Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 414831, 35 pages
doi:10.1155/2012/414831
Research Article
A System of Mixed Equilibrium Problems,
a General System of Variational Inequality
Problems for Relaxed Cocoercive, and Fixed Point
Problems for Nonexpansive Semigroup and Strictly
Pseudocontractive Mappings
Poom Kumam1, 2 and Phayap Katchang2, 3
1
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi
(KMUTT), Bangkok 10140, Thailand
2
Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand
3
Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology,
Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand
Correspondence should be addressed to Phayap Katchang, p.katchang@hotmail.com
Received 17 November 2011; Accepted 23 January 2012
Academic Editor: Giuseppe Marino
Copyright q 2012 P. Kumam and P. Katchang. 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 introduce an iterative algorithm for finding a common element of the set of solutions of a
system of mixed equilibrium problems, the set of solutions of a general system of variational
inequalities for Lipschitz continuous and relaxed cocoercive mappings, the set of common
fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite
family of strictly pseudocontractive mappings in Hilbert spaces. Furthermore, we prove a strong
convergence theorem of the iterative sequence generated by the proposed iterative algorithm
under some suitable conditions which solves some optimization problems. Our results extend and
improve the recent results of Chang et al. 2010 and many others.
1. Introduction
Let H be a real Hilbert space with inner product ·, · and norm · . Let C be a nonempty
closed convex subset of H. Recall that a mapping T : C → C is nonexpansive if
T x − T y ≤ x − y,
∀x, y ∈ C.
1.1
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We denote the set of fixed points of T by FT , that is FT {x ∈ C : x T x}. A mapping
f : C → C is said to be an α-contraction if there exists a coefficient α ∈ 0, 1 such that
fx − f y ≤ αx − y,
∀x, y ∈ C.
1.2
Let B : C → H be a mapping. Then B is called:
1 monotone if
Bx − By, x − y ≥ 0,
∀x, y ∈ C;
1.3
2 d-strongly monotone if there exists a positive real number d such that
2
Bx − By, x − y ≥ dx − y ,
∀x, y ∈ C,
1.4
for constant d > 0, this implies that
Bx − By ≥ dx − y,
1.5
that is, B is d-expansive and when d 1, it is expansive;
3 L-Lipschitz continuous if there exists a positive real number L such that
Bx − By ≤ Lx − y,
∀x, y ∈ C;
1.6
4 c-cocoercive 1, 2 if there exists a positive real number c such that
2
Bx − By, x − y ≥ cBx − By ,
∀x, y ∈ C,
1.7
Clearly, every c-cocoercive map B is 1/c-Lipschitz continuous;
5 relaxed c-cocoercive, if there exists a positive real number c such that
2
Bx − By, x − y ≥ −cBx − By ,
∀x, y ∈ C;
1.8
6 relaxed c, d-cocoercive, if there exists a positive real number c, d such that
2
2
Bx − By, x − y ≥ −cBx − By dx − y ,
∀x, y ∈ C,
1.9
for c 0, B is d-strongly monotone. This class of mapping is more general than the
class of strongly monotone mapping. It is easy to see that we have the following
implication: d-strongly monotonicity implying relaxed c, d-cocoercivity,
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3
7 k-strictly pseudocontractive, if there exists a constant k ∈ 0, 1 such that
Bx − By2 ≤ x − y2 kI − Bx − I − By2 ,
∀x, y ∈ C.
1.10
Remark 1.1 see 3, Remark 1.1 pages 135-136. If B : C → H is a LB -Lipschitz continuous
and relaxed c, d-cocoercive mapping with d > cL2B and 0 < τ < 2d − cL2B /L2B , then I − τB
satisfies the following:
I − τBx − I − τBy ≤ 1 − τξx − y,
∀x, y ∈ C,
1.11
where ξ L2B /22d − cL2B /L2B − τ.
Similarly, if D : C → H is LD -Lipschitz continuous and relaxed c , d -cocoercive
mapping with d > c L2D and 0 < δ < 2d − c L2D /L2D , then the mapping I − δD satisfies the
following:
I − δDx − I − δDy ≤ 1 − δξ x − y,
1.12
where ξ L2D /22d − c L2D /L2D − δ.
Let A be a strongly positive linear bounded operator on H if there is a constant γ > 0 with
the property
Ax, x ≥ γx2 ,
∀x ∈ H.
1.13
We recall optimization problem for short, OP as the following
min
x∈F
μ
1
Ax, x x − u2 − hx,
2
2
1.14
∞
where F ∩∞
∅,
n1 Cn , C1 , C2 , . . . are infinitely closed convex subsets of H such that ∩n1 Cn /
u ∈ H, μ ≥ 0 is a real number, A is a strongly positive linear bounded operator on H, and h is
a potential function for γf i.e., h x γfx for x ∈ H. This kind of optimization problem
has been studied extensively by many authors, see, for example, 4–7 when F ∩∞
n1 Cn and
hx x, b, where b is a given point in H.
On the other hand, a family S {Ss : 0 ≤ s < ∞} of mappings of C into itself is
called a nonexpansive semigroup on C if it satisfies the following conditions:
i S0x x for all x ∈ C;
ii Ss t SsSt for all s, t ≥ 0;
iii Ssx − Ssy ≤ x − y for all x, y ∈ C and s ≥ 0;
iv for all x ∈ C, s → Ssx is continuous.
We denote by FS the set of all common fixed points of S {Ss : s ≥ 0}, that is, FS ∩s≥0 FSs. It is known that FS is closed and convex.
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Let φ : C → R be a real-valued function and let {Θk : C × C → R, k 1, 2, . . . , N}
be a finite family of equilibrium functions, that is, Θk u, u 0 for each u ∈ C. The system of
mixed equilibrium problems for short, SMEP for function Θ1 , Θ2 , . . . , ΘN , φ is to find z ∈ C
such that
Θ1 z, y φ y − φz ≥ 0, ∀y ∈ C,
Θ2 z, y φ y − φz ≥ 0, ∀y ∈ C,
..
.
ΘN z, y φ y − φz ≥ 0, ∀y ∈ C.
1.15
MEPΘk , φ, where MEPΘk , φ is the set of
The set of solutions of 1.15 is denoted by ∩N
k1
solutions of the mixed equilibrium problem for short, MEP, which is to find z ∈ C such that
Θk z, y φ y − φz ≥ 0,
∀y ∈ C.
1.16
In particular, if φ ≡ 0, and N 1, then the problem 1.15 reduces to the equilibrium problem
for short, EP, which is to find z ∈ C such that
Θ z, y ≥ 0,
∀y ∈ C.
1.17
It is well known that the SMEP includes fixed point problem, optimization problem,
variational inequality problem, and Nash equilibrium problem as its special cases see 8–
13 for more details.
For solving the solutions of a nonexpansive semigroup and the solutions of the system
of mixed equilibrium problems were studied by many authors see 14–23 and reference
therein. In 2010, Chang et al. 24 studied the following approximation method:
1
1
1
1
1
K un − K xn , η x, un
≥ 0, ∀x ∈ C,
Θ1 un , x φx − φ un r1
1
2
2
2
2
K un − K xn , η x, un
≥ 0, ∀x ∈ C,
Θ2 un , x φx − φ un r2
..
1.18
.
1
N
N
N
N
ΘN un , x φx − φ un
K un
− K xn , η x, un
≥ 0, ∀x ∈ C,
rN
1 tn
N
xn1 αn fWn xn βn xn γn
SsWn un ds,
tn 0
where
1
un JrΘ1 1 xn ,
k
k−1
un JrΘk k un
k−2
k−1
JrΘk k JrΘk−1
un
JrΘk k · · · JrΘ2 2 JrΘ1 1 xn ,
1
JrΘk k · · · JrΘ2 2 un ,
k 2, 3, . . . , N,
1.19
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JrΘk k : C → C, k 1, 2, . . . , N is the mapping defined by 2.22 below, Wn is the mapping
defined by 2.12, and S {Ss : 0 ≤ s < ∞} is a nonexpansive semigroup. They proved that
MEPΘk , φ under control
{xn } converges strongly to a fixed point of FS ∩ FW ∩ ∩N
k1
conditions on the parameters.
Let B, D : C → H be two mappings. The general system of variational inequalities problem
see 25 is to find x∗ , y∗ ∈ C × C such that
τBy∗ x∗ − y∗ , x − x∗ ≥ 0,
δDx∗ y∗ − x∗ , x − y∗ ≥ 0,
∀x ∈ C,
∀x ∈ C,
1.20
where τ and δ are two positive real numbers. The set of solutions of the general system of
variational inequalities problem is denoted by SVIC, B, D. In particular, if B D, then the
problem 1.20 reduces to the following equation:
τBy∗ x∗ − y∗ , x − x∗ ≥ 0,
δBx∗ y∗ − x∗ , x − y∗ ≥ 0,
∀x ∈ C,
∀x ∈ C,
1.21
which is defined by Verma 26 see also Verma 27, and is called the new system of variational
inequalities. Further, if we set D 0, then problem 1.20 reduces to the classical variational
inequality is to find x∗ ∈ C such that
Bx∗ , x − x∗ ≥ 0,
∀x ∈ C.
1.22
We denoted by VIC, B the set of solutions of the variational inequality problem. The
variational inequality problem has been extensively studied in literature, see, for example,
28–31 and references therein. In order to find the solutions of the general system of
variational inequality problem 1.20, Wangkeeree and Kamraksa 32 considered the
following iterative algorithm:
1
K un − K xn , ηx, un ≥ 0, ∀x ∈ C,
r
zn PC un − δDun ,
αn γfxn βn xn 1 − βn I − αn A Wn PC zn − τBzn ,
Θun , x φx − φun xn1
1.23
where B, D : C → H is a LB -Lipschitz continuous and relaxed c, d-cocoercive mapping and
LD -Lipschitz continuous and relaxed c , d -cocoercive mapping, respectively. They proved
that {xn } converges strongly to a fixed point of FWn ∩ MEPΘ, φ ∩ SVIC, B, D which
is a solution of general system of variational inequality 1.20. Very recently, Jaiboon and
Kumam 33 studied a new general iterative method for finding a common element of the
set of solution of a mixed equilibrium problem, the set of fixed points of an infinite family
of nonexpansive mappings, and the set of solutions of variational inequalities for an inversestrongly monotone mapping in Hilbert spaces, which solves some optimization problems.
Inspired and motivated by Chang et al. 24, Jaiboon and Kumam 33, Kumam and
Jaiboon 34 and Wangkeeree and Kamraksa 32, the purpose of this paper is to introduce
an iterative algorithm for finding a common element of the set of solutions of 1.15, the
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Journal of Applied Mathematics
set of solutions of 1.20 for Lipschitz continuous and relaxed cocoercive mappings, the set
of common fixed points for nonexpansive semigroup, and the set of common fixed points
for an infinite family of strictly pseudocontractive mappings. Consequently, we prove the
strong convergence theorem in Hilbert spaces under control conditions on the parameters.
Furthermore, we can apply our results for solving some optimization problems. Our results
extend and improve the corresponding results in Chang et al. 24, Kumam and Jaiboon 34,
Wangkeeree and Kamraksa 32, and many others.
2. Preliminaries
Let H a real Hilbert space and C a nonempty closed convex subset of H. We denote strong
convergence weak convergence by notation → . In a real Hilbert space H, it is well
known that
x − y2 x2 − y2 − 2 x − y, y ,
x y2 ≤ x2 2 y, x y ,
x y2 ≥ x2 2 y, x ,
λx 1 − λy2 λx2 1 − λy2 − λ1 − λx − y2
2.1
2.2
2.3
2.4
for all x, y ∈ H and λ ∈ R.
Recall that for every point x ∈ H, there exists a unique nearest point in C, denoted by
PC x, such that
x − PC x ≤ x − y,
∀y ∈ C.
2.5
PC is called the metric projection of H onto C. It is well known that PC is a nonexpansive
mapping of H onto C and satisfies
2
x − y, PC x − PC y ≥ PC x − PC y
2.6
for every x, y ∈ H. Obviously, this immediately implies that
x − y − PC x − PC y 2 ≤ x − y2 − PC x − PC y2 ,
∀x, y ∈ H.
2.7
Moreover, PC x is characterized by the following properties: PC x ∈ C and
x − PC x, y − PC x ≤ 0,
x − y2 ≥ x − PC x2 y − PC x2
for all x ∈ H, y ∈ C.
2.8
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In order to prove our main results, we need the following lemmas.
Lemma 2.1 see 35. Let V : C → H be a k-strict pseudo-contraction, then
1 the fixed point set FV of V is closed convex so that the projection PFV is well defined;
2 define a mapping T : C → H by
T x tx 1 − tV x,
∀x ∈ C.
2.9
If t ∈ k, 1, then T is a nonexpansive mapping such that FV FT .
A family of mappings {Vi : C → H}∞
i1 is called a family of uniformly k-strict pseudocontractions, if there exists a constant k ∈ 0, 1 such that
Vi x − Vi y2 ≤ x − y2 kI − Vi x − I − Vi y2 ,
∀x, y ∈ C, ∀i ≥ 1.
2.10
Let {Vi : C → C}∞
i1 be a countable family of uniformly k-strict pseudo-contractions. Let
be
the
sequence of nonexpansive mappings defined by 2.9, that is,
{Ti : C → C}∞
i1
Ti x tx 1 − tVi x,
∀x ∈ C, ∀i ≥ 1, t ∈ k, 1.
2.11
Let {Ti } be a sequence of nonexpansive mappings of C into itself defined by 2.11 and
let {μi } be a sequence of nonnegative numbers in 0, 1. For each n ≥ 1, define a mapping Wn
of C into itself as follows:
Un,n1 I,
Un,n μn Tn Un,n1 1 − μn I,
Un,n−1 μn−1 Tn−1 Un,n 1 − μn−1 I,
..
.
Un,k μk Tk Un,k1 1 − μk I,
Un,k−1 μk−1 Tk−1 Un,k 1 − μk−1 I,
..
.
Un,2 μ2 T2 Un,3 1 − μ2 I,
Wn Un,1 μ1 T1 Un,2 1 − μ1 I.
2.12
Such a mapping Wn is nonexpansive from C to C and it is called the W-mapping generated
by T1 , T2 , . . . , Tn and μ1 , μ2 , . . . , μn .
For each n, k ∈ N, let the mapping Un,k be defined by 2.12. Then we can have the
following crucial conclusions concerning Wn . You can find them in 36. Now we only need
the following similar version in Hilbert spaces.
Lemma 2.2 see 36. Let C be a nonempty closed convex subset of a real Hilbert space H. Let
T1 , T2 , . . . be nonexpansive mappings of C into itself such that ∩∞
n1 FTn is nonempty, let μ1 , μ2 , . . .
be real numbers such that 0 ≤ μn ≤ b < 1 for every n ≥ 1. Then,
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1 Wn is nonexpansive and FWn ∩ni1 FTi , for all n ≥ 1;
2 for every x ∈ C and k ∈ N, the limit limn → ∞ Un,k x exists;
3 a mapping W : C → C defined by
Wx : lim Wn x lim Un,1 x,
n→∞
n→∞
∀x ∈ C
2.13
is a nonexpansive mapping satisfying FW ∩∞
i1 FTi and it is called the W-mapping
generated by T1 , T2 , . . . and μ1 , μ2 , . . . .
Lemma 2.3 see 37. Let C be a nonempty closed convex subset of a Hilbert space H, {Ti : C →
C} a countable family of nonexpansive mappings with ∩∞
/ ∅, {μi } a real sequence such that
i1 FTi 0 < μi ≤ b < 1, for all i ≥ 1. If D is any bounded subset of C, then
lim supWx − Wn x 0.
n → ∞ x∈D
2.14
Lemma 2.4 see 38. Each Hilbert space H satisfies Opial’s condition, that is, for any sequence
{xn } ⊂ H with xn
x, the inequality
lim inf xn − x < lim inf xn − y
n→∞
n→∞
2.15
holds for each y ∈ H with y /
x.
Lemma 2.5 see 39. Assume A is a strongly positive linear bounded operator on H with coefficient
γ > 0 and 0 < ρ ≤ A−1 . Then, I − ρA ≤ 1 − ργ.
For solving the system of mixed equilibrium problems 1.15, let us assume that
function Θk : H × H → R, k 1, 2, . . . , N satisfies the following conditions:
H1 Θk is monotone, that is, Θk x, y Θk y, x ≤ 0, for all x, y ∈ H;
H2 for each fixed y ∈ H, x → Θk x, y is convex and upper semicontinuous;
H3 for each x ∈ H, y → Θk x, y is convex.
Let η : H × H → H and B : H → H be two mappings. B is said to be
1 monotone if
Bx − By, η x, y ≥ 0,
∀x, y ∈ H;
2.16
2 d-strongly monotone if there exists a positive real number d such that
2
Bx − By, η x, y ≥ dx − y ,
∀x, y ∈ H;
2.17
3 L-Lipschitz continuous if there exists a constant L > 0 such that
η x, y ≤ Lx − y,
∀x, y ∈ H.
2.18
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Let K : H → R be a differentiable functional on H, which is called:
1 η-convex 40 if
K y − Kx ≥ K x, η y, x ,
∀x, y ∈ H,
2.19
where K x is the Fréchet derivative of K at x;
2 η-strongly convex 41 if there exists a constant σ > 0 such that
2
σ K y − Kx − K x, η y, x ≥ x − y ,
2
∀x, y ∈ H.
2.20
In particular, if ηx, y x − y for all x, y ∈ H, then K is said to be strongly convex.
Lemma 2.6 see 42. Let H be a real Hilbert space and let φ be a lower semicontinuous and convex
functional from H to R. Let Θ be a bifunction from H × H to R satisfying (H1)–(H3). Assume that
i η : H × H → H is λ-Lipschitz continuous with constant λ > 0 such that
a ηx, y ηy, x 0, for all x, y ∈ H,
b η·, · is affine in the first variable,
c for each fixed x ∈ H, y → ηx, y is sequentially continuous from the weak topology
to the weak topology;
ii K : H → R is η-strongly convex with constant σ > 0 and its derivative K is sequentially
continuous from the weak topology to the strong topology;
iii for each x ∈ H, there exist bounded subsets Ex ⊂ H and zx ∈ H such that for any
y ∈ H \ Ex ,
1 Θ y, zx φzx − φ y K y − K x, η zx , y < 0.
r
2.21
For given r > 0, let JrΘ : H → H be the mapping defined by
JrΘ x 1 y ∈ H : Θ y, z φz − φ y K y − K x, η z, y ≥ 0, ∀z ∈ H
r
2.22
for all x ∈ H. Then
1 JrΘ is single-valued.
2 FJrΘ MEPΘ, Œ, where MEPΘ, Œ is the set of solution of the mixed equilibrium
problem,
Θ x, y φ y − φx ≥ 0,
3 MEPΘ, Œ is closed and convex.
∀y ∈ H.
2.23
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Journal of Applied Mathematics
Lemma 2.7 see 43. Let {xn } and {vn } be bounded sequences in a Banach space X and let {βn } be
a sequence in 0, 1 with 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1. Suppose xn1 1 − βn vn βn xn
for all integers n ≥ 0 and lim supn → ∞ vn1 − vn − xn1 − xn ≤ 0. Then, limn → ∞ vn − xn 0.
Lemma 2.8 see 44. Assume {xn } is a sequence of nonnegative real numbers such that
xn1 ≤ 1 − an xn bn ,
∀n ≥ 0,
2.24
where {an } is a sequence in 0, 1 and {bn } is a sequence in R such that
1
∞
n1
an ∞,
2 lim supn → ∞ bn /an ≤ 0 or
∞
n1
|bn | < ∞.
Then, limn → ∞ xn 0.
Lemma 2.9 see 45. Let C be a nonempty closed convex subset of a real Hilbert space H and
g : C → R ∪ {∞} a proper lower-semicontinuous differentiable convex function. If z is a solution to
the minimization problem
2.25
gz inf gx,
x∈C
then
g x, x − z ≥ 0,
x ∈ C.
2.26
In particular, if z solves problem OP , then
u γf − I μA z, x − z ≤ 0.
2.27
Lemma 2.10 see 46. Let C be a nonempty bounded closed convex subset of a Hilbert space H and
let S {Ss : 0 ≤ s < ∞} be a nonexpansive semigroup on C, then for any h ≥ 0,
1
lim sup
t → ∞ x∈C t
t
0
T sx ds − T h
1
t
T sx ds 0.
0
t
2.28
Lemma 2.11 see 47. Let C be a nonempty bounded closed convex subset of H, {xn } a sequence
in C, and S {Ss : 0 ≤ s < ∞} a nonexpansive semigroup on C. If the following conditions are
satisfied:
i xn
z;
ii lim sups → ∞ lim supn → ∞ Ssxn − xn 0, then z ∈ S.
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11
Lemma 2.12 see 25. For given x∗ , y∗ ∈ C and x∗ , y∗ is a solution of the problem 1.20 if and
only if x∗ is a fixed point of the mapping G : C → C is defined by
Gx PC PC x − δDx − τBPC x − δDx,
∀x ∈ H,
2.29
where y∗ PC x − δDx, δ and τ are positive constants and B, D : H → H are two mappings.
Throughout this paper, the set of fixed points of the mapping G is denoted by
SVIC, B, D.
Lemma 2.13 see 32. Let G : C → C be defined in Lemma 2.12. If B : H → H is a
LB -Lipschitzian and relaxed c, d-cocoercive mapping and D : H → H is a LD -Lipschitz and
relaxed c , d -cocoercive mapping where τ ≤ 2d − cL2B /L2B and δ ≤ 2d − c L2D /L2D , then G is
nonexpansive.
Lemma 2.14 demiclosedness principle 48. Assume that S is a nonexpansive self-mapping of
a nonempty closed convex subset C of a real Hilbert space H. If S has a fixed point, then I − S is
demiclosed; that is, whenever {xn } is a sequence in C converging weakly to some x ∈ C (for short,
x ∈ C), and the sequence {I − Sxn } converges strongly to some y (for short, I − Sxn → y),
xn
it follows that I − Sx y. Here I is the identity operator of H.
3. Main Results
In this section, we prove a strong convergence theorem of an iterative algorithm 3.1 for
finding the solutions of a common element of the set of solutions of 1.15, the set of solutions
of 1.20 for Lipschitz continuous and relaxed cocoercive mappings, the set of common fixed
points for nonexpansive semigroups, and the set of common fixed points for an infinite family
of strictly pseudocontractive mappings in a real Hilbert space.
Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H which C C ⊂ C
and let f be a contraction of C into itself with α ∈ 0, 1. Let φ be a lower semicontinuous and convex
functional from H to R and let {Θk : H × H → R, k 1, 2, . . . , N} be a finite family of equilibrium
functions satisfying conditions (H1)–(H3). Let S {Ss : 0 ≤ s < ∞} be a nonexpansive semigroup
on C and let {tn } be a positive real divergent sequence. Let {Vi : C → C}∞
i1 be a countable family of
uniformly k-strict pseudo-contractions, let {Ti : C → C}∞
i1 be the countable family of nonexpansive
mappings defined by Ti x tx 1 − tVi x, for all x ∈ C, for all i ≥ 1, t ∈ k, 1, let Wn be the
W-mapping defined by 2.12, and let W be a mapping defined by 2.13 with FW /
∅. Let A be a
strongly positive linear bounded operator on H with coefficient γ > 0 and let 0 < γ < 1 μγ/α,
B : H → H be a LB -Lipschitz continuous and relaxed c, d-cocoercive mapping with d > cL2B ,
and let D : H → H be a LD -Lipschitz continuous and relaxed c , d -cocoercive mapping with
MEPΘk , φ.
d > c L2D . Suppose that Ω : FS ∩ FW ∩ F ∩ SVIC, B, D / ∅, where F ∩N
k1
Let μ > 0, γ > 0 and rk > 0, k 1, 2, . . . , N, which are constants. For given x1 ∈ H arbitrarily
k
and fixed u ∈ H, suppose {xn }, {yn }, {zn } and {un }, k 1, 2, . . . , N are the sequences generated
iteratively by
1
1
1
1
1
K un − K xn , η x, un
Θ1 un , x φx − φ un r1
≥ 0,
∀x ∈ H,
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Journal of Applied Mathematics
1
2
2
2
2
K un − K xn , η x, un
≥ 0, ∀x ∈ H,
Θ2 un , x φx − φ un r2
..
.
1
N
N
N
N
ΘN un , x φx − φ un
K un
− K xn , η x, un
≥ 0, ∀x ∈ H,
rN
N
N
zn PC un − δDun ,
yn PC zn − τBzn ,
xn1 αn u γfWn xn βn xn 1
1 − βn I − αn I μA
tn
tn
SsWn yn ds,
0
3.1
where
1
un JrΘ1 1 xn ,
k
k−1
un JrΘk k un
JrΘk k
k−2
k−1
JrΘk k JrΘk−1
un
· · · JrΘ2 2 JrΘ1 1 xn ,
1
JrΘk k · · · JrΘ2 2 un ,
3.2
k 2, 3, . . . , N,
JrΘk k : H → H, k 1, 2, . . . , N is the mapping defined by 2.22 and {αn } and {βn } are two
sequences in 0, 1 for all n ∈ N. Assume the following conditions are satisfied:
C1 η : H × H → H is λ-Lipschitz continuous with constant λ > 0 such that
a ηx, y ηy, x 0, for all x, y ∈ H,
b x → ηx, y is affine,
c for each fixed y ∈ H, y → ηx, y is sequentially continuous from the weak topology
to the weak topology;
C2 K : H → 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 a Lipschitz constant ν > 0 such that σ > λν;
C3 for each k ∈ {1, 2, . . . , N} and for all x ∈ H, there exist bounded subsets Ex ⊂ H and
zx ∈ H such that for any y ∈ H \ Ex ,
1 K y − K x, η zx , y < 0;
Θk y, zx φzx − φ y rk
C4 limn → ∞ αn 0 and
∞
n1
3.3
αn ∞;
C5 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1;
C6 0 < τ < 2d − cL2B /L2B and 0 < δ < 2d − c L2D /L2D .
Then, {xn } converges strongly to x∗ ∈ Ω, which solves the following optimization problem (OP):
min
∗
x ∈Ω
μ
1
Ax∗ , x∗ x∗ − u2 − hx∗ ,
2
2
3.4
Journal of Applied Mathematics
13
and x∗ , y∗ is a solution of the general system of variational inequality problem 1.20 such that
y∗ PC x∗ − δDx∗ .
Proof. By the condition C4 and C5, we may assume, without loss of generality, that αn ≤
1 − βn 1 μA−1 for all n ∈ N. Indeed, A is a strongly positive bounded linear operator on
H, we have
A sup{|Ax, x| : x ∈ H, x 1}.
3.5
Observe that
1 − βn I − αn I μA x, x 1 − βn − αn − αn μAx, x
≥ 1 − βn − αn − αn μA
3.6
≥ 0,
so this shows that 1 − βn I − αn I μA is positive. It follows that
1 − βn I − αn I μA sup 1 − βn I − αn I μA x, x : x ∈ H, x 1
sup 1 − βn − αn − αn μAx, x : x ∈ H, x 1
3.7
≤ 1 − βn − αn − αn μγ.
We shall divide the proofs into several steps.
Step 1. We show that {xn } is bounded.
MEPΘk , φ ∩ SVIC, B, D. In fact, by the
Let x∗ ∈ Ω : FS ∩ FW ∩ ∩N
k1
assumption that for each k ∈ {1, 2, . . . , N}, JrΘk k is nonexpansive. Let AN : JrΘNN · · · JrΘ2 2 JrΘ1 1
N
and A0 I. Then, we have x∗ AN x∗ and un AN xn . Since x∗ ∈ SVIC, B, D, then
x∗ PC PC x∗ − δDx∗ − τBPC x∗ − δDx∗ PC PC I − δDAN x∗ − τBPC I − δDAN x∗ .
3.8
Putting y∗ PC x∗ − δDx∗ PC I − δDAN x∗ , we have x∗ PC y∗ − τBy∗ . Since x∗ Ssx∗ , for all s ≥ 0 and x∗ Wn x∗ , for all n ≥ 1, therefore, we have
x∗ AN x∗ PC y∗ − τBy∗ Wn PC y∗ − τBy∗ SsWn PC y∗ − τBy∗ .
Because PC and AN are nonexpansive mappings and from Remark 1.1, we have
yn − x∗ PC zn − τBzn − PC y∗ − τBy∗ ≤ I − τBzn − I − τBy∗ ≤ zn − y∗ N
N
− PC x∗ − δDx∗ PC un − δDun
3.9
14
Journal of Applied Mathematics
N
≤ I − δDun − I − δDx∗ N
≤ un − x∗ AN xn − AN x∗ ≤ xn − x∗ 3.10
which yields that
xn1 − x αn u αn γfWn xn − I μA x∗ βn xn − x∗ 1 tn
∗ 1 − βn I − αn I μA
SsWn yn ds − x tn 0
∗
≤ αn u αn γfWn xn − I μA x βn xn − x∗ 1 − βn − αn 1 μγ xn − x∗ ≤ αn u αn γfWn xn − γfx∗ αn γfx∗ − I μA x∗ βn xn − x∗ 1 − βn − αn 1 μγ xn − x∗ ≤ αn u αn γαxn − x∗ αn γfx∗ − I μA x∗ βn xn − x∗ 1 − βn − αn 1 μγ xn − x∗ αn u γfx∗ − I μA x∗ 1 − αn 1 μγ αn γα xn − x∗ 1 − αn 1 μγ − γα xn − x∗ u γfx∗ − I μA x∗ .
αn 1 μγ − γα
1 μγ − γα
3.11
∗
It follows from 3.11 and induction that
u γfx∗ − I μA x∗ ,
xn − x ≤ max x1 − p ,
1 μγ − γα
∗
n ≥ 1.
k
3.12
Hence, {xn } is bounded, so are {yn }, {zn }, {Wn xn }, {fWn xn }, {un } for all k 1, 2, . . . , N
t
and {Kn Wn yn }, where Kn 1/tn 0n Ssds.
N
N
Step 2. We prove that limn → ∞ xn1 − xn 0 and limn → ∞ un1 − un 0.
Again, from Remark 1.1, we have the following estimates:
yn1 − yn PC zn1 − τBzn1 − PC zn − τBzn ≤ zn1 − τBzn1 − zn − τBzn ≤ zn1 − zn N
N
N
N PC un1 − δDun1 − PC un − δDun Journal of Applied Mathematics
N
N
N
N ≤ un1 − δDun1 − un − δDun N
N ≤ un1 − un AN xn1 − AN xn 15
≤ xn1 − xn .
3.13
On the other hand, since Ti and Un,i are nonexpansive, we have
Wn1 yn − Wn yn μ1 T1 Un1,2 yn − μ1 T1 Un,2 yn ≤ μ1 Un1,2 yn − Un,2 yn μ1 μ2 T2 Un1,3 yn − μ2 T2 Un,3 yn ≤ μ1 μ2 Un1,3 yn − Un,3 yn 3.14
..
.
≤ μ1 μ2 · · · μn Un1,n1 yn − Un,n1 yn n
≤ M1
μi ,
i1
where M1 ≥ 0 is a constant such that Un1,n1 yn − Un,n1 yn ≤ M1 for all n ≥ 0. It follows
from 3.13 and 3.14 that we have
Wn1 yn1 − Wn yn ≤ Wn1 yn1 − Wn1 yn Wn1 yn − Wn yn n
≤ yn1 − yn M1
μi
i1
3.15
n
≤ xn1 − xn M1
μi .
i1
It follows that
1
Kn1 Wn1 yn1 − Kn Wn yn tn1
≤
1
tn1
0
1
SsWn1 yn1 ds −
tn
tn
0
SsWn yn ds
tn1 tn1 0
1
tn1
SsWn1 yn1 − SsWn yn ds
1 tn
SsWn yn ds −
SsWn yn ds
t
n
0
0
tn1
1
1
≤ Wn1 yn1 −Wn yn SsWn yn ds
tn1 tn
tn1
tn1
1
≤ Wn1 yn1 − Wn yn tn1
tn1 SsWn yn ds
tn
tn
0
1
SsWn yn ds−
tn
tn
0
SsWn yn ds
16
Journal of Applied Mathematics
1
1 tn − SsWn yn ds
tn1 tn 0
tn
≤ Wn1 yn1 − Wn yn 2 1 −
M2
tn1
n
tn
≤ xn1 − xn M1
μi 2 1 −
M2 ,
tn1
i1
3.16
where M2 max{SsWn yn }.
Setting xn1 1 − βn vn βn xn , for all n ≥ 1, we have
xn1 − βn xn αn u γfWn xn 1 − βn I − αn I μA Kn Wn yn
.
vn 1 − βn
1 − βn
3.17
Then, we obtain
αn1 u γfWn1 xn1 1 − βn1 I − αn1 I μA Kn1 Wn1 yn1
vn1 − vn 1 − βn1
αn u γfWn xn 1 − βn I − αn I μA Kn Wn yn
−
1 − βn
αn1 αn u γfWn1 xn1 −
u γfWn xn Kn1 Wn1 yn1 − Kn Wn yn
1 − βn1
1 − βn
αn αn1 I μA Kn Wn yn −
I μA Kn1 Wn1 yn1
1 − βn
1 − βn1
αn1 u γfWn1 xn1 − I μA Kn1 Wn1 yn1
1 − βn1
αn I μA Kn Wn yn − u − γfWn xn Kn1 Wn1 yn1 − Kn Wn yn .
1 − βn
3.18
It follows from 3.16 and 3.18 that
vn1 − vn − xn1 − xn ≤
αn1 u γfWn1 xn1 I μA Kn1 Wn1 yn1 1 − βn1
αn I μA Kn Wn yn u γfWn xn 1 − βn
n
tn
M1
μi 2 1 −
M2 .
tn1
i1
3.19
By the conditions C4, C5 and from tn ∈ 0, ∞, tn → ∞ and 0 < μi ≤ b < 1, for all i ≥ 1,
we have
lim supvn1 − vn − xn1 − xn ≤ 0.
n→∞
3.20
Journal of Applied Mathematics
17
Hence, by Lemma 2.7, we obtain
lim vn − xn 0.
3.21
lim xn1 − xn lim 1 − βn vn − xn 0.
3.22
n→∞
It follows that
n→∞
n→∞
Applying 3.22 into 3.13, we obtain that
N
N lim yn1 − yn lim zn1 − zn lim un1 − un 0.
n→∞
n→∞
n→∞
3.23
Step 3. We show that limn → ∞ Kn Wn yn − yn 0, limn → ∞ yn − Ssyn 0, and
t
k1
k
limn → ∞ un
− un 0, where Kn 1/tn 0n Ssds.
Since xn1 αn u γfWn xn βn xn 1 − βn I − αn I μAKn Wn yn , we have
xn − Kn Wn yn ≤ xn − xn1 xn1 − Kn Wn yn xn − xn1 αn u γfWn xn βn xn 1 − βn I −αn I μA Kn Wn yn −Kn Wn yn xn − xn1 αn u γfWn xn − I μA Kn Wn yn βn xn − Kn Wn yn ≤ xn − xn1 αn u γfWn xn I μA Kn Wn yn βn xn − Kn Wn yn ,
3.24
that is
xn − Kn Wn yn ≤
1
αn xn − xn1 u γfWn xn I μA Kn Wn yn .
1 − βn
1 − βn
3.25
By C4, C5, and 3.22 it follows that
lim Kn Wn yn − xn 0.
n→∞
3.26
18
Journal of Applied Mathematics
N
Since JrΘNN : C → C is firmly nonexpansive, un
and x ∈ Ω, we have
∗
AN xn , where AN : JrΘNN · · · JrΘ2 2 JrΘ1 1
2 2
N
un − x∗ AN xn − AN x∗ ≤ AN xn − AN x∗ , xn − x∗
N
un − x∗ , xn − x∗
2
1 N
N 2
2
un − x∗ xn − x∗ − xn − un ,
2
3.27
and hence
2
N
N 2
2
un − x∗ ≤ xn − x∗ − xn − un .
3.28
Observe that
xn1 − x∗ 2 1 − βn I − αn I μA Kn Wn yn − x∗ βn xn − x∗ 2
αn u γfWn xn − I μA x∗ 2
1 − βn I − αn I μA Kn Wn yn − x∗ βn xn − x∗ 2
α2n u γfWn xn − I μA x∗ 2βn αn xn − x∗ , u γfWn xn − I μA x∗
2αn 1 − βn I − αn I μA Kn Wn yn − x∗ , u γfWn xn − I μA x∗
2
≤ 1 − βn − αn − αn μγ Kn Wn yn − x∗ βn xn − x∗ 2
α2n u γfWn xn − I μA x∗ 2βn αn xn − x∗ , u γfWn xn − I μA x∗
2αn 1 − βn I − αn I μA Kn Wn yn − x∗ , u γfWn xn − I μA x∗
2
1 − βn − αn − αn μγ Kn Wn yn − x∗ βn xn − x∗ cn
2
2 ≤ 1 − βn − αn − αn μγ Kn Wn yn − x∗ βn2 xn − x∗ 2
2 1 − βn − αn − αn μγ βn Kn Wn yn − x∗ xn − x∗ cn
2
2 ≤ 1 − βn − αn − αn μγ Kn Wn yn − x∗ βn2 xn − x∗ 2
2
1 − βn − αn − αn μγ βn Kn Wn yn − x∗ xn − x∗ 2 cn
1 − αn − αn μγ
2
2
− 2 1 − αn − αn μγ βn βn2 Kn Wn yn − x∗ βn2 xn − x∗ 2
2
1 − αn − αn μγ βn − βn2 Kn Wn yn − x∗ xn − x∗ 2 cn
Journal of Applied Mathematics
2
2 1 − αn − αn μγ − 1 − αn − αn μγ βn Kn Wn yn − x∗ 19
1 − αn − αn μγ βn xn − x∗ 2 cn
2
≤ 1 − αn − αn μγ 1 − βn − αn − αn μγ yn − x∗ 1 − αn − αn μγ βn xn − x∗ 2 cn ,
3.29
where
2
cn α2n u γfWn xn − I μAx∗ 2βn αn xn − x∗ , u γfWn xn − I μA x∗
2αn 1 − βn I − αn I μA Kn Wn yn − x∗ , u γfWn xn − I μA x∗ .
3.30
It follows from condition C4 that
lim cn 0.
n→∞
3.31
Putting 3.28 into 3.29 and using also 3.10, we have
2
xn1 − x∗ 2 ≤ 1 − αn − αn μγ 1 − βn − αn − αn μγ yn − x∗ 1 − αn − αn μγ βn xn − x∗ 2 cn
2
N
≤ 1 − αn − αn μγ 1 − βn − αn − αn μγ un − x∗ 1 − αn − αn μγ βn xn − x∗ 2 cn
N 2
∗ 2
≤ 1 − αn − αn μγ 1 − βn − αn − αn μγ xn − x − xn − un 1 − αn − αn μγ βn xn − x∗ 2 cn
2
1 − αn − αn μγ xn − x∗ 2
N 2
− 1 − αn − αn μγ 1 − βn − αn − αn μγ xn − un cn
N 2
≤ xn − x∗ 2 − 1 − αn − αn μγ 1 − βn − αn − αn μγ xn − un cn .
3.32
It follows that
1 − αn − αn μγ
N 2
1 − βn − αn − αn μγ xn − un ≤ xn − x∗ 2 − xn1 − x∗ 2 cn
≤ xn − xn1 xn − x∗ xn1 − x∗ cn .
3.33
20
Journal of Applied Mathematics
Therefore, by 3.22 and 3.31, we get
N lim xn − un 0.
3.34
N
N
un − Kn Wn yn ≤ un − xn xn − Kn Wn yn ,
3.35
n→∞
Since
and by 3.26 and 3.70, we have
N
lim un − Kn Wn yn 0.
3.36
n→∞
Since B is a LB -Lipschitz continuous and relaxed c, d-cocoercive mapping on B and 0 < τ <
2d − cL2B /L2B for any x∗ ∈ Ω, we have
yn − x∗ 2 PC zn − τBzn − PC y∗ − τBy∗ 2
2
≤ zn − y∗ − τ Bzn − By∗ 2
2
zn − y∗ − 2τ zn − y∗ , Bzn − By∗ τ 2 Bzn − By∗ 2
2 2
≤ xn − x∗ 2 − 2τ −cBzn − By∗ dzn − y∗ τ 2 Bzn − By∗ 2
2
2
xn − x∗ 2 2τcBzn − By∗ − 2τdzn − y∗ τ 2 Bzn − By∗ 2
2 2τd 2
≤ xn − x∗ 2 2τcBzn − By∗ − 2 Bzn − By∗ τ 2 Byn − Bp
L
B
2
2τd xn − x∗ 2 2τc τ 2 − 2 Bzn − By∗ .
LB
3.37
Similarly, since D is a LD -Lipschitz continuous and relaxed c , d -cocoercive mapping on D
and 0 < δ < 2d − c L2D /L2D , we also have
zn − y∗ 2 ≤ xn − x∗ 2 2δd
2δc δ − 2
LD
2
2
N
Dun − Dx∗ .
3.38
Journal of Applied Mathematics
21
Substituting 3.37 into 3.29, we have
xn1 − x∗ 2 ≤ 1 − αn − αn μγ 1 − βn − αn − αn μγ
2τd ∗ 2
2
∗ 2
× xn − x 2τc τ − 2
Bzn − By
LB
1 − αn − αn μγ βn xn − x∗ 2 cn
2
1 − αn − αn μγ xn − x∗ 2
2
2τd 2
1 − αn − αn μγ 1 − βn − αn − αn μγ 2τc τ − 2 Bzn − By∗ cn
LB
2
2τd ≤ xn − x∗ 2 2τc τ 2 − 2 Bzn − By∗ cn .
LB
3.39
Again, substituting 3.38 into 3.29 and using also 3.10, we get
2
xn1 − x∗ 2 ≤ 1 − αn − αn μγ 1 − βn − αn − αn μγ yn − x∗ 1 − αn − αn μγ βn xn − x∗ 2 cn
2
≤ 1 − αn − αn μγ 1 − βn − αn − αn μγ zn − y∗ 1 − αn − αn μγ βn xn − x∗ 2 cn
≤ 1 − αn − αn μγ 1 − βn − αn − αn μγ
2
2δd N
∗ 2
2
∗
× xn − x 2δc δ − 2
Dun − Dx LD
1 − αn − αn μγ βn xn − x∗ 2 cn
2
1 − αn − αn μγ xn − x∗ 2 1 − αn − αn μγ 1 − βn − αn − αn μγ
2
2δd N
2
× 2δc δ − 2
Dun − Dx∗ cn
LD
2
2δd N
∗ 2
2
≤ xn − x 2δc δ − 2
Dun − Dx∗ cn .
LD
3.40
Therefore, by 3.39 and 3.40, we have
2
2τd
2 Bzn − By∗ ≤ xn − x∗ 2 − xn1 − x∗ 2 cn
− 2τc − τ
2
LB
≤ xn − xn1 xn − x∗ xn1 − x∗ cn ,
2
2δd
N
2
2
2 ∗
−
2δc
−
δ
−
Dx
Du
≤ xn − x∗ − xn1 − x∗ cn
n
L2D
≤ xn − xn1 xn − x∗ xn1 − x∗ cn .
3.41
22
Journal of Applied Mathematics
It follows from 3.22 and 3.31 that we obtain
lim Bzn − By∗ 0,
3.42
N
lim Dun − Dx∗ 0.
3.43
n→∞
n→∞
From 2.6, we have
2
N
N
zn − y∗ 2 − PC x∗ − δDx∗ PC un − δDun
≤
N
un
N
− δDun
− x∗ − δDx∗ , zn − y∗
2 2
1 N
N
− x∗ − δDx∗ zn − y∗ un − δDun
2
2
N
N
− x∗ − δDx∗ − zn − y∗ − un − δDun
≤
2 2 2 1 N
N
N
un − x∗ zn − y∗ − un − zn − x∗ − y∗ − δ Dun − Dx∗ 2
2 2 1 2
N
N
un − x∗ zn − y∗ − un − zn − x∗ − y∗ 2
2δ
N
un
2 N
N
− zn − x∗ − y∗ , Dun − Dx∗ − δ2 Dun − Dx∗ 2 1
2
N
≤
xn − x∗ 2 zn − y∗ − un − zn − x∗ − y∗ 2
2 N
N
N
∗
∗ ∗
2
∗
2δ un − zn − x − y Dun − Dx − δ Dun − Dx .
3.44
So, we obtain
2
N
zn − y∗ 2 ≤ xn − x∗ 2 − un − zn − x∗ − y∗ 2
N
N
N
2δ un − zn − x∗ − y∗ Dun − Dx∗ − δ2 Dun − Dx∗ .
3.45
Journal of Applied Mathematics
23
By 3.29, we get
2
xn1 − x∗ 2 ≤ 1 − αn − αn μγ 1 − βn − αn − αn μγ yn − x∗ 1 − αn − αn μγ βn xn − x∗ 2 cn
2
≤ 1 − αn − αn μγ 1 − βn − αn − αn μγ zn − y∗ 1 − αn − αn μγ βn xn − x∗ 2 cn
≤ 1 − αn − αn μγ 1 − βn − αn − αn μγ
2
N
× xn − x∗ 2 − un − zn − x∗ − y∗ 2 N
N
N
2δ un − zn − x∗ − y∗ Dun − Dx∗ − δ2 Dun − Dx∗ 1 − αn − αn μγ βn xn − x∗ 2 cn
2
1 − αn − αn μγ xn − x∗ 2 1 − αn − αn μγ 1 − βn − αn − αn μγ
N
2
× − un − zn − x∗ − y∗ 2 N
N
N
2δ un − zn − x∗ − y∗ Dun − Dx∗ − δ2 Dun − Dx∗ cn
N
2
≤ xn − x∗ 2 − 1 − αn − αn μγ 1 − βn − αn − αn μγ un − zn − x∗ − y∗ N
N
2δ un − zn − x∗ − y∗ Dun − Dx∗ 2
N
− δ2 1 − αn − αn μγ 1 − βn − αn − αn μγ Dun − Dx∗ cn
3.46
which implies that
1 − αn − αn μγ
N
2
1 − βn − αn − αn μγ un − zn − x∗ − y∗ ≤ xn − x∗ 2 − xn1 − x∗ 2
N
N
2δ un − zn − x∗ − y∗ Dun − Dx∗ 2
N
− δ2 1 − αn − αn μγ 1 − βn − αn − αn μγ Dun − Dx∗ cn
≤ xn − xn1 xn − x∗ xn1 − x∗ N
N
2δ un − zn − x∗ − y∗ Dun − Dx∗ 2
N
− δ2 1 − αn − αn μγ 1 − βn − αn − αn μγ Dun − Dx∗ cn .
3.47
24
Journal of Applied Mathematics
From 3.22, 3.31, and 3.43, we have
N
lim un − zn − x∗ − y∗ 0.
n→∞
3.48
Now, from 2.2 and 2.7, we observe that
zn − yn x∗ − y∗ 2 zn − τBzn − y∗ − τBy∗
2
− PC zn − τBzn − PC y∗ − τBy∗ τ Bzn − By∗ 2
≤ zn − τBzn − y∗ − τBy∗ − PC zn − τBzn − PC y∗ − τBy∗ 2τ Bzn − By∗ , zn − yn x∗ − y∗
2 2
≤ zn − τBzn − y∗ − τBy∗ − PC zn − τBzn −PC y∗ − τBy∗ 2τ Bzn − By∗ zn − yn x∗ − y∗ 2
≤ zn − τBzn − y∗ − τBy∗ 2
− Kn Wn PC zn − τBzn − Kn Wn PC y∗ − τBy∗ 2τ Bzn − By∗ zn − yn x∗ − y∗ 2
2 zn − τBzn − y∗ − τBy∗ − Kn Wn yn − Kn Wn x∗ 2τ Bzn − By∗ zn − yn x∗ − y∗ zn − τBzn − y∗ − τBy∗ − Kn Wn yn − x∗ × zn − τBzn − y∗ − τBy∗ Kn Wn yn − x∗ 2τ Bzn − By∗ zn − yn x∗ − y∗ ≤ zn − τBzn − y∗ − τBy∗ − Kn Wn yn − x∗ × zn − τBzn − y∗ − τBy∗ Kn Wn yn − x∗ 2τ Bzn − By∗ zn − yn x∗ − y∗ N
N
un − Kn Wn yn x∗ − y∗ − un − zn − τ Bzn − By∗ × zn − τBzn − y∗ − τBy∗ Kn Wn yn − x∗ 2τ Bzn − By∗ zn − yn x∗ − y∗ .
3.49
It follows from 3.36, 3.42, and 3.48 that we have
lim zn − yn x∗ − y∗ 0,
n→∞
3.50
since
N
N Kn Wn yn − yn ≤ Kn Wn yn − un un − zn − x∗ − y∗ zn − yn x∗ − y∗ .
3.51
Journal of Applied Mathematics
25
It follows from 3.36, 3.48 and 3.50, we get
lim Kn Wn yn − yn 0,
3.52
n→∞
and from 3.26, and 3.52 that we have
lim xn − yn 0.
3.53
n→∞
Since {Wn yn } is a bounded sequence in C, from Lemma 2.10 for all s ≥ 0, we have
1
lim Kn Wn yn − SsKn Wn yn lim n→∞
n → ∞ tn
tn
SsWn yn ds − Ss
0
1
tn
tn
0
SsWn yn ds 0,
3.54
and since
yn − Ssyn ≤ yn − Kn Wn yn Kn Wn yn − SsKn Wn yn SsKn Wn yn − Ss yn ≤ 2yn − Kn Wn yn Kn Wn yn − SsKn Wn yn ,
3.55
it follows from 3.52 and 3.54 that we get
lim yn − Ssyn 0.
n→∞
3.56
On the other hand, since JrΘk k : H → H is firmly nonexpansive, Ak : JrΘk k · · ·
JrΘ2 2 JrΘ1 1 , k 1, 2, . . . , N and x∗ ∈ Ω, we have
2 2
k1
Θk1 k
k1 ∗ A xn − JrΘk1
x A xn − x∗ Jrk1
k1
≤ JrΘk1
Ak xn − x∗ , Ak xn − x∗
3.57
2
2
2
1 Θk1 k
Θk1 k
A xn − Ak xn ,
Jrk1 A xn − x∗ Ak xn − x∗ − Jrk1
2
and hence
2
2
k1
2
A xn − x∗ ≤ xn − x∗ − Ak1 xn − Ak xn .
3.58
26
Journal of Applied Mathematics
From 3.10, 3.29, and 3.58, for each k 1, 2, . . . , N − 1, we have
2
xn1 − x∗ 2 ≤ 1 − αn − αn μγ 1 − βn − αn − αn μγ yn − x∗ 1 − αn − αn μγ βn xn − x∗ 2 cn
2
≤ 1 − αn − αn μγ 1 − βn − αn − αn μγ Ak xn − x∗ 1 − αn − αn μγ βn xn − x∗ 2 cn
2
≤ 1 − αn − αn μγ 1 − βn − αn − αn μγ Ak1 xn − x∗ 1 − αn − αn μγ βn xn − x∗ 2 cn
2 k1
∗ 2
k
≤ 1 − αn − αn μγ 1 − βn − αn − αn μγ xn − x − A xn − A xn 1 − αn − αn μγ βn xn − x∗ 2 cn
2
1 − αn − αn μγ xn − x∗ 2
2
− 1 − αn − αn μγ 1 − βn − αn − αn μγ Ak1 xn − Ak xn cn
≤ xn − x∗ 2
2
− 1 − αn − αn μγ 1 − βn − αn − αn μγ Ak1 xn − Ak xn cn .
3.59
It follows that
1 − αn − αn μγ
2
1 − βn − αn − αn μγ Ak1 xn − Ak xn ≤ xn − x∗ 2 − xn1 − x∗ 2 cn
3.60
≤ xn − xn1 xn − x∗ xn1 − x∗ cn .
Therefore, by 3.22 and 3.31, we get
k1
k lim Ak1 xn − Ak xn 0 that is lim un
− un 0.
n→∞
n→∞
3.61
Step 4. We prove that
lim sup u γf − I μA x∗ , xn − x∗ ≤ 0,
n→∞
3.62
where x∗ is a solution of the optimization problem:
min
x∈Ω
μ
1
Ax∗ , x∗ x∗ − u2 − hx∗ .
2
2
3.63
Journal of Applied Mathematics
27
To show this inequality, we can choose a subsequence {yni } of {yn } such that
lim u γf − I μA x∗ , yni − x∗ lim sup u γf − I μA x∗ , yn − x∗ .
i→∞
n→∞
3.64
Since {yni } is bounded, there exists a subsequence {ynij } of {yni } which converges
z. From 3.53, we get
weakly to z ∈ C. Without loss of generality, we can assume that yni
z.
xni
Next, we show that z ∈ Ω : FS ∩ FW ∩ F ∩ SVIC, B, D, where F MEPΘ
∩N
k , φ.
k1
1 First, we prove that z ∈ FS. Indeed, from Lemma 2.11 and 3.56, we get z ∈
FS, that is, z Ssz, for all s ≥ 0.
n
2 Next, we show that z ∈ FW ∩∞
n1 FWn , where FWn ∩i1 FTi , for all n ≥ 1
/ FW, then there exists a positive integer m such
and FWn1 ⊂ FWn . Assume that z ∈
/ ∩m
FT
.
Hence
for any n ≥ m, z ∈
/ ∩ni1 FTi FWn , that is,
that z ∈
/ FTm and so z ∈
i
i1
SsWn z, for all s ≥ 0;
z/
Wn z. This together with z Ssz, for all s ≥ 0, shows z Ssz /
therefore, we have z /
Kn Wn z, for all n ≥ m. It follows from the Opial’s condition and 3.52
that
lim infyni − z < lim infyni − Kni Wni z
i→∞
i→∞
≤ lim inf yni − Kni Wni yni Kni Wni yni − Kni Wni z
i→∞
≤ lim infyni − z,
3.65
i→∞
which is a contradiction. Thus, we get z ∈ FW.
k1
k1
3 Now, we prove that z ∈ F. Since Ak1 JrΘk1
Ak , k 1, 2, . . . , N − 1, and un
Ak1 xn , we have
1
K Ak1 xn − K Ak xn , η x, Ak1 xn
≥ 0,
Θ Ak1 xn , x φx − φ Ak1 xn rk1
∀x ∈ H.
3.66
It follows that
1
rk1
K Ak1 xni − K Ak xni , η x, Ak1 xni
≥ −Θ Ak1 xni , x − φx φ Ak1 xni
3.67
for all x ∈ H. From 3.61 and by conditions C1c and C2, we get
lim
1
ni → ∞ rk1
K Ak1 xni − K Ak xni , η x, Ak1 xni
0.
3.68
By the assumption that φ is lower semicontinuous, then it is weakly lower semicontinuous
and by the condition H2 that x → −Θi x, y is lower semicontinuous, then it is weakly
28
Journal of Applied Mathematics
k
lower semicontinuous. Since yni
z, it follows from 3.36, 3.52, and 3.61 that uni
for each k 1, 2, . . . , N − 1. Taking the lower limit ni → ∞ in 3.67, we have
Θk1 z, x φx − φz ≥ 0,
∀x ∈ H, ∀k 0, 1, 2, . . . , N − 1.
z
3.69
MEPΘk , φ.
Therefore, z ∈ ∩N
k1
4 Next, we show that z ∈ SVIC, B, D. By 3.36 and 3.52, we have
N
N
un − yn ≤ un − Kn Wn yn Kn Wn yn − yn −→ 0 as n −→ ∞.
3.70
By Lemma 2.13 that G is a nonexpansive, we obtain
N
N
N
N
yn − G yn − τBPC un − δDun
− G yn PC PC un − δDun
N
G un
− G yn N
≤ un − yn .
3.71
Thus,
lim yn − G yn 0.
n→∞
3.72
By Lemma 2.14, we obtain that z ∈ SVIC, B, D. Hence z ∈ Ω is proved.
Now, from Lemma 2.9, 3.64, and 3.53, we have
lim sup u γf − I μA x∗ , xn − x∗ lim sup u γf − I μA x∗ , yn − x∗
n→∞
n→∞
lim u γf − I μA x∗ , yni − x∗
i→∞
u γf − I μA x∗ , z − x∗ ≤ 0.
3.73
By 3.52, 3.53, and 3.73, we obtain
lim sup u γf − I μA x∗ , Kn Wn yn − x∗ ≤ 0.
n→∞
3.74
Journal of Applied Mathematics
29
Step 5. Finally, we show that xn → x∗ . From 3.1, we obtain
xn1 − x∗ 2
2
αn u γfWn xn βn xn 1 − βn I − αn I μA Kn Wn yn − x∗ 1 − βn I − αn I μA Kn Wn yn − x∗ βn xn − x∗ 2
αn u γfWn xn − I μA x∗ 2
1 − βn I − αn I μA Kn Wn yn − x∗ βn xn − x∗ 2αn 1 − βn I − αn I μA Kn Wn yn − x∗ , u γfWn xn − I μA x∗
2αn βn xn − x∗ , u γfWn xn − I μA x∗ α2n u γfWn xn − I μA x∗ 2
2
≤ 1 − βn − αn 1 μγ Kn Wn yn − x∗ βn xn − x∗ 2αn 1 − βn γ Kn Wn yn − x∗ , fWn xn − fx∗ 2αn 1 − βn Kn Wn yn − x∗ , u γfx∗ − I μA x∗
− 2α2n γ I μA Kn Wn yn − x∗ , fWn xn − fx∗ − 2α2n I μA Kn Wn yn − x∗ , u γfx∗ − I μA x∗
2αn βn γ xn − x∗ , fWn xn − fx∗ 2αn βn xn − x∗ , u γfx∗ − I μA x∗
2
α2n u γfWn xn − I μA x∗ ≤
2
1 − βn − αn 1 μγ Kn Wn yn − x∗ βn xn − x∗ 2αn 1 − βn γ Kn Wn yn − x∗ fWn xn − fx∗ 2αn 1 − βn Kn Wn yn − x∗ , u γfx∗ − I μA x∗
− 2α2n γ I μA Kn Wn yn − x∗ fWn xn − fx∗ − 2α2n I μA Kn Wn yn − x∗ u γfx∗ − I μA x∗ 2αn βn γxn − x∗ fWn xn − fx∗ 2αn βn xn − x∗ , u γfx∗ − I μA x∗
2
α2n u γfWn xn − I μA x∗ ≤ 1 − βn − αn 1 μγxn − x∗ βn xn − x∗ 2αn 1 − βn γαxn − x∗ 2
2αn 1 − βn Kn Wn yn − x∗ , u γfx∗ − I μA x∗
− 2α2n γα I μA Kn Wn yn − x∗ xn − x∗ − 2α2n I μA Kn Wn yn − x∗ u γfx∗ − I μA x∗ 2αn βn γαxn − x∗ 2 2αn βn xn − x∗ , u γfx∗ − I μA x∗
2
30
Journal of Applied Mathematics
2
α2n u γfWn xn − I μA x∗ 1 − αn 1 μγ
2αn γα xn − x∗ 2
αn 2 1 − βn Kn Wn yn − x∗ , u γfx∗ − I μA x∗
− 2αn γα I μA Kn Wn yn − x∗ xn − x∗ − 2αn I μA Kn Wn yn − x∗ u γfx∗ − I μA x∗ 2βn xn − x∗ , u γfx∗ − I μA x∗
2 αn u γfWn xn − I μA x∗ 2
1 − 2αn 1 μγ α2n 1 μγ 2αn γα xn − x∗ 2
αn 2 1 − βn Kn Wn yn − x∗ , u γfx∗ − I μA x∗
− 2αn γα I μA Kn Wn yn − x∗ xn − x∗ − 2αn I μA Kn Wn yn − x∗ u γfx∗ − I μA x∗ 2βn xn − x∗ , u γfx∗ − I μA x∗
2 αn u γfWn xn − I μA x∗ .
1 − 2αn 1 μγ − γα xn − x∗ 2
αn 2 1 − βn Kn Wn yn − x∗ , u γfx∗ − I μA x∗
2βn xn − x∗ , u γfx∗ − I μA x∗
2
αn 1 μγ xn − x∗ 2
− 2γα I μA Kn Wn yn − x∗ xn − x∗ − 2 I μA Kn Wn yn − x∗ u γfx∗ − I μA x∗ 2 u γfWn xn − I μAx∗ .
3.75
Since {xn }, {fWn xn }, and {Kn Wn yn } are bounded, there exist M > 0 such that
2
1 μγ xn − x∗ 2
− 2γα I μA Kn Wn yn − x∗ xn − x∗ − 2 I μA Kn Wn yn − x∗ u γfx∗ − I μA x∗ 2
u γfWn xn − I μA x∗ ≤ M
3.76
Journal of Applied Mathematics
31
for all n ≥ 0. It follows that
xn1 − x∗ 2 ≤ 1 − αn an xn − x∗ 2 αn bn ,
3.77
an 2 1 μγ − γα ,
bn 2 1 − βn Kn Wn yn − x∗ , u γfx∗ − I μA x∗
2βn xn − x∗ , u γfx∗ − I μA x∗ αn M.
3.78
where
Applying Lemma 2.8 to 3.77, we conclude that xn → x∗ . This completes the proof.
Remark 3.2. For example, of the control conditions C4–C6, we set αn 1/10n, βn n/n 1. We set B, D is a 1-Lipschitz continuous and relaxed 0, 1-cocoercive mapping, i.e., LB 1 LD and c 0 c , d 1 d .
Then, we can choose τ ∈ 0, 2 and δ ∈ 0, 2 which satisfies the condition C6 in
Theorem 3.1.
Corollary 3.3. Let C be a nonempty closed convex subset of a real Hilbert space H which C C ⊂ C
and let f be a contraction of C into itself with α ∈ 0, 1. Let φ be a lower semicontinuous and convex
functional from H to R and let Θ : H × H → R be a finite family of equilibrium functions satisfying
conditions (H1)–(H3). Let S {Ss : 0 ≤ s < ∞} be a nonexpansive semigroup on C and let {tn } be
a positive real divergent sequence. Let {Vi : C → C}∞
i1 be a countable family of uniformly k- strict
be
the
countable
family of nonexpansive mappings defined
pseudo-contractions, let {Ti : C → C}∞
i1
by Ti x tx 1 − tVi x, for all x ∈ C, for all i ≥ 1, t ∈ k, 1, let Wn be the W-mapping defined
by 2.12, and let W be a mapping defined by 2.13 with FW / ∅. Let A be a strongly positive
linear bounded operator on H with coefficient γ > 0 and let 0 < γ < 1 μγ/α, B : H → H be a
LB -Lipschitz continuous and relaxed c, d-cocoercive mapping with d > cL2B , and let D : H → H
be a LD -Lipschitz continuous and relaxed c , d -cocoercive mapping with d > c L2D . Suppose that
Ω : FS∩FW∩MEPΘ, Œ∩SVIC, B, D /
∅. Let μ > 0, γ > 0 and r > 0, which are constants.
For given x1 ∈ H arbitrarily and fixed u ∈ H, suppose {xn }, {yn },{zn }, and{un } are the sequences
generated iteratively by
1
K un − K xn , ηx, un ≥ 0,
r
zn PC un − δDun ,
Θun , x φx − φun ∀x ∈ H,
3.79
yn PC zn − τBzn ,
xn1 αn u γfWn xn βn xn 1
1 − βn I − αn I μA
tn
tn
SsWn yn ds,
0
where un JrΘ xn such that JrΘ : H → H is the mapping defined by 2.22 and {αn } and {βn } are
two sequences in 0, 1 for all n ∈ N. If the functions η : H × H → H and K : H → R satisfy the
32
Journal of Applied Mathematics
conditions (C1)–(C6) as given in Theorem 3.1, then {xn } converges strongly to x∗ ∈ Ω, which solves
the following optimization problem (OP):
min
∗
x ∈Ω
μ
1
Ax∗ , x∗ x∗ − u2 − hx∗ ,
2
2
3.80
and x∗ , y∗ is a solution of the general system of variational inequality problem 1.20 such that
y∗ PC x∗ − δDx∗ .
Proof. Taking N 1 in Theorem 3.1. Hence, the conclusion follows. This completes the proof.
Corollary 3.4. Let C be a nonempty closed convex subset of a real Hilbert space H which C C ⊂ C
and let f be a contraction of C into itself with α ∈ 0, 1. Let S {Ss : 0 ≤ s < ∞} be a
nonexpansive semigroup on C and let {tn } be a positive real divergent sequence. Let {Vi : C → C}∞
i1
be a countable family of uniformly k-strict pseudo-contractions, let {Ti : C → C}∞
i1 be the countable
family of nonexpansive mappings defined by Ti x tx 1 − tVi x, for all x ∈ C, for all i ≥ 1, t ∈
k, 1, let Wn be the W-mapping defined by 2.12, and let W be a mapping defined by 2.13 with
FW /
∅. Let A be a strongly positive linear bounded operator on H with coefficient γ > 0 and
let 0 < γ < 1 μγ/α, B : H → H be a LB -Lipschitz continuous and relaxed c, d-cocoercive
mapping with d > cL2B , and let D : H → H be a LD -Lipschitz continuous and relaxed c , d cocoercive mapping with d > c L2D . Suppose that Ω : FS ∩ FW ∩ SVIC, B, D / ∅. Let μ > 0
and γ > 0, which are constants. For given x1 ∈ H arbitrarily and fixed u ∈ H, suppose {xn }, {yn },
and{zn } are the sequences generated iteratively by
zn PC xn − δDxn ,
yn PC zn − τBzn ,
xn1 αn u γfWn xn βn xn 1
1 − βn I − αn I μA
tn
3.81
tn
SsWn yn ds,
0
where {αn } and {βn } are two sequences in 0, 1 for all n ∈ N. If the sequence {xn } satisfy the
conditions (C1)–(C6) as given in Theorem 3.1, then {xn } converges strongly to x∗ ∈ Ω, which solves
the following optimization problem (OP):
min
∗
x ∈Ω
μ
1
Ax∗ , x∗ x∗ − u2 − hx∗ ,
2
2
3.82
and x∗ , y∗ is a solution of the general system of variational inequality problem 1.20 such that
y∗ PC x∗ − δDx∗ .
Proof. Put Θx, y ≡ φx ≡ 0 for all x, y ∈ H and r 1. Take Kx x2 /2 and ηy, x y − x, for all x, y ∈ H. Then, we get un PC xn xn in Corollary 3.3. Hence, the conclusion
follows. This completes the proof.
Corollary 3.5. Let C be a nonempty closed convex subset of a real Hilbert space H and let f be a
contraction of H into itself with α ∈ 0, 1. Let S {Ss : 0 ≤ s < ∞} be a nonexpansive semigroup
on C and let {tn } be a positive real divergent sequence. Let A be a strongly positive linear bounded
Journal of Applied Mathematics
33
operator on H with coefficient γ > 0 and let 0 < γ < 1 μγ/α, B : H → H be a LB -Lipschitz
∅.
continuous and relaxed c, d-cocoercive mapping with d > cL2B . Suppose that Ω : FS ∩ B−1 0 /
Let μ > 0 and γ > 0, which are constants. For given x1 ∈ H arbitrarily and fixed u ∈ H, suppose the
{xn }, {yn }, and {zn } are the sequences generated iteratively by
zn xn − τBxn ,
yn zn − τBzn ,
xn1 αn u γfxn βn xn 1
1 − βn I − αn I μA
tn
3.83
tn
Ssyn ds,
0
where {αn } and {βn } are two sequences in 0, 1 for all n ∈ N. If the sequence {xn } satisfy the
conditions (C1)–(C6) as given in Theorem 3.1, then {xn } converges strongly to x∗ ∈ Ω.
Proof. Setting τ δ, C ≡ H, D ≡ B and Wn ≡ PH ≡ I in Corollary 3.4, it follows from the proof
of Theorem 4.1 in 25 that B−1 0 VIH, B. Hence, the conclusion follows. This completes
the proof.
Acknowledgments
The authors would like to thank the “Centre of Excellence in Mathematics” under the
Commission on Higher Education, Ministry of Education, Thailand. Moreover, the authors
are grateful to the reviewers for the careful reading of the paper and for the suggestions which
improved the quality of this work.
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