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Eigenvalues of quasibounded maximal monotone operators
Journal of Inequalities and Applications volume 2014, Article number: 21 (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
A systematic theory of compact operators emerged from the theory of integral equations of the form
Here, $\lambda \in \mathbb{R}$ is a parameter, 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 selfadjoint 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 LeraySchauder degree and the KartsatosSkrypnik degree; see [11–15].
Let 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 multivalued operator and $C:D(C)\subset X\to {X}^{\ast}$ is a singlevalued 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 LeraySchauder 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].
In the present paper, the first goal is to study the above eigenvalue problem (E) for strongly quasibounded maximal monotone operators, provided that the operator 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 KartsatosSkrypnik 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 KartsatosQuarcoo 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 $A{x}_{n}\rightharpoonup A{x}_{0}$.
An operator $T:D(T)\subset X\to {2}^{{X}^{\ast}}$ is said to be monotone if
where $D(T)=\{x\in X:Tx\ne \mathrm{\varnothing}\}$ denotes the effective domain of T.
The operator 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 ${u}^{\ast}\in Tx$.
An operator $T:D(T)\subset X\to {2}^{{X}^{\ast}}$ is said to be 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 ${x}_{n}\to {x}_{0}$.
We say that $T:D(T)\subset X\to {X}^{\ast}$ satisfies the condition $({S}_{+})$ on a set $M\subset D(T)$ if for every sequence $\{{x}_{n}\}$ in M with
we have ${x}_{n}\to {x}_{0}$.
Throughout this paper, X will always be an infinitedimensional real reflexive separable Banach space which has been renormed so that X and its dual ${X}^{\ast}$ are locally uniformly convex.
An operator ${J}_{\phi}:X\to {X}^{\ast}$ is said to be a 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 ${X}^{\ast}$.
Let L be a dense subspace of X and let $\mathcal{F}(L)$ denote the class of all finitedimensional 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 singlevalued 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 ${C}_{h}x:=Cxh$, 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 multivalued operator and $C:D(C)\subset X\to {X}^{\ast}$ is a singlevalued 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
(c3) 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:
Considering a multivalued map 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 $0\in (T+\epsilon J)(D(T)\cap \mathrm{\Omega})$.
Now we consider a singlevalued map ${H}_{1}$ given by
We will first show that there exists a positive number ${t}_{0}$ such that the equation
has no solution 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
where ${s}_{0}\in [0,1]$, ${x}_{0}\in X$, and ${j}^{\ast}\in {X}^{\ast}$. Let 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)
and so ${x}_{n}\to 0\in \mathrm{\Omega}$; but ${x}_{n}\in \partial \mathrm{\Omega}$, which is a contradiction. Since we have the inequality
Lemma 1.3 implies that the sequence $\{{T}_{{t}_{n}}{x}_{n}\}$ is bounded in the reflexive Banach space ${X}^{\ast}$. Passing to a subsequence, if necessary, we may suppose that ${T}_{{t}_{n}}{x}_{n}\rightharpoonup {v}^{\ast}$ for some ${v}^{\ast}\in {X}^{\ast}$. Set
By equation (2.3), we have $C{x}_{n}\rightharpoonup {u}^{\ast}$ and hence
Recall that if two operators ${A}_{1}:D({A}_{1})\subset X\to {2}^{{X}^{\ast}}$ and ${A}_{2}:D({A}_{2})\subset X\to {2}^{{X}^{\ast}}$ are maximal monotone and $D({A}_{1})\cap intD({A}_{2})\ne \mathrm{\varnothing}$, then the sum ${A}_{1}+{A}_{2}:D({A}_{1})\cap D({A}_{2})\to {2}^{{X}^{\ast}}$ is also maximal monotone; see [[5], Theorem 32.I]. Since $T+\epsilon J$ is thus maximal monotone and ${T}_{{t}_{n}}{x}_{n}+\epsilon J{x}_{n}\rightharpoonup {v}^{\ast}+\epsilon {j}^{\ast}$, Lemma 1.2(a) says that
From equations (2.3), (2.5), and the equality
it follows that
Since the operator 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
which contradicts our boundary condition equation (2.1). Consequently, we have proven our first assertion: that there exists a number ${t}_{0}>0$ such that
In the next step, we want to show that for each fixed $t\in (0,{t}_{0}]$, the degree $d({H}_{1}(t,s,\cdot ),\mathrm{\Omega},0)$ is independent of $s\in [0,1]$, where d denotes the KartsatosSkrypnik 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
where $D({A}_{s})=X$ for $s=0$ and $D({A}_{s})=D(C)$ for $s\in (0,1]$. First of all, for every finitedimensional space $F\subset L\{{F}_{j}\}$ and every $v\in L\{{F}_{j}\}$, the function $\tilde{a}(F,v):F\times [0,1]\to \mathbb{R}$, defined by $\tilde{a}(F,v)(x,s)=\u3008{A}_{s}x,v\u3009$, is continuous on $F\times [0,1]$ because the operators ${T}_{t}$ and 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
for every $y\in L\{{F}_{n}\}$, where ${s}_{0}\in [0,1]$ and ${x}_{0}\in X$. By Lemma 1.1(a), the sequence $\{{T}_{t}{x}_{n}\}$ is bounded in ${X}^{\ast}$. So we may suppose without loss of generality that ${T}_{t}{x}_{n}\rightharpoonup {v}^{\ast}$ and $J{x}_{n}\rightharpoonup {j}^{\ast}$ for some ${v}^{\ast},{j}^{\ast}\in {X}^{\ast}$. There are two cases to consider. If ${s}_{0}=0$, then we have
which implies along with equation (2.7)
where 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
From the second part of equation (2.8), it is obvious that
By the monotonicity of the operator ${T}_{t}+\epsilon J$, we have
Hence it follows from the first part of equation (2.8) and from equation (2.10) that
Since the operator C satisfies the condition ${({S}_{+})}_{L}$, we find from equations (2.9) and (2.11) that
By the demicontinuity of the operators ${T}_{t}$ and J, we have
and hence
Consequently, the family $\{{A}_{s}\}$ satisfies the condition ${({S}_{+})}_{0,L}^{(s)}$, as required.
Since ${A}_{s}(x)\ne 0$ for all $(s,x)\in [0,1]\times (D({A}_{s})\cap \partial \mathrm{\Omega})$, we see, in view of Theorem A of [18], that the degree $d({A}_{s},\mathrm{\Omega},0)$ is independent of the choice of $s\in [0,1]$. Until now, we have shown that for each fixed $t\in (0,{t}_{0}]$, the degree $d({H}_{1}(t,s,\cdot ),\mathrm{\Omega},0)$ is constant for all $s\in [0,1]$. Notice that $T+\epsilon J$ is maximal monotone and strongly quasibounded, $0\in (T+\epsilon J)(0)$, and
Combining this with our first assertion above, Theorem 2 of [18] says that for each fixed $s\in [0,1]$, the degree $d({T}_{t}+s\mathrm{\Lambda}C+\epsilon J,\mathrm{\Omega},0)$ is constant for all $t\in (0,{t}_{0}]$. If deg denotes the degree introduced in [18], then for every $s\in [0,1]$, we have
and hence
where the last equality follows from Theorem 3 in [23]. Thus, for all $s\in (0,1]$, the inclusion
has a solution in $D(T+C)\cap \mathrm{\Omega}$, which contradicts property $(\mathcal{P})$. We conclude that statement (a) is true.

(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
$${u}_{{\epsilon}_{n}}^{\ast}+{\lambda}_{{\epsilon}_{n}}C{x}_{{\epsilon}_{n}}+{\epsilon}_{n}J{x}_{{\epsilon}_{n}}=0,$$
where ${u}_{{\epsilon}_{n}}^{\ast}\in T{x}_{{\epsilon}_{n}}$. If we set ${\lambda}_{n}:={\lambda}_{{\epsilon}_{n}}$, ${x}_{n}:={x}_{{\epsilon}_{n}}$, and ${u}_{n}^{\ast}:={u}_{{\epsilon}_{n}}^{\ast}$, it can be rewritten in the form
Notice that the sequence $\{{u}_{n}^{\ast}\}$ is bounded in ${X}^{\ast}$. This follows from the strong quasiboundedness of the operator T together with the inequality
where 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
where ${\lambda}_{0}\in [0,\mathrm{\Lambda}]$, ${x}_{0}\in X$, and ${u}_{0}^{\ast}\in {X}^{\ast}$. Note that the limit ${\lambda}_{0}$ belongs to $(0,\mathrm{\Lambda}]$. In fact, if ${\lambda}_{0}=0$, then the boundedness of the set $C(D(C)\cap \partial \mathrm{\Omega})$ implies that ${\lambda}_{n}C{x}_{n}\to 0$ and so by equation (2.12) ${u}_{n}^{\ast}\to 0$. Since the maximal monotone operator 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
From equation (2.12) it follows that
where the last inequality follows from Lemma 1.2(a). Since the operator 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 KartsatosSkrypnik 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
(c3) 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
for every $y\in L\{{F}_{n}\}$. Then $\{\u3008C{x}_{n},{x}_{n}\u3009\}$ is obviously bounded from above. By the strong quasiboundedness of the operator 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)
Since C satisfies the condition $({S}_{+})$ on X and is demicontinuous, we have
Thus, the operator 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)
where ${u}_{n}^{\ast}\in T{x}_{n}$. Notice that the sequence $\{C{x}_{n}\}$ is bounded in ${X}^{\ast}$ and so is $\{{u}_{n}^{\ast}\}$. This follows from the strong quasiboundedness of the operator 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 ${({S}_{+})}_{D(A)}$.
Let 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 KartsatosQuarcoo 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 $p1$ 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
(c3) 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
where $D({A}_{t})=X$ for $t=0$ and $D({A}_{t})=L$ for $t\in (0,1]$. The first aim is to prove that the set of all solutions of the equation $H(t,x)=0$ is bounded, independent of $t\in [0,1]$. If $t=0$, then $H(0,x)=\epsilon {J}_{\phi}x=0$ implies $x=0$. It suffices to show that $\{(t,x)\in (0,1]\times L:H(t,x)=0\}$ is bounded. Assume the contrary; then there exist sequences $\{{t}_{n}\}$ in $(0,1]$ and $\{{x}_{n}\}$ in L such that ${t}_{n}\to {t}_{0}\in [0,1]$, $\parallel {x}_{n}\parallel \to \mathrm{\infty}$, and
which can be written as
We may suppose that $\parallel {x}_{n}\parallel \ge 1$ for all $n\in \mathbb{N}$. Since the operators T, C, and ${J}_{\phi}$ are positively homogeneous of degree γ, it follows from equation (3.1) that
Setting ${u}_{n}:={x}_{n}/\parallel {x}_{n}\parallel $ and ${q}_{n}:=1/{t}_{n}$, we have $\parallel {u}_{n}\parallel =1$, ${q}_{n}>0$, and
Then we obtain from equation (3.2) and (c3)
Hence the strong quasiboundedness of 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
which is a contradiction. Now let ${t}_{0}>0$ and set ${q}_{0}:=1/{t}_{0}$. Without loss of generality, we may suppose that
where ${u}_{0}\in X$, ${v}^{\ast}\in {X}^{\ast}$, and ${j}^{\ast}\in {X}^{\ast}$. By equation (3.2), we have $C{u}_{n}\rightharpoonup \lambda {v}^{\ast}{q}_{0}\epsilon {j}^{\ast}$ and hence
Since the operator $\lambda T+{q}_{0}\epsilon {J}_{\phi}$ is maximal monotone, we have
In fact, if equation (3.4) is false, then there is a subsequence of $\{{u}_{n}\}$, denoted again by $\{{u}_{n}\}$, such that
Hence it is clear that
For every $u\in D(T)$, we have, by the monotonicity of the operator $\lambda T+{q}_{n}\epsilon {J}_{\phi}$,
which implies along with equation (3.5)
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.
Furthermore, equation (3.4) implies, because of $(1/{\parallel {x}_{n}\parallel}^{\gamma}){p}^{\ast}\to 0$, that
From equations (3.2), (3.7), and the equality
it follows that
Since the operator C satisfies the condition ${({S}_{+})}_{L}$, we obtain from equations (3.3) and (3.8)
Since T is maximal monotone and ${J}_{\phi}$ is continuous, Lemma 1.2(b) implies that
Therefore, we obtain
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.
So we can choose an open ball ${B}_{r}(0)$ in 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 $x\in X$.
Using the homotopy invariance property of the degree stated in [[18], Theorem 3], we have
Applying equation (3.9) with $\epsilon =1/n$, there exists a sequence $\{{x}_{n}\}$ in L such that
Next, we show that the sequence $\{{x}_{n}\}$ is bounded in 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
and so ${w}_{n}^{\ast}\to 0$. Since $\lambda \u3008T{u}_{n},{u}_{n}\u3009=\u3008C{u}_{n},{u}_{n}\u3009+\u3008{w}_{n}^{\ast},{u}_{n}\u3009\le \psi (1)+\parallel {w}_{n}^{\ast}\parallel $ for all $n\in \mathbb{N}$, it follows from (t1) that the sequence $\{T{u}_{n}\}$ is bounded in ${X}^{\ast}$. We may suppose that ${u}_{n}\rightharpoonup {u}_{0}$ and $T{u}_{n}\rightharpoonup {v}^{\ast}$ for some ${u}_{0}\in X$ and some ${v}^{\ast}\in {X}^{\ast}$. As in the proof of equations (3.3) and (3.8) above, we can show that
for every $y\in L\{{F}_{n}\}$. Since the operator C satisfies the condition ${({S}_{+})}_{L}$, we obtain
By Lemma 1.2(b), we have ${u}_{0}\in D(T)$ and $T{u}_{0}={v}^{\ast}$ and hence
which contradicts hypothesis (t2) with $\mu =0$. Therefore, the sequence $\{{x}_{n}\}$ is bounded in X.
Combining this with equation (3.10), we know from (c3) and (t1) that the sequence $\{T{x}_{n}\}$ is also bounded in ${X}^{\ast}$. Thus we may suppose that ${x}_{n}\rightharpoonup {x}_{0}$ and $T{x}_{n}\rightharpoonup {v}_{0}^{\ast}$ for some ${x}_{0}\in X$ and some ${v}_{0}^{\ast}\in {X}^{\ast}$. From $C{x}_{n}\rightharpoonup \lambda {v}_{0}^{\ast}+{p}^{\ast}$ and the maximal monotonicity of the operator T, we get as before
for every $y\in L\{{F}_{n}\}$. Since the operator 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 KartsatosSkrypnik 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. □
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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 (NRF20120008345).
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KI conceived of the study and drafted the manuscript. BI participated in coordination. All authors approved the final manuscript.
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Kim, I., Bae, J. Eigenvalues of quasibounded maximal monotone operators. J Inequal Appl 2014, 21 (2014). https://doi.org/10.1186/1029242X201421
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Keywords
 Maximal Monotone
 Maximal Monotone Operator
 Degree Theory
 Duality Operator
 Dense Subspace