Volume 2007, Article ID 29653,6pages doi:10.1155/2007/29653
Research Article
Generalized Nonlinear Variational Inclusions Involving (A, η)-Monotone Mappings in Hilbert Spaces
Yeol Je Cho, Xiaolong Qin, Meijuan Shang, and Yongfu Su Received 30 July 2007; Accepted 12 November 2007
Recommended by Mohamed Amine Khamsi
A new class of generalized nonlinear variational inclusions involving (A,η)-monotone mappings in the framework of Hilbert spaces is introduced and then based on the gen- eralized resolvent operator technique associated with (A,η)-monotonicity, the approxi- mation solvability of solutions using an iterative algorithm is investigated. Since (A,η)- monotonicity generalizesA-monotonicity andH-monotonicity, results obtained in this paper improve and extend many others.
Copyright © 2007 Yeol Je Cho 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.
1. Introduction and preliminaries
Variational inequalities and variational inclusions are among the most interesting and important mathematical problems and have been studied intensively in the past years since they have wide applications in mechanics, physics, optimization and control, non- linear programming, economics and transportation equilibrium, engineering sciences, and so on. There exists a vast literature [1–6] on the approximation solvability of nonlin- ear variational inequalities as well as nonlinear variational inclusions using projection- type methods, resolvent-operator-type methods, or averaging techniques. In most of the resolvent operator methods, the maximal monotonicity has played a key role, but more recently introduced notions ofA-monotonicity [4] andH-monotonicity [1,2] have not only generalized the maximal monotonicity, but gave a new edge to resolvent operator methods.
Recently, Verma [5] generalized the recently introduced and studied notion ofA- monotonicity to the case of (A,η)-monotonicity. Furthermore, these developments added a new dimension to the existing notion of the maximal monotonicity and its applications to several other fields such as convex programming and variational inclusions.
In this paper, we explore the approximation solvability of a generalized class of non- linear variational inclusion problems based on (A,η)-resolvent operator techniques.
Now, we explore some basic properties derived from the notion of (A,η)-monotonicity.
LetH denote a real Hilbert space with the norm·and inner product·,·. Letη: H×H:→Hbe a single-valued mapping. The mappingηis calledτ-Lipschitz continuous if there is a constantτ >0 such thatη(u,v) ≤ τ y−v for allu,v∈H.
Definition 1.1. Letη:H×H→Hbe a single-valued mapping andM:H→2H be a multi- valued mapping onH.
(i) The mappingMis said to be (r,η)-strongly monotone if u∗−v∗,η(u,v)≥ru−v, ∀
u,u∗,v,v∗∈Graph(M), (1.1) (ii) the mappingM is said to be (m,η)-relaxed monotone if there exists a positive
constantmsuch that
u∗−v∗,η(u,v)≥ −mu−v2, ∀u,u∗,v,v∗∈Graph(M). (1.2)
Definition 1.2 [3]. A mappingM:H→2His said to be maximal (m,η)-relaxed monotone if
(i)Mis (m,η)-relaxed monotone,
(ii) for (u,u∗)∈H×Handu∗−v∗,η(u,v)≥−mu−v2, for all (v,v∗)∈Graph(M), andu∗∈M(u).
Definition 1.3 [3]. LetA:H→Handη:H×H→H be two single-valued mappings. The mappingM:H→2His said to be (A,η)-monotone if
(i)Mis (m,η)-relaxed monotone, (ii)R(A+ρM)=Hforρ >0.
Note that, alternatively, the mappingM:H→2His said to be (A,η)-monotone if (i)Mis (m,η)-relaxed monotone,
(ii)A+ρMisη-pseudomonotone forρ >0.
Remark 1.4. The (A,η)-monotonicity generalizes the notion of theA-monotonicity in- troduced by Verma [4] and theH-monotonicity introduced by Fang and Huang [1,2].
Definition 1.5. LetA:H→H be an (r,η)-strong monotone mapping andM:H→H be an (A,η)-monotone mapping. Then the generalized resolvent operatorJM,ρA,η:H→His de- fined byJM,ρA,η(u)=(A+ρM)−1(u) for allu∈H.
Definition 1.6. The mappingT:H×His said to be relaxed (α,β)-cocoercive with respect toAin the first argument if there exist two positive constantsα,βsuch that
T(x,u)−T(y,u),Ax−Ay≥(−α)T(x,u)−T(y,u)2+βx−y2, ∀x,y,u∈H.
(1.3) Proposition 1.7 [5]. Letη:H× →Hbe a single-valued mapping,A:H→Hbe an (r,η)- strongly monotone mapping andM:H→2H an (A,η)-monotone mapping. Then the map- ping (A+ρM)−1is single-valued.
2. Results on algorithmic convergence analysis
LetN:H×H→H,g:H→H,η:H×H→Hbe three nonlinear mappings andM:H→2H be an (A,η)-monotone mapping. Then the nonlinear variational inclusion (NVI) prob- lem: determine an elementu∈Hfor a given element f ∈Hsuch that
f ∈N(u,u) +Mg(u). (2.1) A special cases of the NVI (2.1) problem is to find an elementu∈Hsuch that
0∈N(u,u) +Mg(u). (2.2) Ifg=Iin (2.1), then NVI (2.1) reduces to the following nonlinear variational inclu- sion problem: determine an elementu∈Hfor a given element f ∈Hsuch that
f ∈N(u,u) +M(u). (2.3)
The solvability of the NVI problem (2.1) depends on the equivalence between (2.1) and the problem of finding the fixed point of the associated generalized resolvent oper- ator. Note that, ifM is (A,η)-monotone, then the corresponding generalized resolvent operatorJM,ρA,η is defined byJM,ρA,η(u)=(A+ρM)−1(u) for allu∈H, whereρ >0 andAis an (r,η)-strongly monotone mapping.
In order to prove our main results, we need the following lemmas.
Lemma 2.1. Assume that{an}is a sequence of nonnegative real numbers such that an+1≤
1−λn
an+bn, ∀n≥n0, (2.4)
wheren0is some nonnegative integer,{λn}is a sequence in (0, 1) with∞n=1λn= ∞,bn=
◦(λn), then limn→∞an=0.
Lemma 2.2. LetH be a real Hilbert space andη:H×H→H be aτ-Lipschitz continuous nonlinear mapping. LetA:H→Hbe a (r,η)-strongly monotone andM:H→2H be (A,η)- monotone. Then the generalized resolvent operatorJM,ρA,η :H→Hisτ/(r−ρm)-Lipschitz con- tinuous, that is,
JM,ρA,η(x)−JM,ρA,η(y) ≤ τ
r−ρmx−y, ∀x,y∈H. (2.5) Lemma 2.3. Let H be a real Hilbert space, A:H→H be (r,η)-strongly monotone and M:H→2H be (A,η)-monotone. Letη:H×H→H be aτ-Lipschitz continuous nonlinear mapping. Then the following statements are mutually equivalent:
(i) An elementu∈His a solution to the NVI (2.1).
(ii)g(u)=JM,ρA,η[Ag(u)−ρN(u,u) +ρ f].
FromLemma 2.3, we have the following:
u=u−g(u) +JM,ρA,ηAg(u)−ρN(u,u) +ρ f, (2.6)
whereuis a solution to the NVI problem (2.1). LetSbe a nonexpansive mapping onH.
Ifuis also a fixed point ofS, we have
u=S{u−g(u) +JM,ρA,η(Ag(u)−ρN(u,u) +ρ f)}. (2.7) Next, we consider the following algorithms and denote the solution to the NVI prob- lem (2.1) byΩ1, the NVI problem (2.3) byΩ2, respectively.
Algorithm 2.4. For anyu0∈H, compute the sequence{un}by the iterative processes un+1=
1−αnun+αnSun−gun
+JM,ρA,ηAgun
−ρNun,un
+ρ f, (2.8) where{αn}is a sequence in [0, 1] andSis a nonexpansive mapping onH.
IfS=g=Iand{αn} =1 inAlgorithm 2.4, then we have the following algorithm.
Algorithm 2.5. For anyu0∈H, compute the sequence{un}by the iterative processes un+1=JM,ρA,η
Aun−ρNun,un
+ρ f. (2.9)
We remark thatAlgorithm 2.5gives the approximate solution to the NVI problem (2.3).
Now, we are in the position to prove our main results.
Theorem 2.6. LetHbe a real Hilbert space,A:H×H be (r,η)-strongly monotone ands- Lipschitz continuous andM:H→2Hbe (A,η)-monotone. Letη:H×H→Hbe aτ-Lipschitz continuous nonlinear mapping andN:H×H→H be relaxed (α1,β1)-cocoercive (with re- spect toAg) andμ1-Lipschitz coninuous in the first variable andN beν1-Lipschitz contin- uous in the second variable. Letg:H→H be relaxed (α2,β2)-cocoercive andμ2-Lipschitz continuous onH,S:H→H be a nonexpansive mapping and{un}be a sequence generated byAlgorithm 2.4. Suppose the following conditions are satisfied:
(i)αn⊂(0, 1), ∞n=0αn= ∞;
(ii)τ(θ1+ρν1)<(r−ρm)(1−θ2), where θ1=
μ22s2−2ρβ1+2ρα1μ21+ρ2μ21 and θ2= 1 + 2μ22α2−2β2+μ22.
Then the sequence{un}converges strongly tou∗∈F(S)∩Ω1. Proof. Letu∗∈Cbe the common element ofF(S)∩Ω1. Then we have
u∗= 1−αn
u∗+αnSu∗−gu∗+JM,ρA,ηAgu∗−ρNu∗,u∗+ρ f. (2.10) It follows that
un+1−u∗ ≤(1−αn) un−u∗ +αn un−u∗− gun
−gu∗ + ταn
r−ρm Agun
−Agu∗−ρNun,un
−Nu∗,un
−ρNu∗,un
−Nu∗,u∗ .
(2.11)
It follows from relaxed (α1,β1)-cocoercive monotonicity andμ1-Lipschitz continuity of Nin the first variable, thes-Lipschitz continuity ofAand theμ2-Lipschitz continuity of gthat
Agun
−Agu∗−ρNun,un
−Nu∗,un 2
= Agun
−Agu∗ 2−2ρNun,un
−Nu∗,un,Agun
−Agu∗ +ρ2 Nun,un
−Nu∗,un 2≤θ21 un−u∗ 2,
(2.12)
whereθ1=
μ22s2−2ρβ1+ 2ρα1μ21+ρ2μ21.Observe that theν1-Lipschitz continuity ofN in the second argument yields that
Nu∗,un
−Nu∗,u ≤ν1 un−u∗ . (2.13) Now, we consider the second term of the right side of (2.11). It follows from the relaxed (α2,β2)-cocoercive monotonicity andμ2-Lipschitz continuity ofgthat
un−u∗−gun
−gu∗ 2
= un−u∗ 2−2gun
−gu∗,un−u∗+ gun
−gu∗ 2
≤ un−u∗ 2−2−α2 gun
−gu∗ 2+β2 un−u∗ 2+ gun
−gu∗ 2
≤θ22 un−u∗ 2,
(2.14) whereθ2=
1 + 2μ22α2−2β2+μ22. Substituting (2.12), (2.13), and (2.14) into (2.11), we arrive at
un+1−u
≤
1−αn un−u∗ +αnθ2 un−u∗ + ταn
r−ρmθ1 un−u∗ +ταnρν1
r−ρm un−u∗
=
1−αn
1−θ2− τ
r−ρmθ1− τρν1
r−ρm un−u∗ .
(2.15) Using the conditions (i)-(ii) and applyingLemma 2.1to (2.15), we can obtain the desired
conclusion. This completes the proof.
Remark 2.7. Theorem 2.6mainly improves the results of Verma [5,6].
Corollary 2.8. LetHbe a real Hilbert space,A:H×Hbe (r,η)-strongly monotone, ands- Lipschitz continuous andM:H→2Hbe (A,η)-monotone. Letη:H×H→Hbe aτ-Lipschitz continuous nonlinear mapping andN:H×H→H be relaxed (α1,β1)-cocoercive (with re- spect toA) andμ1-Lipschitz coninuous in the first variable andNbeν1-Lipschitz continuous in the second variable. Let{un}be a sequence generated byAlgorithm 2.5. Suppose the fol- lowing condition is satisfied:τ(θ1+ρν1)< r−ρm, whereθ1=
μ22s2−2ρβ1+ 2ρα1μ21+ρ2μ21, then the sequence{un}converges strongly tou∗∈Ω2.
Acknowledgment
The authors are extremely grateful to the referees for useful suggestions that improved the content of the paper.
References
[1] Y. P. Fang and N. J. Huang, “H-monotone operator and resolvent operator technique for varia- tional inclusions,” Applied Mathematics and Computation, vol. 145, no. 2-3, pp. 795–803, 2003.
[2] Y. P. Fang and N. J. Huang, “H-monotone operators and system of variational inclusions,” Com- munications on Applied Nonlinear Analysis, vol. 11, no. 1, pp. 93–101, 2004.
[3] R. U. Verma, “Sensitivity analysis for generalized strongly monotone variational inclusions based on the (A,η)-resolvent operator technique,” Applied Mathematics Letters, vol. 19, no. 12, pp.
1409–1413, 2006.
[4] R. U. Verma, “A-monotonicity and applications to nonlinear variational inclusion problems,”
Journal of Applied Mathematics and Stochastic Analysis, no. 2, pp. 193–195, 2004.
[5] R. U. Verma, “Approximation solvability of a class of nonlinear set-valued variational inclu- sions involving (A,η)-monotone mappings,” Journal of Mathematical Analysis and Applications, vol. 337, no. 2, pp. 969–975, 2008.
[6] R. U. Verma, “A-monotone nonlinear relaxed cocoercive variational inclusions,” Central Euro- pean Journal of Mathematics, vol. 5, no. 2, pp. 386–396, 2007.
Yeol Je Cho: Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Korea
Email address:[email protected]
Xiaolong Qin: Department of Mathematics Education, Gyeongsang National University, Chinju 660-701, Korea
Email address:[email protected]
Meijuan Shang: Department of Mathematics, Shijiazhuang University, Shijiazhuang 050035, China Email address:[email protected]
Yongfu Su: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China Email address:[email protected]
