Implicit eigenvalue problems for maximal monotone operators

Abstract

We study the implicit eigenvalue problem of the form

$$0\in Tx+C(\lambda ,x),$$

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

1 Introduction and preliminaries

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

Let X be a real reflexive Banach space with dual space $X^{\ast }$ and $T:D(T)\subset X\to 2^{X^{\ast }}$ be a maximal monotone multi-valued operator. When the resolvents of T or the single-valued operator C are compact, the solvability of the nonlinear inclusion

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

was studied in [5, 6, 10, 12] by applying the Leray-Schauder degree theory for compact operators. More generally, implicit eigenvalue problems were considered in [7, 10], where the single-valued and compact case was dealt with in [7]. In direction of [10], we study the implicit eigenvalue problem of the form

$$0\in Tx+C(\lambda ,x),$$

where $C:[0,\mathrm{\Lambda }]\times M\to X^{\ast }$ satisfies condition ($S_{+}$) or (${\tilde{S}}_{+}$). As in [12], we adopt property ($\mathcal{P}$) about the solvability of the related equation

$$T_{s}x+C(\lambda ,x)+\epsilon J_{\psi }x=0,$$

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

In the present paper, we divide our investigation into two cases to apply suitable degree theory. One case deals with demicontinuous bounded operators satisfying condition ($S_{+}$) with the aid of the most elementary degree theory of Skrypnik [14], in comparison with [10]. The other case is concerned with quasibounded densely defined operators satisfying condition (${\tilde{S}}_{+}$), where the degree theory of Kartsatos and Skrypnik [8, 9] for densely defined operators is used. In more concrete situations, the eigenvalue problem $0\in Tx+\lambda Cx$ is discussed. We point out that Browder’s degree in [3] for the reduced simple operator $T_s+\epsilon J_{\psi }$ under the homotopy plays a crucial role in the proof of our results presented here.

Let X be a real Banach space with its dual $X^{\ast }$, Ω a nonempty subset of X, and Y another real Banach space. An operator $F:\mathrm{\Omega }\to Y$ is said to be bounded if F maps bounded subsets of Ω into bounded subsets of Y. F is said to be demicontinuous if for every $x_0\in \mathrm{\Omega }$ and for every sequence $\{x_n\}$ in Ω with $x_n\to x_0$, we have $F{x}_n\rightharpoonup F{x}_0$. Here the symbol → (⇀) stands for strong (weak) convergence.

A multi-valued 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. 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 $u^{\ast }\in Tx$.

We say that an operator $T:D(T)\subset X\to X^{\ast }$ satisfies condition ($S_{+}$) if for every sequence $\{x_n\}$ in $D(T)$ with $x_n\rightharpoonup x_0$ and

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

we have $x_n\to x_0$.

We say that $T:D(T)\subset X\to X^{\ast }$ satisfies condition (${\tilde{S}}_{+}$) if for every sequence $\{x_n\}$ in $D(T)$ with $x_n\rightharpoonup x_0$, $T{x}_n\rightharpoonup h_0^{\ast }$ and

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

we have $x_n\to x_0$, $x_0\in D(T)$ and $T{x}_0=h_0^{\ast }$.

We say that a multi-valued operator $T:D(T)\subset X\to 2^{X^{\ast }}$ satisfies condition ($S_q$) on a set $M\subset D(T)$ if for every sequence $\{x_n\}$ in M with $x_n\rightharpoonup x_0$ and every sequence $\{u_n^{\ast }\}$ in $X^{\ast }$ with $u_n^{\ast }\to u^{\ast }$ where $u_n^{\ast }\in T{x}_n$, we have $x_n\to x_0$.

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

A function $\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is called a gauge function if ψ is continuous, strictly increasing, $\psi (0)=0$ and $\psi (t)\to \mathrm{\infty }$ as $t\to \mathrm{\infty }$. An operator $J_{\psi }:X\to X^{\ast }$ is called a duality operator with a gauge function ψ if

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

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

Given a maximal monotone operator $T:D(T)\subset X\to 2^{X^{\ast }}$, $x\in X$ and $s>0$, there exist unique elements $x_s\in D(T)$ and $u_s^{\ast }\in T{x}_s$ such that

$$J(x_s-x)+s{u}_s^{\ast }=0.$$

Two operators $J_s:X\to D(T)$ and $T_s:X\to X^{\ast }$ defined by

$$J_{s}x:=x_s\text{and}{T}_{s}x:=u_s^{\ast }\text{for }x\in X$$

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

Let $C:[0,\mathrm{\Lambda }]\times M\to X^{\ast }$ be an operator, where M is a subset of X. Then $C(t,x)$ is said to be continuous in t uniformly with respect to $x\in M$ if for every $t_0\in [0,\mathrm{\Lambda }]$ and for every sequence $\{t_n\}$ in $[0,\mathrm{\Lambda }]$ with $t_n\to t_0$, we have $C(t_n,x)\to C(t_0,x)$ uniformly with respect to $x\in M$.

We say that C satisfies condition ($S_{+}$) if for every $\lambda \in (0,\mathrm{\Lambda }]$ and for every sequence $\{x_n\}$ in M with $x_n\rightharpoonup x_0$ and

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C(\lambda ,x_n),x_n-x_0〉\le 0,$$

we have $x_n\to x_0$.

We say that C satisfies condition (${\tilde{S}}_{+}$) if for every $\lambda \in (0,\mathrm{\Lambda }]$ and for every sequence $\{x_n\}$ in M with $x_n\rightharpoonup x_0$, $C(\lambda ,x_n)\rightharpoonup h_0^{\ast }$ and

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C(\lambda ,x_n),x_n-x_0〉\le 0,$$

we have $x_n\to x_0$, $x_0\in M$ and $C(\lambda ,x_0)=h_0^{\ast }$.

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

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

2 Implicit eigenvalue problem about demicontinuous operators

In this section, we are concerned with the implicit eigenvalue problem for perturbed maximal monotone operators in reflexive Banach spaces by applying the degree theories of Skrypnik and Browder for nonlinear operators of monotone type.

In what follows, for a bounded subset Ω of X, let $\overline{\mathrm{\Omega }}$ and ∂ Ω denote the closure and the boundary of Ω in X, respectively. Following Browder [2, 3], a homotopy $H:[0,1]\times \overline{\mathrm{\Omega }}\to X^{\ast }$ is said to be of class ($S_{+}$) if the following condition holds:

For every sequence $\{u_j\}$ in $\overline{\mathrm{\Omega }}$ with $u_j\rightharpoonup u_0$ and every sequence $\{t_j\}$ in $[0,1]$ with $t_j\to t_0$ such that

$$\underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈H(t_j,u_j),u_j-u_0〉\le 0,$$

we have $u_j\to u_0$ and $H(t_j,u_j)\rightharpoonup H(t_0,u_0)$.

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

Theorem 2.1 Let Ω be a bounded open set in X with $0\in \mathrm{\Omega }$. Let $T:D(T)\subset X\to 2^{X^{\ast }}$ be a maximal monotone multi-valued operator with $0\in D(T)$ and $0\in T(0)$. Let Λ, $s_0$, and ${\epsilon }_0$ be three positive numbers. Suppose that $C:[0,\mathrm{\Lambda }]\times \overline{\mathrm{\Omega }}\to X^{\ast }$ is demicontinuous, bounded and satisfies condition ($S_{+}$) such that $C(0,x)=0$ for all $x\in \overline{\mathrm{\Omega }}$ and $C(t,x)$ is continuous in t uniformly with respect to $x\in \overline{\mathrm{\Omega }}$.

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

($\mathcal{P}$) For every $s\in (0,s_0)$, there exists a $\lambda \in (0,\mathrm{\Lambda }]$ such that the equation

$$T_{s}x+C(\lambda ,x)+\epsilon Jx=0$$

has no solution in Ω. Then there exists a $({\lambda }_{\epsilon },x_{\epsilon })\in (0,\mathrm{\Lambda }]\times (D(T)\cap \partial \mathrm{\Omega })$ such that

$$0\in T{x}_{\epsilon }+C({\lambda }_{\epsilon },x_{\epsilon })+\epsilon J{x}_{\epsilon }.$$

• (b) If $0\notin T(D(T)\cap \partial \mathrm{\Omega })$, T satisfies condition ($S_q$) on $D(T)\cap \partial \mathrm{\Omega }$, and property ($\mathcal{P}$) is fulfilled for all $\epsilon \in (0,{\epsilon }_0]$, then there exists a $({\lambda }_0,x_0)\in (0,\mathrm{\Lambda }]\times (D(T)\cap \partial \mathrm{\Omega })$ such that

  $$0\in T{x}_0+C({\lambda }_0,x_0).$$

Proof (a) We first claim that for any $s\in (0,s_0)$, there exists a $({\lambda }_0,x_0)\in (0,\mathrm{\Lambda }]\times \partial \mathrm{\Omega }$ such that

$$T_s{x}_0+C({\lambda }_0,x_0)+\epsilon J{x}_0=0.$$

(2.1)

Assume on the contrary that for some $s\in (0,s_0)$ and for every $\lambda \in (0,\mathrm{\Lambda }]$, the following holds:

$$T_{s}x+C(\lambda ,x)+\epsilon Jx\ne 0\text{for all }x\in \partial \mathrm{\Omega }.$$

(2.2)

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

Consider a homotopy $H:[0,1]\times \overline{\mathrm{\Omega }}\to X^{\ast }$ defined by

$$H(t,x)=T_{s}x+C(t\mathrm{\Lambda },x)+\epsilon Jx\text{for }(t,x)\in [0,1]\times \overline{\mathrm{\Omega }}.$$

Then H is of class ($S_{+}$). To prove this, let $\{u_j\}$ be a sequence in $\overline{\mathrm{\Omega }}$ with $u_j\rightharpoonup u_0$ and $\{t_j\}$ be a sequence in $[0,1]$ with $t_j\to t_0$ such that

$$\underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈H(t_j,u_j),u_j-u_0〉\le 0.$$

(2.3)

Since $T_s$ and J are monotone, it follows from

$$〈H(t_j,u_j),u_j-u_0〉=〈T_s{u}_j,u_j-u_0〉+〈C(t_j\mathrm{\Lambda },u_j),u_j-u_0〉+\epsilon 〈J{u}_j,u_j-u_0〉$$

that

$$〈H(t_j,u_j),u_j-u_0〉\ge 〈T_s{u}_0,u_j-u_0〉+〈C(t_j\mathrm{\Lambda },u_j),u_j-u_0〉+\epsilon 〈J{u}_j,u_j-u_0〉$$

(2.4)

and

$$〈H(t_j,u_j),u_j-u_0〉\ge 〈T_s{u}_0,u_j-u_0〉+〈C(t_j\mathrm{\Lambda },u_j),u_j-u_0〉+\epsilon 〈J{u}_0,u_j-u_0〉.$$

(2.5)

By (2.3) and (2.5), we have

$$\underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C(t_j\mathrm{\Lambda },u_j),u_j-u_0〉\le 0.$$

(2.6)

There are two cases to consider. If $t_0=0$, then $C(t_j\mathrm{\Lambda },u_j)\to 0$ and so

$$\underset{j\to \mathrm{\infty }}{lim}〈C(t_j\mathrm{\Lambda },u_j),u_j-u_0〉=0.$$

(2.7)

Using (2.3), (2.4), and (2.7), we obtain

$$\underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈J{u}_j,u_j-u_0〉\le 0.$$

Since J satisfies condition ($S_{+}$), we have

$$u_j\to u_0,$$

which implies by the demicontinuity of $T_s$ and C and the continuity of J

$$T_s{u}_j\rightharpoonup T_s{u}_0,C(t_j\mathrm{\Lambda },u_j)\rightharpoonup C(0,u_0),\text{and}J{u}_j\to J{u}_0.$$

This means that $H(t_j,u_j)\rightharpoonup H(0,u_0)$. Now, let $t_0>0$. We have

$$〈C(t_j\mathrm{\Lambda },u_j),u_j-u_0〉=〈C(t_j\mathrm{\Lambda },u_j)-C(t_0\mathrm{\Lambda },u_j),u_j-u_0〉+〈C(t_0\mathrm{\Lambda },u_j),u_j-u_0〉,$$

which implies

$$\begin{array}{c}\underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C(t_0\mathrm{\Lambda },u_j),u_j-u_0〉\hfill \\ \le \underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C(t_j\mathrm{\Lambda },u_j),u_j-u_0〉+\underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}sup}[-〈C(t_j\mathrm{\Lambda },u_j)-C(t_0\mathrm{\Lambda },u_j),u_j-u_0〉]\hfill \end{array}$$

and hence by (2.6)

$$\underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C(t_0\mathrm{\Lambda },u_j),u_j-u_0〉\le \underset{j\to \mathrm{\infty }}{lim}\parallel C(t_j\mathrm{\Lambda },u_j)-C(t_0\mathrm{\Lambda },u_j)\parallel \parallel u_j-u_0\parallel =0.$$

Since C satisfies condition ($S_{+}$), we get $u_j\to u_0$ from which $T_s{u}_j\rightharpoonup T_s{u}_0$, $C(t_j\mathrm{\Lambda },u_j)\rightharpoonup C(t_0\mathrm{\Lambda },u_0)$, and $J{u}_j\to J{u}_0$. Consequently, $H(t_j,u_j)\rightharpoonup H(t_0,u_0)$. We have just shown that the homotopy H is of class ($S_{+}$).

We are now ready to apply the degree theory of Skrypnik [14, 15]. If $d_S$ denotes the Skrypnik degree, the homotopy invariance property of $d_S$ implies that

$$\begin{array}{rcl}{d}_S(T_s+C(\lambda ,\cdot )+\epsilon J,\mathrm{\Omega },0)& =& d_S(H(t,\cdot ),\mathrm{\Omega },0)\\ =& d_S(T_s+\epsilon J,\mathrm{\Omega },0)\\ =& 1\end{array}$$

for all $\lambda \in [0,\mathrm{\Lambda }]$. The last equality follows from Theorem 3 in [3], based on the fact that $T_s+\epsilon J$ is demicontinuous, bounded, injective, and satisfies condition ($S_{+}$) and $〈T_{s}x+\epsilon Jx,x〉\ge 0$ for all $x\in \partial \mathrm{\Omega }$. For every $\lambda \in (0,\mathrm{\Lambda }]$, the existence property of $d_S$ implies that there is a point x in Ω such that

$$T_{s}x+C(\lambda ,x)+\epsilon Jx=0,$$

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

In view of (2.1), let $\{s_n\}$ be a sequence in $(0,s_0)$ with $s_n\to 0$ and $\{({\lambda }_n,x_n)\}$ be the corresponding sequence in $(0,\mathrm{\Lambda }]\times \partial \mathrm{\Omega }$ such that

$$T_{s_n}{x}_n+C({\lambda }_n,x_n)+\epsilon J{x}_n=0.$$

(2.8)

Without loss of generality, we may suppose that

$${\lambda }_n\to {\lambda }_0,x_n\rightharpoonup x_0,C({\lambda }_n,x_n)\rightharpoonup c^{\ast }\text{and}J{x}_n\rightharpoonup j^{\ast },$$

where ${\lambda }_0\in [0,\mathrm{\Lambda }]$, $x_0\in X$, $c^{\ast }\in X^{\ast }$ and $j^{\ast }\in X^{\ast }$. To arrive at the conclusion of (a) in the next step, we may suppose that ${\lambda }_0\in (0,\mathrm{\Lambda }]$. In fact, when ${\lambda }_0=0$, we know that $0\notin Tx+C(0,x)+\epsilon Jx$ for all $x\in D(T)\cap \partial \mathrm{\Omega }$ because $T+\epsilon J$ is injective on $D(T)\cap \overline{\mathrm{\Omega }}$ and $0\in (T+\epsilon J)(D(T)\cap \mathrm{\Omega })$. For our aim, we will now show that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C({\lambda }_n,x_n),x_n-x_0〉\le 0.$$

(2.9)

Assume on the contrary that there exists a subsequence of $\{x_n\}$, denoted again by $\{x_n\}$, such that

$$\underset{n\to \mathrm{\infty }}{lim}〈C({\lambda }_n,x_n),x_n-x_0〉>0.$$

Note by (2.8) and the monotonicity of J that

$$〈T_{s_n}{x}_n,x_n-x_0〉\le -〈C({\lambda }_n,x_n),x_n-x_0〉-\epsilon 〈J{x}_0,x_n-x_0〉$$

and so

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈T_{s_n}{x}_n,x_n-x_0〉<0.$$

Hence, it follows from $T_{s_n}{x}_n\rightharpoonup -c^{\ast }-\epsilon j^{\ast }$ that

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈T_{s_n}{x}_n,x_n〉& =& \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}[〈T_{s_n}{x}_n,x_n-x_0〉+〈T_{s_n}{x}_n,x_0〉]\\ <& 〈-c^{\ast }-\epsilon j^{\ast },x_0〉.\end{array}$$

(2.10)

Let $x\in D(T)$ and $u^{\ast }\in Tx$ be arbitrary. Since T is monotone, $T_{s_n}{x}_n\in T{J}_{s_n}{x}_n$, and $J_{s_n}=I-s_n{J}^{-1}{T}_{s_n}$, we have

$$〈T_{s_n}{x}_n-u^{\ast },x_n-s_n{J}^{-1}{T}_{s_n}{x}_n-x〉\ge 0,$$

which implies

$$〈T_{s_n}{x}_n,x_n〉\ge s_n{\parallel T_{s_n}{x}_n\parallel }^2+〈T_{s_n}{x}_n,x〉+〈u^{\ast },x_n-x〉-s_n〈u^{\ast },J^{-1}{T}_{s_n}{x}_n〉.$$

Since $\{T_{s_n}{x}_n\}$ and $J^{-1}$ are bounded and $s_n\to 0$, we have

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}〈T_{s_n}{x}_n,x_n〉\ge 〈-c^{\ast }-\epsilon j^{\ast },x〉+〈u^{\ast },x_0-x〉.$$

(2.11)

Combining (2.10) and (2.11), we get

$$〈-c^{\ast }-\epsilon j^{\ast }-u^{\ast },x_0-x〉>0\text{for all }x\in D(T)\text{ and }{u}^{\ast }\in Tx.$$

(2.12)

Since T is maximal monotone, we have $x_0\in D(T)$ and $-c^{\ast }-\epsilon j^{\ast }\in T{x}_0$. Letting $x=x_0\in D(T)$ in (2.12) yields a contradiction. Thus, (2.9) holds.

Since $C(t,x)$ is continuous in t uniformly with respect to $x\in \overline{\mathrm{\Omega }}$ and ${\lambda }_n\to {\lambda }_0$, we have by (2.9)

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C({\lambda }_0,x_n),x_n-x_0〉\le 0,$$

(2.13)

by observing that

$$〈C({\lambda }_n,x_n),x_n-x_0〉=〈C({\lambda }_n,x_n)-C({\lambda }_0,x_n),x_n-x_0〉+〈C({\lambda }_0,x_n),x_n-x_0〉.$$

Since C satisfies condition ($S_{+}$), it follows from (2.13) that

$$x_n\to x_0\in \partial \mathrm{\Omega },$$

which implies by (2.8)

$$T_{s_n}{x}_n\rightharpoonup -C({\lambda }_0,x_0)-\epsilon J{x}_0$$

and as above

$$J_{s_n}{x}_n=x_n-s_n{J}^{-1}{T}_{s_n}{x}_n\to x_0.$$

Since T is maximal monotone and $T_{s_n}{x}_n\in T{J}_{s_n}{x}_n$, Lemma 1.1 implies that $x_0\in D(T)$ and

$$0\in T{x}_0+C({\lambda }_0,x_0)+\epsilon J{x}_0.$$

• (b) According to the statement (a), for a sequence $\{{\epsilon }_n\}$ in $(0,{\epsilon }_0]$ with ${\epsilon }_n\to 0$, we can choose sequences $\{{\lambda }_n\}$ in $(0,\mathrm{\Lambda }]$, $\{x_n\}$ in $D(T)\cap \partial \mathrm{\Omega }$, and $\{u_n^{\ast }\}$ in $X^{\ast }$ with $u_n^{\ast }\in T{x}_n$ such that

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

  (2.14)

We may suppose that

$${\lambda }_n\to {\lambda }_0,x_n\rightharpoonup x_0,C({\lambda }_n,x_n)\rightharpoonup c^{\ast }\text{and}J{x}_n\rightharpoonup j^{\ast },$$

where ${\lambda }_0\in [0,\mathrm{\Lambda }]$, $x_0\in X$, $c^{\ast }\in X^{\ast }$, and $j^{\ast }\in X^{\ast }$. Note that ${\lambda }_0\in (0,\mathrm{\Lambda }]$. Indeed, if ${\lambda }_0=0$, then (2.14) implies $u_n^{\ast }\to 0$ which gives by condition ($S_q$) on $D(T)\cap \partial \mathrm{\Omega }$, $x_n\to x_0\in \partial \mathrm{\Omega }$ and therefore $x_0\in D(T)$ and $0\in T{x}_0$, which contradicts the hypothesis that $0\notin T(D(T)\cap \partial \mathrm{\Omega })$.

To show that $x_n\to x_0$, we first claim that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C({\lambda }_n,x_n),x_n-x_0〉\le 0.$$

(2.15)

Assume the contrary. So, we may suppose that

$$\underset{n\to \mathrm{\infty }}{lim}〈C({\lambda }_n,x_n),x_n-x_0〉>0.$$

Hence, it follows from (2.14) that $u_n^{\ast }\rightharpoonup -c^{\ast }$ and

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈u_n^{\ast },x_n-x_0〉<0,$$

which imply

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈u_n^{\ast },x_n〉<〈-c^{\ast },x_0〉.$$

(2.16)

For every $x\in D(T)$ and every $u^{\ast }\in Tx$, we obtain from the monotonicity of T that

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}〈u_n^{\ast },x_n〉& \ge & \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}[〈u_n^{\ast },x〉+〈u^{\ast },x_n-x〉]\\ \ge & 〈-c^{\ast },x〉+〈u^{\ast },x_0-x〉,\end{array}$$

which implies along with (2.16)

$$〈-c^{\ast }-u^{\ast },x_0-x〉>0.$$

(2.17)

Since T is maximal monotone, we have $x_0\in D(T)$ and $-c^{\ast }\in T{x}_0$. Letting $x=x_0$ in (2.17), we have a contradiction. Thus, (2.15) is true.

As above, we can deduce from (2.15) that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C({\lambda }_0,x_n),x_n-x_0〉\le 0.$$

Since C satisfies condition ($S_{+}$) and is demicontinuous, we obtain from (2.14) that

$$x_n\to x_0\text{and}{u}_n^{\ast }\rightharpoonup -C({\lambda }_0,x_0).$$

We conclude that $x_0\in D(T)\cap \partial \mathrm{\Omega }$ and

$$0\in T{x}_0+C({\lambda }_0,x_0).$$

This completes the proof. □

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

Corollary 2.2 Let T, Ω, Λ, $s_0$, and ${\epsilon }_0$ be as in Theorem  2.1. Suppose that $C:\overline{\mathrm{\Omega }}\to X^{\ast }$ is a demicontinuous bounded operator which satisfies condition ($S_{+}$).

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

($\mathcal{P}$) For every $s\in (0,s_0)$, there exists a $\lambda \in (0,\mathrm{\Lambda }]$ such that the equation

$$T_{s}x+\lambda Cx+\epsilon Jx=0$$

has no solution in Ω. Then there exists a $({\lambda }_{\epsilon },x_{\epsilon })\in (0,\mathrm{\Lambda }]\times (D(T)\cap \partial \mathrm{\Omega })$ such that

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

• (b) If $0\notin T(D(T)\cap \partial \mathrm{\Omega })$, T satisfies condition ($S_q$) on $D(T)\cap \partial \mathrm{\Omega }$, and property ($\mathcal{P}$) is fulfilled for all $\epsilon \in (0,{\epsilon }_0]$, then there exists a $({\lambda }_0,x_0)\in (0,\mathrm{\Lambda }]\times (D(T)\cap \partial \mathrm{\Omega })$ such that

  $$0\in T{x}_0+{\lambda }_{0}C{x}_0.$$

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

$$\tilde{C}(\lambda ,x)=\lambda Cx\text{for }(\lambda ,x)\in [0,\mathrm{\Lambda }]\times \overline{\mathrm{\Omega }}.$$

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

3 Implicit eigenvalue problem about densely defined operators

In this section, we study the implicit eigenvalue problem for densely defined perturbations of maximal monotone operators, based on the degree theories of Kartsatos and Skrypnik.

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

In a regularization method by the duality operator, we establish a new result on the existence of eigenvalues, by applying topological degree for densely defined operators in [8, 9].

Theorem 3.1 Let Ω be a bounded open set in X with $0\in \mathrm{\Omega }$ and L a dense subspace of X. Let $T:D(T)\subset X\to 2^{X^{\ast }}$ be a maximal monotone multi-valued operator with $0\in D(T)$ and $0\in T(0)$. Let Λ, $s_0$, and ${\epsilon }_0$ be positive numbers. Assume that $C:[0,\mathrm{\Lambda }]\times D(C)\to X^{\ast }$ is a single-valued operator with $L\subset D(C)\subset X$ such that $C(0,x)=0$ for all $x\in D(C)$ and $C(t,x)$ is continuous in t uniformly with respect to $x\in D(C)$. Assume further that

(c1) C is uniformly quasibounded,

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

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

Then the following statements hold:

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

($\mathcal{P}$) For every $s\in (0,s_0)$, there exists a $\lambda \in (0,\mathrm{\Lambda }]$ such that the equation

$$T_{s}x+C(\lambda ,x)+\epsilon J_{\psi }x=0$$

has no solution in $D(C)\cap \mathrm{\Omega }$. Then there is a $({\lambda }_{\epsilon },x_{\epsilon })\in (0,\mathrm{\Lambda }]\times (D(T)\cap D(C)\cap \partial \mathrm{\Omega })$ such that

$$0\in T{x}_{\epsilon }+C({\lambda }_{\epsilon },x_{\epsilon })+\epsilon J_{\psi }{x}_{\epsilon }.$$

• (b) If $0\notin T(D(T)\cap \partial \mathrm{\Omega })$, T satisfies condition ($S_q$) on $D(T)\cap \partial \mathrm{\Omega }$, and property ($\mathcal{P}$) is fulfilled for all $\epsilon \in (0,{\epsilon }_0]$, then there exists a $({\lambda }_0,x_0)\in (0,\mathrm{\Lambda }]\times (D(T)\cap D(C)\cap \partial \mathrm{\Omega })$ such that

  $$0\in T{x}_0+C({\lambda }_0,x_0).$$

Proof (a) First, we prove that for every $s\in (0,s_0)$, there exists a $({\lambda }_0,x_0)\in (0,\mathrm{\Lambda }]\times (D(C)\cap \partial \mathrm{\Omega })$ such that

$$T_s{x}_0+C({\lambda }_0,x_0)+\epsilon J_{\psi }{x}_0=0.$$

(3.1)

Assume on the contrary that for some $s\in (0,s_0)$ and for every $\lambda \in (0,\mathrm{\Lambda }]$, the following holds:

$$T_{s}x+C(\lambda ,x)+\epsilon J_{\psi }x\ne 0\text{for all }x\in D(C)\cap \partial \mathrm{\Omega }.$$

(3.2)

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

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

($c_1^t$) $\{C^t\}$ is uniformly strongly quasibounded with respect to $T^t$, i.e., for every $\ell >0$ there exists a constant $K(\ell )>0$ such that for all $u\in L$ with $\parallel u\parallel \le \ell$ and all $t\in [0,1]$,

$$〈T^{t}u+C^{t}u,u〉\le 0\text{implies}\parallel C^{t}u\parallel \le K(\ell ).$$

This follows trivially from (c1) and the fact that $T_s$ and $J_{\psi }$ are monotone and $J_{\psi }$ is bounded.

($c_2^t$) For every pair of sequences $\{t_j\}$ in $[0,1]$ and $\{u_j\}$ in L such that $t_j\to t_0$, $u_j\rightharpoonup u_0$, $C^{t_j}{u}_j\rightharpoonup h_0^{\ast }$ and

$$\underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C^{t_j}{u}_j,u_j-u_0〉\le 0,〈T^{t_j}{u}_j+C^{t_j}{u}_j,u_j〉\le 0,$$

(3.3)

where $t_0\in [0,1]$, $u_0\in X$, and $h_0^{\ast }\in X^{\ast }$, we have $u_j\to u_0$, $u_0\in D(C)$ and $C^{t_0}{u}_0=h_0^{\ast }$.

For this, there are two cases to consider. If $t_0=0$, then the second inequality in (3.3) implies

$$\epsilon \psi (\parallel u_j\parallel )\parallel u_j\parallel =\epsilon 〈J_{\psi }{u}_j,u_j〉\le 〈T_s{u}_j,u_j〉+\epsilon 〈J_{\psi }{u}_j,u_j〉\le -〈C(t_j\mathrm{\Lambda },u_j),u_j〉\to 0$$

and hence $u_j\to 0$, $u_0=0\in D(C)$, and $C^0{u}_0=0=h_0^{\ast }$. Now let $t_0>0$. From the first inequality in (3.3) it follows that

$$〈C^{t_j}{u}_j,u_j-u_0〉\ge 〈C(t_j\mathrm{\Lambda },u_j),u_j-u_0〉+\epsilon 〈J_{\psi }{u}_0,u_j-u_0〉$$

and so

$$\underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C(t_j\mathrm{\Lambda },u_j),u_j-u_0〉\le 0.$$

Combining this with

$$〈C(t_j\mathrm{\Lambda },u_j),u_j-u_0〉=〈C(t_j\mathrm{\Lambda },u_j)-C(t_0\mathrm{\Lambda },u_j),u_j-u_0〉+〈C(t_0\mathrm{\Lambda },u_j),u_j-u_0〉,$$

we have

$$\begin{array}{rcl}\underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C(t_0\mathrm{\Lambda },u_j),u_j-u_0〉& \le & \underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}sup}[-〈C(t_j\mathrm{\Lambda },u_j)-C(t_0\mathrm{\Lambda },u_j),u_j-u_0〉]\\ \le & \underset{j\to \mathrm{\infty }}{lim}\parallel C(t_j\mathrm{\Lambda },u_j)-C(t_0\mathrm{\Lambda },u_j)\parallel \parallel u_j-u_0\parallel =0.\end{array}$$

In view of $C^{t_j}{u}_j\rightharpoonup h_0^{\ast }$, we can find a subsequence of $\{u_j\}$, denoted again by $\{u_j\}$, such that $C(t_j\mathrm{\Lambda },u_j)\rightharpoonup h_1^{\ast }$ and $J_{\psi }{u}_j\rightharpoonup h_2^{\ast }$ for some $h_1^{\ast },h_2^{\ast }\in X^{\ast }$. Note that $C(t_0\mathrm{\Lambda },u_j)\rightharpoonup h_1^{\ast }$ and $h_0^{\ast }=h_1^{\ast }+\epsilon h_2^{\ast }$. Hence (c2) implies that $u_j\to u_0$, $u_0\in D(C)$, and $C(t_0\mathrm{\Lambda },u_0)=h_1^{\ast }$ and so $C^{t_0}{u}_0=C(t_0\mathrm{\Lambda },u_0)+\epsilon J_{\psi }{u}_0=h_0^{\ast }$. Thus, condition ($c_2^t$) is satisfied in both cases.

($c_3^t$) For every $F\in \mathcal{F}(L)$ and $v\in L$, the function , $\tilde{c}(F,v)(t,u)=〈C^{t}u,v〉$, is continuous.

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

Following Kartsatos and Skrypnik [8, 9], we can use the homotopy invariance property of the degree d with respect to the bounded open set Ω as follows:

$$\begin{array}{rcl}d(T_s+C(\lambda ,\cdot )+\epsilon J_{\psi },\mathrm{\Omega },0)& =& d(T_s+\epsilon J_{\psi },\mathrm{\Omega },0)\\ =& 1\end{array}$$

for all $\lambda \in [0,\mathrm{\Lambda }]$. The latter follows from Theorem 3 in [3], noticing that $T_s+\epsilon J_{\psi }$ is demicontinuous, bounded, strictly monotone and satisfies condition ($S_{+}$). For every $\lambda \in (0,\mathrm{\Lambda }]$, the existence property of the degree implies that

$$T_{s}x+C(\lambda ,x)+\epsilon J_{\psi }x=0\text{for some }x\in D(C)\cap \mathrm{\Omega },$$

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

According to assertion (3.1), let $\{s_n\}$ be a sequence in $(0,s_0)$ with $s_n\to 0$ and $\{({\lambda }_n,x_n)\}$ be a sequence in $(0,\mathrm{\Lambda }]\times (D(C)\cap \partial \mathrm{\Omega })$ such that

$$T_{s_n}{x}_n+C({\lambda }_n,x_n)+\epsilon J_{\psi }{x}_n=0.$$

(3.4)

Without loss of generality, we may suppose that

$${\lambda }_n\to {\lambda }_0,x_n\rightharpoonup x_0,C({\lambda }_n,x_n)\rightharpoonup c^{\ast }\text{and}{J}_{\psi }{x}_n\rightharpoonup j^{\ast },$$

where ${\lambda }_0\in [0,\mathrm{\Lambda }]$, $x_0\in X$, $c^{\ast }\in X^{\ast }$ and $j^{\ast }\in X^{\ast }$. We may consider the case that ${\lambda }_0\in (0,\mathrm{\Lambda }]$ as the conclusion of (a) does not hold for ${\lambda }_0=0$. In order to apply the condition (${\tilde{S}}_{+}$), we first show that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C({\lambda }_n,x_n),x_n-x_0〉\le 0.$$

(3.5)

Assume on the contrary that there exists a subsequence of $\{x_n\}$, denoted again by $\{x_n\}$, such that

$$\underset{n\to \mathrm{\infty }}{lim}〈C({\lambda }_n,x_n),x_n-x_0〉>0.$$

Note by (3.4) and the monotonicity of $J_{\psi }$ that

$$〈T_{s_n}{x}_n,x_n-x_0〉\le -〈C({\lambda }_n,x_n),x_n-x_0〉-\epsilon 〈J_{\psi }{x}_0,x_n-x_0〉$$

and so

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈T_{s_n}{x}_n,x_n-x_0〉<0.$$

Hence, it follows from $T_{s_n}{x}_n\rightharpoonup -c^{\ast }-\epsilon j^{\ast }$ that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈T_{s_n}{x}_n,x_n〉<〈-c^{\ast }-\epsilon j^{\ast },x_0〉.$$

(3.6)

For every $x\in D(T)$ and every $u^{\ast }\in Tx$, it is shown as in the proof of part (a) of Theorem 2.1 that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}〈T_{s_n}{x}_n,x_n〉\ge 〈-c^{\ast }-\epsilon j^{\ast },x〉+〈u^{\ast },x_0-x〉,$$

which implies in view of (3.6)

$$〈-c^{\ast }-\epsilon j^{\ast }-u^{\ast },x_0-x〉>0.$$

(3.7)

Since T is maximal monotone, we have $x_0\in D(T)$ and $-c^{\ast }-\epsilon j^{\ast }\in T{x}_0$. Letting $x=x_0$ in (3.7), we have a contradiction. Thus, (3.5) holds.

As before, we can deduce from (3.5) that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C({\lambda }_0,x_n),x_n-x_0〉\le 0.$$

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

$$x_n\to x_0,x_0\in D(C),\text{and}C({\lambda }_0,x_0)=c^{\ast }.$$

(3.8)

Combining (3.4) and (3.8), we obtain

$$T_{s_n}{x}_n\rightharpoonup -C({\lambda }_0,x_0)-\epsilon J_{\psi }{x}_0$$

and

$$J_{s_n}{x}_n\to x_0.$$

Since T is maximal monotone and $T_{s_n}{x}_n\in T{J}_{s_n}{x}_n$, Lemma 1.1 implies that $x_0\in D(T)\cap D(C)\cap \partial \mathrm{\Omega }$ and

$$0\in T{x}_0+C({\lambda }_0,x_0)+\epsilon J_{\psi }{x}_0.$$

• (b) In view of (a), for a sequence $\{{\epsilon }_n\}$ in $(0,{\epsilon }_0]$ with ${\epsilon }_n\to 0$, we take sequences $\{{\lambda }_n\}$ in $(0,\mathrm{\Lambda }]$, $\{x_n\}$ in $D(T)\cap D(C)\cap \partial \mathrm{\Omega }$ and $\{u_n^{\ast }\}$ in $X^{\ast }$ with $u_n^{\ast }\in T{x}_n$ such that

  $$u_n^{\ast }+C({\lambda }_n,x_n)+{\epsilon }_n{J}_{\psi }{x}_n=0.$$

  (3.9)

We may suppose that

$${\lambda }_n\to {\lambda }_0,x_n\rightharpoonup x_0,C({\lambda }_n,x_n)\rightharpoonup c^{\ast }\text{and}{J}_{\psi }{x}_n\rightharpoonup j^{\ast },$$

where ${\lambda }_0\in [0,\mathrm{\Lambda }]$, $x_0\in X$, $c^{\ast }\in X^{\ast }$ and $j^{\ast }\in X^{\ast }$. Note that ${\lambda }_0\in (0,\mathrm{\Lambda }]$. As in the proof of part (b) of Theorem 2.1, a similar argument shows that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C({\lambda }_n,x_n),x_n-x_0〉\le 0$$

(3.10)

and so in a usual way

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈C({\lambda }_0,x_n),x_n-x_0〉\le 0.$$

Hence, it follows from (c2) that

$$x_n\to x_0\in \partial \mathrm{\Omega },x_0\in D(C),\text{and}C({\lambda }_0,x_0)=c^{\ast }$$

which implies by (3.9)

$$u_n^{\ast }\rightharpoonup -C({\lambda }_0,x_0).$$

Consequently, we obtain from the maximal monotonicity of T that $x_0\in D(T)$ and

$$0\in T{x}_0+C({\lambda }_0,x_0).$$

This completes the proof. □

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