 3. Every Hilbert space H has Radon-Riesz property or Kadec-Klee property; that is, for a sequence {xn } ⊂ H with xn x and xn → x then xn → x. Lemma 2.5 see 16. Let C be a nonempty closed convex subset of a Hilbert space H, and let S {Tn : n ≥ 0} be an asymptotically κ-strictly pseudocontractive semigroup on C. Let {xn } be a sequence in C such that limn → ∞ xn −xn1 0 and limn → ∞ xn −Tn xn 0. Then ωw xn ⊂ FS.

3. Main Results Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H, let Φ be a bifunction from C × C into R satisfying the conditions (A1)–(A5), and let ϕ : C → R ∪ {∞} be a proper lower semicontinuous and convex function with either (B1) or (B2) holds. Let S {Tn : n ≥ 0} be an asymptotically κ-SPC semigroup on C for some κ ∈ 0, 1 and a bounded sequence {Ln } ⊂ 1, ∞ such that limn → ∞ Ln 1. Assume that Ω : FS ∩ MEP Φ, ϕ is a nonempty bounded subset of C. For x0 x ∈ C chosen arbitrarily, suppose that {xn }, {yn }, and {un } are generated iteratively by 1 un ∈ C such that Φ un , y ϕ y − ϕun y − un , un − xn ≥ 0, rn

Cn1

∀y ∈ C,

yn αn un 1 − αn Tn un , 2 z ∈ Cn ∩ Qn : yn − z ≤ xn − z2 1 − αn θn κ − αn un − Tn un 2 , 3.1 Qn1 {z ∈ Cn ∩ Qn : xn − z, x0 − xn ≥ 0}, C0 Q0 C, xn1 PCn1 ∩Qn1 x0 ,

∀n ∈ N ∪ {0},

where θn Ln − 1 · sup{xn − z2 : z ∈ Ω} < ∞ satisfying the following conditions: C1 {αn } ⊂ a, b such that κ < a < b < 1, C2 {rn } ⊂ r, ∞ for some r > 0,

C3 ∞ n0 |rn1 − rn | < ∞. Then the sequences {xn }, {yn }, and {un } converge strongly to w PΩ x0 . Proof. Pick p ∈ Ω. Therefore, by 3.1 and the definition of Srn in Lemma 2.3, we have un Srn xn ∈ dom ϕ,

3.2

8


and, by FS

∞

n0

FTn and Lemma 2.34, we have Tn p p Srn p.

3.3

By 3.2, 3.3, and the nonexpansiveness of Srn , we have un − p Sr xn − Sr p ≤ xn − p. n n

3.4

By 3.3, 3.4, Lemma 2.1, and the asymptotically κ-SPC semigroupness of S, we have yn − p2 αn un − p 1 − αn Tn un − p 2 2 2 αn un − p 1 − αn Tn un − p − αn 1 − αn un − Tn un 2 2 2 ≤ αn un − p 1 − αn Ln un − p κun − Tn un 2 −αn 1 − αn un − Tn un 2 2 1 1 − αn Ln − 1un − p 1 − αn κ − αn un − Tn un 2 2 ≤ xn − p 1 − αn θn κ − αn un − Tn un 2 , 3.5 where θn : Ln − 1 · sup{xn − z2 : z ∈ Ω} for all n ∈ N ∪ {0}. Firstly, we prove that Cn ∩ Qn is closed and convex for all n ∈ N ∪ {0}. It is obvious that C0 ∩ Q0 is closed and by mathematical induction that Cn ∩ Qn is closed for all n ≥ 1; that is Cn ∩ Qn is closed for all n ∈ N ∪ {0}. Let n 1 − αn θn k − αn un − Tn un 2 since, for any z ∈ C, yn − z2 ≤ xn − z2 n is equivalent to yn − xn 2 2 yn − xn , xn − z − n ≤ 0,

3.6

for all n ∈ N ∪ {0}. Therefore, for any z1 , z2 ∈ Cn1 ∩ Qn1 ⊂ Cn ∩ Qn and ∈ 0, 1, we have yn −xn 2 2 yn −xn , xn − z1 1 − z2 − n yn −xn 2 2 yn −xn , xn −z1 − n 1− 2 × yn − xn 2 yn − xn , xn − z2 − n ≤ 0, 3.7 for all n ∈ N ∪ {0}, and we have xn − z1 1 − z2 , x0 − xn xn − z1 , x0 − xn 1 − xn − z2 , x0 − xn ≥ 0,

3.8


9

for all n ∈ N ∪ {0}. Since C0 ∩ Q0 is convex and by putting n 0 in 3.6, 3.7, and 3.8, we have that C1 ∩ Q1 is convex. Suppose that xk is given and Ck ∩ Qk is convex for some k ≥ 1. It follows by putting n k in 3.6, 3.7, and 3.8 that Ck1 ∩ Qk1 is convex. Therefore, by mathematical induction, we have that Cn ∩ Qn is convex for all n ≥ 1; that is, Cn ∩ Qn is convex for all n ∈ N ∪ {0}. Hence, we obtain that Cn ∩ Qn is closed and convex for all n ∈ N ∪ {0}. Next, we prove that Ω ⊂ Cn ∩ Qn for all n ∈ N ∪ {0}. It is obvious that p ∈ Ω ⊂ C C0 ∩ Q0 . Therefore, by 3.1 and 3.5, we have p ∈ C1 and note that p ∈ C Q1 , and so p ∈ C1 ∩ Q1 . Hence, we have Ω ⊂ C1 ∩ Q1 . Since C1 ∩ Q1 is a nonempty closed convex subset of C, there exists a unique element x1 ∈ C1 ∩ Q1 such that x1 PC1 ∩Q1 x0 . Suppose that xk ∈ Ck ∩ Qk is given such that xk PCk ∩Qk x0 and p ∈ Ω ⊂ Ck ∩ Qk for some k ≥ 1. Therefore, by 3.1 and 3.5, we have p ∈ Ck1 . Since xk PCk ∩Qk x0 ; therefore, by Lemma 2.2, we have xk − z, x0 − xk ≥ 0,

3.9

for all z ∈ Ck ∩ Qk . Thus, by 3.1, we have p ∈ Qk1 , and so p ∈ Ck1 ∩ Qk1 . Hence, we have Ω ⊂ Ck1 ∩ Qk1 . Since Ck1 ∩ Qk1 is a nonempty closed convex subset of C, there exists a unique element xk1 ∈ Ck1 ∩ Qk1 such that xk1 PCk1 ∩Qk1 x0 . Therefore, by mathematical induction, we obtain Ω ⊂ Cn ∩ Qn for all n ≥ 1, and so Ω ⊂ Cn ∩ Qn for all n ∈ N ∪ {0}, and we can define xn1 PCn1 ∩Qn1 x0 for all n ∈ N ∪ {0}. Hence, we obtain that the iteration 3.1 is well defined. Next, we prove that {xn } is bounded. Since xn PCn ∩Qn x0 for all n ∈ N ∪ {0}, we have xn − x0 ≤ z − x0 ,

3.10

for all z ∈ Cn ∩ Qn . It follows by p ∈ Ω ⊂ Cn ∩ Qn that xn − x0 ≤ p − x0 for all n ∈ N ∪ {0}. This implies that {xn } is bounded and so are {yn } and {un }. Next, we prove that xn − xn1 → 0 and un − un1 → 0 as n → ∞. Since xn1 PCn1 ∩Qn1 x0 ∈ Cn1 ∩ Qn1 ⊂ Cn ∩ Qn ; therefore, by 3.10, we have xn − x0 ≤ xn1 − x0 for all n ∈ N ∪ {0}. This implies that {xn − x0 } is a bounded nondecreasing sequence; there exists the limit of xn − x0 ; that is, lim xn − x0 m,

n→∞

3.11

for some m ≥ 0. Since xn1 ∈ Qn1 ; therefore, by 3.1, we have xn − xn1 , x0 − xn ≥ 0.

3.12

It follows by 3.12 that xn − xn1 2 xn − x0 x0 − xn1 2 xn − x0 2 2xn − x0 , x0 − xn 2xn − x0 , xn − xn1 xn1 − x0 2 ≤ xn1 − x0 2 − xn − x0 2 .

3.13

10


Therefore, by 3.11, we obtain xn − xn1 −→ 0 as n −→ ∞.

3.14

Indeed, from 3.1, we have 1 Φ un , y ϕ y − ϕun y − un , un − xn ≥ 0, rn 1 Φ un1 , y ϕ y − ϕun1 y − un1 , un1 − xn1 ≥ 0, rn1

∀y ∈ C, ∀y ∈ C.

3.15 3.16

Substituting y un1 into 3.15 and y un into 3.16, we have Φun , un1 ϕun1 − ϕun Φun1 , un ϕun − ϕun1

1 un1 − un , un − xn ≥ 0, rn

1 rn1

3.17

un − un1 , un1 − xn1 ≥ 0.

Therefore, by the condition A2, we get un − xn un1 − xn1 0 ≤ Φun , un1 Φun1 , un un1 − un , − rn rn1 un − xn un1 − xn1 − . ≤ un1 − un , rn rn1

3.18

It follows that 0≤

un1 − un , un − un1 un1 − xn −

rn un1 − xn1 rn1

rn un1 − un , un − un1 un1 − un , un1 − xn1 xn1 − xn − un1 − xn1 . rn1 3.19 Thus, we have rn un1 − xn1 un1 − un ≤ un1 − un , xn1 − xn 1 − rn1 rn ≤ un1 − un xn1 − xn 1 − un1 − xn1 . rn1 2

3.20


11

It follows by the condition C2 that |rn1 − rn | un1 − xn1 rn1 L ≤ xn1 − xn |rn1 − rn |, r

un1 − un ≤ xn1 − xn

3.21

where L supn≥0 un − xn < ∞. Therefore, by the condition C3 and 3.14, we obtain un − un1 −→ 0

as n −→ ∞.

3.22

Next, we prove that un − Tn un → 0, yn − xn → 0, and un − xn → 0 as n → ∞. Since xn1 ∈ Cn1 , by 3.1, we have yn − xn1 2 ≤ xn − xn1 2 1 − αn θn κ − αn un − Tn un 2 .

3.23

It follows that 2 1 − αn αn − κun − Tn un 2 ≤ xn − xn1 2 1 − αn θn − yn − xn1 ≤ xn − xn1 2 θn .

3.24

Therefore, by the condition C1, 3.14, and limn → ∞ θn 0, we obtain un − Tn un −→ 0 as n −→ ∞.

3.25

From 3.23 and the condition C1, we have yn − xn1 2 ≤ xn − xn1 2 θn .

3.26

yn − xn 2 yn − xn1 xn1 − xn 2 2 yn − xn1 2 yn − xn1 , xn1 − xn xn1 − xn 2 2 ≤ yn − xn1 2yn − xn1 xn1 − xn xn1 − xn 2 ≤ 2xn1 − xn xn1 − xn yn − xn1 θn .

3.27

Therefore,

Hence, by 3.14 and limn → ∞ θn 0, we obtain yn − xn −→ 0 as n −→ ∞.

3.28

12


By 3.2, 3.3, and the firmly nonexpansiveness of Srn , we have un − p2 ≤ Sr xn − Sr p, xn − p un − p, xn − p n n

3.29

1 un − p2 xn − p2 − un − xn 2 . 2

It follows that un − p2 ≤ xn − p2 − un − xn 2 .

3.30

Therefore, by the condition C1 and 3.5, we have yn − p2 ≤ 1 1 − αn Ln − 1un − p2 1 − αn κ − αn un − Tn un 2 2 2 ≤ Ln − αn Ln − 1un − p ≤ Ln un − p 2 ≤ Ln xn − p − un − xn 2 .

3.31

It follows that 2 2 Ln un − xn 2 ≤ Ln xn − p − yn − p 2 2 2 Ln − 1xn − p xn − p − yn − p ≤ θn xn − yn xn − p yn − p .

3.32

Hence, by 3.28 and limn → ∞ θn 0, we obtain un − xn −→ 0

as n −→ ∞.

3.33

Since {un } is bounded, there exists a subsequence {uni } of {un } which converges weakly to w. Next, we prove that w ∈ Ω. From 3.22 and 3.25, we have uni − uni 1 → 0, and uni − Tni uni → 0 as i → ∞; therefore, by Lemma 2.5, we obtain w ∈ FS. From 3.1, we have 1 y − un , un − xn , 0 ≤ Φ un , y ϕ y − ϕun rn

∀y ∈ C.

3.34

It follows by the condition A2 that 1 Φ y, un ≤ Φ y, un Φ un , y ϕ y − ϕun y − un , un − xn , rn 1 y − un , un − xn , ≤ ϕ y − ϕun rn

∀y ∈ C.

∀y ∈ C 3.35


13

Hence, uni − xni ϕ y − ϕuni y − uni , ≥ Φ y, uni , rni

∀y ∈ C.

3.36

Therefore, from 3.33 and by uni w as i → ∞, we obtain Φ y, w ϕw − ϕ y ≤ 0,

∀y ∈ C.

3.37

For a constant t with 0 < t < 1 and y ∈ C, let yt ty 1 − tw. Since y, w ∈ C, thus, yt ∈ C. So, from 3.37, we have Φ yt , w ϕw − ϕ yt ≤ 0.

3.38

By 3.38, the conditions A1 and A4, and the convexity of ϕ, we have 0 Φ yt , yt ϕ yt − ϕ yt ≤ tΦ yt , y 1 − tΦ yt , w tϕ y 1 − tϕw − ϕ yt t Φ yt , y ϕ y − ϕ yt 1 − t Φ yt , w ϕw − ϕ yt ≤ t Φ yt , y ϕ y − ϕ yt .

3.39

It follows that Φ yt , y ϕ y − ϕ yt ≥ 0.

3.40

Therefore, by the condition A3 and the weakly lower semicontinuity of ϕ, we have Φw, y ϕy − ϕw ≥ 0 as t → 0 for all y ∈ C, and; hence, we obtain w ∈ MEP Φ, ϕ, and so w ∈ Ω. Since Ω is a nonempty closed convex subset of C, there exists a unique w ∈ Ω such that w PΩ x0 . Next, we prove that xn → w as n → ∞. Since w PΩ x0 , we have x0 − w ≤ x0 − z for all z ∈ Ω; it follows that x0 − w ≤ x0 − w.

3.41

Since w ∈ Ω ⊂ Cn ∩ Qn ; therefore, by 3.10, we have x0 − xn ≤ x0 − w.

3.42

Since xni − uni → 0 by 3.33 and uni w, we have xni w as i → ∞. Therefore, by 3.41, 3.42, and the weak lower semicontinuity of norm, we have x0 − w ≤ x0 − w ≤ lim infx0 − xni ≤ lim supx0 − xni ≤ x0 − w. i→∞

i→∞

3.43

14


It follows that x0 − w lim x0 − xni x0 − w.

3.44

i→∞

Since xni w as i → ∞; therefore, we have x0 − xni x0 − w as i −→ ∞.

3.45

Hence, from 3.44, 3.45, the Kadec-Klee property, and the uniqueness of w PΩ x0 , we obtain xni −→ w w

as i −→ ∞.

3.46

It follows that {xn } converges strongly to w and so are {yn } and {un }. This completes the proof. Remark 3.2. The iteration 3.1 is the difference with the iterative scheme of Kim 16 as the following. 1 The sequence {xn } is a projection sequence of x0 onto Cn ∩ Qn for all n ∈ N ∪ {0} such that C0 ∩ Q0 ⊃ C1 ∩ Q1 ⊃ · · · ⊃ Cn ∩ Qn ⊃ · · · ⊃ Ω.

3.47

2 A solving of a common element of the set of common fixed point for a discrete asymptotically κ-SPC semigroup and the set of solutions of the mixed equilibrium problems by iteration is obtained. For solving the equilibrium problem, let us define the condition B3 as the condition B1 such that ϕ 0. We have the following result. Corollary 3.3. Let C be a nonempty closed convex subset of a real Hilbert space H, and let Φ be a bifunction from C × C into R satisfying the conditions (A1)–(A5) with either B2 or B3 holds. Let S {Tn : n ≥ 0} be an asymptotically κ-SPC semigroup on C for some κ ∈ 0, 1 and a bounded sequence {Ln } ⊂ 1, ∞ such that limn → ∞ Ln 1. Assume that Ω : FS ∩ EP Φ is a nonempty bounded subset of C. For x0 x ∈ C chosen arbitrarily, suppose that {xn }, {yn }, and {un } are generated iteratively by 1 un ∈ C such that Φ un , y y − un , un − xn ≥ 0, rn

Cn1

∀y ∈ C,

yn αn un 1 − αn Tn un , 2 z ∈ Cn ∩ Qn : yn − z ≤ xn − z2 1 − αn θn κ − αn un − Tn un 2 , Qn1 {z ∈ Cn ∩ Qn : xn − z, x0 − xn ≥ 0},


15 C0 Q0 C,

xn1 PCn1 ∩Qn1 x0 ,

∀n ∈ N ∪ {0}, 3.48


C3 ∞ n0 |rn1 − rn | < ∞. Then the sequences {xn }, {yn }, and {un } converge strongly to w PΩ x0 . Proof. It is concluded from Theorem 3.1 immediately, by putting ϕ 0. Corollary 3.4. Let C be a nonempty closed convex subset of a real Hilbert space H. Let S {Tn : n ≥ 0} be an asymptotically κ-SPC semigroup on C for some κ ∈ 0, 1 and a bounded sequence {Ln } ⊂ 1, ∞ such that limn → ∞ Ln 1. Assume that FS is a nonempty bounded subset of C. For x0 x ∈ C chosen arbitrarily, suppose that {xn } and {yn } are generated iteratively by yn αn xn 1 − αn Tn xn , 2 Cn1 z ∈ Cn ∩ Qn : yn − z ≤ xn − z2 1 − αn θn κ − αn xn − Tn xn 2 , Qn1 {z ∈ Cn ∩ Qn : xn − z, x0 − xn ≥ 0},

3.49

C0 Q0 C, xn1 PCn1 ∩Qn1 x0 ,

∀n ∈ N ∪ {0},

where θn Ln − 1 · sup{xn − z2 : z ∈ FS} < ∞ and {αn } ⊂ a, b such that κ < a < b < 1. Then the sequences {xn } and {yn } converge strongly to w PFS x0 . Proof. It is concluded from Corollary 3.3 immediately, by putting Φ 0.

4. Applications We introduce the equilibrium problem to the optimization problem: minζx, x∈C

4.1

where C is a nonempty closed convex subset of a real Hilbert space H and ζ : C → R ∪ {∞} is a proper convex and lower semicontinuous. We denote by Argminζ the set of solutions of problem 4.1. We define the condition B4 as the condition B3 such that Φ : C×C → R is a bifunction defined by Φx, y ζy − ζx for all x, y ∈ C. Observe that EP Φ Argminζ. We give the interesting result as the following theorem.

16


Theorem 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H, and let ζ : C → R ∪ {∞} be a proper lower semicontinuous and convex function with either B2 or B4 holds. Let S {Tn : n ≥ 0} be an asymptotically κ-SPC semigroup on C for some κ ∈ 0, 1 and a bounded sequence {Ln } ⊂ 1, ∞ such that limn → ∞ Ln 1. Assume that Ω : FS ∩ Argminζ is a nonempty bounded subset of C. For x0 x ∈ C chosen arbitrarily, suppose that {xn }, {yn } and {un } are generated iteratively by 1 un ∈ C such that ζ y − ζun y − un , un − xn ≥ 0, rn

Cn1

∀y ∈ C,

yn αn un 1 − αn Tn un , 2 z ∈ Cn ∩ Qn : yn − z ≤ xn − z2 1 − αn θn κ − αn un − Tn un 2 , 4.2 Qn1 {z ∈ Cn ∩ Qn : xn − z, x0 − xn ≥ 0}, C0 Q0 C, xn1 PCn1 ∩Qn1 x0 ,

∀n ∈ N ∪ {0},


C3 ∞ n0 |rn1 − rn | < ∞. Then the sequences {xn }, {yn }, and {un } converge strongly to w PΩ x0 . Proof. It is concluded from Corollary 3.3 immediately, by defining Φx, y ζy − ζx for all x, y ∈ C.

References 1 E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994. 2 P. Hartman and G. Stampacchia, “On some non-linear elliptic differential-functional equations,” Acta Mathematica, vol. 115, pp. 271–310, 1966. 3 W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000. 4 W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953. 5 F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 20, pp. 197–228, 1967. 6 Q. Liu, “Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings,” Nonlinear Analysis, vol. 26, no. 11, pp. 1835–1842, 1996. 7 K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 35, pp. 171–174, 1972. 8 M. O. Osilike, A. Udomene, D. I. Igbokwe, and B. G. Akuchu, “Demiclosedness principle and convergence theorems for k-strictly asymptotically pseudocontractive maps,” Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 1334–1345, 2007. 9 T. H. Kim and H. K. Xu, “Strong convergence of modified Mann iterations,” Nonlinear Analysis, vol. 61, no. 1-2, pp. 51–60, 2005. 10 S. Reich, “Weak convergence theorems for nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 274–276, 1979.


17

11 G. Marino and H.-K. Xu, “Convergence of generalized proximal point algorithms,” Communications on Pure and Applied Analysis, vol. 3, no. 4, pp. 791–808, 2004. 12 T. H. Kim and H. K. Xu, “Convergence of the modified Mann’s iteration method for asymptotically strict pseudo-contractions,” Nonlinear Analysis, vol. 68, no. 9, pp. 2828–2836, 2008. 13 A. Genel and J. Lindenstrauss, “An example concerning fixed points,” Israel Journal of Mathematics, vol. 22, no. 1, pp. 81–86, 1975. 14 K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372– 379, 2003. 15 G. Marino and H.-K. Xu, “Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 336–346, 2007. 16 T. H. Kim, “Convergence of modified Mann’s iteration methods for discrete asymptotically strict pseudo-contractive semigroups,” in Proceedings of the Proceedings of the Asian Conference on Nonlinear Analysis and Optimization, pp. 139–162, 2008. 17 W. Takahashi, Introduction to Nonlinear and Convex Analysis, Yokohama Publishers, Yokohama, Japan, 2009. 18 J.-W. Peng and J.-C. Yao, “Two extragradient methods for generalized mixed equilibrium problems, nonexpansive mappings and monotone mappings,” Computers & Mathematics with Applications, vol. 58, no. 7, pp. 1287–1301, 2009.

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