R E S E A R C H
Open Access
Some eigenvalue results for perturbations of
maximal monotone operators
In-Sook Kim
*and Jung-Hyun Bae
*Correspondence: [email protected]
Department of Mathematics, Sungkyunkwan University, Suwon, 440-746, Republic of Korea
Abstract
We study the solvability of a nonlinear eigenvalue problem for maximal monotone operators under a normalization observation. The investigation is based on degree theories for appropriate classes of operators, and a regularization method by the duality operator is used. LetXbe a real reflexive Banach space with its dualX*and
be a bounded open set inX. Suppose thatT:D(T)⊂X→X*is a maximal monotone operator andC: (0,∞)×
→X*is a bounded demicontinuous operator satisfying condition (S+). Applying the Browder degree theory, we solve a nonlinear eigenvalue problem of the formTx+C(
λ
,x) = 0. In the case whereTx+λCx
= 0, an eigenvalue result for generalized pseudomonotone densely defined perturbations is obtained by the Kartsatos-Skrypnik degree theory.MSC: 47J10; 47H05; 47H14; 47H11
Keywords: eigenvalue; maximal monotone operator; perturbation; degree theory
1 Introduction and preliminaries
Eigenvalue theory is closely related to the problem of solving nonlinear equations which was initiated by Krasnosel’skii [] for compact operators. Regarding maximal monotone operators, it has been extensively investigated by many researchers in various aspects, with applications to evolution equations and differential equations; see,e.g., [–]. The study was mostly based on degree theories for appropriate classes of operators and the usual method of regularization by means of the duality operator; see [–].
LetXbe a real reflexive Banach space with its dualX*andbe a bounded open set
inX. Suppose thatT:D(T)⊂X→X*is a maximal monotone operator. We consider the
nonlinear eigenvalue problem
Tx+λCx= , (E)
whereC:D(C)⊂X→X*is an operator. When the operatorCor the resolvents of the
operatorT are compact, this problem was studied, for instance, by Guan-Kartsatos [], Kartsatos [], and Li-Huang [], where the method is to use the Leray-Schauder degree theory. More generally, an implicit eigenvalue problem of the form
Tx+C(λ,x) = (IE)
was investigated in [, ], whereC: (,∞)×→X*is a bounded operator to be specified
later. Kartsatos and Skrypnik [] observed the above eigenvalue problem provided that the
following property (P) is fulfilled: Forε> , there exists aλ> such that the equation
Tx+C(λ,x) +εJx=
has no solution inD(T)∩, whereJ denotes the duality operator. This property has a close relation to the use of topological degree in eigenvalue theory by the regularization method. It is shown in [] that two conditions about the weak closure of a certain set con-sisting of normalized vectors and the asymptotic behavior of the operatorCat infinity of λ, callednormalized conditions, are main ingredients in solving an eigenvalue problem. In this connection, we are now interested in finding eigenvectors under normalized condi-tions more concrete than property (P).
The purpose of this paper is to establish the existence of solutions for the above eigen-value problems under normalized conditions, motivated by the works of Kartsatos and Skrypnik [, ]. We first study implicit eigenvalue problem (IE), whereCis assumed to be a bounded demicontinuous operator satisfying condition (S+). For this, a key tool is the Browder degree given in []. Next, we consider two types of the operatorCfor eigenvalue problem (E). For the one, we apply the Browder degree theory for nonlinear operators of the formT+f withTmaximal monotone andf bounded with condition (S+) introduced
in []. In the other case, whereCis a generalized pseudomonotone, quasibounded, and densely defined operator, we solve this problem by using the Kartsatos-Skrypnik degree theory for densely defined (S+˜ )-perturbations of maximal monotone operators developed in [].
This paper is organized as follows. In Section , we study the solvability of implicit eigen-value problem (IE) with normalized conditions based on the Browder degree theory. Sec-tion contains an eigenvalue result for problem (E) as a special case of (IE). In SecSec-tion , we deal with eigenvalue problem (E) for densely defined perturbations of maximal mono-tone operators under normalized conditions.
LetXbe a real Banach space with dual spaceX*,be a nonempty subset ofX, and Y be another real Banach space. Let,int, and ∂denote the closure, the interior, and the boundary ofinX, respectively. The symbol→() stands for strong (weak) convergence. An operatorF:→Y is said to beboundedifFmaps bounded subsets of into bounded subsets ofY.Fis said to bedemicontinuousif for everyx∈and for
every sequence{xn}inwithxn→x, we haveFxnFx.
LetT:D(T)⊂X→X*be an operator. ThenTis said to bemonotoneif
Tx–Ty,x–y ≥ for everyx,y∈D(T).
Tis said to bemaximal monotoneif it is monotone and it follows from (x,x*)∈X×X*and
x*–Ty,x–y≥ for everyy∈D(T)
thatx∈D(T) andTx=x*.
Tis said to begeneralized pseudomonotoneif for every sequence{xn}inD(T) withxn x,Txnh*and
lim sup
n→∞
Txn,xn–x ≤,
we havex∈D(T),Tx=h*, andlim
We say thatT 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 thatT satisfies condition (S+) if for every sequence{xn}inD(T) withxnx
and
lim sup
n→∞
Txn,xn–x ≤,
we havexn→x.
We say thatTsatisfies condition (S) on a setM⊂D(T) if for every sequence{xn}inM
withxnxandTxnh*and
lim
n→∞Txn,xn=
h*,x,
we havexn→x.
We say thatTsatisfies condition (Sq) on a setM⊂D(T) if for every sequence{xn}inM
withxnxandTxn→h*, we havexn→x.
We say thatT satisfies condition (T∞()) on a setM⊂D(T) if for every sequence{xn}in MwithTxn → ∞, we haveTxn–Txn.
It is obvious from the definitions that (S+) implies (S) and (S) implies (Sq). Note that if T satisfies condition (S˜+) andXis reflexive, thenT is generalized pseudomonotone. For
the above definitions, we refer to,e.g., [, , ], and [].
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.
Throughout this paper,Xwill always be an infinite dimensional real reflexive Banach space which has been renormed so thatXandX*are locally uniformly convex.
An operatorJψ:X→X*is said to be aduality operatorif
Jψx,x=ψxx and Jψx=ψx forx∈X,
whereψ: [,∞)→[,∞) is continuous, strictly increasing,ψ() = , andψ(t)→ ∞as
t→ ∞called agauge function. Ifψis the identity mapI, thenJ:=JIis called anormalized duality operator. It is known in [] thatJψ is continuous, bounded, surjective, strictly monotone, maximal monotone and satisfies condition (S+).
Lemma . Let T:D(T)⊂X→X*be a maximal monotone operator.Then for every se-quence{xn}in D(T),xn→x in X and Txnx*in X*imply that x∈D(T)and Tx=x*.
2 Implicit eigenvalue problem
In this section, we establish the existence of a solution for a nonlinear implicit eigenvalue problem under normalized conditions by using the Browder degree theory for class (S+).
Recall that a mappingH: [, ]×→X*is of class (S+) if the following condition holds:
For any sequence{uj}inwithujuand any sequence{tj}in [, ] withtj→tfor
which
lim sup
j→∞
H(tj,uj),uj–u
≤,
we haveuj→uandH(tj,uj)H(t,u); see [, ].
As mentioned in the introduction, Kartsatos and Skrypnik [] gave the following result provided that property (P) is fulfilled. In a more concrete situation, we adopt the normal-ization method considered in [].
Theorem . Letbe a bounded open set in X with∈.Let T:D(T)⊂X→X*be a maximal monotone operator such that the closure ofis included in the interior of D(T)
with T() = and T satisfies condition(T∞())on.Assume 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∈.Further assume that
(c) There exists a positive numberN such that the weak sequential closure of the set
G=
C(λ,x)
C(λ,x) :λ≥N,x∈,Jx+Tx ≤M(λ)
does not contain the origin,where
M(λ) =sup C(λ,x) :x∈.
(c) limλ→∞m(λ) =∞,wherem(λ) =inf{C(λ,x):x∈}.
Then the following statements hold:
(a) For eachε> ,there exists a point(λε,xε)in(,∞)×∂such that
Txε+C(λε,xε) +εJxε= .
(b) If /∈T(∂)andT satisfies condition(Sq)on∂,then the implicit eigenvalue
problem
Tx+C(λ,x) =
has a solution(λ,x)in(,∞)×∂.
Proof (a) For our aim, we use the Browder degreedBgiven in []. Letεbe any positive
number. We first prove that there is a number∈(,∞) such that
dB
Assume the contrary. For a sequence{n}in (,∞) withn→ ∞, the following occurs:
For eachn∈N, either there exists a pointxn∈such thatTxn+C(n,xn) +εJxn= ,
in view ofdB(T+C(n,·) +εJ,, )= , or there exists a pointxn∈∂such thatTxn+ C(n,xn) +εJxn= . Thus, we get a sequence{xn}insuch that
Txn+C(n,xn) +εJxn= . (.)
This implies
Jxn+Txn ≤ ( –ε)Jxn + C(n,xn) ≤M(n)
for sufficiently largen. Since the sequence{C(n,xn)–C(n,xn)}is bounded in the
re-flexive Banach spaceX*, we may suppose thatC(
n,xn)–C(n,xn) converges weakly to
someh∈X*. Using (c), it is clear thath= . It follows from (.) that
Txn Txn
–h. (.)
However,C(n,xn) → ∞by (c) impliesTxn → ∞, which is a contradiction to
con-dition (T∞()). Hence assertion (.) holds.
Next, we consider a mappingH: [, ]×→X*given by
H(t,x) =Tx+C(t,x) +εJx for (t,x)∈[, ]×.
ThenHis of class (S+). To prove this, let{uj}be any sequence inwithujuand{tj}
be any sequence in [, ] withtj→tsuch that
lim sup
j→∞
H(tj,uj),uj–u
≤. (.)
Since the operatorsT andJare monotone, it follows from
H(tj,uj),uj–u
=Tuj,uj–u+
C(tj,uj),uj–u
+εJuj,uj–u (.)
that
H(tj,uj),uj–u
≥ Tu,uj–u+
C(tj,uj),uj–u
+εJu,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
Since (.) implies
H(tj,uj),uj–u
≥ Tu,uj–u+
C(tj,uj),uj–u
+εJuj,uj–u,
it follows from (.) and (.) that
lim sup
j→∞ Juj,uj–u ≤.
SinceJsatisfies condition (S+), we obtain
uj→u,
which implies
TujTu, C(tj,uj)C(,u), and Juj→Ju
on observing thatTis demicontinuous on. This means thatH(tj,uj)H(,u). Ift>
, we have
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 sup
j→∞
C(tj,uj) –C(t,uj) uj–u
= .
Since C satisfies condition (S+), we getuj→u from which TujTu, C(tj,uj) C(t,u), andJuj→Ju. Consequently,H(tj,uj)H(t,u). We have just shown that
the mappingHis of class (S+).
We are now ready to apply the degree theory of Browder [, ]. Then we have
dB
H(,·),, =dB(T+εJ,, ) = .
The last equality is based on Theorem in [] because the operatorT+εJis strictly mono-tone and demicontinuous onand satisfies condition (S+). On the other hand, it is shown in (.) that
dB
H(,·),, =dB
T+C(,·) +εJ,, = .
Hence, in view of Theorem in [], there existt∈[, ] andx∈∂such that
Tx+C(t,x) +εJx= .
(b) Let{εn}be a sequence in (,∞) such thatεn→. According to statement (a), there
exists a sequence{(λεn,xεn)}in (,∞)×∂such that
Txεn+C(λεn,xεn) +εnJxεn= .
If we setxn:=xεnandλn:=λεn, it can be written in the form
Txn+C(λn,xn) +εnJxn= . (.)
Without loss of generality, we may suppose that
λn→λ, xnx, and C(λn,xn)c*, (.)
whereλ∈[,∞],x∈X, andc*∈X*. Note thatλbelongs to (,∞). In fact, ifλ= ,
thenC(λn,xn)→ impliesTxn→. SinceTsatisfies condition (Sq) on∂, we obtain that xn→x∈∂and thereforeTx= , which contradicts the hypothesis that /∈T(∂). If
λ=∞, then (c) impliesC(λn,xn) → ∞and soTxn → ∞. As in (.), a similar
argu-ment proves thatTxn–Txnconverges weakly to some nonzero vector, which contradicts
condition (T∞()). Thus we have shown thatλ∈(,∞).
For the next aim, we now show that
lim sup
n→∞
C(λn,xn),xn–x
≤. (.)
Assume that (.) is false. Then there exists a subsequence of{xn}, denoted again by{xn},
such that
lim
n→∞
C(λn,xn),xn–x
> .
Hence we obtain from (.) that
lim sup
n→∞ Txn,xn–x< .
Noticing by (.) and (.) thatTxn–c*, we get
lim sup
n→∞ Txn,xn<nlim→∞Txn,x=
–c*,x. (.)
For everyx∈D(T), we have by the monotonicity ofT
lim inf
n→∞ Txn,xn ≥lim infn→∞
Txn,x+Tx,xn–x
=–c*,x+Tx,x–x,
which implies along with (.)
–c*–Tx,x–x> . (.)
By the maximal monotonicity ofT, we havex∈D(T) andTx= –c*. Lettingx=xin
SinceC(λn,xn) –C(λ,xn)→, it follows from (.) and
C(λn,xn),xn–x
=C(λn,xn) –C(λ,xn),xn–x
+C(λ,xn),xn–x
that
lim sup
n→∞
C(λ,xn),xn–x
≤.
SinceCsatisfies condition (S+) andλ∈(,∞), we havexn→x∈∂, which implies C(λn,xn)C(λ,x). Hence we obtain from (.) thatTxn–C(λ,x). By Lemma .,
we conclude thatTx+C(λ,x) = . This completes the proof.
Remark . In the proof of Theorem ., the demicontinuity ofTonis needed to show that H is of class (S+). This is guaranteed under an additional condition ⊂intD(T).
Actually, local boundedness ofTonintD(T) implies the demicontinuity ofT; see,e.g., [].
3 Eigenvalue problem with condition (S+)
In this section, we study a multiplicative eigenvalue problem as a special case of the im-plicit eigenvalue problem in the previous section. As a key tool, we employ the Browder degree for nonlinear operators of the formT+f withTmaximal monotone andfbounded with condition (S+).
We give a variant of Corollary in [] under normalized conditions.
Theorem . Letbe a bounded open set in X with∈.Let T:D(T)⊂X→X*be a maximal monotone operator with∈D(T)and T() = such that T satisfies condition
(T∞())on D(T)∩.Assume that C:→X*is a demicontinuous bounded operator which satisfies condition(S+)and the two additional conditions:
(c) There is a positive numberN such that the weak sequential closure of the set
G=
Cx
Cx:λ≥N,x∈,Jx+Tx ≤M(λ)
does not contain zero vector,where
M(λ) =|λ|supCx:x∈.
(c) inf{Cx:x∈}is not equal to. Then we have the following properties:
(a) For eachε> ,there exists(λε,xε)∈(,∞)×(D(T)∩∂)such that
Txε+λεCxε+εJxε= .
(b) If /∈T(D(T)∩∂)andTsatisfies condition(Sq)onD(T)∩∂,then the eigenvalue
problem
Tx+λCx=
Proof (a) Fixε> . LetdB denote the Browder degree in the sense of []. We first claim
that there exists a numberin (,∞) such that
dB(T+C+εJ,, ) = . (.)
Assume the contrary. As in the proof of (.), a similar argument establishes that for a sequence{n}in (,∞) withn→ ∞, there is a sequence{xn}inD(T)∩such that
Txn+nCxn+εJxn= . (.)
This implies
Jxn+Txn ≤ ( –ε)Jxn +nCxn ≤M(n)
for sufficiently largen. By the boundedness of the sequence{Cxn–Cxn}inX*, we may
suppose thatCxn–Cxnhfor someh∈X*. It follows from (c) and (.) thath=
and
Txn Txn
–h.
But (c) impliesTxn → ∞, which contradicts condition (T∞()). Hence assertion (.)
holds.
Now we consider a mappingH: [, ]×(D(T)∩)→X*given by
H(t,x) :=Tx+εJx+tCx for (t,x)∈[, ]×D(T)∩.
Using the normalization property of the Browder degreedB,e.g., Theorem in [], we
have
dB
H(,·),, =dB(T+εJ,, ) = .
Moreover, (.) means that
dB
H(,·),, =dB(T+εJ+C,, ) = .
Note thatT+εJis maximal monotone and satisfies condition (S+). Hence, there aret∈
[, ] andx∈D(T)∩∂such that
Tx+tCx+εJx= .
By the injectivity of the strictly monotone operatorT +εJ, we know thatt> .
Conse-quently, if we letλε:=tandxε:=x, thenλε∈(,∞) andTxε+λεCxε+εJxε= .
(b) Let{εn} be a sequence in (,∞) such thatεn→. By (a), there exists a sequence {(λn,xn)}in (,∞)×(D(T)∩∂) such that
We may suppose that
λn→λ, xnx, and Cxnc*, (.)
whereλ∈[,∞],x∈X, andc*∈X*. Note thatλbelongs to (,∞). Indeed, ifλ= ,
then it follows fromTxn→ and condition (Sq) thatxn→x∈∂and thereforex∈ D(T) andTx= , which contradicts the hypothesis that /∈T(D(T)∩∂). Ifλ=∞, then
(c) impliesλnCxn → ∞and soTxn → ∞. But we can show as above thatTxn–Txn
converges weakly to some nonzero vector, which contradicts condition (T∞()). To prove that
lim sup
n→∞
λnCxn,xn–x ≤, (.)
we assume to the contrary that there exists a subsequence of{xn}, denoted again by{xn},
such that
lim
n→∞λnCxn,xn–x> .
Hence we obtain from (.) that
lim sup
n→∞
Txn,xn–x< .
Noticing by (.) and (.) thatTxn–λc*, we get
lim sup
n→∞ Txn,xn<nlim→∞Txn,x=
–λc*,x. (.)
For everyx∈D(T), we have by the monotonicity ofT
lim inf
n→∞ Txn,xn ≥
–λc*,x+Tx,x–x,
which implies along with (.)
–λc*–Tx,x–x> .
By the maximal monotonicity ofT, we get a contradiction. Therefore, (.) holds. It follows from (.) and
λnCxn,xn–x= (λn–λ)Cxn,xn–x+λCxn,xn–x
that
λlim sup n→∞
Cxn,xn–x ≤.
SinceC satisfies condition (S+) and is demicontinuous, we havexn→x∈∂, which
impliesCxnCx. Hence we obtain from (.) thatTxn–λCx. By Lemma ., we
Remark . We point out that in Theorem ., the condition⊂intD(T) is not neces-sary to be assumed.
4 Densely defined perturbations
This section is devoted to the eigenvalue problem for densely defined quasibounded per-turbations of maximal monotone operators in reflexive Banach spaces. To do this, we apply the Kartsatos-Skrypnik degree for densely defined (S˜+)-perturbations of maximal
mono-tone operators developed in [].
Recall that an operator C:D(C)⊂X→X* isquasiboundedif for every S> there
exists a constantK(S) > such that for allu∈D(C) withu ≤SandCu,u ≤, we have
Cu ≤K(S).
As in Section , we employ a normalization method to obtain an eigenvalue result for generalized pseudomonotone operators.
Theorem . Letbe a bounded open set in X with∈and L be a dense subspace
of X.Let T:D(T)⊂X→X*be a maximal monotone operator with∈D(T)and T() =
which satisfies condition(T∞())on D(T)∩.Assume that C:D(C)⊂X→X*is a
general-ized pseudomonotone quasibounded operator with L⊂D(C).Furthermore,assume that (h) There exists a positive numberN such that the weak sequential closure of the set
G=
Cx
Cx:λ≥N,x∈D(C)∩,Jψx+Tx ≤M(λ)
does not contain zero vector,where
M(λ) =|λ|supCx:x∈D(C)∩.
(h) inf{Cx:x∈D(C)∩}is not equal to.
(h) For everyF∈F(L)andv∈L,the functionc(F,v) :F→R,c(F,v)(u) =Cu,v,is continuous onF,whereF(L)denotes the set of all finite-dimensional subspaces ofL. Then the following statements hold:
(a) For eachε> ,there exists a point(λε,xε)in(,∞)×(D(T+C)∩∂)such that
Txε+λεCxε+εJψxε= .
HereD(T+C)denotes the intersection ofD(T)andD(C).
(b) If /∈T(D(T)∩∂)andTsatisfies condition(S)onD(T)∩∂,then the eigenvalue problem
Tx+λCx=
has a solution(λ,x)in(,∞)×(D(T+C)∩∂).
Proof (a) We will apply the Kartsatos-Skrypnik degreedSgiven in []. Letεbe an arbitrary
such that
dS(T+C+εJψ,, ) = . (.)
Fort∈[, ], we setTt:=TandCt:=tC+εJψ, whereD(Tt) andD(Ct) denote the
do-main ofTt andCt, respectively. In this case,D(Tt) =D(T) fort∈[, ],D(C) =Xfor
t= andD(Ct) =D(C) fort∈(, ]. Notice that the operatorsC=εJψandC=C+εJψ
satisfy condition (S˜+), based on the facts thatCis generalized pseudomonotone andJψis
bounded and satisfies condition (S+).
In the sense of Definition . in [], we check the following conditions on two families
{Tt}and{Ct}. In fact, conditions on{Tt}are obviously satisfied, withTt independent of t, due to maximal monotonicity ofT, ∈D(T), andT() = .
(ct) SinceJψis monotone and bounded, it follows from the quasiboundedness ofCthat
{Ct}is uniformly quasibounded.
(ct) Let{tn}be any sequence in [, ] and{un}be any sequence inLsuch thattn→t, unu,Ctnunh*and
lim sup
n→∞
Ctnu
n,un–u
≤, Ctnu n,un
≤, (.)
wheret∈[, ],u∈X, andh*∈X*. Ift= , then the second inequality in(.) implies
εψun
un=εJψun,un ≤–tnCun,un →
and henceun→,u= ∈D(C), andCu=h*. Now lett> . From the first inequality
in (.) and the following inequality
Ctnu
n,un–u
≥tnCun,un–u+εJψu,un–u,
we obtain that
lim sup
n→∞ Cun,un–u ≤. (.)
In view ofCtnu
nh*, there exists a subsequence of{un}, denoted again by{un}, such that Cunh*andJψunh*for someh*,h*∈X*. SinceCis generalized pseudomonotone,
we obtain from (.) thatu∈D(C),Cu=h*, andCun,un → Cu,u. Thus,
lim
n→∞Cun,un–u=Cu,u–
h*,u= .
Hence it follows from the first inequality in (.) that
εlim sup
n→∞ Jψun,un–u=lim supn→∞ tnCun+εJψun,un–u ≤.
SinceJψ satisfies condition (S+), we haveun→uand soJψun→Jψu. Consequently, u∈D(Ct) andCtu=th*
(ct) For every F∈F(L) andv∈L, the function˜c(F,v) : [, ]×F→R,c˜(F,v)(t,u) =
Ctu,v, is continuous on [, ]×Fbecausec(F,v) is continuous onFandJψis continuous
onX.
We can now consider a mappingH: [, ]×(D(T+C)∩)→X*given by
H(t,x) :=Tx+tCx+εJψx.
By Theorem . in [] and (.), we have
dS
H(,·),, =dS(T+εJψ,, ) =
and
dS
H(,·),, =dS(T+C+εJψ,, ) = .
According to Theorem . in [], there existt∈[, ] andx∈D(T+C)∩∂such that
Tx+tCx+εJψx= .
The injectivity ofT +εJψ implies thatt> . If we let λε:=tandxε:=x, then the
conclusion follows.
(b) Let{εn} be a sequence in (,∞) such thatεn→. According to (a), there are
se-quences{λn}in (,∞) and{xn}inD(T+C)∩∂such that
Txn+λnCxn+εnJψxn= . (.)
Then it follows from the monotonicity ofTwithT() = that
Cxn,xn ≤–
εn
λn
ψxn
xn ≤.
Hence the quasiboundedness ofCimplies that{Cxn}is bounded. Without loss of
gener-ality, we may suppose that
λn→λ, xnx, and Cxnc*, (.)
whereλ∈[,∞],x∈X, andc*∈X*. Since T satisfies conditions (Sq) and (T∞()) and since
/∈T(D(T)∩∂), it is easily verified thatλbelongs to (,∞).
As in the proof of Theorem ., we can show that
lim sup
n→∞
λnCxn,xn–x ≤. (.)
Then it follows from (.) andλn→λthat
lim sup
n→∞
SinceCis generalized pseudomonotone, we have by (.)
x∈D(C), Cx=c*, and lim
n→∞Cxn,xn=Cx,x.
Hence we obtain from (.) thatTxn–λCxand
lim
n→∞Txn,xn=nlim→∞–λnCxn,xn=–λCx,x.
SinceT satisfies condition (S) onD(T)∩∂, we havexn→x. By the maximal
mono-tonicity ofT, we conclude thatx∈D(T+C)∩∂andTx+λCx= . This completes
the proof.
As a consequence of Theorem ., we get another eigenvalue result for densely defined operators satisfying condition (S+˜ ) in comparison with Theorem in [].
Corollary . Let T,,and L be as in Theorem..Assume that C:D(C)⊂X→X*is a quasibounded operator with L⊂D(C)and satisfies condition(S+˜ ).Furthermore,assume that conditions(h), (h),and(h)in Theorem.are satisfied.Then:
(a) For eachε> ,there exists(λε,xε)∈(,∞)×(D(T+C)∩∂)such that Txε+λεCxε+εJψxε= .
(b) If /∈T(D(T)∩∂)andTsatisfies condition(Sq)onD(T)∩∂,then there exists
(λ,x)∈(,∞)×(D(T+C)∩∂)such thatTx+λCx= .
Proof Statement (a) follows from part (a) of Theorem . by noting that ifCsatisfies con-dition (S˜+), thenCis generalized pseudomonotone.
(b) Let{εn}be a sequence in (,∞) such thatεn→. In view of (a), we can choose a
sequence{(λn,xn)}in (,∞)×(D(T+C)∩∂) such that
Txn+λnCxn+εnJψxn= .
We may suppose that
λn→λ, xnx, and Cxnc*,
whereλ∈[,∞],x∈X, andc*∈X*. Obviously,λ∈(,∞). As before, the same
argu-ment shows that
lim sup
n→∞
Cxn,xn–x ≤.
SinceCsatisfies condition (S+˜ ), we have
xn→x, x∈D(C), and Cx=c*.
Combining this withTxn–λCx, we obtain thatx∈D(T+C)∩∂andTx+λCx=
as required.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
BJ participated in the sequence alignment and coordination. KI conceived of the study and drafted the manuscript. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank anonymous referees for valuable comments and suggestions. This research was supported by the 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: 19 September 2012 Accepted: 9 February 2013 Published: 28 February 2013
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doi:10.1186/1029-242X-2013-72
