D O I: 1 0 .1 5 0 1 / C o m m u a 1 _ 0 0 0 0 0 0 0 7 6 9 IS S N 1 3 0 3 –5 9 9 1
SENSITIVITY ANALYSIS FOR A PARAMETRIC MULTI-VALUED IMPLICIT QUASI VARIATIONAL-LIKE
INCLUSION
K. R. KAZMI AND SHAKEEL A. ALVI
Abstract. In this paper, using proximal-point mapping of strongly maximal P- -monotone mapping and the property of the …xed-point set of multi-valued contractive mapping, we study the behaviour and sensitivity analysis of the solution set of a parametric generalized implicit quasi-variational-like inclu-sion involving strongly maximal P - -monotone mapping in real Hilbert space. Further, under suitable conditions, we discuss the Lipschitz continuity of the solution set with respect to the parameter. The technique and results presented in this paper can be viewed as extension of the techniques and corresponding results given in [2,7-10,20,21].
1. Introduction
Variational inequality theory has become very e¤ective and powerful tool for studying a wide range of problems arising in mechanics, contact problems in elas-ticity, optimization and control problems, management science, operation research, general equilibrium problems in economics and transportation, unilateral, obstacle, moving boundary valued problems etc. Variational inequalities have been general-ized and extended in di¤erent directions using novel and innovative techniques.
In recent years, much attention has been given to develop general techniques for the sensitivity analysis of solution set of various classes of variational inequali-ties (inclusions). From the mathematical and engineering point of view, sensitivity properties of various classes of variational inequalities can provide new insight con-cerning the problem being studied and stimulate ideas for solving problems. The
Received by the editors: Jan 01, 2012, Accepted: July 30, 2016.
2010 Mathematics Subject Classi…cation. 49J40, 47H05, 47J25; 47J20, 49J53.
Key words and phrases. Parametric generalized implicit quasi-variational-like inclusion; sensi-tivity analysis; strongly maximal P - -monotone mapping; P - -proximal-point mapping; strongly mixed monotone mapping; generalized mixed pseudocontractive mapping; mixed Lipschitz con-tinuous.
This work has been done under a Major Research Project No. F.36-7/2008 (SR) sanctioned by the University Grants Commission, Government of India, New Delhi.
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sensitivity analysis of solution set for variational inequalities have been studied ex-tensively by many authors using quite di¤erent techniques. By using the projection technique, Dafermos [5], Mukherjee and Verma [17], Noor [19] and Yen [23] studied the sensitivity analysis of solution of some classes of variational inequalities with single-valued mappings. By using the implicit function approach that makes use of called normal mappings, Robinson [22] studied the sensitivity analysis of so-lutions for variational inequalities in …nite-dimensional spaces. By using resolvent operator technique, Adly [1], Noor [20], and Agarwal et al. [2] studied the sensi-tivity analysis of solution of some classes of quasi-variational inclusions involving single-valued mappings.
Recently, by using projection and resolvent techniques, Ding and Luo [8], Liu et al. [16], Park and Jeong [21], Ding [6,7], Khan [11] Kazmi and Khan [13,14] and Huang et al. [10] studied the behaviour and sensitivity analysis of solution set for some classes of generalized variational inequalities (inclusions) involving multi-valued mappings.
The method based on proximal-point mapping is a generalization of projection method and has been widely used to study the existence of solution and to develop iterative schemes of variational (-like) inclusions. Recently Chidume, Kazmi and Zegeye [4], Fang and Huang [9], and Kazmi and Khan [12] has introduced the notion of -proximal point mapping, P -proximal point mapping and P - -proximal point mapping and used these to study the various classes of variational (-like) inclusions. Motivated by recent work going in this direction, we introduce the notion of strongly maximal P - -monotone mapping and discuss some of its properties. Fur-ther, we de…ne strongly P - -proximal mapping for strongly maximal P - -monotone, an extension of proximal mappings [4], P proximal mappings [9], strongly P -proximal mappings [24], P - --proximal mapping [12] and classical -proximal map-ping in Hilbert space, and prove that it is single-valued and Lipschitz continuous. Further, we consider a parametric multi-valued implicit quasi-variational-like inclu-sion problem (in short PMIQVLIP) in real Hilbert space. Further, using strongly P - -proximal mapping technique and the property of the …xed point set of multi-valued mapping, we study the behaviour and sensitivity analysis of the solution set for PMIQVLIP. Further, the Lipschitz continuity of solution set of PMIQVLIP is proved under some suitable conditions. The results presented in this chapter gener-alize and improve the results given by many authors, see for example [2,7,10,20,21].
2. Strongly P - -proximal mappings
We assume that H is a real Hilbert space equipped with inner product h ; i and norm k k; CB(H) is the family of all nonempty closed and bounded subsets of H; C(H) is the family of all nonempty compact subsets of H; 2H is the power set of
H; H( ; ) is the Hausdro¤ metric on C(H), de…ned by H(A; B) = maxfsup x2A inf y2Bd(x; y); supy2B inf x2Ad(x; y)g; A; B 2 C(H):
First, we review and de…ne the following concepts:
De…nition 2.1[12]. Let : H H ! H be a mapping. Then a mapping P : H ! H is said to be
(i) -monotone if
hP (x) P (y); (x; y)i 0; 8 x; y 2 H; (ii) strictly -monotone if
hP (x) P (y); (x; y)i > 0; 8 x; y 2 H and equality holds if and only if x = y;
(iii) -strongly -monotone if there exists a constant > 0 such that hP (x) P (y); (x; y)i kx yk2; 8 x; y 2 H:
De…nition 2.2[4]. A mapping : H H ! H is said to be -Lipschitz continuous, if there exists a constant > 0 such that
k (x; y)k kx yk; 8 x; y 2 H:
De…nition 2.3[4]. Let : H H ! H be a single-valued mapping. Then a multi-valued mapping M : H ! 2H is said to be
(i) -monotone if
hu v; (x; y)i 0; 8 x; y 2 H; 8 u 2 M(x); 8 v 2 M(y); (ii) strictly -monotone if
hu v; (x; y)i 0; 8 x; y 2 H; 8 u 2 M(x); 8 v 2 M(y); and equality holds if and only if x = y.
(iii) -strongly -monotone if there exists a constant > 0 such that hu v; (x; y)i kx yk2; 8 x; y 2 H; 8 u 2 M(x); 8 v 2 M(y); (iv) maximal -monotone if M is -monotone and (I + M )(H) = H for any
> 0, where I stands for identity mapping.
Remark 2.1. If (x; y) x y, 8 x; y 2 H, then from De…nitions 2.1 and 2.3, we recover the usual de…nitions of monotonicity of mappings P and M .
De…nition 2.4[12]. Let : H H ! H and P : H ! H be mappings. Then a multi-valued mapping M : H ! 2H is said to be maximal P - -monotone if M is
De…nition 2.5. Let : H H ! H and P : H ! H be mappings. A multi-valued mapping M : H ! 2H is said to be -strongly maximal P - -monotone if M
is -strongly -monotone and (P + M )H = H for any > 0: Remark 2.2.
(i) If M is -monotone, De…nition 2.5 reduces to De…nition 2.4 given by Kazmi and Khan [12] in Banach space.
(ii) If M is -monotone and P I, De…nition 2.5 reduces to de…nition of maximal -monotone mapping given by Chidume et al. [4] in Banach space. (iii) If (x; y) x y for all x; y 2 H then De…nition 2.5 reduces to the de…nition
of -strongly maximal monotone mapping given by Zeng et al. [24]. (iv) If (x; y) x y for all x; y 2 H and M is monotone, De…nition 2.5
reduces to the de…nition of maximal P -monotone mapping given by Fang and Huang [9].
The following theorem gives some properties of -strongly maximal P - -monotone mappings.
Theorem 2.1. Let : H H ! H be a mapping; let P : H ! H be a strictly -monotone mapping and let M : H ! 2H be a -strongly maximal P - -monotone
multi-valued mapping. Then
(a) hu v; (x; y)i 0; 8 (v; y) 2 Graph(M) implies (u; x) 2 Graph(M); where Graph(M ) := f(u; x) 2 H H : u 2 M(x)g;
(b) the mapping (P + M ) 1 is single-valued for all > 0.
Proof (a). Proof is on similar lines of proof of Theorem 2.1(a)[12].
Proof (b). For any given z 2 H and constant > 0, let x; y 2 (P + M) 1(z).
Then 1(z P (x)) 2 M(x) and 1(z P (y)) 2 M(y):
Now
0 = h 1(z P (x)) 1(z P (y)); (x; y)i +hP (x) P (y); (x; y)i
kx yk2+ hP (x) P (y); (x; y)i kx yk2;
using -strongly -monotonicity of M and strictly -monotonicity of P . Hence, preceeding inequality implies that x = y, which implies (P + M ) 1 is
single-valued. This completes the proof of (b).
By Theorem 2.1, we de…ne strongly P - -proximal mapping for a -strongly max-imal -monotone mapping M as follows:
RP;M (z) = (P + M ) 1; 8 z 2 H; (2:3)
where > 0 is a constant, : H H ! H is a mapping and P : H ! H is a strictly -monotone mapping.
Remark 2.3.
(i) If (x; y) x y for all x; y 2 H and M is -monotone, then strongly P - -proximal mapping reduces to P -proximal mapping given by Fang and Huang [9].
(ii) If P I and M is -monotone, then strongly P - -proximal mapping re-duces to -proximal mapping given by Chidume et al. [4] in Banach space. Next, we prove that strongly P - -proximal mapping is Lipschitz continuous. Theorem 2.2. Let P : H ! H be a -strongly -accretive mapping; let : H H ! H be a -Lipschitz continuous mapping and let M : H ! 2H be a
-strongly maximal -monotone multi-valued mapping. Then -strongly P - -proximal mapping RMP; is
+ -Lipschitz continuous, that is,
kRMP; (x) RMP; (y)k + kx yk; 8 x; y 2 H:
Proof. Let x; y 2 H. From de…nition of RM
P; , we have RMP; (x) = (P + M ) 1(x).
This implies that
1(x P (RM
P; (x))) 2 M(RP;M (x)):
Similarly, we have
1(y P (RM
P; (y))) 2 M(RMP; (y)):
Since M is -strongly -monotone, we obtain kRP;M (x) RMP; (y)k
1
= 1hx y; (RMP; (x); RMP; (y))i 1hP (RMP; (x)) P (RMP; (y)); (RMP; (x); RP;M (y))i: Since P is -strongly -monotone and is -Lipschitz continuous, then the pre-ceding inequality implies
kRMP; (x) RMP; (y)k kx yk kRMP; (x) RMP; (y)k kRMP; (x) RP;M (y)k2;
which gives
kRMP; (x) RMP; (y)k + kx yk 8 x; y 2 H:
This completes the proof. Remark 2.4.
(i) Theorems 2.1-2.2 generalize Proposition 2.1 and Theorems 2.1-2.2 [9] and corresponding result in [24].
(ii) Lemmas 2.6 and 2.8 [4] and Theorems 2.1-2.2 [12] can be extended using the technique of Theorems 2.1-2.2.
3. Formulation of problem
Let and be open subsets of a real Hilbert space H such that ( ; d1) and
( ; d2) are metric spaces, in which the parameters and takes values, respectively.
Let P : H ! H; : H H ! H; N; M : H H ! H; g; m : H ! H
be single-valued mappings such that g 6 0 and let A; B; C; D : H ! C(H) and F : H ! C(H) be multi-valued mappings. Suppose that W : H H ! 2H
is a multi-valued mapping such that for each (t; ) 2 H , W ( ; t; ) : H ! 2H is
strongly maximal P - -monotone and range(g m)(H f g)\domain W ( ; y; ) 6= ;, where (g m)(x; ) = g(x; ) m(x; ) for any (x; ) 2 H . For each (f; ; ) 2 H , we consider the parametric multi-valued implicit quasi-variational-like inclusion problem (PMIQVLIP):
Find x = x( ; ) 2 H, u = u(x; ) 2 A(x; ), v = v(x; ) 2 B(x; ), w = w(x; ) = c(x; ), y = y(x; ) 2 D(x; ) and z = z(x; ) 2 F (x; ) such that (g m)(x; ) 2 domain W ( ; z; ) and
f 2 N(u; v; ) M (w; y; ) + W ((g m)(x; ); z; ): (3:1) Some special cases:
(1) If E H; ( ; d1) ( ; d2); P I, an identity mapping; (x; t) x t
for all x; t 2 H; and W ( ; z; ) is maximal monotone for each …xed (z; ) 2 H , then PMIQVLIP (3.1) reduces to the problem:
Find x = x( ) 2 H, u = u(x; ) 2 A(x; ); v = v(x; ) 2 B(x; ), w = w(x; ) 2 C(x; ), y = y(x; ) 2 D(x; ) and z = z(x; ) 2 F (x; ) such that (g m)(x; ) 2 domain W ( ; z; ) and
which has been studied by Liu et al. [16].
(2) If E H; ( ; d1) ( ; d2); P I; (x; t) x t for all x; t 2 H; M 0;
f 0, and W ( ; z; ) is maximal monotone for each …xed (z; ) 2 H , then PMIQVLIP (3.1) reduces to the problem:
Find x = x( ) 2 H, u = u(x; ) 2 A(x; ), v = v(x; ) 2 B(x; ) and z = z(x; ) 2 F (x; ) such that (g m)(x; ) 2 domain W ( ; z; ) and
0 2 N(u; v; ) + W ((g m)(x; ); z; ); which has been studied by Ding [6].
(3) If E H; ( ; d1) ( ; d2); P I; (x; t) x t for all x; t 2 H; f 0;
M C D m 0; g I and A(x; ) B(x; ) F (x; ) x
for all (x; ) 2 H and W ( ; z; ) is maximal monotone for each …xed (z; ) 2 H . Then PMIQVLIP (3.1) reduces to the problem:
Find x 2 H such that
0 2 N(x; x; ) + W (x; x; ); which has been studied by Agarwal et al. [2].
(4) If E H; ( ; d1) ( ; d2); P I; (x; t) x t for all x; t 2 H; f 0;
M B C D E m 0; A(x; ) x for all (x; ) 2 H ;
N (x; t; ) N1(x; ) and W (x; y; ) W1(x; ), for all (x; y; ) 2 H H
, where N1; W1: H ! 2H, be such that W1( ; ) is maximal monotone
for each …xed 2 , then PMIQVLIP (3.1) reduces to the problem: 0 2 N1(x; ) + W1(g(x; ); );
which has been studied by Adly [1].
For a suitable choices of the mappings A; B; C; D; F; N; M; W; g; P; m; , it is easy to see that PMIQVLIP (3.1) includes a number of known classes of quasi-variational-like inequalities (inclusions) studied by many authors as special cases see for example [1-6,8,16,19-21] and the references therein.
Now, for each …xed ( ; ) 2 , the solution set S( ; ) of PMIQVLIP (3.1) is denoted as
S( ; ) := n
x = x( ; ) 2 H : u = u(x; ) 2 A(x; ); v = v(x; ) 2 B(x; ) w = w(x; ) 2 C(x; ); y = y(x; ) 2 D(x; ); z = z(x; ) 2 F (x; ); such that f 2 N(u; v; ) M (w; y; ) + W ((g m)(x; ); z; )
o
: (3:2)
The aim of the paper is to study the behaviour and sensitivity analysis of the solution set S( ; ), and the conditions on mappings A; B; C; D; F; N; M; W; g; P; m; under which the solution set S( ; ) of PMIQVLIP (3.1) is nonempty and Lipschitz continuous with respect to the parameters 2 , 2 .
Lemma 3.1[18]. Let (X; d) be a complete metric space. Suppose that Q : X ! C(X) satis…es
H(Q(x); Q(t) d(x; y); 8 x; t 2 X;
where 2 (0; 1) is a constant. Then the mapping Q has …xed point in E.
Lemma 3.2[15]. Let (X; d) be a complete metric space and let T1; T2: X ! C(X)
be -H-contraction mappings, then
H(F (T1); F (T2)) (1 ) 1sup
x2XH(T
1(x); T2(x));
where F (T1) and F (T2) are the sets of …xed points of T1 and T2, respectively.
4. Sensitivity analysis of the solution set S( ; ) First, we de…ne the following concepts.
De…nition 4.1[13]. A mapping g : H ! H is said to be
(i) (Lg; lg)-mixed Lipschitz continuous, if there exist constants Lg; lg> 0 such
that
kg(x1; 1) g(x2; 2)k Lgkx1 x2k + lgk 1 2k; 8 (x1; 1); (x2; 2) 2 H ;
(ii) s-strongly monotone, if there exists a constant s > 0 such that hg(x1; ) g(x2; ); x1 x2i skx1 x2k2; 8 (x1; ); (x2; ) 2 H :
De…nition 4.2[13]. A multi-valued mapping A : H ! C(H) is said to be (LA; lA)-H-mixed Lipschitz continuous, if there exist constants LA; lA> 0 such that
H(A(x1; 1); A(x2; 2)) LAkx1 x2k+lAk 1 2k; 8 (x1; 1); (x2; 2) 2 H :
De…nition 4.3[13]. Let A; B : H ! C(H) be multi-valued mappings. A mapping N : H H ! H is said to be
(i) (L(N;1); L(N;2); lN)-mixed Lipschitz continuous, if there exist constants L(N;1);
L(N;2); lN > 0 such that
kN(x1; y1; 1) N (x2; y2; 2)k L(N;1)kx1 x2k + L(N;2)ky1 y2k + lNk 1 2k;
8 (x1; y1; 1); (x2; y2; 2) 2 H H ;
(ii) -strongly mixed monotone with respect to A and B, if there exists a con-stant > 0 such that
hN(u1; v1; ) N (u2; v2; ); x yi kx yk2;
(iii) -generalized mixed pseudocontractive with respect to A and B, if there exists a constant > 0 such that
hN(u1; v1; ) N (u2; v2; ); x yi kx yk2;
8 x; y 2 H; 2 ; u12 A(x; ); u22 A(y; ); v12 B(x; ); v22 B(y; );
Now, we have the following …xed-point formulation of PMIQVLIP (3.1). Lemma 4.1. For each (f; ; ) 2 H ; (x; u; v; w; y; z) with x 2 x( ; ) 2 H, u = u(x; ) 2 A(x; ), v = v(x; ) 2 B(x; ), w = w(x; ) 2 C(x; ), y = y(x; ) 2 D(x; ) and z = z(x; ) 2 F (x; ) such that (g m)(x; ) 2 domain W ( ; z; ) is a solution of PMIQVLIP (3.1) if and only if the multi-valued mapping G : H ! 2H de…ned by G(t; ; ) = [ u2A(t; );v2B(t; );w2C(t; );y2D(t; );z2F (x; ) h t (g m)(t; ) +RW ( ;z; )P; [P (g m)(t; ) N (u; v; ) + M (w; y; ) + f ] i ; t 2 H; (4:1) has a …xed point, where P : H ! H; P (g m) denotes P composition of (g m); RW ( ;z; )P; = (P + W ( ; z; )) 1 and > 0 is a constant.
Proof. For each (f; ; ) 2 H , PMIQVLIP (3.1) has a solution (x; u; v; w; y; z) if and only if
f 2 N(u; v; ) M(w; y; )+W ((g m)(x; ); z; )
, P (g m)(x; ) N (u; v; )+ M (w; y; )+ f 2 (P + W ( ; z; ))((g m)(x; )): Since for each (z; ) 2 H , W ( ; z; ) is maximal strongly P - -monotone, by de…nition of strongly P - -proximal mapping RW ( ;z; )P; of W ( ; z; ); preceding inclusion holds if and only if
(g m)(x; ) = RW ( ;z; )P; [P (g m)(x; ) N (u; v; ) + M (w; y; ) + f ]; that is x 2 G(x; ; ). This completes the proof.
Theorem 4.1. Let the multi-valued mappings A; B; C; D : H ! C(H) and F : H ! C(H) be H-Lipschitz continuous in the …rst arguments with constant LA; LB; LC; LD and LF, respectively; let the mappings : H H ! H be
-Lipschitz continuous and P : H ! H be -strongly -monotone. Let the mappings g; m : H ! H be such that (g m) is s-strongly monotone and L(g m)
-Lipschitz continuous in the …rst argument and let the mapping P (g m) be r-strongly monotone and LP (g m)-Lipschitz continuous in the …rst argument; let
the mapping N : H H ! H be -strongly mixed monotone with respect to A and B and (L(N;1); L(N;2))-mixed Lipschitz continuous in …rst two arguments and
let the mapping M : H H ! H be -generalized mixed pseudocontractive with respect to C and D, and (L(M;1); L(M;2))-mixed Lipschitz continuous in …rst
two arguments. Suppose that the multi-valued mapping W : H H ! 2H
is such that for each (t; ) 2 H , W ( ; t; ) : H ! 2H is -strongly maximal
P - -monotone with range (g m)(H f g) \ domain W ( ; t; ) 6= ;: Suppose that there exist constants k1; k2> 0 such that
kRW ( ;x1; 1)
P; (t) R
W ( ;x2; 2)
P; (t)k k1kx1 x2k + k2k 1 2k; (4:2)
8 x1; x2; t 2 H; 1; 22 ;
and suppose for > 0, the following condition holds:
= q + ( ); (4:3) where q := k1LF+ q 1 2s + L2 (g m); ( ) := + [p+ q 1 2 ( ) + 2 2(L2 N+ L2M)]; p := q 1 2r + L2 p (g m); LN := LAL(N;1)+LBL(N;2); LM := LCL(M;1)+LDL(M;2); + (e p)e 2(L2 N + L2M) e2 2 < p [ (e p)e 2] [2(L2 N+ L2M e2 2][1 (e p)2] 2(L2 N+ L2M) e2 2 ; > (e p)e + q [2(L2 N+ L2M) e2 2][1 (e p)2]; > ; (4:4) 2(L2N+L2M) > e2 2; (e p); e := (1 q)= ; q 2 (0; 1):
Then, for each …xed f 2 H, the multi-valued mapping G de…ned by (4.1) is a compact-valued uniform -H-contraction mapping with respect to ( ; ) 2 , where is given by (4.3)-(4.4). Moreover, for each ( ; ) 2 , the solution set S( ; ) of PMIQVLIP (3.1) is nonempty and closed.
Proof. Let (x; ; ) be an arbitrary element of H :. Since A; B; C; D; F are compact-valued, then for any sequences fung A(x; ), fvng B(x; ), fwng
C(x; ), fyng D(x; ), fzng F (x; ), there exist subsequences funig fung,
fvnig fvng, fwnig fwng, fynig fyng, fznig fzng and elements u 2
A(x; ), v 2 B(x; ), w 2 C(x; ), y 2 D(x; ), z 2 F (x; ) such that uni ! u;
vni ! v, wni ! w, yni ! y, zni ! z as i ! 1. By using Theorem 2.2, (4.2) and
the mixed Lipschitz continuity of N and M , we estimate kRW ( ;zni; ) P; [P (g m)(x; ) N (uni; vni; )+ M (wni; yni; )+ f ] RW ( ;z; )P; [P (g m)(x; ) N (u; v; )+ M (w; y; )+ f ]k kRW (:;zni; ) P; [P (g m)(x; ) N (uni; vni; )+ M (wni; yni; )+ f ] RW ( ;z; )P; [P (g m)(x; ) N (uni; vni; )+ M (wni; yni; )+ f ]k +kRP;W ( ;z; )[P (g m)(x; ) N (uni; vni; )+ M (wni; yni; )+ f ] RW ( ;z; )P; [P (g m)(x; ) N (u; v; )+ M (w; y; )+ f ]k
k1kzni zk+ + h kN(uni; vni; ) N (u; v; )k+kM(wni; yni; ) M (w; y; )k i k1kzni zk+ + h L(N;1)kuni uk+L(N;2)kvni vk+L(M;1)kwni wk+L(M;2)kyni yk i ! 0; as i ! 1: (4:5) Thus (4.1) and (4.5) yield that G(x; ; ) 2 C(H):
Now, for each …xed ( ; ) 2 , we prove that G(x; ; ) is a uniform -H-contraction mapping. Let (x1; ; ), (x2; ; ) be arbitrary elements of H
and any t1 2 G(x1; ; ), there exist u1 = u1(x1; ) 2 A(x1; ), v1 =
v1(x1; ) 2 B(x1; ), w1 = w1(x1; ) 2 C(x1; ), y1 = y1(x1; ) 2 D(x1; ) and
z1= z1(x1; ) 2 F (x1; ) such that
t1 = x1 (g m)(x1; )+RW ( ;zP; 1; )[P (g m)(x1; ) N (u1; v1; )+ M (w1; y1; )+ f ]:
(4:6) It follows from the compactness of A(x2; ), B(x2; ), C((x2; ), D(x2; ) and
F (x2; ) and H-Lipschitz continuity of A; B; C; D; F that there exist u2= u2(x2; ) 2
A(x2; ), v2 = v2(x2; ) 2 B(x2; ), w2 = w2(x2; ) 2 C(x2; ), y2 = y2(x2; ) 2
D(x2; ) and z2= z2(x2; ) 2 F (x2; ) such that
ku1 u2k H(A(x1; ); A(x2; )) LAkx1 x2k; kv1 v2k H(B(x1; ); B(x2; )) LBkx1 x2k; kw1 w2k H(C(x1; ); C(x2; )) LCkx1 x2k; ky1 y2k H(D(x1; ); D(x2; )) LDkx1 x2k; kz1 z2k H(F (x1; ); F (x2; )) LFkx1 x2k: (4:7) Let t2= x2 (g m)(x2; )+RW ( ;zP; 2; )[P (g m)(x2; ) N (u2; v2; )+ M (w2; y2; )+ f ]; (4:8) then we have t22 G(x2; ; ).
Next, using Theorem 2.2 and (4.1), we estimate kt1 t2k kx1 x2 ((g m)(x1; ) (g m)(x2; ))k +kRW ( ;z1; ) P; [P (g m)(x1; ) N (u1; v1; ) + M (w1; y1; ) + f ] RW ( ;z2; ) P; [P (g m)(x1; ) N (u1; v1; ) + M (w1; y1; ) + f ]k +kRW ( ;z2; ) P; [P (g m)(x1; ) N (u1; v1; ) + M (w1; y1; ) + f ] RW ( ;z2; ) P; [P (g m)(x2; ) N (u2; v2; ) + M (w2; y2; ) + f ]k
kx1 x2 ((g m)(x1; ) (g m)(x2; ))k + k1kz1 z2k + + kx1 x2 (P (g m)(x1; ) P (g m)(x2; ) +kx1 x2 (N (u1; v1; ) N (u2; v2; ) M (w1; y1; ) + M (w2; y2; ))k i : (4:9) Since (g m) is s-strongly monotone and L(g m)-Lipschitz continuous, we have
kx1 x2 ((g m)(x1; ) (g m)(x2; ))k2
kx1 x2k2 2h(g m)(x1; ) (g m)(x2; ); x1 x2i+k(g m)(x1; ) (g m)(x2; )k2
(1 2s+L2(g m))kx1 x2k2: (4:10)
Similarly, since P (g m) is r-strongly monotone and LP (g m)-Lipschitz
con-tinuous, we have
kx1 x2 (P (g m)(x1; ) P (g m)(x2; ))k2 (1 2r+L2P (g m))kx1 x2k2:
(4:11) Since N is (L(N;1); L(N;2))-mixed Lipschitz continuous; M is (L(M;1); L(M;2)
)-mixed Lipschitz continuous and the multi-valued mappings A; B; C; D are H-Lipschitz continuous, we have kN(u1; v1; ) N (u2; v2; )k L(N;1)ku1 u2k + L(N;2)kv1 v2k L(N;1)H(A(x1; ); A(x2; )) + L(N;2)H(B(x1; ); B(x2; )) (LAL(N;1)+ LBL(N;2))kx1 x2k; (4:12) and kM(w1; y1; ) N (w2; y2; )k (LCL(M;1)+ LDL(M;2))kx1 x2k: (4:13)
Further, since N is -strongly mixed monotone with respect to A and B, M is -generalized mixed pseudocontractive with respect to C and D then, using ka + bk2 2(kak2+ kbk2), we have kx1 x2 (N (u1; v1; ) N (u2; v2; ) M (w1; y1; )+M (w2; y2; ))k2 kx1 x2k2 2 h hN(u1; v1; ) N (u2; v2; ); x1 x2i hM(w1; y1; ) M (w2; y2; ); x1 x2i i + 2 2 h kN(u1; v1; ) N (u2; v2; )k2 +kM(w1; y1; ) M (w2; y2; )k2 i kx1 x2k2 2 ( )kx1 x2k2 +2 2[(LAL(N;1)+ LBL(N;2))2+ (LCL(M;1)+ LDL(M;2))2 i kx1 x2k2 1 2 ( )+2 2h(LAL(N;1)+L(N;2))2+(LCL(M;1)+LDL(M;2))2 i kx1 x2k2: (4:14)
Now, from (4.9)-(4.14), we have kt1 t2k kx1 x2k; (4:15) where = q + ( ); q := k1LF+ q 1 2s + L2 (g m); ( ) := + hq 1 2s + L2 P (g m)+ q 1 2 ( ) + 2 2(L2 N+ L2M); LN := (LAL(N;1)+ LBL(N;2)); LM := (LCL(M;1)+ LDL(M;2)): Hence, we have d(t1; G(x2; ; )) = inf t22G(x2; ; )kt 1 t2k kx1 x2k:
Since t12 G(x1; ; ) is arbitrary, we obtain
sup
t12G(x1; ; )
d(t1; G(x2; ; )) kx1 x2k:
By using same argument, we can prove sup
t22G(x2; ; )
d(G(x1; G(x1; ; ); t2) kx1 x2k:
By the de…nition of the Hausdor¤ metric H on C(H), we have
H G(x1; ; ); G(x2; ; ) kx1 x2k; (4:16)
that is, G(x; ; ) is a uniform -H-contraction mapping with respect to ( ; ) 2 . Also, it follows from condition (4.3)-(4.4) that < 1 and hence G(x; ; ) is a multi-valued contraction mapping which is uniform with respect to ( ; ) 2 . By Lemma 3.1 for each ( ; ) 2 , G(x; ; ) has a …xed point x = x( ; ) 2 H, that is, x = x( ; ) 2 G(x; ; ) and hence Lemma 4.1 ensure that S( ; ) 6= ;. Further, for any sequence fxng S( ; ) with lim
n!1xn= x0, we have
xn2 G(xn; ; ) for all n 1. By virtue of (4.16), we have
d(x0; G(x0; ; )) kx0 xnk + H G(xn; ; ); G(x0; ; )
(1 + )kxn x0k ! 0 as n ! 1;
that is, x02 G(x0; ; ) and hence x02 S( ; ). Thus S( ; ) is closed in H. This
5. Lipschitz continuity
Now, we prove that the solution set S( ; ) of PMIQVLIP (3.1) is H-Lipschitz continuity for each ( ; ) 2 .
Theorem 5.1. Let the multi-valued mappings A; B; C; D and F be H-mixed Lip-schitz continuous with pairs of constants (LA; lA), (LB; lB), (LC; lC), (LD; lD) and
(LF; lF), respectively. Let the mappings ; P be the same as in Theorem 4.1; let the
mapping (g m) be s-strongly monotone and (L(g m); l(g m))-Lipschitz
continu-ous; let the mapping P (g m) be r-strongly monotone and (LP (g m); lP (g m)
)-Lipschitz continuous. Let the mapping N be -strongly mixed monotone with respect to A and B and (L(N;1); L(N;2); lN)-mixed Lipschitz continuous, and let
the mapping M be -generalized mixed pseudomonotone with respect to C and D, and (L(M;1); L(M;2); lM)-mixed Lipschitz continuous. Suppose that the
multi-valued mapping W is same as in Theorem 4.1 and conditions (4.2),(4.3),(4.4) hold, then for each ( ; ) 2 , the solution set S( ; ) of PMIQVLIP (3.1) is a H-Lipschitz continuous mapping from into H.
Proof. For each ( ; ); ( ; ) 2 , it follows from Theorem 4.1 that S( ; ) and S( ; ) are both nonempty and closed subsets of H. It also follows from Theorem 4.1 that G(x; ; ) and G(x; ; ) both are multi-valued -H-contraction mappings with same contractive constant 2 (0; 1). By Lemma 3.2, we obtain
H(S( ; ); S( ; )) 1 1 sup
x2H H(G(x; ; ); G(x; ; ));
(5:1) where is given by (4.3)-(4.4).
Now, for any a 2 G(x; ; ), there exist u = u(x; ) 2 A(x; ), v = v(x; ) 2 B(x; ), w = w(x; ) 2 C(x; ), y = y(x; ) 2 D(x; ) and z = z(x; ) 2 F (x; ) satisfying
a = x (g m)(x; )+RW ( ;z; )P; [P (g m)(x; ) N (u; v; )+ M (w; y; )+ f ]: (5:2) It is easy to see that there exist u = u(x; ) 2 A(x; ), v = v(x; ) 2 B(x; ), w = w(x; ) 2 C(x; ), y = y(x; ) 2 D(x; ) and z = z(x; ) 2 F (x; ) such that
ku uk H(A(x; ); A(x; )) lAk k;
kv vk H(B(x; ); B(x; )) lBk k;
kw wk H(C(x; ); C(x; )) lCk k;
ky yk H(D(x; ); D(x; )) lDk k;
Let b = x (g m)(x; )+RW ( ;z; )P; h P (g m)(x; ) N (u; v; )+ M (w; y; )+ f i : (5:4) Clearly, b 2 G(x; ; ):
Since N and M are mixed Lipschitz continuous and in view of (4.3) and (5.1)-(5.4) and with t = P (g m)(x; ) N (u; v; ) + M (w; y; ); we have
ka bk k(g m)(x; ) (g m)(x; )k +kRW ( ;z; )P; [P (g m)(x; ) N (u; v; ) + M (w; y; ) + f ] R W ( ;z; ) P; (t)k +kRW ( ;z; )P; (t) R W ( ;z; ) P; (t)k+kR W ( ;z; ) P; (t) R W ( ;z; ) P; (t)k k(g m)(x; ) (g m)(x; )k+ + hkP (g m)(x; ) P (g m)(x; )k + kN(u; v; ) N (u; v; )k + kM(w; y; ) M (w; y; )k i +k1kz zk+k2k k l(g m)k k + + lP (g m)k k + lAL(N;1)+ lBL(N;2)+ lN +lCL(M;1)+ lDL(M;2)+ lM k k i + k1lFk k + k2k k 1(k k+k k); where 1:= maxf(l(g m)+k1lF+k2+ + LP (g m)); + (lAL(N;1)+ lBL(N;2)+ lN + lCL(M;1)+ lDL(M;2)+ lM)g Hence, we obtain sup a2G(x; ; ) d(a; G(x; ; )) 1k ; ) ( ; )k ; where k( ; )k = k k + k k.
By using similar argument, we have sup
b2G(x; ; )
d(G(x; ; ); b) 1k( ; ) ( ; )k :
Hence, it follows that
H(G(x; ; ); G(x; ; )) 1k( ; ) ( ; )k ; 8 (x; ; ); (x; ; ) 2 H :
By Lemma 5.1, we obtain
H(S( ; ); S( ; )) 1
which implies that S( ; ) is H-Lipschitz continuous in ( ; ) 2 , and this completes the proof.
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Current address : K. R. Kazmi: Department of Mathematics, Aligarh Muslim University Ali-garh 202002, India
E-mail address : krkazmi@gmail.com
Current address : Department of Mathematics, Faculty of Science, King Faisal University Al-Hasa, Kingdom of Saudi Arabia
