Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 979870, 30 pages
doi:10.1155/2012/979870
Research Article
An Iterative Method for Solving a System
of Mixed Equilibrium Problems, System of
Quasivariational Inclusions, and Fixed
Point Problems of Nonexpansive Semigroups
with Application to Optimization Problems
Pongsakorn Sunthrayuth1, 2 and Poom Kumam1, 2
1
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi
(KMUTT), Bangmod, Bangkok 10140, Thailand
2
Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand
Correspondence should be addressed to Poom Kumam, poom.kum@kmutt.ac.th
Received 11 September 2011; Revised 22 October 2011; Accepted 24 November 2011
Academic Editor: Donal O’Regan
Copyright q 2012 P. Sunthrayuth and P. Kumam. 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 a general implicit iterative scheme base on viscosity approximation method with
a φ-strongly pseudocontractive mapping for finding a common element of the set of solutions
for a system of mixed equilibrium problems, the set of common fixed point for a nonexpansive
semigroup, and the set of solutions of system of variational inclusions with set-valued maximal
monotone mapping and Lipschitzian relaxed cocoercive mappings in Hilbert spaces. Furthermore,
we prove that the proposed iterative algorithm converges strongly to a common element of the
above three sets, which is a solution of the optimization problem related to a strongly positive
bounded linear operator.
1. Introduction
Throughout this paper we denoted by N and R the set of all positive integers and all positive
real numbers, respectively. We always assume that H be a real Hilbert space with inner
product ·, · and norm · , respectively, C is a nonempty closed convex subset of H. Let
ϕ : C → R be a real-valued function and Θ : C × C → R be an equilibrium bifunction. The
mixed equilibrium problem for short, MEP is to find x∗ ∈ C such that
Θ x∗ , y ϕ y − ϕx∗ ≥ 0,
∀y ∈ C.
1.1
2
Abstract and Applied Analysis
The set of solutions of 1.1 is denoted by MEPΘ, ϕ, that is,
MEP Θ, ϕ x∗ ∈ C : Θ x∗ , y ϕ y − ϕx∗ ≥ 0, ∀y ∈ C .
1.2
In particular, if ϕ 0, this problem reduces to the equilibrium problem, that is, to find
x ∈ C such that
∗
Θ x∗ , y ≥ 0,
∀y ∈ C.
1.3
The set of solution of 1.3 is denoted by EPΘ.
Mixed equilibrium problems include fixed point problems, optimization problems,
variational inequality problems, Nash equilibrium problems, and equilibrium problems as
special cases see, e.g., 1–6. Some methods have been proposed to solve the equilibrium
problem, see, for instance, 7–21.
Let A be a strongly positive bounded linear operator on H, that is, there exists a
constant γ > 0 such that
Ax, x ≥ γx2 ,
1.4
∀x ∈ H.
Recall that, a mapping f : H → H is said to be contractive if there exists a constant
α ∈ 0, 1 such that
fx − f y ≤ αx − y,
∀x, y ∈ H.
1.5
A mapping T : H → H is said to be
i nonexpansive if
T x − T y ≤ x − y,
∀x, y ∈ H,
1.6
ii pseudocontractive if
2
T x − T y, x − y ≤ x − y ,
1.7
∀x, y ∈ H,
iii φ-strongly pseudocontractive if there exists a continuous and strictly increasing
function φ : R → R with φ0 0 such that
2
T x − T y, x − y ≤ x − y − φ x − y x − y,
∀x, y ∈ H.
1.8
It is obvious that pseudocontractive mapping is more general than φ-strongly
pseudocontractive mapping. If φt αt with 0 < α < 1, then φ-strongly pseudocontractive
mapping reduces to β-strongly pseudocontractive mapping with 1 − α β ∈ 0, 1, which is
more general than contractive mapping.
Abstract and Applied Analysis
3
Definition 1.1. A one-parameter family mapping S {Tt : t ∈ R } from C into itself is said
to be a nonexpansive semigroup on C if it satisfies the following conditions:
i T 0x x for all x ∈ C,
ii T s t T s ◦ T t for all s, t ∈ R ,
iii for each x ∈ C the mapping t → T tx is continuous,
iv T tx − T ty ≤ x − y for all x, y ∈ C and t ∈ R .
Remark 1.2. We denote by FS the set of all common fixed points of S, that is, FS :
t∈R FT t {x ∈ C : T tx x}.
Recall the following definitions of a nonlinear mapping B : C → H, the following are
mentioned.
Definition 1.3. The nonlinear mapping B : C → H is said to be
i monotone if
Bx − By, x − y ≥ 0,
∀x, y ∈ C,
1.9
ii β-strongly monotone if there exists a constant β > 0 such that
2
Bx − By, x − y ≥ βx − y ,
1.10
∀x, y ∈ C,
iii L-Lipschitz continuous if there exists a constant L > 0 such that
Bx − By ≤ Lx − y,
∀x, y ∈ C,
1.11
iv ν-inverse-strongly monotone if there exists a constant ν > 0 such that
2
Bx − By, x − y ≥ νBx − By ,
∀x, y ∈ C,
1.12
v relaxed c, d-cocoercive if there exists a constants c, d > 0 such that
2
2
Bx − By, x − y ≥ −cBx − By ddx − y ,
∀x, y ∈ C.
1.13
The resolvent operator technique for solving variational inequalities and variational
inclusions is interesting and important. The resolvent equation technique is used to
develop powerful and efficient numerical techniques for solving various classes of
variational inequalities, inclusions, and related optimization problems.
Definition 1.4. Let M : H → 2H be a multivalued maximal monotone mapping. The singlevalued mapping JM,ρ : H → H, defined by
−1
JM,ρ u I ρM u,
∀u ∈ H,
1.14
4
Abstract and Applied Analysis
is called resolvent operator associated with M, where ρ is any positive number and I is the
identity mapping.
Next, we consider a system of quasivariational inclusions problem is to find x∗ , y∗ ∈
H × H such that
0 ∈ x∗ − y∗ ρ1 B1 y∗ M1 x∗ ,
0 ∈ y∗ − x∗ ρ2 B2 x∗ M2 y∗ ,
1.15
where Bi : H → H and Mi : H → 2H are nonlinear mappings for each i 1, 2.
As special cases of the problem 1.15, we have the following results.
1 If B1 B2 B and M1 M2 M, then the problem 1.15 is reduces to the following. Find x∗ , y∗ ∈ H × H such that
0 ∈ x∗ − y∗ ρ1 By∗ Mx∗ ,
0 ∈ y∗ − x∗ ρ2 Bx∗ My∗ .
1.16
2 Further, if x∗ y∗ in problem 1.16, then the problem 1.16 is reduces to the
following. Find x∗ ∈ H such that
0 ∈ Bx∗ Mx∗ .
1.17
The problem 1.17 is called variational inclusion problem. We denote by VIH, B, M the set of
solutions of the variational inclusion problem 1.17. Next, we consider two special cases of
the problem 1.17.
1 M ∂φ : H → 2H , where φ : H → R ∪ {∞} is a proper convex lower
semicontinuous function and ∂φ is the subdifferential of φ then the quasivariational
inclusion problem 1.17 is equivalent to finding x∗ ∈ H such that Bx∗ , x − x∗ φx−φx∗ ≥ 0, for all x ∈ H, which is said to be the mixed quasivariational inequality.
2 If M ∂δC , where C is a nonempty closed convex subset of H, and δC : H →
0, ∞ is the indicator function of C, that is,
δC x ⎧
⎨0,
x ∈ C,
⎩∞, x ∈
/ C,
1.18
then the quasivariational inclusion problem 1.17 is equivalent to the classical
variational inequality problem denoted by VIC, B which is to find x∗ ∈ C such
that
Bx∗ , x − x∗ ≥ 0,
∀x ∈ C.
1.19
This problem is called Hartman-Stampacchia variational inequality problem see e.g., 22–24.
Abstract and Applied Analysis
5
It is known that problem 1.17 provides a convenient framework for the unified
study of optimal solutions in many optimization related areas including mathematical
programming, complementarity, variational inequalities, optimal control, mathematical
economics, equilibria, and game theory. Also various types of variational inclusions problems
have been extended and generalized see 25–40 and the references therein.
On the other hand, the following optimization problem has been studied extensively
by many authors:
μ
1
min Ax, x x − u2 − hx,
2
x∈Ω 2
1.20
where Ω ∞
n1 Cn , C1 , C2 , . . . are infinitely many closed convex subsets of H such that
∞
∅, u ∈ H, μ ≥ 0 is a real number, A is a strongly positive linear bounded operator
n1 Cn /
on H and h is a potential function for γf i.e., h x γfx for all x ∈ H. This kind of
optimization problem has been studied extensively by many authors see, e.g. 41–44 when
Ω ∞
n1 Cn and hx x, b, where b is a given point in H.
Li et al. 45 introduced two steps of iterative procedures for the approximation of
common fixed point of a nonexpansive semigroup S {T t : t ∈ R } on a nonempty closed
convex subset C in a Hilbert space. Recently, Liu et al. 46 introduced a hybrid iterative
scheme for finding a common element of the set of solutions of system of mixed equilibrium
problems, the set of common fixed points for nonexpansive semigroup and the set of solution
of quasivariational inclusions with multivalued maximal monotone mappings and inversestrongly monotone mappings. Very recently, Hao 47 introduced a general iterative method
for finding a common element of solution set of quasivariational inclusion problems and the
set of common fixed points of an infinite family of nonexpansive mappings.
In this paper, motivated and inspired by Li et al. 45, Liu et al. 46, and Hao 47, we
introduce a general implicit iterative algorithm base on viscosity approximation methods
with a φ-strongly pseudocontractive mapping which is more general than a contraction
mapping for finding a common element of the set of solutions for a system of mixed
equilibrium problems, the set of common fixed point for a nonexpansive semigroup, and the
set of solutions of system of variational inclusions 1.15 with set-valued maximal monotone
mapping and Lipschitzian relaxed cocoercive mappings in Hilbert spaces. We prove that the
proposed iterative algorithm converges strongly to a common element of the above three sets,
which is a solution of the optimization problem related to a strongly positive bounded linear
operator. The results obtained in this paper extend and improve several recent results in this
area.
2. Preliminaries
In the sequel, we use xn x and xn → x to denote the weak convergence and strong
convergence of the sequence {xn } in H, respectively.
This collects some results that will be used in the proofs of our main results.
Proposition 2.1 see 21. i The resolvent operator JM,ρ associated with M is single-valued and
nonexpansive for all ρ > 0, that is,
JM,ρ x − JM,ρ y ≤ x − y,
∀x, y ∈ H, ∀ρ > 0.
2.1
6
Abstract and Applied Analysis
ii The resolvent operator JM,ρ is 1-inverse-strongly monotone, that is,
JM,ρ x − JM,ρ y 2 ≤ x − y, JM,ρ x − JM,ρ y ,
∀x, y ∈ H.
2.2
Obviously, this immediately implies that
x − y − JM,ρ x − JM,ρ y 2 ≤ x − y2 − JM,ρ x − JM,ρ y 2 ,
∀x, y ∈ H.
2.3
For solving the equilibrium problem for bifunction Θ : H × H → R, let us assume that
satisfies the following conditions:
H1 Θx, x 0 for all x ∈ H;
H2 Θ is monotone, that is, Θx, y Θy, x ≤ 0 for all x, y ∈ H;
H3 for each y ∈ H, x → Θx, y is concave and upper semicontinuous;
H4 for each y ∈ H, x → Θx, y is convex;
H5 for each y ∈ H, x → Θx, y is lower semicontinuous.
Definition 2.2. A map η : H × H → H is called Lipschitz continuous, if there exists a constant
L > 0 such that
η x, y ≤ Lx − y,
∀x, y ∈ H.
2.4
A differentiable function K : H → R on a convex set H is called
i η—convex 7 if
K y − Kx ≥ K x, η y, x ,
∀x, y ∈ H,
2.5
where K x is the Fréchet differentiable of K at x,
i η—strongly convex 7 if there exists a constant ν > 0 such that
ν x − y2 ,
K y − Kx − K x, η x, y ≥
2
∀x, y ∈ H.
2.6
Let Θ : H × H → R be an equilibrium bifunction satisfying the conditions H1–H5. Let
r be any given positive number. For a given point x ∈ H, consider the following auxiliary
problem for MEP for short, MEP x, y to find y ∈ H such that
1 Θ y, z ϕz − ϕ y K y − K x, η z, y ≥ 0,
r
∀z ∈ H,
2.7
Abstract and Applied Analysis
7
where η : H × H → H is a mapping, and K x is the Fréchet derivative of a functional
Θ,ϕ
Θ,ϕ
: H → H be the mapping such that for each x ∈ H, Vr
x is
K : H → R at x. Let Vr
the set of solutions of MEP x, y, that is,
Θ,ϕ
Vr
x y ∈ H : Θ y, z ϕz − ϕ y
1 K y − K x, η z, y ≥ 0, ∀z ∈ H ,
r
2.8
∀x ∈ H.
Then the following conclusion holds.
Proposition 2.3 see 7. Let H be a real Hilbert space, ϕ : H → R be a a lower semicontinuous
and convex functional. Let Θ : H × H → R be an equilibrium bifunction satisfying conditions
(H1)–(H5). Assume that
i η : H × H → R is Lipschitz continuous with constant σ > 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 y ∈ H, x → ηy, x is continuous from the weak topology to the weak
topology;
ii K : H → R is η-strongly convex with constant μ > 0, and its derivative K is continuous
from the weak topology to the strong topology;
iii for each x ∈ H, there exists a bounded subset Dx ⊂ H and zx ∈ H such that for all
y∈
/ Dx ,
1 Θ y, zx ϕzx − ϕ y K y − K x, η zx , y < 0.
r
2.9
Then the following hold:
Θ,ϕ
i Vr
ii
is single valued;
Θ,ϕ
FVr
MEPΘ, ’;
iii MEPΘ, ϕ is closed and convex.
Lemma 2.4 see 48. Let C be a nonempty bounded closed and convex subset of a real Hilbert space
H. Let S {T t : t ∈ R } be a nonexpansive semigroup on C, then for all h > 0,
t
1 t
1
lim sup
T sx ds − T h
T sx ds 0.
t → ∞ x∈C t 0
t 0
2.10
Lemma 2.5 see 49. Let X be a uniformly convex Banach space, C be a nonempty closed and
convex subset of X, and T : C → X be a nonexpansive mapping. Then I − T is demiclosed at zero.
8
Abstract and Applied Analysis
Lemma 2.6 see 50. Assume that A is a strongly positive linear bounded operator on H with
coefficient γ > 0 and 0 < ρ ≤ A−1 . Then I − ρA ≤ 1 − ργ.
Lemma 2.7 see 51. Let X be a Banach space and f : X → X be a φ-strongly pseudocontractive
and continuous mapping. Then f has a unique fixed point in X.
Lemma 2.8. In a real Hilbert space H, the following inequality holds:
x y2 ≤ x2 2 y, x y ,
∀x, y ∈ H.
2.11
The following lemma can be found in 52, 53 see also Lemma 2.2 in 54.
Lemma 2.9. Let C be a nonempty closed and convex subset of a real Hilbert space H and g : C → R ∪
{∞} be a proper lower semicontinuous differentiable convex function differentiable convex function.
If x∗ is a solution to the minimization problem
gx∗ inf gx,
2.12
x∈C
then
g x, x − x∗ ≥ 0,
x ∈ C.
2.13
In particular, if x∗ solves the optimization problem
μ
1
min Ax, x x − u2 − hx,
x∈Ω 2
2
2.14
then
u γf − I μA x∗ , x − x∗ ≤ 0,
x ∈ C,
2.15
where h is a potential function for γf.
The following lemmas can be found in 55, 56. For the sake of the completeness,
one includes its proof in a Hilbert space version. Without loss of generality, one assumes that
c, d ∈ 0, 1 and LB ∈ 1, ∞.
Lemma 2.10. Let H be a real Hilbert space, B : H → H be an LB -Lipschitzian and relaxed c, dcocoercive mapping. Then, one has
I − ρB x − I − ρB y2 ≤ 1 2ρcL2 − 2ρd ρ2 L2 x − y2 ,
B
B
where ρ > 0. In particular, if 0 < ρ ≤ 2d − cL2B /L2B , then I − ρB is nonexpansive.
2.16
Abstract and Applied Analysis
9
Proof. For all x, y ∈ H, we have
I − ρBx − I − ρBy2 x − y − ρBx − ρBy2
2
2
x − y − 2ρ Bx − By, x − y ρ2 Bx − By
2
2
2 2
≤ x − y − 2ρ −cBx − By dx − y ρ2 Bx − By
2
2
2
2
x − y − 2ρdx − y 2ρcBx − By ρ2 Bx − By
2
≤ 1 2ρcL2B − 2ρd ρ2 L2B x − y .
2.17
It is clear that, if 0 < ρ ≤ 2d − cL2B /L2B , then I − ρB is nonexpansive. This completes the
proof.
Lemma 2.11. Let H be a real Hilbert space, Mi : H → 2H be the a maximal monotone mapping
and Bi : H → H be an Li -Lipschitzian and relaxed ci , di -cocoercive mapping for all i 1, 2. Let
Q : H → H be a mapping defined by
Qx : JM1 ,ρ1 JM2 ,ρ2 x − ρ2 B2 x − ρ1 B1 JM2 ,ρ2 x − ρ2 B2 x ,
∀x ∈ H.
2.18
If 0 < ρi ≤ 2di − ci L2i /L2i for all i 1, 2, then Q : H → H is nonexpansive.
Proof. By Lemma 2.10, we know that I − ρ2 B2 and I − ρ1 B1 are nonexpansive, for all x, y ∈
H, we have
Qx − Qy JM ,ρ JM ,ρ x − ρ2 B2 x − ρ1 B1 JM ,ρ x − ρ2 B2 x
1
1
2
2
2
2
−JM1 ,ρ1 JM2 ,ρ2 y − ρ2 B2 y − ρ1 B1 JM2 ,ρ2 y − ρ2 B2 y ≤ JM2 ,ρ2 x − ρ2 B2 x − ρ1 B1 JM2 ,ρ2 x − ρ2 B2 x
− JM2 ,ρ2 y − ρ2 B2 y − ρ1 B1 JM2 ,ρ2 y − ρ2 B2 y JM2 ,ρ2 I − ρ2 B2 I − ρ1 B1 x − JM2 ,ρ2 I − ρ2 B2 I − ρ1 B1 y
≤ I − ρ2 B2 I − ρ 1 B1 x − I − ρ 2 B2 I − ρ 1 B1 y ≤ I − ρ2 B2 x − I − ρ2 B2 y ≤ x − y,
2.19
which implies that Q is nonexpansive. This completes the proof.
Lemma 2.12. For all x∗ , y∗ ∈ H × H, where y∗ JM2 ,ρ2 x∗ − ρ2 B2 x∗ , x∗ , y∗ is a solution
of the problem 1.15 if and only if x∗ is a fixed point of the mapping Q : H → H defined as in
Lemma 2.11.
10
Abstract and Applied Analysis
Proof. Let x∗ , y∗ ∈ H × H be a solution of the problem 1.15. Then, we have
y∗ − ρ1 B1 y∗ ∈ I ρ1 M1 x∗ ,
x∗ − ρ2 B2 x∗ ∈ I ρ2 M2 y∗ ,
2.20
x∗ JM1 ,ρ1 y∗ − ρ1 B1 y∗ ,
y∗ JM2 ,ρ2 x∗ − ρ2 B2 x∗ .
2.21
which implies that
We can deduce that 2.21 is equivalent to
x∗ JM1 ,ρ1 JM2 ,ρ2 x∗ − ρ2 B2 x∗ − ρ1 B1 JM2 ,ρ2 x∗ − ρ2 B2 x∗ .
2.22
This completes the proof.
3. Main Results
Now, in this section, we prove our main results of this article. Before proving the main result
we need the following lemma.
Lemma 3.1. Let H be a real Hilbert space. Let S {T t : t ∈ R } be a nonexpansive semigroup
t
from H into itself. Then I − σt · is monotone, where σt x : 1/t 0 T sx ds for all x ∈ H and
t ≥ 0.
Proof. For all x, y ∈ H, we have
x − y, I − σt ·x − I − σt ·y x − y,
1
I−
t
2
x − y −
t
T sds x −
0
1
x − y,
t
1
2 ≥ x − y − x − y
t
2 2
≥ x − y − x − y
t
1
I−
t
1
T sx ds −
t
0
t
t
T sds y
0
t
T sy ds
0
T sx − T syds
0
0,
3.1
which implies that I − σt · is monotone. This completes the proof.
Theorem 3.2. Let H be a real Hilbert space. Let ϕi : H → R i 1, 2, . . . , N be a finite family of
lower semicontinuous and convex function, Θi : H × H → R i 1, 2, . . . , N be a finite family of
Abstract and Applied Analysis
11
bifunctions satisfying (H1)–(H5), and ηi : H × H → H be a finite family of Lipschitz continuous
mappings with a constant σi i 1, 2, . . . , N. Let S {T t : t ∈ R } be a nonexpansive semigroup
from H into itself, Bi : H → Hi 1, 2 be an Li -Lipschitzian and relaxed ci , di -cocoercive
mapping with ρi ∈ 0, 2di − ci L2i /L2i for all i 1, 2 and Mi : H → 2H i 1, 2 be a maximal
∅, where Q is defined as in
monotone mapping. Assume that Ω : FS∩ N
k1 MEP Θk , ϕk ∩FQ /
Lemma 2.11. Let f : H → H be a φ-strongly pseudocontractive mapping with limt → ∞ Œt ∞
and A be a strongly positive linear bounded operator on H with a coefficient γ > 0. Let μ > 0 and
γ > 0 be two constants such that 0 < γ < 1 μγ. Let {ri,n }i 1, 2, . . . , N be a finite family of
positive real sequence such that lim infn → ∞ ri,n > 0, {αn } and {βn } be two sequences in 0, 1, and
{tn } be a positive real divergent sequence. For any fixed u ∈ H, let {xn } be the sequence defined by
1 1 1
1
1
K1 un − K1 xn , η1 x, un
≥ 0, ∀x ∈ H,
Θ1 un , x ϕ1 x − ϕ1 un r1,n
1 2 2
2
1
2
Θ2 un , x ϕ2 x − ϕ2 un K2 un − K2 un , η1 x, un
≥ 0, ∀x ∈ H,
r2,n
..
.
1 N
N
N
N−1
N
KN
un
−KN
un
, ηN x, un
≥ 0, ∀x ∈ H,
ΘN un , x ϕN x−ϕN un
rN,n
N
N
yn JM2 ,ρ2 un − ρ2 B2 un ,
xn αn u γfxn βn xn 1
1 − βn I − αn I μA
tn
tn
T sJM1 ,ρ1 yn − ρ1 B1 yn ds,
0
3.2
where
1
Θ ,ϕ1 i
Θ ,ϕi i−1
un
un Vr1,n 1
un Vri,n i
Θ ,ϕi Vri,n i
Θ ,ϕi and Vri,n i
xn ,
Θ ,ϕi Vri,n i
Θ ,ϕ2 · · · Vr2,n 2
Θ
Vri−1,ni−1
Θ ,ϕ1 Vr1,n 1
,ϕi−1 i−2
un
xn ,
Θ ,ϕi Vri,n i
Θ ,ϕ2 1
un
· · · Vr2,n 2
3.3
i 2, 3, . . . , N,
: H → H, i 1, 2, . . . , N is the mapping defined by 2.8. Assume the following.
i ηi : H × H → H is Lipschitz continuous with constant σi > 0 i 1, 2, . . . , N such that
a ηi x, y ηi y, x 0 for all x, y ∈ H,
b ηi ·, · is affine in the first variable,
c for each fixed y ∈ H, x → ηi y, x is sequentially continuous from the weak topology
to the weak topology.
ii Ki : H → R is ηi -strongly convex with constant μi > 0, and its derivative Ki is not only
continuous from the weak topology to the strong topology but also Lipschitz continuous
with constant νi such that μi ≥ σi νi .
12
Abstract and Applied Analysis
iii For all i 1, 2, ..., N and for all x ∈ H, there exists a bounded subset Dx ⊂ H and zx ∈ H
such that for all y ∈
/ Dx ,
1
Ki y − Ki x, ηi zx , y < 0.
Θi y, zn ϕi zx − ϕi y ri,n
3.4
If the following conditions are satisfied:
C1 limn → ∞ αn 0,
C2 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1,
then the sequence {xn } defined by 3.2 converges strongly to x∗ ∈ Ω : FS ∩ N
k1 MEPΘk , ϕk ∩
Θi ,ϕi ∗
FQ, provided Vri,n
is firmly nonexpansive, where x is the unique solution of the
variational inequality
u γf − I μA x∗ , z − x∗ ≤ 0,
∀z ∈ Ω,
3.5
or, equivalently, x∗ is the unique solution of the optimization problem
μ
1
min Ax, x x − u2 − hx,
x∈Ω 2
2
3.6
where h is a potential function for γf and x∗ , y∗ is the solution of the problem 1.15, where
y∗ JM2 ,ρ2 x∗ − ρ2 B2 x∗ .
Proof. By the conditions C1 and C2, we may assume, without loss of generality, that αn ≤
1 − βn 1 μA−1 for all n ∈ N. Since A is a linear bounded self-adjoint operator on H, by
1.4, we have
A sup{|Au, u| : u ∈ H, u 1}.
3.7
Observe that
1 − βn I − αn I μA u, u 1 − βn − αn − αn μAu, u
≥ 1 − βn − αn − αn μA
3.8
≥ 0.
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 u, u : u ∈ H, u 1
sup 1 − βn − αn − αn μAu, u : u ∈ H, u 1
≤ 1 − βn − αn − αn μγ.
3.9
Abstract and Applied Analysis
13
Θ ,ϕk In fact, by the assumption that for all k ∈ {1, 2, . . . , N}, Vrk,nk
Θ ,ϕk Vrk,nk
by
Θ ,ϕ2 · · · Vr2,n 2
Θ ,ϕ1 Vr1,n 1
is nonexpansive. Setting Vnk :
for k ∈ {1, 2, . . . , N} and Vn0 : I. Define a mapping Wn : H → H
1
Wn x :
tn
tn
0
T sQVnN x ds,
∀x ∈ H.
3.10
Hence, by Lemma 2.11 and nonexpansiveness semigroup of T s and Vnk , for all x, y ∈ C,
we have
1 tn
1 tn
N
N
Wn x − Wn y T sQVn x ds −
T sQVn y ds
tn 0
tn 0
≤ QVnN x − QV N
n y
≤ VnN x − VnN y
3.11
≤ x − y,
which implies that Wn is nonexpansive.
f
First, we show that {xn } defined by 3.2 is well defined. Define a mapping Tn : H →
H by
f
Tn x : αn u γfx βn x 1 − βn I − αn I μA Wn x,
∀x ∈ H.
3.12
Indeed, by Lemma 2.6, and from 3.11, for all x, y ∈ H, we have
f
f
Tn x − Tn y, x − y αn γ fx − f y , x − y βn x − y, x − y
1 − βn I − αn I μA Wx − Wy , x − y
2
≤ αn γ x − y − φ x − y x − y
2
2 βn x − y 1 − βn − αn − αn μγ x − y
1 αn γ − 1 μγ x − y2 − αn γφ x − y x − y
≤ x − y2 − αn γφ x − y x − y.
f
3.13
This shows that Tn is a φ-strongly pseudocontractive and strongly continuous. It follows
f
from Lemma 2.7 that Tn has a unique fixed point xn ∈ H, that is, {xn } defined by 3.2 is well
defined.
14
Abstract and Applied Analysis
Next, we show that uniqueness of the solution of the variational inequality 3.5.
Suppose that x,
x∗ ∈ Ω satisfy 3.5, then
u γf − I μA x∗ , x − x∗ ≤ 0,
u γf − I μA x,
x∗ − x ≤ 0.
3.14
, x∗ − x
I μA x∗ − I μA x − γ fx∗ − fx
I μA x∗ − I μA x,
x∗ − x − γ fx∗ − fx,
x∗ − x
∗
≥ 1 μγ x∗ − x
2 − γx∗ − x
2 φx∗ − xx
− x
∗
∗
1 μγ − γ x − x
2 φx∗ − xx
− x.
3.15
Adding up 3.14, we have
0≥
It follows that
1 μγ − γ x∗ − x
φx∗ − x
≤ 0,
3.16
which is a contradiction. Hence, x x∗ and the uniqueness is proved.
Next, we show that {xn } is bounded. Taking x ∈ Ω, it follows from Lemma 2.11 that
x JM1 ,ρ1 JM2 ,ρ2 x − ρ2 B2 x − ρ1 B1 JM2 ,ρ2 x − ρ2 B2 x .
3.17
Putting y JM2 ,ρ2 x − ρ2 B2 x, we have x JM1 ,ρ1 y − ρ1 B1 y. Setting zn : VnN xn and
vn : JM1 ,ρ1 yn − ρ1 B1 yn , then
1
xn αn u γfxn βn xn 1 − βn I − αn I μA
tn
Θ ,ϕk Since for all k ∈ {1, 2, . . . , N}, Vrk,nk
and x VnN x, then
tn
T svn ds.
3.18
0
is nonexpansive, we also have that VnN is nonexpansive
zn − x VnN xn − VnN x ≤ xn − x,
∀n ∈ N.
By nonexpansiveness of JMi ,ρi and I − ρi Bi i 1, 2, we have
vn − x JM1 ,ρ1 yn − ρ1 B1 yn − JM1 ,ρ1 y − ρ1 B1 y ≤ yn − ρ1 B1 yn − y − ρ1 B1 y ≤ yn − y
JM2 ,ρ2 zn − ρ2 B2 zn − JM2 ,ρ2 x − ρ2 B2 x ≤ zn − ρ2 B2 zn − x − ρ2 B2 x 3.19
Abstract and Applied Analysis
15
≤ zn − x
≤ xn − x.
3.20
It follows from 3.20 that
xn − x2 xn − x, xn − x
αn u γfxn − I μA x, xn − x βn xn − x, xn − x
1 tn
1 − βn I − αn I μA
T svn − xds , xn − x
tn 0
αn γ fxn − fx, xn − x αn u γfx − I μA x, xn − x
βn xn − x, xn − x
1 tn
1 − βn I − αn I μA
T svn − xds , xn − x
tn 0
≤ αn γ xn − x2 − φxn − xxn − x
3.21
αn u γfx − I μA x, xn − x βn xn − x2
1 − βn − αn − αn μγ xn − x2 ,
and, so
γφxn − xxn − x 1 − γ μγ xn − x2 ≤ u γfx − I μA x, xn − x .
3.22
It follows that
γφxn − xxn − x ≤ u γfx − I μA x, xn − x
≤ u γfx − I μA xxn − x.
3.23
u γfx − I μA x
,
xn − x ≤ φ
γ
3.24
Hence
−1
which implies that {xn } is bounded, so are {zn }, {yn }, and {vn }. Since f is φ-strongly pseudocontractive, we have
fxn − fx, xn − x ≤ xn − x2 − φxn − xxn − x
≤ xn − x2 .
Thus {fxn } is bounded.
3.25
16
Abstract and Applied Analysis
Next, we show that limn → ∞ xn − T hxn 0, for all h ≥ 0. From 3.18, we observe
that
1 tn
1 tn
T svn ds ≤ αn u γfxn − I μA
T svn ds
xn −
tn 0
tn 0
1 tn
βn xn −
T svn ds.
tn 0
3.26
It follows that
1 tn
αn 1 tn
T svn ds ≤
T svn ds.
xn −
u γfxn − I μA
1 − βn tn 0
tn 0
3.27
By the conditions C1 and C2, we obtain
1 tn
lim xn −
T svn ds 0.
n → ∞
tn 0
3.28
Let B {ω ∈ H : ω − x ≤ φ−1 u γfx − I μAx/γ}, then B is nonempty bounded
closed and convex subset of H, which is T h-invariant for all h ≥ 0 and contains {xn }. It
follows from Lemma 2.4 that
t
1 tn
1 n
T svn ds − T h
T svn ds 0.
lim n → ∞ tn 0
tn 0
3.29
On the other hand, we note that
t
1 tn
1 n
1 tn
T svn ds T svn ds − T h
T svn ds xn − T hxn ≤ xn −
tn 0
tn 0
tn 0
t
1 n
T h
T svn ds − T hxn tn 0
t
1 tn
1 n
1 tn
≤ 2xn −
T svn ds T svn ds − T h
T svn ds .
tn 0
tn 0
tn 0
3.30
From 3.28 and 3.29, for all h ≥ 0, we have
lim xn − T hxn 0.
n→∞
3.31
Abstract and Applied Analysis
17
Θ ,ϕk Next, we show that limn → ∞ Vnk xn − Vnk−1 xn 0 for all k ∈ {1, 2, . . . , N}. Since Vrk,nk
firmly nonexpansive for all k ∈ {1, 2, . . . , N} and
{1, 2, . . . , N}, hence for x ∈ Ω, we have
Vnk
Θ ,ϕ Θk−1 ,ϕk−1 Vrk,nk k Vrk−1,n
···
Θ ,ϕ Vr1,n 1 1
is
for k ∈
2 Θ ,ϕ 2
k
Θk ,ϕk k−1
Vn xn − Vrk,nk k x
Vn xn − x Vrk,n
Θ ,ϕ Θ ,ϕ ≤ Vrk,nk k Vnk−1 xn − Vrk,nk k x, Vnk−1 xn − x
Vnk xn − x, Vnk−1 xn − x
3.32
2 2 2 1 k
k−1
k
k−1
x
−
x
x
−
x
−
x
−
V
x
.
Vn n
Vn
Vn n
n
n
n
2
It follows that
2
2
k
2
Vn xn − x ≤ xn − x − Vnk xn − Vnk−1 xn .
3.33
Now, by Lemma 2.8, we have
1 tn
T svn ds
xn − x αn u αn γfxn − I μA x βn xn −
tn 0
2
I − αn I μA
1
tn
tn
0
2
T svn ds − x 2
1 tn
1 tn
≤ I − αn I μA
T svn ds − x βn xn −
T svn ds tn 0
tn 0
2αn u γfxn − I μA x, xn − x
2
1 tn
1 tn
≤ I − αn I μA
T svn ds − x βn xn −
T svn ds
tn 0
tn 0
2αn u γfxn − I μA x, xn − x
2
1 tn
≤
T svn ds
1 − αn − αn μγ vn − x βn xn −
tn 0
2αn u γfxn − I μA xxn − x
2
1 − αn − αn μγ vn − x2 cn ,
3.34
18
Abstract and Applied Analysis
where
2
1 tn
T svn ds 2 1 − αn − αn μγ βn vn − xxn −
T svn ds
cn :
t
n 0
0
3.35
2αn u γfxn − I μA xxn − x.
βn2 xn
1
−
tn
tn
From the condition C1 and 3.28, we have
lim cn 0.
n→∞
3.36
From 3.20, we observe that
vn − x ≤ yn − x
≤ VnN xn − x
≤ Vnk xn − x,
3.37
∀k ∈ {1, 2, . . . , N}.
Substituting 3.33 into 3.34, we have
2 2
xn − x2 − Vnk xn − Vnk−1 xn cn
xn − x2 ≤ 1 − αn − αn μγ
2
1 − 2αn 1 μγ α2n 1 μγ
xn − x2
3.38
2
2 − 1 − αn − αn μγ Vnk xn − Vnk−1 xn cn ,
which in turn implies that
2 2 2
1 − αn − αn μγ Vnk xn − Vnk−1 xn ≤ 1 α2n 1 μγ
xn − x2 − xn − x2 cn .
3.39
From the condition C1 and from 3.36, we obtain that
lim Vnk xn − Vnk−1 xn 0.
n→∞
3.40
Abstract and Applied Analysis
19
On the other hand, we observe that
xn − zn xn − VnN xn N !
k
k−1
V x − Vn xn k1 n n
≤
3.41
N !
k
Vn xn − Vnk−1 xn .
k1
Then, we have
lim xn − zn 0.
3.42
1 tn
1 tn
T svn ds ≤ zn − xn xn −
T svn ds,
zn −
tn 0
tn 0
3.43
1 tn
lim zn −
T svn ds 0.
n → ∞
tn 0
3.44
n→∞
Moreover, we observe that
and hence
Next, we show that for all x, y ∈ Ω, limn → ∞ B1 yn − B1 y 0 and limn → ∞ B2 zn − B2 x 0.
By the cocoercivity of the mapping B1 , we have
2
vn − x2 JM1 ,ρ1 yn − ρ1 B1 yn − JM1 ,ρ1 y − ρ1 B1 y 2
≤ yn − ρ1 B1 yn − y − ρ1 B1 y 2
yn − y − ρ1 B1 yn − B1 y 2
2
yn − y − 2ρ1 B1 yn − B1 y, yn − y ρ12 B1 yn − B1 y 2
2 2
≤ xn − x2 − 2ρ1 −c1 B1 yn − B1 y d1 yn − y ρ12 B1 yn − B1 y 2
≤ xn − x 2ρ1 c1 ρ12
−
2ρ1 d1
L21
B1 yn − B1 y2 .
Similarly, we have
yn − y 2 JM ,ρ zn − ρ2 B2 zn − JM ,ρ x − ρ2 B2 x 2
2 2
2 2
2
≤ zn − ρ2 B2 zn − x − ρ2 B2 x 3.45
20
Abstract and Applied Analysis
2
zn − x − ρ2 B2 zn − B2 x
zn − x2 − 2ρ2 B2 zn − B2 x, zn − x ρ22 B2 zn − B2 x2
≤ xn − x2 − 2ρ2 −c2 B2 zn − B2 x2 d2 zn − x2 ρ22 B2 zn − B2 x2
2
≤ xn − x 2ρ2 c2 ρ22
−
2ρ2 d2
L22
B2 zn − B2 x2 .
3.46
Substituting 3.45 into 3.34, we have
2
xn − x ≤ 1 − αn − αn μγ
2
"
2
xn − x 2ρ1 c1 ρ12 −
2ρ1 d1
L21
B1 yn − B1 y2
#
cn .
3.47
Again, from 3.34, we obtain
2
xn − x2 ≤ 1 − αn − αn μγ vn − x2 cn
2
2 ≤ 1 − αn − αn μγ yn − y cn
3.48
"
#
2
2ρ2 d2
≤ 1 − αn − αn μγ
xn − x2 2ρ2 c2 ρ22 −
B2 zn − B2 x2 cn .
L22
Therefore, from 3.47 and 3.48, we obtain
1 − αn − αn μγ
2
−2ρ1 c1 −
ρ12
2ρ1 d1
B1 yn − B1 y 2
L21
2
≤ 1 − αn − αn μγ xn − x2 − xn − x2 cn
2
≤ 1 α2n 1 μγ
xn − x2 − xn − x2 cn ,
1 − αn − αn μγ
2
−2ρ2 c2 −
ρ22
2ρ2 d2
L22
3.49
2
B2 zn − B2 x
2
≤ 1 − αn − αn μγ xn − x2 − xn − x2 cn
2
≤ 1 α2n 1 μγ
xn − x2 − xn − x2 cn .
From the condition C1 and from 3.36, we obtain that
lim B1 yn − B1 y 0,
n→∞
lim B2 zn − B2 x 0.
n→∞
3.50
Abstract and Applied Analysis
21
Next, we show that limn → ∞ zn − vn 0 and limn → ∞ xn − vn 0. By the nonexpansiveness
of I − ρ2 B2 , we have
yn − y2 JM ,ρ zn − ρ2 B2 zn − JM ,ρ x − ρ2 B2 x 2
2 2
2 2
≤ zn − ρ2 B2 zn − x − ρ2 B2 x , yn − y
1 zn − ρ2 B2 zn − x − ρ2 B2 x 2 yn − y 2
2
2
− zn − ρ2 B2 zn − x − ρ2 B2 x − yn − y 2 2
1
≤
zn − x2 yn − y − zn − yn − ρ2 B2 zn − B2 x − x − y 2
2 2
1
zn − x2 yn − y − zn − yn − x − y 2
2ρ2 zn − yn − x − y , B2 zn − B2 x − ρ22 B2 zn − B2 x2 .
3.51
So, we obtain
yn − y2 ≤ zn − x2 − zn − yn − x − y 2 2ρ2 zn − yn − x − y , B2 zn − B2 x .
3.52
Substituting 3.52 into 3.34, we have
2 2
xn − x2 ≤ 1 − αn − αn μγ
zn − x2 − zn − yn − x − y 2ρ2 zn − yn − x − y , B2 zn − B2 x cn ,
3.53
which in turn implies that
2 2 2
1 − αn − αn μγ zn − yn − x − y ≤ 1 − αn − αn μγ xn − x2 − xn − x2
2 1 − αn − αn μγ 2ρ2 zn − yn − x − y × B2 zn − B2 x cn
2
≤ 1 α2n 1 μγ
xn − x2 − xn − x2
2 1 − αn − αn μγ 2ρ2 zn − yn − x − y × B2 zn − B2 x cn .
3.54
From the condition C1 and from 3.36, 3.50, we obtain that
lim zn − yn − x − y 0.
n→∞
3.55
22
Abstract and Applied Analysis
On the other hand, by Lemma 2.8 and from 2.3, we have,
yn − vn x − y 2 yn − ρ1 B1 yn − y − ρ1 B1 y
2
− JM1 ,ρ1 yn − ρ1 B1 yn − JM1 ,ρ1 y − ρ1 B1 y ρ1 B1 yn − B1 y ≤ yn − ρ1 B1 yn − y − ρ1 B1 y
2
− JM1 ,ρ1 yn − ρ1 B1 yn − JM1 ,ρ1 y − ρ1 B1 y 2ρ1 B1 yn − B1 y, yn − vn x − y
2
≤ yn − ρ1 B1 yn − y − ρ1 B1 y 2
−JM1 ,ρ1 yn − ρ1 B1 yn − JM1 ,ρ1 y − ρ1 B1 y 2ρ1 B1 yn − B1 y yn − vn x − y 2
≤ yn − ρ1 B1 yn − y − ρ1 B1 y 1 tn
−
T sJM1 ,ρ1 yn − ρ1 B1 yn ds
tn 0
2
1 tn
T sJM1 ,ρ1 y − ρ1 B1 y ds
−
tn 0
2ρ1 B1 yn − B1 y yn − vn x − y 2
yn − ρ1 B1 yn − y − ρ1 B1 y 2
1 tn
1 tn
−
T svn ds −
T s xds
tn 0
tn 0
2ρ1 B1 yn − B1 y yn − vn x − y t
1 n
T svn ds − x ≤ yn − ρ1 B1 yn − y − ρ1 B1 y −
tn 0
#
"
tn
1
× yn − ρ1 B1 yn − y − ρ1 B1 y T svn ds − x
tn 0
2ρ1 B1 yn − B1 y yn − vn x − y 1 tn
zn −
T svn ds x − y − zn − yn − ρ1 B1 yn − B1 y tn 0
#
"
t
1 n
× yn − ρ1 B1 yn − y − ρ1 B1 y T svn ds − x
tn 0
2ρ1 B1 yn − B1 y yn − vn x − y .
3.56
Abstract and Applied Analysis
23
we obtain that
lim yn − vn x − y 0.
3.57
zn − vn ≤ zn − yn − x − y yn − vn x − y ,
3.58
lim zn − vn 0.
3.59
xn − vn ≤ xn − zn zn − vn ,
3.60
lim xn − vn 0.
3.61
n→∞
In fact, since
so
n→∞
And since
and so
n→∞
Next, we show that x ∈ Ω : FS ∩
N
k1 MEP
Θk , ϕk ∩ FQ.
i We first show that x ∈ FS. Since {xn } is bounded, there exists a subsequence {xnj }
of {xn } such that xnj x ∈ H as j → ∞. From 3.31 and Lemma 2.5, we obtain
that x ∈ FS.
Θk ,ϕk k−1
k
Vn for k ∈
ii Now, we show that x ∈ N
k1 MEP Θk , ϕk . Since Vn Vrk,n
{1, 2, . . . , N}. Hence, for all x ∈ H and for all k ∈ {1, 2, . . . , N}, we obtain
1 k Kk Vn xn − Kk Vnk−1 xn , ηk x, Vnk xn
≥ 0.
Θk Vnk xn , x ϕk x − ϕk Vnk xn rk,n
3.62
And hence
Kk Vnk xnj − Kk Vnk−1 xnj
rk,n
, ηk x, Vnk xnj
≥ −Θk Vnk xnj , x − ϕk x
ϕk Vnk xnj ,
3.63
∀x ∈ H.
By the assumptions that ϕk is lower semicontinuous and by conditions H4, H5,
the mapping x → −Θk x, y is lower semicontinuous. So, they are weakly lower
24
Abstract and Applied Analysis
we have that Vnk xn x for all k ∈ {1, 2, . . . , N} and from
semicontinuous. Since xnj x,
ic, ii, 3.40. Now, taking lower limit as j → ∞ in 3.63, we obtain that
Θk x,
x ϕk x − ϕk x
≥ 0,
∀x ∈ H, ∀k ∈ {1, 2, . . . , N}.
3.64
Therefore x ∈ N
k1 MEP Θk , ϕk .
iii Now, we show that x ∈ FQ, where Q is defined as in Lemma 2.11. Since Q is
nonexpansive. Then, we have
vn − Qvn JM1 ,ρ1 yn − ρ1 B1 yn − Qvn JM1 ,ρ1 JM2 ,ρ2 zn − ρ2 B2 zn − ρ1 B1 JM2 ,ρ2 zn − ρ2 B2 zn − Qvn Qzn − Qvn 3.65
≤ zn − vn .
From 3.59, we have limn → ∞ vn − Qvn 0. Since xnj x and from 3.61, we also have
Hence, we obtain by Lemma 2.5 that x ∈ FQ.
vnj x.
Next, we show that {xn } is sequentially compact, namely, there is a subsequence
{xnj } ⊂ {xn } that converges strongly to x ∈ Ω as j → ∞. From 3.23, replacing x by x
to obtain
γφxn − xx
≤ u γfx
− I μA x,
xn − x .
n − x
3.66
we obtain that xnj → x
Now, replacing n with nj in 3.66 and letting j → ∞, since xnj x,
as j → ∞.
Next, we show that x is the unique solution of the variational inequality 3.5. Since
xn αn u γfxn βn xn 1
1 − βn I − αn I μA
tn
tn
T svn ds,
3.67
0
we derive that
−u I μA − γf xn
1 − βn
1 tn
xn −
T svn ds
−
αn
tn 0
1 tn
T svn ds .
I μA xn −
tn 0
3.68
Abstract and Applied Analysis
25
For all z ∈ Ω, it follows that
−u I μA − γf xn , xn − z
1 − βn
1 tn
1 tn
N
N
I−
−
T sQVn ds xn − I −
T sQVn ds z, xn − z
αn
tn 0
tn 0
3.69
1 tn
T svn ds , xn − z .
I μA xn −
tn 0
By Lemma 2.10, we obtain that I − σt · is monotone, where σt · : 1/t
Then, from 3.69, we have
−u I μA − γf xn , xn − z ≤ I μA xn − σt vn , xn − z .
t
0
T s ds.
3.70
Now, replacing n by nj in 3.70 and letting j → ∞ and xnj → x,
we notice that
1
xnj −
tnj
tn
j
T svnj ds → 0.
3.71
0
Then, we have
u γf − I μA x,
z − x ≤ 0,
∀z ∈ Ω.
3.72
That is, x is the solution of variational inequality 3.5.
Finally, we show that {xn } converges strongly to x ∈ Ω. Suppose that there exists
another subsequence xnk → x$ as k → ∞. We note Lemma 2.5 that x$ ∈ Ω is the solution
of the variational inequality 3.5. Hence x x$ x∗ by uniqueness. In summary, we have
shown that {xn } is sequentially compact and each cluster point of the sequence {xn } is equal
to x∗ . Then, we conclude that xn → x∗ as n → ∞. This completes the proof.
From Theorem 3.2, we can deduce the following result.
Theorem 3.3. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let ϕi : C →
R i 1, 2, . . . , N be a finite family of lower semicontinuous and convex function, Θi : C × C →
R i 1, 2, . . . , N be a finite family of bifunctions satisfying (H1)–(H5), ηi : C × H → H be a finite
family of Lipschitz continuous mappings with a constant σi i 1, 2, . . . , N. Let S {T t : t ∈ R }
be a nonexpansive semigroup from C into H and Bi : H → H i 1, 2 be an Li -Lipschitzian and
relaxed ci , di -cocoercive mapping with ρi ∈ 0, 2di − ci L2i /L2i for all i 1, 2. Let Q : C → C be
a mapping defined by
Qx : PC PC x − ρ2 B2 x − ρ1 B1 PC x − ρ2 B2 x .
3.73
Assume that Ω : FS ∩ N
∅. Let f : H → H be a φ-strongly
k1 MEPΘk , ϕk ∩ FQ /
pseudocontractive mapping with limt → ∞ φt ∞ and A be a strongly positive linear
26
Abstract and Applied Analysis
bounded operator on H with a coefficient γ > 0. Let μ > 0 and γ > 0 be two constants
such that 0 < γ < 1 μγ. Let {ri,n }i 1, 2, . . . , N be a finite family of positive real sequence
such that lim infn → ∞ ri,n > 0, {αn } and {βn } be two sequences in 0, 1, and {tn } be a positive
real divergent sequence. For any fixed u ∈ H, let {xn } be the sequence defined by
1 1 1
1
1
K1 un − K1 xn , η1 x, un
≥ 0, ∀x ∈ C,
Θ1 un , x ϕ1 x − ϕ1 un r1,n
1 2 2
2
1
2
Θ2 un , x ϕ2 x − ϕ2 un K2 un − K2 un , η1 x, un
≥ 0, ∀x ∈ C,
r2,n
..
.
1 N
N
N
N−1
N
KN
un
−KN
un
, ηN x, un
≥ 0, ∀x ∈ C,
ΘN un , x ϕN x−ϕN un
rN,n
N
N
yn PC un − ρ2 B2 un ,
1
xn αn u γfxn βn xn 1 − βn I − αn I μA
tn
tn
T sPC yn − ρ1 B1 yn ds,
0
3.74
where
1
Θ ,ϕ1 i
Θ ,ϕi i−1
un
un Vr1,n 1
un Vri,n i
Θ ,ϕi Vri,n i
Θ ,ϕi and Vri,n i
xn ,
Θ ,ϕi Vri,n i
Θ ,ϕ2 · · · Vr2,n 2
Θ
Vri−1,ni−1
Θ ,ϕ1 Vr1,n 1
,ϕi−1 i−2
un
xn ,
Θ ,ϕi Vri,n i
Θ ,ϕ2 1
un
· · · Vr2,n 2
3.75
i 2, 3, . . . , N,
: C → C, i 1, 2, . . . , N is the mapping defined by 2.8. Assume the following.
i ηi : C × C → R is Lipschitz continuous with constant σi > 0 i 1, 2, . . . , N such
that
a ηi x, y ηi y, x 0 for all x, y ∈ C,
b ηi ·, · is affine in the first variable,
c for each fixed y ∈ C, x → ηi y, x is sequentially continuous from the weak
topology to the weak topology.
ii Ki : C → R is ηi —strongly convex with constant μi > 0, and its derivative Ki is not
only continuous from the weak topology to the strong topology but also Lipschitz
continuous with constant νi such that μi ≥ σi νi .
iii For all i 1, 2, . . . , N and for all x ∈ C, there exists a bounded subset Dx ⊂ C and
/ Dx ,
zx ∈ C such that for all y ∈
1 Θi y, zn ϕi zx − ϕi y K y − Ki x, ηi zx , y < 0.
ri,n i
3.76
Abstract and Applied Analysis
27
If the following conditions are satisfied:
C1 limn → ∞ αn 0,
C2 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1,
then the sequence {xn } defined by 3.74 converges strongly to x∗ ∈ Ω : FS ∩
N
Θi ,ϕi is firmly nonexpansive, where x∗ is the unique
k1 MEPΘk , ϕk ∩ FQ, provided Vri,n
solution of the variational inequality
u γf − I μA x∗ , z − x∗ ≤ 0,
∀z ∈ Ω,
3.77
or, equivalently, x∗ is the unique solution of the optimization problem
μ
1
min Ax, x x − u2 − hx,
x∈Ω 2
2
3.78
where h is a potential function for γf and x∗ , y∗ is the solution of general system of
variational inequality problem
ρ1 B1 y∗ x∗ − y∗ , x − x∗ ≥ 0,
ρ2 B2 x∗ y∗ − x∗ , x − y∗ ≥ 0,
∀x ∈ C,
∀x ∈ C.
3.79
Proof. From Theorem 3.2, taking M1 M2 ∂δC , where C is a nonempty closed convex
subset of H and δC : H → 0, ∞ is the indicator function of C, then, we have JM1 ,ρ1 JM2 ,ρ2 PC and the quasivariational inclusion problem 1.17 is equivalent to the classical
variational inequality 1.19. Thus, we can get the desired conclusion immediately.
Acknowledgments
This paper was supported by the Center of Excellence in Mathematics, the Commission on
Higher Education, Thailand under Project no. RG-1-54-02-1. P. Sunthrayuth was partially
supported by the Centre of Excellence in Mathematics for the Ph.D. Program at KMUTT.
Moreover, the authors are greatly indebted to the reviewers for their extremely constructive
comments and valuable suggestions leading to the revised version.
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