Eigenvalues of quasibounded maximal monotone operators

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

$$0\in Tx+\lambda Cx,$$

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

$$Tx+\lambda x=y,\text{where }Tx(t)={\int }_a^{b}k(t,s,x(s))ds.$$

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

$$Tx+\lambda Cx=0$$

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].

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

$$0\in Tx+\lambda Cx,$$

(E)

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].

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 $ϵ>0$, there exists a $\lambda >0$ such that the inclusion

$$0\in Tx+\lambda Cx+\epsilon Jx$$

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 $〈\cdot ,\cdot 〉$, 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

$$〈u^{\ast }-v^{\ast },x-y〉\ge 0\text{for every }x,y\in D(T)\text{ and every }{u}^{\ast }\in Tx,v^{\ast }\in Ty,$$

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

$$〈u^{\ast }-v^{\ast },x-y〉\ge 0\text{for every }y\in D(T)\text{ and every }{v}^{\ast }\in Ty$$

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

$$\parallel x\parallel \le S\text{and}〈u^{\ast },x〉\le S,$$

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

$$x_n\rightharpoonup x_0\text{and}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈T{x}_n,x_n-x_0〉\le 0,$$

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.

An operator $J_{\phi }:X\to X^{\ast }$ is said to be a duality operator if

$$〈J_{\phi }x,x〉=\phi (\parallel x\parallel )\parallel x\parallel \text{and}\parallel J_{\phi }x\parallel =\phi (\parallel x\parallel )\text{for }x\in X,$$

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 ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}〈T_{t_n}{x}_n+w_n^{\ast },x_n-x_0〉\ge 0$ is true.

• (b) If ${lim}_{n\to \mathrm{\infty }}〈T_{t_n}{x}_n+w_n^{\ast },x_n-x_0〉=0$, 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

$$\parallel x_n\parallel \le S\mathit{\text{and}}〈T_{t_n}{x}_n,x_n〉\le S_1,$$

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 finite-dimensional subspaces of L. Let $\{F_n\}$ be a sequence in the class $\mathcal{F}(L)$ such that for each $n\in \mathbb{N}$

$$F_n\subset F_{n+1},dim{F}_n=n,\text{and}\overline{\bigcup _{n\in \mathbb{N}}{F}_n}=X.$$

(1.1)

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

$$x_n\rightharpoonup x_0,\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C{x}_n,x_n〉\le 0,\text{and}\underset{n\to \mathrm{\infty }}{lim}〈C{x}_n,y〉=0$$

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 $c(F,v)(x)=〈Cx,v〉$, is continuous on F, and

(c3) there exists a nondecreasing function $\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ such that

$$〈Cx,x〉\ge -\psi (\parallel x\parallel )\mathit{\text{for all }}x\in D(C).$$

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

$$0\in Tx+\lambda Cx+\epsilon Jx$$

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

$$0\in T{x}_{\epsilon }+{\lambda }_{\epsilon }C{x}_{\epsilon }+\epsilon J{x}_{\epsilon }.$$

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:

$$0\notin Tx+\lambda Cx+\epsilon Jx\text{for all }x\in D(T+C)\cap \partial \mathrm{\Omega }.$$

(2.1)

Considering a multi-valued map H given by

$$H(s,x):=Tx+s\mathrm{\Lambda }Cx+\epsilon Jx\text{for }s\in [0,1],$$

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 single-valued map $H_1$ given by

$$H_1(t,s,x):=T_{t}x+s\mathrm{\Lambda }Cx+\epsilon Jx\text{for }t\in (0,\mathrm{\infty })\text{ and }s\in [0,1].$$

We will first show that there exists a positive number $t_0$ such that the equation

$$H_1(t,s,x)=0$$

(2.2)

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

$$T_{t_n}{x}_n+s_n\mathrm{\Lambda }C{x}_n+\epsilon J{x}_n=0,$$

(2.3)

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)

$$\begin{array}{rcl}\epsilon {\parallel x_n\parallel }^2& \le & \epsilon 〈J{x}_n,x_n〉+〈T_{t_n}{x}_n,x_n〉=-s_n\mathrm{\Lambda }〈C{x}_n,x_n〉\\ \le & s_n\mathrm{\Lambda }\psi (\parallel x_n\parallel )\le s_n\mathrm{\Lambda }\psi (S)\end{array}$$

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

$$〈T_{t_n}{x}_n,x_n〉=-s_n\mathrm{\Lambda }〈C{x}_n,x_n〉-\epsilon 〈J{x}_n,x_n〉\le \mathrm{\Lambda }\psi (S),$$

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

$$u^{\ast }:=\frac{1}{s_0\mathrm{\Lambda }}(v^{\ast }+\epsilon j^{\ast }).$$

By equation (2.3), we have $C{x}_n\rightharpoonup -u^{\ast }$ and hence

$$\underset{n\to \mathrm{\infty }}{lim}〈C{x}_n+u^{\ast },y〉=0\text{for every }y\in L\{F_n\}.$$

(2.4)

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

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}〈T_{t_n}{x}_n+\epsilon J{x}_n,x_n-x_0〉\ge 0.$$

(2.5)

From equations (2.3), (2.5), and the equality

$$\begin{array}{rcl}〈C{x}_n+u^{\ast },x_n〉& =& 〈C{x}_n+\frac{1}{s_n\mathrm{\Lambda }}(T_{t_n}{x}_n+\epsilon J{x}_n),x_n〉-〈\frac{1}{s_n\mathrm{\Lambda }}(T_{t_n}{x}_n+\epsilon J{x}_n),x_n-x_0〉\\ -〈\frac{1}{s_n\mathrm{\Lambda }}(T_{t_n}{x}_n+\epsilon J{x}_n),x_0〉+〈u^{\ast },x_n〉\end{array}$$

it follows that

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C{x}_n+u^{\ast },x_n〉& \le & -\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\frac{1}{s_n\mathrm{\Lambda }}〈T_{t_n}{x}_n+\epsilon J{x}_n,x_n-x_0〉\\ \le & 0.\end{array}$$

(2.6)

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 ${lim}_{n\to \mathrm{\infty }}〈T_{t_n}{x}_n,x_n-x_0〉=0$, 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

$$v^{\ast }+s_0\mathrm{\Lambda }C{x}_0+\epsilon j^{\ast }=0\text{or}0\in T{x}_0+s_0\mathrm{\Lambda }C{x}_0+\epsilon J{x}_0,$$

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

$$H_1(t,s,x)\ne 0\text{for any }(t,s)\in (0,t_0]\times [0,1]\text{ and all }x\in D(C)\cap \partial \mathrm{\Omega }.$$

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 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

$$A_{s}x:=H_1(t,s,x)=T_{t}x+s\mathrm{\Lambda }Cx+\epsilon Jx,$$

where $D(A_s)=X$ for $s=0$ and $D(A_s)=D(C)$ for $s\in (0,1]$. First of all, for every finite-dimensional 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 $\tilde{a}(F,v)(x,s)=〈A_{s}x,v〉$, 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

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈A_{s_n}{x}_n,x_n〉\le 0\text{and}\underset{n\to \mathrm{\infty }}{lim}〈A_{s_n}{x}_n,y〉=0$$

(2.7)

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

$$\epsilon {\parallel x_n\parallel }^2\le \epsilon 〈J{x}_n,x_n〉+〈T_t{x}_n,x_n〉\le 〈A_{s_n}{x}_n,x_n〉+s_n\mathrm{\Lambda }\psi (S),$$

which implies along with equation (2.7)

$$\epsilon \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\parallel x_n\parallel }^2\le \underset{n\to \mathrm{\infty }}{lim}{s}_n\mathrm{\Lambda }\psi (S)=0,$$

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

$$\begin{array}{r}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C{x}_n+{\tilde{s}}_n(T_t+\epsilon J)x_n,x_n〉\le 0,\\ \underset{n\to \mathrm{\infty }}{lim}〈C{x}_n+{\tilde{s}}_n(T_t+\epsilon J)x_n,y〉=0\text{for every }y\in L\{F_n\}.\end{array}$$

(2.8)

From the second part of equation (2.8), it is obvious that

$$\underset{n\to \mathrm{\infty }}{lim}〈C{x}_n+\tilde{s}(v^{\ast }+\epsilon j^{\ast }),y〉=0\text{for every }y\in L\{F_n\}.$$

(2.9)

By the monotonicity of the operator $T_t+\epsilon J$, we have

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}〈(T_t+\epsilon J)x_n,x_n-x_0〉\ge \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}〈(T_t+\epsilon J)x_0,x_n-x_0〉=0.$$

(2.10)

Hence it follows from the first part of equation (2.8) and from equation (2.10) that

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C{x}_n+\tilde{s}(v^{\ast }+\epsilon j^{\ast }),x_n〉& \le & -\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{\tilde{s}}_n〈(T_t+\epsilon J)x_n,x_n-x_0〉\\ \le & 0.\end{array}$$

(2.11)

Since the operator C satisfies the condition ${(S_{+})}_L$, we find from equations (2.9) and (2.11) that

$$x_n\to x_0,x_0\in D(C)=D(A_{s_0})\text{and}C{x}_0+\tilde{s}(v^{\ast }+\epsilon j^{\ast })=0.$$

By the demicontinuity of the operators $T_t$ and J, we have

$$T_t{x}_n\rightharpoonup T_t{x}_0=v^{\ast }\text{and}J{x}_n\rightharpoonup J{x}_0=j^{\ast }$$

and hence

$$A_{s_0}{x}_0=T_t{x}_0+s_0\mathrm{\Lambda }C{x}_0+\epsilon J{x}_0=0.$$

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

$$H(s,x)=(T+\epsilon J)x+s\mathrm{\Lambda }Cx\not\ni 0\text{for all }s\in [0,1]\text{ and all }x\in D(T+C)\cap \partial \mathrm{\Omega }.$$

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

$$deg(T+s\mathrm{\Lambda }C+\epsilon J,\mathrm{\Omega },0)=\mathrm{d}(T_t+s\mathrm{\Lambda }C+\epsilon J,\mathrm{\Omega },0)\text{for }t\in (0,t_0]$$

and hence

$$\begin{array}{rcl}deg(T+s\mathrm{\Lambda }C+\epsilon J,\mathrm{\Omega },0)& =& \mathrm{d}(A_s,\mathrm{\Omega },0)=\mathrm{d}(A_0,\mathrm{\Omega },0)\\ =& \mathrm{d}(T_t+\epsilon J,\mathrm{\Omega },0)=1,\end{array}$$

where the last equality follows from Theorem 3 in [23]. Thus, for all $s\in (0,1]$, the inclusion

$$0\in Tx+s\mathrm{\Lambda }Cx+\epsilon Jx$$

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

$$u_n^{\ast }+{\lambda }_{n}C{x}_n+{\epsilon }_{n}J{x}_n=0.$$

(2.12)

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

$$〈u_n^{\ast },x_n〉=-{\lambda }_n〈C{x}_n,x_n〉-{\epsilon }_n〈J{x}_n,x_n〉\le \mathrm{\Lambda }\psi (S),$$

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

$${\lambda }_n\to {\lambda }_0,x_n\rightharpoonup x_0,\text{and}{u}_n^{\ast }\rightharpoonup u_0^{\ast },$$

(2.13)

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

$$\underset{n\to \mathrm{\infty }}{lim}〈C{x}_n+\frac{1}{{\lambda }_0}{u}_0^{\ast },y〉=0\text{for every }y\in L\{F_n\}.$$

(2.14)

From equation (2.12) it follows that

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C{x}_n+\frac{1}{{\lambda }_0}{u}_0^{\ast },x_n〉& \le & -\frac{1}{{\lambda }_0}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}〈u_n^{\ast }+{\epsilon }_{n}J{x}_n,x_n-x_0〉\\ \le & 0,\end{array}$$

(2.15)

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

$$0\in T{x}_0+{\lambda }_{0}C{x}_0\text{and}{x}_0\in D(T+C)\cap \partial \mathrm{\Omega }.$$

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 $c(F,v)(x)=〈Cx,v〉$, is continuous on F, and

(c3) there exists a nondecreasing function $\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ such that

$$〈Cx,x〉\ge -\psi (\parallel x\parallel )\mathit{\text{for all }}x\in X.$$

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

$$x_n\rightharpoonup x_0,\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C{x}_n-h,x_n〉\le 0,\text{and}\underset{n\to \mathrm{\infty }}{lim}〈C{x}_n-h,y〉=0$$

(2.16)

for every $y\in L\{F_n\}$. Then $\{〈C{x}_n,x_n〉\}$ 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)

$$\begin{array}{c}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C{x}_n,x_n-x_0〉\hfill \\ \le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C{x}_n-h,x_n〉-\underset{n\to \mathrm{\infty }}{lim}〈C{x}_n-h,x_0〉+\underset{n\to \mathrm{\infty }}{lim}〈h,x_n-x_0〉\le 0.\hfill \end{array}$$

Since C satisfies the condition $(S_{+})$ on X and is demicontinuous, we have

$$x_n\to x_0\in X\text{and}C{x}_0-h=0.$$

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

$$〈C{x}_n,x_n〉=-\frac{1}{{\lambda }_n}〈u_n^{\ast },x_n〉-\frac{{\epsilon }_n}{{\lambda }_n}〈J{x}_n,x_n〉\le 0.$$

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

$$\begin{array}{c}〈A_{1}u,v〉=\sum _{i=1}^N{\int }_G|\frac{\partial u}{\partial x_i}{|}^{p-2}\frac{\partial u}{\partial x_i}\frac{\partial v}{\partial x_i}dx,\hfill \\ 〈A_{2}u,v〉={\int }_G{|u|}^{p-2}uvdx.\hfill \end{array}$$

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 $c(F,v)(x)=〈Cx,v〉$, is continuous on F, and

(c3) there exists a nondecreasing function $\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ such that

$$〈Cx,x〉\ge -\psi (\parallel x\parallel )\mathit{\text{for all }}x\in D(C).$$

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

$$A_t(x):=H(t,x):=t(\lambda Tx+Cx+\epsilon J_{\phi }x-p^{\ast })+(1-t)\epsilon J_{\phi }x,$$

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

$$t_n(\lambda T{x}_n+C{x}_n+\epsilon J_{\phi }{x}_n-p^{\ast })+(1-t_n)\epsilon J_{\phi }{x}_n=0,$$

which can be written as

$$\lambda T{x}_n+C{x}_n-p^{\ast }+\frac{\epsilon }{t_n}{J}_{\phi }{x}_n=0.$$

(3.1)

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

$$\lambda T\left(\frac{x_n}{\parallel x_n\parallel }\right)+C\left(\frac{x_n}{\parallel x_n\parallel }\right)-\frac{1}{{\parallel x_n\parallel }^{\gamma }}{p}^{\ast }+\frac{\epsilon }{t_n}{J}_{\phi }\left(\frac{x_n}{\parallel x_n\parallel }\right)=0.$$

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

$$\lambda T{u}_n+C{u}_n-\frac{1}{{\parallel x_n\parallel }^{\gamma }}{p}^{\ast }+q_n\epsilon J_{\phi }{u}_n=0.$$

(3.2)

Then we obtain from equation (3.2) and (c3)

$$\begin{array}{rcl}\lambda 〈T{u}_n,u_n〉& =& -〈C{u}_n,u_n〉+\frac{1}{{\parallel x_n\parallel }^{\gamma }}〈p^{\ast },u_n〉-q_n\epsilon 〈J_{\phi }{u}_n,u_n〉\\ \le & \psi (1)+\parallel p^{\ast }\parallel .\end{array}$$

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 $q_n\to \mathrm{\infty }$, $〈J_{\phi }{u}_n,u_n〉=1$, and the monotonicity of T with $T(0)=0$ implies

$$0\le \lambda 〈T{u}_n,u_n〉\le \psi (1)+\parallel p^{\ast }\parallel -q_n\epsilon \to -\mathrm{\infty },$$

which is a contradiction. Now let $t_0>0$ and set $q_0:=1/t_0$. Without loss of generality, we may suppose that

$$u_n\rightharpoonup u_0,T{u}_n\rightharpoonup v^{\ast },\text{and}{J}_{\phi }{u}_n\rightharpoonup j^{\ast },$$

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

$$\underset{n\to \mathrm{\infty }}{lim}〈C{u}_n+\lambda v^{\ast }+q_0\epsilon j^{\ast },y〉=0\text{for every }y\in L\{F_n\}.$$

(3.3)

Since the operator $\lambda T+q_0\epsilon J_{\phi }$ is maximal monotone, we have

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}〈\lambda T{u}_n+q_n\epsilon J_{\phi }{u}_n,u_n-u_0〉\ge 0.$$

(3.4)

In fact, if equation (3.4) is false, then there is a subsequence of $\{u_n\}$, denoted again by $\{u_n\}$, such that

$$\underset{n\to \mathrm{\infty }}{lim}〈\lambda T{u}_n+q_n\epsilon J_{\phi }{u}_n,u_n-u_0〉<0.$$

Hence it is clear that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈\lambda T{u}_n+q_n\epsilon J_{\phi }{u}_n,u_n〉<〈\lambda v^{\ast }+q_0\epsilon j^{\ast },u_0〉.$$

(3.5)

For every $u\in D(T)$, we have, by the monotonicity of the operator $\lambda T+q_n\epsilon J_{\phi }$,

$$\begin{array}{c}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}〈\lambda T{u}_n+q_n\epsilon J_{\phi }{u}_n,u_n〉\hfill \\ \ge \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}[〈\lambda T{u}_n+q_n\epsilon J_{\phi }{u}_n,u〉+〈\lambda Tu+q_n\epsilon J_{\phi }u,u_n-u〉]\hfill \\ =〈\lambda v^{\ast }+q_0\epsilon j^{\ast },u〉+〈\lambda Tu+q_0\epsilon J_{\phi }u,u_0-u〉,\hfill \end{array}$$

which implies along with equation (3.5)

$$〈\lambda v^{\ast }+q_0\epsilon j^{\ast }-(\lambda Tu+q_0\epsilon J_{\phi }u),u_0-u〉>0.$$

(3.6)

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

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}〈\lambda T{u}_n-\frac{1}{{\parallel x_n\parallel }^{\gamma }}{p}^{\ast }+q_n\epsilon J_{\phi }{u}_n,u_n-u_0〉\ge 0.$$

(3.7)

From equations (3.2), (3.7), and the equality

$$\begin{array}{c}〈C{u}_n+\lambda v^{\ast }+q_0\epsilon j^{\ast },u_n〉\hfill \\ =〈C{u}_n+\lambda T{u}_n-\frac{1}{{\parallel x_n\parallel }^{\gamma }}{p}^{\ast }+q_n\epsilon J_{\phi }{u}_n,u_n〉\hfill \\ -〈\lambda T{u}_n-\frac{1}{{\parallel x_n\parallel }^{\gamma }}{p}^{\ast }+q_n\epsilon J_{\phi }{u}_n,u_n-u_0〉\hfill \\ -〈\lambda T{u}_n-\frac{1}{{\parallel x_n\parallel }^{\gamma }}{p}^{\ast }+q_n\epsilon J_{\phi }{u}_n,u_0〉+〈\lambda v^{\ast }+q_0\epsilon j^{\ast },u_n〉\hfill \end{array}$$

it follows that

$$\begin{array}{c}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C{u}_n+\lambda v^{\ast }+q_0\epsilon j^{\ast },u_n〉\hfill \\ \le -\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}〈\lambda T{u}_n-\frac{1}{{\parallel x_n\parallel }^{\gamma }}{p}^{\ast }+q_n\epsilon J_{\phi }{u}_n,u_n-u_0〉\hfill \\ \le 0.\hfill \end{array}$$

(3.8)

Since the operator C satisfies the condition ${(S_{+})}_L$, we obtain from equations (3.3) and (3.8)

$$u_n\to u_0,u_0\in D(C),\text{and}C{u}_0+\lambda v^{\ast }+q_0\epsilon j^{\ast }=0.$$

Since T is maximal monotone and $J_{\phi }$ is continuous, Lemma 1.2(b) implies that

$$u_0\in D(T),T{u}_0=v^{\ast },\text{and}{J}_{\phi }{u}_0=j^{\ast }.$$

Therefore, we obtain

$$\lambda T{u}_0+C{u}_0+q_0\epsilon J_{\phi }{u}_0=0\text{and}\parallel u_0\parallel =1,$$

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

$$\{x\in L:H(t,x)=0\text{ for some }t\in [0,1]\}\subset B_r(0).$$

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 $\tilde{c}(F,v)(x):=〈\tilde{C}x,v〉$ for $x\in F$ and $〈\tilde{C}x,x〉\ge -\tilde{\psi }(\parallel x\parallel )$ 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 $〈\epsilon J_{\phi }x,x〉=\epsilon {\parallel x\parallel }^{\gamma +1}$ for $x\in X$.

Using the homotopy invariance property of the degree stated in [[18], Theorem 3], we have

$$deg(\lambda T+C+\epsilon J_{\phi }-p^{\ast },B_r(0),0)=deg(\epsilon J_{\phi },B_r(0),0)=1.$$

(3.9)

Applying equation (3.9) with $\epsilon =1/n$, there exists a sequence $\{x_n\}$ in L such that

$$\lambda T{x}_n+C{x}_n+\frac{1}{n}{J}_{\phi }{x}_n=p^{\ast }.$$

(3.10)

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

$$\lambda T{u}_n+C{u}_n+\frac{1}{n}{J}_{\phi }{u}_n=\frac{1}{{\parallel x_n\parallel }^{\gamma }}{p}^{\ast }$$

and so $w_n^{\ast }\to 0$. Since $\lambda 〈T{u}_n,u_n〉=-〈C{u}_n,u_n〉+〈w_n^{\ast },u_n〉\le \psi (1)+\parallel w_n^{\ast }\parallel$ 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

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C{u}_n+\lambda v^{\ast },u_n〉\le 0\text{and}\underset{n\to \mathrm{\infty }}{lim}〈C{u}_n+\lambda v^{\ast },y〉=0$$

for every $y\in L\{F_n\}$. Since the operator C satisfies the condition ${(S_{+})}_L$, we obtain

$$u_n\to u_0,u_0\in D(C),\text{and}C{u}_0+\lambda v^{\ast }=0.$$

By Lemma 1.2(b), we have $u_0\in D(T)$ and $T{u}_0=v^{\ast }$ and hence

$$\lambda T{u}_0+C{u}_0=0\text{and}\parallel u_0\parallel =1,$$

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

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C{x}_n+\lambda v_0^{\ast }-p^{\ast },x_n〉\le 0\text{and}\underset{n\to \mathrm{\infty }}{lim}〈C{x}_n+\lambda v_0^{\ast }-p^{\ast },y〉=0$$

for every $y\in L\{F_n\}$. Since the operator C satisfies the condition ${(S_{+})}_L$ and T is maximal monotone, we conclude that

$$x_0\in D(\lambda T+C)\text{and}\lambda T{x}_0+C{x}_0=p^{\ast }.$$

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′) $〈\lambda Tx+Cx,x〉\ge 0$ 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

$$〈\lambda Tx+Cx+\mu J_{\phi }x,x〉\ge \mu 〈J_{\phi }x,x〉=\mu {\parallel x\parallel }^{\gamma +1}\ge 0$$

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