Some eigenvalue results for perturbations of maximal monotone operators

Eigenvalue theory is closely related to the problem of solving nonlinear equations which was initiated by Krasnosel’skii [1] for compact operators. Regarding maximal monotone operators, it has been extensively investigated by many researchers in various aspects, with applications to evolution equations and differential equations; see, e.g., [2–5]. The study was mostly based on degree theories for appropriate classes of operators and the usual method of regularization by means of the duality operator; see [6–10].

has no solution in $D(T)\cap \mathrm{\Omega }$, where J denotes the duality operator. This property has a close relation to the use of topological degree in eigenvalue theory by the regularization method. It is shown in [3] that two conditions about the weak closure of a certain set consisting of normalized vectors and the asymptotic behavior of the operator C at infinity of λ, called normalized conditions, are main ingredients in solving an eigenvalue problem. In this connection, we are now interested in finding eigenvectors under normalized conditions more concrete than property ($\mathcal{P}$).

The purpose of this paper is to establish the existence of solutions for the above eigenvalue problems under normalized conditions, motivated by the works of Kartsatos and Skrypnik [3, 4]. We first study implicit eigenvalue problem (IE), where C is assumed to be a bounded demicontinuous operator satisfying condition ($S_{+}$). For this, a key tool is the Browder degree given in [6]. Next, we consider two types of the operator C for eigenvalue problem (E). For the one, we apply the Browder degree theory for nonlinear operators of the form $T+f$ with T maximal monotone and f bounded with condition ($S_{+}$) introduced in [7]. In the other case, where C is a generalized pseudomonotone, quasibounded, and densely defined operator, we solve this problem by using the Kartsatos-Skrypnik degree theory for densely defined (${\tilde{S}}_{+}$)-perturbations of maximal monotone operators developed in [8].

This paper is organized as follows. In Section 2, we study the solvability of implicit eigenvalue problem (IE) with normalized conditions based on the Browder degree theory. Section 3 contains an eigenvalue result for problem (E) as a special case of (IE). In Section 4, we deal with eigenvalue problem (E) for densely defined perturbations of maximal monotone operators under normalized conditions.

Let X be a real Banach space with dual space $X^{\ast }$, Ω be a nonempty subset of X, and Y be another real Banach space. 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 $F:\mathrm{\Omega }\to Y$ 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 $F{x}_n\rightharpoonup F{x}_0$.

that $x\in D(T)$ and $Tx=x^{\ast }$.

we have $x_0\in D(T)$, $T{x}_0=h^{\ast }$, and ${lim}_{n\to \mathrm{\infty }}〈T{x}_n,x_n〉=〈T{x}_0,x_0〉$.

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

we have $x_n\to x_0$.

we have $x_n\to x_0$.

We say that T 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 $T{x}_n\to h^{\ast }$, we have $x_n\to x_0$.

We say that T satisfies condition ($T_{\mathrm{\infty }}^{(0)}$) on a set $M\subset D(T)$ if for every sequence $\{x_n\}$ in M with $\parallel T{x}_n\parallel \to \mathrm{\infty }$, we have ${\parallel T{x}_n\parallel }^{-1}T{x}_n\rightharpoonup 0$.

It is obvious from the definitions that ($S_{+}$) implies ($S_0$) and ($S_0$) implies ($S_q$). Note that if T satisfies condition (${\tilde{S}}_{+}$) and X is reflexive, then T is generalized pseudomonotone. For the above definitions, we refer to, e.g., [3, 4, 8], and [5].

Let $C:[0,\mathrm{\infty })\times M\to X^{\ast }$ 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 $t_0\in [0,\mathrm{\infty })$ and for every sequence $\{t_n\}$ in $[0,\mathrm{\infty })$ 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 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 $X^{\ast }$ are locally uniformly convex.

where $\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is continuous, strictly increasing, $\psi (0)=0$, and $\psi (t)\to \mathrm{\infty }$ as $t\to \mathrm{\infty }$ called a gauge function. If ψ is the identity map I, then $J:=J_I$ is called a normalized duality operator. It is known in [14] that $J_{\psi }$ is continuous, bounded, surjective, strictly monotone, maximal monotone and satisfies condition ($S_{+}$).

The following demiclosedness property of maximal monotone operators will be frequently used; see [5].

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

In this section, we establish the existence of a solution for a nonlinear implicit eigenvalue problem under normalized conditions by using the Browder degree theory for class ($S_{+}$).

Recall that a mapping $H:[0,1]\times \overline{\mathrm{\Omega }}\to X^{\ast }$ is of class ($S_{+}$) if the following condition holds:

we have $u_j\to u_0$ and $H(t_j,u_j)\rightharpoonup H(t_0,u_0)$; see [2, 6].

As mentioned in the introduction, Kartsatos and Skrypnik [4] gave the following result provided that property ($\mathcal{P}$) is fulfilled. In a more concrete situation, we adopt the normalization method considered in [3].

Theorem 2.1 Let Ω be a bounded open set in X with $0\in \mathrm{\Omega }$. Let $T:D(T)\subset X\to X^{\ast }$ be a maximal monotone operator such that the closure of Ω is included in the interior of $D(T)$ with $T(0)=0$ and T satisfies condition ($T_{\mathrm{\infty }}^{(0)}$) on $\overline{\mathrm{\Omega }}$. Assume that $C:[0,\mathrm{\infty })\times \overline{\mathrm{\Omega }}\to X^{\ast }$ 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 $x\in \overline{\mathrm{\Omega }}$. Further assume that

(c2) ${lim}_{\lambda \to \mathrm{\infty }}m(\lambda )=\mathrm{\infty }$, where $m(\lambda )=inf\{\parallel C(\lambda ,x)\parallel :x\in \overline{\mathrm{\Omega }}\}$.

has a solution $({\lambda }_0,x_0)$ in $(0,\mathrm{\infty })\times \partial \mathrm{\Omega }$.

Assume the contrary. For a sequence $\{{\mathrm{\Lambda }}_n\}$ in $(0,\mathrm{\infty })$ with ${\mathrm{\Lambda }}_n\to \mathrm{\infty }$, the following occurs:

However, $\parallel C({\mathrm{\Lambda }}_n,x_n)\parallel \to \mathrm{\infty }$ by (c2) implies $\parallel T{x}_n\parallel \to \mathrm{\infty }$, which is a contradiction to condition ($T_{\mathrm{\infty }}^{(0)}$). Hence assertion (2.1) holds.

Since C satisfies condition ($S_{+}$), we get $u_j\to u_0$ from which $T{u}_j\rightharpoonup T{u}_0$, $C(t_j\mathrm{\Lambda },u_j)\rightharpoonup C(t_0\mathrm{\Lambda },u_0)$, 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 mapping H is of class ($S_{+}$).

where ${\lambda }_0\in [0,\mathrm{\infty }]$, $x_0\in X$, and $c^{\ast }\in X^{\ast }$. Note that ${\lambda }_0$ belongs to $(0,\mathrm{\infty })$. In fact, if ${\lambda }_0=0$, then $C({\lambda }_n,x_n)\to 0$ implies $T{x}_n\to 0$. Since T satisfies condition ($S_q$) on ∂ Ω, we obtain that $x_n\to x_0\in \partial \mathrm{\Omega }$ and therefore $T{x}_0=0$, which contradicts the hypothesis that $0\notin T(\partial \mathrm{\Omega })$. If ${\lambda }_0=\mathrm{\infty }$, then (c2) implies $\parallel C({\lambda }_n,x_n)\parallel \to \mathrm{\infty }$ and so $\parallel T{x}_n\parallel \to \mathrm{\infty }$. As in (2.3), a similar argument proves that ${\parallel T{x}_n\parallel }^{-1}T{x}_n$ converges weakly to some nonzero vector, which contradicts condition ($T_{\mathrm{\infty }}^{(0)}$). Thus we have shown that ${\lambda }_0\in (0,\mathrm{\infty })$.

By the maximal monotonicity of T, we have $x_0\in D(T)$ and $T{x}_0=-c^{\ast }$. Letting $x=x_0$ in (2.13), we get a contradiction. Therefore, (2.11) is true.

Since C satisfies condition ($S_{+}$) and ${\lambda }_0\in (0,\mathrm{\infty })$, we have $x_n\to x_0\in \partial \mathrm{\Omega }$, which implies $C({\lambda }_n,x_n)\rightharpoonup C({\lambda }_0,x_0)$. Hence we obtain from (2.9) that $T{x}_n\rightharpoonup -C({\lambda }_0,x_0)$. By Lemma 1.1, we conclude that $T{x}_0+C({\lambda }_0,x_0)=0$. This completes the proof. □

Remark 2.2 In the proof of Theorem 2.1, the demicontinuity of T on $\overline{\mathrm{\Omega }}$ is needed to show that H is of class ($S_{+}$). This is guaranteed under an additional condition $\overline{\mathrm{\Omega }}\subset intD(T)$. Actually, local boundedness of T on $intD(T)$ implies the demicontinuity of T; see, e.g., [5].

In this section, we study a multiplicative eigenvalue problem as a special case of the implicit eigenvalue problem in the previous section. As a key tool, we employ the Browder degree for nonlinear operators of the form $T+f$ with T maximal monotone and f bounded with condition ($S_{+}$).

We give a variant of Corollary 1 in [4] under normalized conditions.

Theorem 3.1 Let Ω be a bounded open set in X with $0\in \mathrm{\Omega }$. Let $T:D(T)\subset X\to X^{\ast }$ be a maximal monotone operator with $0\in D(T)$ and $T(0)=0$ such that T satisfies condition ($T_{\mathrm{\infty }}^{(0)}$) on $D(T)\cap \overline{\mathrm{\Omega }}$. Assume that $C:\overline{\mathrm{\Omega }}\to X^{\ast }$ is a demicontinuous bounded operator which satisfies condition ($S_{+}$) and the two additional conditions:

(c2) $inf\{\parallel Cx\parallel :x\in \overline{\mathrm{\Omega }}\}$ is not equal to 0.

has a solution $({\lambda }_0,x_0)$ in $(0,\mathrm{\infty })\times (D(T)\cap \partial \mathrm{\Omega })$.

But (c2) implies $\parallel T{x}_n\parallel \to \mathrm{\infty }$, which contradicts condition ($T_{\mathrm{\infty }}^{(0)}$). Hence assertion (3.1) holds.

where ${\lambda }_0\in [0,\mathrm{\infty }]$, $x_0\in X$, and $c^{\ast }\in X^{\ast }$. Note that ${\lambda }_0$ belongs to $(0,\mathrm{\infty })$. Indeed, if ${\lambda }_0=0$, then it follows from $T{x}_n\to 0$ and condition ($S_q$) that $x_n\to x_0\in \partial \mathrm{\Omega }$ and therefore $x_0\in D(T)$ and $T{x}_0=0$, which contradicts the hypothesis that $0\notin T(D(T)\cap \partial \mathrm{\Omega })$. If ${\lambda }_0=\mathrm{\infty }$, then (c2) implies $\parallel {\lambda }_{n}C{x}_n\parallel \to \mathrm{\infty }$ and so $\parallel T{x}_n\parallel \to \mathrm{\infty }$. But we can show as above that ${\parallel T{x}_n\parallel }^{-1}T{x}_n$ converges weakly to some nonzero vector, which contradicts condition ($T_{\mathrm{\infty }}^{(0)}$).

By the maximal monotonicity of T, we get a contradiction. Therefore, (3.5) holds.

Since C satisfies condition ($S_{+}$) and is demicontinuous, we have $x_n\to x_0\in \partial \mathrm{\Omega }$, which implies $C{x}_n\rightharpoonup C{x}_0$. Hence we obtain from (3.3) that $T{x}_n\rightharpoonup -{\lambda }_{0}C{x}_0$. By Lemma 1.1, we conclude that $x_0\in D(T)\cap \partial \mathrm{\Omega }$ and $T{x}_0+{\lambda }_{0}C{x}_0=0$, what we wanted to prove. □

Remark 3.2 We point out that in Theorem 3.1, the condition $\overline{\mathrm{\Omega }}\subset intD(T)$ is not necessary to be assumed.

This section is devoted to the eigenvalue problem for densely defined quasibounded perturbations of maximal monotone operators in reflexive Banach spaces. To do this, we apply the Kartsatos-Skrypnik degree for densely defined (${\tilde{S}}_{+}$)-perturbations of maximal monotone operators developed in [8].

Recall that an operator $C:D(C)\subset X\to X^{\ast }$ is quasibounded if for every $S>0$ there exists a constant $K(S)>0$ such that for all $u\in D(C)$ with $\parallel u\parallel \le S$ and $〈Cu,u〉\le 0$, we have $\parallel Cu\parallel \le K(S)$.

As in Section 3, we employ a normalization method to obtain an eigenvalue result for generalized pseudomonotone operators.

Theorem 4.1 Let Ω be a bounded open set in X with $0\in \mathrm{\Omega }$ and L be a dense subspace of X. Let $T:D(T)\subset X\to X^{\ast }$ be a maximal monotone operator with $0\in D(T)$ and $T(0)=0$ which satisfies condition ($T_{\mathrm{\infty }}^{(0)}$) on $D(T)\cap \overline{\mathrm{\Omega }}$. Assume that $C:D(C)\subset X\to X^{\ast }$ is a generalized pseudomonotone quasibounded operator with $L\subset D(C)$. Furthermore, assume that

(h2) $inf\{\parallel Cx\parallel :x\in D(C)\cap \overline{\mathrm{\Omega }}\}$ is not equal to 0.

(h3) For every $F\in \mathcal{F}(L)$ and $v\in L$, the function $c(F,v):F\to \mathbb{R}$, $c(F,v)(u)=〈Cu,v〉$, is continuous on F, where $\mathcal{F}(L)$ denotes the set of all finite-dimensional subspaces of L.

has a solution $({\lambda }_0,x_0)$ in $(0,\mathrm{\infty })\times (D(T+C)\cap \partial \mathrm{\Omega })$.

For $t\in [0,1]$, we set $T^t:=T$ and $C^t:=t\mathrm{\Lambda }C+\epsilon J_{\psi }$, where $D(T^t)$ and $D(C^t)$ denote the domain of $T^t$ and $C^t$, respectively. In this case, $D(T^t)=D(T)$ for $t\in [0,1]$, $D(C^0)=X$ for $t=0$ and $D(C^t)=D(C)$ for $t\in (0,1]$. Notice that the operators $C^0=\epsilon J_{\psi }$ and $C^1=\mathrm{\Lambda }C+\epsilon J_{\psi }$ satisfy condition (${\tilde{S}}_{+}$), based on the facts that C is generalized pseudomonotone and $J_{\psi }$ is bounded and satisfies condition ($S_{+}$).

In the sense of Definition 4.2 in [8], we check the following conditions on two families $\{T^t\}$ and $\{C^t\}$. In fact, conditions on $\{T^t\}$ are obviously satisfied, with $T^t$ independent of t, due to maximal monotonicity of T, $0\in D(T)$, and $T(0)=0$.

($c_1^t$) Since $J_{\psi }$ is monotone and bounded, it follows from the quasiboundedness of C that $\{C^t\}$ is uniformly quasibounded.