Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 261534, 12 pages
doi:10.1155/2011/261534
Research Article
H·, ·-Cocoercive Operator and an Application for
Solving Generalized Variational Inclusions
Rais Ahmad,1 Mohd Dilshad,1
Mu-Ming Wong,2 and Jen-Chin Yao3, 4
1
Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
Department of Applied Mathematics, Chung Yuan Christian University, Chung Li 32023, Taiwan
3
Center for General Education, Kaohsiung Medical University, Kaohsiung 807, Taiwan
4
Department of Applied Mathematics, National Sun-Yat Sen University, Kaohsiung 804, Taiwan
2
Correspondence should be addressed to Mu-Ming Wong, mmwong@cycu.edu.tw
Received 15 April 2011; Accepted 25 June 2011
Academic Editor: Ngai-Ching Wong
Copyright q 2011 Rais Ahmad et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The purpose of this paper is to introduce a new H·, ·-cocoercive operator, which generalizes
many existing monotone operators. The resolvent operator associated with H·, ·-cocoercive
operator is defined, and its Lipschitz continuity is presented. By using techniques of resolvent
operator, a new iterative algorithm for solving generalized variational inclusions is constructed.
Under some suitable conditions, we prove the convergence of iterative sequences generated by the
algorithm. For illustration, some examples are given.
1. Introduction
Various concepts of generalized monotone mappings have been introduced in the literature.
Cocoercive mappings which are generalized form of monotone mappings are defined by
Tseng 1, Magnanti and Perakis 2, and Zhu and Marcotte 3. The resolvent operator
techniques are important to study the existence of solutions and to develop iterative schemes
for different kinds of variational inequalities and their generalizations, which are providing
mathematical models to some problems arising in optimization and control, economics,
and engineering sciences. In order to study various variational inequalities and variational
inclusions, Fang and Huang, Lan, Cho, and Verma investigated many generalized operators
such as H-monotone 4, H-accretive 5, H, η-accretive 6, H, η-monotone7, 8, A, ηaccretive mappings 9. Recently, Zou and Huang 10 introduced and studied H·, ·accretive operators and Xu and Wang 11 introduced and studied H·, ·, η-monotone
operators.
2
Abstract and Applied Analysis
Motivated and inspired by the excellent work mentioned above, in this paper, we
introduce and discuss new type of operators called H·, ·-cocoercive operators. We define
resolvent operator associated with H·, ·-cocoercive operators and prove the Lipschitz
continuity of the resolvent operator. We apply H·, ·-cocoercive operators to solve a
generalized variational inclusion problem. Some examples are constructed for illustration.
2. Preliminaries
Throughout the paper, we suppose that X is a real Hilbert space endowed with a norm · and an inner product ·, ·, d is the metric induced by the norm · , 2X resp., CBX is the
family of all nonempty resp., closed and bounded subsets of X, and D·, · is the Hausdorff
metric on CBX defined by
DP, Q max sup dx, Q, sup d P, y ,
2.1
x∈P
y∈Q
where dx, Q infy∈Q dx, y and dP, y infx∈P dx, y.
Definition 2.1. A mapping g : X → X is said to be
i Lipschitz continuous if there exists a constant λg > 0 such that
gx − g y ≤ λg x − y,
∀x, y ∈ X;
2.2
ii monotone if
gx − g y , x − y ≥ 0,
∀x, y ∈ X;
2.3
iii strongly monotone if there exists a constant ξ > 0 such that
2
gx − g y , x − y ≥ ξx − y ,
∀x, y ∈ X;
2.4
iv α-expansive if there exists a constant α > 0 such that
gx − g y ≥ αx − y,
∀x, y ∈ X,
2.5
if α 1, then it is expansive.
Definition 2.2. A mapping T : X → X is said to be cocoercive if there exists a constant μ
> 0
such that
2
T x − T y , x − y ≥ μ
T x − T y ,
∀x, y ∈ X.
2.6
Note 1. Clearly T is 1/μ
-Lipschitz continuous and also monotone but not necessarily strongly
monotone and Lipschitz continuous consider a constant mapping. Conversely, strongly
Abstract and Applied Analysis
3
monotone and Lipschitz continuous mappings are cocoercive, and it follows that cocoercivity
is an intermediate concept that lies between simple and strong monotonicity.
Definition 2.3. A multivalued mapping M : X → 2X is said to be cocoercive if there exists a
constant μ
> 0 such that
u − v, x − y ≥ μ
u − v2 ,
∀x, y ∈ X, u ∈ Mx, v ∈ M y .
2.7
Definition 2.4. A mapping T : X → X is said to be relaxed cocoercive if there exists a constant
γ > 0 such that
2
T x − T y , x − y ≥ −γ T x − T y ,
∀x, y ∈ X.
2.8
Definition 2.5. Let H : X × X → X and A, B : X → X be the mappings.
i HA, · is said to be cocoercive with respect to A if there exists a constant μ > 0 such
that
2
HAx, u − H Ay, u , x − y ≥ μAx − Ay ,
∀x, y ∈ X;
2.9
ii H·, B is said to be relaxed cocoercive with respect to B if there exists a constant
γ > 0 such that
2
Hu, Bx − H u, By , x − y ≥ −γ Bx − By ,
∀x, y ∈ X;
2.10
iii HA, · is said to be r1 -Lipschitz continuous with respect to A if there exists a
constant r1 > 0 such that
HAx, · − H Ay, · ≤ r1 x − y,
∀x, y ∈ X;
2.11
iv H·, B is said to be r2 -Lipschitz continuous with respect to B if there exists a
constant r2 > 0 such that
H·, Bx − H ·, By ≤ r2 x − y,
∀x, y ∈ X.
2.12
Example 2.6. Let X R2 with usual inner product. Let A, B : R2 → R2 be defined by
Ax 2x1 − 2x2 , −2x1 4x2 ,
By −y1 y2 , −y2 , ∀x1 , x2 , y1 , y2 ∈ R2 .
2.13
Suppose that HA, B : R2 × R2 → R2 is defined by
H Ax, By Ax By,
∀x, y ∈ R2 .
2.14
4
Abstract and Applied Analysis
Then HA, B is 1/6-cocoercive with respect to A and 1/2-relaxed cocoercive with respect
to B since
HAx, u − H Ay, u , x − y Ax − Ay, x − y
2x1 − 2x2 , −2x1 4x2 − 2y1 − 2y2 , −2y1 4y2 ,
x1 − y1 , x2 − y2
2 x1 − y1 − 2 x2 − y2 , −2 x1 − y1 4 x2 − y2 ,
x1 − y1 , x2 − y2
2
2
2 x1 − y1 4 x2 − y2 − 4 x1 − y1 x2 − y2 ,
Ax − Ay2 2x1 − 2x2 , −2x1 4x2 − 2y1 − 2y2 , −2y1 4y2 ,
2x1 − 2x2 , −2x1 4x2 − 2y1 − 2y2 , −2y1 4y2
2
2
8 x1 − y1 20 x2 − y2 − 24 x1 − y1 x2 − y2
2
2
≤ 12 x1 − y1 24 x2 − y2 − 24 x1 − y1 x2 − y2
2
2
6 2 x1 − y1 4 x2 − y2 − 4 x1 − y1 x2 − y2
6 Hu, Ax − H u, Ay , x − y ,
2.15
which implies that
2
1
HAx, u − H Ay, u , x − y ≥ Ax − Ay ,
6
2.16
That is, HA, B is 1/6-cocoercive with respect to A.
Hu, Bx − H u, By , x − y Bx − By, x − y
−x1 x2 , −x2 − −y1 y2 , −y2 , x1 − y1 , x2 − y2
− x1 − y1 x2 − y2 , − x2 − y2 , x1 − y1 , x2 − y2
2 2 − x1 − y1 − x2 − y2 x1 − y1 x2 − y2
−
x1 − y1
2
2 x2 − y2 − x1 − y1 x2 − y2 ,
Bx − By2 − x1 − y1 x2 − y2 , − x2 − y2 ,
− x1 − y1 x2 − y2 , − x2 − y2
Abstract and Applied Analysis
5
2
2
x1 − y1 2 x2 − y2 − 2 x1 − y1 x2 − y2
2 2 ≤ 2 x1 − y1 x2 − y2 − x1 − y1 x2 − y2
2−1 HBx, u − H By, u , x − y
2.17
which implies that
2
1
Hu, Bx − H u, By , x − y ≥ − Bx − By ,
2
2.18
that is, HA, B is 1/2-relaxed cocoercive with respect to B.
3. H·, ·-Cocoercive Operator
In this section, we define a new H·, ·-cocoercive operator and discuss some of its properties.
Definition 3.1. Let A, B : X → X, H : X × X → X be three single-valued mappings. Let
M : X → 2X be a set-valued mapping. M is said to be H·, ·-cocoercive with respect
to mappings A and B or simply H·, ·-cocoercive in the sequel if M is cocoercive and
HA, B λMX X, for every λ > 0.
Example 3.2. Let X, A, B, and H be the same as in Example 2.6, and let M : R2 → R2 be
define by Mx1 , x2 0, x2 , ∀x1 , x2 ∈ R2 . Then it is easy to check that M is cocoercive and
HA, B λMR2 R2 , ∀λ > 0, that is, M is H·, ·-cocoercive with respect to A and B.
Remark 3.3. Since cocoercive operators include monotone operators, hence our definition
is more general than definition of H·, ·-monotone operator 10. It is easy to check that
H·, ·-cocoercive operators provide a unified framework for the existing H·, ·-monotone,
H-monotone operators in Hilbert space and H·, ·-accretive, H-accretive operators in Banach
spaces.
Since H·, ·-cocoercive operators are more general than maximal monotone operators,
we give the following characterization of H·, ·-cocoercive operators.
Proposition 3.4. Let HA, B be μ-cocoercive with respect to A, γ-relaxed cocoercive with respect
to B, A is α-expansive, B is β-Lipschitz continuous, and μ > γ, α > β. Let M : X → 2X be
H·, ·-cocoercive operator. If the following inequality
x − y, u − v ≥ 0
3.1
holds for all v, y ∈ GraphM, then x ∈ Mu, where
GraphM {x, u ∈ X × X : u ∈ Mx}.
3.2
6
Abstract and Applied Analysis
Proof. Suppose that there exists some u0 , x0 such that
x0 − y, u0 − v ≥ 0,
∀ v, y ∈ GraphM.
3.3
Since M is H·, ·-cocoercive, we know that HA, B λMX X holds for every λ > 0,
and so there exists u1 , x1 ∈ GraphM such that
HAu1 , Bu1 λx1 HAu0 , Bu0 λx0 ∈ X.
3.4
It follows from 3.3 and 3.4 that
0 ≤ λx0 HAu0 , Bu0 − λx1 − HAu1 , Bu1 , u0 − u1 ,
0 ≤ λx0 − x1 , u0 − u1 −HAu0 , Bu0 − HAu1 , Bu1 , u0 − u1 −HAu0 , Bu0 − HAu1 , Bu0 , u0 − u1 −HAu1 , Bu0 − HAu1 , Bu1 , u0 − u1 3.5
≤ −μAu0 − Au1 2 γBu0 − Bu1 2
≤ −μα2 u0 − u1 2 γβ2 u0 − u1 2
− μα2 − γβ2 u0 − u1 2 ≤ 0,
which gives u1 u0 since μ > γ, α > β. By 3.4, we have x1 x0 . Hence u0 , x0 u1 , x1 ∈
GraphM and so x0 ∈ Mu0 .
Theorem 3.5. Let X be a Hilbert space and M : X → 2X a maximal monotone operator. Suppose
that H : X × X → X is a bounded cocoercive and semicontinuous with respect to A and B. Let
H : X × X → X be also μ-cocoercive with respect to A and γ-relaxed cocoercive with respect to
B. The mapping A is α-expansive, and B is β-Lipschitz continuous. If μ > γ and α > β, then M is
H·, ·-cocoercive with respect to A and B.
Proof. For the proof we refer to 10.
Theorem 3.6. Let HA, B be a μ-cocoercive with respect to A and γ-relaxed cocoercive with respect
to B, A is α-expansive, and B is β-Lipschitz continuous, μ > γ and α > β. Let M be an H·, ·cocoercive operator with respect to A and B. Then the operator HA, B λM−1 is single-valued.
Proof. For any given u ∈ X, let x, y ∈ HA, B λM−1 u. It follows that
−HAx, Bx u ∈ λMx,
−H Ay, By u ∈ λMy.
3.6
Abstract and Applied Analysis
7
As M is cocoercive thus monotone, we have
0 ≤ −HAx, Bx u − −H Ay, By u , x − y
− HAx, Bx − H Ay, By , x − y
− HAx, Bx − H Ay, Bx H Ay, Bx − H Ay, By , x − y
− HAx, Bx − H Ay, Bx , x − y − H Ay, Bx − H Ay, By , x − y .
3.7
Since H is μ-cocoercive with respect to A and γ-relaxed cocoercive with respect to B, A is
α-expansive and B is β-Lipschitz continuous, thus 3.7 becomes
2
2
2
0 ≤ −μα2 x − y γβ2 x − y − μα2 − γβ2 x − y ≤ 0
3.8
since μ > γ, α > β. Thus, we have x y and so HA, B λM−1 is single-valued.
Definition 3.7. Let HA, B be μ-cocoercive with respect to A and γ-relaxed cocoercive with
respect to B, A is α-expansive, B is β-Lipschitz continuous, and μ > γ, α > β. Let M be an
H·,·
H·, ·-cocoercive operator with respect to A and B. The resolvent operator Rλ,M : X → X is
defined by
Rλ,M u HA, B λM−1 u,
H·,·
∀u ∈ X.
3.9
Now, we prove the Lipschitz continuity of resolvent operator defined by 3.9 and
estimate its Lipschitz constant.
Theorem 3.8. Let HA, B be μ-cocoercive with respect to A, γ-relaxed cocoercive with respect to
B, A is α-expansive, B is β-Lipschitz continuous, and μ > γ, α > β. Let M be an H·, ·-cocoercive
H·,·
operator with respect to A and B. Then the resolvent operator Rλ,M : X → X is 1/μα2 − γβ2 Lipschitz continuous, that is,
H·,·
H·,·
Rλ,M u − Rλ,M v ≤
1
u − v,
μα2 − γβ2
∀u, v ∈ X.
3.10
Proof. Let u and v be any given points in X. It follows from 3.9 that
Rλ,M u HA, B λM−1 u,
H·,·
Rλ,M v HA, B λM−1 v.
H·,·
3.11
8
Abstract and Applied Analysis
This implies that
1
H·,·
H·,·
H·,·
∈ M Rλ,M u ,
u − H A Rλ,M u , B Rλ,M u
λ
1
H·,·
H·,·
H·,·
∈ M Rλ,M v .
v − H A Rλ,M v , B Rλ,M v
λ
3.12
For the sake of clarity, we take
H·,·
P u Rλ,M u,
H·,·
P v Rλ,M v.
3.13
Since M is cocoercive hence monotone, we have
1
u − HAP u, BP u − v − HAP v, BP v, P u − P v ≥ 0,
λ
1
u − v − HAP u, BP u HAP v, BP v, P u − P v ≥ 0,
λ
3.14
which implies that
u − v, P u − P v ≥ HAP u, BP u − HAP v, BP v, P u − P v.
3.15
Further, we have
u − vP u − P v ≥ u − v, P u − P v
≥ HAP u, BP u − HAP v, BP v, P u − P v
HAP u, BP u − HAP v, BP u HAP v, BP u
−HAP v, BP v, P u − P v
HAP u, BP u − HAP v, BP u, P u − P v
HAP v, BP u − HAP v, BP v, P u − P v
≥ μAP u − AP v2 − γBP u − BP v2
≥ μα2 P u − P v2 − γβ2 P u − P v2 ,
3.16
and so
u − vP u − P v ≥ μα2 − γβ2 P u − P v2 ,
3.17
Abstract and Applied Analysis
9
thus,
1
u − v,
μα2 − γβ2
1
H·,·
H·,·
that is Rλ,M u − Rλ,M v ≤
u − v,
2
μα − γβ2
P u − P v ≤
3.18
∀u, v ∈ X.
This completes the proof.
4. Application of H·, ·-Cocoercive Operators for
Solving Variational Inclusions
We apply H·, ·-cocoercive operators for solving a generalized variational inclusion problem.
We consider the problem of finding u ∈ X and w ∈ T u such that
0 ∈ w M gu ,
4.1
where g : X → X, M : X → 2X , and T : X → CBX are the mappings. Problem 4.1 is
introduced and studied by Huang 12 in the setting of Banach spaces.
Lemma 4.1. The u, w, where u ∈ X, w ∈ T u, is a solution of the problem 4.1, if and only if
u, w is a solution of the following:
H·,· gu Rλ,M H A gu , B gu − λw ,
4.2
where λ > 0 is a constant.
H·,·
Proof. By using the definition of resolvent operator Rλ,M , the conclusion follows directly.
Based on 4.2, we construct the following algorithm.
Algorithm 4.2. For any u0 ∈ X, w0 ∈ T u0 , compute the sequences {un } and {wn } by iterative
schemes such that
H·,· gun1 Rλ,M H A gun , B gun − λwn ,
1
DT un , T un1 ,
wn ∈ T un , wn − wn1 ≤ 1 n1
4.3
for all n 0, 1, 2, . . ., and λ > 0 is a constant.
Theorem 4.3. Let X be a real Hilbert space and A, B, g : X → X, H : X ×X → X the single-valued
mappings. Let T : X → CBX be a multi-valued mapping and M : X → 2X the multi-valued
H·, ·-cocoercive operator. Assume that
i T is δ-Lipschitz continuous in the Hausdorff metric D·, ·;
ii HA, B is μ-cocoercive with respect to A and γ-relaxed cocoercive with respect to B;
10
Abstract and Applied Analysis
iii A is α-expansive;
iv B is β-Lipschitz continuous;
v g is λg -Lipschitz continuous and ξ-strongly monotone;
vi HA, B is r1 -Lipschitz continuous with respect to A and r2 -Lipschitz continuous with
respect to B;
vii r1 r2 λg < μα2 − γβ2 ξ − λδ; μ > γ, α > β.
Then the generalized variational inclusion problem 4.1 has a solution u, w with
u ∈ X, w ∈ T u, and the iterative sequences {un } and {wn } generated by Algorithm 4.2
converge strongly to u and w, respectively.
Proof. Since T is δ-Lipschitz continuous, it follows from Algorithm 4.2 that
wn − wn1 ≤
1
1
1
DT un , T un1 ≤ 1 δun − un1 ,
n1
n1
4.4
for n 0, 1, 2, . . ..
Using the ξ-strong monotonicity of g, we have
gun1 − gun un1 − un ≥ gun1 − gun , un1 − un
≥ ξun1 − un 2
4.5
which implies that
un1 − un ≤
1
gun1 − gun .
ξ
4.6
H·,·
Now we estimate gun1 − gun by using the Lipschitz continuity of Rλ,M ,
H·,· gun1 − gun Rλ,M H A gun , B gun − λwn
H·,· −Rλ,M H A gun−1 , B gun−1 − λwn−1 ≤
μα2
≤
1
H A gun , B gun − H A gun−1 , B gun−1 2
− γβ
λ
wn − wn−1 μα2 − γβ2
1
H A gun , B gun − H A gun−1 , B gun μα2 − γβ2
μα2
1
H A gun−1 , B gun − H A gun−1 , B gun−1 2
− γβ
μα2
λ
wn − wn−1 .
− γβ2
4.7
Abstract and Applied Analysis
11
Since HA, B is r1 -Lipschitz continuous with respect to A and r2 -Lipschitz continuous with
respect to B, g is λg -Lipschitz continuous and using 4.4, 4.7 becomes
gun1 − gun ≤
r1 λg
μα2
−
γβ2
un − un−1 r2 λg
μα2 − γβ2
un − un−1 1
λ
δun − un−1 1
n
μα2 − γβ2
4.8
or
gun1 − gun ≤
λ
1
δ un − un−1 .
1
n
μα2 − γβ2 μα2 − γβ2 μα2 − γβ2
r1 λg
r2 λg
4.9
Using 4.9, 4.6 becomes
un1 − un ≤ θn un − un−1 ,
4.10
where
θn r1 r2 λg λδ1 1/n
.
2
μα − γβ2 ξ
4.11
r1 r2 λg λδ
θ 2
.
μα − γβ2 ξ
4.12
Let
We know that θn → θ and n → ∞. From assumption vii, it is easy to see that θ < 1.
Therefore, it follows from 4.10 that {un } is a Cauchy sequence in X. Since X is a Hilbert
space, there exists u ∈ X such that un → u as n → ∞. From 4.4, we know that {wn } is
also a Cauchy sequence in X, thus there exists w ∈ X such that wn → w and n → ∞. By the
H·,·
continuity of g, Rλ,M H, A, B, and T and Algorithm 4.2, we have
H·,· gu Rλ,M H A gu , B gu − λw .
4.13
Now, we prove that w ∈ T u. In fact, since wn ∈ T un , we have
dw, T u ≤ w − wn dwn , T u
≤ w − wn DT un , T u
≤ w − wn δun − u −→ 0,
4.14
as n −→ ∞,
which implies that dw, T u 0. Since T u ∈ CBX, it follows that w ∈ T u. By
Lemma 4.1, we know that u, w is a solution of problem 4.1. This completes the proof.
12
Abstract and Applied Analysis
Acknowledgments
This work is partially done during the visit of the first author to National Sun-Yat Sen
University, Kaohsiung, Taiwan. The first and second authors are supported by Department
of Science and Technology, Government of India under Grant no. SR/S4/MS: 577/09. The
fourth author is supported by Grant no. NSC 99-2221-E-037-007-MY3.
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Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014