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Volume 2011, Article ID 840319, 25 pagesdoi:10.1155/2011/840319 Research Article The Shrinking Projection Method for Common Solutions of Generalized Mixed Equilibrium Problems and Fixed

Volume 2011, Article ID 840319, 25 pages

doi:10.1155/2011/840319

Research Article

The Shrinking Projection Method for Common

Solutions of Generalized Mixed Equilibrium

Problems and Fixed Point Problems for Strictly

Pseudocontractive Mappings

Thanyarat Jitpeera and Poom Kumam

Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand

Correspondence should be addressed to Poom Kumam,poom.kum@kmutt.ac.th

Received 21 September 2010; Revised 14 December 2010; Accepted 20 January 2011

Academic Editor: Jewgeni Dshalalow

Copyrightq 2011 T Jitpeera and P Kumam This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited

We introduce the shrinking hybrid projection method for finding a common element of the set offixed points of strictly pseudocontractive mappings, the set of common solutions of the variationalinequalities with inverse-strongly monotone mappings, and the set of common solutions ofgeneralized mixed equilibrium problems in Hilbert spaces Furthermore, we prove strongconvergence theorems for a new shrinking hybrid projection method under some mild conditions.Finally, we apply our results to Convex Feasibility ProblemsCFP The results obtained in thispaper improve and extend the corresponding results announced by Kim et al.2010 and thepreviously known results

2 strictly pseudocontractive with the coefficient k ∈ 0, 1 if

k ∈ 0, 1, and define a mapping S kby

S k x  kx  1 − kSx, ∀x ∈ E, 1.4

where S is strictly pseudocontractive mappings Under appropriate restrictions on k, it is proved that the mapping S k is nonexpansive Therefore, the techniques of studying nonex-pansive mappings can be applied to study more general strictly pseudocontractive mappings

Recall that letting A : E → H be a mapping, then A is called

E × E intoÊsuch that E ∩ dom ϕ / ∅, whereÊis the set of real numbers

There exists the generalized mixed equilibrium problem for finding x ∈ E such that

We see that x is a solution of a problem 1.7 which implies that x ∈ dom ϕ  {x ∈ E : ϕx <

∞}

In particular, if A ≡ 0, then the problem 1.7 is reduced into the mixed equilibrium

problem2 for finding x ∈ E such that

F

x, y

 ϕy

− ϕx ≥ 0, ∀y ∈ E. 1.9

The set of solutions of1.9 is denoted by MEPF, ϕ.

If A ≡ 0 and ϕ ≡ 0, then the problem 1.7 is reduced into the equilibrium problem 3

for finding x ∈ E such that

F

x, y

≥ 0, ∀y ∈ E. 1.10

The set of solutions of1.10 is denoted by EPF This problem contains fixed point problems

and includes as special cases numerous problems in physics, optimization, and economics.Some methods have been proposed to solve the equilibrium problem; please consult4,5

If F ≡ 0 and ϕ ≡ 0, then the problem 1.7 is reduced into the Hartmann-Stampacchia

variational inequality6 for finding x ∈ E such that



Ax, y − x

≥ 0, ∀y ∈ E. 1.11

The set of solutions of 1.11 is denoted by VIE, A The variational inequality has been

extensively studied in the literature See, for example,7 10 and the references therein.Many authors solved the problems GMEPF, ϕ, A, MEPF, ϕ, and EPF based oniterative methods; see, for instance,4,5,11–25 and reference therein

In 2007, Tada and Takahashi26 introduced a hybrid method for finding the commonelement of the set of fixed point of nonexpansive mapping and the set of solutions ofequilibrium problems in Hilbert spaces Let{x n } and {u n} be sequences generated by thefollowing iterative algorithm:

and the set of solutions of variational inequalities with inverse-strongly monotone mappings

Then, they proved that, under certain appropriate conditions imposed on{ n }, {β n }, {γ n},

{α1n }, {α2n }, and {α3n }, the sequence {x n} generated by 1.13 converges strongly to q ∈

FS ∩ VIE, B ∩ VIE, C, where q  P FS∩VIE,B∩VIE,C fq.

In 2010, Kumam and Jaiboon28 introduced a new method for finding a commonelement of the set of fixed point of strictly pseudocontractive mappings, the set of commonsolutions of variational inequalities with inverse-strongly monotone mappings, and the set ofcommon solutions of a system of generalized mixed equilibrium problems in Hilbert spaces.Then, they proved that, under certain appropriate conditions imposed on { n }, {β n}, and

{α i n }, where i  1, 2, 3, 4, 5 The sequence {x n } converges strongly to q ∈ Θ : FS∩VIE, B∩

VIE, C ∩ GMEPF1, ϕ, A1 ∩ GMEPF2, ϕ, A2, where q  PΘI − A  γfq.

In this paper, motivate, by Tada and Takahashi26, Qin and Kang 27, and Kumamand Jaiboon28, we introduce a new shrinking projection method for finding a commonelement of the set of fixed points of strictly pseudocontractive mappings, the set of commonsolutions of generalized mixed equilibrium problems, and the set of common solutions of thevariational inequalities for inverse-strongly monotone mappings in Hilbert spaces Finally,

we apply our results to Convex Feasibility ProblemsCFP The results obtained in this paperimprove and extend the corresponding results announced by the previously known results

for all x, y ∈ H and λ ∈ 0, 1.

For any x ∈ H, there exists a unique nearest point in E, denoted by P E x, such that

x − P E x ≤x − y, ∀y ∈ E. 2.2

The mapping P E is called the metric projection of H onto E.

It is well known that P E is a firmly nonexpansive mapping of H onto E, that is,



x − y, P E x − P E y

≥P E x − P E y2

, ∀x, y ∈ H. 2.3

Moreover, P E x is characterized by the following properties: P E x ∈ E and

Lemma 2.2 Let H be a Hilbert space, let E be a nonempty closed convex subset of H, and let B be a

mapping of E into H Let u ∈ E Then, for λ > 0,

u ∈ VI E, B ⇐⇒ u  P E u − λBu, 2.6

where P E is the metric projection of H onto E.

Lemma 2.3 see 1 Let E be a nonempty closed convex subset of a real Hilbert space H, and let

S : E → E be a k-strictly pseudocontractive mapping with a fixed point Then FS is closed and convex Define S k : E → E by S k  kx  1 − kSx for each x ∈ E Then S k is nonexpansive such that FS k   FS.

Lemma 2.4 see 29 Let E be a closed convex subset of a real Hilbert space H, and let S : E → E

be a nonexpansive mapping Then I − S is demiclosed at zero; that is,

x n  x, x n − Sx n−→ 0 2.7

implies x  Sx.

Lemma 2.5 see 30 Each Hilbert space H satisfies the Kadec-Klee property, for any sequence {x n}

with x n  x and x n  → x together implying x n − x → 0.

Lemma 2.6 see 31 Let E be a closed convex subset of H Let {x n } be a bounded sequence in H.

Assume that

1 the weak ω-limit set ω w x n  ⊂ E,

2 for each z ∈ E, lim n → ∞ x n − z exists.

Then {x n } is weakly convergent to a point in E.

Lemma 2.7 see 32 Let E be a closed convex subset of H Let {x n } be a sequence in H and u ∈ H.

Let q  P E u If {x n } is ω w x n  ⊂ E and satisfies the condition

x n − u ≤u − q 2.8

for all n, then x n → q.

Lemma 2.8 see 33 Let E be a nonempty closed convex subset of a strictly convex Banach space

X Let {T n : n ∈Æ} be a sequence of nonexpansive mappings on E Suppose∞

n1 FT n  is nonempty.

Let δ n be a sequence of positive number with∞

n1 δ n  1 Then a mapping S on E defined by

For solving the mixed equilibrium problem, let us give the following assumptions for

the bifunction F, the function A, and the set E:

A1 Fx, x  0 for all x ∈ E

A2 F is monotone, that is, Fx, y  Fy, x ≤ 0 for all x, y ∈ E

A3 for each x, y, z ∈ E, lim t → 0 Ftz  1 − tx, y ≤ Fx, y

A4 for each x ∈ E, y → Fx, y is convex and lower semicontinuous

A5 for each y ∈ E, x → Fx, y is weakly upper semicontinuous

B1 for each x ∈ H and r > 0, there exists a bounded subset D x ⊆ E and y x ∈ E such that, for any z ∈ E \ D x,

Lemma 2.10 Let H be a Hilbert space, let E be a nonempty closed convex subset of H, and let

A : E → H be ρ-inverse-strongly monotone If 0 < r ≤ 2ρ, then I − ρA is a nonexpansive mapping

in H.

Proof For all x, y ∈ E and 0 < r ≤ 2ρ, we have

I − rAx −I −rAy2x − y

Theorem 3.1 Let E be a nonempty closed convex subset of a real Hilbert space H Let F1 and F2

be two bifunctions from E × E toÊsatisfying (A1)–(A5), and let ϕ : E → Ê∪ {∞} be a proper

lower semicontinuous and convex function with either (B1) or (B2) Let A1, A2, B, C be four ρ, ω,

β, ξ-inverse-strongly monotone mappings of E into H, respectively Let S : E → E be a k-strictly pseudocontractive mapping with a fixed point Define a mapping S k : E → E by S k x  kx1−kSx, for all x ∈ E Suppose that

t n  α1n S k x n  α2n y n  α3n z n  α4n u n  α5n v n ,

E n1  {w ∈ E n : t n − w ≤ x n − w},

x n1  P E n1 x0, ∀n ≥ 0,

3.2

where {α i n } are sequences in 0, 1, where i  1, 2, 3, 4, 5, r n ∈ 0, 2ρ, s n ∈ 0, 2ω, and {λ n }, {μ n}

are positive sequences Assume that the control sequences satisfy the following restrictions:

C15

i1 α i n  1,

C2 limn → ∞ α i n  α i ∈ 0, 1, where i  1, 2, 3, 4, 5,

C3 a ≤ r n ≤ 2ρ and b ≤ s n ≤ 2ω, where a, b are two positive constants,

C4 c ≤ λ n ≤ 2β and d ≤ μ n ≤ 2ξ, where c, d are two positive constants,

C5 limn → ∞ |λ n1 − λ n|  limn → ∞ |μ n1 − μ n |  0.

Then, {x n } converges strongly to PΘx0.

Proof Letting p ∈ Θ and byLemma 2.9, we obtain

Step 1 We show that {x n } is well defined and E n is closed and convex for any n ≥ 1.

From the assumption, we see that E1  E is closed and convex Suppose that E k is

closed and convex for some k ≥ 1 Next, we show that E k1 is closed and convex for some k For any p ∈ E k, we obtain

t k − p ≤ x k − p 3.5

is equivalent to

t k − p2 2t k − x k , x k − p≤ 0. 3.6

Thus, E k1 is closed and convex Then, E n is closed and convex for any n ≥ 1 This implies

that{x n} is well defined

Step 2 We show that Θ ⊂ E n for each n ≥ 1 From the assumption, we see that Θ ⊂ E  E1.Suppose Θ ⊂ E k for some k ≥ 1 For any p ∈ Θ ⊂ E k , since y n  P E x n − λ n Bx n and

z n  P E x n − μ n Cx n , for each λ n ≤ 2β and μ n ≤ 2ξ byLemma 2.10, we have I − λ n B and

I − μ n C are nonexpansive Thus, we obtain

It follows that p ∈ E k1 This implies thatΘ ⊂ E n for each n ≥ 1.

Step 3 We claim that lim n → ∞ x n1 − x n  0 and limn → ∞ x n − t n  0

Hence, for p ∈ Θ, we obtain

Thus, the sequence{x n −x0} is a bounded and nondecreasing sequence, so limn → ∞ x n −x0

exists; that is, there exists m such that

m  lim

Similarly, we also have

t n − p2≤ α1n S k x n − p2 α2n y n − p2 α3n z n − p2 α4n u n − p2 α5n v n − p2

≤ α1n x n − p2 α2n y n − p2 α3n z n − p2 α4n u n − p2 α5n v n − p2

.

3.25Substituting3.21, 3.22, 3.23, and 3.24 into 3.25, we obtain

On the other hand, letting p ∈ Θ for each n ≥ 1, we get p  T F1

r n I − r n A1p Since T F1

r n is firmlynonexpansive, we have

I − r

n A1x n − I − r n A1p2u n − p2

−I − rn A1x n − I − r n A1p −u n − p2

≤ 12

y n − p2P E x n − λ n Bx n  − P E

p − λ n Bp2

≤I − λ n B x n − I − λ n B p, y n − p

 12

I − λ

n B x n − I − λ n B p2y n − p2

−I − λn B x n − I − λ n B p −y n − p2

≤ 12

By using the same argument in3.33 and 3.35, we can get

v n − p2≤x n − p2

− x n − v n2 2s n x n − v nA2x n − A2p,

z n − p2 ≤x n − p2− x n − z n2 2μ n x n − z nCx n − Cp. 3.36Substituting3.33, 3.35, and 3.36 into 3.25, we obtain

FromC2, 3.20, 3.28, 3.30, and 3.31, we have

Since{x n i } is bounded, there exists a subsequence {x n i } of {x n} which converges weakly to

z Without loss of generality, we may assume that {x n i }  z It follows from 3.47, that

lim

It follows fromLemma 2.4that z ∈ FP By 3.44, we have z ∈ Θ.

Step 6 Finally, we show that x n → z, where z  PΘx0

SinceΘ is nonempty closed convex subset of H, there exists a unique z∈ Θ such that

z PΘx0 Since z∈ Θ ⊂ E n and x n  P E n x0, we have

x0− x n   x0− P E n x0 ≤x0− z 3.49

for all n ≥ 1 From 3.49, {x n } is bounded, so ω w x n  / ∅ By the weak lower semicontinuity

of the norm, we have

x0− z ≤ lim inf

Since z ∈ ω w x n ⊂ Θ, we obtain

x0− z  x0− PΘx0 ≤ x0− z. 3.51Using3.49 and 3.50, we obtain z z Thus, ω w x n   {z} and x n  z So we have

x0− z ≤ x0− z ≤ lim inf

i → ∞ x0− x n ≤ lim sup

i → ∞

x0− x n ≤x0− z. 3.52Thus,

x0− z  lim

From x n  z, we obtain x0− x n   x0− z UsingLemma 2.5, we obtain that

x n − z  x n − x0 − z − x0 −→ 0 3.54

as n → ∞ and hence x n → z in norm This completes the proof.

If the mapping S is nonexpansive, then S k  S0 S We can obtain the following result

Corollary 3.2 Let E be a nonempty closed convex subset of a real Hilbert space H Let F1and F2

be two bifunctions from E × E toÊsatisfying (A1)–(A5), and let ϕ : E → Ê∪ {∞} be a proper

lower semicontinuous and convex function with either (B1) or (B2) Let A1, A2, B, C be four ρ, ω, β,

ξ-inverse-strongly monotone mappings of E into H, respectively Let S : E → E be a nonexpansive mapping with a fixed point Suppose that

Let {x n } be a sequence generated by the following iterative algorithm 3.1, where {α i n } are sequences

in 0, 1, where i  1, 2, 3, 4, 5, r n ∈ 0, 2ρ, s n ∈ 0, 2ω, and {λ n }, {μ n } are positive sequences.

Assume that the control sequences satisfy (C1)–(C5) in Theorem 3.1 Then, {x n } converges strongly

Θ : FS ∩ EPF1 ∩ EPF2 ∩ VIE, B ∩ VIE, C / ∅. 3.56

Let {x n } be a sequence generated by the following iterative algorithm:

where {α i n } are sequences in (0,1), where i  1, 2, 3, 4, 5, r n ∈ 0, ∞, s n ∈ 0, ∞ and {λ n }, {μ n } are

positive sequences Assume that the control sequences satisfy the condition (C1)–(C5) in Theorem 3.1 Then, {x n } converges strongly to PΘx0.

If B  0, C  0, and F1u n , u  F1v n , v  0 inCorollary 3.3, then P E  I and we get

u n  y n  x n and v n  z n  x n; hence, we can obtain the following result immediately

Corollary 3.4 Let E be a nonempty closed convex subset of a real Hilbert space H Let S : E → E be

a k-strictly pseudocontractive mapping with a fixed point Define a mapping S k : E → E by S k x  kx1−kSx, for all x ∈ E Suppose that FS /  ∅ Let {x n } be a sequence generated by the following

where {α n } are sequences in 0, 1 Assume that the control sequences satisfy the condition

limn → ∞ α n  α ∈ 0, 1 in Theorem 3.1 Then, {x n } converges strongly to a point P FS x0.

4 Convex Feasibility Problem

Finally, we consider the following Convex Feasibility Problem CFP: finding an x ∈ M

j1 C j,

where M ≥ 1 is an integer and each C iis assumed to be the solutions of equilibrium problem

with the bifunction F j , j  1, 2, 3, , M and the solution set of the variational inequality

problem There is a considerable investigation on CFP in the setting of Hilbert spaces whichcaptures applications in various disciplines such as image restoration 35, 36, computertomography37, and radiation therapy treatment planning 38

The following result can be obtained fromTheorem 3.1 We, therefore, omit the proof

Theorem 4.1 Let E be a nonempty closed convex subset of a real Hilbert space H Let {F j}M

j1 be a family of bifunction from E × E toÊsatisfying (A1)–(A5), and let ϕ : E → Ê∪ {∞} be a proper

lower semicontinuous and convex function with either (B1) or (B2) Let A j : E → H be ρ j strongly monotone mapping for each j ∈ {1, 2, 3, , M} Let B i : E → H be β i -inverse-strongly monotone mapping for each i ∈ {1, 2, 3, , N} Let S : E → E be a k-strictly pseudocontractive mapping with a fixed point Define a mapping S k : E → E by S k x  kx  1 − kSx, for all x ∈ E Suppose that

For solving the mixed equilibrium problem, let us give the following assumptions for

the bifunction F, the function A, and the set... 3.22, 3.23, and 3.24 into 3.25, we obtain