# Eigenvalues of quasibounded maximal monotone operators

- In-Sook Kim
^{1}Email author and - Jung-Hyun Bae
^{1}

**2014**:21

https://doi.org/10.1186/1029-242X-2014-21

© Kim and Bae; licensee Springer. 2014

**Received: **11 October 2013

**Accepted: **16 December 2013

**Published: **14 January 2014

## Abstract

Let *X* be a real reflexive separable Banach space with dual space ${X}^{\ast}$ and let *L* be a dense subspace of *X*. We study a nonlinear eigenvalue problem of the type

where $T:D(T)\subset X\to {2}^{{X}^{\ast}}$ is a strongly quasibounded maximal monotone operator and $C:D(C)\subset X\to {X}^{\ast}$ satisfies the condition ${({S}_{+})}_{D(C)}$ with $L\subset D(C)$. The method of approach is to use a topological degree theory for ${({S}_{+})}_{L}$-perturbations of strongly quasibounded maximal monotone operators, recently developed by Kartsatos and Quarcoo. Moreover, applying degree theory, a variant of the Fredholm alternative on the surjectivity of the operator $\lambda T+C$ is discussed, where we assume that *λ* is not an eigenvalue for the pair $(T,C)$, *T* and *C* are positively homogeneous, and *C* satisfies the condition ${({S}_{+})}_{L}$.

## 1 Introduction and preliminaries

*y*and

*k*are given functions, and

*x*is the unknown function. Such equations play a role in the theory of differential equations. The study goes back to Krasnosel’skii [1]. Moreover, the eigenvalue problem of the form

could be solved with the Galerkin method, where *C* is continuous, bounded, and of type $(S)$; see, *e.g.*, [2].

From now on, we concentrate on the class of maximal monotone operators, as a generalization of linear self-adjoint operators. The theory of nonlinear maximal monotone operators started with a pioneer work of Minty [3] and has been extensively developed, with applications to evolution equations and to variational inequalities of elliptic and parabolic type; see [4, 5]. The eigenvalue problem for various types of nonlinear operators was investigated in [6–10]. As a key tool, topological degree theory was made frequent use of; for instance, the Leray-Schauder degree and the Kartsatos-Skrypnik degree; see [11–15].

*X*be a real reflexive Banach space with dual space ${X}^{\ast}$. We consider a nonlinear eigenvalue problem of the form

where $T:D(T)\subset X\to {2}^{{X}^{\ast}}$ is a maximal monotone multi-valued operator and $C:D(C)\subset X\to {X}^{\ast}$ is a single-valued operator. In the case where the operator *C* or the resolvents of *T* are compact, it was studied in [6, 7, 10] by using the Leray-Schauder degree for compact operators. When the operator *C* is densely defined and quasibounded and satisfies the condition $({\tilde{S}}_{+})$, Kartsatos and Skrypnik [9] solved the above problem (E) via the topological degree for these operators given in [13].

We are now focused on the quasiboundedness of the operator *T* instead of that of the operator *C*. Actually, a strongly quasibounded operator due to Browder and Hess [16] may not necessarily be bounded. One more thing to be considered is the condition ${({S}_{+})}_{L}$, where *L* is a dense subspace of *X* with $L\subset D(C)$. In fact, the condition ${({S}_{+})}_{0,L}$ was first introduced in [12] and the structure of the class ${({S}_{+})}_{L}$ or ${({S}_{+})}_{D(C)}$ was discussed in [17], as a natural extension of the class $({S}_{+})$; see [4, 14].

*C*satisfies the condition ${({S}_{+})}_{D(C)}$. In addition, we assume the following property $(\mathcal{P})$: For $\u03f5>0$, there exists a $\lambda >0$ such that the inclusion

has no solution in $D(T)\cap D(C)\cap \mathrm{\Omega}$, where Ω is a bounded open set in *X* and *J* is a normalized duality operator. This property is closely related to the use of a topological tool for finding the eigensolution on the boundary of Ω; see [9, 10]. To solve the above problem (E), we thus use the degree theory for densely defined ${({S}_{+})}_{L}$-perturbations of maximal monotone operators introduced by Kartsatos and Quarcoo in [18]. Roughly speaking, the degree function is based on the Kartsatos-Skrypnik degree [8] of the densely defined operators ${T}_{t}+C$, which is constant for all small values of *t*, where ${T}_{t}$ is the approximant introduced by Brézis *et al.* [19]. Such an approach was first used by Browder in [11]. The second goal is to establish a variant of a Fredholm alternative result on the surjectivity for the operator $\lambda T+C$, where $\lambda \ge 1$ is not an eigenvalue for the pair $(T,C)$ and the operator *C* satisfies the condition ${({S}_{+})}_{L}$; see [9, 20].

This paper is organized as follows: In Section 2, we give some eigenvalue results for strongly quasibounded maximal monotone operators by applying the Kartsatos-Quarcoo degree theory. Section 3 contains a version of the Fredholm alternative for positively homogeneous operators, with a regularization method by means of a duality operator ${J}_{\phi}$.

Let *X* be a real Banach space, ${X}^{\ast}$ its dual space with the usual dual pairing $\u3008\cdot ,\cdot \u3009$, and Ω a nonempty subset of *X*. Let $\overline{\mathrm{\Omega}}$, intΩ, and *∂* Ω denote the closure, the interior, and the boundary of Ω in *X*, respectively. The symbol → (⇀) stands for strong (weak) convergence. An operator $A:\mathrm{\Omega}\to {X}^{\ast}$ is said to be *bounded* if *A* maps bounded subsets of Ω into bounded subsets of ${X}^{\ast}$. *A* 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 $A{x}_{n}\rightharpoonup A{x}_{0}$.

*monotone*if

where $D(T)=\{x\in X:Tx\ne \mathrm{\varnothing}\}$ denotes the *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 ${u}^{\ast}\in Tx$.

*strongly quasibounded*if for every $S>0$ there exists a constant $K(S)>0$ such that for all $x\in D(T)$ with

where ${u}^{\ast}\in Tx$, we have $\parallel {u}^{\ast}\parallel \le K(S)$.

We say that $T:D(T)\subset X\to {2}^{{X}^{\ast}}$ satisfies the 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 $\{{u}_{n}^{\ast}\}$ with ${u}_{n}^{\ast}\to {u}^{\ast}$ where ${u}_{n}^{\ast}\in T{x}_{n}$, we have ${x}_{n}\to {x}_{0}$.

*M*with

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

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

*duality operator*if

where $\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ is continuous, strictly increasing, $\phi (0)=0$ and $\phi (t)\to \mathrm{\infty}$ as $t\to \mathrm{\infty}$. When *φ* is the identity map *I*, $J:={J}_{I}$ is called a *normalized duality operator*.

It is described in [21] that ${J}_{\phi}$ is continuous, bounded, surjective, strictly monotone, maximal monotone, and that it satisfies the condition $({S}_{+})$ on *X*.

The following properties as regards maximal monotone operators will often be used, taken from [[19], Lemma 1.3], [[13], Lemma 3.1], [[22], Lemma 1], and [[18], Lemma D] in this order.

**Lemma 1.1**

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

*be a maximal monotone operator*.

*Then the following statements hold*:

- (a)
*For each*$t\in (0,\mathrm{\infty})$,*the operator*${T}_{t}\equiv {({T}^{-1}+t{J}^{-1})}^{-1}:X\to {X}^{\ast}$*is bounded*,*demicontinuous*,*and maximal monotone*. - (b)
*If*,*in addition*, $0\in D(T)$*and*$0\in T(0)$,*then the operator*$(0,\mathrm{\infty})\times X\to {X}^{\ast}$, $(t,x)\mapsto {T}_{t}x$*is continuous on*$(0,\mathrm{\infty})\times X$.

**Lemma 1.2**

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

*and*$S:D(S)\subset X\to {X}^{\ast}$

*be two maximal monotone operators with*$0\in D(T)\cap D(S)$

*and*$0\in T(0)\cap S(0)$

*such that*$T+S$

*is maximal monotone*.

*Assume that there is a sequence*$\{{t}_{n}\}$

*in*$(0,\mathrm{\infty})$

*with*${t}_{n}\downarrow 0$

*and a sequence*$\{{x}_{n}\}$

*in*$D(S)$

*such that*${x}_{n}\rightharpoonup {x}_{0}\in X$

*and*${T}_{{t}_{n}}{x}_{n}+{w}_{n}^{\ast}\rightharpoonup {y}_{0}^{\ast}\in {X}^{\ast}$,

*where*${w}_{n}^{\ast}\in S{x}_{n}$.

*Then the following hold*:

- (a)
*The inequality*${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}\u3008{T}_{{t}_{n}}{x}_{n}+{w}_{n}^{\ast},{x}_{n}-{x}_{0}\u3009\ge 0$*is true*. - (b)
*If*${lim}_{n\to \mathrm{\infty}}\u3008{T}_{{t}_{n}}{x}_{n}+{w}_{n}^{\ast},{x}_{n}-{x}_{0}\u3009=0$,*then*${x}_{0}\in D(T+S)$*and*${y}_{0}^{\ast}\in (T+S){x}_{0}$.

**Lemma 1.3**

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

*be a strongly quasibounded maximal monotone operator such that*$0\in D(T)$

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

*If*$\{{t}_{n}\}$

*is a sequence in*$(0,\mathrm{\infty})$

*and*$\{{x}_{n}\}$

*is a sequence in*

*X*

*such that*

*where* *S*, ${S}_{1}$ *are positive constants*, *then the sequence* $\{{T}_{{t}_{n}}{x}_{n}\}$ *is bounded in* ${X}^{\ast}$.

*L*be a dense subspace of

*X*and let $\mathcal{F}(L)$ denote the class of all finite-dimensional subspaces of

*L*. Let $\{{F}_{n}\}$ be a sequence in the class $\mathcal{F}(L)$ such that for each $n\in \mathbb{N}$

Set $L\{{F}_{n}\}:={\bigcup}_{n\in \mathbb{N}}{F}_{n}$.

**Definition 1.4**Let $C:D(C)\subset X\to {X}^{\ast}$ be a single-valued operator with $L\subset D(C)$. We say that

*C*satisfies the condition ${({S}_{+})}_{0,L}$ if for every sequence $\{{F}_{n}\}$ in $\mathcal{F}(L)$ satisfying equation (1.1) and for every sequence $\{{x}_{n}\}$ in

*L*with

for every $y\in L\{{F}_{n}\}$, we have ${x}_{n}\to {x}_{0},{x}_{0}\in D(C)$, and $C{x}_{0}=0$.

We say that *C* satisfies the condition ${({S}_{+})}_{L}$ if the operator ${C}_{h}:D(C)\to {X}^{\ast}$, defined by ${C}_{h}x:=Cx-h$, satisfies the condition ${({S}_{+})}_{0,L}$ for every $h\in {X}^{\ast}$.

We say that the operator *C* satisfies the condition ${({S}_{+})}_{0,D(C)}$ if it satisfies the condition ${({S}_{+})}_{0,L}$ with ‘$\{{x}_{n}\}\subset L$’ replaced by ‘$\{{x}_{n}\}\subset D(C)$’. We say that *C* satisfies the condition ${({S}_{+})}_{D(C)}$ if the operator ${C}_{h}$ satisfies the condition ${({S}_{+})}_{0,D(C)}$ for every $h\in {X}^{\ast}$.

It is obvious from Definition 1.4 that if the operator *C* satisfies the condition ${({S}_{+})}_{D(C)}$, then *C* satisfies the condition ${({S}_{+})}_{L}$. However, the converse is not true in general, as we see in Example 3.2 of [17].

## 2 The existence of eigenvalues

In this section, we deal with some eigenvalue results for strongly quasibounded maximal monotone operators in reflexive separable Banach spaces, based on a topological degree theory for ${({S}_{+})}_{L}$-perturbations of maximal monotone operators due to Kartsatos and Quarcoo [18].

We establish the existence of an eigenvalue concerning ${({S}_{+})}_{D(C)}$-perturbations of strongly quasibounded maximal monotone operators.

**Theorem 2.1** *Let* Ω *be a bounded open set in* *X* *with* $0\in \mathrm{\Omega}$ *and let* *L* *be a dense subspace of* *X*. *Suppose that* $T:D(T)\subset X\to {2}^{{X}^{\ast}}$ *is a multi*-*valued operator and* $C:D(C)\subset X\to {X}^{\ast}$ *is a single*-*valued operator with* $L\subset D(C)$ *such that*

(t1) *T* *is maximal monotone and strongly quasibounded with* $0\in D(T)$ *and* $0\in T(0)$,

(c1) *C* *satisfies the condition* ${({S}_{+})}_{D(C)}$,

(c2) *for every* $F\in \mathcal{F}(L)$ *and* $v\in L$, *the function* $c(F,v):F\to \mathbb{R}$, *defined by* $c(F,v)(x)=\u3008Cx,v\u3009$, *is continuous on* *F*, *and*

*there exists a nondecreasing function*$\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$

*such that*

*Let*Λ

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

*be two given positive numbers*.

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

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

*such that the inclusion*

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

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

*such that*

*Here*, $D(T+C)$

*denotes the intersection of*$D(T)$

*and*$D(C)$.

- (b)
*If property*$(\mathcal{P})$*is fulfilled for every*$\epsilon \in (0,{\epsilon}_{0}]$,*T**satisfies the condition*$({S}_{q})$*on*$D(T)\cap \partial \mathrm{\Omega}$, $0\notin T(D(T)\cap \partial \mathrm{\Omega})$,*and the set*$C(D(C)\cap \partial \mathrm{\Omega})$*is bounded*,*then the inclusion*$0\in Tx+\lambda Cx$

*has a solution* $({\lambda}_{0},{x}_{0})$ *in* $(0,\mathrm{\Lambda}]\times (D(T+C)\cap \partial \mathrm{\Omega})$.

*Proof*(a) Assume that the conclusion of (a) is not true. Then for every $\lambda \in (0,\mathrm{\Lambda}]$, the following boundary condition holds:

*H*given by

the inclusion $0\in H(s,x)$ has no solution *x* in $D(T+C)\cap \partial \mathrm{\Omega}$ for all $s\in [0,1]$. Actually, this holds for $s=0$, in view of the injectivity of the operator $T+\epsilon J$ with $0\in (T+\epsilon J)(D(T)\cap \mathrm{\Omega})$.

*x*in $D(C)\cap \partial \mathrm{\Omega}$ for all $t\in (0,{t}_{0}]$ and all $s\in [0,1]$. For $s=0$, assertion (2.2) is obvious because $({T}_{t}+\epsilon J)x=0$ implies $x=0$. Assume that assertion (2.2) does not hold for any $s\in (0,1]$. Then there exist sequences $\{{t}_{n}\}$ in $(0,\mathrm{\infty})$, $\{{s}_{n}\}$ in $(0,1]$, and $\{{x}_{n}\}$ in $D(C)\cap \partial \mathrm{\Omega}$ such that ${t}_{n}\downarrow 0$, ${s}_{n}\to {s}_{0}$, ${x}_{n}\rightharpoonup {x}_{0}$, $J{x}_{n}\rightharpoonup {j}^{\ast}$, and

*S*be a positive upper bound for the bounded sequence $\{\parallel {x}_{n}\parallel \}$. Note that ${s}_{0}\in (0,1]$. Indeed, if ${s}_{0}=0$, then we have by the monotonicity of ${T}_{{t}_{n}}$ with ${T}_{{t}_{n}}(0)=0$, equation (2.3), and (c3)

*C*satisfies the condition ${({S}_{+})}_{D(C)}$, we find from equations (2.4) and (2.6) that ${x}_{n}\to {x}_{0}\in D(C)$ and $C{x}_{0}+{u}^{\ast}=0$. Since ${lim}_{n\to \mathrm{\infty}}\u3008{T}_{{t}_{n}}{x}_{n},{x}_{n}-{x}_{0}\u3009=0$, Lemma 1.2(b) tells us that ${x}_{0}\in D(T)$ and ${v}^{\ast}\in T{x}_{0}$. From $J{x}_{n}\rightharpoonup J{x}_{0}={j}^{\ast}$, we get

*d*denotes the Kartsatos-Skrypnik degree from [12]. Fix $t\in (0,{t}_{0}]$. For $s\in [0,1]$, let ${A}_{s}:D({A}_{s})\subset X\to {X}^{\ast}$ be defined by

*J*are continuous and

*C*satisfies the condition (c2). To show that the family $\{{A}_{s}\}$ satisfies the condition ${({S}_{+})}_{0,L}^{(s)}$, we assume that $\{{s}_{n}\}$ is a sequence in $[0,1]$ and $\{{x}_{n}\}$ is a sequence in $L\{{F}_{n}\}$ such that ${s}_{n}\to {s}_{0}$, ${x}_{n}\rightharpoonup {x}_{0}$, and

*S*is an upper bound for the sequence $\{\parallel {x}_{n}\parallel \}$. Hence it follows that ${x}_{n}\to 0$, ${x}_{0}=0\in X=D({A}_{{s}_{0}})$, and ${A}_{{s}_{0}}{x}_{0}=0$. Now let ${s}_{0}\in (0,1]$. We may suppose that ${s}_{n}>0$ for all $n\in \mathbb{N}$. Set ${\tilde{s}}_{n}:=1/({s}_{n}\mathrm{\Lambda})$ and $\tilde{s}:=1/({s}_{0}\mathrm{\Lambda})$. The relation (2.7) can be expressed in the form

*C*satisfies the condition ${({S}_{+})}_{L}$, we find from equations (2.9) and (2.11) that

*J*, we have

Consequently, the family $\{{A}_{s}\}$ satisfies the condition ${({S}_{+})}_{0,L}^{(s)}$, as required.

- (b)Let $\{{\epsilon}_{n}\}$ be a sequence in $(0,{\epsilon}_{0}]$ such that ${\epsilon}_{n}\to 0$. According to statement (a), there exists a sequence $\{({\lambda}_{{\epsilon}_{n}},{x}_{{\epsilon}_{n}})\}$ in $(0,\mathrm{\Lambda}]\times (D(T+C)\cap \partial \mathrm{\Omega})$ such that${u}_{{\epsilon}_{n}}^{\ast}+{\lambda}_{{\epsilon}_{n}}C{x}_{{\epsilon}_{n}}+{\epsilon}_{n}J{x}_{{\epsilon}_{n}}=0,$

*T*together with the inequality

*S*is an upper bound for the sequence $\{\parallel {x}_{n}\parallel \}$. From equation (2.12), $\{{\lambda}_{n}C{x}_{n}\}$ is bounded in ${X}^{\ast}$. Without loss of generality, we may suppose that

*T*satisfies the condition $({S}_{q})$ on $D(T)\cap \partial \mathrm{\Omega}$, we find from equation (2.13) and Lemma 1.2(b) that ${x}_{n}\to {x}_{0}\in \partial \mathrm{\Omega}$, ${x}_{0}\in D(T)$, and $0\in T{x}_{0}$, which contradicts the hypothesis that $0\notin T(D(T)\cap \partial \mathrm{\Omega})$. As $C{x}_{n}\rightharpoonup (-1/{\lambda}_{0}){u}_{0}^{\ast}$, we have

*C*satisfies the condition ${({S}_{+})}_{D(C)}$, we obtain from equations (2.14) and (2.15) ${x}_{n}\to {x}_{0}\in D(C)$ and ${\lambda}_{0}C{x}_{0}+{u}_{0}^{\ast}=0$. By the maximal monotonicity of the operator

*T*, we have ${x}_{0}\in D(T)$ and ${u}_{0}^{\ast}\in T{x}_{0}$. We conclude that

This completes the proof. □

**Remark 2.2**(a) In Theorem 2.1, it is inevitable that the set $C(D(C)\cap \partial \mathrm{\Omega})$ is assumed to be bounded because it does not hold in general that if ${\lambda}_{n}\to 0$ then ${\lambda}_{n}C{x}_{n}\to 0$.

- (b)
When

*C*is quasibounded and satisfies the condition $({\tilde{S}}_{+})$, it was studied in [[9], Theorem 4] by using Kartsatos-Skrypnik degree theory for $({\tilde{S}}_{+})$-perturbations of maximal monotone operators developed in [13]. For the case where*C*is generalized pseudomonotone in place of the condition $({\tilde{S}}_{+})$, we refer to [[20], Theorem 2.1].

From Theorem 2.1, we get the following eigenvalue result in the case when the operator *C* satisfies the condition $({S}_{+})$.

**Corollary 2.3** *Let* *T*, Ω, *L*, Λ, ${\epsilon}_{0}$ *be as in Theorem * 2.1. *Suppose that* $C:X\to {X}^{\ast}$ *is a strongly quasibounded demicontinuous operator such that*

(c1′) *C* *satisfies the condition* $({S}_{+})$ *on* *X*,

(c2) *for every* $F\in \mathcal{F}(L)$ *and* $v\in L$, *the function* $c(F,v):F\to \mathbb{R}$, *defined by* $c(F,v)(x)=\u3008Cx,v\u3009$, *is continuous on* *F*, *and*

*there exists a nondecreasing function*$\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$

*such that*

*Then the following statements hold*:

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

*Proof*Statement (a) follows immediately from Theorem 2.1 if we only show that the operator

*C*satisfies the condition ${({S}_{+})}_{D(C)}$ with $D(C)=X$. To do this, let $h\in {X}^{\ast}$ be given and suppose that $\{{x}_{n}\}$ is any sequence in

*X*such that

*C*, the sequence $\{C{x}_{n}\}$ is bounded in ${X}^{\ast}$. Since $L\{{F}_{n}\}$ is dense in the reflexive Banach space

*X*, it follows from the third one of equation (2.16) that $C{x}_{n}\rightharpoonup h$. Hence we obtain from the first and second one of equation (2.16)

*C*satisfies the condition $({S}_{+})$ on

*X*and is demicontinuous, we have

*C*satisfies the condition ${({S}_{+})}_{D(C)}$ with $D(C)=X$.

- (b)Let $\{{\epsilon}_{n}\}$ be a sequence in $(0,{\epsilon}_{0}]$ such that ${\epsilon}_{n}\to 0$. In view of (a), there exists a sequence $\{({\lambda}_{n},{x}_{n})\}$ in $(0,\mathrm{\Lambda}]\times (D(T)\cap \partial \mathrm{\Omega})$ such that${u}_{n}^{\ast}+{\lambda}_{n}C{x}_{n}+{\epsilon}_{n}J{x}_{n}=0,$(2.17)

*C*and the inequality

We may suppose that ${\lambda}_{n}\to {\lambda}_{0}$, ${x}_{n}\rightharpoonup {x}_{0}$, and ${u}_{n}^{\ast}\rightharpoonup {u}_{0}^{\ast}$, where ${\lambda}_{0}\in [0,\mathrm{\Lambda}]$, ${x}_{0}\in X$, and ${u}_{0}^{\ast}\in {X}^{\ast}$. Note that ${\lambda}_{0}$ belongs to $(0,\mathrm{\Lambda}]$. Indeed, if ${\lambda}_{0}=0$, then we have by the boundedness of $\{C{x}_{n}\}$ and equation (2.17) ${u}_{n}^{\ast}\to 0$ and hence by the condition $({S}_{q})$ ${x}_{n}\to {x}_{0}\in D(T)$ and $0\in T{x}_{0}$, which contradicts the hypothesis $0\notin T(D(T)\cap \partial \mathrm{\Omega})$. The rest of the proof proceeds analogously as in the proof of Theorem 2.1. □

**Remark 2.4**(a) The boundedness assumption on the set $C(D(C)\cap \partial \mathrm{\Omega})$ is unnecessary in Corollary 2.3, provided that the operator

*C*is strongly quasibounded.

- (b)
An analogous result to Corollary 2.3 can be found in [[9], Corollary 1], where the operator

*C*is supposed to be bounded.

We close this section by exhibiting a simple example of operators *A* satisfying the condition ${({S}_{+})}_{D(A)}$.

*G*be a bounded open set in ${\mathbb{R}}^{N}$. Let $1<p<\mathrm{\infty}$ and $X={W}_{0}^{1,p}(G)$. Define the two operators ${A}_{1},{A}_{2}:X\to {X}^{\ast}$ by

Then the operator ${A}_{1}$ is clearly bounded and continuous, and it satisfies the condition $({S}_{+})$ on *X*. The operator ${A}_{2}$ is compact; see [[24], Theorem 2.2] and [[5], Proposition 26.10]. In particular, the sum $A:={A}_{1}+{A}_{2}$ satisfies the condition ${({S}_{+})}_{D(A)}$ with $D(A)=X$.

## 3 Fredholm alternative

In this section, we present a variant of the Fredholm alternative for strongly quasibounded maximal monotone operators, by applying Kartsatos-Quarcoo degree theory as in Section 2.

Given $\gamma >0$, an operator $A:D(A)\subset X\to {X}^{\ast}$ is said to be *positively homogeneous* of degree *γ* on a set $M\subset D(A)$ if $A(rx)={r}^{\gamma}Ax$ for all $x\in M$ and all $r>0$. For example, the duality operator ${J}_{\phi}:X\to {X}^{\ast}$ is positively homogeneous of degree *γ* on *X* if $\phi (t)={t}^{\gamma}$ for $t\in [0,\mathrm{\infty})$. In addition, the operators ${A}_{1}$ and ${A}_{2}$ given at the end of Section 2 are positively homogeneous of degree $p-1$ on $X={W}_{0}^{1,p}(G)$.

**Theorem 3.1** *Let* *L* *be a dense subspace of* *X* *and let* $\lambda ,\gamma \in [1,\mathrm{\infty})$ *be given*. *Suppose that* $T:D(T)=L\to {X}^{\ast}$ *is an operator and* $C:D(C)\subset X\to {X}^{\ast}$ *is an operator with* $L\subset D(C)$ *and* $C(0)=0$ *such that*

(t1) *T* *is maximal monotone and strongly quasibounded with* $T(0)=0$,

(t2) $\lambda Tx+Cx+\mu {J}_{\phi}x=0$ *implies* $x=0$ *for every* $\mu \ge 0$, *where* $\phi (t)={t}^{\gamma}$,

(c1) *C* *satisfies the condition* ${({S}_{+})}_{L}$,

(c2) *for every* $F\in \mathcal{F}(L)$ *and* $v\in L$, *the function* $c(F,v):F\to \mathbb{R}$, *defined by* $c(F,v)(x)=\u3008Cx,v\u3009$, *is continuous on* *F*, *and*

*there exists a nondecreasing function*$\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$

*such that*

*If the operators* *T* *and* *C* *are positively homogeneous of degree* *γ* *on* *L*, *then the operator* $\lambda T+C$ *is surjective*.

*Proof*Let ${p}^{\ast}$ be an arbitrary but fixed element of ${X}^{\ast}$. For each fixed $\epsilon >0$, consider a family of operators ${A}_{t}:D({A}_{t})\subset X\to {X}^{\ast}$, $t\in [0,1]$ given by

*L*such that ${t}_{n}\to {t}_{0}\in [0,1]$, $\parallel {x}_{n}\parallel \to \mathrm{\infty}$, and

*T*,

*C*, and ${J}_{\phi}$ are positively homogeneous of degree

*γ*, it follows from equation (3.1) that

*T*implies that the sequence $\{T{u}_{n}\}$ is bounded in ${X}^{\ast}$. There are two cases to consider. If ${t}_{0}=0$, then ${q}_{n}\to \mathrm{\infty}$, $\u3008{J}_{\phi}{u}_{n},{u}_{n}\u3009=1$, and the monotonicity of

*T*with $T(0)=0$ implies

By the maximal monotonicity of $\lambda T+{q}_{0}\epsilon {J}_{\phi}$, we have ${u}_{0}\in D(T)$ and $(\lambda T+{q}_{0}\epsilon {J}_{\phi}){u}_{0}=\lambda {v}^{\ast}+{q}_{0}\epsilon {j}^{\ast}$. Letting $u={u}_{0}\in D(T)$ in equation (3.6), we get a contradiction. Thus, equation (3.4) is true.

*C*satisfies the condition ${({S}_{+})}_{L}$, we obtain from equations (3.3) and (3.8)

*T*is maximal monotone and ${J}_{\phi}$ is continuous, Lemma 1.2(b) implies that

which contradicts hypothesis (t2) with $\mu ={q}_{0}\epsilon $. Thus, we have shown that $\{(t,x)\in [0,1]\times L:H(t,x)=0\}$ is bounded.

*X*of radius $r>0$ centered at the origin 0 so that

This means that $H(t,x)={A}_{t}(x)\ne 0$ for all $(t,x)\in [0,1]\times (D({A}_{t})\cap \partial {B}_{r}(0))$. Note that the operator ${\tilde{T}}_{\epsilon}:=\lambda T+\epsilon {J}_{\phi}$ is maximal monotone, strongly quasibounded, ${\tilde{T}}_{\epsilon}(0)=0$, and the operator $\tilde{C}:=C-{p}^{\ast}$ satisfies the condition ${({S}_{+})}_{L}$ and other conditions with $\tilde{c}(F,v)(x):=\u3008\tilde{C}x,v\u3009$ for $x\in F$ and $\u3008\tilde{C}x,x\u3009\ge -\tilde{\psi}(\parallel x\parallel )$ for $x\in D(\tilde{C})$, where $\tilde{\psi}(t):=(1+\parallel {p}^{\ast}\parallel )max\{\psi (t),t\}$. Moreover, we know from Section 1 that the operator $\epsilon {J}_{\phi}$ is continuous, bounded and strictly monotone, and that it satisfies the condition $({S}_{+})$, and $\u3008\epsilon {J}_{\phi}x,x\u3009=\epsilon {\parallel x\parallel}^{\gamma +1}$ for $x\in X$.

*L*such that

*X*. Indeed, assume on the contrary that there is a subsequence of $\{{x}_{n}\}$, denoted by $\{{x}_{n}\}$, such that $\parallel {x}_{n}\parallel \to \mathrm{\infty}$. Dividing both sides of equation (3.10) by ${\parallel {x}_{n}\parallel}^{\gamma}$ and setting ${u}_{n}:={x}_{n}/\parallel {x}_{n}\parallel $ and ${w}_{n}^{\ast}:=\lambda T{u}_{n}+C{u}_{n}$, we get

*C*satisfies the condition ${({S}_{+})}_{L}$, we obtain

which contradicts hypothesis (t2) with $\mu =0$. Therefore, the sequence $\{{x}_{n}\}$ is bounded in *X*.

*T*, we get as before

*C*satisfies the condition ${({S}_{+})}_{L}$ and

*T*is maximal monotone, we conclude that

As ${p}^{\ast}\in {X}^{\ast}$ was arbitrary, this says that the operator $\lambda T+C$ is surjective. This completes the proof. □

**Remark 3.2** An analogous result to Theorem 3.1 was investigated in [[20], Theorem 4.1], where the method was to use Kartsatos-Skrypnik degree theory for quasibounded densely defined $({\tilde{S}}_{+})$-perturbations of maximal monotone operators, developed in [13]; see also [[9], Theorem 5].

As a particular case of Theorem 3.1, we have another surjectivity result.

**Corollary 3.3** *Let* *L*, *T*, *and* *C* *be the same as in Theorem * 3.1, *except that hypothesis* (t2) *is replaced by*

(t2′) $\u3008\lambda Tx+Cx,x\u3009\ge 0$ *for all* $x\in L$.

*If* *λ* *is not an eigenvalue for the pair* $(T,C)$, *that is*, $\lambda Tx+Cx=0$ *implies* $x=0$, *then the operator* $\lambda T+C$ *is surjective*.

*Proof*Noting that

for every $x\in L$ and $\mu >0$, it is clear that hypothesis (t2) in Theorem 3.1 is satisfied. Apply Theorem 3.1. □

## Declarations

### Acknowledgements

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-2012-0008345).

## Authors’ Affiliations

## References

- Krasnosel’skii MA:
*Topological Methods in the Theory of Nonlinear Integral Equations*. Pergamon, New York; 1964.Google Scholar - Zeidler E:
*Nonlinear Functional Analysis and Its Applications III: Variational Methods and Optimization*. Springer, New York; 1985.View ArticleMATHGoogle Scholar - Minty G:
**Monotone operators in Hilbert spaces.***Duke Math. J.*1962,**29:**341-346. 10.1215/S0012-7094-62-02933-2MathSciNetView ArticleMATHGoogle Scholar - Browder FE:
**Nonlinear operators and nonlinear equations of evolution in Banach spaces.**In*Nonlinear Functional Analysis*. Am. Math. Soc., Providence; 1976:1-308. Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill 1968Google Scholar - Zeidler E:
*Nonlinear Functional Analysis and Its Applications II/B: Nonlinear Monotone Operators*. Springer, New York; 1990.View ArticleMATHGoogle Scholar - Guan Z, Kartsatos AG:
**On the eigenvalue problem for perturbations of nonlinear accretive and monotone operators in Banach spaces.***Nonlinear Anal.*1996,**27:**125-141. 10.1016/0362-546X(95)00016-OMathSciNetView ArticleMATHGoogle Scholar - Kartsatos AG:
**New results in the perturbation theory of maximal monotone and**m**-accretive operators in Banach spaces.***Trans. Am. Math. Soc.*1996,**348:**1663-1707. 10.1090/S0002-9947-96-01654-6MathSciNetView ArticleMATHGoogle Scholar - Kartsatos AG, Skrypnik IV:
**Normalized eigenvectors for nonlinear abstract and elliptic operators.***J. Differ. Equ.*1999,**155:**443-475. 10.1006/jdeq.1998.3592MathSciNetView ArticleMATHGoogle Scholar - Kartsatos AG, Skrypnik IV:
**On the eigenvalue problem for perturbed nonlinear maximal monotone operators in reflexive Banach spaces.***Trans. Am. Math. Soc.*2006,**358:**3851-3881. 10.1090/S0002-9947-05-03761-XMathSciNetView ArticleMATHGoogle Scholar - Li H-X, Huang F-L:
**On the nonlinear eigenvalue problem for perturbations of monotone and accretive operators in Banach spaces.***Sichuan Daxue Xuebao (J. Sichuan Univ.)*2000,**37:**303-309.MathSciNetMATHGoogle Scholar - Browder FE:
**Fixed point theory and nonlinear problems.***Bull. Am. Math. Soc.*1983,**9:**1-39. 10.1090/S0273-0979-1983-15153-4MathSciNetView ArticleMATHGoogle Scholar - Kartsatos AG, Skrypnik IV:
**Topological degree theories for densely defined mappings involving operators of type**$({S}_{+})$.*Adv. Differ. Equ.*1999,**4:**413-456.MathSciNetMATHGoogle Scholar - Kartsatos AG, Skrypnik IV:
**A new topological degree theory for densely defined quasibounded**$({\tilde{S}}_{+})$-perturbations of multivalued maximal monotone operators in reflexive Banach spaces.*Abstr. Appl. Anal.*2005,**2005:**121-158. 10.1155/AAA.2005.121MathSciNetView ArticleMATHGoogle Scholar - Skrypnik IV:
*Nonlinear Higher Order Elliptic Equations*. Naukova Dumka, Kiev; 1973. (Russian)MATHGoogle Scholar - Skrypnik IV
**Transl., Ser. II. 139.**In*Methods for Analysis of Nonlinear Elliptic Boundary Value Problems*. Am. Math. Soc., Providence; 1994.Google Scholar - Browder FE, Hess P:
**Nonlinear mappings of monotone type in Banach spaces.***J. Funct. Anal.*1972,**11:**251-294. 10.1016/0022-1236(72)90070-5MathSciNetView ArticleMATHGoogle Scholar - Berkovits J:
**On the degree theory for densely defined mappings of class**${({S}_{+})}_{L}$.*Abstr. Appl. Anal.*1999,**4:**141-152. 10.1155/S1085337599000111MathSciNetView ArticleMATHGoogle Scholar - Kartsatos AG, Quarcoo J:
**A new topological degree theory for densely defined**${({S}_{+})}_{L}$-perturbations of multivalued maximal monotone operators in reflexive separable Banach spaces.*Nonlinear Anal.*2008,**69:**2339-2354. 10.1016/j.na.2007.08.017MathSciNetView ArticleMATHGoogle Scholar - Brézis H, Crandall MG, Pazy A:
**Perturbations of nonlinear maximal monotone sets in Banach space.***Commun. Pure Appl. Math.*1970,**23:**123-144. 10.1002/cpa.3160230107View ArticleMathSciNetMATHGoogle Scholar - Kim I-S, Bae J-H:
**Eigenvalue results for pseudomonotone perturbations of maximal monotone operators.***Cent. Eur. J. Math.*2013,**11:**851-864. 10.2478/s11533-013-0211-2MathSciNetMATHGoogle Scholar - Petryshyn WV:
*Approximation-Solvability of Nonlinear Functional and Differential Equations*. Dekker, New York; 1993.MATHGoogle Scholar - Adhikari DR, Kartsatos AG:
**Topological degree theories and nonlinear operator equations in Banach spaces.***Nonlinear Anal.*2008,**69:**1235-1255. 10.1016/j.na.2007.06.026MathSciNetView ArticleMATHGoogle Scholar - Browder FE:
**Degree of mapping for nonlinear mappings of monotone type.***Proc. Natl. Acad. Sci. USA*1983,**80:**1771-1773. 10.1073/pnas.80.6.1771MathSciNetView ArticleMATHGoogle Scholar - Schmitt K, Sim I:
**Bifurcation problems associated with generalized Laplacians.***Adv. Differ. Equ.*2004,**9:**797-828.MathSciNetMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.