R E S E A R C H
Open Access
Implicit eigenvalue problems for maximal
monotone operators
In-Sook Kim
**Correspondence: [email protected] Department of Mathematics, Sungkyunkwan University, Suwon, 440–746, Republic of Korea
Abstract
We study the implicit eigenvalue problem of the form
0∈Tx+C(
λ
,x),whereTis a maximal monotone multi-valued operator and the operatorCsatisfies condition (S+) or (S˜+). In a regularization method by the duality operator, we use the degree theories of Kartsatos and Skrypnik upon conditions ofCas well as Browder’s degree. There are two cases to consider: One is thatCis demicontinuous and bounded with condition (S+); and the other is thatCis quasibounded and densely defined with condition (S˜+). Moreover, the eigenvalue problem 0∈Tx+
λ
Cxis also discussed.1 Introduction and preliminaries
A general eigenvalue theory for maximal monotone operators has been developed in var-ious ways with applications to partial differential equations; see [–, –]. A key tool was topological degrees for appropriate classes of operators in,e.g., [–, , , –] and the method of approach was in many cases to use regularization by means of the duality operator, while in [, ] the eigenvalue problem was solved by a transformed equation in terms of the approximant without using the regularization method.
LetXbe a real reflexive Banach space with dual spaceX*andT:D(T)⊂X→X*be a
maximal monotone multi-valued operator. When the resolvents ofTor the single-valued operatorCare compact, the solvability of the nonlinear inclusion
∈Tx+λCx
was studied in [, , , ] by applying the Leray-Schauder degree theory for compact operators. More generally, implicit eigenvalue problems were considered in [, ], where the single-valued and compact case was dealt with in []. In direction of [], we study the implicit eigenvalue problem of the form
∈Tx+C(λ,x),
whereC: [,]×M→X*satisfies condition (S
+) or (S˜+). As in [], we adopt property
(P) about the solvability of the related equation
Tsx+C(λ,x) +εJψx= ,
whereTsis the Brezis-Crandall-Pazy approximant introduced in [] andJψis the
(normal-ized) duality operator with a gauge functionψ.
In the present paper, we divide our investigation into two cases to apply suitable degree theory. One case deals with demicontinuous bounded operators satisfying condition (S+)
with the aid of the most elementary degree theory of Skrypnik [], in comparison with []. The other case is concerned with quasibounded densely defined operators satisfying condition (S˜+), where the degree theory of Kartsatos and Skrypnik [, ] for densely
de-fined operators is used. In more concrete situations, the eigenvalue problem ∈Tx+λCx
is discussed. We point out that Browder’s degree in [] for the reduced simple operator
Ts+εJψunder the homotopy plays a crucial role in the proof of our results presented here.
LetXbe a real Banach space with its dualX*,a nonempty subset ofX, andYanother
real Banach space. An operatorF:→Yis said to beboundedifFmaps bounded subsets ofinto bounded subsets ofY.Fis said to bedemicontinuousif for everyx∈and for
every sequence{xn}in withxn→x, we haveFxnFx. Here the symbol →()
stands for strong (weak) convergence.
A multi-valued operatorT:D(T)⊂X→X*is said to bemonotoneif
u*–v*,x–y≥ for everyx,y∈D(T) and everyu*∈Tx,v*∈Ty,
whereD(T) ={x∈X:Tx=∅}denotes theeffective domainofT.T is said to bemaximal monotoneif it is monotone and it follows from (x,u*)∈X×X*and
u*–v*,x–y≥ for everyy∈D(T) and everyv*∈Ty
thatx∈D(T) andu*∈Tx.
We say that an operatorT:D(T)⊂X→X*satisfies condition (S+) if for every sequence
{xn}inD(T) withxnxand
lim sup
n→∞ Txn,xn–x ≤,
we havexn→x.
We say thatT:D(T)⊂X→X*satisfies condition (S˜
+) if for every sequence{xn}inD(T)
withxnx,Txnh*and
lim sup
n→∞ Txn,xn–x ≤,
we havexn→x,x∈D(T) andTx=h*.
We say that a multi-valued operatorT:D(T)⊂X→X* satisfies condition (Sq) on a
setM⊂D(T) if for every sequence{xn}inMwithxnxand every sequence{u*n}inX*
withu*
Throughout this paper,Xwill always be an infinite dimensional real reflexive Banach space which has been renormed so thatXand its dualX*are locally uniformly convex.
A function ψ : [,∞)→[,∞) is called agauge functionif ψ is continuous, strictly increasing,ψ() = andψ(t)→ ∞ast→ ∞. An operatorJψ:X→X*is called aduality
operatorwith a gauge functionψif
Jψx,x=ψ
xx and Jψx=ψ
x forx∈X.
Ifψis the identity mapI, thenJ:=JI is called anormalized duality operator. It is known in [] thatJψ is continuous, bounded, surjective, strictly monotone, maximal monotone
and satisfies condition (S+).
Given a maximal monotone operatorT:D(T)⊂X→X*,x∈Xands> , there exist
unique elementsxs∈D(T) andu*
s∈Txssuch that
J(xs–x) +su*s= .
Two operatorsJs:X→D(T) andTs:X→X*defined by
Jsx:=xs and Tsx:=u*s forx∈X
are called theBrezis-Crandall-Pazy approximants. It is known in [] thatJsis continuous and bounded andTsis demicontinuous, bounded, and maximal monotone. It is easy to
see thatJs=I–sJ–TsandTsx∈TJsxforx∈X.
LetC: [,]×M→X*be an operator, whereMis a subset ofX. ThenC(t,x) is said
to becontinuous in t uniformlywith respect tox∈Mif for everyt∈[,] and for every
sequence{tn}in [,] withtn→t, we haveC(tn,x)→C(t,x) uniformly with respect to
x∈M.
We say thatCsatisfies condition (S+) if for everyλ∈(,] and for every sequence{xn}
inMwithxnxand
lim sup n→∞
C(λ,xn),xn–x
≤,
we havexn→x.
We say thatCsatisfies condition (S˜+) if for everyλ∈(,] and for every sequence{xn}
inMwithxnx,C(λ,xn)h*and
lim sup n→∞
C(λ,xn),xn–x
≤,
we havexn→x,x∈MandC(λ,x) =h*.
We often need the following demiclosedness property of maximal monotone operators given in [].
Lemma . Let T:D(T)⊂X→X*
be a maximal monotone multi-valued operator.Then for every sequence{xn}in D(T),xn→x in X and u*
2 Implicit eigenvalue problem about demicontinuous operators
In this section, we are concerned with the implicit eigenvalue problem for perturbed max-imal monotone operators in reflexive Banach spaces by applying the degree theories of Skrypnik and Browder for nonlinear operators of monotone type.
In what follows, for a bounded subsetofX, letand∂denote the closure and the boundary ofinX, respectively. Following Browder [, ], a homotopyH: [, ]×→X*
is said to be of class (S+) if the following condition holds:
For every sequence{uj}inwithujuand every sequence{tj}in [, ] withtj→t
such that
lim sup j→∞
H(tj,uj),uj–u
≤,
we haveuj→uandH(tj,uj)H(t,u).
Kartsatos and Skrypnik [] obtained the following result by using Browder’s degree in [] for multi-valued operators. As in [], we adopt property (P) in terms of the Brezis-Crandall-Pazy approximant so that we can apply the most essential degree theory of Skrypnik [] for single-valued operators to prove in a direct method.
Theorem . Letbe a bounded open set in X with∈.Let T:D(T)⊂X→X* be a maximal monotone multi-valued operator with∈D(T)and∈T().Let,s,andε
be three positive numbers.Suppose that C: [,]×→X*is demicontinuous,bounded
and satisfies condition(S+)such that C(,x) = for all x∈and C(t,x)is continuous in t
uniformly with respect to x∈.
(a) For a givenε> ,assume the following property:
(P)For everys∈(,s),there exists aλ∈(,]such that the equation
Tsx+C(λ,x) +εJx=
has no solution in.Then there exists a(λε,xε)∈(,]×(D(T)∩∂)such that
∈Txε+C(λε,xε) +εJxε.
(b) If /∈T(D(T)∩∂),T satisfies condition(Sq)onD(T)∩∂,and property(P)is fulfilled for allε∈(,ε],then there exists a(λ,x)∈(,]×(D(T)∩∂)such that
∈Tx+C(λ,x).
Proof (a) We first claim that for anys∈(,s), there exists a (λ,x)∈(,]×∂such
that
Tsx+C(λ,x) +εJx= . (.)
Assume on the contrary that for somes∈(,s) and for everyλ∈(,], the following
holds:
Since (Ts+εJ)() = andTs+εJis injective by the strict monotonicity ofJ, we haveTsx+
C(,x) +εJx= for allx∈∂. Thus, (.) holds for allλ∈[,]. Consider a homotopyH: [, ]×→X*defined by
H(t,x) =Tsx+C(t,x) +εJx for (t,x)∈[, ]×.
ThenHis of class (S+). To prove this, let{uj}be a sequence inwithujuand{tj}be
a sequence in [, ] withtj→tsuch that
lim sup j→∞
H(tj,uj),uj–u
≤. (.)
SinceTsandJare monotone, it follows from
H(tj,uj),uj–u
= Tsuj,uj–u+
C(tj,uj),uj–u
+εJuj,uj–u
that
H(tj,uj),uj–u
≥ Tsu,uj–u+
C(tj,uj),uj–u
+ε Juj,uj–u (.)
and
H(tj,uj),uj–u
≥ Tsu,uj–u+
C(tj,uj),uj–u
+ε Ju,uj–u. (.)
By (.) and (.), we have
lim sup j→∞
C(tj,uj),uj–u
≤. (.)
There are two cases to consider. Ift= , thenC(tj,uj)→ and so lim
j→∞
C(tj,uj),uj–u
= . (.)
Using (.), (.), and (.), we obtain
lim sup
j→∞ Juj,uj–u ≤.
SinceJsatisfies condition (S+), we have
uj→u,
which implies by the demicontinuity ofTsandCand the continuity ofJ
TsujTsu, C(tj,uj)C(,u), and Juj→Ju.
This means thatH(tj,uj)H(,u). Now, lett> . We have
C(tj,uj),uj–u
=C(tj,uj) –C(t,uj),uj–u
+C(t,uj),uj–u
which implies
lim sup j→∞
C(t,uj),uj–u
≤lim sup j→∞
C(tj,uj),uj–u
+lim sup j→∞
–C(tj,uj) –C(t,uj),uj–u
and hence by (.)
lim sup j→∞
C(t,uj),uj–u
≤ lim
j→∞C(tj,uj) –C(t,uj)uj–u= .
SinceCsatisfies condition (S+), we getuj→ufrom whichTsujTsu, C(tj,uj) C(t,u), andJuj→Ju. Consequently,H(tj,uj)H(t,u). We have just shown that
the homotopyHis of class (S+).
We are now ready to apply the degree theory of Skrypnik [, ]. IfdS denotes the Skrypnik degree, the homotopy invariance property ofdSimplies that
dSTs+C(λ,·) +εJ,, =dSH(t,·),,
=dS(Ts+εJ,, )
=
for allλ∈[,]. The last equality follows from Theorem in [], based on the fact thatTs+ εJis demicontinuous, bounded, injective, and satisfies condition (S+) and Tsx+εJx,x ≥
for allx∈∂. For everyλ∈(,], the existence property ofdSimplies that there is a point
xinsuch that
Tsx+C(λ,x) +εJx= ,
which contradicts property (P). Therefore, the first claim (.) is true.
In view of (.), let{sn}be a sequence in (,s) withsn→ and{(λn,xn)}be the
corre-sponding sequence in (,]×∂such that
Tsnxn+C(λn,xn) +εJxn= . (.)
Without loss of generality, we may suppose that
λn→λ, xnx, C(λn,xn)c* and Jxnj*,
whereλ∈[,],x∈X,c*∈X*andj*∈X*. To arrive at the conclusion of (a) in the next
step, we may suppose thatλ∈(,]. In fact, whenλ= , we know that /∈Tx+C(,x) +
εJxfor allx∈D(T)∩∂becauseT+εJis injective onD(T)∩and ∈(T+εJ)(D(T)∩). For our aim, we will now show that
lim sup n→∞
C(λn,xn),xn–x
Assume on the contrary that there exists a subsequence of{xn}, denoted again by{xn}, such that
lim n→∞
C(λn,xn),xn–x
> .
Note by (.) and the monotonicity ofJthat
Tsnxn,xn–x ≤–
C(λn,xn),xn–x
–ε Jx,xn–x
and so
lim sup
n→∞ Tsnxn,xn–x< .
Hence, it follows fromTsnxn–c*–εj*that
lim sup
n→∞ Tsnxn,xn=lim supn→∞
Tsnxn,xn–x+ Tsnxn,x
<–c*–εj*,x
. (.)
Let x∈D(T) andu*∈Txbe arbitrary. SinceT is monotone, T
snxn∈TJsnxn, andJsn=
I–snJ–Tsn, we have
Tsnxn–u
*,xn–snJ–Ts
nxn–x
≥,
which implies
Tsnxn,xn ≥snTsnxn
+ Ts
nxn,x+
u*,xn–x–snu*,J–Tsnxn
.
Since{Tsnxn}andJ–are bounded andsn→, we have
lim inf
n→∞ Tsnxn,xn ≥
–c*–εj*,x+u*,x–x
. (.)
Combining (.) and (.), we get
–c*–εj*–u*,x–x
> for allx∈D(T) andu*∈Tx. (.) SinceTis maximal monotone, we havex∈D(T) and –c*–εj*∈Tx. Lettingx=x∈D(T)
in (.) yields a contradiction. Thus, (.) holds.
SinceC(t,x) is continuous intuniformly with respect tox∈andλn→λ, we have by
(.)
lim sup n→∞
C(λ,xn),xn–x
≤, (.)
by observing that
C(λn,xn),xn–x
=C(λn,xn) –C(λ,xn),xn–x
+C(λ,xn),xn–x
SinceCsatisfies condition (S+), it follows from (.) that
xn→x∈∂,
which implies by (.)
Tsnxn–C(λ,x) –εJx
and as above
Jsnxn=xn–snJ
–Ts
nxn→x.
SinceTis maximal monotone andTsnxn∈TJsnxn, Lemma . implies thatx∈D(T) and
∈Tx+C(λ,x) +εJx.
(b) According to the statement (a), for a sequence{εn}in (,ε] withεn→, we can
choose sequences{λn}in (,],{xn}inD(T)∩∂, and{u*n}inX*withu*
n∈Txn such
that
u*n+C(λn,xn) +εnJxn= . (.)
We may suppose that
λn→λ, xnx, C(λn,xn)c* and Jxnj*,
where λ∈[,],x∈X,c*∈X*, andj*∈X*. Note thatλ∈(,]. Indeed, ifλ= ,
then (.) impliesu*
n→ which gives by condition (Sq) onD(T)∩∂,xn→x∈∂
and thereforex∈D(T) and ∈Tx, which contradicts the hypothesis that /∈T(D(T)∩
∂).
To show thatxn→x, we first claim that
lim sup n→∞
C(λn,xn),xn–x
≤. (.)
Assume the contrary. So, we may suppose that
lim n→∞
C(λn,xn),xn–x
> .
Hence, it follows from (.) thatu*
n–c*and lim sup
n→∞
u*n,xn–x
< ,
which imply
lim sup n→∞
u*
n,xn
<–c*,x
For everyx∈D(T) and everyu*∈Tx, we obtain from the monotonicity ofTthat
lim inf n→∞
u*n,xn≥lim inf n→∞
u*n,x+u*,xn–x ≥–c*,x+u*,x–x
,
which implies along with (.)
–c*–u*,x–x
> . (.)
SinceT is maximal monotone, we havex∈D(T) and –c*∈Tx. Lettingx=xin (.),
we have a contradiction. Thus, (.) is true. As above, we can deduce from (.) that
lim sup n→∞
C(λ,xn),xn–x
≤.
SinceCsatisfies condition (S+) and is demicontinuous, we obtain from (.) that
xn→x and u*n–C(λ,x).
We conclude thatx∈D(T)∩∂and
∈Tx+C(λ,x).
This completes the proof.
As a consequence of Theorem ., we have the following result. WhenCis a compact operator, it was proved by Li and Huang [] with the aid of the Leray-Schauder degree for compact operators.
Corollary . Let T,,,s,andεbe as in Theorem..Suppose that C:→X*is a
demicontinuous bounded operator which satisfies condition(S+).
(a) For a givenε> ,assume the following property:
(P)For everys∈(,s),there exists aλ∈(,]such that the equation
Tsx+λCx+εJx=
has no solution in.Then there exists a(λε,xε)∈(,]×(D(T)∩∂)such that
∈Txε+λεCxε+εJxε.
(b) If /∈T(D(T)∩∂),T satisfies condition(Sq)onD(T)∩∂,and property(P)is fulfilled for allε∈(,ε],then there exists a(λ,x)∈(,]×(D(T)∩∂)such that
Proof Define an operatorC˜ : [,]×→X*by
˜
C(λ,x) =λCx for (λ,x)∈[,]×.
By hypotheses onC, the operatorC˜ is obviously demicontinuous, bounded, and satisfies condition (S+). Moreover,C˜(t,x) is continuous intuniformly with respect tox∈
be-causeC() is bounded. Apply Theorem . withC=C˜.
3 Implicit eigenvalue problem about densely defined operators
In this section, we study the implicit eigenvalue problem for densely defined perturbations of maximal monotone operators, based on the degree theories of Kartsatos and Skrypnik. An operatorC: [,]×D(C)→X*is said to beuniformly quasiboundedif for every
S> there exists a constantK(S) > such that for allλ∈[,] and allu∈D(C) with
u ≤Sand C(λ,u),u ≤, we haveC(λ,u) ≤K(S).
In a regularization method by the duality operator, we establish a new result on the existence of eigenvalues, by applying topological degree for densely defined operators in [, ].
Theorem . Letbe a bounded open set in X with∈and L a dense subspace of X.
Let T:D(T)⊂X→X*be a maximal monotone multi-valued operator with∈D(T)and
∈T().Let,s,andεbe positive numbers.Assume that C: [,]×D(C)→X*is a
single-valued operator with L⊂D(C)⊂X such that C(,x) = for all x∈D(C)and C(t,x)
is continuous in t uniformly with respect to x∈D(C).Assume further that
(c) Cis uniformly quasibounded, (c) Csatisfies condition(S˜+),and
(c) for everyλ∈[,]and for everyF∈F(L)andv∈L,the functionc(λ,F,v) :F→R,
c(λ,F,v)(u) = C(λ,u),v,is continuous onF,whereF(L)denotes the set of all finite-dimensional subspaces ofL.
Then the following statements hold:
(a) For a givenε> ,assume the following property:
(P)For everys∈(,s),there exists aλ∈(,]such that the equation
Tsx+C(λ,x) +εJψx=
has no solution inD(C)∩.Then there is a(λε,xε)∈(,]×(D(T)∩D(C)∩∂)
such that
∈Txε+C(λε,xε) +εJψxε.
(b) If /∈T(D(T)∩∂),T satisfies condition(Sq)onD(T)∩∂,and property(P)is fulfilled for allε∈(,ε],then there exists a(λ,x)∈(,]×(D(T)∩D(C)∩∂)
such that
Proof (a) First, we prove that for everys∈(,s), there exists a (λ,x)∈(,]×(D(C)∩
∂) such that
Tsx+C(λ,x) +εJψx= . (.)
Assume on the contrary that for somes∈(,s) and for everyλ∈(,], the following
holds:
Tsx+C(λ,x) +εJψx= for allx∈D(C)∩∂. (.)
Then (.) holds for allλ∈[,]. Forλ= , the assertion is obvious. Fort∈[, ], we setTt=T
sandCt=C(t,·) +εJψ. To show that{Tt+Ct}is an
ad-missible homotopy in the sense of Definition . in [], we have to check the following conditions on two families {Tt} and{Ct}. In fact, conditions on{Tt} are automatically
satisfied, withTtindependent oft, due to the monotonicity ofTsandTs() = . (ct
){Ct}is uniformly strongly quasibounded with respect toTt,i.e., for every > there
exists a constantK( ) > such that for allu∈Lwithu ≤ and allt∈[, ],
Ttu+Ctu,u≤ implies Ctu≤K( ).
This follows trivially from (c) and the fact thatTsandJψare monotone andJψis bounded.
(ct
) For every pair of sequences{tj}in [, ] and{uj}inLsuch thattj→t,uju,
Ctju
jh*and
lim sup j→∞
Ctju
j,uj–u
≤, Ttju
j+Ctjuj,uj
≤, (.)
wheret∈[, ],u∈X, andh*∈X*, we haveuj→u,u∈D(C) andCtu=h*.
For this, there are two cases to consider. Ift= , then the second inequality in (.)
implies
εψujuj=ε Jψuj,uj ≤ Tsuj,uj+ε Jψuj,uj ≤–
C(tj,uj),uj→
and henceuj→,u= ∈D(C), andCu= =h*. Now lett> . From the first
in-equality in (.) it follows that
Ctjuj,uj–u
≥C(tj,uj),uj–u
+ε Jψu,uj–u
and so
lim sup j→∞
C(tj,uj),uj–u
≤.
Combining this with
C(tj,uj),uj–u
=C(tj,uj) –C(t,uj),uj–u
+C(t,uj),uj–u
we have
lim sup j→∞
C(t,uj),uj–u
≤lim sup j→∞
–C(tj,uj) –C(t,uj),uj–u
≤ lim
j→∞C(tj,uj) –C(t,uj)uj–u= .
In view ofCtju
jh*, we can find a subsequence of{uj}, denoted again by{uj}, such that
C(tj,uj)h* andJψujh*for someh*,h*∈X*. Note thatC(t,uj)h* andh*=
h*
+εh*. Hence (c) implies thatuj→u,u∈D(C), andC(t,u) =h*and soCtu=
C(t,u) +εJψu=h*. Thus, condition (ct) is satisfied in both cases.
(ct) For every F∈F(L) andv∈L, the function˜c(F,v) : [, ]×F→R,c˜(F,v)(t,u) =
Ctu,v, is continuous.
Actually,˜c(F,v) is continuous on [, ]×Fbecausec(t,F,v) in the notation of (c) is continuous onFandJψis continuous onX. Consequently, we have shown that{Tt+Ct},
t∈[, ], is an admissible homotopy.
Following Kartsatos and Skrypnik [, ], we can use the homotopy invariance property of the degreedwith respect to the bounded open setas follows:
dTs+C(λ,·) +εJψ,,
=d(Ts+εJψ,, )
=
for allλ∈[,]. The latter follows from Theorem in [], noticing thatTs+εJψ is
demi-continuous, bounded, strictly monotone and satisfies condition (S+). For everyλ∈(,],
the existence property of the degree implies that
Tsx+C(λ,x) +εJψx= for somex∈D(C)∩,
which contradicts property (P). Therefore, the first assertion (.) is true.
According to assertion (.), let{sn}be a sequence in (,s) withsn→ and{(λn,xn)}
be a sequence in (,]×(D(C)∩∂) such that
Tsnxn+C(λn,xn) +εJψxn= . (.)
Without loss of generality, we may suppose that
λn→λ, xnx, C(λn,xn)c* and Jψxnj*,
whereλ∈[,],x∈X,c*∈X*andj*∈X*. We may consider the case thatλ∈(,]
as the conclusion of (a) does not hold forλ= . In order to apply the condition (S˜+), we
first show that
lim sup n→∞
C(λn,xn),xn–x
≤. (.)
Assume on the contrary that there exists a subsequence of{xn}, denoted again by{xn}, such that
lim n→∞
C(λn,xn),xn–x
Note by (.) and the monotonicity ofJψ that
Tsnxn,xn–x ≤–
C(λn,xn),xn–x
–ε Jψx,xn–x
and so
lim sup
n→∞ Tsnxn,xn–x< .
Hence, it follows fromTsnxn–c*–εj*that
lim sup
n→∞ Tsnxn,xn<
–c*–εj*,x
. (.)
For everyx∈D(T) and everyu*∈Tx, it is shown as in the proof of part (a) of Theorem . that
lim inf
n→∞ Tsnxn,xn ≥
–c*–εj*,x+u*,x–x
,
which implies in view of (.)
–c*–εj*–u*,x–x
> . (.)
SinceT is maximal monotone, we havex∈D(T) and –c*–εj*∈Tx. Lettingx=xin
(.), we have a contradiction. Thus, (.) holds. As before, we can deduce from (.) that
lim sup n→∞
C(λ,xn),xn–x
≤.
SinceCsatisfies condition (S˜+), we have
xn→x, x∈D(C), and C(λ,x) =c*. (.)
Combining (.) and (.), we obtain
Tsnxn–C(λ,x) –εJψx
and
Jsnxn→x.
SinceT is maximal monotone and Tsnxn∈TJsnxn, Lemma . implies thatx∈D(T)∩
D(C)∩∂and
(b) In view of (a), for a sequence{εn}in (,ε] withεn→, we take sequences{λn}in
(,],{xn}inD(T)∩D(C)∩∂and{u*
n}inX*withu*n∈Txnsuch that
u*n+C(λn,xn) +εnJψxn= . (.)
We may suppose that
λn→λ, xnx, C(λn,xn)c* and Jψxnj*,
whereλ∈[,],x∈X,c*∈X*andj*∈X*. Note thatλ∈(,]. As in the proof of part
(b) of Theorem ., a similar argument shows that
lim sup n→∞
C(λn,xn),xn–x
≤ (.)
and so in a usual way
lim sup n→∞
C(λ,xn),xn–x
≤.
Hence, it follows from (c) that
xn→x∈∂, x∈D(C), and C(λ,x) =c*
which implies by (.)
u*n–C(λ,x).
Consequently, we obtain from the maximal monotonicity ofT thatx∈D(T) and
∈Tx+C(λ,x).
This completes the proof.
Remark . In an analogous way to Theorem ., we can observe the eigenvalue problem ∈Tx+λCxfor quasibounded perturbations of maximal monotone operators; see [].
Competing interests
The author declares that she has no competing interests.
Acknowledgements
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2011-0021-829).
Received: 24 March 2012 Accepted: 3 October 2012 Published: 17 October 2012 References
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doi:10.1186/1687-1812-2012-178
