 Research
 Open Access
 Published:
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. □
References
Krasnosel’skii MA: Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon, New York; 1964.
Zeidler E: Nonlinear Functional Analysis and Its Applications III: Variational Methods and Optimization. Springer, New York; 1985.
Minty G: Monotone operators in Hilbert spaces. Duke Math. J. 1962, 29: 341346. 10.1215/S0012709462029332
Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. In Nonlinear Functional Analysis. Am. Math. Soc., Providence; 1976:1308. Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill 1968
Zeidler E: Nonlinear Functional Analysis and Its Applications II/B: Nonlinear Monotone Operators. Springer, New York; 1990.
Guan Z, Kartsatos AG: On the eigenvalue problem for perturbations of nonlinear accretive and monotone operators in Banach spaces. Nonlinear Anal. 1996, 27: 125141. 10.1016/0362546X(95)00016O
Kartsatos AG: New results in the perturbation theory of maximal monotone and m accretive operators in Banach spaces. Trans. Am. Math. Soc. 1996, 348: 16631707. 10.1090/S0002994796016546
Kartsatos AG, Skrypnik IV: Normalized eigenvectors for nonlinear abstract and elliptic operators. J. Differ. Equ. 1999, 155: 443475. 10.1006/jdeq.1998.3592
Kartsatos AG, Skrypnik IV: On the eigenvalue problem for perturbed nonlinear maximal monotone operators in reflexive Banach spaces. Trans. Am. Math. Soc. 2006, 358: 38513881. 10.1090/S000299470503761X
Li HX, Huang FL: On the nonlinear eigenvalue problem for perturbations of monotone and accretive operators in Banach spaces. Sichuan Daxue Xuebao (J. Sichuan Univ.) 2000, 37: 303309.
Browder FE: Fixed point theory and nonlinear problems. Bull. Am. Math. Soc. 1983, 9: 139. 10.1090/S027309791983151534
Kartsatos AG, Skrypnik IV:Topological degree theories for densely defined mappings involving operators of type({S}_{+}). Adv. Differ. Equ. 1999, 4: 413456.
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: 121158. 10.1155/AAA.2005.121
Skrypnik IV: Nonlinear Higher Order Elliptic Equations. Naukova Dumka, Kiev; 1973. (Russian)
Skrypnik IV Transl., Ser. II. 139. In Methods for Analysis of Nonlinear Elliptic Boundary Value Problems. Am. Math. Soc., Providence; 1994.
Browder FE, Hess P: Nonlinear mappings of monotone type in Banach spaces. J. Funct. Anal. 1972, 11: 251294. 10.1016/00221236(72)900705
Berkovits J:On the degree theory for densely defined mappings of class{({S}_{+})}_{L}.Abstr. Appl. Anal. 1999, 4: 141152. 10.1155/S1085337599000111
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: 23392354. 10.1016/j.na.2007.08.017
Brézis H, Crandall MG, Pazy A: Perturbations of nonlinear maximal monotone sets in Banach space. Commun. Pure Appl. Math. 1970, 23: 123144. 10.1002/cpa.3160230107
Kim IS, Bae JH: Eigenvalue results for pseudomonotone perturbations of maximal monotone operators. Cent. Eur. J. Math. 2013, 11: 851864. 10.2478/s1153301302112
Petryshyn WV: ApproximationSolvability of Nonlinear Functional and Differential Equations. Dekker, New York; 1993.
Adhikari DR, Kartsatos AG: Topological degree theories and nonlinear operator equations in Banach spaces. Nonlinear Anal. 2008, 69: 12351255. 10.1016/j.na.2007.06.026
Browder FE: Degree of mapping for nonlinear mappings of monotone type. Proc. Natl. Acad. Sci. USA 1983, 80: 17711773. 10.1073/pnas.80.6.1771
Schmitt K, Sim I: Bifurcation problems associated with generalized Laplacians. Adv. Differ. Equ. 2004, 9: 797828.
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
KI conceived of the study and drafted the manuscript. BI participated in coordination. All authors approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Kim, IS., Bae, JH. Eigenvalues of quasibounded maximal monotone operators. J Inequal Appl 2014, 21 (2014). https://doi.org/10.1186/1029242X201421
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029242X201421
Keywords
 Maximal Monotone
 Maximal Monotone Operator
 Degree Theory
 Duality Operator
 Dense Subspace