A -MONOTONICITY AND APPLICATIONS TO NONLINEAR VARIATIONAL INCLUSION PROBLEMS
RAM U. VERMA Received 2 March 2004
A new notion of theA-monotonicity is introduced, which generalizes theH-monoto- nicity. Since theA-monotonicity originates from hemivariational inequalities, and hemi- variational inequalities are connected with nonconvex energy functions, it turns out to be a useful tool proving the existence of solutions of nonconvex constrained problems as well.
Recently, Fang and Huang [1] introduced a new class of mappings—h-monotone mappings—in the context of solving a system of variational inclusions involving a com- bination ofh-monotoneand strongly monotone mappings based on the resolvent oper- ator technique. The notion of theh-monotonicityhas revitalized the theory of maximal monotone mappings in several directions, especially in the domain of applications. Here we announce the notion of theA-monotonemappings and its applications to the solvabil- ity of systems of nonlinear variational inclusions. The class of theA-monotonemappings generalizes theh-monotonicity. On the top of that,A-monotonicityoriginates from hemi- variational inequalities, and emerges as a major contributor to the solvability of nonlinear variational problems on nonconvex settings. As a matter of fact, some nice examples on A-monotone(or generalized maximal monotone) mappings can be found in Naniewicz and Panagiotopoulos [2] and Verma [4]. Hemivariational inequalities—initiated and de- veloped by Panagiotopoulos [3]—are connected with nonconvex energy functions and turned out to be useful tools proving the existence of solutions of nonconvex constrained problems. We note that the A-monotonicity is defined in terms of relaxed monotone mappings—a more general notion than the monotonicity/strong monotonicity—which gives a significant edge over theh-monotonicity.
Definition 1[1]. Leth:H→HandM:H→2Hbe any two mappings onH. The mapM is said to beh-monotoneifMis monotone and (h+ρM)(H)=Hholds forρ >0. This is equivalent to stating thatM ish-monotone ifMis monotone and (h+ρM) is maximal monotone.
Copyright©2004 Hindawi Publishing Corporation
Journal of Applied Mathematics and Stochastic Analysis 2004:2 (2004) 193–195 2000 Mathematics Subject Classification: 49J40, 47J20
URL:http://dx.doi.org/10.1155/S1048953304403013
194 Variational inclusion problems
LetXdenote a reflexive Banach space andX∗its dual. Inspired by [2,4], we introduce the notion of theA-monotonicityas follows.
Definition 2. LetA:X→X∗andM:X→2X∗be any mappings onX. The mapMis said to beA-monotoneifM ism-relaxedmonotone and (A+ρM) is maximal monotone for ρ >0.
Lemma3. LetA:H→Hber-strongly monotone andM:H→2H beA-monotone. Then the resolvent operatorJA,Mρ :H→His(1/(r−ρm))-Lipschitz continuous for0< ρ < r/m. Example 4[2, Lemma 7.11]. LetA:X→X∗be (m)-stronglymonotone andf :X→Rbe locally Lipschitz such that∂ f is (α)-relaxedmonotone. Then∂ f isA-monotone, that is, A+∂ f is maximal monotone form−α >0, wherem,α >0.
Example 5[4, Theorem 4.1]. LetA:X→X∗be (m)-stronglymonotone and letB:X→ X∗be (c)-stronglyLipschitz continuous. Let f :X→Rbe locally Lipschitz such that∂ f is (α)-relaxedmonotone. Then∂ f is (A−B)-monotone.
LetH1 andH2be two real Hilbert spaces andK1andK2, respectively, be nonempty closed convex subsets ofH1 andH2. LetA:H1→H1,B:H2→H2,M:H1→2H1, and N:H2→2H2be nonlinear mappings. LetS:H1×H2→H1andT:H1×H2→H2be any two multivalued mappings. Then the problem of finding (a,b)∈H1×H2such that
0∈S(a,b) +M(a), 0∈T(a,b) +N(b) (1) is called the system of nonlinear variational inclusion (SNVI) problems.
WhenM(x)=∂K1(x) andN(y)=∂K2(y) for allx∈K1andy∈K2, whereK1andK2, respectively, are nonempty closed convex subsets ofH1andH2, and∂K1and∂K2denote indicator functions of K1 andK2, respectively, the SNVI (1) reduces to determine an element (a,b)∈K1×K2such that
S(a,b),x−a≥0 ∀x∈K1, (2)
T(a,b),y−b≥0 ∀y∈K2. (3)
Lemma6. LetH1andH2be two real Hilbert spaces. LetA:H1→H1andB:H2→H2be strictly monotone, letM:H1→2H1 beA-monotone, and letN:H2→2H2beB-monotone.
Let S:H1×H2→H1 and T:H1×H2→H2 be any two multivalued mappings. Then a given element(a,b)∈H1×H2 is a solution to the SNVI (1) problem if and only if(a,b) satisfies
a=JA,Mρ A(a)−ρS(a,b), b=JB,Nη B(b)−ηT(a,b). (4) Theorem7. LetH1 andH2 be two real Hilbert spaces. LetA:H1→H1 be(r1)-strongly monotone and(α1)-Lipschitz continuous, and letB:H2→H2 be (r2)-strongly monotone and(α2)-Lipschitz continuous. LetM:H1→2H1 beA-monotone and letN:H2→2H2 be B-monotone. LetS:H1×H2→H1be such thatS(·,y)is(γ,r)-relaxed cocoercive and(µ)- Lipschitz continuous in the first variable andS(x,·)is(ν)-Lipschitz continuous in the second variable for all(x,y)∈H1×H2. LetT:H1×H2→H2be such thatT(u,·)is(λ,s)-relaxed
Ram U. Verma 195 cocoercive and(β)-Lipschitz continuous in the second variable andT(·,v)is(τ)-Lipschitz continuous in the first variable for all(u,v)∈H1×H2. If, in addition, there exist positive constantsρandηsuch that
α1−2ρr+ 2ργµ2+ρ2µ2+ητ < r1, α2−2ηs+ 2ηλβ2+η2β2+ρν< r2,
(5)
then the SNVI (1) problem has a unique solution.
References
[1] Y. P. Fang and N. J. Huang,H-monotone operators and system of variational inclusions, Comm.
Appl. Nonlinear Anal.11(2004), no. 1, 93–101.
[2] Z. Naniewicz and P. D. Panagiotopoulos,Mathematical Theory of Hemivariational Inequalities and Applications, Monographs and Textbooks in Pure and Applied Mathematics, vol. 188, Marcel Dekker, New York, 1995.
[3] P. D. Panagiotopoulos,Hemivariational Inequalities. Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993.
[4] R. U. Verma,Nonlinear variational and constrained hemivariational inequalities involving re- laxed operators, ZAMM Z. Angew. Math. Mech.77(1997), no. 5, 387–391.
Ram U. Verma: International Publications, 5066 Jamieson Drive, Suite B-9, Toledo, OH 43613, USA
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