R E S E A R C H
Open Access
Solution sensitivity of generalized nonlinear
parametric
(A,
η,
m)
-proximal operator
system of equations in Hilbert spaces
Jong Kyu Kim
1*, Heng-you Lan
2and Yeol Je Cho
3,4Dedicated to Professor Shih-Sen Chang’s 80th birthday.
*Correspondence:
1Department of Mathematics
Education, Kyungnam University, Changwon, Gyeongnam 631-701, Korea
Full list of author information is available at the end of the article
Abstract
By using the new parametric resolvent operator technique associated with
(A,
η
,m)-monotone operators, the purpose of this paper is to analyze and establish an existence theorem for a new class of generalized nonlinear parametric(A,
η
,m)-proximal operator system of equations with non-monotone multi-valued operators in Hilbert spaces. The results presented in this paper generalize the sensitivity analysis results of recent work on strongly monotone quasi-variational inclusions, nonlinear implicit quasi-variational inclusions, and nonlinear mixed quasi-variational inclusion systems in Hilbert spaces.MSC: 49J40; 47H05; 90C33
Keywords: sensitivity analysis; relaxed cocoercive operator; (A,
η
,m)-monotone variational inclusion system; (A,η
,m)-proximal operator equation; generalized resolvent operator technique1 Introduction
Recently, since the study of the sensitivity (analysis) of solutions for variational inclusion (operator equation) problems involving strongly monotone and relaxed cocoercive map-pings under suitable second order and regularity assumptions is an increasing interest, there are many motivated researchers basing their work on the generalized resolvent op-erator (equation) techniques, which is used to develop powerful and efficient numerical techniques for solving (mixed) variational inequalities, related optimization, control the-ory, operations research, transportation network modeling, and mathematical program-ming problems. It is well known that the project technique and the resolvent operator technique can be used to establish an equivalence between (mixed) variational inequali-ties, variational inclusions, and resolvent equations. See, for example, [–] and the ref-erences therein.
In this paper, we consider the following system of (A,η,m)-proximal operator equations: For each fixed (ω,λ)∈×, find (z,t), (x,y)∈H×Hsuch thatu∈S(x,ω) and
⎧ ⎨ ⎩
E(x,y,ω) +ρ–RρM,A(·,x,ω)
(z) = ,
F(u,y,λ) +–RN(·,y,λ)
,A (t) = ,
(.)
whereandare two nonempty open subsets of real Hilbert spacesHandH, in which the parameterωandλtake values, respectively,S:H×→His a set-valued opera-tor,E:H×H×→H,F:H×H×→H,f :H×→H,g:H×→H,
η:H×H×→H, andη:H×H×→H are nonlinear single-valued op-erators, A :H →H, A:H→H, M:H×H ×→H and N:H×H×
→ H are any nonlinear operators such that for all (z,)∈ H
×, M(·,z,ω) :
H→H is an (A,η,m)-monotone operator withf(H,ω)∩dom(M(·,z,ω))=∅and for all (t,λ)∈H×,N(·,t,λ) :H→H is an (A,η,m)-monotone operator with
g(H,λ)∩dom(N(·,t,λ))=∅, respectively,RρM,A(·,x,ω)=I–A(J M(·,x,ω)
ρ,A ),Iis the identity oper-ator,RN,(A·,y,λ)=I–A(J
N(·,y,λ)
,A ),A(J M(·,x,ω)
ρ,A (z)) =A(J M(·,x,ω)
ρ,A )(z),A(J N(·,y,λ)
,A (t)) =A(J N(·,y,λ)
,A )(t),
JρM,A(·,x,ω)= (A+ρM(·,x,ω))–andJN(·,y,
λ)
,A = (A+N(·,y,λ))
–for allx,z∈H
,y,t∈H, and (ω,λ)∈×.
For appropriate and suitable choices ofS,E,F,M, N,f,g,Ai,ηi, andHi fori= , , one sees that problem (.) is a generalized version of some problems, which includes a number (systems) of (parametric) quasi-variational inclusions, (parametric) generalized quasi-variational inclusions, (parametric) quasi-variational inequalities, (parametric) im-plicit quasi-variational inequalities studied by many authors as special cases; see, [, , , , , –, –, , , –] and the references therein.
Example . IfS:H×→His a single-valued operator, then for each fixed (ω,λ)∈
×, problem (.) reduces to the following problem of finding (x,y), (z,t)∈H×H such that:
⎧ ⎨ ⎩
E(x,y,ω) +ρ–RρM,A(·,x,ω)(z) = ,
F(S(x,ω),y,λ) +–RN,(A·,y,λ)
(t) = .
(.)
Example . If H=H=H,A=A=A,S=I,E=F,M=N, x=yandω=λ, then problem (.) reduces to findingx,z∈Hsuch that
E(x,x,ω) +ρ–RρM,A(·,x,ω)(z) = . (.)
Problem (.) is equivalent to the following nonlinear equation:
x=JρM,A(·,x,ω)(z), z=A(x) –ρE(x,x,ω),
which can be rewritten as the following generalized strongly monotone mixed quasi-variational inclusion:
∈E(x,x,ω) +M(x,x,ω),
and studied by Verma [, ] whenMisA-monotone and (A,η)-monotone with respect to first variable.
continuous functionu: [, )→Hsuch that the following first-order evolution equation: ⎧
⎨ ⎩
u(t) +Mu(t) = , <t<∞,
u() =u,
where the derivativeu(t) exists in the sense of weak convergence, that is,
u(t+h) –u(t)
h u
(t) ash→
holds for allt∈(,∞).
On the other hand, Lan [] introduced a new concept of (A,η)-monotone operators, which generalizes the (H,η)-monotonicity and A-monotonicity in Hilbert spaces and other existing monotone operators as special cases, and studied some properties of (A,η )-monotone operators and applied resolvent operators associated with (A,η)-monotone op-erators to approximate the solutions of a new class of nonlinear (A,η)-monotone operator inclusion problems with relaxed cocoercive operators in Hilbert spaces. Lanet al.[] and Verma [] introduced and studied a new class of parametric generalized relaxed coco-ercive implicit quasi-variational inclusions withA-monotone operators, respectively. By using the parametric implicit resolvent operator technique forA-monotone, we analyzed solution sensitivity for this kind of generalized relaxed cocoercive inclusions in Hilbert spaces. In [, ], based on the (A,η)-resolvent operator technique, Verma and Lan intro-duced and investigated a sensitivity analysis for a class of generalized strongly monotone variational inclusions in Hilbert spaces, respectively. Furthermore, using the concept and technique of resolvent operators, Agarwalet al.[] and Jeong [] introduced and studied a new system of parametric generalized nonlinear mixed quasi-variational inclusions in a Hilbert space and inLp(p≥) spaces, respectively.
In this paper, we shall generalize the resolvent equations by introducing (A,η,m )-proximal operator equations in Hilbert spaces and establish a relationship between a class of parametric (A,η,m)-monotone variational inclusion systems and a class of generalized nonlinear parametric (A,η,m)-proximal operator system of equations. Further, we study sensitivity analysis of the solution set for the system (.) of (A,η,m)-proximal operator equations with non-monotone set-valued operators in Hilbert spaces.
Our results improve and generalize the results on the sensitivity analysis for generalized nonlinear mixed quasi-variational inclusions [, , , , –] and others. For more details, we recommend [, , , , , , , , , , ].
2 Preliminaries
In the sequel, letbe a nonempty open subset of a real Hilbert spaceHin which the parameterλtake values.
Definition . An operatorT:H×H×→His said to be
(i) m-relaxed monotone in the first argument if there exists a positive constantmsuch that
T(x,u,λ) –T(y,u,λ),x–y≥–mx–y
(ii) s-cocoercive in the first argument if there exists a constants> such that
T(x,u,λ) –T(y,u,λ),x–y≥sT(x,u,λ) –T(y,u,λ)
for all(x,y,u,λ)∈H×H×H×;
(iii) γ-relaxed cocoercive with respect toAin the first argument if there exists a positive constantγ such that
T(x,u,λ) –T(y,u,λ),A(x) –A(y)≥–γT(x,u,λ) –T(y,u,λ)
for all(x,y,u,λ)∈H×H×H×;
(iv) (,α)-relaxed cocoercive with respect toAin the first argument if there exist positive constantsandαsuch that
T(x,u,λ) –T(y,u,λ),A(x) –A(y)≥–αT(x,u,λ) –T(y,u,λ)+x–y
for all(x,y,u,λ)∈H×H×H×.
In a similar way, we can define (relaxed) cocoercivity of the operatorT(·,·,·) in the sec-ond argument.
Definition . An operatorT:H×H×→His said to beμ-Lipschitz continuous in the first argument if there exists a constantμ> such that
T(x,u,λ) –T(y,u,λ)≤μx–y, ∀(x,y,u,λ)∈H×H×H×.
In a similar way, we can define Lipschitz continuity of the operatorT(·,·,·) in the second and third argument.
Definition . LetF :H×→H be a multi-valued operator. Then F is said to be
τ-Hˆ-Lipschitz continuous in the first argument if there exists a constant τ > such that
ˆ
HF(x,λ),F(y,λ) ≤τx–y, ∀x,y∈H,λ∈,
whereHˆ : H×H→(–∞, +∞)∪ {+∞}is the Hausdorff metric,i.e.,
ˆ
H(A,B) =max
sup
x∈A
inf
y∈Bx–y,supx∈Byinf∈Ax–y
, ∀A,B∈H.
In a similar way, we can defineHˆ-Lipschitz continuity of the operatorF(·,·) in the second argument.
Lemma .([]) Let(X,d)be a complete metric space and T,T:X →CB(X)be two
set-valued contractive operators with same contractive constant t∈(, ),i.e.,
ˆ
Then
ˆ
HF(T),F(T) ≤ –tsupx∈X
ˆ
HT(x),T(x),
where F(T)and F(T)are fixed point sets of Tand T,respectively.
Definition . LetA:H→H,η:H×H→Hbe two single-valued operators. Then a multi-valued operator M: H→ H is called (A,η,m)-monotone (so-called (A,η )-monotonicity [, ], (A,η)-maximal relaxed monotonicity []) if
(i) Mism-relaxedη-monotone, (ii) (A+ρM)(H) =Hfor everyρ> .
Remark . For appropriate and suitable choices ofm,A,η, andH, it is easy to see that Definition . includes a number of definitions of monotone operators and monotone mappings (see [, , , ]).
Proposition .([]) Let A:H→Hbe a r-stronglyη-monotone operator,M:H→H
be an(A,η)-monotone operator.Then the operator(A+ρM)–is single-valued.
Definition . LetA:H→Hbe a strictlyη-monotone operator andM:H→Hbe an (A,η,m)-monotone operator. The resolvent operatorJηρ,,MA:H→His defined by
Jηρ,,MA(u) = (A+ρM)–(u), ∀u∈H.
Proposition .([]) LetHbe a q-uniformly smooth Banach space andη:H×H→H
beτ-Lipschitz continuous,A:H→Hbe a r-stronglyη-monotone operator and M:H→
Hbe an(A,η,m)-monotone operator.Then the resolvent operator Jηρ,,MA:H→His
τ
r–ρm
-Lipschitz continuous,i.e.,
Jρ,A
η,M(x) –J
ρ,A
η,M(y)≤
τ
r–ρmx–y, ∀x,y∈H, whereρ∈(,mr)is a constant.
In connection with the(A,η,m)-proximal operator equations system(.),we consider the following generalized parametric(A,η,m)-monotone variational inclusion system:
For each fixed(ω,λ)∈×,find(x,y)∈H×Hsuch that u∈S(x,ω)and ⎧
⎨ ⎩
∈E(x,y,ω) +M(x,x,ω),
∈F(u,y,λ) +N(y,y,λ). (.)
Now, for each fixed (ω,λ)∈×, the solution setQ(ω,λ) of problem (.) is denoted by
Q(ω,λ) =(z,t,x,y)∈H×H:∃u∈S(x,ω) such thatE(x,y,ω) +ρ–RρM,A(·,x,ω)(z) =
andF(u,y,λ) +–RN,(A·,y,λ)(t) = .
In this paper, our aim is to study the behavior of the solution setQ(ω,λ) and the condi-tions on these operatorsS,E,F,M,N,η,η,A,Aunder which the functionQ(ω,λ) is continuous or Lipschitz continuous with respect to the parameter (ω,λ)∈×.
3 Sensitivity analysis results
In the sequel, we first transfer problem (.) into a problem of finding the parametric fixed point of the associated (A,η,m)-resolvent operator.
Lemma . For each fixed(ω,λ)∈×,an element(x,y)∈Q(ω,λ)is a solution of prob-lem(.)if and only if there are(x,y)∈H×Hand u∈S(x,ω)such that
⎧ ⎨ ⎩
x=JρM,A(·,x,ω)(A(x) –ρE(x,y,ω)),
y=JN,A(·,y,λ)
(A(y) –F(u,y,λ)),
(.)
where JρM,A(·,x,ω)= (A+ρM(·,x,ω))–and JN,A(·,y,λ)= (A+N(·,y,λ))
–are the
correspond-ing resolvent operator in first argument of an(A,η)-monotone operator M(·,·,·), (A,η)
-monotone operator N(·,·,·),respectively,Aiis an ri-strongly monotone operator for i= ,
andρ,> .
Proof For each fixed (ω,λ)∈×, by the definition of the resolvent operatorsJρM,A(·,x,ω)
= (A+ρM(·,x,ω))–ofM(·,x,ω) andJN,A(·,y,λ)= (A+N(·,y,λ))
–ofN(·,y,λ), respectively, we know that there existx∈H,y∈H, andu∈S(x(ω),ω) such that (.) holds if and only if
A(x) –ρE(x,y,ω)∈A(x) +ρM(x,x,ω),
A(y) –F(u,y,λ)∈A(y) +N(y,y,λ),
i.e.,
∈E(x,y,ω) +M(x,x,ω), ∈F(u,y,λ) +N(y,y,λ).
It follows from the definition ofQ(ω,λ) that (x,y)∈Q(ω,λ) is a solution of problem (.) if and only if there exist (x,y)∈H×H, andu∈S(x,ω) such that equation (.) holds.
Lemma . The problem(.) has a solution(z,t,x,y,u)with u∈S(x,ω)if and only if problem(.)has a solution(x,y,u)with u∈S(x,ω),where
x=JρM,A(·,x,ω)(z), y=J N(·,y,λ)
,A (t) (.)
and
z=A(x) –ρE(x,y,ω),
t=A(y) –F(u,y,λ).
Proof Let (x,y,u) withu∈S(x,ω) be a solution of problem (.). Then, by Lemma ., it is a solution of the following system of equations:
⎧ ⎨ ⎩
x=JρM,A(·,x,ω)(A(x) –ρE(x,y,ω)),
y=JN,A(·,y,λ)
(A(y) –F(u,y,λ)).
By using the factRMρ,A(·,x,ω)
=I–A(J M(·,x,ω)
ρ,A ),R N(·,y,λ)
,A =I–A(J N(·,y,λ)
,A ) and (.), we have
RMρ,A(·,x,ω)
A(x) –ρE(x,y,ω)
=A(x) –ρE(x,y,ω) –A
JρM,A(·,x,ω)
A(x) –ρE(x,y,ω)
=A(x) –ρE(x,y,ω) –A(x)
= –ρE(x,y,ω)
and
RN,A(·,y,λ)
A(y) –F(u,y,λ)
=A(y) –F(u,y,λ) –A
JN,A(·,y,λ)A(y) –F(u,y,λ)
=A(y) –F(u,y,λ) –A(y)
= –F(u,y,λ),
which imply that
E(x,y,ω) +ρ–RρM,A(·,x,ω)(z) = ,
F(u,y,λ) +–RN,(A·,y,λ)(t) =
withz=A(x) –ρE(x,y,ω) andt=A(y) –F(u,y,λ),i.e.(z,t,x,y,u) withu∈S(x,ω) is a solution of problem (.).
Conversely, letting (z,t,x,y,u) withu∈S(x,ω) is a solution of problem (.), then ⎧
⎨ ⎩
ρE(x,y,ω) = –RρM,A(·,x,ω)
(z) =A(J M(·,x,ω)
ρ,A (z)) –z,
F(u,y,λ) = –RN,(A·,y,λ)(t) =A(J N(·,y,λ)
,A (t)) –t.
It follows from (.) and (.) that
ρE(x,y,ω) =A
JρM,A(·,x,ω)A(x) –ρE(x,y,ω)
–A(x) +ρE(x,y,ω),
F(u,y,λ) =A
JN,A(·,y,λ)A(y) –F(u,y,λ)
–A(y) +F(u,y,λ),
which imply that
A(x) =A
JρM,A(·,x,ω)
A(x) –ρE(x,y,ω)
,
A(y) =A
JN,A(·,y,λ)A(y) –F(u,y,λ)
and so
x=JρM,A(·,x,ω)
A(x) –ρE(x,y,ω) ,
y=JN,A(·,y,λ)
A(y) –F(u,y,λ),
i.e., (x,y,u) withu∈S(x,ω) is a solution of problem (.).
Alternative proof Let
z=A(x) –ρE(x,y,ω), t=A(y) –F(u,y,λ).
Then, by (.), we know
x=JρM,A(·,x,ω)(z), y=JN,A(·,y,λ)(t)
and
z=A
JρM,A(·,x,ω)(z) –ρE(x,y,ω), t=A
JN,A(·,y,λ)(t) –F(u,y,λ).
SinceA(JρM,A(·,x,ω)(z)) =A(J M(·,x,ω)
ρ,A )(z) andA(J N(·,y,λ)
,A (t)) =A(J N(·,y,λ)
,A )(t), we have
E(x,y,ω) +ρ–RρM,A(·,x,ω)
(z) = , F(u,y,λ) +
–RN(·,y,λ)
,A (t) = ,
the required problem (.).
We now invoke Lemmas . and . to suggest the following sensitivity analysis results for the system of (A,η,m)-proximal operator equations (.).
Theorem . Let Ai:Hi→Hi be ri-strongly monotone and si-Lipschitz continuous for
all i= , , S:H×→CB(H)beκ-Hˆ-Lipschitz continuous in the first variable, M:
H×H×→H be(A,η)-monotone with constant min the first variable,and N :
H×H×→H be(A,η)-monotone with constant min the first variable.Letη:
H×H→Hbeτ-Lipschitz continuous,η:H×H→Hbeτ-Lipschitz continuous,
E:H×H×→H be(γ,α)-relaxed cocoercive with respect to A andμ-Lipschitz
respect to Aandμ-Lipschitz continuous in the second variable,and let E beβ-Lipschitz
continuous in the second variable,and F beβ-Lipschitz continuous in the first variable.If
JρM,A(·,x,ω)
(z) –J M(·,y,ω)
ρ,A (z)
≤νx–y, ∀(x,y,z,ω)∈H×H×H×, (.) JN,A(·,x,λ)(z) –J
N(·,y,λ)
,A (z)
≤νx–y, ∀(x,y,z,λ)∈H×H×H×, (.)
withνi< for i= , and there exist constantsρ∈(,mr),∈(,mr)such that ⎧
⎨ ⎩
τ(s– ργ+ρμ+ ραμ) < (r–ρm)( –ν–rβ–κτm) ,
τ
(s– γ+μ+ αμ) < (r–m)( –ν–rρβ–ρτm),
(.)
then,for each(ω,λ)∈×,the following results hold: () the solution setQ(ω,λ)of problem(.)is nonempty; () Q(ω,λ)is a closed subset inH×H.
Proof In the sequel, from (.), we first define operatorsρ:H×H××→Hand
:H×H××→Has follows:
ρ(x,y,ω,λ) =JρM,A(·,x,ω)
A(x) –ρE(x,y,ω),
(x,y,ω,λ) =JN,A(·,y,λ)
A(y) –F(u,y,λ)
(.)
for all (x,y,ω,λ)∈H×H××. Now, define a norm · onH×Hby
(x,y)=x+y, ∀(x,y)∈H×H.
It is easy to see that (H×H, · ) is a Banach space (see []). By (.), for any given
ρ> and> , define an operatorG:H×H××→H×H by
Gρ,(x,y,ω,λ) =
ρ(x,y,ω,λ),λ(x,y,ω,λ) :u∈S(x,ω),
∀(x,y,ω,λ)∈H×H××
.
For any (x,y,ω,λ)∈H×H××, sinceS(x,ω)∈CB(H),A,A,η,η,E,F,JρM,A(·,x,ω),
JρM,A(·,x,λ)are continuous, we haveGρ,(x,y,ω,λ)∈CB(H×H). Now, for each fixed (ω,λ)∈
×, we prove thatGρ,(x,y,ω,λ) is a multi-valued contractive operator.
In fact, for any (x,y,ω,λ), (xˆ,yˆ,ω,λ)∈H×H××and (a,a)∈Gρ,(x,y,ω,λ),
there existsu∈S(x,ω) such that
a=JρM,A(·,x,ω)
A(x) –ρE(x,y,ω) ,
a=JN(·,y,
λ)
,A
Note thatS(xˆ,ω)∈CB(H), it follows from Nadler’s result [] that there existsuˆ∈S(xˆ,ω) such that
u–uˆ ≤ ˆHS(x,ω),S(xˆ,ω) . (.)
Setting
b=JρM,A(·,xˆ,ω)
A(xˆ) –ρE(xˆ,yˆ,ω),
b=JN(·,yˆ,
λ)
,A
A(ˆy) –F(uˆ,yˆ,λ) ,
then we have (b,b)∈Gρ,(xˆ,yˆ,ω,λ). It follows from (.) and Proposition . that
a–b =JρM,A(·,x,ω)
A(x) –ρE(x,y,ω) –JρM,A(·,ˆx,ω)
A(xˆ) –ρE(xˆ,yˆ,ω)
≤JρM,A(·,x,ω)
A(x) –ρE(x,y,ω) –JρM,A(·,xˆ,ω)
A(x) –ρE(x,y,ω)
+JρM,A(·,ˆx,ω)
A(x) –ρE(x,y,ω) –JM(·,xˆ,
ω)
ρ,A
A(xˆ) –ρE(xˆ,yˆ,ω)
≤νx–xˆ+
τ
r–ρm
A(x) –ρE(x,y,ω) –
A(xˆ) –ρE(xˆ,yˆ,ω)
≤νx–xˆ+
ρτ
r–ρm
E(xˆ,y,ω) –E(xˆ,yˆ,ω)
+ τ
r–ρm
A(x) –A(xˆ) –ρ
E(x,y,ω) –E(xˆ,y,ω). (.)
By the assumptions ofE,A, we have
E(xˆ,y,ω) –E(xˆ,yˆ,ω)≤βy–yˆ (.)
and
A(x) –A(xˆ) –ρE(x,y,ω) –E(xˆ,y,ω)
≤A(x) –A(xˆ)
+ρE(x,y,ω) –E(xˆ,y,ω) – ρE(x,y,ω) –E(xˆ,y,ω),A(x) –A(xˆ)
≤A(x) –A(xˆ)
+ρE(x,y,ω) –E(xˆ,y,ω) – ρ–αE(x,y,ω) –E(xˆ,y,ω)+γx–xˆ
≤s– ργ+ρμ+ ραμ x–xˆ. (.)
Combining (.)-(.), we have
a–b ≤θx–xˆ+ϑy–yˆ, (.)
where
θ=ν+
τ
r–ρm
s
– ργ+ρμ+ ραμ, ϑ=
ρβτ
Similarly, by the assumptions ofS,A,F, and (.), we obtain
a–b
=JN,A(·,y,λ)A(y) –F(u,y,λ) –JN(·,ˆy,
λ)
,A
A(yˆ) –F(uˆ,yˆ,λ)
≤JN,A(·,y,λ)
A(y) –F(u,y,λ) –JN(·,yˆ,
λ)
,A
A(y) –F(u,y,λ)
+JN,A(·,yˆ,λ)A(y) –F(u,y,λ) –JN,A(·,yˆ,λ)
A(yˆ) –F(uˆ,ˆy,λ)
≤νy–yˆ+
τ
r–m
A(y) –F(u,y,λ) –
A(yˆ) –F(uˆ,yˆ,λ)
≤νy–yˆ+
τ
r–m
F(u,y,λ) –F(uˆ,y,λ)
+ τ
r–m
A(y) –A(yˆ) –
F(uˆ,y,λ) –F(uˆ,yˆ,λ)
≤θx–xˆ+ϑy–yˆ, (.)
where
θ=
βκτ
r–m
, ϑ=ν+
τ
r–m
s
– γ+μ+ αμ.
It follows from (.) and (.) that
a–b+a–b ≤(θ+θ)x–xˆ+ (ϑ+ϑ)y–yˆ
≤σx–xˆ+y–yˆ , (.)
where
σ=max{θ+θ,ϑ+ϑ}.
It follows from condition (.) thatσ< . Hence, from (.), we get
d(a,a),Gρ,(xˆ,yˆ,ω,λ) = inf
(b,b)∈Gρ,(xˆ,yˆ,ω,λ)
a–b+a–b
≤–σ(x,y) – (xˆ–yˆ).
Since (a,a)∈Gρ,(x,y,ω,λ) is arbitrary, we obtain sup
(a,a)∈Gρ,(x,y,ω,λ)
d(a,a),Gρ,(xˆ,yˆ,ω,λ) ≤–σ(x,y) – (xˆ–yˆ).
By the same argument, we can prove
sup
(b,b)∈Gρ,(xˆ,yˆ,ω,λ)
d(b,b),Gρ,(x,y,ω,λ) ≤–σ(x,y) – (xˆ–yˆ).
It follows from the definition of the Hausdorff metricHˆ onCB(H×H) that
ˆ
for all (x,xˆ,ω)∈H ×H ×, (y,yˆ,λ)∈H ×H ×, i.e., Gρ,(x,y,ω,λ) is a
multi-valued contractive operator, which is uniform with respect to (ω,λ)∈×. By a fixed point theorem of Nadler [], for each (ω,λ)∈×,Gρ,(x,y,ω,λ) has a fixed point
(x(λ),y(λ))∈H×H,i.e., (x,y)∈Gρ,(x,y,ω,λ). By the definition ofG, we know that there
existsu∈S(x,ω) such that (.) holds. Thus, it follows from Lemma . that (x,y,u) with
u∈S(x,ω) is a solution of problem (.). Hence, it follows from Lemma . that (z,t,x,y,u) withu∈S(x,ω) is a solution of problem (.). Therefore,Q(ω,λ)=∅for all (ω,λ)∈×. Next, we prove the conclusion (). For each (ω,λ)∈×, let{(zn,tn,xn,yn)} ⊂Q(ω,λ) andzn→z,tn→t,xn→x,yn→yasn→ ∞. Then we know that there existsun∈
S(xn,ω) and
(xn,yn)∈Gρ,(xn,yn,ω,λ),
zn=A(xn) –ρE(xn,yn,ω), tn=A(yn) –F(un,yn,λ), ∀n= , , . . . ,
and
z=A(x) –ρE(x,y,ω), t=A(y) –F(u,y,λ).
By the proof of conclusion (), we have
ˆ
HGρ,(xn,yn,ω,λ),Gρ,(x,y,ω,λ) ≤σ(xn,yn) – (x,y), ∀(ω,λ)∈×.
It follows that
d(x,y),Gρ,(x,y,ω,λ) ≤(x,y) – (xn,yn)
+d(xn,yn),Gρ,(xn,yn,ω,λ)
+HˆGρ,(xn,yn,ω,λ),Gρ,(x,y,ω,λ)
≤( +σ)(xn,yn) – (x,y).
Hence, we have (x,y)∈Gρ,(x,y,ω,λ) and (x,y)∈Q(ω,λ). Therefore,Q(ω,λ) is a closed subset ofH×H.
Theorem . Under the hypotheses of Theorem.,further assume that
(i) for anyx∈H,ω→S(x,ω)islS-Hˆ-Lipschitz continuous(or continuous); (ii) for anyx,z∈H,y,t∈H,ω→E(x,y,ω),ω→JM(·,x,
ω)
ρ,A (z),λ→F(x,y,λ)and
λ→JN,A(·,y,λ)
(t)both are Lipschitz continuous(or continuous)with Lipschitz constants
lE,lJ,lF,andlJ,respectively.
Then the solution set Q(ω,λ)of problem(.)is Lipschitz continuous(or continuous)from ×toH×H.
with the same contraction constantσ∈(, ) and have fixed points (x(ω,λ),y(ω,λ)) and (x(ω¯,λ¯),y(ω¯,λ¯)), respectively. It follows from Lemmas . and . that
ˆ
HQ(ω,λ),Q(ω¯,λ¯)
≤
–σ(x,y)∈Hsup×H
ˆ
HGρ,
x(ω,λ),y(ω,λ),ω,λ ,Gρ,
x(ω¯,λ¯),y(ω¯,λ¯),ω¯,λ¯ . (.)
Setting (a,a)∈Gρ,(x(ω,λ),y(ω,λ),ω,λ), there existsu(ω,λ)∈S(x(ω,λ),ω) such that a=JρM,A(·,x(ω,λ),ω)
A
x(ω,λ) –ρEx(ω,λ),y(ω,λ),ω ,
a=JN(·,y(
ω,λ),λ)
,A
A
y(ω,λ) –Fu(ω,λ),y(ω,λ),λ .
SinceS(x(ω,λ),ω),S(x(ω¯,λ¯),ω¯)∈CB(H), it follows from Nadler’s result [] that there existsu(ω¯,λ¯)∈S(x(ω¯,λ¯),ω¯) such that
u(ω,λ) –u(ω¯,λ¯)≤ ˆHSx(ω,λ),ω ,Sx(ω¯,λ¯),ω¯ . (.)
Let
b=JM(·,x(ω¯,
¯ λ),ω¯)
ρ,A
A
x(ω¯,λ¯) –ρEx(ω¯,λ¯),y(ω¯,λ¯),ω¯ ,
b=JN(·,y(¯
ω,¯λ),λ¯)
,A
A
y(ω¯,λ¯) –Fu(ω¯,λ¯),y(ω¯,λ¯),λ¯ .
Then we have (b,b)∈Gρ,(x(ω¯,λ¯),y(ω¯,λ¯),ω¯,λ¯). It follows from the assumptions on JρM,A(·,·,·),E,A, andSthat
a–b=JM(·,x(
ω,λ),ω)
ρ,A
A
x(ω,λ) –ρEx(ω,λ),y(ω,λ),ω
–JρM,A(·,x(ω¯,λ¯),ω¯)
A
x(ω¯,λ¯) –ρEx(ω¯,λ¯),y(ω¯,λ¯),ω¯
≤JρM,A(·,x(ω,λ),ω)
A
x(ω,λ) –ρEx(ω,λ),y(ω,λ),ω
–JρM,A(·,x(ω¯,λ¯),ω)
A
x(ω¯,λ¯) –ρEx(ω¯,λ¯),y(ω¯,λ¯),ω
+JρM,A(·,x(ω¯,λ¯),ω)
A
x(ω¯,λ¯) –ρEx(ω¯,λ¯),y(ω¯,λ¯),ω
–JρM,A(·,x(ω¯,λ¯),ω¯)
A
x(ω¯,λ¯) –ρEx(ω¯,λ¯),y(ω¯,λ¯),ω
+JρM,A(·,x(ω¯,λ¯),ω¯)
A
x(ω¯,λ¯) –ρEx(ω¯,λ¯),y(ω¯,λ¯),ω
–JρM,A(·,x(ω¯,λ¯),ω¯)
A
x(ω¯,λ¯) –ρEx(ω¯,λ¯),y(ω¯,λ¯),ω¯
≤θx(ω,λ) –x(ω¯,λ¯)+ϑy(ω,λ) –y(ω¯,λ¯)+lJω–ω¯) + ρτ
r–ρm
Ex(ω¯,λ¯),y(ω¯,λ¯),ω –Ex(ω¯,λ¯),y(ω¯,λ¯),ω¯
≤θx(ω,λ) –x(ω¯,λ¯)+ϑy(ω,λ) –y(ω¯,λ¯)+kω–ω¯, (.)
whereθandϑare the constants of (.) and
k=lJ+
ρτlE
Similarly, by the assumptions ong,JρN,A(·,·,·),F,AandS,
a–b=JN(·,y(
ω,λ),λ)
,A
A
y(ω,λ) –Fu(ω,λ),y(ω,λ),λ
–JN,A(·,y(ω¯,λ¯),λ¯)
A
y(ω¯,λ¯) –Fu(ω¯,λ¯),y(ω¯,λ¯),λ¯
≤JN,A(·,y(ω,λ),λ)
A
y(ω,λ) –Fu(ω,λ),y(ω,λ),λ
–JN,A(·,y(ω¯,λ¯),λ)
A
y(ω¯,λ¯) –Fu(ω¯,λ¯),y(ω¯,λ¯),λ
+JN,A(·,y(ω¯,λ¯),λ)A
y(ω¯,λ¯) –Fu(ω¯,λ¯),y(ω¯,λ¯),λ
–JN,A(·,y(ω¯,λ¯),λ¯)A
y(ω¯,λ¯) –Fu(ω¯,λ¯),y(ω¯,λ¯),λ
+JN,A(·,y(ω¯,λ¯),λ¯)
A
y(ω¯,λ¯) –Fu(ω¯,λ¯),y(ω¯,λ¯),λ
–JN,A(·,y(ω¯,λ¯),λ¯)
A
y(ω¯,λ¯) –Fu(ω¯,λ¯),y(ω¯,λ¯),λ¯
≤θx(ω,λ) –x(ω¯,λ¯)+ϑy(ω,λ) –y(ω¯,λ¯)+lJλ–λ¯ + τ
r–ρm
Fu(ω¯,λ¯),y(ω¯,λ¯),λ –Fu(ω¯,λ¯),y(ω¯,λ¯),λ¯
≤θx(ω,λ) –x(ω¯,λ¯)+ϑy(ω,λ) –y(ω¯,λ¯)+kλ–λ¯, (.)
whereθandϑare the constants of (.) and
k=lJ+
τlF
r–ρm .
It follows from (.), (.) and (.) that
a–b+a–b ≤(θ+θ)x(ω,λ) –x(ω¯,λ¯) + (ϑ+ϑ)y(ω,λ) –y(ω¯,λ¯) +kω–ω¯+kλ–λ¯
≤σa–b+a–b +kω–ω¯+kλ–λ¯,
whereσ is the constant of (.), which implies that
a–b+a–b ≤
ω–ω¯+λ–λ¯ , (.)
where
=
–σ max{k,k}.
Hence, from (.), we obtain
sup
(a,a)∈Gρ,(x,y,ω,λ
By using a similar argument as above, we get
sup
(b,b)∈Gρ,(x,y,ω¯,λ¯)
dGρ,(x,y,ω,λ), (b,b) ≤(ω,λ) – (ω¯,λ¯).
It follows that
ˆ
HGρ,(x,y,ω,λ),Gρ,(x,y,ω¯,λ¯) ≤(ω,λ) – (ω¯,λ¯)
for all (x,y,ω,ω¯,λ,λ¯)∈H×H××××. Thus, (.) implies
ˆ
HQ(ω,λ),Q(ω¯,λ¯) ≤
–σ(ω,λ) – (ω¯,λ¯).
This proves thatQ(ω,λ) is Lipschitz continuous in (ω,λ)∈×. If each operator under conditions (i) and (ii) is assumed to be continuous in (ω,λ)∈×, then by a similar argument as above, we can show thatS(λ) is continuous in (ω,λ)∈×.
Remark . In Theorems . and ., ifE,Fare strongly monotone in the first and second variable,i.e., whenαi= (i= , ) in Theorems . and ., respectively, then we can obtain the corresponding results. Our results improve and generalize the well-known results in [, , –].
4 Application
In this section, we give an application.
Lemma .([]) Letφ:H→R∪ {+∞}be a proper convex lower semi-continuous func-tion.Then J∂φ
α = (I+α∂φ)–is nonexpansive for any constantα> .
Theorem . LetHibe a real Hilbert space andφi:Hi→R∪ {+∞}be a proper convex
lower semi-continuous function for i= , . Suppose that E:H×H×→H isγ
-strongly monotone and μ-Lipschitz continuous in the first variable,and isβ-Lipschitz
continuous in the second variable,F:H×H×→H isγ-strongly monotone and
μ-Lipschitz continuous in the second variable,and isβ-Lipschitz continuous in the first
variable.If there exist positive constantsρandsuch that
⎧ ⎨ ⎩
β+
s– ργ+ρμ< ,
ρβ+
s
– γ+μ< ,
then,for each(ω,λ)∈×:
() (x∗,y∗)∈Ris the unique solution of the following nonlinear problem: ⎧
⎨ ⎩
E(x∗,y∗,ω),x–x∗ ≥ρφ(x∗,ω) –ρφ(x,ω),
F(x∗,y∗,λ),y–y∗ ≥φ(y∗,λ) –φ(y,λ).
(.)
() Moreover,the solution(x∗,y∗)of problem(.)is continuous(or Lipschitz continuous)
from×toR,if in addition,for anyx,z∈H
,y,t∈H,ω→E(x,y,ω),
ω→J∂φ(·,ω)
ρ (z),λ→F(x,y,λ)andλ→J∂φ(·,λ)(t)both are Lipschitz continuous(or
Proof Letting
ρ(x,y,ω,λ) =J∂φ(·,ω)
ρ
x–ρE(x,y,ω) ,
(x,y,ω,λ) =J∂φ(·,λ)
y–F(u,y,λ)
(.)
for all (x,y,ω,λ)∈H×H××and defining · onH×Hby (x,y)=x+y, ∀(x,y)∈H×H,
then it is easy to see that (H×H,·) is a Banach space (see []). Further, one can show thatGρ,(x,y,ω,λ) = (ρ(x,y,ω,λ),λ(x,y,ω,λ)) is a contractive operator and the rest of
proof can be carried out by Theorems . and ., and so it is omitted.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The main idea of this paper was proposed by JKK. JKK and HYL prepared the manuscript initially and performed all the steps of proof in this research. All authors read and approved the final manuscript.
Author details
1Department of Mathematics Education, Kyungnam University, Changwon, Gyeongnam 631-701, Korea.2Department of
Mathematics, Sichuan University of Science & Engineering, and Artificial Intelligence Key Laboratory of Sichuan Province, Zigong, Sichuan 643000, P.R. China.3Department of Mathematics Education and RINS, Gyeongsang National University,
Chinju, 660-701, Korea. 4Department of Mathematics, King Abdulaziz University, Jeddah, 21589, Saudi Arabia.
Acknowledgements
This work was supported by the Basic Science Research Program through the National Research Foundation (NRF) Grant funded by Ministry of Education of the republic of Korea (2013R1A1A2054617), and partially supported by the Open Foundation of Artificial Intelligence Key Laboratory of Sichuan Province (2012RYY04). The authors are grateful to the referee for the helpful suggests and comments.
Received: 25 March 2014 Accepted: 18 August 2014 Published:24 Sep 2014
References
1. Agarwal, RP, Cho, YJ, Huang, NJ: Sensitivity analysis for strongly nonlinear quasivariational inclusions. Appl. Math. Lett. 13(6), 19-24 (2000)
2. Agarwal, RP, Huang, NJ, Tan, MY: Sensitivity analysis for a new system of generalized nonlinear mixed quasi-variational inclusions. Appl. Math. Lett.17, 345-352 (2004)
3. Agarwal, RP, Verma, RU: General system of (A,η)-maximal relaxed monotone variational inclusion problems based on generalized hybrid algorithms. Commun. Nonlinear Sci. Numer. Simul.15(2), 238-251 (2010)
4. Ahmad, R, Siddiqi, AH, Khan, Z: Proximal point algorithm for generalized multivalued nonlinear quasi-variational-like inclusions in Banach spaces. Appl. Math. Comput.163, 295-308 (2005)
5. Anh, NLH, Khanh, PQ: Variational sets of perturbation maps and applications to sensitivity analysis for constrained vector optimization. J. Optim. Theory Appl.158(2), 363-384 (2013)
6. Anh, PN, Kim, JK: A new method for solving monotone generalized variational inequalities. J. Inequal. Appl.2010, Article ID 657192 (2010)
7. Balooee, J, Kim, JK: Some remarks on regularized nonconvex variational inequalities. J. Inequal. Appl.2013, 531 (2013) 8. Cao, HW: Sensitivity analysis for a system of generalized nonlinear mixed quasi variational inclusions with
H-monotone operators. J. Appl. Math.2011, Article ID 921835 (2011)
9. Chang, SS, Kim, JK, Kim, KH: On the existence and iterative approximation problems of solutions for set-valued variational inclusion in Banach spaces. J. Math. Anal. Appl.268(1), 89-108 (2002)
10. Ding, K, He, ZQ, Li, J: Sensitivity analysis for a new class of generalized strongly nonlinear projection equations. Nonlinear Funct. Anal. Appl.8(3), 423-431 (2003)
11. Ding, XP, Wang, ZB: Sensitivity analysis for a system of parametric generalized mixed quasi-variational inclusions involving (K,η)-monotone mappings. Appl. Math. Comput.214(1), 318-327 (2009)
12. Ding, XP: Sensitivity analysis for a system of generalized mixed implicit equilibrium problems in uniformly smooth Banach spaces. Nonlinear Anal. TMA73(5), 1264-1276 (2010)
13. Dong, H, Lee, BS, Huang, NJ: Sensitivity analysis for generalized parametric implicit quasi-variational inequalities. Nonlinear Anal. Forum6(2), 313-320 (2001)
15. Farajzadeh, A, Salahuddin: Sensitivity analysis for nonlinear set-valued variational equations in Banach framework. J. Funct. Spaces Appl.2013, Article ID 258543 (2013)
16. Gao, CJ, Cho, YJ, Huang, NJ: Solution sensitivity of nonlinear implicit quasi-variational inclusions. Arch. Inequal. Appl. 1(3-4), 453-462 (2003)
17. Hu, RX: Sensitivity analysis for generalized mixed quasivariational inequalities. J. Eng. Math.17(3), 23-30 (2000) 18. Huang, NJ, Lan, HY, Cho, YJ: Sensitivity analysis for nonlinear generalized mixed implicit equilibrium problems with
non-monotone set-valued mappings. J. Comput. Appl. Math.196(2), 608-618 (2006)
19. Jeong, JU: A system of parametric generalized nonlinear mixed quasi-variational inclusions inLpspaces. J. Appl. Math.
Comput.19(1-2), 493-506 (2005)
20. Jeong, JU, Kim, SH: Sensitivity analysis for a system of generalized nonlinear mixed quasi-variational inclusions in q-uniformly smooth Banach spaces. Appl. Math. Comput.202(1), 178-187 (2008)
21. Kazmi, KR, Komal, BS, Khan, FA, Mansotra, P: Sensitivity analysis for a system of parametric general
quasi-variational-like inequality problems in uniformly smooth Banach spaces. Appl. Math. Comput.215(2), 716-726 (2009)
22. Kim, JK, Fang, YP, Huang, NJ: An existence result for a system of inclusion problems with applications. Appl. Math. Lett. 21, 1209-1214 (2008)
23. Kim, JK, Kim, KS: A new system of generalized nonlinear mixed quasivariational inequalities and iterative algorithms in Hilbert spaces. J. Korean Math. Soc.44(4), 823-834 (2007)
24. Kim, JK, Kim, KS: On new systems of generalized nonlinear mixed quasivariational inequalities with two-variable operators. Taiwan. J. Math.11(3), 867-881 (2007)
25. Kim, JK, Kim, DS: A new system of generalized nonlinear mixed variational inequalities in Hilbert spaces. J. Convex Anal.11(1), 235-243 (2004)
26. Kim, JK, Tuyen, TM: Regularization proximal point algorithm for finding a common fixed point of a finite family of nonexpansive mappings in Banach spaces. Fixed Point Theory Appl.2011, 52 (2011)
27. Lan, HY: A class of nonlinear (A,η)-monotone operator inclusion problems with relaxed cocoercive mappings. Adv. Nonlinear Var. Inequal.9(2), 1-11 (2006)
28. Lan, HY: Nonlinear parametric multi-valued variational inclusion systems involving (A,η)-accretive mappings in Banach spaces. Nonlinear Anal. TMA69(5-6), 1757-1767 (2008)
29. Lan, HY, Cho, YJ, Verma, RU: On solution sensitivity of generalized relaxed cocoercive implicit quasivariational inclusions withA-monotone mappings. J. Comput. Anal. Appl.8(1), 75-89 (2006)
30. Lan, HY, Kang, JI, Cho, YJ: Nonlinear (A,η)-monotone operator inclusion systems involving non-monotone set-valued mappings. Taiwan. J. Math.11(3), 683-701 (2007)
31. Lan, HY: Sensitivity analysis for generalized nonlinear parametric (A,η,m)-maximal monotone operator inclusion systems with relaxed cocoercive type operators. Nonlinear Anal. TMA74(2), 386-395 (2011)
32. Moudafi, A: Mixed equilibrium problems: sensitivity analysis and algorithmic aspect. Comput. Math. Appl.44, 1099-1108 (2002)
33. Verma, RU: Sensitivity analysis for relaxed cocoercive nonlinear quasivariational inclusions. J. Appl. Math. Stoch. Anal. 2006, Article ID 52041 (2006)
34. Verma, RU: Role ofA-monotonicity in sensitivity analysis for relaxed cocoercive nonlinear quasivariational inclusions and resolvent operator technique. Nonlinear Funct. Anal. Appl.11(4), 577-588 (2006)
35. Verma, RU: Sensitivity analysis for generalized strongly monotone variational inclusions based on the (A,η)-resolvent operator technique. Appl. Math. Lett.19(12), 1409-1413 (2006)
36. Komura, Y: Nonlinear semigroups in Hilbert space. J. Math. Soc. Jpn.19, 493-507 (1967)
37. Lim, TC: On fixed point stability for set-valued contractive mappings with application to generalized differential equations. J. Math. Anal. Appl.110, 436-441 (1985)
38. Nadler, SB: Multi-valued contraction mappings. Pac. J. Math.30, 475-488 (1969)
39. Minty, GJ: Monotone (nonlinear) operators in Hilbert space. Duke Math. J.29, 341-346 (1962) 10.1186/1029-242X-2014-362
