Fixed Point Theory and Applications Volume 2010, Article ID 794503,22pages doi:10.1155/2010/794503
Research Article
Algorithm for Solving a Generalized
Mixed Equilibrium Problem with Perturbation
in a Banach Space
Lu-Chuan Ceng,
1Sangho Kum,
2and Jen-Chih Yao
31Department of Mathematics, Shanghai Normal University, Scientific Computing Key Laboratory of
Shanghai Universities, Shanghai 200234, China
2Department of Mathematics Education, Chungbuk National University, Cheongju 361763, Republic of
Korea
3Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan
Correspondence should be addressed to Jen-Chih Yao,[email protected]
Received 22 February 2010; Accepted 11 April 2010
Academic Editor: Wataru Takahashi
Copyrightq2010 Lu-Chuan Ceng 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.
LetBbe a real Banach space with the dual spaceB∗. Letφ:B → R∪ {∞}be a proper functional and letΘ : B×B → Rbe a bifunction. In this paper, a new concept ofη-proximal mapping ofφwith respect toΘ is introduced. The existence and Lipschitz continuity of theη-proximal mapping ofφ with respect to Θ are proved. By using properties of theη-proximal mapping ofφwith respect toΘ, a generalized mixed equilibrium problem with perturbation for short, GMEPPis introduced and studied in Banach spaceB. An existence theorem of solutions of the GMEPP is established and a new iterative algorithm for computing approximate solutions of the GMEPP is suggested. The strong convergence criteria of the iterative sequence generated by the new algorithm are established in a uniformly smooth Banach spaceB, and the weak convergence criteria of the iterative sequence generated by this new algorithm are also derived inB Ha Hilbert space.
1. Introduction
LetXbe a real Banach space with norm · and letX∗be its dual space. The value off ∈B∗ atx∈Bwill be denoted byf, x . The normalized duality mappingJfromBinto the family of nonemptyby Hahn-Banach theoremweak-star compact subsets of its dual spaceB∗ is defined by
It is known that the norm ofBis said to be Gateaux differentiableandBis said to be smooth if
lim t→0
xty − x
t 1.2
exists for every x, y in U {x ∈ B : x 1}, the unit sphere of B. It is said to be uniformly Gateaux differentiable if for each y ∈ U, this limit is attained uniformly for x ∈ U. The norm of B is said to be uniformly Frechet differentiable and B is said to be uniformly smooth if the limit in 1.2 is attained uniformly for x, y ∈ U× U. Every uniformly smooth Banach space Bis reflexive and has a uniformly Gateaux differentiable norm.
Recall also that ifBis smooth, thenJis single-valued and continuous from the norm topology ofBto the weak star topology ofB∗, that is, norm-to-weak∗ continuous. It is also well known that if B has a uniformly Gateaux differentiable norm, then J is uniformly continuous on bounded subsets ofBfrom the strong topology ofBto the weak star topology ofB∗, that is, uniformly norm-to-weak∗continuous on any bounded subset ofB. Moreover, ifBis uniformly smooth, thenJis uniformly continuous on bounded subsets ofBfrom the strong topology ofBto the strong topology ofB∗, that is, uniformly norm-to-norm continuous on any bounded subset ofB. See1 for more details.
It is well known that the variational inequality theory has played an important and powerful role in the studying of a wide class of linear and nonlinear problems arising in many diverse fields of pure and applied sciences, such as mathematical programming, optimization theory, engineering, elasticity theory and equilibrium problems of mathematical economics, and game theory; see, for instance, 1–6 and the references therein.
One of the most interesting and important problems in the theory of variational inequalities is the development of an efficient iterative algorithm to compute approximate solutions. In the setting of Hilbert spaces, one of the most efficient numerical techniques is the projection method and its variant forms; see 4,6–15 . Since the standard projection
method strictly depends on the inner product property of Hilbert spaces, it can no longer be applied for general mixed type variational inequalities in Banach spaces. This fact motivates us to develop alterative methods to study the existence and iterative algorithms of solutions for general mixed variational inequalities in Banach spaces. Recently, 16–
18 extended the auxiliary principle technique to study the existence of solutions and to suggest the iterative algorithms for solving various mixed type variational inequalities in Banach spaces. For some related work, we refer to 19–21 and the references therein.
LetB be a real Banach space with the dual spaceB∗. Let φ : B → R∪ {∞}be a proper functional and letΘ : B×B → Rbe a bifunction. In this paper, motivated by Xia and Huang14 , we first introduce a new concept ofη-proximal mapping ofφwith respect toΘ. We prove an existence theorem and Lipschitz continuity of theη-proximal mapping of φwith respect toΘ. Utilizing the properties of theη-proximal mapping ofφwith respect to Θ, we introduce and consider a generalized mixed equilibrium problem with perturbation for short, GMEPPwhich includes as a special case the general mixed variational inequality studied by Xia and Huang14 . We show an existence theorem of solutions for this problem
under some appropriate conditions. In order to compute approximate solutions of the GMEPP, we propose a new iterative algorithm which includes as a special case the iterative algorithm considered by Xia and Huang14 . Finally, we establish the strong convergence
criteria of the iterative sequence generated by the new algorithm in a uniformly smooth Banach space B, and also derive the weak convergence criteria of the iterative sequence generated by this new algorithm in B H a Hilbert space. Our results are new and represent the improvement, extension, and development of Xia and Huang’s results in 14 .
2. Preliminaries
LetBbe a real Banach space with the topological dual spaceB∗and letf, x be the pairing betweenf ∈ B∗ andx ∈ B. We writexn xto indicate that the sequence{xn}converges weakly tox.xn → ximplies that{xn}converges strongly tox. Let 2B∗andCBB∗denote the family of all subsets ofB∗and the family of all nonempty closed bounded subsets ofB∗, respectively. LetΘ:B×B → Rbe a bifunction, letT, A:B → B∗andg :B → Bbe single-valued mappings, and letφ:B → R∪{∞}be a proper lower semicontinuous functional. We consider the following generalized mixed equilibrium problem with perturbationfor short, GMEPP: findu∈Bsuch that
Θgu, vφv−φguTu−Au, v−gu ≥0, ∀v∈B. 2.1
Some special cases of problem2.1are the following.
1If B H a real Hilbert space, g I an identity mapping on H, and φ is a lower semicontinuous and convex functional, then GMEPP 2.1 reduces to the generalized mixed equilibrium problem with perturbation considered by Ceng et al.22 .
2IfB Ha real Hilbert space,g I an identity mapping onH,A 0, andφ is a lower semicontinuous and convex functional, then GMEPP2.1reduces to the generalized mixed equilibrium problem considered by Peng and Yao23 .
3If Θ 0, then GMEPP2.1reduces to the general mixed variational inequality problemfor short, GMVIPconsidered by Xia and Huang14 .
4IfBHa real Hilbert space,Θ 0, andφis a proper convex lower semicontinuous functional, then GMEPP2.1was studied by many authorssee, e.g.,1,17–19 .
Definition 2.1. LetT :B → 2B∗be a set-valued mapping, and letA:B → B∗andg :B → B be two single-valued mappings. We say that
iAisα-strongly monotone with constantα >0 if, for anyx, y∈B,
Ax−Ay, x−y ≥αx−y2; 2.2
iiT isλ-strongly monotone if, for anyx, y∈B, u∈Tx, andv∈Ty,
u−v, x−y≥λx−y2; 2.3
iiiT isβ-Lipschitz continuous with constantβ >0 if, for allx, y∈B,
HTx, Ty≤βx−y, 2.4
whereH·,·is the Hausdorffmetric on 2B∗;
ivgisk-strongly accretivewherek∈0,1if, for anyx, y∈B, there existsjx−y∈ Jx−ysuch that
jx−y, gx−gy ≥kx−y2, 2.5
whereJ:B → 2B∗is the normalized duality mapping defined by
Jx f ∈B∗:f, x f · xf2x2, ∀x∈B. 2.6
Definition 2.2. Let B be a Banach space with the dual spaceB∗, let Θ : B×B → R be a bifunction, and letφ : B → R∪ {∞}be a proper functional. If, for x ∈ B, there exists a f∈B∗such that
Θx, yφy−φx≥ f, y−x , ∀y∈B, 2.7
thenφis said to beΘ-subdifferentiable atx. We denote by∂Θφxthe set of such elements f∈B∗, that is,
∂Θφx f ∈B∗:Θx, yφy−φx≥ f, y−x , ∀y∈B . 2.8
The set∂Θφxis said to be theΘ-subdifferential ofφatx. If there exists theΘ-subdifferential
Definition 2.3. LetBbe a Banach space with the dual spaceB∗, and letΘ:B×B → R, η:B → B∗, φ:B → R∪ {∞}be a proper subdifferentiable functional. If for any givenx∗ ∈B∗and any constantρ >0, there is a uniquex∈Bsatisfying
Θx, yφy−φx 1
ρηx−x∗, y−x ≥0, ∀y∈B, 2.9
then the mappingx∗ →x, denoted byxJρΘ,φx∗, is said to be anη-proximal mapping of φwith respect toΘ.
Remark 2.4. From Definitions2.2and2.3it follows thatφisΘ-subdifferentiable at eachx ∈ RJρΘ,φthe range ofJρΘ,φ. Ifφis additionallyΘ-subdifferentiable at eachx∈B\RJρΘ,φ, then there exists theΘ-subdifferential∂Θφxat eachx∈B; that is,φisΘ-subdifferentiable. Observe thatx∗−ηx∈ρ∂Θφxand so
xJρΘ,φx∗ ηρ∂Θφ−1x∗. 2.10
In particular, wheneverΘ 0, then the concept ofη-proximal mapping ofφ with respect toΘreduces to the one ofη-proximal mapping ofφby Xia and Huang14, Definition 2.2 . In this case,JρΘ,φ is rewritten asJρφ. By the definition of the subdifferential, we know that x∗−ηx∈ρ∂φxand so
xJρφx∗ ηρ∂φ−1x∗. 2.11
Lemma 2.5 see24 . Let D be a nonempty convex subset of a topological vector space and let f:D×D → −∞,∞ be such that
ifor eachx∈D, y→fx, yis lower semicontinuous on each nonempty compact subset of D;
iifor each nonempty finite set {x1, . . . , xm} ⊂Dand for eachyim1λixiwithλi≥0and
m i1λi1,
min
1≤i≤mf
xi, y≤0; 2.12
iiithere exist a nonempty compact convex subsetD0ofDand a nonempty compact subsetK
ofDsuch that for eachy∈D\K, there is anx∈coD0∪ {y}withfx, y>0.
Then there existsy∈Ksuch thatfx,y≤0for allx∈D.
Recall now thatBsatisfies Opial’s property25 provided that, for each sequence{xn} inB, the conditionxn ximplies
lim sup
n→ ∞ xn−x<lim supn→ ∞ xn−y, ∀y∈X, y /x. 2.13
It is known26 that any separable Banach space can be equivalently renormed so that it satisfies Opial’s property.
Furthermore, recall that in a Hilbert space, there holds the following equality:
λx 1−λy2λx2 1−λy2−λ1−λx−y2 2.14
for allx, y∈Handλ∈0,1 .
In order to investigate the generalized mixed equilibrium problem 2.1 with perturbation, we will need the following conditions on mappingη:B → B∗and bifunction Θ:B×B → Rin the sequel:
H1 Θx, x 0, for allx∈B;
H2 Θis monotone, that is,Θx, y Θy, x≤0, for allx, y∈B; H3for eachx∈B, y→Θx, yis convex and lower semicontinuous;
H4the following equilibrium problem for short, EP or generalized equilibrium problemfor short, GEPhas a solution yin domφ:
EPFindy∈Bsuch thatΘy, x ≥0,for allx∈B, or
GEPFindy∈Bsuch thatΘy, x ηy, x −y ≥0,for allx∈B.
Now we give some sufficient conditions which guarantee the existence and Lipschitz continuity of anη-proximal mapping ofφwith respect toΘin a reflexive Banach spaceB.
Theorem 2.6. Let B be a reflexive Banach space with the dual space B∗, let Θ : B× B → R be a bifunction satisfying conditions (H1)–(H4), letη : B → B∗ be an α-strongly monotone and continuous mapping, and letφ:B → R∪ {∞}be a lower semicontinuous subdifferentiable proper functional. Then for any givenx∗∈B∗and anyρ >0, there exists a uniquex∈Bsuch that
Θx, y φy−φx 1
ρ
ηx−x∗, y−x≥0, ∀y∈B; 2.15
that is,x JρΘ,φx∗and theη-proximal mapping ofφ with respect toΘis well defined. If φ is additionallyΘ-subdifferentiable at eachx∈B\RJρΘ,φ, thenφisΘ-subdifferentiable andJρΘ,φ ηρ∂Θφ−1.
Proof. For any givenx∗ ∈ B∗and anyρ >0, define a functional f : B×B → R∪ {∞}as follows:
fy, xx∗−ηx, y−x ρφx−ρφyρΘy, x, ∀x, y∈B. 2.16
By the continuity ofηand the lower semicontinuity ofφandx→Θy, x, we know that the functionx→fy, xis lower semicontinuous onBfor each fixedy∈B.
Now, let us show thatfy, xsatisfies conditioniiofLemma 2.5. If it is false, then there exist a finite set{y1, . . . , ym} ⊂Bandx0mi1λiyiwithλi ≥0 and
m
i1λi1 such that
x∗−ηx
0, yi−x0 ρφx0−ρφ
yiρΘyi, x0
Sinceφis subdifferentiable atx0, there exists a pointf∗∈B∗such that
ρφyi−ρφx0≥ρf∗, yi−x0 , ∀i1, . . . , m. 2.18
FromH2it follows that
Θx0, yi≤ −Θyi, x0
, ∀i1, . . . , m. 2.19
Utilizing the convexity ofx→Θy, x, we get fromH1
0x∗−ηx0ρf∗, x0−x0 ρΘx0, x0
m i1
λix∗−ηx0, x0−yi ρf∗, yi−x0
ρΘ
x0,
m
i1
λiyi
≤m i1
λix∗−ηx0, x0−yi ρf∗, yi−x0 ρΘ
x0, yi
≤m i1
λix∗−ηx0, x0−yi ρφ
yi−ρφx0−ρΘ
yi, x0
<0,
2.20
which leads to a contradiction. Therefore,fy, xsatisfies conditioniiofLemma 2.5. From H4 we know that the following equilibrium problem for short, EP or generalized equilibrium problemfor short, GEPhas a solutiony∈dom φ:
EP Θy, x ≥0, for allx∈B,or
GEP Θy, x ηy, x −y ≥0,for all x∈B.
Sinceφis subdifferentiable aty, there exists a pointf∗∈B∗such that
φx−φy≥ f∗, x−y , ∀x∈B. 2.21
It follows that
fy, x x∗−ηx,y−x ρφx−ρφyρΘy, x
≥ηy−ηx,y−xx∗−ηy, y−xρf∗, x−yρΘy, x . 2.22
Case 1. If EP has solutiony∈domφ, then from2.22we have
fy, x x∗−ηx,y−x ρφx−ρφyρΘy, x
≥ ηy−ηx,y−x x∗−ηy, y−x ρf∗, x−y
≥αy−x2−x∗ηyρf∗y−x y−xαy−x −x∗ηyρf∗
2.23
Let
r 1 α
x∗ηyρf∗, Kx∈B:y−x ≤r . 2.24
Then D0{y}andKare both weakly compact convex subset ofB. For eachx∈B\K, there
exists a pointy∈coD0∪ {x}such thatfy, x >0 and so all conditions ofLemma 2.5are
satisfied. ByLemma 2.5, there exists a pointx∈Bsuch thatfy,x≤0 for ally∈B, that is,
ηx−x, y −x ρφy−ρφx ρΘx, y ≥0, ∀y∈B. 2.25
Case 2. If GEP has solutiony∈domφ, then from2.22we have
fy, x x∗−ηx,y−x ρφx−ρφyρΘy, x
≥ ηy−ηx,y−x x∗−ηy, y−x ρf∗, x−y ρΘy, x ≥ ηy−ηx,y−x x∗,y−x ρf∗, x−y
≥αy−x2−x∗ρf∗y−x y−xαy−x −x∗ρf∗.
2.26
Let
r α1x∗ρf∗, Kx∈B:y−x ≤r . 2.27
ThenD0{y}and K are both weakly compact convex subset ofB. For eachx∈B\K, there
exists a pointy∈coD0∪ {x}such thatfy, x >0 and so all conditions ofLemma 2.5are
Now let us show thatxis a unique solution of auxiliary equilibrium problem2.15.
Suppose thatx1, x2 ∈Bare arbitrary two solutions of auxiliary equilibrium problem2.15.
Then,
Θx1, y
φy−φx1
1
ρηx1−x∗, y−x1 ≥0, ∀y∈B, 2.28
Θx2, y
φy−φx2 ρ1ηx2−x∗, y−x2 ≥0, ∀y∈B. 2.29
Takingy x2 in2.28andyx1in2.29and adding these inequalities we have from the
monotonicity ofΘ
−ρ1ηx1−ηx2, x1−x2 ≥Θx1, x2 Θx2, x1−
1
ρηx1−ηx2, x1−x2 ≥0. 2.30
This together with theα-strong monotonicity ofηimplies that
αx1−x22≤ ηx1−ηx2, x1−x2 ≤0, 2.31
and sox1x2. Therefore,xJρΘ,φx∗and theη-proximal mapping ofφwith respect toΘis well defined. In the meantime, it is known thatφisΘ-subdifferentiable at eachx∈RJρΘ,φ. Ifφis additionallyΘ-subdifferentiable at eachx∈B\RJρΘ,φ, then byRemark 2.4we obtain thatφisΘ-subdifferentiable andJρΘ,φ ηρ∂Θφ−1. This completes the proof.
Corollary 2.7see14, Theorem 2.1 . LetBbe a reflexive Banach space with the dual spaceB∗, let φ:B → R∪{∞}be a lower semicontinuous subdifferentiable proper functional, and letη:B → B∗ be anα-strongly monotone and continuous mapping. Then for any givenx∗∈B∗and anyρ >0, there exists a uniquex∈Bsuch that
ηx−x∗, y−xρφy−ρφx≥0, ∀y∈B; 2.32
that is,xJρφx∗and theη-proximal mapping ofφis well defined.
Proof. PuttingΘ 0 inTheorem 2.6, we obtain the desired result.
Remark 2.8. Theorem 2.6shows that ifΘ: B×B → Ris a bifunction satisfying conditions H1–H4 and φ : B → R∪ {∞} is a lower semicontinuous subdifferentiable proper functional, then for any strongly monotone and continuous mappingη : B → B∗, theη -proximal mappingJρΘ,φ:B∗ → Bofφwith respect toΘis well defined. Furthermore, ifφis additionallyΘ-subdifferentiable at eachx∈B\RJρΘ,φ, then for eachx∗∈B∗
JρΘ,φx∗ ηρ∂Θφ−1x∗ 2.33
Theorem 2.9. LetBbe a reflexive Banach space with the dual space B∗, letη : B → B∗ be an α-strongly monotone and continuous mapping, letΘ:B×B → Rbe a bifunction satisfying conditions (H1)–(H4), and letφ:B → R∪{∞}be a lower semicontinuous subdifferentiable proper functional. IfφisΘ-subdifferentiable at eachx∈B\RJρΘ,φ, thenφisΘ-subdifferentiable and theη-proximal mappingJρΘ,φ ηρ∂Θφ−1ofφwith respect toΘis1/α-Lipschitz continuous. Furthermore, if
additionally theΘ-subdifferential∂Θφ:B → 2B∗forφisξ-strongly monotone, then theη-proximal
mappingJρΘ,φ ηρ∂Θφ−1ofφwith respect toΘis1/αρξ-Lipschitz continuous.
Proof. First, utilizingTheorem 2.6we know that wheneverφisΘ-subdifferentiable at each x∈ B\RJρΘ,φ,φisΘ-subdifferentiable. For anyx∗, y∗∈B∗, letxJρΘ,φx∗andyJρΘ,φy∗. Thenx∗−ηx∈ρ∂Θφxandy∗−ηy∈ρ∂Θφy. By the definition of theΘ-subdifferential of φ, we have
Θx, u φu−φx 1
ρηx−x∗, u−x ≥0, ∀u∈B, 2.34
Θy, uφu−φyρ1ηy−y∗, u−y ≥0, ∀u∈B. 2.35
Takinguyin2.34anduxin2.35and adding these inequalities, we obtain
1
ρηx−ηy, x−y ≤ 1
ρx∗−y∗, x−y Θ
x, y Θy, x. 2.36
Utilizing conditionH2we getΘx, y Θy, x≤0 and hence
ηx−ηy, x−y ≤ x∗−y∗, x−y . 2.37
Sinceηisα-strongly monotone,
αx−y2≤ ηx−ηy, x−y ≤ x∗−y∗, x−y ≤ x∗−y∗x−y, 2.38
which implies thatJρΘ,φis 1/α-Lipschitz continuous.
Now we suppose that the Θ-subdifferential ∂Θφ : B → 2B∗ for φ is ξ-strongly monotone. Then
x∗−ηx−y∗−ηy, x−y ≥ρξx−y2. 2.39
Sinceηisα-strongly monotone,
x∗−y∗, x−y x∗−ηx−y∗−ηy, x−y ηx−ηy, x−y ≥αρξx−y2.
It follows that
JρΘ,φx∗−JρΘ,φy∗ ≤ α1ρξx∗−y∗. 2.41
Thus,JρΘ,φis 1/αρξ-Lipschitz continuous.
Corollary 2.10see14, Theorem 2.2 . LetBbe a reflexive Banach space with the dual spaceB∗, letη :B → B∗be anα-strongly monotone and continuous mapping, and letφ :B → R∪ {∞} be a lower semicontinuous subdifferentiable proper functional. Then the η-proximal mappingJρφ ηρ∂φ−1 is1/α-Lipschitz continuous. Furthermore, if the subdifferential∂φ : B → 2B∗
forφ is ξ-strongly monotone, then the η-proximal mappingJρφ ηρ∂φ−1 is 1/αρξ-Lipschitz continuous.
Proof. PuttingΘ 0 inTheorem 2.9, we derive the desired result.
3. Existence and Algorithm
We first transfer GMEPP2.1into a fixed point problem.
Theorem 3.1. LetBbe a reflexive Banach space with the dual space B∗, letη : B → B∗ be an α-strongly monotone and continuous mapping, letΘ:B×B → Rbe a bifunction satisfying conditions (H1)–(H4), and letφ:B → R∪{∞}be a lower semicontinuous subdifferentiable proper functional. IfφisΘ-subdifferentiable at eachx∈B\RJρΘ,φ, thenqis a solution of the GMEPP2.1if and only ifqsatisfies the following relation:
gqJρΘ,φηgq−ρTq−Aq, 3.1
whereJρΘ,φ ηρ∂Θφ−1is theη-proximal mapping ofφwith respect toΘandρ >0is a constant.
Proof. SinceφisΘ-subdifferentiable at eachx ∈ B\RJρΘ,φ, in terms ofTheorem 2.6,φis Θ-subdifferentiable.
Assume thatq satisfies relation 3.1. Noting JρΘ,φ η ρ∂Θφ−1, we know that relation3.1holds if and only if
ηgq−ρTq−Aq∈ηgqρ∂Θφgq. 3.2
By the definition of theΘ-subdifferential ofφwith respect toΘ, the above relation holds if and only if
that is,
Θgq, vφv−φgqTq−Aq, v−gq ≥0, ∀v∈B. 3.4
Thus,qis a solution of GMEPP2.1if and only ifqsatisfies3.1. This completes the proof.
Remark 3.2. Relation3.1can be written as
qq−gqJρΘ,φηgq−ρTq−Aq, 3.5
whereJρΘ,φ ηρ∂Θφ−1.
Remark 3.3. ByTheorem 2.6, we can choose a strongly monotone and Lipschitz continuous mappingη :B → B∗such that it is easy to compute the values of theη-proximal mapping JρΘ,φofφwith respect toΘ.Theorem 3.1shows that, by using the mappingJρΘ,φ, GMEPP 2.1can be transferred into a fixed point problem3.5. Based on these observations, we can
suggest the following new and general iterative algorithms for computing the approximate solutions of GMEPP2.1in reflexive Banach spaces.
Lemma 3.4see27 . LetBbe a real Banach space and letJ :B → 2B∗be the normalized duality mapping. Then for anyx, y∈B, the following inequality holds:
xy2≤ x22y, jxy , ∀jxy∈Jxy. 3.6
We now useTheorem 3.1to construct the following algorithms for solving the GMEPP 2.1in Banach spaces.
Algorithm 3.5. LetT, A:B → B∗be two single-valued mappings, letg :B → Bbe a single-valued mapping withgB B,letη :B → B∗be anα-strongly monotone andθ-Lipschitz continuous mapping, letΘ:B×B → Rbe a bifunction satisfying conditionsH1–H4, and letφ:B → R∪{∞}be a lower semicontinuous subdifferentiable proper functional. For any givenx0 ∈B, an iterative sequence{xn}is defined by
xn1 1−αnxnαn
xn−gxn JρΘ,φηgxn−ρTxn−Axn, n0,1, . . . , 3.7
Algorithm 3.6. LetT, A, g, η,Θ,andφbe the same as inAlgorithm 3.5. For any givenx0 ∈B,
the iterative sequences{xn}and{yn}are defined by
yn1−βnxnβn
xn−gxn JρΘ,φηgxn−ρTxn−Axn,
xn1 1−αnxnαn
yn−gynJρΘ,φηgyn−ρTyn−Ayn
n0,1, . . . ,
3.8
where αn, βn ∈ 0,1 for all n ≥ 0. Algorithm 3.6 is called the Ishikawa-type iterative algorithm.
Remark 3.7. Ifβn 0 for alln ≥ 0, thenAlgorithm 3.6reduces toAlgorithm 3.5. Whenever Θ 0, Algorithms3.5 and 3.6reduce to Algorithms 3.1 and 3.2 by Xia and Huang 14 ,
respectively.
Now we prove an existence theorem of solutions for GMEPP2.1.
Theorem 3.8. LetBbe a reflexive Banach space with the dual spaceB∗,letT, A :B → B∗be two continuous mappings, and letg :B → Bbe a continuous mapping. Letη :B → B∗beα-strongly monotone and continuous, letΘ:B×B → Rbe a bifunction satisfying conditions (H1)–(H4), and letφ : B → R∪ {∞} be a lower semicontinuous subdifferentiable proper functional. Ifφ isΘ -subdifferentiable at eachx∈B\RJρΘ,φ, and the rangesRI−g, RηgandRT−Aare totally bounded, then there existsq∈Bwhich is a solution of GMEPP2.1.
Proof. DefineF:B → Bby
Fx x−gx JρΘ,φηgx−ρTx−Ax, ∀x∈B. 3.9
ByTheorem 2.9, the mappingJρΘ,φ ηρ∂Θφ−1is Lipschitz continuous. Since the ranges
RI−g, RηgandRT−Aare totally bounded, we know that the rangeRFis also totally bounded inB; that is,FBis totally bounded inB. Thus,FBis a compact subset ofB. Since T, A, g, η,andJρΘ,φ are continuous, so doesF : B → B. By Schauder fixed point theorem, F:B → Bhas a fixed pointq∈B. It follows fromTheorem 3.1thatqis a solution of GMEPP 2.1. This completes the proof.
Remark 3.9. From the proof of Theorem 3.8, we know that in 14, Theorem 3.2 , the assumption of the boundedness of the rangesRI−g, Rηg,andRT−Acannot guarantee that all the conditions of Schauder fixed point theorem are satisfied. Thus, in14, Theorem 3.2 , the assumption “the ranges RI −g, Rηg, and RT −A are bounded” should be replaced by the stronger condition “the ranges RI −g, Rηg,and RT −A are totally bounded”. Here the well-known Schauder fixed point theorem is stated as follows.
In order to give some sufficient conditions, which guarantee the convergence of the iterative sequences generated by Algorithm 3.6, we will need the following lemma in the sequel.
Lemma 3.10see28 . Let{sn}be a sequence of nonnegative real numbers satisfying the condition
sn1≤
1−μnsnμnνn, ∀n≥1, 3.10
where{μn}and{νn}are sequences of real numbers such that
i{μn} ⊂0,1 and∞n1μn∞, or equivalently,
∞
n1
1−μn:nlim → ∞
n
k1
1−μk0; 3.11
iilim supn→ ∞νn≤0, or ii∞
n1μnνnis convergent.
Then,limn→ ∞sn0.
Theorem 3.11. LetBbe a uniformly smooth Banach space with the dual spaceB∗,letT :B → B∗ beδ-Lipschitz continuous, let A : B → B∗ beγ-Lipschitz continuous, and letg : B → Bbe k-strongly accretive andε-Lipschitz continuous. Suppose thatη:B → B∗isα-strongly monotone and θ-Lipschitz continuous,Θ:B×B → Ris a bifunction satisfying conditions (H1)–(H4), andφ:B → R∪ {∞}is a lower semicontinuous subdifferentiable proper functional which isΘ-subdifferentiable at eachx∈B\RJρΘ,φ. Let{αn}and{βn}be two sequences in0,1 withαn → 0, βn → 0, and
∞
n0αn ∞. Assume that theΘ-subdifferential∂Θφ :B → 2B ∗
ofφisξ-strongly monotone, the rangesRI−g, Rηg, andRT−Aare totally bounded, andρ >θε−kα/kξ−δγwhere
kξ > δγ, θε > kα. 3.12
Then for any givenx0 ∈ B, the sequence{xn} defined by Algorithm 3.6 converges strongly to a
solutionqof GMEPP2.1.
Proof. ByTheorem 3.8and the assumptions inTheorem 3.11, we know that the solution set of GMEPP2.1is nonempty. Letqbe a solution of GMEPP2.1. Sincekξ > δγandθε > kα, we can choose a constantρsuch that
ρ > θε−kα
kξ−δγ. 3.13
ByAlgorithm 3.6, we have
yn1−βnxnβn
xn−gxn JρΘ,φηgxn−ρTxn−Axn,
xn1 1−αnxnαn
yn−gynJρΘ,φηgyn−ρTyn−Ayn.
Let
pnyn−gynJρΘ,φηgyn−ρTyn−Ayn,
rn xn−gxn JρΘ,φηgxn−ρTxn−Axn.
3.15
Then
xn1 1−αnxnαnpn, yn1−βnxnβnrn. 3.16
It follows from Theorem 2.9 that JρΘ,φ is Lipschitz continuous. Since the ranges RI − g, Rηg, andRT−Aare totally bounded, we know that the set
x−gx JρΘ,φηgx−ρTx−Ax:x∈B
3.17
is bounded. Let
Msupw−q:wx−gx JρΘ,φηgx−ρTx−Ax, x∈B
x0−q<∞.
3.18
This implies that
pn−q ≤M, rn−q ≤M, ∀n≥0. 3.19
Sincex0−q ≤M,
y0−q
1−β0
x0−q
β0
r0−q
≤1−β0
x0−qβ0r0−q
≤M.
3.20
It follows that
x1−q ≤1−α0x0−qα0p0−q ≤M,
y1−q ≤
1−β1
x1−qβ1r1−q ≤M.
3.21
By induction we can prove that
On the other hand, byLemma 3.4,
xn1−q21−αnxn−q αnpn−q2
≤1−αn2x
n−q22αnpn−q, Jxn1−q
1−αn2x
n−q22αnpn−q, Jyn−q 2αnpn−q, Jxn1−q
−Jyn−q.
3.23
Now we consider the third term on the right-hand side of3.23. From3.19and3.22it follows that
xn1−q
−yn−qxn1−yn
1−αnxn−ynαnpn−yn
≤1−αnβnxn−rnαnpn−qyn−q
≤1−αnβnxn−qrn−qαnpn−qyn−q ≤21−αnβnαnM−→0 n−→ ∞.
3.24
SinceBis a uniformly smooth Banach space, the normalized duality mappingJ:B → B∗is uniformly norm-to-norm continuous on any bounded subset ofB. Hence it is easy to see that
Jxn1−q
−Jyn−q−→0 n−→ ∞. 3.25
Let
δnpn−q, Jxn1−q
−Jyn−q . 3.26
Since{pn−q}is bounded,
δnpn−q, Jxn1−q
−Jyn−q −→0 n−→ ∞. 3.27
Next we consider the second term on the right-hand side of3.23. Since qis a solution of GMEPP2.1, byTheorem 3.1, we have
It follows that
2αnpn−q, Jyn−q
2αn
yn−gynJρΘ,φηgyn−ρTyn−Ayn
− q−gqJρΘ,φηgq−ρTq−Aq, Jyn−q
2αnyn−q, Jyn−q −2αngyn−gq, Jyn−q
2αnJρΘ,φηgyn−ρTyn−Ayn
−JρΘ,φηgq−ρTq−Aq, Jyn−q
≤2αn
yn−q2−kyn−q2λyn−q2
2αn1−kλyn−q2,
3.29
where
λ α1ρξθερδργ. 3.30
Substituting3.27and3.29into3.23, we have
xn1−q2≤1−αn2xn−q22αn1−kλyn−q22αnδn. 3.31
Next we make an estimation foryn−q2. Indeed,
yn−q21−βnxn−q βnrn−q2
≤1−βn2x
n−q22βnrn−q, Jyn−q ≤1−βn2x
n−q22βnrn−qyn−q
≤ xn−q22βnM2.
3.32
Substituting3.32into3.31and simplifying it, we have
xn1−q2≤1−αn2xn−q22αn1−kλ
xn−q22βnM2
2αnδn
1−2αnk−λ xn−q2α2nxn−q24αnβn1−kλM22αnδn
≤1−2αnk−λ xn−q2
2αnk−λ· 1 2k−λ
αnM24βn1−kλM22δn
.
From condition3.13, we getk > λ. Sinceαn → 0, βn → 0, and∞n0αn ∞, we deduce
that∞n02αnk−λ ∞and
lim n→ ∞
1 2k−λ
αnM24βn1−kλM22δn
0. 3.34
Thus, utilizingLemma 3.10we conclude thatxn−q → 0 asn → ∞. This completes the proof.
Remark 3.12. By the careful analysis of the proof ofTheorem 3.11, we can see readily that the following conditions are used to derive the following conclusion:
ithe normalized duality mapping J : B → B∗ is uniformly norm-to-norm continuous on any bounded subset ofB;
iithe constantρinAlgorithm 3.6satisfies the inequalityρ >θε−kα/kξ−δγ.
In addition, we apply Lemma 3.4 to derive the strong convergence of the iterative sequence generated byAlgorithm 3.6. Moreover, we simplify the original proof by Xia and Huang14,Theorem 3.11 to a great extent. Therefore,Theorem 3.11is a generalization and modification of Xia and Huang’s Theorem 3.314 .
Remark 3.13. We would like to point out that, inTheorem 3.11, the functionalφmay not be convex, the mappingsTandAmay not have any monotonicity, and their domains and ranges are reflexive Banach spaceBand the dual spaceB∗ ofB, respectively. Hence Theorem 3.11
improves and generalizes some known results in8,9,11,16,17 . Furthermore, the argument
methods presented in this paper are quite different from those in8–11,13,14,16,21 .
Finally, we give a weak convergence theorem for the iterative sequence generated by
Algorithm 3.6inBHa real Hilbert space. However, we first recall the following lemmas.
Lemma 3.14 see 29, page 303 . Let {an}∞n1 and {bn}∞n1 be sequences of nonnegative real numbers satisfying the inequality
an1 ≤anbn, ∀n≥1. 3.35
If∞n1bnconverges, thenlimn→ ∞anexists.
Lemma 3.15see30 . Demiclosedness Principle. Assume thatT is a nonexpansive self-mapping of a nonempty closed convex subsetC of a Hilbert spaceH. IfT has a fixed point, then I −T is demiclosed. That is, whenever{xn} is a sequence inCweakly converging to some x ∈ C and the sequence{I−Txn}strongly converges to somey, it follows thatI−Txy. HereIis the identity operator ofH.
lim infn→ ∞βn ≤ lim supn→ ∞βn < 1. Assume that the Θ-subdifferential ∂Θφ : H → 2H of
φ is ξ-strongly monotone, the ranges RI − g, Rηg, and RT −A are totally bounded, and ρ≥θε−kα /kξ −δγwhere
k1−1−2kε21/2, kξ > δ γ, θε >kα. 3.36
Then for any givenx0∈H, the sequence{xn}defined byAlgorithm 3.6converges weakly to a solution
of GMEPP2.1.
Proof. ByTheorem 3.8and the assumptions inTheorem 3.16, we know that the solution set of GMEPP2.1is nonempty. Letqbe a solution of GMEPP2.1. It is easy to see that
ρ≥ θε−kα
kξ−δγ ⇐⇒k≥λ. 3.37
Define a mappingΓ:H → Has follows:
Γx x−gx JρΘ,φηgx−ρTx−Ax, ∀x∈H, 3.38
whereJρΘ,φ ηρ∂Θφ−1. Since theΘ-subdifferential∂Θφ : H → 2H ofφ isξ-strongly monotone, by Theorem 2.9 we conclude that JρΘ,φ : B∗ → B is 1/α ρξ-Lipschitz continuous.
Observe that for allx, y∈H
Γx−Γy ≤ x−y−gx−gy
JρΘ,φηgx−ρTx−Ax−JρΘ,φηgy−ρTy−Ay
≤1−2kε21/2x−y
α1ρξηgx−ρTx−Ax−ηgy−ρTy−Ay
≤1−2kε21/2x−y 1 αρξ
θερδργx−y
1−kλx−y
≤ x−y,
3.39
wherek1−1−2kε21/2andλ 1/αρξθερδργ. This implies thatΓ:H → H
Note that
xn1−q21−αn
xn−qαnΓyn−Γq2
≤1−αnxn−q2αnΓyn−Γq2
≤1−αnxn−q2αnyn−q2
1−αnxn−q2αn1−βnxn−qβnΓxn−Γq2
1−αnxn−q2αn1−βnxn−q2βnΓxn−Γq2
−βn1−βnΓxn−xn2
≤1−αnxn−q2αn1−βnxn−q2βnxn−q2
−βn1−βnΓxn−xn2
xn−q2−αnβn1−βnΓxn−xn2
≤ xn−q2.
3.40
This together withLemma 3.14implies that limn→ ∞xn−qexists. And also, from3.40we have
αnβn1−βnΓxn−xn2≤ xn−q2− xn1−q2. 3.41
Since 0<lim infn→ ∞αn,0<lim infn→ ∞βn≤lim supn→ ∞βn<1, we get
lim
n→ ∞xn−Γxn0. 3.42
Now, let us show thatωwxn ⊂ FixΓ. Indeed, letx ∈ ωwxn. Then there exists a subsequence{xni}of{xn}such thatxni x. SinceI−Γxn → 0, byLemma 3.15we know
thatx∈FixΓ.
Further, let us show that ωwxn is a singleton. Indeed, let {xmj} be another
subsequence of{xn}such thatxmj x. Thenxis also a fixed point ofΓ. Ifx /x, by Opial’s
property ofH, we reach the following contradiction:
lim
n→ ∞xn−xilim→ ∞xni−x
< lim
i→ ∞xni−xjlim→ ∞xmj−x < lim
j→ ∞xmj−x
lim
n→ ∞xn−x.
This implies thatωwxnis a singleton. Consequently,{xn}converges weakly to a fixed point ofΓ, that is, a solution of GMEPP2.1. This completes the proof.
Acknowledgments
This research was partially supported by the Leading Academic Discipline Project of Shanghai Normal University DZL707, Innovation Program of Shanghai Municipal Education Commission Grant09ZZ133, National Science Foundation of China10771141, Ph D Program Foundation of Ministry of Education of China 20070270004, Science and Technology Commission of Shanghai Municipality Grant075105118, and Shanghai Leading Academic Discipline Project S30405 to Lu-Chuan Ceng. It was supported by Basic Science Research Program through KOSEF 2009-0077742to Sangho Kumand was partially supported by the Grant NSC 98-2115-M-110-001to Jen-Chih Yao.
References
1 K. Goebel and W. A. Kirk,Topics in Metric Fixed Point Theory, vol. 28 ofCambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990.
2 C. Baiocchi and A. Capelo,Variational and Quasivariational Inequalities, Applications to Free Boundary Problems, John Wiley & Sons, New York, NY, USA, 1984.
3 F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, NY, USA, 2003.
4 R. Glowinski, J.-L. Lions, and R. Tr´emoli`eres,Numerical Analysis of Variational Inequalities, vol. 8 of
Studies in Mathematics and Its Applications, North-Holland, Amsterdam, The Netherlands, 1981.
5 D. Kinderlehrer and G. Stampacchia,An Introduction to Variational Inequalities and Their Applications, vol. 88 ofPure and Applied Mathematics, Academic Press, New York, NY, USA, 1980.
6 N. Xiu and J. Zhang, “Some recent advances in projection-type methods for variational inequalities,”
Journal of Computational and Applied Mathematics, vol. 152, no. 1-2, pp. 559–585, 2003.
7 Y. J. Cho, J. K. Kim, and R. U. Verma, “A class of nonlinear variational inequalities involving partially relaxed monotone mappings and general auxiliary problem principle,”Dynamic Systems and Applications, vol. 11, no. 3, pp. 333–337, 2002.
8 G. Cohen, “Auxiliary problem principle extended to variational inequalities,”Journal of Optimization Theory and Applications, vol. 59, no. 2, pp. 325–333, 1988.
9 S. Schaible, J. C. Yao, and L.-C. Zeng, “Iterative method for set-valued mixed quasivariational inequalities in a Banach space,”Journal of Optimization Theory and Applications, vol. 129, no. 3, pp. 425–436, 2006.
10 L.-C. Zeng, S.-M. Guu, and J.-C. Yao, “An iterative method for generalized nonlinear set-valued mixed quasi-variational inequalities withH-monotone mappings,” Computers & Mathematics with Applications, vol. 54, no. 4, pp. 476–483, 2007.
11 L.-C. Zeng, “Iterative algorithms for finding approximate solutions for general strongly nonlinear variational inequalities,”Journal of Mathematical Analysis and Applications, vol. 187, no. 2, pp. 352–360, 1994.
12 L.-C. Zeng, “Iterative algorithm for finding approximate solutions to completely generalized strongly nonlinear quasivariational inequalities,”Journal of Mathematical Analysis and Applications, vol. 201, no. 1, pp. 180–194, 1996.
13 N.-J. Huang and C.-X. Deng, “Auxiliary principle and iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities,”Journal of Mathematical Analysis and Applications, vol. 256, no. 2, pp. 345–359, 2001.
14 F.-Q. Xia and N.-J. Huang, “Algorithm for solving a new class of general mixed variational inequalities in Banach spaces,”Journal of Computational and Applied Mathematics, vol. 220, no. 1-2, pp. 632–642, 2008.
16 X. P. Ding, “General algorithm of solutions for nonlinear variational inequalities in Banach space,”
Computers & Mathematics with Applications, vol. 34, no. 9, pp. 131–137, 1997.
17 X. P. Ding, “General algorithm for nonlinear variational-like inequalities in reflexive Banach spaces,”
Indian Journal of Pure and Applied Mathematics, vol. 29, no. 2, pp. 109–120, 1998.
18 X. P. Ding, J.-C. Yao, and L.-C. Zeng, “Existence and algorithm of solutions for generalized strongly nonlinear mixed variational-like inequalities in Banach spaces,” Computers & Mathematics with Applications, vol. 55, no. 4, pp. 669–679, 2008.
19 S. S. Chang, “Set-valued variational inclusions in Banach spaces,”Journal of Mathematical Analysis and Applications, vol. 248, no. 2, pp. 438–454, 2000.
20 N.-J. Huang and Y.-P. Fang, “Iterative processes with errors for nonlinear set-valued variational inclusions involving accretive type mappings,”Computers & Mathematics with Applications, vol. 47, no. 4-5, pp. 727–738, 2004.
21 N.-J. Huang and G. X.-Z. Yuan, “Approximating solution of nonlinear variational inclusions by Ishikawa iterative process with errors in Banach spaces,”Journal of Inequalities and Applications, vol. 6, no. 5, pp. 547–561, 2001.
22 L. C. Zeng, H. Y. Hu, and M. M. Wong, “Strong and weak convergence theorems for generalized mixed equilibrium problem with perturbation and fixed point problem of infinitely many nonexpansive mappings,” to appear inTaiwanese Journal of Mathematics.
23 J.-W. Peng and J.-C. Yao, “A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems,”Taiwanese Journal of Mathemat-ics, vol. 12, no. 6, pp. 1401–1432, 2008.
24 X. P. Ding and K.-K. Tan, “A minimax inequality with applications to existence of equilibrium point and fixed point theorems,”Colloquium Mathematicum, vol. 63, no. 2, pp. 233–247, 1992.
25 Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,”Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967.
26 D. van Dulst, “Equivalent norms and the fixed point property for nonexpansive mappings,”The Journal of the London Mathematical Society, vol. 25, no. 1, pp. 139–144, 1982.
27 W. V. Petryshyn, “A characterization of strict convexity of Banach spaces and other uses of duality mappings,”Journal of Functional Analysis, vol. 6, pp. 282–291, 1970.
28 H. K. Xu and T. H. Kim, “Convergence of hybrid steepest-descent methods for variational inequalities,”Journal of Optimization Theory and Applications, vol. 119, no. 1, pp. 185–201, 2003.
29 K.-K. Tan and H. K. Xu, “Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process,”Journal of Mathematical Analysis and Applications, vol. 178, no. 2, pp. 301–308, 1993.
