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# Implicit eigenvalue problems for maximal monotone operators

- In-Sook Kim
^{1}Email author

**2012**:178

https://doi.org/10.1186/1687-1812-2012-178

© Kim; licensee Springer 2012

**Received:**24 March 2012**Accepted:**3 October 2012**Published:**17 October 2012

## Abstract

We study the implicit eigenvalue problem of the form

where *T* is a maximal monotone multi-valued operator and the operator *C* satisfies condition (${S}_{+}$) or (${\tilde{S}}_{+}$). In a regularization method by the duality operator, we use the degree theories of Kartsatos and Skrypnik upon conditions of *C* as well as Browder’s degree. There are two cases to consider: One is that *C* is demicontinuous and bounded with condition (${S}_{+}$); and the other is that *C* is quasibounded and densely defined with condition (${\tilde{S}}_{+}$). Moreover, the eigenvalue problem $0\in Tx+\lambda Cx$ is also discussed.

## Keywords

- Satisfy Condition
- Compact Operator
- Regularization Method
- Maximal Monotone
- Real Banach Space

## 1 Introduction and preliminaries

A general eigenvalue theory for maximal monotone operators has been developed in various ways with applications to partial differential equations; see [5–7, 10–12]. A key tool was topological degrees for appropriate classes of operators in, *e.g.*, [2–4, 8, 9, 14–16] and the method of approach was in many cases to use regularization by means of the duality operator, while in [6, 11] the eigenvalue problem was solved by a transformed equation in terms of the approximant without using the regularization method.

*X*be a real reflexive Banach space with dual space ${X}^{\ast}$ and $T:D(T)\subset X\to {2}^{{X}^{\ast}}$ be a maximal monotone multi-valued operator. When the resolvents of

*T*or the single-valued operator

*C*are compact, the solvability of the nonlinear inclusion

where ${T}_{s}$ is the Brezis-Crandall-Pazy approximant introduced in [1] and ${J}_{\psi}$ is the (normalized) duality operator with a gauge function *ψ*.

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 [14], in comparison with [10]. The other case is concerned with quasibounded densely defined operators satisfying condition (${\tilde{S}}_{+}$), where the degree theory of Kartsatos and Skrypnik [8, 9] for densely defined operators is used. In more concrete situations, the eigenvalue problem $0\in Tx+\lambda Cx$ is discussed. We point out that Browder’s degree in [3] for the reduced simple operator ${T}_{s}+\epsilon {J}_{\psi}$ under the homotopy plays a crucial role in the proof of our results presented here.

Let *X* be a real Banach space with its dual ${X}^{\ast}$, Ω a nonempty subset of *X*, and *Y* another real Banach space. An operator $F:\mathrm{\Omega}\to Y$ is said to be *bounded* if *F* maps bounded subsets of Ω into bounded subsets of *Y*. *F* is said to be *demicontinuous* if for every ${x}_{0}\in \mathrm{\Omega}$ and for every sequence $\{{x}_{n}\}$ in Ω with ${x}_{n}\to {x}_{0}$, we have $F{x}_{n}\rightharpoonup F{x}_{0}$. Here the symbol → (⇀) stands for strong (weak) convergence.

*monotone*if

*effective domain*of

*T*.

*T*is said to be

*maximal monotone*if it is monotone and it follows from $(x,{u}^{\ast})\in X\times {X}^{\ast}$ and

that $x\in D(T)$ and ${u}^{\ast}\in Tx$.

we have ${x}_{n}\to {x}_{0}$.

we have ${x}_{n}\to {x}_{0}$, ${x}_{0}\in D(T)$ and $T{x}_{0}={h}_{0}^{\ast}$.

We say that a multi-valued operator $T:D(T)\subset X\to {2}^{{X}^{\ast}}$ satisfies condition (${S}_{q}$) on a set $M\subset D(T)$ if for every sequence $\{{x}_{n}\}$ in *M* with ${x}_{n}\rightharpoonup {x}_{0}$ and every sequence $\{{u}_{n}^{\ast}\}$ in ${X}^{\ast}$ with ${u}_{n}^{\ast}\to {u}^{\ast}$ where ${u}_{n}^{\ast}\in T{x}_{n}$, we have ${x}_{n}\to {x}_{0}$.

Throughout this paper, *X* will always be an infinite dimensional real reflexive Banach space which has been renormed so that *X* and its dual ${X}^{\ast}$ are locally uniformly convex.

*gauge function*if

*ψ*is continuous, strictly increasing, $\psi (0)=0$ and $\psi (t)\to \mathrm{\infty}$ as $t\to \mathrm{\infty}$. An operator ${J}_{\psi}:X\to {X}^{\ast}$ is called a

*duality operator*with a gauge function

*ψ*if

If *ψ* is the identity map *I*, then $J:={J}_{I}$ is called a *normalized duality operator*. It is known in [13] that ${J}_{\psi}$ is continuous, bounded, surjective, strictly monotone, maximal monotone and satisfies condition (${S}_{+}$).

are called the *Brezis-Crandall-Pazy approximants*. It is known in [1] that ${J}_{s}$ is continuous and bounded and ${T}_{s}$ is demicontinuous, bounded, and maximal monotone. It is easy to see that ${J}_{s}=I-s{J}^{-1}{T}_{s}$ and ${T}_{s}x\in T{J}_{s}x$ for $x\in X$.

Let $C:[0,\mathrm{\Lambda}]\times M\to {X}^{\ast}$ be an operator, where *M* is a subset of *X*. Then $C(t,x)$ is said to be *continuous in* *t* *uniformly* with respect to $x\in M$ if for every ${t}_{0}\in [0,\mathrm{\Lambda}]$ and for every sequence $\{{t}_{n}\}$ in $[0,\mathrm{\Lambda}]$ with ${t}_{n}\to {t}_{0}$, we have $C({t}_{n},x)\to C({t}_{0},x)$ uniformly with respect to $x\in M$.

*C*satisfies condition (${S}_{+}$) if for every $\lambda \in (0,\mathrm{\Lambda}]$ and for every sequence $\{{x}_{n}\}$ in

*M*with ${x}_{n}\rightharpoonup {x}_{0}$ and

we have ${x}_{n}\to {x}_{0}$.

*C*satisfies condition (${\tilde{S}}_{+}$) if for every $\lambda \in (0,\mathrm{\Lambda}]$ and for every sequence $\{{x}_{n}\}$ in

*M*with ${x}_{n}\rightharpoonup {x}_{0}$, $C(\lambda ,{x}_{n})\rightharpoonup {h}_{0}^{\ast}$ and

we have ${x}_{n}\to {x}_{0}$, ${x}_{0}\in M$ and $C(\lambda ,{x}_{0})={h}_{0}^{\ast}$.

We often need the following demiclosedness property of maximal monotone operators given in [17].

**Lemma 1.1** *Let* $T:D(T)\subset X\to {2}^{{X}^{\ast}}$ *be a maximal monotone multi*-*valued operator*. *Then for every sequence* $\{{x}_{n}\}$ *in* $D(T)$, ${x}_{n}\to x$ *in* *X* *and* ${u}_{n}^{\ast}\rightharpoonup {u}^{\ast}$ *in* ${X}^{\ast}$, *where* ${u}_{n}^{\ast}\in T{x}_{n}$, *imply that* $x\in D(T)$ *and* ${u}^{\ast}\in Tx$.

## 2 Implicit eigenvalue problem about demicontinuous operators

In this section, we are concerned with the implicit eigenvalue problem for perturbed maximal 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 subset Ω of *X*, let $\overline{\mathrm{\Omega}}$ and *∂* Ω denote the closure and the boundary of Ω in *X*, respectively. Following Browder [2, 3], a homotopy $H:[0,1]\times \overline{\mathrm{\Omega}}\to {X}^{\ast}$ is said to be of class (${S}_{+}$) if the following condition holds:

we have ${u}_{j}\to {u}_{0}$ and $H({t}_{j},{u}_{j})\rightharpoonup H({t}_{0},{u}_{0})$.

Kartsatos and Skrypnik [10] obtained the following result by using Browder’s degree in [4] for multi-valued operators. As in [12], we adopt property ($\mathcal{P}$) in terms of the Brezis-Crandall-Pazy approximant so that we can apply the most essential degree theory of Skrypnik [14] for single-valued operators to prove in a direct method.

**Theorem 2.1**

*Let*Ω

*be a bounded open set in*

*X*

*with*$0\in \mathrm{\Omega}$.

*Let*$T:D(T)\subset X\to {2}^{{X}^{\ast}}$

*be a maximal monotone multi*-

*valued operator with*$0\in D(T)$

*and*$0\in T(0)$.

*Let*Λ, ${s}_{0}$,

*and*${\epsilon}_{0}$

*be three positive numbers*.

*Suppose that*$C:[0,\mathrm{\Lambda}]\times \overline{\mathrm{\Omega}}\to {X}^{\ast}$

*is demicontinuous*,

*bounded and satisfies condition*(${S}_{+}$)

*such that*$C(0,x)=0$

*for all*$x\in \overline{\mathrm{\Omega}}$

*and*$C(t,x)$

*is continuous in*

*t*

*uniformly with respect to*$x\in \overline{\mathrm{\Omega}}$.

- (a)
*For a given*$\epsilon >0$,*assume the following property*:

*For every*$s\in (0,{s}_{0})$,

*there exists a*$\lambda \in (0,\mathrm{\Lambda}]$

*such that the equation*

*has no solution in*Ω.

*Then there exists a*$({\lambda}_{\epsilon},{x}_{\epsilon})\in (0,\mathrm{\Lambda}]\times (D(T)\cap \partial \mathrm{\Omega})$

*such that*

- (b)
*If*$0\notin T(D(T)\cap \partial \mathrm{\Omega})$,*T**satisfies condition*(${S}_{q}$)*on*$D(T)\cap \partial \mathrm{\Omega}$,*and property*($\mathcal{P}$)*is fulfilled for all*$\epsilon \in (0,{\epsilon}_{0}]$,*then there exists a*$({\lambda}_{0},{x}_{0})\in (0,\mathrm{\Lambda}]\times (D(T)\cap \partial \mathrm{\Omega})$*such that*$0\in T{x}_{0}+C({\lambda}_{0},{x}_{0}).$

*Proof*(a) We first claim that for any $s\in (0,{s}_{0})$, there exists a $({\lambda}_{0},{x}_{0})\in (0,\mathrm{\Lambda}]\times \partial \mathrm{\Omega}$ such that

Since $({T}_{s}+\epsilon J)(0)=0$ and ${T}_{s}+\epsilon J$ is injective by the strict monotonicity of *J*, we have ${T}_{s}x+C(0,x)+\epsilon Jx\ne 0$ for all $x\in \partial \mathrm{\Omega}$. Thus, (2.2) holds for all $\lambda \in [0,\mathrm{\Lambda}]$.

*H*is of class (${S}_{+}$). To prove this, let $\{{u}_{j}\}$ be a sequence in $\overline{\mathrm{\Omega}}$ with ${u}_{j}\rightharpoonup {u}_{0}$ and $\{{t}_{j}\}$ be a sequence in $[0,1]$ with ${t}_{j}\to {t}_{0}$ such that

*J*are monotone, it follows from

*J*satisfies condition (${S}_{+}$), we have

*C*and the continuity of

*J*

Since *C* satisfies condition (${S}_{+}$), we get ${u}_{j}\to {u}_{0}$ from which ${T}_{s}{u}_{j}\rightharpoonup {T}_{s}{u}_{0}$, $C({t}_{j}\mathrm{\Lambda},{u}_{j})\rightharpoonup C({t}_{0}\mathrm{\Lambda},{u}_{0})$, and $J{u}_{j}\to J{u}_{0}$. Consequently, $H({t}_{j},{u}_{j})\rightharpoonup H({t}_{0},{u}_{0})$. We have just shown that the homotopy *H* is of class (${S}_{+}$).

*x*in Ω such that

which contradicts property ($\mathcal{P}$). Therefore, the first claim (2.1) is true.

*J*that

*T*is monotone, ${T}_{{s}_{n}}{x}_{n}\in T{J}_{{s}_{n}}{x}_{n}$, and ${J}_{{s}_{n}}=I-{s}_{n}{J}^{-1}{T}_{{s}_{n}}$, we have

Since *T* is maximal monotone, we have ${x}_{0}\in D(T)$ and $-{c}^{\ast}-\epsilon {j}^{\ast}\in T{x}_{0}$. Letting $x={x}_{0}\in D(T)$ in (2.12) yields a contradiction. Thus, (2.9) holds.

*t*uniformly with respect to $x\in \overline{\mathrm{\Omega}}$ and ${\lambda}_{n}\to {\lambda}_{0}$, we have by (2.9)

*C*satisfies condition (${S}_{+}$), it follows from (2.13) that

*T*is maximal monotone and ${T}_{{s}_{n}}{x}_{n}\in T{J}_{{s}_{n}}{x}_{n}$, Lemma 1.1 implies that ${x}_{0}\in D(T)$ and

- (b)According to the statement (a), for a sequence $\{{\epsilon}_{n}\}$ in $(0,{\epsilon}_{0}]$ with ${\epsilon}_{n}\to 0$, we can choose sequences $\{{\lambda}_{n}\}$ in $(0,\mathrm{\Lambda}]$, $\{{x}_{n}\}$ in $D(T)\cap \partial \mathrm{\Omega}$, and $\{{u}_{n}^{\ast}\}$ in ${X}^{\ast}$ with ${u}_{n}^{\ast}\in T{x}_{n}$ such that${u}_{n}^{\ast}+C({\lambda}_{n},{x}_{n})+{\epsilon}_{n}J{x}_{n}=0.$(2.14)

where ${\lambda}_{0}\in [0,\mathrm{\Lambda}]$, ${x}_{0}\in X$, ${c}^{\ast}\in {X}^{\ast}$, and ${j}^{\ast}\in {X}^{\ast}$. Note that ${\lambda}_{0}\in (0,\mathrm{\Lambda}]$. Indeed, if ${\lambda}_{0}=0$, then (2.14) implies ${u}_{n}^{\ast}\to 0$ which gives by condition (${S}_{q}$) on $D(T)\cap \partial \mathrm{\Omega}$, ${x}_{n}\to {x}_{0}\in \partial \mathrm{\Omega}$ and therefore ${x}_{0}\in D(T)$ and $0\in T{x}_{0}$, which contradicts the hypothesis that $0\notin T(D(T)\cap \partial \mathrm{\Omega})$.

*T*that

Since *T* is maximal monotone, we have ${x}_{0}\in D(T)$ and $-{c}^{\ast}\in T{x}_{0}$. Letting $x={x}_{0}$ in (2.17), we have a contradiction. Thus, (2.15) is true.

*C*satisfies condition (${S}_{+}$) and is demicontinuous, we obtain from (2.14) that

This completes the proof. □

As a consequence of Theorem 2.1, we have the following result. When *C* is a compact operator, it was proved by Li and Huang [12] with the aid of the Leray-Schauder degree for compact operators.

**Corollary 2.2**

*Let*

*T*, Ω, Λ, ${s}_{0}$,

*and*${\epsilon}_{0}$

*be as in Theorem*2.1.

*Suppose that*$C:\overline{\mathrm{\Omega}}\to {X}^{\ast}$

*is a demicontinuous bounded operator which satisfies condition*(${S}_{+}$).

- (a)
*For a given*$\epsilon >0$,*assume the following property*:

*For every*$s\in (0,{s}_{0})$,

*there exists a*$\lambda \in (0,\mathrm{\Lambda}]$

*such that the equation*

*has no solution in*Ω.

*Then there exists a*$({\lambda}_{\epsilon},{x}_{\epsilon})\in (0,\mathrm{\Lambda}]\times (D(T)\cap \partial \mathrm{\Omega})$

*such that*

- (b)
*If*$0\notin T(D(T)\cap \partial \mathrm{\Omega})$,*T**satisfies condition*(${S}_{q}$)*on*$D(T)\cap \partial \mathrm{\Omega}$,*and property*($\mathcal{P}$)*is fulfilled for all*$\epsilon \in (0,{\epsilon}_{0}]$,*then there exists a*$({\lambda}_{0},{x}_{0})\in (0,\mathrm{\Lambda}]\times (D(T)\cap \partial \mathrm{\Omega})$*such that*$0\in T{x}_{0}+{\lambda}_{0}C{x}_{0}.$

*Proof*Define an operator $\tilde{C}:[0,\mathrm{\Lambda}]\times \overline{\mathrm{\Omega}}\to {X}^{\ast}$ by

By hypotheses on *C*, the operator $\tilde{C}$ is obviously demicontinuous, bounded, and satisfies condition (${S}_{+}$). Moreover, $\tilde{C}(t,x)$ is continuous in *t* uniformly with respect to $x\in \overline{\mathrm{\Omega}}$ because $C(\overline{\mathrm{\Omega}})$ is bounded. Apply Theorem 2.1 with $C=\tilde{C}$. □

## 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 operator $C:[0,\mathrm{\Lambda}]\times D(C)\to {X}^{\ast}$ is said to be *uniformly quasibounded* if for every $S>0$ there exists a constant $K(S)>0$ such that for all $\lambda \in [0,\mathrm{\Lambda}]$ and all $u\in D(C)$ with $\parallel u\parallel \le S$ and $\u3008C(\lambda ,u),u\u3009\le 0$, we have $\parallel C(\lambda ,u)\parallel \le K(S)$.

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 [8, 9].

**Theorem 3.1** *Let* Ω *be a bounded open set in* *X* *with* $0\in \mathrm{\Omega}$ *and* *L* *a dense subspace of* *X*. *Let* $T:D(T)\subset X\to {2}^{{X}^{\ast}}$ *be a maximal monotone multi*-*valued operator with* $0\in D(T)$ *and* $0\in T(0)$. *Let* Λ, ${s}_{0}$, *and* ${\epsilon}_{0}$ *be positive numbers*. *Assume that* $C:[0,\mathrm{\Lambda}]\times D(C)\to {X}^{\ast}$ *is a single*-*valued operator with* $L\subset D(C)\subset X$ *such that* $C(0,x)=0$ *for all* $x\in D(C)$ *and* $C(t,x)$ *is continuous in* *t* *uniformly with respect to* $x\in D(C)$. *Assume further that*

(c1) *C* *is uniformly quasibounded*,

(c2) *C* *satisfies condition* (${\tilde{S}}_{+}$), *and*

(c3) *for every* $\lambda \in [0,\mathrm{\Lambda}]$ *and for every* $F\in \mathcal{F}(L)$ *and* $v\in L$, *the function*
, $c(\lambda ,F,v)(u)=\u3008C(\lambda ,u),v\u3009$, *is continuous on* *F*, *where* $\mathcal{F}(L)$ *denotes the set of all finite*-*dimensional subspaces of* *L*.

*Then the following statements hold*:

- (a)
*For a given*$\epsilon >0$,*assume the following property*:

*For every*$s\in (0,{s}_{0})$,

*there exists a*$\lambda \in (0,\mathrm{\Lambda}]$

*such that the equation*

*has no solution in*$D(C)\cap \mathrm{\Omega}$.

*Then there is a*$({\lambda}_{\epsilon},{x}_{\epsilon})\in (0,\mathrm{\Lambda}]\times (D(T)\cap D(C)\cap \partial \mathrm{\Omega})$

*such that*

- (b)
*If*$0\notin T(D(T)\cap \partial \mathrm{\Omega})$,*T**satisfies condition*(${S}_{q}$)*on*$D(T)\cap \partial \mathrm{\Omega}$,*and property*($\mathcal{P}$)*is fulfilled for all*$\epsilon \in (0,{\epsilon}_{0}]$,*then there exists a*$({\lambda}_{0},{x}_{0})\in (0,\mathrm{\Lambda}]\times (D(T)\cap D(C)\cap \partial \mathrm{\Omega})$*such that*$0\in T{x}_{0}+C({\lambda}_{0},{x}_{0}).$

*Proof*(a) First, we prove that for every $s\in (0,{s}_{0})$, there exists a $({\lambda}_{0},{x}_{0})\in (0,\mathrm{\Lambda}]\times (D(C)\cap \partial \mathrm{\Omega})$ such that

Then (3.2) holds for all $\lambda \in [0,\mathrm{\Lambda}]$. For $\lambda =0$, the assertion is obvious.

For $t\in [0,1]$, we set ${T}^{t}={T}_{s}$ and ${C}^{t}=C(t\mathrm{\Lambda},\cdot )+\epsilon {J}_{\psi}$. To show that $\{{T}^{t}+{C}^{t}\}$ is an admissible homotopy in the sense of Definition 2.4 in [9], we have to check the following conditions on two families $\{{T}^{t}\}$ and $\{{C}^{t}\}$. In fact, conditions on $\{{T}^{t}\}$ are automatically satisfied, with ${T}^{t}$ independent of *t*, due to the monotonicity of ${T}_{s}$ and ${T}_{s}(0)=0$.

*i.e.*, for every $\ell >0$ there exists a constant $K(\ell )>0$ such that for all $u\in L$ with $\parallel u\parallel \le \ell $ and all $t\in [0,1]$,

This follows trivially from (c1) and the fact that ${T}_{s}$ and ${J}_{\psi}$ are monotone and ${J}_{\psi}$ is bounded.

*L*such that ${t}_{j}\to {t}_{0}$, ${u}_{j}\rightharpoonup {u}_{0}$, ${C}^{{t}_{j}}{u}_{j}\rightharpoonup {h}_{0}^{\ast}$ and

where ${t}_{0}\in [0,1]$, ${u}_{0}\in X$, and ${h}_{0}^{\ast}\in {X}^{\ast}$, we have ${u}_{j}\to {u}_{0}$, ${u}_{0}\in D(C)$ and ${C}^{{t}_{0}}{u}_{0}={h}_{0}^{\ast}$.

In view of ${C}^{{t}_{j}}{u}_{j}\rightharpoonup {h}_{0}^{\ast}$, we can find a subsequence of $\{{u}_{j}\}$, denoted again by $\{{u}_{j}\}$, such that $C({t}_{j}\mathrm{\Lambda},{u}_{j})\rightharpoonup {h}_{1}^{\ast}$ and ${J}_{\psi}{u}_{j}\rightharpoonup {h}_{2}^{\ast}$ for some ${h}_{1}^{\ast},{h}_{2}^{\ast}\in {X}^{\ast}$. Note that $C({t}_{0}\mathrm{\Lambda},{u}_{j})\rightharpoonup {h}_{1}^{\ast}$ and ${h}_{0}^{\ast}={h}_{1}^{\ast}+\epsilon {h}_{2}^{\ast}$. Hence (c2) implies that ${u}_{j}\to {u}_{0}$, ${u}_{0}\in D(C)$, and $C({t}_{0}\mathrm{\Lambda},{u}_{0})={h}_{1}^{\ast}$ and so ${C}^{{t}_{0}}{u}_{0}=C({t}_{0}\mathrm{\Lambda},{u}_{0})+\epsilon {J}_{\psi}{u}_{0}={h}_{0}^{\ast}$. Thus, condition (${c}_{2}^{t}$) is satisfied in both cases.

(${c}_{3}^{t}$) For every $F\in \mathcal{F}(L)$ and $v\in L$, the function , $\tilde{c}(F,v)(t,u)=\u3008{C}^{t}u,v\u3009$, is continuous.

Actually, $\tilde{c}(F,v)$ is continuous on $[0,1]\times F$ because $c(t\mathrm{\Lambda},F,v)$ in the notation of (c3) is continuous on *F* and ${J}_{\psi}$ is continuous on *X*. Consequently, we have shown that $\{{T}^{t}+{C}^{t}\}$, $t\in [0,1]$, is an admissible homotopy.

*d*with respect to the bounded open set Ω as follows:

which contradicts property ($\mathcal{P}$). Therefore, the first assertion (3.1) is true.

Since *T* is maximal monotone, we have ${x}_{0}\in D(T)$ and $-{c}^{\ast}-\epsilon {j}^{\ast}\in T{x}_{0}$. Letting $x={x}_{0}$ in (3.7), we have a contradiction. Thus, (3.5) holds.

*C*satisfies condition (${\tilde{S}}_{+}$), we have

*T*is maximal monotone and ${T}_{{s}_{n}}{x}_{n}\in T{J}_{{s}_{n}}{x}_{n}$, Lemma 1.1 implies that ${x}_{0}\in D(T)\cap D(C)\cap \partial \mathrm{\Omega}$ and

- (b)In view of (a), for a sequence $\{{\epsilon}_{n}\}$ in $(0,{\epsilon}_{0}]$ with ${\epsilon}_{n}\to 0$, we take sequences $\{{\lambda}_{n}\}$ in $(0,\mathrm{\Lambda}]$, $\{{x}_{n}\}$ in $D(T)\cap D(C)\cap \partial \mathrm{\Omega}$ and $\{{u}_{n}^{\ast}\}$ in ${X}^{\ast}$ with ${u}_{n}^{\ast}\in T{x}_{n}$ such that${u}_{n}^{\ast}+C({\lambda}_{n},{x}_{n})+{\epsilon}_{n}{J}_{\psi}{x}_{n}=0.$(3.9)

*T*that ${x}_{0}\in D(T)$ and

This completes the proof. □

**Remark 3.2** In an analogous way to Theorem 3.1, we can observe the eigenvalue problem $0\in Tx+\lambda Cx$ for quasibounded perturbations of maximal monotone operators; see [10].

## Declarations

### 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).

## Authors’ Affiliations

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