Volume 2009, Article ID 806727,33pages doi:10.1155/2009/806727
Research Article
A New System of Nonlinear Fuzzy Variational
Inclusions Involving
A, η
-Accretive Mappings in
Uniformly Smooth Banach Spaces
M. Alimohammady,
1, 2J. Balooee,
1Y. J. Cho,
3and M. Roohi
11Department of Mathematics, Faculty of Basic Sciences, University of Mazandaran, Babolsar 47416-1468, Iran
2Department of Mathematics, King’s College London Strand, London WC2R 2LS, UK 3Department of Mathematics Education and the RINS, Gyeongsang National University,
Chinju 660-701, South Korea
Correspondence should be addressed to Y. J. Cho,yjcho@gnu.ac.kr
Received 4 October 2009; Accepted 4 November 2009 Recommended by Charles E. Chidume
A new system of nonlinear fuzzy variational inclusions involvingA, η-accretive mappings in uniformly smooth Banach spaces is introduced and studied many fuzzy variational and variational inequalityinclusionproblems as special cases of this system. By using the resolvent operator technique associated withA, η-accretive operator due to Lan et al. and Nadler’s fixed points theorem for set-valued mappings, an existence theorem of solutions for this system of fuzzy variational inclusions is proved. We also construct some new iterative algorithms for the solutions of this system of nonlinear fuzzy variational inclusions in uniformly smooth Banach spaces and discuss the convergence of the sequences generated by the algorithms in uniformly smooth Banach spaces. Our results extend, improve, and unify many known results in the recent literatures. Copyrightq2009 M. Alimohammady 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
Variational inequality was initially studied by Stampacchia 1 in 1964. In order to study many kinds of problems arising in industrial, physical, regional, economical, social, pure, and applied sciences, the classical variational inequality problems have been extended and generalized in many directions. Among these generalizations, variational inclusion introduced and studied by Hassouni and Moudafi 2 is of interest and importance. It provides us with a unified, natural, novel innovative, and general technique to study a wide class of the problems arising in different branches of mathematical and engineering sciences
Next, the development of variational inequality is to design efficient iterative algorithms to compute approximate solutions for variational inequalities and their gen-eralizations. Up to now, many authors have presented implementable and significant numerical methods such as projection method, and its variant forms, linear approximation, descent method, Newton’s method and the method based on the auxiliary principle technique. In particular, the method based on the resolvent operator technique is a generalization of the projection method and has been widely used to solve variational inclusions.
Some new and interesting problems, which are called the systems of variational inequality problems, were introduced and studied. Pang 8, Cohen and Chaplais 9, Bianchi 10, and Ansari and Yao 11 considered some systems of scalar variational inequalities and Pang showed that the traffic equilibrium problem, the spatial equilibrium problem, the Nash equilibrium, and the general equilibrium programming problems can be modelled as variational inequalities. He decomposed the original variational inequality into a system of variational inequalities which are easy to solve and studied the convergence of such methods. Ansari et al. 12 introduced and studied a system of vector variational inequalities by a fixed point theorem. Allevi et al. 13 considered a system of generalized vector variational inequalities and established some existence results under relative pseudomonotonicity. Kassay and Kolumb´an 14 introduced a system of variational inequalities and proved an existence theorem by the Ky Fan lemma. Kassay et al.15studied Minty and Stampacchia variational inequality systems with the help of the Kakutani-Fan-Glicksberg fixed point theorem. Peng16,17Peng and Yang18introduced a system of quasivariational inequality problems and proved its existence theorem by maximal element theorems. Verma 19–23 introduced and studied some systems of variational inequalities and developed some iterative algorithms for approximating the solution for this system of generalized nonlinear quasivariational inequalities in Hilbert spaces. J. K. Kim and D. S. Kim 24 introduced a new system of generalized nonlinear quasivariational inequalities and obtained some existence and uniqueness results of solution for this system of generalized nonlinear quasivariational inequalities in Hilbert spaces. Cho et al. 25 introduced a new system of nonlinear variational inequalities and proved some existence and uniqueness theorems of solutions for the system of nonlinear variational inequalities in Hilbert spaces. As generalizations of system of variational inequalities, Agarwal et al. 26 introduced a system of generalized nonlinear mixed quasivariational inclusions and investigated the sensitivity of solutions for this system of generalized nonlinear mixed quasivariational inclusions in Hilbert spaces. Kazmi and Bhat 27 introduced a system of nonlinear variational-like inclusions and gave an iterative algorithm for finding its approximate solution. It is known that accretivity of the underlying operator plays indispensable roles in the theory of variational inequality and its generalizations.
In 2001, Huang and Fang 28 were the first to introduce generalized m-accretive mapping and give the definition of the resolvent operator for generalized m-accretive mappings in Banach spaces. They also proved some properties of the resolvent operator for generalized m-accretive mappings in Banach spaces. Subsequently, Fang and Huang29, Yan et al. 30, Fang et al. 31, Lan et al. 32, 33, Fang and Huang 34, and Peng et al.35introduced and investigated many new systems of variational inclusions involving
In 2004, Verma in 36, 37 introduced new notions of A-monotone and A, η -monotone operators and studied some properties ofA-monotone andA, η-monotone oper-ators in Hilbert spaces. In38, Lan et al. first introduced a new concept ofA, η-accretive mappings, which generalizes the existing monotone or accretive operators and studied some properties of A, η-accretive mappings and defined resolvent operators associated with
A, η-accretive mappings. They also investigated a class of variational inclusions using the resolvent operator associated with A, η-accretive mappings. Subsequently, Lan 39, by using the concept of A, η-accretive mappings and the new resolvent operator technique associated with A, η-accretive mappings, introduced and studied a system of general mixed quasivariational inclusions involvingA, η-accretive mappings in Banach spaces and constructed a perturbed iterative algorithm with mixed errors for this system of nonlinear
A, η-accretive variational inclusions inq-uniformly smooth Banach spaces.
On the other hand, the fuzzy set theory introduced by Zadeh 40has emerged as an interesting and fascinating branch of pure and applied sciences. The application of the fuzzy set theory can be found in many branches of regional, physical, mathematical, and engineering sciencessee41–45and the references therein.
In 1989, Chang and Zhu46first introduced the classes of variational inequalities for fuzzy mappings. In subsequent years, several classes of variational inequalities, variational inclusions, and complementarity problems for fuzzy mappings were investigated by many authors, in particular, by Chang and Haung47,48, Lan et al.49, Noor50–52, Noor and Al-said53, and many others.
Recently, Lan and Verma 54, by using the concept of A, η-accretive mappings, the resolvent operator technique associated withA, η-accretive mappings, introduced and studied a new class of nonlinear fuzzy variational inclusion systems with A, η-accretive mappings in Banach spaces and construct some new iterative algorithms to approximate the solutions of the nonlinear fuzzy variational inclusion systems.
Inspired and motivated by recent research works in these fields, in this paper, we introduce and study a new system of nonlinear fuzzy variational inclusions with
A, η-accretive mappings in Banach spaces. By using the resolvent operator associated with A, η-mappings due to Lan et al. and Nadler’s fixed points theorem, we construct some new iterative algorithms for approximating the solutions of this system of nonlinear fuzzy variational inclusions in Banach spaces and prove the existence of solutions and the convergence of the sequences generated by the algorithms inq-uniformly smooth Banach spaces. The results presented in this paper improve and extend the corresponding results of
29–35,38,39,55–60and many other recent works.
2. Preliminaries
LetX be a real Banach space with dual spaceX∗,·,·be the dual pair betweenX andX∗, 2X denote the family of all nonempty subsets ofX, and let CBXdenote the family of all
nonempty closed bounded subsets ofX. The generalized duality mapping Jq : X → 2X
∗ is defined by
Jqx
f∗∈X∗:x, f∗xq,f∗xq−1, ∀x∈X, 2.1
It is known that, in general,Jqx xq−2J2xfor allx /0 andJq is single valued if
X∗is strictly convex. In the sequel, we always assume thatXis a real Banach space such that
Jqis single-valued. IfXis a Hilbert space, thenJ2becomes the identity mapping onX.
The modulus of smoothnessofXis the functionρX :0,∞ → 0,∞defined by
ρXt sup
1
2x y x−y −1 :x ≤1, y≤t
. 2.2
A Banach spaceXis said to beuniformly smoothif
lim
t→0
ρXt
t 0. 2.3
Xis calledq-uniformly smoothif there exists a constantc >0 such that
ρXt≤ctq, ∀q >1. 2.4
Note that Jq is single-valued if X is uniformly smooth. Concerned with the
characteristic inequalities in q-uniformly smooth Banach spaces, Xu 61 proved the following result.
Lemma 2.1. A real Banach spaceXisq-uniformly smooth if and only if there exists a constantcq>0
such that, for allx, y∈X,
x yq≤ xq qy, Jqx cqyq. 2.5
Definition 2.2. A set-valued mappingT :X → 2Xis said to beξ-H-Lipschitz continuousif there
exists a constantξ >0 such that
HTx, Ty ≤ξx−y, ∀x, y∈X, 2.6
whereH: 2X×2X → R∪ { ∞}is the Hausdorffpseudo-metric, that is,
HA, B max
sup
x∈A
dx, B,sup
y∈B
dy, A
, ∀A, B∈2X, 2.7
wheredu, K infv∈Ku−v.
Lemma 2.3see62. LetX, dbe a complete metric space and letT:X → CBXbe a set-valued mapping satisfying
HTx, Ty ≤kdx, y , ∀x, y∈X, 2.8
wherek∈0,1is a constant. Then the mappingT has a fixed point inX.
Lemma 2.4see62. LetX, dbe a complete metric space and letT :X → CBXba a set-valued mapping. Then for anyε >0and anyx, y∈X,u∈Tx, there existsv∈Tysuch that
du, v≤1 εHTx, Ty . 2.9
Definition 2.5. LetXbe aq-uniformly smooth Banach space,T, A:X → Xand letη:X×X → Xbe single-valued mappings.
iT is said to beaccretiveif
Tx−Ty , Jq
x−y ≥0, ∀x, y∈X; 2.10
iiT is said to bestrictly accretiveifTis accretive and
Tx−Ty , Jq
x−y 0 2.11
if and only ifxy;
iiiT is said to ber-strongly accretiveif there exists a constantr >0 such that
Tx−Ty , Jq
x−y ≥rx−yq, ∀x, y∈X; 2.12
ivT is said to bem-relaxed accretiveif there exists a constantm >0 such that
Tx−Ty , Jq
x−y ≥ −mx−yq, ∀x, y∈X; 2.13
vT is said to beζ, ς-relaxed cocoerciveif there exist constantsζ, ς >0 such that
Tx−Ty , Jq
x−y ≥ −ζTx−Tyq ςx−yq, ∀x, y∈X; 2.14
viT is said to beγ-Lipschitz continuousif there exists a constantγ >0 such that
Tx−T
y ≤γx−y, ∀x, y∈X; 2.15
viiηis said to beτ-Lipschitz continuousif there exists a constantτsuch that
viiiη·,·is said to be-Lipschitz continuous in the first variableif there exists a constant
>0 such that
ηx, u−η
y, u ≤x−y, ∀x, y, u∈X; 2.17
ixη·, uis said to beρ, ξ-relaxed cocoercive with respect toAif there exist constants
ρ, ξ >0 such that
ηx, u−ηy, u , Jq
Ax−Ay ≥ −ρηx, u−ηy, uq ξx−yq, ∀x, y, u∈X.
2.18
In a similar way to viii and ix, we can define the Lipschitz continuity of the mappingη·,·in the second variable and relaxed cocoercivity ofηu,·with respect toA.
Definition 2.6. LetX be aq-uniformly smooth Banach space,η : X×X → X and letH, A:
X → Xbe three single-valued mappings. Set-valued mappingM:X → 2Xis said to be iaccretiveif
u−v, Jq
x−y ≥0, ∀x, y∈X, u∈Mx, v∈My; 2.19
iiη-accretiveif
u−v, Jq
ηx, y ≥0, ∀x, y∈X, u∈Mx, v∈My; 2.20
iiistrictlyη-accretiveifMisη-accretive and the equality holds if and only ifxy;
ivr-stronglyη-accretiveif there exists a constantr >0 such that
u−v, Jq
ηx, y ≥rx−yq, ∀x, y∈X, u∈Mx, v∈My; 2.21
vα-relaxedη-accretiveif there exists a constantα >0 such that
u−v, Jq
ηx, y ≥ −αx−yq, ∀x, y∈X, u∈Mx, v∈My; 2.22
vim-accretiveifMis accretive andI λMX Xfor allλ >0, whereIdenotes the identity operator onX;
viigeneralizedm-accretiveifMisη-accretive andI λMX Xfor allλ >0;
viiiH-accretiveifMis accretive andH λMX Xfor allλ >0;
ix H, η-accretiveifMisη-accretive andH λMX Xfor allλ >0.
Remark 2.7. The following should be noticed.
H-accretive operators was first introduced and studied by Fang and Huang63
and also includes that ofm-accretive operators as a special case.
2WhenX His a Hilbert space,i–ixofDefinition 2.6reduce to the definitions of monotone operators, η-monotone operators, strictly η-monotone operators, stronglyη-monotone operators, relaxedη-monotone operators, maximal monotone operators, maximal η-monotone operators, H-monotone operators, and H, η -monotone operators, respectively.
Definition 2.8. LetA : X → X, η : X ×X → X be two single-valued mappings and let
M:X → 2X be a set-valued mapping. ThenMis said to beA, η-accretivewith constantm
ifMism-relaxedη-accretive andA λMX Xfor allλ >0.
Remark 2.9. For appropriate and suitable choices ofm,A,η, and the spaceX, it is easy to see thatDefinition 2.8includes a number of definitions of monotone operators and accretive operatorssee38.
In38, Lan et al. showed thatA ρM−1is a single-valued operator ifM:X → 2X
is anA, η-accretive mapping andA : X → X an r-stronglyη-accretive mapping. Based on this fact, we can define the resolvent operatorRη,AM,ρ associated with an A, η-accretive mappingMas follows.
Definition 2.10. LetA:X → X be a strictlyη-accretive mapping and letM:X → 2Xbe an A, η-accretive mapping. The resolvent operatorRη,AM,ρ :X → X associated withAandMis defined by
Rη,AM,ρx A ρM −1x, ∀x∈X. 2.23
Proposition 2.11see38. LetXbe aq-uniformly smooth Banach space, letη :X×X → Xbe
τ-Lipschitz continuous, letA:X → X be ar-stronglyη-accretive mapping and letM :X → 2X
be an A, η-accretive mapping with constant m. Then the resolvent operator Rη,AM,ρ : X → X is τq−1/r−ρm-Lipschitz continuous, that is,
Rη,A
M,ρx−R η,A M,ρ
y ≤ τ
q−1
r−ρmx−y, ∀x, y∈X, 2.24
whereρ∈0, r/mis a constant.
In what follows, we denote the collection of all fuzzy sets onXbyFX {A | A :
X → 0,1}. A mappingS fromX toFXis called a fuzzy mapping. IfS : X → FXis a fuzzy mapping, then the setSxfor anyx ∈ Xis a fuzzy set onFX in the sequel we denoteSxbySxandSxyfor anyy∈Xis the degree of membership ofyinSx. For any
A∈FXandα∈0,1, the set
Aα{x∈X :Ax≥α} 2.25
A fuzzy mappingS : X → FXis said to satisfy the condition∗if there exists a functiona:X → 0,1such that for eachx∈Xthe set
Sxax:
y∈X:Sx
y ≥ax 2.26
is a nonempty closed and bounded subset ofX, that is,Sxax∈CBX.
By using the fuzzy mapping S satisfying the condition ∗ with corresponding functiona:X → 0,1, we can define a set-valued mappingSas follows:
S:X−→CBX, x−→Sxax. 2.27
In the sequel,S,T,L,D,G,W, andK are calledthe set-valued mappings induced by the fuzzy mappingsS,T,L,D,G,W, andK, respectively.
3. A New System of Fuzzy Variational Inclusions
In this section, we introduce some systems of fuzzy variational inclusions q-uniformly smooth Banach spacesXand their relations.
LetX1 be aq1-uniformly smooth Banach space withq1 > 1, letX2 be aq2-uniformly
smooth Banach space withq2>1, letE, P :X1×X2 → X1,F, Q:X1×X2 → X2,A1:X1 → X1,
A2 :X2 → X2,f, p, l :X1 → X1,g, h, k :X2 → X2,η1 :X1×X1 → X1,η2 :X2×X2 → X2
be single-valued mappings, and let S,T,L,D : X1 → FX1and G,W,K : X2 → FX2
be fuzzy mappings. Further, suppose thatM : X1 ×X1 → 2X1 and N : X2 ×X2 → 2X2
are any nonlinear operators such that for all z ∈ X1, M·, z : X1 → 2X1 is anA1, η1
-accretive withfx−y∈domM·, zfor allx, y∈X1and for allt∈X2,N·, t:X2 → 2X2
is anA2, η2-accretive withgu ∈ domN·, tfor all u ∈ X2. Now, for given mappings
a,b,c,d:X1 → 0,1ande, f,g:X2 → 0,1, we consider the following system.
System 3.1. For any givena∈X1,b∈X2,λ1>0,λ2>0, our problem is as follows:
Findx, z, u, v, m∈X1andy, w, t, s∈X2such thatSxu≥ax,Txv≥bx,Lxz≥
cx,Dxm≥dx,Gyw≥ey,Wyt≥fy,Kys≥gy, and
⎧ ⎨ ⎩
a∈Epx, w Plz, t λ1M
fx−v, x ,
b∈Fu, hy Qm, ks λ2N
gy , y . 3.1
This system is called asystem of nonlinear fuzzy variational inclusions involvingA, η-accretive mappingsin uniformly smooth Banach spaces.
Remark 3.2. For appropriate and suitable choices ofX1,X2,q1,q2,E,P,F,Q,A1,A2,f,g,h,k,
l,p,η1,η2,S,T,L,D,G,W,K,M,N,a,b,c,d,e,f, andgone can obtain many known and
new classes offuzzyvariational inequalities andfuzzyvariational inclusions as special cases of System3.1.
System 3.3. LetS, T, L, D:X1 → CBX1andG, W, K:X2 → CBX2be classical set-valued
mappings and letM,N,f,g,E,P,F,Q,p,l,h,kbe the mappings as in System3.1. Now, by usingS,T,L,D, G,W, and K,we define fuzzy mappings S,T,L,D : X1 → 2X1 and
G,W,K:X2 → 2X2as follows:
Sx χSx, TxχTx, LxχLx
DxχDx,GxχGx, WxχWx, KxχKx,
3.2
whereχSx,χTx,χLx,χDx,χGx,χWx, andχKxare the characteristic functions of the
setsSx,Tx,Lx,Dx,Gx,Wx, andKx, respectively.
It is easy to see thatS,T,L, andDare fuzzy mappings satisfying the condition∗ with constant functionsax 1,bx 1, cx 1, dx 1 for allx ∈ X1, respectively,
and G,W, and K are fuzzy mappings satisfying the condition∗with constant functions
ey 1,fy 1,gy 1 for ally∈X2, respectively. Also
Sax χSx 1
r ∈X1:χSxr 1Sx,
Tbx
χTx 1 r∈X1:χTxr 1Tx,
Lcx χLx 1r ∈X1:χLxr 1
Lx,
Ddx χDx 1
r∈X1:χDxr 1Dx,
Gey χGy 1t∈X2 :χGyt 1Gy ,
Wfy
χWy 1t∈X2:χWyt 1Wy ,
Kgy χKy 1t∈X2:χKyt 1
Ky .
3.3
Then System3.1is equivalent to the following:
Findx, z, u, v, m∈X1,y, w, t, s∈X2such thatu∈Sx,v∈Tx,z∈Lx,m∈Dx,
w∈Gy,t∈Wy,s∈Ky, and
⎧ ⎨ ⎩
a∈Epx, w Plz, t λ1M
fx−v, x ,
b∈Fu, hy Qm, ks λ2N
gy , y . 3.4
System3.3 is called a system of nonlinear set-valued variational inclusions with A, η -accretive mappings.
System 3.4. IfT : X1 → X1 is a single-valued mapping, then System 3.3collapses to the
following system of nonlinear variational inclusions:
Findx, z, u, m ∈X1,y, w, t, s∈X2such thatu∈Sx,z∈Lx,m∈Dx,w ∈Gy,
t∈Wy,s∈Ky, and
⎧ ⎨ ⎩
a∈Epx, w Plz, t λ1M
fx−Tx, x ,
b∈Fu, hy Qm, ks λ2N
gy , y .
System 3.5. IfXiHi i1,2are two Hilbert spaces,S:H1 → H1andG:H2 → H2are
two single-valued mappings,p hl k ≡I: the identity mapping,λ1 λ2 1,T ≡ 0:
the zero mapping,ab0,Mx, y Mxfor allx, y∈ H1× H1,Nx, y Nxfor
allx, y∈ H2× H2, then System3.4reduces to the following system:
Findx, y, z, m, t, ssuch thatx, y∈ H1×H2,z∈Lx,m∈Dx,t∈Wy,s∈Ky,
and
⎧ ⎨ ⎩
0∈Ex,y Pz, t Mfx ,
0∈Fx, y Qm, s Ngy .
3.6
System3.5was introduced and studied by Peng and Zhu in59.
System 3.6. Ifa b0,λ1 λ2 1, andP Q≡0, then System3.3can be replaced by the
following:
Findu∈Sx,v∈Txandw∈Gysuch that
⎧ ⎨ ⎩
0∈Epx, w Mfx−v, x ,
0∈Fu, hy Ngy , y ,
3.7
which is studied by Lan and Verma54.
System 3.7. IfT : X1 → X1 is a single-valued mapping, then System 3.6collapses to the
following system of nonlinear variational inclusions:
Findx, u∈X1,y, w∈X2such thatu∈Sx,w∈Gyand
⎧ ⎨ ⎩
0∈Epx, w Mfx−Tx, x ,
0∈Fu, hy Ngy , y ,
3.8
which is studied by Lan and Verma54.
System 3.8. Ifph≡I,T ≡0,Mx, y Mxfor allx, y∈X1×X1,Nx, y Nxfor
allx, y∈X2×X2,S :X1 → X1, andG :X2 → X2are identity mappings, then System3.7
reduces to the following system: Findx, y∈X1×X2such that
⎧ ⎨ ⎩
0∈Ex, Gy Mfx ,
0∈FSx, y Ngy . 3.9
System3.8is investigated by Jin55whenSandGare the identity mappings.
System 3.9. Iff−T g ≡I,P Q≡0,λ1 λ2 1,S:X1 → X1, andG:X2 → X2are two
Findx, y∈X1×X2such that
⎧ ⎨ ⎩
a∈Epx, Gy Mx, x,
b∈FSx, hy Ny, y ,
3.10
which is introduced and studied by Lan39whenSandGare identity mappings.
System 3.10. WhenphSG≡I,ab0, System3.9can be replaced to the following: Findx, y∈X1×X2such that
⎧ ⎨ ⎩
0∈Ex, y Mx, x,
0∈Fx, y Ny, y ,
3.11
which is studied by Jin56.
System 3.11. If Xi Hi i 1,2 is two Hilbert spaces, Mx, y Mx for all x, y ∈
H1×H1andNx, y Nxfor allx, y∈ H2×H2, then System3.7reduces to the following
generalized system of set-valued variational inclusions:
Findx, u∈ H1,y, w∈ H2such thatu∈Sx,w∈Gy, and
⎧ ⎨ ⎩
0∈Epx, w Mfx−Tx ,
0∈Fu, hy Ngy ,
3.12
which is studied by Lan et al.57whenM,N areA-monotone mappings and there exists single-valued mappingψ:H1 → H1such thatψx fx−Txfor allx∈ H1.
System 3.12. Ifg p hf−T ≡ I, then System3.11collapses to the following system of nonlinear variational inclusions:
Findx, y∈ H1× H2,u∈Sx,w∈Gysuch that
⎧ ⎨ ⎩
0∈Ex, w Mx,
0∈Fu, y Ny ,
3.13
which is considered by Huang and Fang28.
System 3.13. If S : H1 → H1 and G : H2 → H2 are two single-valued mappings, then
System3.12is equivalent to the following: Findx, y∈ H1× H2such that
⎧ ⎨ ⎩
0∈Ex, Gy Mx,
0∈FSx, y Ny ,
which is investigated by Fang et al.31and Peng et al.35, Fang and Huang34, and Verma
36withSG≡I.
System 3.14. If Mx ∂ϕx andNy ∂φy for allx ∈ H1 and y ∈ H2, where ϕ :
H1 → R∪ { ∞}andφ:H2 → R∪ { ∞}are two proper, convex, and lower semicontinuous
functionals, and∂ϕ and ∂φ denote subdifferential operators ofϕ and φ, respectively, then System3.13reduces to the following system:
Findx, y∈ H1× H2such that
⎧ ⎨ ⎩
Ex, Gy , s−x ϕs−ϕx≥0, ∀s∈ H1,
FSx, y , t−y φt−φy ≥0, ∀t∈ H2,
3.15
which is called asystem of nonlinear mixed variational inequalities. Some special cases of System
3.7can be found in22. Further, ifS G ≡ I, then System3.7 reduces to the system of nonlinear variational inequalities considered by Cho et al.25.
System 3.15. If Mx ∂δK1x and Ny ∂δK2y for all x ∈ K1 and y ∈ K2, where
K1andK2are nonempty closed convex subsets ofH1andH2, respectively, andδK1andδK2
denote indicator functions ofK1andK2, respectively, then System3.14becomes the following
problem:
Findx, y∈K1×K2such that
⎧ ⎨ ⎩
Ex, Gy , s−x ≥0, ∀s∈K1,
FSx, y , t−y ≥0, ∀t∈K2,
3.16
which is the just system in24whenSandGare singlevalued andSG≡I.
System 3.16. IfH1 H2 H,K1 K2 K,Ex, Gy ρ1Gy x−y, andFSx, y
ρ2Sx y−xfor allx, y∈ H, whereρ1>0 andρ2>0 are two constants, then System3.15is
equivalent to the following:
Find an elementx, y∈K×Ksuch that
⎧ ⎨ ⎩
ρ1G
y x−y, s−x ≥0, ∀s∈K,
ρ2Sx y−x, t−y ≥0, ∀t∈K,
3.17
which is the system of nonlinear variational inequalities considered by Verma22withS G.
Remark 3.17. Ifxy,SGandρ1 ρ2, then System3.16reduces to the following classical
nonlinear variational inequality problem: Find an elementx∈Ksuch that
4. Existence Theorems
In this section, we prove the existence theorem for solutions of Systems3.1. For our main results, we have the following lemma which offers a good approach to solve System3.1.
Lemma 4.1. LetXi,Ai,ηi,λi i 1,2,E,F,P,Q,S,T,L,D,G,W,K,M,N,f,g,h,p,l,
k,a, andbbe the same as in System3.1. Then, for any givenx, z, u, v, m∈X1andy, w, t, s∈X2, x, y, z, t, m, s, u, v, wis a solution of System3.1if and only if
fx v Rη1,A1
M·,x,ρ1
A1
fx−v −ρ1
λ1
Epx, w Plz, t−a ,
gy Rη2,A2 N·,y,ρ2
A2
gy − ρ2 λ2
Fu, hy Qm, ks−b ,
4.1
whereρ1>0andρ2>0are two constants.
Proof. The conclusion follows directly fromDefinition 2.10and some simple arguments.
FromLemma 4.1, we have the following.
Theorem 4.2. Let X1 and X2 be the same as inLemma 4.1, let S,T,L,D : X1 → FX1and
G,W,K : X2 → FX2 be fuzzy mappings satisfying the condition ∗ with the corresponding functionsa,b,c,d,e,fandg, respectively,S, T, L, D :X1 → CBX1, and letG, W, K :X2 →
CBX2 be ξ-H1-Lipschitz continuous, ζ-H1-Lipschitz continuous, γ-H1-Lipschitz continuous,
-H1-Lipschitz continuous, ξ-H2-Lipschitz continuous, ζ-H2-Lipschitz continuous, and γ-H2 -Lipschitz continuous, respectively, whereHiis the Hausdorffpseudometric on2Xifori1,2. Assume
thatηi :Xi×Xi → Xiisτi-Lipschitz continuous,Ai :Xi → Xiisri-stronglyηi-accretive andβi
-Lipschitz continuous for i 1,2,p, l : X1 → X1 are δ1-Lipschitz continuous, andδ2-Lipschitz continuous, respectively,h, k :X2 → X2areπ1-Lipschitz continuous andπ2-Lipschitz continuous, respectively,f:X1 → X1isκ, e1-relaxed cocoercive,μ-Lipschitz continuous andg :X2 → X2is σ, e2-relaxed cocoercive,-Lipschitz continuous. Suppose thatM·, z:X1 → 2X1is anA1, η1 -accretive operator with constantm1for allz∈X1andN·, t:X2 → 2X2 is anA2, η2-accretive operator with constantm2for allt∈X2, andE, P :X1×X2 → X1are two single-valued mappings such that E·, y and P·, y areν1-Lipschitz continuous and ν2-Lipschitz continuous in the first variable, respectively,Ex,·,Px,·areι1-Lipschitz continuousι2-Lipschitz continuous in the second variable, respectively, for all x, y ∈ X1 ×X2, andEp1·, yis θ1, s1-relaxed cocoercive with respect tof, wheref : X1 → X1 is defined byfx A1◦fx−v A1fx−vfor allx ∈ X1,b : X1 → 0,1 and Txv ≥ bx. Further, suppose thatF, Q : X1 ×X2 → X2 are two nonlinear mappings such thatF·, y,Q·, yareρ1-Lipschitz continuous andρ2-Lipschitz continuous in the first variable, respectively,Fx,·andQx,·areυ1-Lipschitz continuous andυ2 -Lipschitz continuous in the second variable, respectively, andFx, h·isθ2, s2-relaxed cocoercive with respect tog, whereg:X2 → X2is defined bygx A2◦gx A2gxfor allx∈X2.
In addition, if there exist constantsρ1∈0, r1/m1andρ2∈0, r2/m2such that
Rη1,A1
M·,x,ρ1z−R η1,A1 M·,y,ρ1z
≤ςx−y, ∀x, y, z∈X1, 4.2
Rη2,A2
N·,x,ρ2z−R η2,A2 N·,y,ρ2z
ς ζ q1
1−q1e1
cq1 q1κ μq1<1,
ϑ q2
1−q2e2
cq2 q2σ q2 <1,
q1
βq1 1
μ ζ q1−q
1
ρ1
λ1
−θ1ν1q1δ1q1 s1
cq1ρ1q1ν1q1δ1 q1
λ1q1
< τ
1−q1 1 λ1
ρ1
r1−ρ1m1 χ1−ν2δ2γ, q2
βq2 2 q2−q2
ρ2
λ2−θ2υ1
q2π1q2 s2 cq2ρ2 q2υ
1q2π1q2
λ2q2
< τ
1−q2 2 λ2
ρ2
r2−ρ2m2 χ2−υ2π2γ, 4.4
where
χ1 1−
ς ζ q1
1−q1e1
cq1 q1κ μq1
− ρ2τ
q2−1 2
ρ1ξ ρ2
λ2
r2−ρ2m2
,
χ2 1−
ϑ q2
1−q2e2
cq2 q2σ q2
−ρ1τ
q1−1
1 ι1ξ ι2ζ
λ1
r1−ρ1m1
,
4.5
λ1,λ2 are the same as in System3.1, andcq1,cq2 are two constants guaranteed byLemma 2.1, then System3.1admits a solution.
Proof. For any givenρ1 >0 andρ2 >0, define mappingsΦρ1 :X1×X1×X1×X2×X2 → X1
andΨρ2:X1×X1×X2×X2 → X2as follows:
Φρ1x, z, v, t, w
x−fx v Rη1,A1 M·,x,ρ1
A1
fx−v − ρ1
λ1
Epx, w Plz, t−a ,
Ψρ2
u, m, s, y
y−gy Rη2,A2 N·,y,ρ2
A2
gy −ρ2 λ2
Fu, hy Qm, ks−b
4.6
for allx, y, z, t, m, s, u, v, w∈X1×X2×X1×X2×X1×X2×X1×X1×X2, wherea∈X1and
b∈X2are the same as in System3.1, and leta, b, c, d:X1 → 0,1ande,f,g:X2 → 0,1be
mappings such thatSxu≥ax,Txv≥bx,Lxz≥cx,Dxm≥dx,Gyw≥ey,
Wyt≥fy, andKys≥gy.
Now, define a norm · ∗onX1×X2by
u, v∗u v, ∀u, v∈X1×X2. 4.7
It is easy to see thatX1×X2, · ∗is a Banach spacesee34. For any givenρ1 > 0 and
ρ2>0, define a mappingQρ1,ρ2:X1×X2×X1×X1×X1×X1×X2×X2×X2 → X1×X2by
Qρ1,ρ2
x, y, z, u, v, m, s, t, w Φρ1x, z, v, t, w,Ψρ2
for allx, y, z, u, v, m, s, t, w∈X1×X2×X1×X1×X1×X1×X2×X2×X2and let
Rρ1,ρ2
x, y Qρ1,ρ2
x, y, z, u, v, m, s, t, w :
Sxu≥ax,Txv≥bx,Lxz≥cx,Dxm≥dx,Gyw≥e
y ,
Wyt≥f
y ,Kys≥g
y , wherea, b,c,d:X1 → 0,1,
e,f,g:X2 → 0,1
4.9
for all x, y ∈ X1 × X2. Then, for any given x, y,x, y ∈ X1 × X2, ε > 0 and
Qρ1,ρ2x, y, z, u, v, m, s, t, w ∈ Rρ1,ρ2x, y, there existsz, u, v, m, s, t, w ∈ X1 ×X1×X1 ×
X1×X2×X2×X2such that
Sxu≥ax, Txv≥bx, Lxz≥cx, Dxm≥dx,
Gyw≥e
y , Wyt≥f
y , Kys≥g
y ,
4.10
wherea, b,c,d:X1 → 0,1,e, f,g:X2 → 0,1and4.6holds. SinceSxu≥ax,Txv≥
bx,Lxz ≥ cx,Dxm ≥ dx,Gyw ≥ ey,Wyt ≥ fy,Kys ≥ gy, that is,u ∈
Sx ∈CBX1,v ∈Tx∈ CBX1,z ∈ Lx∈ CBX1,m∈ Dx ∈CBX1,w ∈ Gy ∈
CBX2, t ∈ Wy ∈ CBX2,s ∈ Ky ∈ CBX2, it follows fromLemma 2.4 that there existu ∈ Sx,v ∈ Tx,z ∈ Lx,m ∈ Dx,w ∈ Gy,t ∈ Wy,s ∈ Ky, that is,Sxu ≥ ax,Txv ≥ bx,Lxz ≥ cx,Dxm ≥ dx,Gyw ≥ ey,Wyt ≥
fy,Kys≥gysuch that
u−u≤1 εH
1
Sx, Sx , v−v≤1 εH1
Tx, Tx , z−z≤1 εH1
Lx, Lx , m−m≤1 εH1
Dx, Dx , w−w≤1 εH2
Gy , Gy , t−t≤1 εH2
Wy , Wy , s−s≤1 εH
2
Ky , Ky .
4.11
Letting
Φρ1
x, z, v, t, w
x−fx v Rη1,A1 M·,x,ρ1
A1
fx −v −ρ1
λ1
Epx , w Plz , t −a ,
Ψρ2
u, m, s, y
y−gy Rη2,A2 N·,y,ρ2
A2
gy −ρ2 λ2
Fu, hy Qm, ks −b ,
we have
Φρ1
x, z, v, t, w ,Ψρ2
u, m, s, y Qρ1,ρ2
x, y, z, u, v, m, s, t, w . 4.13
Now, it follows from4.2andProposition 2.11that
Φρ
1x, z, v, t, w−Φρ1
x, z, v, t, w
≤x−x−fx−fx v−v
Rη1,A1 M·,x,ρ1
A1
fx−v − ρ1
λ1
Epx, w Plz, t−a
−Rη1,A1 M·,x,ρ1
A1
fx −v − ρ1
λ1
Epx , w Plz , t −a
≤x−x−fx−fx v−v
Rη1,A1 M·,x,ρ1
A1
fx−v − ρ1
λ1
Epx, w Plz, t−a
−Rη1,A1 M·,x,ρ1
A1
fx −v −ρ1
λ1
Epx , w Plz , t −a
Rη1,A1
M·,x,ρ1
A1
fx −v −ρ1
λ1
Epx , w Plz , t −a
−Rη1,A1 M·,x,ρ1
A1
fx −v − ρ1
λ1
Epx , w Plz , t −a
≤x−x−fx−fx v−v ςx−x
τq1−1 1
r1−ρ1m1
A1
fx−v − ρ1
λ1
Epx, w Plz, t−a
−A1
fx −v −ρ1
λ1
Epx , w Plz , t −a
≤x−x−fx−fx v−v ςx−x
τq1−1 1
r1−ρ1m1
ρ
1
λ1
Epx, w −Epx, w Plz, t−Plz, t Plz, t −Plz , t
A1
fx−v −A1
fx −v − ρ1
λ1
Epx, w −Epx , w
.
Thus, byLemma 2.1, we have
x−x−fx−fxq1
≤x−xq1−q 1
fx−fx , Jq1
x−x cq1fx−fx
q1. 4.15
Sincefisκ, e1-relaxed cocoercive andμ-Lipschitz continuous, we conclude that
x−x−fx−fxq1≤x−xq1−q
1e1x−xq1
cq1 q1κ μ
q1x−xq1
1−q1e1
cq1 q1κ μ
q1 x−xq1. 4.16
By4.11andζ-H1-Lipschitz continuity ofT,
v−v≤1 εH
1
Tx, Tx ≤ζ1 εx−x. 4.17
Since Ex,· is ι1-Lipschitz continuous in the second variable and G is ξ-H2-Lipschitz
continuous, by4.11, we have
Epx, w −Epx, w ≤ι1w−w≤ι11 εH2
Gy , Gy
≤ι1ξ1 εy−y.
4.18
Since Px,· is ι2-Lipschitz continuous in the second variable, W is ζ-H2-Lipschitz
continuous, p2 is δ2-Lipschitz continuous, P·, y is ν2-Lipschitz continuous in the first
variable, andLisγ-H1-Lipschitz continuous, using4.11, we deduce that
Plz, t−Plz, t ≤ι2t−t≤ι21 εH2
Wy , Wy
≤ι2ζ1 εy−y,
4.19
and also
P
lz, t −Plz , t ≤ν2lz−l
z ≤ν2δ2z−z
≤ν2δ21 εH1
Lx, Lx
≤ν2δ2γ1 εx−x.
Again, byLemma 2.1, it follows that
A1fx−v−A1fx−v−ρ1
λ1Epx, w
−Epx, wq1
≤A1fx−v−A1fx−vq1−q 1
ρ1
λ1
×Epx, w −Epx , w , Jq1
A1
fx−v −A1
fx −v
cq1
ρ1q1
λ1q1
Epx, w−Epx, wq1.
4.21
Since A1 is β1-Lipschitz continuous, f is μ-Lipschitz continuous, and T is ζ-Lipschitz
continuous, by4.11, we get
A1
fx−v −A1
fx −v ≤β1fx−f
x −v−v
≤β1fx−f
x v−v
≤β1
μ ζ1 ε x−x.
4.22
SinceEp·, yisθ1, s1-relaxed cocoercive with respect tof, wherefx A1◦fx−
v A1fx−v,E·, yisν1-Lipschitz continuous in the first variable andpisδ1-Lipschitz
continuous, we have
Epx, w −Epx , w , Jq1
A1
fx−v −A1
fx −v
≤ −θ1Epx, w−Epx, wq1 s1x−xq1
≤ −θ1ν1q1px−pxq1 s1x−xq1
≤−θ1ν1q1δ1q1 s1 x−xq1,
4.23
and also
Epx, w −Epx , w ≤ν1px−p
x ≤ν1δ1x−x. 4.24
Hence, using4.21–4.24, we have
A1fx−v−A1fx−v−ρ1 λ1
Epx, w−Epx, w
q1
≤
β1q1
μ ζ1 ε q1−q
1
ρ1
λ1
−θ1ν1q1δ1q1 s1
cq1ρ1q1ν1q1δ1 q1
λ1q1
x−xq1,
4.25
wherecq1is the constant as inLemma 2.1. Using4.14–4.20, and4.25, it follows that
Φρ
1x, z, v, t, w−Φρ1
where
ϕ1ε ς ζ1 ε q1
1−q1e1
cq1 q1κ μq1
ρ1τ1q1−1
ν2δ2γ1 ε ψ1ε
λ1
r1−ρ1m1
,
ψ1ε
q1
β1q1
μ ζ1 ε q1−q
1
ρ1
λ1
−θ1ν1q1δ1q1 s1
cq1ρ1q1ν1q1δ1 q1
λ1q1
,
φ1ε ρ1τ
q1−1
1 ι1ξ ι2ζ1 ε
λ1
r1−ρ1m1
.
4.27
Similarly, for anyu, m, s, y,u, m, s, y∈X1×X1×X2×X2, it follows from4.3and
Proposition 2.11that
Ψρ
2
u, m, s, y −Ψρ2
u, m, s, y
≤y−y−gy −gy
Rη2,A2 N·,y,ρ2
A2
gy −ρ2 λ2
Fu, hy Qm, ks−b
−Rη2,A2 N·,y,ρ2
A2
gy −ρ2 λ2
Fu, hy Qm, ks −b
≤y−y−gy −gy
Rη2,A2 N·,y,ρ2
A2
gy −ρ2 λ2
Fu, hy Qm, ks−b
−Rη2,A2 N·,y,ρ2
A2
gy − ρ2 λ2
Fu, hy Qm, ks −b
Rη2,A2
N·,y,ρ2
A2
gy −ρ2 λ2
Fu, hy Qm, ks −b
−Rη2,A2 N·,y,ρ2
A2
gy −ρ2 λ2
Fu, hy Qm, ks −b
≤y−y−gy −gy ϑy−y
τq2−1 2
r2−ρ2m2
A2
gy −ρ2 λ2
Fu, hy Qm, ks−b
−
A2
gy − ρ2 λ2
≤y−y−gy −gy ϑy−y τ
q2−1 2
r2−ρ2m2
×
ρ
2
λ2
F
u, hy −Fu, hy Qm, ks−Qm, ks
Q
m, ks −Qm, ks
A2
gy −A2
gy −ρ2 λ2
Fu, hy −Fu, hy
.
4.28
Thus, byLemma 2.1, we have
y−y−gy−gyq2
≤y−yq2−q 2g
y −gy , Jq2
y−y cq2gy−gy
q2. 4.29
Sincegisσ, e2-relaxed cocoercive and-Lipschitz continuous, we have
y−y−gy−gyq2≤
y−yq2−
q2e2y−yq2
cq2 q2σ
q2y−yq2
1−q2e2
cq2 q2σ
q2 y−yq2. 4.30
Since F·, y is ρ1-Lipschitz continuous in the first variable and S is ξ-H1-Lipschitz
continuous, by4.11, we obtain
Fu, hy −Fu, hy ≤ρ1u−u≤ρ11 εH1
Sx, Sx
≤ρ1ξ1 εx−x.
4.31
SinceQx,·isυ2-Lipschitz continuous in the second variable,kisπ2-Lipschitz continuous,
Q·, yisρ2-Lipschitz continuous in the first variable,Dis-H1-Lipschitz continuous, and
Kisγ-H2-Lipschitz continuous, using4.11, we conclude that
Qm, ks−Qm, ks ≤ρ2m−m
≤ρ21 εH1
Dx, Dx
≤ρ21 εx−x,
4.32
Qm, ks −Qm, ks ≤υ2ks−k
s
≤υ2π2s−s
≤υ2π21 εH2
Ky , Ky
≤υ2π2γ1 εy−y.
Again, byLemma 2.1, it follows that
A2gy−A2gy−ρ2 λ2Fu
, hy−Fu, hyq2
≤A2gy−A2gyq2
−q2
ρ2
λ2
Fu, hy −Fu, hy , Jq2
A2
gy −A2
gy
cq2
ρ2q2
λ2q2
Fu, hy−Fu, hyq2
.
4.34
SinceA2isβ2-Lipschitz continuous andgis-Lipschitz continuous, we have
A2
gy −A2
gy ≤β2g
y −gy ≤β2y−y. 4.35
SinceFu, h·isθ2, s2-relaxed cocoercive with respect togA2◦g,Fx,·isυ1-Lipschitz
continuous in the second variable, andhisπ1-Lipschitz continuous, we get
Fu, hy −Fu, hy , Jq2
A2
gy −A2
gy
≤ −θ2Fu, hy−Fu, hyq2 s2y−yq2
≤ −θ2υ1q2hy−hyq2 s2y−yq2
≤−θ2υ1q2π1q2 s2y−yq2,
4.36
Fu, hy −Fu, hy ≤υ1h
y −hy
≤υ1π1y−y.
4.37
Therefore, it follows from4.34–4.37that
A2gy−A2gy−ρ2 λ2
Fu, hy−Fu, hy
q2
≤
β2q2q2−q2
ρ2
λ2−
θ2υ1q2π1q2 s2
cq2ρ2 q2υ
1q2π1q2
λ2q2
y−yq2,
4.38
wherecq2is the constant as inLemma 2.1. From4.28–4.33, and4.38, it follows that
Ψρ
2
u, m, s, y −Ψρ2
where
ϕ2ε ρ2τ
q2−1 2
ρ1ξ ρ2 1 ε
λ2
r2−ρ2m2
,
φ2ε ϑ q2
1−q2e2
cq2 q2σ q2
ρ2τ2q2−1
υ2π2γ1 ε ψ2
λ2
r2−ρ2m2
,
ψ2
q2
β2q2q2−q2
ρ2
λ2−
θ2υ1q2π1q2 s2
cq2ρ2q2υ1q2π1q2
λ2q2
.
4.40
It follows from4.26and4.39that
Φρ
1x, z, v, t, w−Φρ1
x, z, v, t, w Ψρ2
u, m, s, y −Ψρ2
u, m, s, y
≤ωεx−x y−y , 4.41
whereωε max{ϕ1ε ϕ2ε, φ1ε φ2ε}. Using4.8and4.41, we deduce that
Qρ1,ρ2x, y, z, u, v, m, s, t, w−Qρ1,ρ2x
, y, z, u, v, m, s, t, w ∗
≤ωεx, y−x, y∗,
4.42
that is,
sup
Qρ1,ρ2x,y,z,u,v,m,s,t,w∈Rρ1,ρ2x,y
dQρ1,ρ2
x, y, z, u, v, m, s, t, w ,Rρ1,ρ2
x, y
≤ωεx, y−x, y∗.
4.43
Similarly, we have
sup
Qρ1,ρ2x,y,z,u,v,m,s,t,w∈Rρ1,ρ2x,y
dQρ1,ρ2
x, y, z, u, v, m, s, t, w ,Rρ1,ρ2
x, y
≤ωεx, y−x, y∗.
4.44
By4.43,4.44, and the definition of Hausdorffpseudo-metric, we have
HRρ1,ρ2
x, y ,Rρ1,ρ2
x, y ≤ωεx, y−x, y∗, ∀x, y ,x, y ∈X1×X2. 4.45
Lettingε → 0, one has
HRρ1,ρ2
x, y ,Rρ1,ρ2
where
ωmaxϕ1 ϕ2, φ1 φ2
, 4.47
ϕ1ς ζ
q1
1−q1e1
cq1 q1κ μq1
ρ1τ1q1−1
ν2δ2γ ψ1
λ1
r1−ρ1m1
,
ψ1
q1
β1q1
μ ζ q1−q
1
ρ1
λ1
−θ1ν1q1δ1q1 s1
cq1ρ1q1ν1q1δ1 q1
λ1q1
,
φ2ϑ
q2
1−q2e2
cq2 q2σ q2
ρ2τ2q2−1
υ2π2γ ψ2
λ2
r2−ρ2m2
,
ϕ2
ρ2τ2q2−1
ρ1ξ ρ2
λ2
r2−ρ2m2
,
φ1
ρ1τ1q1−1ι1ξ ι2ζ
λ1
r1−ρ1m1
4.48
and ψ2 is the constant as in 4.40. From4.4, we know that 0 ≤ ω < 1 and so it follows
from4.46thatRρ1,ρ2 : X1 ×X2 → X1 ×X2 is a contractive mapping. HenceLemma 2.3
implies thatRρ1,ρ2 has a fixed point inX1 ×X2; that is, there exists a pointx∗, y∗ ∈ X1 ×
X2 such thatx∗, y∗ ∈ Rρ1,ρ2x∗, y∗. Now, it follows from4.6,4.8, andLemma 4.1that x∗, y∗, z∗, u∗, v∗, m∗, n∗, t∗, w∗is a solution of System3.1and this is the desired result. This completes the proof.
By usingTheorem 4.2, we can derive the following.
Theorem 4.3. Let Xi,Ai,ηi i1,2,S,T,L,D,G,W,K,S,T,L,D,G,W,K,M,N,E,P,
F,Q,p,l,h,k,f, andgbe the same as inTheorem 4.2. Assume thatf :X1 → X1isκ-strongly accretiveμ-Lipschitz continuous andg :X2 → X2isσ-strongly accretive-Lipschitz continuous.
Further, if there exist constantsρ1∈0, r1/m1andρ2∈0, r2/m2such that4.2and4.3 hold and
ς ζ q1
1−q1κ cq1μq1 <1,
ϑ q2
1−q2σ cq2q2<1,
q1
β1q1
μ ζ q1−q
1
ρ1
λ1
−θ1ν1q1δ1q1 s1
cq1ρ1q1ν1q1δ1 q1
λ1q1
< τ
1−q1 1 λ1
ρ1
r1−ρ1m1 χ1−ν2δ2γ,
q2
β2q2q2−q2
ρ2
λ2−
θ2υ1q2π1q2 s2
cq2ρ2 q2υ
1q2π1q2
λ2q2
< τ
1−q2 2 λ2
ρ2
