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Sensitivity analysis for strongly nonlinear quasi-variational inclusions
PERGAMON
Applied Mathematics Letters 13 (2000) 19-24
Applied Mathematics Letters www.elsevier.nl/locate/anfl
Sensitivity Analysis for Strongly Nonlinear Quasi-Variational Inclusions R. P. AGARWAL Department of Mathematics, National University of Singapore 10 Kent Ridge Crescent, Singapore 119260
YEOL JE CHO Department of Mathematics, Gyeongsang National University Chinju 660-701, Korea NAN JING HUANG Department of Mathematics, Sichuan University Chengdu, Sichuan 610064, P.R. China
(Received October 1999; accepted November 1999) A b s t r a c t - - I n this paper, we use the implicit resolvent operator technique to study the sensitivity analysis for strongly nonlinear quasi-variational inclusions. Our results improve and generalize some of the recent ones. @ 2000 Elsevier Science Ltd. All rights reserved. K e y w o r d s - - N o n l i n e a r quasi-variational inclusion, Sensitivity analysis, Resolvent operator. Hilbert space.
1. I N T R O D U C T I O N It is well recognized that variational inequalities play a fundamental role in the current mathematical technology. Thus, in recent years, the theory of classical variational inequalities has been extended and generalized to study a wide class of problems arising in mechanics, physics, optimization and control, nonlinear programming, economics and transportation equilibrium, and engineering sciences, etc. For details, we refer to [1-15] and the references therein. A useflfl and important generalization of variational inequalities is called the quasi-variational inclusion. Ding [16] arid Huang [17,18] have used the implicit resolvent operator techniques to study the existence of quasi-variational inclusions. Sensitivity analysis of solutions for variational inequalities have been studied by many authors using quite different methods. By using tile projection technique, Dafermos [19], Mukherjee and Verma [20], Noor [21] and Yen [22] dealt with the sensitivity analysis of solutions for variational inequalities. By using the implicit function approach that makes use of so-called normal mappings, Robinson [23] studied the sensitivity analysis of solutions for variational inequalities. Recently, Noor and Noor [24] dealt with the sensitivity The second author wishes to acknowledge the financial support of the Korea Research Foundation made ill the program year of 1998, Project No. 1998-015-D00020 and the third author was supported ill part by '98 APEC Post-Doctor Fellowship (KOSEF) while he visited Gyeongsang National University. 0893-9659/00/$ - see front matter @ 2000 Elsevier Science Ltd. All rights reserved. PIE S0893-9659(00)00048-3
Typeset by A2~[S-TFj~
20
R.P. AGARWALet al.
analysis of solutions for the quasi-variational inclusions by using the implicit resolvent equations technique without assuming the differentiability of the given data. Inspired and motivated by recent research works in this field, in this paper, by using the implicit resolvent operator technique, we study the sensitivity analysis for strongly nonlinear quasi-variational inclusions. Our results improve and generalize the known results of [19,24].
2. P R E L I M I N A R I E S Let H be a real Hilbert space endowed with the norm II" II and inner product (-,-), respectively. Let N : H x H ~ H be a nonlinear mapping and M : H x H ~ 2 H be a maximal monotone mapping with respect to the first argument. We consider the problem of finding u E H such that 0 E N ( u , u) + M ( u , u).
(2.1)
Problem (2.1) is called the generalized strongly nonlinear mixed quasi-variational inclusion. Now, we give some special cases of problem (2.1) as follows. (I) If M(., y) = 0~(., y) for each y E H, where ~ : H x H ---, R U {+c~} such that, for each fixed y E H, ~(., y) : H --* R U {+oe} is a proper convex lower semicontinuous function on H and O~(.,y) denotes the subdifferential of function ~(.,y), then problem (2.1) is equivalent to finding u E H such that
(N(u,u),v-u) >~(u,~)-~(v,u),
VvcH,
which is called the generalized strongly nonlinear mixed quasi-variational inequality. (II) If N ( u , v) = T u + S v for all u, v E H, where S, T : H --* H are two nonlinear mappings, then problem (2.1) is equivalent to finding u E H such that 0 E T u + S u + M ( u , u),
(2.2)
which is called the generalized nonlinear mixed quasi-variational inclusion. (III) If S = 0, then problem (2.3) is equivalent to finding u E H such that
(2.3)
0 ~ Tu + M(u, u), which is called the mixed quasi-variational inclusion (see [24]).
We recall that, if T is a maximal monotone mapping, then the resolvent operator j T associated with T is defined by JT(u) = (I + pT)-l(u), ~/u E H, (2.4) where p > 0 is a constant and I denotes the identity mapping. It is well known that j T : H --~ H is a single-valued nonexpansive mapping. REMARK 2.1. Since M is a maximal monotone mapping with respect to the first argument, for any fixed y E H , we define JM("Y)(u) = (I + pM(., y ) ) - l ( u ) ,
V u E H,
which is called the implicit resolvent operator associated with M(., y). We now consider the parametric version of problem (2.1). 'To this end, let A be a nonempty open subset of H in which the parametric ~ takes values. Let N : H x H x A -~ H be a nonlinear mapping and M : H x H x A ~ 2 H be a maximal monotone mapping with respect to the first argument. The parametric generalized strongly nonlinear mixed quasi-variational inclusion problem is to find u E H such that 0 e N ( u , u, :~)+ M ( u , u, :~).
(2.5)
We now establish the equivalent between problem (2.5) and the problem of finding fixed point for the nonlinear mapping associated with the implicit resolvent operator, which is the main motivation of our next results.
Sensitivity Analysis
21
LEMMA 2.1. u • H is a solution o f p r o b l e m (2.5) if and only if there exists u • H such that
(2.6)
u = F ( u , A) de__fj2i(. ~,x)( u _ p N ( u , u, A)),
where p > 0 is a constant and jM(.,~,~) = (I + p M ( . , u , £))-1 PROOF. Let u • H be a solution of problem (2.6). From the definition of the implicit resoivent IM(.,u,A) operator ~o , we have u - p N ( u , u, A) • u + p M ( u , u, A),
and hence, 0 • N ( u , u, A) + M ( u , u, A).
Thus, u • H is a solution of problem (2.5). Conversely, if u • H is a solution of problem (2.6), we have u • H such t h a t
o • N(~, ~., ~) + M(~, ~, ~). Hence, u - p N ( u , u, A) E u + p M ( u , u, A).
From the definition of the implicit resolvent operator jpAi(.,~,x) we know that u E H is a solution of problem (2.6). This completes the proof. DEFINITION 2.1. L e t N : H x H x A --~ H be a nonlinear mapping. (1) N is said to be a-strongly m o n o t o n e with respect to the first argument if there exists some a > 0 such that
(N(x,u,A)-N(y,u,A),z-y)>_~llx-Yll
2,
V(x,y,u,A)
EHxHxH×A.
(2) N is said to be ~-Lipschitg continuous with respect to the first a r g u m e n t if there exists some ~ > 0 such that []N(x,u,A)-g(y,u,A)ii<~iix-yi],
V(x,y,u,A)
egxg×HxA.
In a similar way, we can define strong monotonicity and Lipschitz continuity of N with respect to the second argument. We also need the following condition for the implicit resolvent operator jM(.,,.X).
ASSUMPTION 2.1. T h e r e is a constant ~/ > 0 such that
jy(.,u,~) w _ jy(.,v,~)w
<_ ~Llu - ~lI,
for all ( u , v , w , A ) E H x H x H x A.
3. M A I N
RESULTS
In this section, we first show t h a t F ( u , A) defined by (2.6) is a contraction mapping.
22
R.P. AGARWALet al.
LEMMA 3.1. L e t N : H × H x A --+ H be a nonlinear m a p p i n g such that N is a - s t r o n g l y m o n o t o n e a n d / 3 - L i p s c h i t z continuous with respect to the first argument, and N is ( - L i p s & i t z continuous with respect to the second argument. If A s s u m p t i o n 2.1 holds, then IIF(u,~)-F(v,~)ll<_Ollu-vll,
V(u,v,A)~H×H×A,
where
0= ~/1--2pa+p2/32+p~+7<
1,
for P
a --_(1_~_~){
/32 _ {u
V/(a - (1 - .y){)2 _ (/32 _ {2) 7( 2 _ 7)
<
,
(3.1)
/3 > ~,
(3.2)
/35 _ {2
> (1 - 7){ + V/(/3 2 - {2) 7(2 - 7), p~
7<1.
(3.3)
PROOF. For any (u, v, A) c H x H x A, we have F ( u , A) = J~M(. ,~,~) ( ~ -
pN(~,~,A))
and
F ( ~ , ~) = J ~ ( " ~ ' ~ ' ) ( v - p N ( v , ~, ~,)). From the definition of jM(.,~A) and A s s u m p t i o n 2.1, it follows t h a t ]IF(u, A) - F ( v , A)]] =
J~i("u';~)(u - p N ( u , u, A)) - J~pS("v"X)(v - p N ( v , v, A))
<_ J ~ ( , u , ~ ) ( u
+
- pN(u,,~,,X)) - J~""~,~)(v
J~("~'~)(,
- pN(,,v,
- pX(v,v,A))
~)) - J ~ ( " ~ ' ~ ) ( v - p N ( v , v , A))
(3.4)
_< Ilu - v - p ( N ( u , ~ , ~ ) - N ( v , v , > , ) ) l l + ~ l l u - vii _< ][u - v - p ( N f u , u,A) - N ( v , u , A ) ) l l + P [ I N ( v , u , A ) - N ( v , v , A ) l l + T ] [ u - vtt.
Since N is a - s t r o n g l y m o n o t o n e and/3-Lipschitz continuous with respect to the first a r g u m e n t , we have [lu- v-
p ( N ( u , u , A ) - N ( v , u , A ) ) H 2 = ] ] u - vii 2 - 2 p ( N ( u , u , A ) - N ( v , u , A ) , u -
+p2llN(u, u, ~) - N ( v , u, A)II ~ _< (1 - 2p~ + p2920 Ilu - vii 2,
v)
(3,5)
for all ( u , v , A ) E H × H × A. Further, since N is ~-Lipschitz continuous with respect to the second a r g u m e n t , we obtain IIN(v,u,.k) - N(v,v,),))ll
< ~llu - vii,
(3.6)
for all (u, v, A) E H x H x A. From (3.4)-(3.6), it follows t h a t IIF(u,,~)-F(v,A)ll
<_Ollu-vll,
V(u,v,A)
EH×H×A,
where O= V/1-epa+p2132+p~+7. F r o m (3.1)-(3.3), we know t h a t 0 < 1. This completes the proof. REMARK 3.1. F r o m L e m m a 3.1, we see that, if the conditions of L e m m a 3.1 hold, then, for each c A, t h e m a p p i n g F ( u , A) defined by (3.1) has a unique fixed point z(A) and so z(A) is a unique solution of p r o b l e m (2.5). It follows fi'Oln (3.1) t h a t
~(;~) = F ( ~ ( ~ ) , ;~). Using L e m m a 3.1, we prove the continuity (or Lipschitz continuity) of the solution z(A) of the p a r a m e t r i c generalized strongly nonlinear mixed quasi-variational inclusion p r o b l e m (2.5) as follows.
Sensitivity Analysis
23
THEOREM 3.1. Let N : H x H x A --* H be a nonlinear mapping such that N is (t-strongly monotone and 3-Lipschitz continuous with respect to the first argument, and N is ~-Lipsc.hitz continuous with respect to the second argument. Suppose that Assumption 2.1 and conditions (3.1)-(3.3) of Lemma 3.1 hold. If, for any u , v , w ~ H, the mappings A --~ N ( u , v , A ) ~md X ~ JM("u'X)w are both continuous (or Lipsehitz continuous) from A to H, then the solution z(A) of the parametric generaBzed strongly nonlinear mixed quasi-variational inclusion problem (2.5) is continuous (or Lipschitz continuous) from A to H. PROOF. For all A, ~ ~ A, from Lemma 3.1 and the definition of F(u, A), we have
IIz(a) - z (X)l[ = IIF(z(~), ~) - r (z (x),X)ll _< IIF(w,), :,) - ~ (z (x), :~)11 + II. (~ (x),:~) - ~ (~ (x),x)H
(3.7)
<_ 0 IIz(~)- z (X)ll + IIF (z (~),~) - F (z (X),X)II and
IIF (~ (x),~) - F (z (X),X)II = J~("~(~)'~) (z (X) - ,N (z (X),z (X),~)) _j~(.,z(~),~) (~ (~) _ ~N (~ (~),~ (x),~))
_< J~("~(~)'~)(z(~)-yv(~(~),z(~),A)) _j~1(.,z(~),~) (~ (:,) _ pN (~ (:,) ,~ (:,) ~:,) )
+ .~(,~(~),~) (~ (~) - p~v (~ (x),z (x),x))
< pll N ( z ( X ) , z ( 5 , ) , : ~ )
- N
(3.8)
(z(X),z(X),X)I [
It follows from (3.7) and (3.8) that
IIz(:,)- z(Y,)ll <_ 1-oP
]lN(z(A),z(~),A)-N(z(~),z(~) 1
+1--~
~)11
j~/(.,z(~),~)(z(~)-~v(~(~),~(~),~))
_j~(.,4~),~) (z (x) - pN (~ (~) ,~ (~), X)) . This completes the proof. REMARK 3.2. Since the strongly nonlinear mixed quasi-variational inclusions include the classical variational inequalities, mixed (quasi) variational inequalities, variational inclusions and complementarity problems as special cases, we can use the technique developed in this paper to study the sensitivity analysis of these problems. Our results improve and generalize the known results of [19,24].
24
R . P . AGARWALet al.
REFERENCES 1. C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities, Application to Free Boundary Problems, Wiley, New York, (1984). 2. A. Bensoussan, Stochastic Control by Functional Analysis Method, North-Holland, Amsterdam, (1982). 3. A. Bensoussan and J.L. Lions, Impulse Control and Quasivariational Inequalities, Gauthiers-Villers, Bordas, Paris, (1984). 4. S.S. Chang, Variational Inequality and Complementarity Problem Theory with Applications, Shanghai Scientific and Tech. Literature Publishing House, Shanghai, (1991). 5. R.W. Cottle, J.P. Pang and R.E. Stone, The Linear Complementarity Problem, Academic Press, London, (1992). 6. J. Crank, Free and Moving Boundary Problems, Clarendon, Oxford, (1984). 7. V.F. Demyanov, G.E. Stavroulakis, L.N. Polyakova and P.D. Panagiotopoulos, Quasidifferentiability and Nonsmooth Modeling in Mechanics, Engineering and Economics, Kluwer Academic, Holland, (1996). 8. G. Duvaut and J.L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, (1976). 9. F. Giannessi and A. Maugeri, Variational Inequalities and Network Equilibrium Problems, Plenum, New York, (1995). 10. P.T. Harker and J.S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Math. Programming 48, 161-220, (1990). 11. G. Isac, Complementarity problems, In Lecture Notes in Math., Volume 1528, Springer-Verlag, Berlin, (1992). 12. U. Mosco, Implicit variational problems and quasi-variational inequalities, In Lecture Notes in Math., Volume 543, Springer-Verlag, Berlin, (1976). 13. M.A. Noor, Some recent advances in variational inequalities (I), New Zealand J. Math. 26, 53-80, (1997). 14. M.A. Noor, Some recent advances in variational inequalities (II), New Zealand Y. Math. 26, 229-255, (1997). 15. G.X.Z. Yuan, K K M Theory and Applications in Nonlinear Analysis, Marcel Dekker, (1999). 16. X.P. Ding, Perturbed proximal point for generalized quasivariational inclusions, J. Math. Anal. Appl. 210, 88-101, (1997). 17. N.J. Huang, Mann and Ishikawa type perturbed iterative algorithms for generalized nonlinear implicit quasivariational inclusions, Computers Math. Applic. 35 (10), 1-7, (1998). 18. N.J. Huang, Generalized nonlinear implicit quasivariational inclusion and an application to implicit variational inequalities, Z. Angew. Math. Mech. 79, 569-575, (1999). 19, S. Dafermos, Sensitivity in variational inequalities, Math. Oper. Res. 13, 421-434, (1988). 20, R.N. Mukherjee and H.L. Verma, Sensitivity analysis of generalized variational inequalities, J. Math. Anal. Appl. 167, 299-304, (1992). 21. M.A. Noor, Sensitivity analysis for quasi-variational inequalities, J. Optim. Theory Appl. 95, 399-407, (1997). 22. N.D. Yen, Lipschitz continuity of solutions of variational inequalities with a parametric polyhedral constraint, Math. Oper. Res. 20, 695-708, (1995). 23. S.M. Robinson, Sensitivity analysis for variational inequalities by normal-map techniques, In Variational Inequalities and Network Equilibrium Problems, (Edited by F. Giannessi and A. Maugeri), Plenum Press, New York, (1995). 24. M.A. Noor and K.I. Noor, Sensitivity analysis for quasi-variational inclusions, J. Math. Anal. Appl. 236, 290-299, (1999). 25. S. Adly, Perturbed algorithm and sensitivity analysis for a general class of variational inclusions, J. Math. Anal. Appl. 201,609-630, (1996). 26. A. Hassouni and A. Moudafi, A perturbed algorithm for variational inclusions, J. Math. Anal. Appl. 185, 706-712, (1994).
Applied Mathematics Letters 13 (2000) 19-24
Applied Mathematics Letters www.elsevier.nl/locate/anfl
Sensitivity Analysis for Strongly Nonlinear Quasi-Variational Inclusions R. P. AGARWAL Department of Mathematics, National University of Singapore 10 Kent Ridge Crescent, Singapore 119260
YEOL JE CHO Department of Mathematics, Gyeongsang National University Chinju 660-701, Korea NAN JING HUANG Department of Mathematics, Sichuan University Chengdu, Sichuan 610064, P.R. China
(Received October 1999; accepted November 1999) A b s t r a c t - - I n this paper, we use the implicit resolvent operator technique to study the sensitivity analysis for strongly nonlinear quasi-variational inclusions. Our results improve and generalize some of the recent ones. @ 2000 Elsevier Science Ltd. All rights reserved. K e y w o r d s - - N o n l i n e a r quasi-variational inclusion, Sensitivity analysis, Resolvent operator. Hilbert space.
1. I N T R O D U C T I O N It is well recognized that variational inequalities play a fundamental role in the current mathematical technology. Thus, in recent years, the theory of classical variational inequalities has been extended and generalized to study a wide class of problems arising in mechanics, physics, optimization and control, nonlinear programming, economics and transportation equilibrium, and engineering sciences, etc. For details, we refer to [1-15] and the references therein. A useflfl and important generalization of variational inequalities is called the quasi-variational inclusion. Ding [16] arid Huang [17,18] have used the implicit resolvent operator techniques to study the existence of quasi-variational inclusions. Sensitivity analysis of solutions for variational inequalities have been studied by many authors using quite different methods. By using tile projection technique, Dafermos [19], Mukherjee and Verma [20], Noor [21] and Yen [22] dealt with the sensitivity analysis of solutions for variational inequalities. By using the implicit function approach that makes use of so-called normal mappings, Robinson [23] studied the sensitivity analysis of solutions for variational inequalities. Recently, Noor and Noor [24] dealt with the sensitivity The second author wishes to acknowledge the financial support of the Korea Research Foundation made ill the program year of 1998, Project No. 1998-015-D00020 and the third author was supported ill part by '98 APEC Post-Doctor Fellowship (KOSEF) while he visited Gyeongsang National University. 0893-9659/00/$ - see front matter @ 2000 Elsevier Science Ltd. All rights reserved. PIE S0893-9659(00)00048-3
Typeset by A2~[S-TFj~
20
R.P. AGARWALet al.
analysis of solutions for the quasi-variational inclusions by using the implicit resolvent equations technique without assuming the differentiability of the given data. Inspired and motivated by recent research works in this field, in this paper, by using the implicit resolvent operator technique, we study the sensitivity analysis for strongly nonlinear quasi-variational inclusions. Our results improve and generalize the known results of [19,24].
2. P R E L I M I N A R I E S Let H be a real Hilbert space endowed with the norm II" II and inner product (-,-), respectively. Let N : H x H ~ H be a nonlinear mapping and M : H x H ~ 2 H be a maximal monotone mapping with respect to the first argument. We consider the problem of finding u E H such that 0 E N ( u , u) + M ( u , u).
(2.1)
Problem (2.1) is called the generalized strongly nonlinear mixed quasi-variational inclusion. Now, we give some special cases of problem (2.1) as follows. (I) If M(., y) = 0~(., y) for each y E H, where ~ : H x H ---, R U {+c~} such that, for each fixed y E H, ~(., y) : H --* R U {+oe} is a proper convex lower semicontinuous function on H and O~(.,y) denotes the subdifferential of function ~(.,y), then problem (2.1) is equivalent to finding u E H such that
(N(u,u),v-u) >~(u,~)-~(v,u),
VvcH,
which is called the generalized strongly nonlinear mixed quasi-variational inequality. (II) If N ( u , v) = T u + S v for all u, v E H, where S, T : H --* H are two nonlinear mappings, then problem (2.1) is equivalent to finding u E H such that 0 E T u + S u + M ( u , u),
(2.2)
which is called the generalized nonlinear mixed quasi-variational inclusion. (III) If S = 0, then problem (2.3) is equivalent to finding u E H such that
(2.3)
0 ~ Tu + M(u, u), which is called the mixed quasi-variational inclusion (see [24]).
We recall that, if T is a maximal monotone mapping, then the resolvent operator j T associated with T is defined by JT(u) = (I + pT)-l(u), ~/u E H, (2.4) where p > 0 is a constant and I denotes the identity mapping. It is well known that j T : H --~ H is a single-valued nonexpansive mapping. REMARK 2.1. Since M is a maximal monotone mapping with respect to the first argument, for any fixed y E H , we define JM("Y)(u) = (I + pM(., y ) ) - l ( u ) ,
V u E H,
which is called the implicit resolvent operator associated with M(., y). We now consider the parametric version of problem (2.1). 'To this end, let A be a nonempty open subset of H in which the parametric ~ takes values. Let N : H x H x A -~ H be a nonlinear mapping and M : H x H x A ~ 2 H be a maximal monotone mapping with respect to the first argument. The parametric generalized strongly nonlinear mixed quasi-variational inclusion problem is to find u E H such that 0 e N ( u , u, :~)+ M ( u , u, :~).
(2.5)
We now establish the equivalent between problem (2.5) and the problem of finding fixed point for the nonlinear mapping associated with the implicit resolvent operator, which is the main motivation of our next results.
Sensitivity Analysis
21
LEMMA 2.1. u • H is a solution o f p r o b l e m (2.5) if and only if there exists u • H such that
(2.6)
u = F ( u , A) de__fj2i(. ~,x)( u _ p N ( u , u, A)),
where p > 0 is a constant and jM(.,~,~) = (I + p M ( . , u , £))-1 PROOF. Let u • H be a solution of problem (2.6). From the definition of the implicit resoivent IM(.,u,A) operator ~o , we have u - p N ( u , u, A) • u + p M ( u , u, A),
and hence, 0 • N ( u , u, A) + M ( u , u, A).
Thus, u • H is a solution of problem (2.5). Conversely, if u • H is a solution of problem (2.6), we have u • H such t h a t
o • N(~, ~., ~) + M(~, ~, ~). Hence, u - p N ( u , u, A) E u + p M ( u , u, A).
From the definition of the implicit resolvent operator jpAi(.,~,x) we know that u E H is a solution of problem (2.6). This completes the proof. DEFINITION 2.1. L e t N : H x H x A --~ H be a nonlinear mapping. (1) N is said to be a-strongly m o n o t o n e with respect to the first argument if there exists some a > 0 such that
(N(x,u,A)-N(y,u,A),z-y)>_~llx-Yll
2,
V(x,y,u,A)
EHxHxH×A.
(2) N is said to be ~-Lipschitg continuous with respect to the first a r g u m e n t if there exists some ~ > 0 such that []N(x,u,A)-g(y,u,A)ii<~iix-yi],
V(x,y,u,A)
egxg×HxA.
In a similar way, we can define strong monotonicity and Lipschitz continuity of N with respect to the second argument. We also need the following condition for the implicit resolvent operator jM(.,,.X).
ASSUMPTION 2.1. T h e r e is a constant ~/ > 0 such that
jy(.,u,~) w _ jy(.,v,~)w
<_ ~Llu - ~lI,
for all ( u , v , w , A ) E H x H x H x A.
3. M A I N
RESULTS
In this section, we first show t h a t F ( u , A) defined by (2.6) is a contraction mapping.
22
R.P. AGARWALet al.
LEMMA 3.1. L e t N : H × H x A --+ H be a nonlinear m a p p i n g such that N is a - s t r o n g l y m o n o t o n e a n d / 3 - L i p s c h i t z continuous with respect to the first argument, and N is ( - L i p s & i t z continuous with respect to the second argument. If A s s u m p t i o n 2.1 holds, then IIF(u,~)-F(v,~)ll<_Ollu-vll,
V(u,v,A)~H×H×A,
where
0= ~/1--2pa+p2/32+p~+7<
1,
for P
a --_(1_~_~){
/32 _ {u
V/(a - (1 - .y){)2 _ (/32 _ {2) 7( 2 _ 7)
<
,
(3.1)
/3 > ~,
(3.2)
/35 _ {2
> (1 - 7){ + V/(/3 2 - {2) 7(2 - 7), p~
7<1.
(3.3)
PROOF. For any (u, v, A) c H x H x A, we have F ( u , A) = J~M(. ,~,~) ( ~ -
pN(~,~,A))
and
F ( ~ , ~) = J ~ ( " ~ ' ~ ' ) ( v - p N ( v , ~, ~,)). From the definition of jM(.,~A) and A s s u m p t i o n 2.1, it follows t h a t ]IF(u, A) - F ( v , A)]] =
J~i("u';~)(u - p N ( u , u, A)) - J~pS("v"X)(v - p N ( v , v, A))
<_ J ~ ( , u , ~ ) ( u
+
- pN(u,,~,,X)) - J~""~,~)(v
J~("~'~)(,
- pN(,,v,
- pX(v,v,A))
~)) - J ~ ( " ~ ' ~ ) ( v - p N ( v , v , A))
(3.4)
_< Ilu - v - p ( N ( u , ~ , ~ ) - N ( v , v , > , ) ) l l + ~ l l u - vii _< ][u - v - p ( N f u , u,A) - N ( v , u , A ) ) l l + P [ I N ( v , u , A ) - N ( v , v , A ) l l + T ] [ u - vtt.
Since N is a - s t r o n g l y m o n o t o n e and/3-Lipschitz continuous with respect to the first a r g u m e n t , we have [lu- v-
p ( N ( u , u , A ) - N ( v , u , A ) ) H 2 = ] ] u - vii 2 - 2 p ( N ( u , u , A ) - N ( v , u , A ) , u -
+p2llN(u, u, ~) - N ( v , u, A)II ~ _< (1 - 2p~ + p2920 Ilu - vii 2,
v)
(3,5)
for all ( u , v , A ) E H × H × A. Further, since N is ~-Lipschitz continuous with respect to the second a r g u m e n t , we obtain IIN(v,u,.k) - N(v,v,),))ll
< ~llu - vii,
(3.6)
for all (u, v, A) E H x H x A. From (3.4)-(3.6), it follows t h a t IIF(u,,~)-F(v,A)ll
<_Ollu-vll,
V(u,v,A)
EH×H×A,
where O= V/1-epa+p2132+p~+7. F r o m (3.1)-(3.3), we know t h a t 0 < 1. This completes the proof. REMARK 3.1. F r o m L e m m a 3.1, we see that, if the conditions of L e m m a 3.1 hold, then, for each c A, t h e m a p p i n g F ( u , A) defined by (3.1) has a unique fixed point z(A) and so z(A) is a unique solution of p r o b l e m (2.5). It follows fi'Oln (3.1) t h a t
~(;~) = F ( ~ ( ~ ) , ;~). Using L e m m a 3.1, we prove the continuity (or Lipschitz continuity) of the solution z(A) of the p a r a m e t r i c generalized strongly nonlinear mixed quasi-variational inclusion p r o b l e m (2.5) as follows.
Sensitivity Analysis
23
THEOREM 3.1. Let N : H x H x A --* H be a nonlinear mapping such that N is (t-strongly monotone and 3-Lipschitz continuous with respect to the first argument, and N is ~-Lipsc.hitz continuous with respect to the second argument. Suppose that Assumption 2.1 and conditions (3.1)-(3.3) of Lemma 3.1 hold. If, for any u , v , w ~ H, the mappings A --~ N ( u , v , A ) ~md X ~ JM("u'X)w are both continuous (or Lipsehitz continuous) from A to H, then the solution z(A) of the parametric generaBzed strongly nonlinear mixed quasi-variational inclusion problem (2.5) is continuous (or Lipschitz continuous) from A to H. PROOF. For all A, ~ ~ A, from Lemma 3.1 and the definition of F(u, A), we have
IIz(a) - z (X)l[ = IIF(z(~), ~) - r (z (x),X)ll _< IIF(w,), :,) - ~ (z (x), :~)11 + II. (~ (x),:~) - ~ (~ (x),x)H
(3.7)
<_ 0 IIz(~)- z (X)ll + IIF (z (~),~) - F (z (X),X)II and
IIF (~ (x),~) - F (z (X),X)II = J~("~(~)'~) (z (X) - ,N (z (X),z (X),~)) _j~(.,z(~),~) (~ (~) _ ~N (~ (~),~ (x),~))
_< J~("~(~)'~)(z(~)-yv(~(~),z(~),A)) _j~1(.,z(~),~) (~ (:,) _ pN (~ (:,) ,~ (:,) ~:,) )
+ .~(,~(~),~) (~ (~) - p~v (~ (x),z (x),x))
< pll N ( z ( X ) , z ( 5 , ) , : ~ )
- N
(3.8)
(z(X),z(X),X)I [
It follows from (3.7) and (3.8) that
IIz(:,)- z(Y,)ll <_ 1-oP
]lN(z(A),z(~),A)-N(z(~),z(~) 1
+1--~
~)11
j~/(.,z(~),~)(z(~)-~v(~(~),~(~),~))
_j~(.,4~),~) (z (x) - pN (~ (~) ,~ (~), X)) . This completes the proof. REMARK 3.2. Since the strongly nonlinear mixed quasi-variational inclusions include the classical variational inequalities, mixed (quasi) variational inequalities, variational inclusions and complementarity problems as special cases, we can use the technique developed in this paper to study the sensitivity analysis of these problems. Our results improve and generalize the known results of [19,24].
24
R . P . AGARWALet al.
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