http://jipam.vu.edu.au/
Volume 7, Issue 2, Article 72, 2006
ITERATIVE ALGORITHM FOR A NEW SYSTEM OF NONLINEAR SET-VALUED VARIATIONAL INCLUSIONS INVOLVING (H, η)-MONOTONE MAPPINGS
MAO-MING JIN
DEPARTMENT OFMATHEMATICS
FULINGNORMALUNIVERSITY
FULING, CHONGQING40800 P. R. CHINA
mmj1898@163.com
Received 05 November, 2005; accepted 28 December, 2005 Communicated by R.U. Verma
ABSTRACT. In this paper, a new system of nonlinear set-valued variational inclusions involving (H, η)-monotone mappings in Hilbert spaces is introduced and studied. By using the resolvent operator method associated with(H, η)-monotone mappings, an existence theorem of solutions for this kind of system of nonlinear set-valued variational inclusion is established and a new iterative algorithm is suggested and discussed. The results presented in this paper improve and generalize some recent results in this field.
Key words and phrases: (H, η)-monotone mapping; System of nonlinear set-valued variational inclusions; Resolvent operator method; Iterative algorithm.
2000 Mathematics Subject Classification. 49J40; 47H10.
1. INTRODUCTION
Variational inclusions are an important generalization of classical variational inequalities and thus, have wide applications to many fields including, for example, mechanics, physics, opti- mization and control, nonlinear programming, economics, and the engineering sciences. For these reasons, various variational inclusions have been intensively studied in recent years. For details, we refer the reader to [1] – [21], [23] – [31] and the references therein.
Verma [24, 25] introduced and studied some systems of variational inequalities and developed some iterative algorithms for approximating the solutions of a system of variational inequali- ties in Hilbert spaces. Recently, Kim and Kim [21] introduced a new system of generalized nonlinear mixed variational inequalities and obtained some existence and uniqueness results for solutions of the system of generalized nonlinear mixed variational inequalities in Hilbert spaces.
Very recently, Fang, Huang and Thompson [9] introduced a system of variational inclusions and developed a Mann iterative algorithm to approximate the unique solution of the system.
ISSN (electronic): 1443-5756 c
2006 Victoria University. All rights reserved.
This work was supported by the National Natural Science Foundation of China(10471151) and the Educational Science Foundation of Chongqing, Chongqing of China (KJ051307).
012-06
On the other hand, monotonicity techniques were extended and applied in recent years be- cause of their importance in the theory of variational inequalities, complementarity problems, and variational inclusions. In 2003, Huang and Fang [16] introduced a class of generalized monotone mappings, maximalη-monotone mappings, and defined an associated resolvent op- erator. Using resolvent operator methods, they developed some iterative algorithms to approx- imate the solution of a class of general variational inclusions involving maximalη-monotone operators. Huang and Fang’s method extended the resolvent operator method associated with an η-subdifferential operator due to Ding and Luo [6]. In [7], Fang and Huang introduced another class of generalized monotone operators,H-monotone operators, and defined an associated re- solvent operator. They also established the Lipschitz continuity of the resolvent operator and studied a class of variational inclusions in Hilbert spaces using the resolvent operator associated withH-monotone operators. In a recent paper [9], Fang, Huang and Thompson further intro- duced a new class of generalized monotone operators,(H, η)-monotone operators, which pro- vide a unifying framework for classes of maximal monotone operators, maximalη-monotone operators, and H-monotone operators. They also studied a system of variational inclusions using the resolvent operator associated with(H, η)-monotone operators.
Inspired and motivated by recent research works in this field, in this paper, we shall intro- duce and study a new system of nonlinear set-valued variational inclusions involving (H, η)- monotone mappings in Hilbert spaces. By using the resolvent operator method associated with (H, η)-monotone mappings, an existence theorem for solutions for this type of system of non- linear set-valued variational inclusion is established and a new iterative algorithm is suggested and discussed. The results presented in this paper improve and generalize some recent results in this field.
2. PRELIMINARIES
LetX be a real Hilbert space endowed with a normk · kand an inner producth·,·i, respec- tively. 2X and C(X) denote the family of all the nonempty subsets of X and the family of all closed subsets of X, respectively. Let us recall the following definitions and some known results.
Definition 2.1. LetT, H :X →X be two single-valued mappings. T is said to be:
(i) monotone, if
hT x−T y, x−yi ≥0 for all x, y ∈X;
(ii) strictly monotone, ifT is monotone and
hT x−T y, x−yi= 0 if and only ifx=y;
(iii) r-strongly monotone, if there exists a constantr >0such that hT(x)−T(y), x−yi ≥rkx−yk2 for all x, y ∈X;
(iv) s-strongly monotone with respect toH, if there exists a constants >0such that hT(x)−T(y), H(x)−H(y)i ≥Skx−yk2 for all x, y ∈X;
(v) t-Lipschitz continuous, if there exists a constantt >0such that kT(x)−T(y)k ≤tkx−yk for all x, y ∈X.
Definition 2.2. A single-valued mappingη:X×X →Xis said to be:
(i) monotone, if
hx−y, η(x, y)i ≥0 for all x, y ∈X;
(ii) strictly monotone, if
hx−y, η(x, y)i ≥0 for all x, y ∈X and equality holds if and only ifx=y;
(iii) δ-srongly monotone, if there exists a constantδ >0such that hx−y, η(x, y)i ≥δkx−yk2 for all x, y ∈X;
(iv) τ-Lipschitz continuous, if there exists a constantτ > 0such that kη(x, y)k ≤τkx−yk, for all x, y ∈X.
Definition 2.3. Let η : X ×X → X and H : X → X be two single-valued mappings. A set-valued mappingM :X →2X is said to be:
(i) monotone, if
hu−v, x−yi ≥0, ∀x, y ∈X, u∈M x, v ∈M y;
(ii) η-monotone, if
hu−v, η(x, y)i ≥0 ∀x, y ∈X, u∈M x, v ∈M y;
(iii) strictlyη-monotone, ifM isη-monotone and equality holds if and only ifx=y;
(iv) r-stronglyη-monotone, if there exists a constantr >0such that
hu−v, η(x, y)i ≥rkx−yk2 ∀x, y ∈X, u∈M x, v ∈M y;
(v) maximal monotone, ifM is monotone and(I+λM)(X) = X, for allλ > 0, whereI denotes the identity mapping onX;
(vi) maximalη-monotone, ifM isη-monotone and(I +λM)(X) =X, for allλ >0;
(vii) H-monotone, ifM is monotone and(H+λM)(X) =X, for allλ >0;
(viii) (H, η)-monotone, ifM isη-monotone and(H+λM)(X) =X, for allλ >0.
Remark 2.1. Maximal η-monotone mappings, H-monotone mappings, and (H, η)-monotone mappings were first introduced in Huang and Fang [16], Fang and Huang [7, 9], respectively.
Obviously, the class of(H, η)- monotone mappings provides a unifying framework for classes of maximal monotone mappings, maximalη-monotone mappings, andH-monotone mappings.
For details about these mappings, we refer the reader to [6, 7, 9, 16] and the references therein.
Lemma 2.2 ([9]). Letη: X×X →X be a single-valued mapping,H :X →Xbe a strictly η-monotone mapping and M : X → 2X an (H, η)-monotone mapping. Then the mapping (H+λM)−1 is single-valued.
By Lemma 2.2, we can define the resolvent operatorRH,ηM,λas follows.
Definition 2.4 ([9]). Letη :X×X → X be a single-valued mapping,H : X → X a strictly η-monotone mapping andM :X →2X an(H, η)-monotone mapping. The resolvent operator RM,λH,η :X →Xis defined by
RH,ηM,λ(z) = (H+λM)−1(z) for all z ∈X, whereλ >0is a constant.
Remark 2.3.
(i) When H = I, Definition 2.4 reduces to the definition of the resolvent operator of a maximalη-monotone mapping, see [16].
(ii) Whenη(x, y) = x−y for allx, y ∈ X, Definition 2.4 reduces to the definition of the resolvent operator of aH-monotone mapping, see [7].
(iii) When H = I and η(x, y) = x −y for all x, y ∈ X, Definition 2.4 reduces to the definition of the resolvent operator of a maximal monotone mapping, see [31].
Lemma 2.4 ([9]). Let η : X ×X → X be aτ-Lipschtiz continuous mapping, H : X → X be an (r, η)-strongly monotone mapping and M : X → 2X be an(H, η)-monotone mapping.
Then the resolvent operatorRH,ηM,λ:X →Xisτ /r-Lipschitz continuous, that is,
RM,λH,η(x)−RH,ηM,λ(y) ≤ τ
rkx−yk for all x, y ∈X.
We define a Hausdorff pseudo-metricD: 2X ×2X →(−∞,+∞)∪ {+∞}by D(·,·) = max
sup
u∈A
v∈Binf ku−vk,sup
u∈B
v∈Ainf ku−vk
for any givenA, B ∈ 2X. Note that if the domain ofDis restricted to closed bounded subsets, thenDis the Hausdorff metric.
Definition 2.5. A set-valued mapping A : X → 2X is said to be D-Lipschitz continuous if there exists a constantη >0such that
D(A(u), A(v))≤ηku−vk, for all u, v ∈X.
3. SYSTEM OFVARIATIONALINCLUSIONS
In this section, we shall introduce a new system of set-valued variational inclusions involving (H, η)-monotone mappings in Hilbert spaces. In what follows, unless other specified, we shall suppose thatX1andX2are two real Hilbert spaces,K1 ⊂X1andK2 ⊂X2are two nonempty, closed and convex sets. Let F : X1 ×X2 → X1, G : X1 × X2 → X2, Hi : Xi → Xi, ηi :Xi×Xi →Xi(i= 1,2)be nonlinear mappings. LetA:X1 →2X1 andB :X2 →2X2 be set-valued mappings,Mi : Xi → 2Xi be (Hi, ηi)-monotone mappings(i = 1,2). The system of nonlinear set-valued variational inclusions is formulated as follows. Find(a, b)∈X1 ×X2, u∈A(a)andv ∈B(b)such that
(3.1)
0∈F(a, v) +M1(a) 0∈G(u, b) +M2(b) Special Cases
Case 1. IfM1(x) =∂ϕ(x)andM2 =∂φ(y)for allx∈X1andy∈X2, whereϕ :X1 →R∪ {+∞}andφ :X2 →R∪{+∞}are two proper, convex and lower semi-continuous functionals,
∂ϕ and ∂φ denote the subdifferential operators of ϕ and φ, respectively, then problem (3.1) reduces to the following problem: find(a, b)∈X1×X2,u∈A(a), andv ∈B(v)such that (3.2)
hF(a, v), x−ai+ϕ(x)−ϕ(a)≥0, ∀x∈X1, hG(u, b), y−ai+φ(y)−φ(b)≥0, ∀y∈X2,
which is called a system of set-valued mixed variational inequalities. Some special cases of problem (3.2) can be found in [26].
Case 2. If A and B are both identity mappings, then problem (3.2) reduces to the following problem: find(a, b)∈X1×X2 such that
(3.3)
hF(a, b), x−ai+ϕ(x)−ϕ(a)≥0, ∀x∈X1, hG(a, b), y−ai+φ(y)−φ(b)≥0, ∀y∈X2,
which is called system of nonlinear variational inequalities considered by Cho, Fang, Huang and Hwang [5]. Some special cases of problem (3.3) were studied by Kim and Kim [21], and Verma [24].
Case 3. If M1(x) = ∂δK1(x) and M2(y) = ∂δK2(y), for all x ∈ K1 and y ∈ K2, where K1 ⊂X1andK2 ⊂X2are two nonempty, closed, and convex subsets, andδK1 andδK2 denote the indicator functions ofK1andK2, respectively. Then problem (3.2) reduces to the following system of variational inequalities: find(a, b)∈K1×K2 such that
(3.4)
hF(a, b), x−ai ≥0, ∀x∈K1, hG(a, b), y−ai ≥0, ∀y∈K2, which is the problem in [20] with bothF andGbeing single-valued.
Case 4. If X1 = X2 = X, K1 = K2 = K, F(X, y) = ρT(y) + x −y, and G(x, y) = γT(x) +y− x, for all x, y ∈ X, where T : K → X is a nonlinear mapping, ρ > 0 and γ > 0 are two constants, then problem (3.4) reduces to the following system of variational inequalities: find(a, b)∈K×K such that
(3.5)
hρT(b) +a−b, x−ai ≥0, ∀x∈K, hγT(a) +b−a, x−bi ≥0, ∀x∈K,
which is the system of nonlinear variational inequalities considered by Verma [25].
Case 5. If A and B are both identity mappings, the problem (3.1) reduces to the following problem:(a, b)∈X1×X2such that
(3.6)
0∈F(a, b) +M1(a) 0∈G(a, b) +M2(b)
which is the system of variational inclusions considered by Fang, Huang and Thompson [9].
4. ITERATIVEALGORITHM ANDCONVERGENCE
In this section, by using the resolvent operator method associated with (H, η)-monotone mappings, a new iterative algorithm for solving problem (3.1) is suggested. The convergence of the iterative sequence generated by the algorithm is proved.
Theorem 4.1. For given(a, b) ∈ X1×X2, u ∈ A(a), v ∈ B(b), (a, b, u, v) is a solution of problem (3.1) if and only if(a, b, u, v)satisfies the relation
(4.1)
a=RHM11,η,ρ11[H1(a)−ρ1F(a, v)], b=RMH2,η2
2,ρ2[H2(b)−ρ2G(u, b)], whereρi >0are two constants fori= 1,2.
Proof. This directly follows from Definition 2.4.
The relation (4.1) and Nadler [22] allows us to suggest the following iterative algorithm.
Algorithm 4.1.
Step 1. Choose(a0, b0)∈X1 ×X2 and chooseu0 ∈A(a0)andv0 ∈B(b0).
Step 2. Let
(4.2)
an+1 = (1−λ)an+λRHM1,η1
1,ρ1[H1(an)−ρ1F(an, vn)], bn+1 = (1−λ)bn+λRHM2,η2
2,ρ2[H2(bn)−ρ2G(un, bn)], where0< λ≤1is a constant.
Step 3. Chooseun+1 ∈A(an+1)andvn+1 ∈B(bn+1)such that (4.3)
kun+1−unk ≤(1 + (1 +n)−1)D1(A(an+1), A(an)), kvn+1−vnk ≤(1 + (1 +n)−1)D2(B(bn+1), B(bn)), whereDi(·,·)is the Hausdorff pseudo-metric on2Xi fori= 1,2.
Step 4. If an+1, bn+1, un+1 and vn+1 satisfy (4.2) to sufficient accuracy, stop; otherwise, set n :=n+ 1and return to Step 2.
Theorem 4.2. Let ηi : Xi ×Xi → Xi be τi-Lipschitz continuous mappings, Hi : Xi → Xi
(ri, η)-strongly monotone andβi-Lipschitz continuous mappings, Mi : Xi → 2Xi be(Hi, ηi)- monotone mappings fori = 1,2.LetA: X1 → C(X1)beD1-γ1-Lipschitz continuous andB : X2 →C(X2)beD2-γ2-Lipschitz continuous. LetF :X1×X2 →X1be a nonlinear mapping such that for any given(a, b) ∈ X1 ×X2, F(·, b)is µ1-strongly monotone with respect to H1 andα1-Lipschitz continuous andF(a,·)isζ1-Lipschitz continuous. LetG:X1×X2 →X2be a nonlinear mapping such that for any given(x, y)∈X1×X2,G(x,·)isµ2-strongly monotone with respect toH2 andα2-Lipschitz continuous andG(·, y)isζ2-Lipschitz continuous. If there exist constantsρi >0fori= 1,2such that
(4.4)
τ1r2p
β12−2ρ1µ1+ρ21α21+τ2r1ζ2γ1 < r1r2, τ2r1p
β22−2ρ2µ2+ρ22α22+τ1r2ζ1γ2 < r1r2,
then problem (3.1) admits a solution (a, b, u, v) and iterative sequences{an},{bn},{un}and {vn}converge strongly toa, b, u andv, respectively, where{an},{bn},{un}and{vn}are the sequences generated by Algorithm 4.1.
Proof. It follows from (4.2) and Lemma 2.4 that kan+1−ank
=
(1−λ)an+λRHM1,η1
1,ρ1(H1(an)−ρ1F(an, vn))
− h
(1−λ)an−1+λRHM1,η1
1,ρ1(H1(an−1)−ρ1F(an−1, vn−1))i
≤(1−λ)kan−an−1k+λ RHM1,η1
1,ρ1(H1(an)−ρ1F(an, vn))
− RHM1,η1
1,ρ1(H1(an−1)−ρ1F(an−1, vn−1))
≤(1−λ)kan−an−1k+λτ1
r1kH1(an)−H1(an−1)−ρ1[F(an, vn)−F(an−1, vn−1)]k
≤(1−λ)kan−an−1k+λτ1
r1(kH1(an)−H1(an−1)−ρ1[F(an, vn)−F(an−1, vn)]k +kF(an−1, vn)−F(an−1, vn−1)k).
(4.5)
Similarly, we can prove that
(4.6) kbn+1−bnk ≤(1−λ)kbn−bn−1k +λτ2
r2(kH2(bn)−H2(bn−1)−ρ2[G(un, bn)−G(un, bn−1)]k
+kG(un, bn−1)−G(un−1, bn−1)k).
SinceHi areβi-Lipschitz continuous fori= 1,2,F(·, b)isµ1-strongly monotone with respect to H1 and α1-Lipschitz continuous, G(x,·) is µ2-strongly monotone with respect to H2 and α2-Lipschitz continuous, we obtain
kH1(an)−H1(an−1)−ρ1[F(an, vn)−F(an−1, vn)]k2
=kH1(an)−H1(an−1)k2−2ρ1hF(an, vn)−F(an−1, vn), H1(an)−H1(an−1)i +ρ21kF(an, vn)−F(an−1, vn)k2
≤(β12−2ρ1µ1+ρ21α21)kan−an−1k2 (4.7)
and
kH2(bn)−H2(bn−1)−ρ2[G(un, bn)−G(un, bn−1)]k2
=kH2(bn)−H2(bn−1)k2−2ρ2hG(un, bn)−G(un, bn−1), H2(bn)−H2(bn−1)i +ρ22kG(un, bn)−G(un, bn−1)k2
≤(β22−2ρ2µ2 +ρ22α22)kbn−bn−1k2. (4.8)
Further, from the assumptions, we have
kF(an−1, vn)−F(an−1, vn−1)k ≤ζ1kvn−vn−1k (4.9)
≤ζ1γ2(1 +n−1)kbn−bn−1k, kG(un, bn−1)−G(un−1, bn−1)k ≤ζ2kun−un−1k
(4.10)
≤ζ2γ1(1 +n−1)kan−an−1k.
It follows from (4.5) – (4.10) that
(4.11)
kan+1−ank ≤
1−λ+λrτ1
1
pβ12−2ρ1µ1+ρ21α21
kan−an−1k +λτr1
1ζ1γ2(1 +n−1)kbn−bn−1k, kbn+1−bnk ≤
1−λ+λτr2
2
pβ22−2ρ2µ2+ρ22α22
kbn−bn−1k +λτr2
2ζ2γ1(1 +n−1)kan−an−1k.
Now (4.11) implies that
kan+1−ank+kbn+1−bnk
≤
1−λ+λτ1
r1 q
β12−2ρ1µ1+ρ21α21+λτ2
r2ζ2γ1(1 +n−1)
kan−an−1k +
1−λ+λτ2 r2
q
β22−2ρ2µ2+ρ22α22+λτ1
r1ζ1γ2(1 +n−1)
kbn−bn−1k
≤(1−λ+λθn)(kan−an−1k+kbn−bn−1k), (4.12)
where θn = max
τ1 r1
q
β12−2ρ1µ1+ρ21α21 +τ2
r2ζ2γ1(1 +n−1), τ2
r2 q
β22−2ρ2µ2+ρ22α22+ τ1
r1ζ1γ2(1 +n−1)
. Letting
θ = max τ1
r1 q
β12−2ρ1µ1+ρ21α21+τ2
r2ζ2γ1 , τ2 r2
q
β22−2ρ2µ2+ρ22α22+τ1 r1ζ1γ2
, we have thatθn →θ asn → ∞. It follows from condition (4.4) that0< θ <1. Therefore, by (4.12) and0< λ≤1,{an}and{bn}are both Cauchy sequences and so there exista∈X1and b∈X2 such thatan →aandbn →basn → ∞.
Now we prove thatun → u∈A(u)andvn →v ∈B(b)asn → ∞. In fact, it follows from (4.9) and (4.10) that{un}and{vn}are also Cauchy sequences. Therefore, there existu ∈ X1
andv ∈X2 such thatun →uandvn→v asn → ∞. Further, d(u, A(u)) = inf{ku−tk:t∈A(a)}
≤ ku−unk+d(un, A(a))
≤ ku−unk+D1(A(an), A(a))
≤ ku−unk+ζ1kan−ak →0.
Hence, sinceA(a)is closed, we haveu∈A(a). Similarly, we can prove thatv ∈B(b).
By continuity,a, b, uandv satisfy the following relation
a=RHM1,η1
1,ρ1[H1(a)−ρ1F(a, v)], b=RMH22,η,ρ22[H2(b)−ρ2G(u, b)].
By Theorem 4.1, we know that(a, b, u, v)is a solution of problem (3.1). This completes the
proof.
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