\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2022 (2022), No. 63, pp. 1--25.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2022. This work is licensed under a CC BY 4.0 license.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2022/63\hfil Positively homogeneous maximal monotone operators] {Solvability of inclusions involving perturbations of positively homogeneous maximal \\ monotone operators} \author[D. Adhikari, A. Aryal, G. Bhatt, I. Kunwar, R. Puri, P. Ranabhat \hfil EJDE-2022/63\hfilneg] {Dhruba R. Adhikari, Ashok Aryal, Ghanshyam Bhatt, \\ Ishwari J. Kunwar, Rajan Puri, Min Ranabhat} \address{Dhruba R. Adhikari \newline Department of Mathematics, Kennesaw State University, Marietta, GA 30060, USA} \email{dadhikar@kennesaw.edu} \address{Ashok Aryal \newline Mathematics Department, Minnesota State University Moorhead, Moorhead, MN 56563, USA} \email{ashok.aryal@mnstate.edu} \address{Ghanshyam Bhatt\newline Department of Mathematical Sciences, Tennessee State University, Nashville, TN 37209, USA} \email{gbhatt@tnstate.edu} \address{Ishwari J. Kunwar \newline Department of Mathematics and Computer Science, Fort Valley State University, Fort Valley, GA 31030, USA} \email{kunwari@fvsu.edu} \address{Rajan Puri \newline Department of Mathematics, Wake Forest University, Winston-Salem, NC 27109, USA} \email{purir@wfu.edu} \address{Min Ranabhat\newline Department of Mathematical Sciences, University of Delaware, EWG 315, Newark, DE 19716, USA} \email{ranabhat@udel.edu} \thanks{Submitted April 1, 2022. Published August 30, 2022.} \subjclass[2020]{47H14, 47H05, 47H11} \keywords{Topological degree theory; operators of type $(S_+)$; \hfill\break\indent monotone operator; duality mapping; Yosida approximant} \begin{abstract} Let $X$ be a real reflexive Banach space and $X^*$ be its dual space. Let $G_1$ and $G_2$ be open subsets of $X$ such that $\overline G_2\subset G_1$, $0\in G_2$, and $G_1$ is bounded. Let $L: X\supset D(L)\to X^*$ be a densely defined linear maximal monotone operator, $A:X\supset D(A)\to 2^{X^*}$ be a maximal monotone and positively homogeneous operator of degree $\gamma>0$, $C:X\supset D(C)\to X^*$ be a bounded demicontinuous operator of type $(S_+)$ with respect to $D(L)$, and $T:\overline G_1\to 2^{X^*}$ be a compact and upper-semicontinuous operator whose values are closed and convex sets in $X^*$. We first take $L=0$ and establish the existence of nonzero solutions of $Ax+ Cx+ Tx\ni 0$ in the set $G_1\setminus G_2$. Secondly, we assume that $A$ is bounded and establish the existence of nonzero solutions of $Lx+Ax+Cx\ni 0$ in $G_1\setminus G_2$. We remove the restrictions $\gamma\in (0, 1]$ for $Ax+ Cx+ Tx\ni 0$ and $\gamma= 1$ for $Lx+Ax+Cx\ni 0$ from such existing results in the literature. We also present applications to elliptic and parabolic partial differential equations in general divergence form satisfying Dirichlet boundary conditions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{application}[theorem]{Application} \allowdisplaybreaks \section{Introduction and preliminaries}\label{S1} Let $X$ be a real reflexive Banach space and $X^*$ be its topological dual space. The symbol $2^{X^*}$ denotes the collection of all subsets of $X^*$. The norm on $X$ is denoted by $\|\cdot\|_X$. When there is no risk of misunderstanding, the norms on $X$ and $X^*$ are both denoted by $\|\cdot\|$. The pairing $\langle x^*,x\rangle$ denotes the value of the functional $x^*\in X^*$ at $x\in X$. The symbols $\partial Z, \textrm{Int} Z, \overline Z$ and $ \operatorname{co}Z$ denote the boundary, interior, closure, and convex hull of the set $Z \subset X$, respectively. The symbol $B_X(0,R)$ denotes the open ball of radius $R>0$ with center at $0$ in $X$. The symbols $\mathbb{R}$ and $\mathbb{R}_+$ denote $ (-\infty, \infty)$ and $[0,\infty)$, respectively. For a sequence $\{x_n\}$ in $X$ and $x_0\in X$, we denote by $x_n\to x_0$ and $x_n\rightharpoonup x_0$ the strong convergence and weak convergence, respectively. Given another real Banach $Y$, an operator $T : X\supset D(T)\to Y$ is said to be \emph{bounded} if it maps bounded subsets of the domain $D(T)$ onto bounded subsets of $Y$. The operator $T$ is said to be \emph{compact} if it maps bounded subsets of $D(T)$ onto relatively compact subsets in $Y$. The operator $T$ is said to be \emph{demicontinuous} if it is strong-to-weak continuous on $D(T)$. A multivalued operator $A$ from $X$ to $X^*$ is written as $A:X\supset D(A) \to 2^{X^*}$, where $D(A)=\{x\in X: Ax\neq\emptyset\}$ is the effective domain of $A$. Here, $Ax$ means $A(x)$, and these notations are used interchangeably in the sequel. We denote the graph of $A$ by $\operatorname{Gr}(A)$, i.e., $\operatorname{Gr}(A)=\{(x,y): x\in D(A), y\in Ax\}$. \begin{definition}\label{Hom} \rm An operator $A: X\supset D(A)\to 2^{X^*}$ is said to be \emph{positively homogeneous of degree $\gamma >0$} if $(x,y)\in \operatorname{Gr}(A)$ implies $sx\in D(A)$ for all $s\ge 0$ and $(sx, s^\gamma y)\in \operatorname{Gr}(A)$. \end{definition} \begin{remark}\rm An equivalent condition for an operator $A: X\supset D(A)\to 2^{X^*}$to be positively homogeneous of degree $\gamma >0$ is that $x\in D(A)$ implies $sx\in D(A)$ for all $s\ge 0$ and $s^\gamma Ax \subset A(sx)$. It follows that a positively homogeneous operator $A$ of degree $\gamma >0$ satisfies $0\in A(0)$. When $A$ is positively homogeneous of degree $\gamma >0$, it can be verified that $x\in D(A)$ implies $sx\in D(A)$ for all $s>0$ and $s^\gamma Ax = A(sx)$. However, in general, the property $s^\gamma Ax = A(sx)$ may not be true for $s=0$. For example, let $A: \mathbb{R}\supset {[0, \infty)} \to 2^\mathbb{R}$ be given by $$ A x=\begin{cases} (-\infty, 0] & \text{for } x=0\\ x^\gamma & \text{for } x>0. \end{cases} $$ Clearly, $A(0) = (-\infty, 0]\ne \{0\}$. \end{remark} A \emph{gauge} function is a strictly increasing continuous function $\varphi: \mathbb{R}_+ \to \mathbb{R}_+$ with $\varphi(0) = 0$ and $\varphi(r) \to \infty$ as $r\to\infty$. The \emph{duality mapping} of $X$ corresponding to a gauge function $\varphi $ is the mapping $J_\varphi:X \supset D(J_\varphi )\to 2^{X^*}$ defined by $$ J_\varphi x = \{x^*\in X^* : \langle x^*, x\rangle = \varphi(\|x\|)\|x\|, \; \|x^*\| = \varphi(\|x\|)\}, \quad x\in X. $$ The Hahn-Banach theorem ensures that $D(J_\varphi) = X$, and therefore $J_\varphi :X\to 2^{X^*}$ is, in general, a multivalued mapping. The duality mapping corresponding to the gauge function $\varphi (r) = r$ is called the \emph{normalized duality} mapping and denoted by $J$. It is well-known that the duality mapping $J_\varphi$ satisfies $$ J_\varphi x = \frac{\varphi(\|x\|)}{\|x\|} Jx, \quad x\in X\setminus \{0\}. $$ Since $J$ is homogeneous of degree $1$, we have \[ J_\varphi(s x) = \frac{\varphi(s\|x\|)}{\|x\|} Jx, \quad (s, x)\in \mathbb{R}_+\times (X\setminus \{0\}). \] In particular, when $\varphi(r) = r^{p-1}$, $1< p< \infty$, we obtain $J_\varphi x = \|x\|^{p-2} Jx$, $x\in X\setminus \{0\}$, which implies \[ J_\varphi(s x) = s^{p-1} J_\varphi x, \quad (s, x)\in \mathbb{R}_+\times X, \] i.e., $J_\varphi$ is positively homogeneous of degree $p-1$. When $X$ is reflexive and both $X$ and $X^*$ are strictly convex, the inverse $J^{-1}_\varphi$ of $J_\varphi$ is the duality mapping of $X^*$ with the gauge function $\varphi^{-1}(r) = r^{q-1}$, where $q$ is given by $1/p + 1/q = 1$. It is easy to verify that \begin{equation}\label{11} J_\varphi^{-1}(s x^*) = s^{q-1} J_\varphi^{-1} x^*, \quad (s, x^*)\in \mathbb{R}_+\times X^*. \end{equation} It is clear that $J_\varphi$ is positively homogeneous of degree $\gamma>0$ if and only if $\varphi$ is positively homogeneous of degree $\gamma>0$. Additional properties of duality mappings in connection with Banach space geometry can be found in Alber and Ryazantseva~\cite{Alber} and Cioranescu~\cite{Cioranescu}. \begin{definition}{\rm An operator $A:X\supset D(A)\to 2^{X^*}$ is said to be \emph{monotone} if for all $(x, u), (y, v)\in \operatorname{Gr}(A)$ we have $ \langle u - v,x-y\rangle \ge 0$. A monotone operator $A:X\supset D(A)\to 2^{X^*}$ is said to be \emph{maximal monotone} if $\operatorname{Gr}(A)$ is maximal in $X\times X^*$, when $X\times X^*$ is partially ordered by set inclusion.} \end{definition} In what follows, we assume that $X$ is reflexive and both $X$ and $X^*$ are strictly convex. It is well-known that the duality mapping $J_\varphi$ is maximal monotone. A monotone operator $A$ is maximal if and only if $R(A+\lambda J_\varphi) = X^*$ for all $\lambda\in (0,\infty)$ and all gauge functions $\varphi$. For a proof of this result for $\varphi(r) = r^{p-1}, 1
0$, $C$ is bounded demicontinuous of type $(S_+)$, and $T$ is of class $(P)$. This result extends an analogous result for $\gamma\in (0, 1]$ established in \cite{Adhikari2017} to an arbitrary degree of homogeneity $\gamma>0$. Another main result established in this section is the existence of nonzero solutions of $Lx+Ax+Cx\ni 0$, where $L$, $C$ are as above, and $A$ is a bounded maximal monotone and positively homogeneous of degree $\gamma>0$. This result extends an analogous result for $\gamma =1$ established in \cite{Adhikari2017} to an arbitrary degree of homogeneity $\gamma>0$. In Section~\ref{S4}, we present some applications of the theories developed in Section~\ref{S3} to elliptic and parabolic partial differential equations, in general, divergence form that include $p$-Laplacian with $1
0$ and each $x\in X$, the inclusion
\begin{equation} \label{Yosida-inclusion}
0\in J_\varphi(x_\lambda- x) + \lambda Ax_\lambda
\end{equation}
has a unique solution $x_\lambda\in D(A)$ (see Proposition~\ref{prop0} (i)).
We define $J_\lambda^\varphi : X\to D(A)\subset X$ and $A_\lambda^\varphi : X\to X^*$ by
\begin{equation}\label{Yosida}
J_\lambda^\varphi x := x_\lambda \quad \text{and}\quad
A_\lambda^\varphi x: = \dfrac{1}{\lambda} J_\varphi (x- J_\lambda^\varphi x), \quad x\in X.
\end{equation}
The operators $A_\lambda^\varphi$ and $J_\lambda^\varphi$ are variants of the standard
Yosida approximant $A_\lambda$ and resolvent $J_\lambda$ of $A$.
For each $x\in X$, we have
$$
A_\lambda^\varphi x\in A (J_\lambda^\varphi x) \quad \text{and} \quad
x = J_\lambda^\varphi x + J^{-1}_\varphi ( \lambda A_\lambda^\varphi x).
$$
When $\varphi(r) = r^{p-1}$, a splitting of $x$ in terms of $A_\lambda^\varphi$ and
$J_\lambda^\varphi$ is
\begin{equation}\label{splitting}
x = J_\lambda^\varphi x + \lambda^{q-1} J^{-1}_\varphi ( A_\lambda^\varphi x),
\end{equation} and therefore
\begin{equation}\label{closed-form}
A_\lambda^\varphi x= \left(A^{-1}+ \lambda^{q-1} J_\varphi^{-1}\right)^{-1} x, \quad x\in X.
\end{equation}
It is easy to verify that $A= A_\lambda^\varphi$ if and only if $A=0$.
In fact, if $A= 0$, then $J_\lambda^\varphi = I$, the identity operator on $X$.
Moreover, if $0\in D(A)$ and $0\in A(0)$, then $A_\lambda^\varphi 0 =0$.
The choice of an appropriate gauge function is essential for the main existence
results in this paper. The following proposition summarizes some important properties
of $A_\lambda^\varphi$ and $J_\lambda^\varphi$ along the lines of analogous properties of
$A_\lambda$ and $J_\lambda$. A complete proof is provided here for the reader's convenience.
\begin{proposition}\label{prop0}
Let $X$ be a strictly convex and reflexive Banach space with strictly convex dual $X^*$ and
$A:X\supset D(A)\to 2^{X^*}$ be a maximal monotone operator.
Then the following statements hold.
\begin{itemize}
\item[(i)] The operator $A_\lambda^\varphi$ is single-valued, monotone, bounded on bounded
subsets of $X$, and demicontinuous from $X$ to $X^*$.
\item[(ii)] For every $x\in D(A)$ and $\lambda >0$, we have
$$
\|A_\lambda^\varphi x\|\le |Ax|:= \inf\{\|x^*\|: x^*\in Ax \}.
$$
\item[(iii)] The operator $J_\lambda^\varphi $ is bounded on bounded subsets of $X$, demicontinuous
from $X$ to $ D(A)$, and
\[ %\label{resolvent}
\lim_{\lambda\to 0} J_\lambda^\varphi x = x \quad \text{for all }
x\in\overline{\operatorname{co}D(A)}.
\]
\item[(iv)] If $\lambda_n\to 0$, $x_n\rightharpoonup x$ in $X$, $A_{\lambda_n}^\varphi x_n \rightharpoonup y$ and
$$
\limsup_{n, m\to\infty} \langle A_{\lambda_n}^\varphi x_n- A_{\lambda_m}^\varphi x_m , x_n - x_m\rangle
\le 0,
$$
then $(x , y) \in \operatorname{Gr} (A)$ and
$$
\lim_{n, m\to\infty} \langle A_{\lambda_n}^\varphi x_n- A_{\lambda_m}^\varphi x_m , x_n - x_m\rangle =0.
$$
\item[(v)] For every sequence $\{\lambda_n\}$ with $\lambda_n\to 0$,
$A_{\lambda_n}^\varphi x\rightharpoonup A^{\{0\}}x$ for all $x\in D(A)$. In addition, if $X^*$
is uniformly convex, then $A_{\lambda_n}^\varphi x \to A^{\{0\}}x$ for all $x\in D(A)$.
\item[(vi)] If $\lambda_n\to 0$ and $x\not\in\overline{D(A)}$, then
$$
\lim_{n\to\infty} \|A_{\lambda_n}^\varphi x\| = \infty.
$$
\end{itemize}
\end{proposition}
\begin{proof}
(i) We first show that $J_\lambda^\varphi$ is single-valued.
Given $x\in X$ and $\lambda>0$, let $x_\lambda$ and $\tilde x_\lambda$ be solutions of
\eqref{Yosida-inclusion}. Take $y \in Ax_\lambda$ and $\tilde y \in A\tilde x_\lambda$
such that
$$
J_\varphi(x_\lambda- x) + \lambda y=0\quad \text{and} \quad
J_\varphi(\tilde x_\lambda- x) + \lambda \tilde y=0.
$$
This along with the monotonicity of $A$ and $J_\varphi$ implies
\begin{equation}\label{21}
\langle J_\varphi(x_\lambda- x) - J_\varphi(\tilde x_\lambda- x),
(x_\lambda- x)-(\tilde x_\lambda- x)\rangle = 0.
\end{equation}
Since $X$ is strictly convex, it follows that $J_\varphi$ is strictly monotone, i.e.,
for $u_1, u_2\in X$, we have
$$
\langle J_\varphi u_1 - J_\varphi u_2, u_1- u_2\rangle > 0 \text{ if and only if } u_1 \ne u_2.
$$
It follows from \eqref{21} that
$x_\lambda =\tilde x_\lambda$. Thus, $J_\lambda^\varphi$ is single-valued, and therefore
$A_\lambda^\varphi$ is also single-valued. It is easy to verify the monotonicity of $ A_\lambda^\varphi$.
To show $A_\lambda^\varphi$ is bounded, let $B\subset X$ be bounded. For each $x\in B$,
let $x_\lambda = J_\lambda^\varphi x$. Let $(u, v) \in \operatorname{Gr} (A)$.
Using \eqref{Yosida-inclusion}, it follows that
$$
\langle J_\varphi(x_\lambda-x)+\lambda y_\lambda, x_\lambda-u\rangle = 0,
$$
where $y_\lambda\in Ax_\lambda$. This implies
$$
\langle J_\varphi(x_\lambda-x), x_\lambda-u\rangle
= - \lambda \langle y_\lambda, x_\lambda - u\rangle \le \lambda \langle v, u-x_\lambda \rangle.
$$
The last inequality follows from the monotonicity of $A$. It then follows that
\begin{equation}\label{22}
\begin{aligned}
\langle J_\varphi(x_\lambda-x), x_\lambda-x\rangle&= \langle J_\varphi(x_\lambda-x), x_\lambda-u\rangle+ \langle J_\varphi(x_\lambda-x), u-x\rangle\\
&\le\lambda \langle v, u-x_\lambda \rangle+\langle J_\varphi(x_\lambda-x), u-x\rangle\\
&=\lambda \langle v, u-x \rangle+\lambda \langle v, x-x_\lambda \rangle+\langle J_\varphi(x_\lambda-x), u-x\rangle.
\end{aligned}
\end{equation}
This implies
\begin{equation}\label{23}
\varphi(\|x_\lambda-x\|)\|x_\lambda-x\|
\le \lambda\|v\| \left(\|u-x\|+\|x_\lambda-x\|\right) +\varphi(\|x_\lambda-x\|)\|u-x\|.
\end{equation}
If $\{x_\lambda : x\in B\}$ is unbounded, the inequality \eqref{23} yields a contradiction.
Thus, $J_\lambda^\varphi$ is bounded on $B$. Since $J_\varphi$ is bounded on $B$,
it follows from \eqref{Yosida} that $A_\lambda^\varphi$ is also bounded on $B$.
Let $\{x_n\}\subset X$ be such that $x_n\to x_0\in X$ as $n\to \infty$.
Denote $u_n = J_\lambda^\varphi x_n$ and $v_n = A_\lambda^\varphi x_n$, so that
\begin{equation} \label{Yosida2}
J_\varphi (u_n - x_n) +\lambda v_n = 0.
\end{equation}
Since $J_\lambda^\varphi$ and $A_\lambda^\varphi$ are bounded on bounded sets, both $\{u_n\}$ and $\{v_n\}$
are bounded. Since $J_\varphi$ and $A$ are monotone, it follows from
\begin{align*}
&\langle J_\varphi(u_n-x_n)- J_\varphi(u_m-x_m), (u_n-x_n) - (u_m-x_m)\rangle\\
&= -\lambda \langle v_n-v_m,(u_n-x_n) - (u_m-x_m)\rangle
\end{align*}
that
\begin{gather*}
\lim_{n, m\to\infty} \langle v_n - v_m, u_n-u_m\rangle = 0, \\
\lim_{n, m\to\infty} \langle J_\varphi(u_n-x_n)- J_\varphi(u_m-x_m), (u_n-x_n) - (u_m-x_m)\rangle = 0.
\end{gather*}
Passing to subsequences, we may assume that $u_n\rightharpoonup u_0$ in $X$, $v_n\rightharpoonup v_0$ in $X^*$,
and $J_\varphi(u_n-x_n) \rightharpoonup w_0$ in $X^*$ for some $u_0\in X$ and some $v_0, w_0\in X^*$.
By \cite[Lemma~2.3]{BA}, it follows that $(u_0, v_0)\in \operatorname{Gr}(A)$ and
$(u_0-x_0, w_0)\in \operatorname{Gr}(J_\varphi)$.
Using all these in \eqref{Yosida2}, we obtain
$J_\varphi(u_0-u_0) + \lambda v_0 = 0$, which implies
$u_0 = J_\lambda^\varphi x_0$ and $v_0 = A_\lambda^\varphi x_0$, i.e., $ J_\lambda^\varphi x_n \rightharpoonup J_\lambda^\varphi x_0$
and $A_\lambda^\varphi x_n \rightharpoonup A_\lambda^\varphi x_0$ as $n\to\infty$. This proves the demicontinuity
of $J_\lambda$ and $A_\lambda$.
\smallskip
(ii) Let $x\in D(A)$ and $\lambda>0$. Let $y\in Ax$ and $x_\lambda = J_\lambda^\varphi x$. Then
\begin{align*}
0&\le \langle y - A_\lambda x, x- x_\lambda\rangle\\
&= \langle y, x- x_\lambda\rangle -\frac{1}{\lambda}\varphi(\|x- x_\lambda\|) \|x- x_\lambda\|\\
&\le \|y\| \|x- x_\lambda\| - \frac{1}{\lambda}\varphi(\|x- x_\lambda\|) \|x- x_\lambda\|,
\end{align*}
which implies
$\varphi(\|x- x_\lambda\|) \le \lambda \|y\|$, and therefore
$$
\|A_\lambda^\varphi x\|= \dfrac{1}{\lambda}\|J_\varphi(x- x_\lambda)\|\le \|y\|.
$$
Consequently, $\|A_\lambda^\varphi x\| \le |Ax|:= \inf\{\|y\|: y\in Ax\}$.
\smallskip
(iii) The boundedness of $J_\lambda^\varphi$ on bounded subsets of $X$ and its demicontinuity
are already proved in (i). Let $ x\in\overline{\textrm{co}D(A)}$ and
$(u, v)\in \operatorname{Gr}(A)$. Following the arguments that lead to \eqref{23},
we find that $\{x_\lambda-x: \lambda>0\}$ is bounded, and therefore $\{J_\varphi(x_\lambda-x): \lambda>0\}$
is bounded. Let $\{\lambda_n\}\subset (0, \infty)$ be such that $\lambda_n \to 0$.
Let $y\in X^*$ be such that $J_\varphi({x_{\lambda_n}}-x)\rightharpoonup y$ in $X^*$.
Then \eqref{22} yields
$$
\limsup_{n\to\infty} \varphi(\|x_{\lambda_n}-x\|)\|x_{\lambda_n}-x\| \le \langle y, u-x\rangle.
$$
It is clear that this argument applies to all $u\in \overline{\textrm{co}D(A)}$.
Taking $u= x$, we obtain
$$
\lim_{n\to\infty} \varphi(\|x_{\lambda_n}-x\|)\|x_{\lambda_n}-x\| =0.
$$
By the homeomorphic property of the gauge function $\varphi$, it follows that we must have
$x_{\lambda_n}\to x$ as $n\to\infty$. This completes the proof of (iii).
\smallskip
(iv) Let $u_n = J_{\lambda_n}^\varphi x_n$ for all $n$.
Since $\{A_{\lambda_n}^\varphi x_n\}$ is bounded, it follows that
$$
\varphi(\|x_n - u_n\|) =\varphi( \|x_n -J_{\lambda_n}^\varphi x_n\|)
= \|J_\varphi(x_n -J_{\lambda_n}^\varphi x_n) \|= \lambda_n \|A_{\lambda_n}^\varphi x_n\|\to 0
$$
as $n\to\infty$. This implies $ \|x_n - u_n\|\to 0$ as $n\to\infty$.
Since
\begin{align*}
&\langle A_{\lambda_n}^\varphi x_n- A_{\lambda_m}^\varphi x_m, x_n - x_m \rangle \\
&=\langle A_{\lambda_n}^\varphi x_n- A_{\lambda_m}^\varphi x_m, u_n - u_m \rangle
+\langle A_{\lambda_n}^\varphi x_n- A_{\lambda_m}^\varphi x_m, (x_n -u_n) - (x_m-u_m)\rangle
\end{align*}
and $A$ is monotone, it follows as in Br\'ezis et al.~\cite{BCP} that
$$
\lim_{n, m\to\infty}\langle A_{\lambda_n}^\varphi x_n- A_{\lambda_m}^\varphi x_m, x_n - x_m \rangle=0
\text{ and }
\lim_{n, m\to\infty}\langle A_{\lambda_n}^\varphi x_n- A_{\lambda_m}^\varphi x_m, u_n - u_m\rangle =0.
$$
The conclusion of (iv) now follows from \cite[Lemma~2.3]{BA}.
\smallskip
(v) Let $x\in D(A)$. Since $X^*$ is reflexive and strictly convex and $Ax$ is a closed
and convex subset of $X^*$, it follows that there exists a unique element of $Ax$,
denoted by $A^{\{0\}}x$, such that $\|A^{\{0\}}x\| = \inf\{\|x^*\|: x^*\in Ax\}$.
Let $\{\lambda_n\}\subset (0, \infty)$ be such that $\lambda_n\to 0 $ and
$A_{\lambda_n}^\varphi x\rightharpoonup y$ in $ X^*$ as $n\to\infty$. As in the proof of (iv), with $x_n = x$,
we have $y\in Ax$. In view of part (ii), it follows that
$$
\|y\| \le \liminf_{n\to\infty}\|A_{\lambda_n}^\varphi x\| \le \limsup_{n\to\infty}\|A_{\lambda_n}^\varphi x\|
\le \|A^{\{0\}}x\|,
$$
and therefore we must have $y = A^{\{0\}}x$ and $A_{\lambda_n}^\varphi x\rightharpoonup A^{\{0\}}x$ in $ X^*$.
Moreover, if $X^*$ is uniformly convex, then, by \cite[Lemma~1.1]{BA}, we obtain
$A_{\lambda_n}^\varphi x\to A^{\{0\}}x$ in $ X^*$.
\smallskip
(vi) Suppose, on the contrary, that there is a sequence $\{\lambda_n\}$ with $\lambda_n\to 0$
and an element $x\not\in\overline{D(A)}$ such that $\{\|A_{\lambda_n}^\varphi x\|\}$ is bounded.
Let $R>0$ be such that $\|A_{\lambda_n}^\varphi x\|\le R$ for all $n$.
Then, by \eqref{Yosida}, we have
$$
\varphi(\|x- J_{\lambda_n}^\varphi x\|) =\|J_\varphi(x- J_{\lambda_n}^\varphi x)\|\le R\lambda_n .
$$
Since $\varphi^{-1}$ is also a gauge function, we obtain
$J_{\lambda_n}^\varphi x \to x\text{ as } n\to\infty$.
This implies $x\in \overline{D(A)}$, a contradiction.
\end{proof}
A proof of the following lemma for $\varphi(r) = r$ can be found in
Boubakari and Kartsatos \cite{BK}. Since we are dealing here with an arbitrary
gauge function $\varphi$, we provide a complete proof.
\begin{lemma}\label{L3}
Let $A:X\supset D(A)\to 2^{X^*}$ be maximal monotone and $G\subset X$ be bounded.
Let $0<\lambda_1<\lambda_2$. Then there exists a constant $K$, independent of $\lambda$,
such that
\[
\|A_\lambda^\varphi x\| \le K
\]
for all $x\in \overline{ G}$ and $\lambda\in [\lambda_1, \lambda_2]$.
\end{lemma}
\begin{proof} For every $x\in X$, we have
$$
A_\lambda^\varphi x = \frac{1}{\lambda}J_\varphi(x- x_\lambda),
$$
where $x_\lambda = J_\lambda^\varphi x$. Let $(u, v) \in \operatorname{Gr}(A)$.
In view of \eqref{23} in the proof of (i) in Proposition~\ref{prop0}, we have
\begin{align*}
\varphi(\|x_\lambda-x\|)\|x_\lambda-x\|
&\leq \lambda\|v\| \left(\|u-x\|+\|x_\lambda-x\|\right) +\varphi(\|x_\lambda-x\|)\|u-x\|\\
&\leq \lambda_2\|v\| \left(\|u-x\|+\|x_\lambda-x\|\right) +\varphi(\|x_\lambda-x\|)\|u-x\|.
\end{align*}
By the properties of the gauge function $\varphi$, it follows that
$\varphi(\|x_\lambda -x\|) $ must be bounded, i.e., there exists a constant $K_0>0$ such that
$$
\varphi(\|x_\lambda -x\|) \le K_0
$$
for all $x\in \overline{G}$ and all $\lambda \in [\lambda_1, \lambda_2]$.
Consequently, we have
$$
\|A_\lambda^\varphi x \| = \frac{1}{\lambda}\varphi(\|x_\lambda -x\|) \le \frac{1}{\lambda_1} K_0 =:K
$$
for all $x\in \overline{ G}$ and all $\lambda \in [\lambda_1, \lambda_2]$.
\end{proof}
By a well-known renorming theorem due to Troyanski \cite{Troyanski},
a reflexive Banach space $X$ can be renormed with an equivalent
norm with respect to which both $X$ and $X^*$ become locally uniformly convex
(therefore strictly convex). With such a renorming, the duality mapping $J_\varphi $
is a homeomorphism from $X$ onto $X^*$.
Henceforth, we assume that both $X$ and $X^*$ are reflexive and locally uniformly convex.
The following lemma involving $A_\lambda^\varphi$ and $J_\lambda^\varphi$ plays an important
role in the sequel. Its proof is omitted here because of its similarity to
\cite[Lemma~1]{AK}, except that, for the general $\varphi$ here, we must make use of
$$
x = J_{\lambda}^\varphi x+ J_\varphi^{-1} (\lambda A_{\lambda}^\varphi x) \text{ and }
\langle A_\lambda^\varphi x, J_\varphi^{-1}(\lambda A_{\lambda}^\varphi x)\rangle
= \varphi^{-1}(\lambda \|A_\lambda^\varphi x\|) \|A_\lambda^\varphi x\|, \quad x\in X.
$$
The lemma for $A_\lambda$ and $J_\lambda$ is essentially due to Br\'ezis et al.\ \cite{BCP}.
\begin{lemma}\label{L1}
Let $A:X\supset D(A)\to 2^{X^*}$ and $S:X\supset D(S)\to 2^{X^*}$
be maximal monotone operators such that $0\in D(A)\cap D(S)$ and
$0\in S(0)\cap A(0)$. Assume that $A+S$ is maximal monotone and
that there is a sequence $\{\lambda_n\}\subset (0,\infty)$ such that $\lambda_n\to 0$,
and a sequence
$\{x_n\}\subset D(S)$ such that $x_n\rightharpoonup x_0\in X$ and
$A^\varphi_{\lambda_n}x_n+w^*_n\rightharpoonup y_0^*\in X^*$, where
$w^*_n\in Sx_n$. Then the following statements are true.
\begin{itemize}
\item[(i)] The inequality
\begin{equation} \label{L12}
\lim_{n\to\infty}\langle A^\varphi_{\lambda_n}x_n+w^*_n,x_n-x_0\rangle < 0
\end{equation}
is impossible.
\item[(ii)] If
\begin{equation} \label{L13}
\lim_{n\to\infty}\langle A^\varphi_{\lambda_n}x_n+w^*_n,x_n-x_0\rangle = 0,
\end{equation}
then $x_0\in D(A+S)$ and $y_0^*\in (A+S)x_0$.
\end{itemize}
\end{lemma}
\begin{definition}\label{D3} \rm
An operator $A:X \supset D(A) \to 2^{X^*}$ is said to be \emph{strongly
quasibounded} if for every $S>0$ there exists $K(S)>0$ such that
$\|x\| \le S$ and $\langle x^*, x \rangle \le S$
for some $x^*\in Ax $ imply $\|x^*\| \le K(S)$.
\end{definition}
It is obvious that a bounded operator is strongly quasibounded.
With regard to possibly unbounded operators, Browder and Hess \cite{BrowderHess1972}
and Pascali and Sburlan \cite{PS} have shown that a monotone operator $A$
with $0\in{\rm Int} D(A)$ is strongly quasibounded. The following lemma with the
particular case $\varphi(r) = r$ addressed in Kartsatos and Quarcoo\cite[Lemma D]{Kartsatos2008}
is needed in the sequel.
\begin{lemma}\label{L2}
Let $A:X \supset D(A) \to 2^{X^*}$ be a strongly quasibounded
maximal monotone operator such that $0 \in A(0)$.
Let $\{\lambda_n\} \subset (0, \infty)$ and $\{x_n\} \subset X$ be such that
$$
\|x_n\| \le S \quad\text{and}\quad
\langle A_{\lambda_n}^\varphi x_n, x_n \rangle \le S_1\quad\text{for all } n,
$$
where $S, S_1$ are positive constants. Then there exists a number
$K>0$ such that $\|A_{\lambda_n}^\varphi x_n\| \le K$ for all $n$.
\end{lemma}
\begin{proof}
Denote $w_n= A_{\lambda_n}^\varphi x_n$ and $u_n =J_{\lambda_n}^\varphi x_n $ for all $n$.
Then we have
$$
w_n \in A u_n \quad\text{and}\quad
x_n = u_n+ J_\varphi^{-1} (\lambda_nw_n).
$$
In view of $0\in A(0)$, we obtain
\begin{align*}
0\le \langle w_n, u_n \rangle
&= \langle w_n , x_n - J_\varphi^{-1}(\lambda_nw_n)\rangle\\
&= \langle w_n , x_n \rangle -\langle w_n, J_\varphi^{-1}(\lambda_n w_n)\rangle\\
&= \langle w_n , x_n \rangle -\varphi^{-1}(\lambda_n\| w_n\|) \|w_n\|\\
&\leq S_1-\varphi^{-1}(\lambda_n\| w_n\|) \|w_n\|.
\end{align*}
This yields $\langle w_n, u_n\rangle \le S_1$ and $\varphi^{-1}(\lambda_n\| w_n\|) \|w_n\|\le S_1$
for all $n$. Suppose $\{w_n\}$ is unbounded. Then there exists a subsequence,
denoted again by $\{w_n\}$, such that $\|w_n\| \to \infty$ and $1\le \|w_n\|$ for all $n$.
Consequently,
$\varphi^{-1}(\lambda_n\| w_n\|)\le S_1$ for all $n$, and since
$x_n = u_n +J_\varphi^{-1} (\lambda_n w_n)$, it follows that
$$
\lambda_n \|w_n\| = \|J_\varphi (x_n-u_n)\|= \varphi(\|x_n -u_n\|) .
$$
This implies
$\|x_n -u_n\| = \varphi^{-1}(\lambda_n\| w_n\|)\le S_1$ for all $n$.
Since $\{x_n\}$ is bounded, we obtain the boundedness of $\{u_n\}$ and
$\{\langle w_n, u_n\rangle \}$, which contradicts the strong quasiboundedness of $A$.
Consequently, $\{w_n\}$ is bounded.
\end{proof}
For the rest of this paper, we take the gauge function $\varphi(r) = r^{p-1}, \;p>1$.
For the special case $\varphi(r) = r$, the reader can find proofs of
Lemma~\ref{L4} in Kartsatos and Skrypnik \cite{KartsatosSkrypnik2005a}
when $0\in A(0)$ and in Asfaw and Kartsatos \cite{ASK2012}, without the condition
$0\in A(0)$. We note that Zhang and Chen in \cite[Lemma~2.7]{ZC} proved the continuity
of $x\mapsto A_\lambda x$ on $ D(A)$ for each $\lambda>0$, also without the condition
$0\in A(0)$. In \cite[Lemma~6]{ASK2012}, however, the continuity of $x\mapsto A_\lambda x$
on $X$ is used with no mention of its validity. We next provide a detailed proof of the
continuity of the mapping $(\lambda, x)\mapsto A_\lambda^\varphi x$ on $(0, \infty) \times X$.
\begin{lemma}\label{L4}
Let $A:X\supset D(A)\to 2^{X^*}$ be a maximal monotone operator.
Then the mapping $(\lambda, x)\mapsto A_\lambda^\varphi x$ is continuous on
$(0, \infty) \times X$.
\end{lemma}
\begin{proof}
We first prove the continuity of $x\mapsto A_{\lambda_0}^\varphi x$ on $X$ for each fixed
$\lambda_0>0$. To this end, let $\{x_n\}\subset X$ be such that $x_n\to x_0\in X$.
By Lemma~\ref{L3}, we have the boundedness of $\{A_{\lambda_0}^\varphi x_n\}$, and therefore
\begin{equation}\label{100}
\lim_{n\to\infty}\langle A_{\lambda_0}^\varphi x_n- A_{\lambda_0}^\varphi x_0, x_n-x_0\rangle = 0.
\end{equation}
We know that
\begin{equation}\label{101}x_n =J_{\lambda_0}^\varphi x_n
+\lambda_0^{q-1} J_\varphi^{-1}(A_{\lambda_0}^\varphi x_n) \quad\text{and}\quad
x_0 =J_{\lambda_0}^\varphi x_0 +\lambda_0^{q-1} J_\varphi^{-1}(A_{\lambda_0}^\varphi x_0).
\end{equation}
Since $A_{\lambda_0}^\varphi x_n \in A(J_{\lambda_0}^\varphi x_n)$ and
$A_{\lambda_0}^\varphi x_0 \in A(J_{\lambda_0}^\varphi x_0)$, the monotonicity of
$A$ together with \eqref{100} and \eqref{101} yields
\begin{equation}\label{102}
\lim_{n\to\infty}\langle A_{\lambda_0}^\varphi x_n
- A_{\lambda_0}^\varphi x_0, J_\varphi^{-1}(A_{\lambda_0}^\varphi x_n)
- J_\varphi^{-1}(A_{\lambda_0}^\varphi x_0)\rangle = 0.
\end{equation}
Since $J_\varphi^{-1}$ is a duality mapping from $X^*$ to $X$, it follows, in view
of \cite[ Proposition~2.17]{Cioranescu}, that
$$
A_{\lambda_0}^\varphi x_n\to A_{\lambda_0}^\varphi x_0 \quad\text{as}\quad
n\to\infty.
$$
This proves the continuity of $A_{\lambda_0}^\varphi$ on $X$.
We now proceed to prove the continuity of $(\lambda, x)\mapsto A_\lambda^\varphi x$ on
$(0, \infty) \times X$. Let $\{\lambda_n\}\subset (0, \infty)$ and
$\{x_n\}\subset X$ be such that $\lambda_n\to \lambda_0\in (0, \infty)$ and
$x_n\to x_0\in X$ as $n\to\infty$.
Let $G\subset X$ be a bounded set that contains $x_n$ for all $n$.
Rename $\lambda_1, \lambda_2>0$ such that $\lambda_n\in [\lambda_1, \lambda_2]$
for all $n$. Since
$$
J_{\lambda_n}^\varphi x_n \in A^{-1}(A_{\lambda_n}^\varphi x_n) \quad\text{and}\quad
x_n =J_{\lambda_n}^\varphi x_n +\lambda_n^{q-1} J_\varphi^{-1}(A_{\lambda_n}^\varphi x_n),
$$
it follows that
\begin{align*}
J_{\lambda_n}^\varphi x_n + \lambda_0^{q-1} J_\varphi^{-1}(A_{\lambda_n}^\varphi x_n)
&\in A^{-1}(A_{\lambda_n}^\varphi x_n) +\lambda_0^{q-1} J_\varphi^{-1}(A_{\lambda_n}^\varphi x_n)\\
& = \left(A^{-1} +\lambda_0^{q-1} J_\varphi^{-1}\right)(A_{\lambda_n}^\varphi x_n).
\end{align*}
This implies
\begin{align*}
A_{\lambda_n}^\varphi x_n
&= \left(A^{-1}+\lambda_0^{q-1}J_\varphi^{-1}\right)^{-1}
\left(J_{\lambda_n}^\varphi x_n + \lambda_0^{q-1} J_\varphi^{-1}(A_{\lambda_n}^\varphi x_n)\right)\\
&= A_{\lambda_0}^\varphi\left(J_{\lambda_n}^\varphi x_n
+ \lambda_0^{q-1} J_\varphi^{-1}(A_{\lambda_n}^\varphi x_n)\right)\\
&= A_{\lambda_0}^\varphi\left(x_n + (\lambda_0^{q-1}
- \lambda_n^{q-1}) J_\varphi^{-1}(A_{\lambda_n}^\varphi x_n)\right).
\end{align*}
By Lemma~\ref{L3}, $\{A_{\lambda_n}^\varphi x_n\}$ is bounded, and so is
$\{J_\varphi^{-1}(A_{\lambda_n}^\varphi x_n)\}$. Since $\lambda_n\to \lambda_0$, we have
$ (\lambda_0^{q-1}- \lambda_n^{q-1}) J_\varphi^{-1}(A_{\lambda_n}^\varphi x_n) \to 0$
as $n\to\infty$. The continuity of $A_{\lambda_0}^\varphi$ implies
$A_{\lambda_n}^\varphi x_n \to A_{\lambda_0}^\varphi x_0$ as $n\to \infty$.
This completes the proof.
\end{proof}
\begin{remark}\rm
We anticipate that Lemma~\ref{L4} holds for any gauge function $\varphi$.
Since the formula \eqref{closed-form} may not hold for $A_\lambda^\varphi $ with a general $\varphi$,
the above proof does not go through and this subject may be of independent
research interest.
\end{remark}
Let $G$ be an open and bounded subset of $X$. Let $L:X\supset D(L)\to X^*$
be densely defined linear maximal monotone, $A:X\supset D(A)\to 2^{X^*}$
maximal monotone, and $C(s) :X\supset \overline{G}\to X^*$, $s\in[0,1]$,
a bounded homotopy of type $(S_+)$ with respect to $D(L)$.
Since $\operatorname{Gr}(L)$ is closed in $X\times X^*$, the space
$Y=D(L)$ associated with the graph norm
$\|x\|_Y = \|x\|_X + \|Lx\|_{X^*}$, $ x\in Y$,
becomes a real reflexive Banach space. We may assume that $Y$ and
its dual $Y^*$ are locally uniformly convex.
Let $j: Y\to X$ be the natural embedding and $j^* : X^*\to Y^*$ its
adjoint. Since $j:Y\to X$ is continuous, we have $D(j^*)=X^*$.
This implies that $j^*$ is also continuous. Since $j^{-1}$ is not necessarily
bounded, we have, in general, $j^*(X^*) \neq Y^*$.
Moreover, $j^{-1}(\overline G)=\overline G\cap D(L)$ is closed and
$j^{-1}(G)=G\cap D(L)$ is open,
$\overline{j^{-1}(G)} \subset j^{-1}(\overline G)$, and
$ \partial (j^{-1}(G))\subset j^{-1}(\partial G)$.
We define $M:Y\to Y^*$ by
$( Mx, y) := \langle Ly, J^{-1}(Lx)\rangle$, $x, y\in Y$, where
the duality pairing $(\cdot, \cdot )$ is in $Y^*\times Y$, and
$J^{-1}$ is the inverse of the duality map $J:X\to X^*$ and is
identified with the duality map from $X^*$ to $X^{**}=X$. Also,
for every $x\in Y$ such that $Mx\in j^*(X^*)$, we have
$J^{-1}(Lx)\in D(L^*)$, $Mx = j^*\circ L^*\circ J^{-1}( Lx)$, %\label{M2}
and
\begin{gather*}
( Mx-My,x-y) = \langle Lx-Ly,J^{-1}(Lx)-J^{-1}(Ly)\rangle \ge 0 \label{M2b}
\end{gather*}
for all $y\in Y$ such that $My\in j^*(X^*)$.
Moreover, it is easy to see that $M$ is continuous on $Y$, and therefore $M$ is
maximal monotone.
We now define $\hat L: Y\to Y^*$ and $\hat C(s): j^{-1}(\overline G)\to Y^*$ by
$\hat L = j^*\circ L \circ j$ and $\hat C(s) = j^*\circ C(s)\circ j$,
respectively, and for each $t >0$, we also define $\hat A_t^\varphi:Y\to Y^*$ by
$\hat A_{t}^\varphi = j^*\circ A_{t}^\varphi\circ j$,
where $A_t^\varphi $ is the Yosida approximant of $A$ corresponding to the gauge function $\varphi$.
The next lemma employs Lemma~\ref{L2} and follows as in \cite[Lemma~5]{AK2008},
and therefore its proof is omitted.
\begin{lemma}\label{L30}
Let $G\subset X$ be open and bounded. Assume the following:
\begin{itemize}
\item[(i)] $L:X\supset D(L)\to X^*$ is linear, maximal monotone with
$\overline{D(L)}=X$;
\item[(ii)] $A:X\supset D(A)\to 2^{X^*}$ is strongly quasibounded, maximal
monotone with $0\in A(0)$; and
\item[(iii)] $C(t):X\supset\overline G\to X^*$ is a bounded homotopy of type
$(S_+)$ with respect to $D(L)$.
\end{itemize}
Then, for a continuous curve $f(s), 0\le s \le 1$, in $X^*$, the set
$$
F=\big\{x\in j^{-1}(\overline G): \hat L + \hat A_t^\varphi +\hat C(s)+tMx = j^*f(s)
\text{ for some }t>0,\; s\in[0,1]\big\}
$$
is bounded in $Y$.
\end{lemma}
The next two propositions are essential for the existence results in Section~\ref{S2}
and Section~\ref{S3}.
\begin{proposition}\label{Prop1}
Let $A:X\supset D(A)\to 2^{X^*}$ be maximal monotone and $C:X\supset D(C)\to X^*$ be bounded,
demicontinuous and of type $(S_+)$. Suppose that $G\subset X$ is open and bounded such
that $0\in A(0)$, $p\in X^*$, and $$p\not\in (A+C)x$$ for all
$x\in \partial G\cap D(A)\cap D(C)$. Then the following statements hold.
\begin{itemize}
\item[(i)] There exists $t_0>0$ such that
$$A_t^\varphi x+Cx\ne p$$ for all $x\in \partial G\cap D(C)$ and $t 0$, $t_0>0$ and
$\epsilon_0>0$ such that the equation
\begin{equation}\label{6*}
A_t^\varphi x + Cx+ q_\epsilon x = \tau v_0^*
\end{equation}
has no solution in $G_1$ for every $\tau\ge \tau_0$, $t\in(0, t_0]$
and $\epsilon\in(0, \epsilon_0]$.
Assuming the contrary, let $\{\tau_n\}\subset(0, \infty)$,
$\{t_n\}\subset(0, \infty)$, $\{\epsilon_n\}\subset(0,\infty)$ and
$\{x_n\}\subset G_1$ be such that $\tau_n\to\infty$, $t_n\to 0$,
$\epsilon_n\to 0$ and
\begin{equation}\label{7}
A_{t_n}^\varphi x_n + Cx_n +q_{\epsilon_n}x_n = \tau_n v_0^*.
\end{equation}
We can assume that $q_{\epsilon_n}x_n\to g^*\in X^*$ in view of
the properties of $T$. Then $\|A_{t_n}^\varphi x_n\|\to \infty$ as
$\|\tau_n v_0^*\|\to\infty$ and $\{Cx_n\}$ is bounded.
Thus, from \eqref{7}, we obtain
\begin{equation}\label{8}
\frac{A_{t_n}^\varphi x_n}{\|A_{t_n}^\varphi x_n\|} +
\frac{Cx_n}{\|A_{t_n}^\varphi x_n\|} + \frac{q_{\epsilon_n}x_n}{\|A_{t_n}^\varphi x_n\|}=
\frac{\tau_n}{\|A_{t_n}^\varphi x_n\|}v_0^*.
\end{equation}
This implies
\begin{equation}\label{10}
\frac{\tau_n\|v_0^*\|}{\|A_{t_n}^\varphi x_n\|}\to 1
\quad\text{so that} \quad
\frac{\tau_n}{\|A_{t_n}^\varphi x_n\|}\to\frac{1}{\|v_0^*\|} \quad\text{as } n\to\infty.
\end{equation}
Since $p-1 =\gamma$, by Lemma~\ref{L5}, $A_t^\varphi$ is also homogeneous of degree
$\gamma = p-1$, and therefore we obtain
\begin{equation} \label{144}
\frac{A_{t_n}^\varphi x_n}{\|A_{t_n}^\varphi x_n\|}
= A_{t_n}^\varphi \Big(\frac{x_n}{\|A_{t_n}^\varphi x_n\|^{1/\gamma}}\Big).
\end{equation}
Let $ u_n = {x_n}/{\|A_{t_n}^\varphi x_n\|^{1/\gamma}}$.
It is clear that $u_n\to 0$. In view of \eqref{8}, \eqref{10}, and \eqref{144}, we obtain
$A_{t_n}^\varphi u_n\to h$ with $ h = {v_0^*}/{\|v_0^*\|}$.
This implies
$$
\lim_{n\to\infty}\langle A_{t_n}^\varphi u_n, u_n\rangle = \langle h,
0\rangle = 0.
$$
Since $t_n\to 0$, by (ii) of Lemma~\ref{L1} with $S=0$ we obtain $0\in D(A)$ and
$h\in A(0)=\{0\}$, a contradiction to $\|h\| = 1$.
We now consider the homotopy mapping
\begin{equation}\label{13}
H_1(s,x, t, \epsilon) = A_t^\varphi x+Cx+q_\epsilon x - s\tau_0v_0^*,
\quad s\in[0,1], \;x\in\overline{G_1},
\end{equation}
where $t\in(0, t_0]$ and $\epsilon\in(0, \epsilon_0]$ are fixed. By following the arguments as in \cite[Theorem 3.1]{Adhikari2017}, we can show that, for
every $s\in[0,1]$ the operator $x\mapsto Cx- s\tau_0v_0^*$ is
bounded, demicontinuous and of type $(S_+)$ on $\overline{G_1}$, and that
the equation $H_1(s, x, t, \epsilon) = 0$
has no solution in $\partial G_1$ for all sufficiently small
$t\in(0, t_0]$, $\epsilon\in(0, \epsilon_0]$
and all $s\in[0,1]$. In doing this, we need to use Lemma~\ref{L1}. The details are omitted.
It follows from Proposition~\ref{Prop1} that the mapping $H_1(s, x, t, \epsilon)$ is an
admissible homotopy for the degree, ${\rm d}_{S_+}$, of $(S_+)$-mappings, and
${\rm d}_{S_+}(H_1(s,\cdot, t, \epsilon), G_1, 0)$ is well-defined and is
a constant for all $s\in[0,1]$ and for all $t\in(0,t_0]$,
$\epsilon\in(0, \epsilon_0]$.
Assume that
$$
{\rm d}_{ S_+}(H_1(1, \cdot,t_1,\epsilon_1), G_1, 0)\ne 0,
$$
for some sufficiently small $t_1\in(0, t_0]$ and
$\epsilon_1\in(0, \epsilon_0]$. Then the equation
$$
A_{t_1}^\varphi x +Cx +g_{\epsilon_1} x = \tau_0v_0^*
$$
has a solution in $ G_1$. However, this contradicts our
choice of the number $\tau_0$ in \eqref{6*}. Consequently,
$$
{\rm d}_{S_+}(A_t^\varphi +C+q_\epsilon, G_1, 0)
= {\rm d}_{S_+}(H_1(0, \cdot,t,\epsilon), G_1, 0)= 0, \quad t\in(0, t_0],\;
\epsilon\in(0, \epsilon_0].
$$
We next consider the homotopy mapping
\begin{equation}\label{18}
H_2(s, x, t,\epsilon) = s(A_t^\varphi x+Cx +q_\epsilon x)+(1-s)Jx, \quad (s,
x)\in[0,1]\times\overline{G_2}.
\end{equation}
We claim that there exist $t_1\in(0, t_0]$ and $\epsilon_1\in(0,
\epsilon_0]$ such that $H_2(s, x, t,\epsilon)= 0$ has
no solution on $\partial G_2$ for any $s\in[0,1]$, any $t\in(0,
t_1]$ and any $\epsilon\in(0, \epsilon_1]$. To prove the claim, we assume the contrary
and then follow the argument used in \cite[Theorem 3.1]{Adhikari2017} along with the
properties of $A_t^\varphi$ established in Lemma~\ref{L1} to arrive at a contradiction to (H2).
For the sake of convenience, we
assume that $t_0$ and $\epsilon_0$ are sufficiently small so that we
may take $t_1 = t_0$ and $\epsilon_1 = \epsilon_0$.
It follows from Proposition~\ref{Prop1} that $H_2(s, x, t,\epsilon)$ is
an admissible homotopy for the degree of $(S_+)$-mappings and
${\rm d}_{\rm S_+}(H_2(s, \cdot, t, \epsilon), G_2, 0)$ is well-defined and
constant for all $s\in[0,1]$, all $t\in(0, t_0]$ and all
$\epsilon\in(0,\epsilon_0]$.
By the invariance of the $(S_+)$-degree, for all $t\in(0, t_0]$ and
$\epsilon\in(0,\epsilon_0]$, we have
\begin{align*}
{\rm d}_{ S_+}(H_2(1, \cdot, t, \epsilon), G_2, 0)
&= {\rm d}_{ S_+}(A_t^\varphi +C+q_\epsilon, G_2, 0)\\
&= {\rm d}_{ S_+}(H_2(0, \cdot, t, \epsilon), G_2, 0)\\
&= {\rm d}_{ S_+}(J, G_2, 0)
= 1.
\end{align*}
Thus, for all $t\in(0, t_0]$, $\epsilon\in(0,\epsilon_0]$, we have
$$
{\rm d}_{ S_+}(A_t^\varphi +C+q_\epsilon, G_1, 0)
\ne {\rm d}_{ S_+}(A_t^\varphi +C+q_\epsilon, G_2,0).
$$
Using the excision property of the $(S_+)$-degree, which is an easy
consequence of its finite-dimensional approximations, for every $t\in(0, t_0]$ and every
$\epsilon\in(0, \epsilon_0]$, there exists a
solution $x_{t, \epsilon}\in G_1\setminus G_2$ of
$A_t^\varphi x+Cx+q_\epsilon x = 0$. Let $t_n\in(0, t_0]$ and
$\epsilon_n\in(0, \epsilon_0]$ be such that $t_n\to 0$,
$\epsilon_n\to 0$ and let $x_n\in G_1\setminus G_2$ be the
corresponding solutions of $A_t^\varphi x +Cx+q_\epsilon x = 0$, i.e.,
$$
A_{t_n}^\varphi x_n +Cx_n+ q_{\epsilon_n}x_n = 0.
$$
We may assume that $x_n\rightharpoonup x_0$ in $X$ and
$q_{\epsilon_n}x_n\to g^*\in X^*$. We observe that
$$
\langle A_{t_n}^\varphi x_n , x_n -x_0\rangle
= -\langle Cx_n + q_{\epsilon_n}x_n, x_n- x_0\rangle.
$$
If
$$
\limsup_{n\to\infty}\langle Cx_n+q_{\epsilon_n}x_n, x_n-
x_0\rangle >0,
$$
then we obtain a contradiction from (i) of
Lemma~\ref{L1} with $S=0$ there. Consequently,
$$
\limsup_{n\to\infty}\langle Cx_n+q_{\epsilon_n}x_n, x_n-
x_0\rangle \le 0,
$$
and hence
$$
\limsup_{n\to\infty}\langle Cx_n, x_n- x_0\rangle \le0.
$$
By the $(S_+)$-property of $C$, we obtain
$x_n\to x_0\in\overline{G_1\setminus G_2}$. Then $Cx_n\rightharpoonup Cx_0$
and $A_{t_n}^\varphi x_n \rightharpoonup -Cx_0-g^*$. Using this in (ii) of
Lemma~\ref{L1} with $S=0$ there, we obtain $x_0\in D(A)$ and $-Cx_0-g^*\in Ax_0$. By
a property of the selection $q_{\epsilon_n}x_n$
as in Hu and Papageorgiou \cite{HP}, we have $g^*\in Tx_0$,
and therefore $Ax_0+Cx_0+Tx_0\ni 0$.
We also have
$$
x_0\in\overline{G_1\setminus G_2}
= (G_1\setminus G_2)\cup\partial(G_1\setminus G_2)
\subset(G_1\setminus G_2)\cup\partial G_1\cup\partial G_2.
$$
By (H1) and (H2), we have $x_0\notin \partial G_1\cup\partial G_2$, and hence
$x_0\in D(A)\cap(G_1\setminus G_2)$.
\end{proof}
\begin{remark}\rm
We point out that the condition $A(0) = \{0\}$ on the homogeneous maximal monotone
operator $A$ used in Theorem~\ref{Th1} is rather mild in view of Rockafellar's result
\cite{Rockafellar} which says that a monotone
map is locally bounded at every point in the interior of its domain.
\end{remark}
The existence of nonzero
solutions of $Lx+Ax+Cx\ni 0$, where the maximal monotone operator $A$ is strongly
quasibounded and positively homogeneous of degree $\gamma =1$, is established in
\cite{Adhikari2017}. In the following theorem, we extend this result to an arbitrary
degree $\gamma >0$ for the same combination of operators in the spirit of the
Berkovits-Mustonen theory in \cite{BM} and the theories developed in \cite{AK}.
We recall that the maximal monotone operator $A$ investigated in \cite{AK} is
strongly quasibounded. However, by a result of Hess \cite{Hess}, a strongly quasibounded
and positively homogeneous operator of degree $\gamma>0$ is necessarily bounded.
Therefore, in the following theorem, we assume that the maximal monotone operator $A$
is bounded.
\begin{theorem}\label{Th2}
Assume that $G_1, G_2\subset X$ are open, bounded with $0\in G_2$
and $\overline{G_2}\subset G_1$.
Let $L: X\supset D(L)\to X^*$ be linear maximal monotone with $\overline{D(L)} =X$,
and $A:X\supset D(A) \to 2^{X^*}$ bounded, maximal monotone
and positively homogeneous of degree $\gamma>0$. Also, let $C:\overline{G_1}\to X^*$ be
bounded, demicontinuous and of type $(S_+)$ with respect to $D(L)$.
Moreover, assume that
\begin{itemize}
\item[ (H3)] there exists $v^*\in X^*\setminus\{0\}$ such that
$Lx+Ax+Cx\not\ni \lambda v^*$ for all $(\lambda,x)\in\mathbb{R}_+
\times( D(L)\cap D(A)\cap \partial G_1)$, and
\item[(H4)] $Lx+Ax+Cx+ \lambda Jx \not\ni 0$ for all $(\lambda,x)\in
\mathbb{R}_+\times( D(L)\cap D(A)\cap \partial G_2)$.
\end{itemize}
Then the inclusion $Lx+Ax+Cx\ni 0$ has a solution $x\in
D(L)\cap D(A)\cap(G_1\setminus G_2)$.
\end{theorem}
% page 17
\begin{proof}
We begin by observing that a positively homogeneous and bounded maximal monotone
operator $A$ of degree $\gamma>0$ satisfies $0\in D(A)$ and $A(0) =\{0\}$.
To solve the inclusion
\begin{equation}\label{T1}
Lx+Ax+Cx\ni 0, \quad x\in\overline{G_1},
\end{equation}
let us consider the associated equation
\begin{equation}\label{T2}
\hat Lx+\hat A_t^\varphi x+\hat Cx +t Mx=0, \quad t\in (0, \infty),\;
x\in j^{-1}(\overline{G_1}).
\end{equation}
Here, the gauge function is $\varphi(r) = r^{p-1}$, $1 0$ such that the open
ball $B_Y(0, R) $ contains all the solutions of \eqref{T2}. We recall that $Y= D(L)$.
We shall prove that \eqref{T2} has a solution $x_t\in j^{-1}(G_1\setminus G_2)$
for all sufficiently small $t >0$.
We first claim that there exist $\tau_0>0$ , $t_0>0$ such that
\begin{equation}\label{T3}
\hat Lx+\hat A_t^\varphi x+\hat Cx +t Mx=\tau j^*v^*
\end{equation}
has no solution in $ G^1_R(Y):=j^{-1}(G_1)\cap B_Y(0, R)$ for all
$t \in (0, t_0]$ and all $\tau \in [\tau_0, \infty)$.
Assume the contrary and let $\{\tau_n\}\subset (0, \infty)$,
$\{t_n\}\subset (0, 1)$ and $\{x_n\} \subset G^1_R(Y)$ such that
$\tau_n\to \infty$, $t_n\to 0$ and
\begin{equation}\label{T4}
\hat Lx_n+\hat A_{t_n}^\varphi x_n+\hat Cx_n +t_n Mx_n=\tau_n j^*v^*.
\end{equation}
We note that $j^*$ is one-to-one because $j(Y) = Y$, which is dense in $X$.
This implies that $j^*v^*$ is nonzero, and therefore
$\|\tau_n j^*v^*\|_{Y^*}\to+\infty$. Also, the sequence $\{x_n\}$
is bounded in $Y$ and so we may assume that $x_n\rightharpoonup x_0$
in $X$ and $Lx_n\rightharpoonup Lx_0$ in $X^*$. In particular, $\{Lx_n\}$
is bounded in $X^*$. Since $Mx_n \in j^*(X^*)$, we have $J^{-1}(Lu)\in D(L^*)$ and
$$
Mx_n = j^* L^* J^{-1}(Lx_n).
$$
Since $j^*$, $L^*$, $J^{-1}$ are bounded, we have the boundedness of
$\{Mx_n\}$. It is clear that $\hat Lx_n$ and $\hat Cx_n$ are bounded in $Y^*$, and
therefore \eqref{T4} implies that $\| \hat A_{t_n}^\varphi x_n\|_{Y^*} \to \infty$.
Since $A$ is positively homogeneous of degree $\gamma$, applying Lemma~\ref{L5} for $\gamma = p-1$ shows that each $A_{t_n}^\varphi$ is also positively homogeneous of $\gamma=p-1$.
Consequently,
\begin{equation}\label{T55}
\frac{\hat A_{t_n}^\varphi x_n}{\|\hat A_{t_n}^\varphi x_n\|_{Y^*}}
= \hat A_{t_n}^\varphi\bigg(\frac{ x_n}{\|\hat A_{t_n}^\varphi x_n\|_{Y^*}^{1/\gamma}}\bigg)
\end{equation}
for all $n$. Define $\beta_n := 1/{\|\hat A_{t_n}^\varphi x_n\|_{Y^*}} \text{ and } \delta_n:= \beta_n^{1/\gamma}. $ Since $\| \hat A_{t_n}^\varphi x_n\|_{Y^*} \to \infty$, it follows that $\beta_n x_n \to 0$ and $\delta_n x_n \to 0$ in $X$ as $n\to\infty$.
From \eqref{T4} and \eqref{T55}, we find
\begin{equation}\label{T5}
\hat L (\beta_n x_n)+\hat A_{t_n}^\varphi (\delta_n x_n)+\beta_n\hat Cx_n+t_n\beta_n Mx_n
=\tau_n\beta_n j^*v^*.
\end{equation}
Because $\|\hat A_{t_n}^\varphi (\delta_n x_n)\|_{Y^*} = 1$ and the remaining
terms on the left in \eqref{T5} converge to $0$ in $X^*$ as $n\to\infty$, we
obtain $ \tau_n\beta_n \to 1/{\|j^*v^*\|_{Y^*}}$,
and therefore
$\hat A_{t_n}^\varphi (\delta_n x_n )\to y_0$,
where
$y_0 = j^*v_*/{\|j^*v^*\|_{Y^*}}. $
Since $u_n:= \delta_n x_n\to 0$ as $n\to\infty$, we have
$ \langle \hat A_{t_n}^\varphi u_n, u_n\rangle
\to\langle y_0, 0\rangle =0 $ as $n\to\infty$.
By Lemma \ref{L1}, (ii), we have
$ y_0\in A(0) =\{0\}$,
which is a contradiction to $\|y_0\|_{Y^*} = 1$.
We now consider the homotopy $H: [0,1]\times Y \to Y^*$ defined by
\begin{equation}\label{T6}
H(s, x) = \hat Lx +\hat A_t^\varphi x+\hat Cx+t Mx - s\tau_0 j^*v^*, \quad
s\in [0, 1], \; x\in j^{-1}(\overline {G_1}),
\end{equation}
where $t\in (0, t _0]$ is fixed. It can be easily seen that
$C-s\tau_0 v^*$ is bounded demicontinuous on $\overline {G_1}$ and of type
$(S_+)$ with respect to $D(L)$.
We now show that the equation $H(s, x) =0$ has no solution on the boundary
$\partial G_R^1(Y)$. Here, the number $R>0$ is increased, if necessary, so
that the ball $B_Y(0, R)$ now also contains all the solutions $x$ of $H(s, x) = 0$.
To this end, assume the contrary so that there exist $\{t_n\}\subset (0, t_0]$,
$\{s_n\}\subset [0, 1]$, and $\{x_n\}\subset \partial G_R^1(Y)$ such that
$t_n\to 0$, $s_n\to s_0$, $x_n\rightharpoonup x_0$ in $Y$,
$A_{t_n}^\varphi x_n\rightharpoonup w^*$ in $X^*$, $Cx_n\rightharpoonup c^*$ and
\begin{equation}\label{T7}
\hat Lx_n +\hat A_{t_n}^\varphi x_n+\hat Cx_n+t_n Mx_n =s_n\tau_0 j^*v^*.
\end{equation}
Here, the boundedness of $\{A_{t_n}^\varphi x_n\}$ follows as in Step I of
\cite[Proposition~1]{AK2016}, except that we now use $A_{t_n}^\varphi $
in place of the operators $T_{s_n}$ used in \cite{AK2016}.
Since $x_n \rightharpoonup x_0$ in $Y$, we have
$x_n \rightharpoonup x_0$ in $X$ and $Lx_n \rightharpoonup Lx_0$ in $X^*$.
Also, since $x_n\in B_Y(0, R)$ and
$$
\partial(j^{-1}(G_1)\cap B_Y(0, R)) \subset \partial(j^{-1}(G_1))
\cup \partial B_Y(0, R) \subset j^{-1}(\partial G_1) \cup \partial B_Y(0, R),
$$
we have $x_n\in j^{-1}(\partial G_1) = \partial G_1\cap Y \subset \partial G_1$.
We now follow the arguments as in \cite[Theorem 2.2]{Adhikari2017}
in conjunction with Lemma~\ref{L1} to arrive at
$$
\langle Lx_0 + w^*+ Cx_0 - s_0\tau_0 v^*, u\rangle = 0
$$
for all $u\in Y$, where $x_0\in D(A)$ and $w^*\in Ax_0$. Since $Y$ is dense
in $X$, we have $ Lx_0 + Tx_0+ Cx_0 \ni s_0\tau_0 v^*$,
which contradicts the hypothesis (H3) because
$x_0\in D(L)\cap D(T)\cap \partial G_1$.
We shrink $t_0$, if necessary, so that
$$
H(s, x) =0,\quad s\in [0, 1], \; x\in \overline{G_R^1(Y)}
$$
has no solution on the boundary $\partial G_R^1(Y)$ for all $t\in (0, t_0]$
and all $s\in [0, 1]$. It now follows from Proposition~\ref{Prop2} that
$H(s, x)$ is an admissible homotopy for
the $(S_+)$-degree, ${\rm d}_{S_+}$, and therefore
${\rm d}_{S_+} (H(s, \cdot), G_R^1(Y), 0)$, is well-defined and remains
constant for all $s\in [0, 1]$. Also, by Proposition~\ref{Prop2},
the limit
$$
\lim_{t \to0+}{\rm d}_{S_+} (H(1, \cdot), G_R^1(Y), 0)
$$
exists. By shrinking $t_0$ further, if necessary,
we find that
$ {\rm d}_{S_+} (H(1, \cdot), G_R^1(Y), 0) = \text{a constant}
\text{ for all } t\in (0, t_0]$.
Suppose, if possible, that
$$
{\rm d}_{ S_+} (H(1, \cdot), G_R^1(Y), 0)\ne 0
$$
for some $t_1\in (0, t_0]$. Then there exists $x_0\in G_R^1(Y)$ such that
$$
\hat Lx +\hat A_{t_1}^\varphi x+\hat Cx+t_1 Mx = \tau_0 j^*v^*.
$$
This contradicts the choice of $\tau_0$ as stated in \eqref{T3}. Since
$$
{\rm d}_{S_+} (H(0, \cdot), G_R^1(Y), 0)= {\rm d}_{S_+} (H(1, \cdot),
G_R^1(Y), 0),
$$
we have
\begin{equation}\label{D1}
{\rm d}_{S_+}(\hat L+ \hat A_t^\varphi + \hat C + t M, G_R^1(Y), 0)
={\rm d}_{S_+}(H(0, \cdot), G_R^1(Y), 0) = 0
\end{equation}
for all $t\in (0, t_0]$.
Next, we consider the homotopy $\widetilde{H}: [0, 1]\times Y\to Y^*$ defined by
$$
\widetilde{H}(s, x)= s(\hat Lx +\hat A_t^\varphi x+\hat Cx)
+t Mx+ (1-s)\hat Jx, \quad s\in [0, 1], \; x\in j^{-1}(\overline {G_2}).
$$
As in \cite[Step III, p. 29]{AK2016}, it can be shown that there exists $t_0>0$
(shrink it to a smaller number if necessary) such that all
the solutions of
$$
\widetilde{H} (s, x) = 0, \quad t\in (0, t_0], \;s\in [0, 1]
$$
are bounded in $Y$. We enlarge the previous number $R>0$, if necessary,
so that all solutions of $\widetilde{H}(s, x) = 0$ as described above
are contained in $B_Y(0, R)$ in $Y$.
Again, by following arguments similar to that in \cite[Theorem~2.2]{Adhikari2017}, we can show the existence of $t_1\in (0, t_0]$ such that the equation
$\widetilde{H}(s, x) = 0$ has no solutions on $\partial G_R^2(Y) $
for any $t\in (0, t_1]$ and any $s\in [0, 1]$.
Here, $G_R^2(Y) := j^{-1}(G_2)\cap B_Y(0, R)$.
In fact, if we assume the contrary, we can arrive at a situation that contradicts (H4).
At this point, we replace the number $t_0$ chosen previously with $t_1$ and
call it $t_0$ again. Let us fix $t\in (0, t_0]$ and consider the homotopy equation
\begin{equation}\label{T15}
\widetilde{H} (s, x) = s(\hat Lx +\hat A_t^\varphi x+\hat Cx)+t Mx
+ (1-s)\hat Jx =0, \;\; s\in [0, 1], \; x\in \overline {G_R^2(Y)}.
\end{equation}
It is already discussed that \eqref{T15} has no solution on $\partial {G_R^2(Y)}$.
We note that $\widetilde{H}$ is an affine homotopy of bounded
demicontinuous operators of type $(S_+)$ on $\overline {G_R^2(Y)}$;
namely, $\hat L +\hat A_t^\varphi +\hat C+t M$ and $t M+ \hat J$. We also note here
that $t M + \hat J$ is strictly monotone. In view of Proposition~\ref{Prop2}, it follows that $\widetilde{H}(s, x)$
is an admissible homotopy for the $(S_+)$-degree, ${\rm d}_{S_+}$, which satisfies
\begin{equation}\label{T16}
{\rm d}_{S_+}(\widetilde{H} (1, \cdot), G_R^2(Y), 0)
= {\rm d}_{ S_+}(\widetilde{H} (0, \cdot), G_R^2(Y), 0).
\end{equation}
This implies
\begin{equation}\label{D2}
{\rm d}_{S_+}(\hat L+\hat A_t^\varphi +\hat C+t M, G_R^2(Y), 0)
= {\rm d}_{S_+}(t M+ \hat J, G_R^2(Y), 0)=1
\end{equation}
for all $t \in (0, t _0]$.
The last equality follows from \cite[Theorem 3, (iv)]{BR1983}.
From \eqref{D1} and \eqref{D2}, we obtain
$$
{\rm d}_{S_+}(\hat L+\hat A_t ^\varphi +\hat C+t M, G_R^1(Y), 0)
\ne {\rm d}_{S_+}(\hat L+\hat A_t^\varphi +\hat C+t M, G_R^2(Y), 0)
$$
for all $t \in (0,t_0]$.
By the excision property of the $(S_+)$-degree, for each $t \in (0, t _0]$,
there exists a solution $x_t\in G_R^1(Y)\setminus G_R^2(Y)$ of the equation
$$
\hat Lx+\hat A_t^\varphi x+\hat Cx+t Mx=0.
$$
We now pick a sequence $\{t_n\}\subset (0, t_0]$ such that $t_n\to 0$
and denote the corresponding solution $x_t$ by $x_n$, i.e.,
$$
\hat Lx_n+\hat A_{t_n}^\varphi x_n +\hat Cx_n+t_n Mx_n=0.
$$
Since $Y$ is reflexive, we have $x_n\rightharpoonup x_0\in Y$ by
passing to a subsequence. This implies $x_n\rightharpoonup x_0$ in $X$ and
$Lx_n \rightharpoonup Lx_0$ in $X^*$. By the boundedness (therefore
strong quasiboundedness) of $A$,
we may assume, in view of Lemma~\ref{L2}, that $A_{t_n}^\varphi x_n\rightharpoonup w^*\in X^*$.
By a standard argument in conjunction with Lemma~\ref{L1} and the
$(S_+)$-property of $C$ with respect to $D(L)$,
we obtain $x_n\to x_0\in\overline{G_R^1(Y)\setminus G_R^2(Y)}$. By
Lemma~\ref{L1} and the demicontinuity of $C$, we have $x_0\in D(A)$,
$w^*\in Ax_0$, and $Cx_n \rightharpoonup Cx_0$ in $X^*$. Thus,
$Lx_0 +Ax_0+Cx_0 \ni 0$.
Finally, to show $x_0\in G_1\setminus G_2$, we note that
\[
G_R^1(Y)\setminus G_R^2(Y) =(G_1\setminus G_2) \cap Y\cap B_Y(0, R)
\subset G_1\setminus G_2.
\]
Consequently,
$x_n\in G_1\setminus G_2$ for all $n$, and therefore
$$
x_0\in\overline{G_1\setminus G_2} \subset (G_1\setminus G_2)
\cup \partial (G_1\setminus G_2)\subset (G_1\setminus G_2)
\cup \partial G_1 \cup \partial G_2.
$$
By (H3) and (H4), $x_0\not\in \partial G_1\cup\partial G_2$ and hence
$x_0\in D(L)\cap D(T)\cap (G_1\setminus G_2)$.
\end{proof}
\subsection{Open Problem}
Does Theorem~\ref{Th2} hold true if the boundedness of $A$ is dropped?
Since a positively homogeneous operator that is strongly quasibounded is
necessarily bounded, it is desirable to determine whether
Theorem~\ref{Th2} holds if $A$ is assumed to be ``quasibounded".
An operator $A:X \supset D(A) \to 2^{X^*}$ is said to be \emph{quasibounded}
if for every $S>0$ there exists $K(S)>0$ such that
$\|x\| \le S$ and $\langle x^*, x \rangle \le S\|x\|$ for some
$x^*\in Ax$ imply $\|x^*\| \le K(S)$. The notions of quasibounded and
strongly quasibounded operators were introduced in Hess~\cite{Hess}.
\section{Applications}\label{S4}
In this section, we apply Theorems~\ref{Th1} and \ref{Th2} to elliptic and
parabolic boundary value problems in general divergence form which are
obtained by modifying relevant examples from Berkovits and
Mustonen ~\cite{BM}, Kittil\"a ~\cite{Kittila}, and Adhikari~\cite{Adhikari2017}.
\begin{application} \rm
We consider the space $X= W_0^{m,p}(\Omega)$ with the
integer $m\ge 1$, the number $p\in(1,\infty)$, and the
domain $\Omega \subset \mathbb{R}^N$ with smooth boundary. We let $N_0$ denote
the number of all multi-indices
$\alpha=(\alpha_1,\dots,\alpha_N)$ such that
$|\alpha | = \alpha_1 +\cdots +\alpha_N\le m$.
For $\xi = (\xi_\alpha)_{|\alpha|\le m}\in\mathbb{R}^{N_0}$,
we have a representation $\xi=(\eta,\zeta)$, where
$\eta=(\eta_\alpha)_{|\alpha|\le m-1}\in\mathbb{R}^{N_1}$,
$\zeta =(\zeta_\alpha)_{|\alpha|=m}\in\mathbb{R}^{N_2}$
and $N_0=N_1+N_2$. We let
$$
\xi(u)= (D^\alpha u)_{|\alpha|\le m} ,\quad
\eta(u)= (D^\alpha u)_{|\alpha|\le m-1}, \quad \text{and}\quad
\zeta(u)= (D^\alpha u)_{|\alpha|= m},
$$
where
$
D^\alpha =\prod_{i=1}^N\Big(\frac{\partial}{\partial x_i}\Big)^{\alpha_i}.
$
We write $\nabla u := (D^\alpha u)_{|\alpha|= 1}$, and when $|\alpha| = k\in\{1, 2, \dots, m\}$, we simply write $D^k u:= (D^\alpha u)_{|\alpha|=k}$. Also, define $q:= p/(p-1)$.
We now consider the partial differential expression in divergence form
$$
\sum_{|\alpha|\le m}(-1)^{|\alpha|}
D^\alpha A_\alpha(x, \xi(u)),\quad x\in\Omega.
$$
The functions $A_\alpha :\Omega\times\mathbb{R}^{N_0}\to \mathbb{R}$
are assumed to be Carath\'eodory, i.e.,
each $A_\alpha(x, \xi)$ is measurable in $x$ for fixed
$ \xi\in\mathbb{R}^{N_0}$ and continuous in $\xi$ for
almost all $x\in\Omega$.
We assume the following conditions on $A_\alpha$:
\begin{itemize}
\item[(H5)] There exist $p\in(1,\infty)$, $c_1 >0$,
and $\kappa_1\in L^q(\Omega)$ such that
$$
|A_\alpha(x, \xi)|\le c_1|\xi|^{p-1}+
\kappa_1(x),\quad x\in\Omega,\;\;\xi\in\mathbb{R}^{N_0},\;\;|\alpha| \le m.
$$
\item[(H6)] The Leray-Lions condition
$$
\sum_{|\alpha|=m} [A_\alpha(x, \eta, \zeta_1)-
A_\alpha(x, \eta, \zeta_2)](\zeta_{1_\alpha}-\zeta_ {2_\alpha})>0
$$
is satisfied for every $x\in \Omega$, $\eta\in\mathbb{R}^{N_1}$ and
$\zeta_1, \zeta_2\in\mathbb{R}^{N_2}$ with
$\zeta_1\ne \zeta_2$.
\item[(H7)]
$$
\sum_{|\alpha|\le m} [A_\alpha(x, \xi_1)-
A_\alpha(x, \xi_2)](\xi_{1_\alpha}-\xi_ {2_\alpha})\ge0
$$
is satisfied for every $x\in \Omega$ and
$\xi_1, \xi_2\in\mathbb{R}^{N_0}$.
\item[(H8)] There exist $c_2>0$, $\kappa_2\in
L^1(\Omega)$ such that
$$
\sum_{|\alpha|\le m}A_\alpha(x,\xi)\xi_\alpha \ge
c_2|\xi|^p-\kappa_2(x),\quad x\in\Omega,\;
\xi\in\mathbb{R}^{N_0}.
$$
\item[(H9)] Each $A_\alpha(x,\xi)$ is homogeneous of degree $\gamma>0$ with respect to $\xi$.
\end{itemize}
If an operator $A: W_0^{m,p}(\Omega)\to W^{-m, q}(\Omega)$ is
given by
\begin{equation} \label{176}
\langle Au, v\rangle = \int_\Omega\sum_{|\alpha|\le m}
A_\alpha(x, \xi(u))D^\alpha v,\quad u,\,v\in
W_0^{m,p}(\Omega),
\end{equation}
then the conditions (H5), (H7) imply that $A$ is bounded,
continuous, and monotone as discussed in Kittil\"a
\cite[pp. 25-26]{Kittila} and Pascali and Sburlan
\cite[pp. 274-275]{PS}. Since $A$ is continuous,
it is maximal monotone.
Moreover, the condition (H9) implies that $A$ is positively homogeneous of degree $\gamma>0$.
For example, for $m=1$, we have $|\alpha| \le 1$, and when
$$
A_\alpha(x,\eta,\zeta) =\begin{cases}
|\zeta|^{p -2}\zeta_\alpha&\text{for } |\alpha| = 1\\
0 &\text{for } |\alpha| = 0,\\
\end{cases}
$$
the operator $A$ in \eqref{176} is given by $A := -\Delta_p$, where
$\Delta_p $ is the $p-$Laplacian from $W_{0}^{1, p}(\Omega)$ to
$W^{-1, q}(\Omega)$
defined as
$$
\Delta_p u:= {\rm div}\left(|\nabla u|^{p-2} \nabla u\right),\quad u\in W_{0}^{1, p}(\Omega).$$
It is clear that $ \Delta_p$ is positively homogeneous of degree $p-1$ .
Similarly, the condition (H5), with $A_\alpha$ replaced by $C_\alpha$,
implies that the operator
\begin{equation} \label{177}
\langle Cu, v\rangle
= \int_\Omega\sum_{|\alpha|\le m} C_\alpha(x, \xi(u))D^\alpha v,\quad\quad u,\,v\in
W_0^{m,p}(\Omega),
\end{equation}
is a bounded continuous mapping. We also know that
conditions (H5), (H6), and (H8), with $C_\alpha$ in place
of $A_\alpha$ everywhere, imply that the operator $C$ is of
type $(S_+)$ (see Kittil\"a \cite[ p. 27]{Kittila}).
We also consider a multifunction $H:\Omega\times \mathbb{R}^{N_1}\to 2^{\mathbb{R}}$
such that
\begin{itemize}
\item[(H10)] $H(x, r) = [\varphi(x, r), \psi(x, r)]$ is measurable in $x$
and upper semicontinuous in $r$, where
$\varphi, \psi :\Omega\times \mathbb{R}^{N_1}\to \mathbb{R}$
are measurable functions; and
\item[(H11)] $|H(x, r)| = \max[|\varphi(x, r)|, |\psi(x, r)|]
\le a(x) + c_2|r| $
a.e.\ on $\Omega\times\mathbb{R}^{N_1}$, where $a(\cdot) \in L^q(\Omega)$, $c_2>0$.
\end{itemize}
Define $T:W_0^{m,p}\to 2^{W^{-m, q}(\Omega)}$
by
\begin{align*}
Tu= \Big\{& h\in W^{-m, q}(\Omega) : \exists w\in L^q(\Omega) \text{ such that }
w(x)\in H(x, u(x)) \\
&\text{ and } \langle h, v\rangle = \int_{\Omega} w(x) v(x)
\text{ for all } v\in W_0^{m, p}(\Omega)\Big\}.
\end{align*}
It is well-known that $T$ is upper-semicontinuous and compact with closed
and convex values (see \cite[p. 254]{HP}), and
therefore $T$ is of class $(P)$.
We now state the following theorem as an application of Theorem~\ref{Th1}.
\begin{theorem}\label{Th3}
Assume that the operators $A$, $C$, and $T$ are defined as above. Assume, further, that the rest of the conditions
of Theorem~\ref{Th1} are satisfied for two
balls $G_1 = B_{\delta_1}(0)$ and $G_2= B_{\delta_2}(0)$, where $0<\delta_2<\delta_1$. Then
the Dirichlet boundary value problem
\begin{gather*}
\sum_{|\alpha|\le m}(-1)^{|\alpha|}
D^\alpha \Big(A_\alpha(x, \xi(u))+
C_\alpha(x, \xi(u))\Big)+ H(x, u) \ni 0,\quad x\in\Omega,\\
D^\alpha u(x) = 0,\quad
x\in\partial\Omega,\quad |\alpha| \le m-1,
\end{gather*}
has a ``weak" nonzero solution
$u\in B_{\delta_1}(0) \setminus B_{\delta_2}(0)\subset W_0^{m,p}(\Omega)$, which satisfies the
inclusion $Au + Cu +Tu\ni 0$.
\end{theorem}
\end{application}
\begin{application} \rm
Let $\Omega$ be a bounded open set in $\mathbb{R}^N$ with smooth boundary,
$m\ge 1$ an integer, and $a>0$. Set $Q= \Omega\times [0, a]$.
Consider differential operators of the form
\begin{equation}\label{IV1}
\frac{\partial u}{\partial t}(x,t) +\sum_{|\alpha|\le m}(-1)^{|\alpha|}D^\alpha \Big(A_\alpha(x,t,\xi(u(x,t))\\
+C_\alpha(x,t,\xi(u(x,t))\Big)\\
\end{equation}
in $Q$. The functions $A_\alpha=A_\alpha
(x,t,\xi)$ and $C_\alpha=C_\alpha
(x,t,\xi)$ are defined for $(x,t)\in Q$,
$\xi=(\xi_\alpha)_{ |\alpha|\le m}=(\eta,\zeta)\in\mathbb{R}^{N_0}$ with
$\eta=(\eta_\gamma)_{ |\alpha|\le m-1}\in\mathbb{R}^{N_1}$,
$\zeta=(\zeta_\alpha)_{ |\alpha|=m}\in\mathbb{R}^{N_2}$, and $N_1+N_2 = N_0$.
We assume that each $A_\alpha(x,t,\xi)$ satisfies the usual
Carath\'eodory condition. We consider the following conditions.
\begin{itemize}
\item[(H12)] (Continuity) For some $p > 1$, $c_1>0$, $g\in L^q(Q)$ with
$q=p/(p-1)$, we have
\[
|A_\alpha(x,t,\eta,\zeta)| \le c_1(|\zeta|^{p-1}+|\eta|^{p-1}+g(x,t)),
\]
for $(x,t)\in Q$, $\xi=(\eta,\zeta)\in\mathbb{R}^{N_0}$ and $|\alpha |\le m$.
\item[(H13)] (Monotonicity)
$$
\sum_{|\alpha| \le m}(A_\alpha(x,t,\xi_1)-
A_\alpha(x,t,\xi_2))(\xi_{1_\alpha}-\xi_{2_\alpha}) \ge 0 \text{ for }
(x,t)\in Q \text{ and } \xi_1,\xi_2\in\mathbb{R}^{N_0}.
$$
\item[(H14)] (Leray-Lions)
\[
\sum_{|\alpha| = m}(A_\alpha(x,t,\eta,\zeta)-
A_\alpha(x,t,\eta,\zeta^*))(\zeta_\alpha-\zeta^*_\alpha) > 0,
\]
for $(x,t)\in Q$, $\eta\in\mathbb{R}^{N_1}$ and $\zeta,\zeta^*\in\mathbb{R}^{N_2}$.
\item[(H15)] (Coercivity) There exist $c_0>0$ and $h\in L^1(Q)$ such that
$$
\sum_{|\alpha|\le m}A_\alpha(x,t,\xi) \ge c_0|\xi|^p-h(x,t),\quad (x,t)\in Q \text{ and }
\xi\in\mathbb{R}^{N_0}.
$$
\item[(H16)] Each $A_\alpha(x, t, \xi)$ is homogeneous of degree $\gamma>0$ with respect to $\xi$.
\end{itemize}
Under condition (H12), the second term of \eqref{IV1} with $C_\alpha =0$ generates a
continuous bounded operator $ A:X\to X^*$
defined by
$$
\langle Au,v\rangle=\sum_{|\alpha|\le m}\int_QA_\alpha(x,t,\xi(u(x, t)))D^\alpha v,
\quad u,v\in X,
$$
where $X=L^p(0,a;V), X^*=L^q(0,a;V^*)$,
and $V=W_0^{m,p}(\Omega)$.
With the additional conditions (H13) and (H16), the operator $A$ is maximal monotone and positively homogeneous of degree $\gamma$.
Under (H12), (H14), and (H15) with $A_\alpha$ replaced by $C_\alpha$ and other obvious
changes, the second term in \eqref{IV1} with $A_\alpha = 0$ generates a continuous, bounded
operator $C$ defined as
$$
\langle Cu,v\rangle=\sum_{|\alpha| \le m}\int_QC_\alpha(x,t,\xi(u(x, t)))D^\alpha v,
\quad u,v\in X,
$$
which satisfies the condition $(S_+)$ with respect to $D(L)$, where the
operator $L$ is defined as follows. The operator $\partial/\partial t$ generates an operator $L:X\supset D(L)\to X^*$,
where
$$
D(L) = \{v\in X: v'\in X^*,\; v(0)=0\},
$$
via the relation
$$
\langle Lu,v\rangle = \int_0^a\langle u'(t),v(t)\rangle_V \,\text{d}t,\quad u\in D(L),\; v\in X,
$$
where $\langle \cdot, \cdot \rangle_V$ is the duality pairing in $V^*\times V$.
The symbol $u'(t)$ is the generalized derivative of $u(t)$, i.e.,
$$
\int_0^a u'(t) \varphi(t)\text{d}t
=-\int_0^a \varphi'(t) u(t) \,\text{d}t,\quad \varphi\in C_0^\infty(0,a).
$$
We can verify, as in Zeidler \cite{Zeidler1}, that $L$ is densely
defined, linear and maximal monotone.
Given $h\in L^q(Q)$, define $h^*\in X^*$ by
\[
\langle h^*, v\rangle =\int_Q h v, \quad v\in X.
\]
\end{application}
As an application of Theorem~\ref{Th2}, we obtain the following theorem.
\begin{theorem} \label{thm4}
Assume that the operators $L, A$, and $C$ are as above, with
$A_\alpha$ satisfying {\rm (H12), (H13),} and {\rm (H16)}, and
$C_\alpha$ in place of $A_\alpha$ satisfying {\rm (H12), (H14),}
and {\rm (H15)}. Assume, for a given $h \in L^q(Q), $ that the
rest of the conditions of Theorem~\ref{Th2} are satisfied when $C$ is
replaced with $C-h^*$ for two
balls $G_1 = B_{\delta_1}(0)$ and $G_2= B_{\delta_2}(0)$ in
$X = L^p(0, a; V)$, where
$0<\delta_2<\delta_1$ and $V= W_0^m(\Omega)$.
Then the initial-boundary value problem
\begin{gather*}
\frac{\partial u}{\partial t}
+\sum_{|\alpha|\le m}(-1)^{|\alpha|}D^\alpha \Big(A_\alpha(x,t,\xi(u))+C_\alpha(x,t,\xi(u))\Big)=h(x, t),\\
D^\alpha u(x, t) = 0,\quad
(x, t)\in\partial\Omega\times [0, a],\quad |\alpha| \le m-1,\\
u(x,0) = 0, \quad x\in \Omega,
\end{gather*}
has a ``weak" nonzero solution $u\in B_{\delta_1}(0) \setminus B_{\delta_2}(0) \subset L^p(0, a;V)$ satisfying
$$
Lu + Au + Cu = h^*.
$$
\end{theorem}
\subsection*{Acknowledgments}
This research was carried out by members of the Analysis Group of the
Association of Nepalese Mathematicians in America (ANMA) within the
\emph{Collaborative Research in Mathematical Sciences} program.
Ghanshyam Bhatt acknowledges Tennessee State University for supporting
his engagement in this research with a Non-Instructional Assignment (NIA)
grant for the years 2021-2022. The authors express their gratitude to the
anonymous referees for providing feedback which helped to improve the article.
\begin{thebibliography}{00}
\bibitem{AM} A.~Addou, B.~Mermri;
Topological degree and application to a parabolic
variational inequality problem, \emph{Int. J. Math. . Sci.,} \textbf{25} (2001), no. 4, 273--287.
\bibitem{Adhikari2017} D. R. Adhikari; Nontrivial solutions of inclusions involving perturbed maximal monotone operators, \emph{Electron. J. Differential Equations,} \textbf{2017} (2017), no.~151, 1--21.
\bibitem{AK2016} D. R. Adhikari, A. G. Kartsatos;
Invariance of domain and eigenvalues for perturbations of
densely defined linear maximal monotone operators, \emph{Appl. Anal.,}
\textbf{95} (2016), no.~1, 24--43.
\bibitem{AK1} D. R. Adhikari, A. G. Kartsatos;
A new topological degree theory for perturbations of the sum of
two maximal monotone operators, \emph{Nonlinear Anal.,} \textbf{74} (2011), no. 14,
4622--4641.
\bibitem{AK2008} D. R. Adhikari, A. G. Kartsatos;
Strongly quasibounded maximal monotone
perturbations for the {Berkovits-Mustonen} topological degree theory, \emph{J.
Math. Anal. Appl.,} \textbf{348} (2008), no.~1, 12--136.
\bibitem{AK} D. R. Adhikari, A. G. Kartsatos;
Topological degree theories and nonlinear operator equations in
{Banach} spaces, \emph{Nonlinear Anal.,} \textbf{69} (2008), no. 4, 1235--1255.
\bibitem{Alber} Y. Alber, I. Ryazantseva; Nonlinear ill-posed problems of monotone type, \emph{Springer, Dordrecht,} 2006.
\bibitem{ASK2012} T. M. Asfaw, A. G. Kartsatos; A {B}rowder topological degree theory for multi-valued pseudomonotone perturbations of maximal monotone operators, \emph{Adv. Math. Sci. Appl.,} \textbf{22} (2012), no.~1, 91--148.
\bibitem{Aubin1984} J. P. Aubin, A.~Cellina;
Differential inclusions, \emph{Springer-Verlag,} 1984.
\bibitem{BA} V. Barbu;
Nonlinear semigroups and differential equations in {Banach}
spaces, \emph{Noordhoff Int. Publ.,} Leyden (The Netherlands), 1975.
\bibitem{BM} J.~Berkovits, V.~Mustonen;
Topological degree for perturbations
of linear maximal monotone mappings and applications to a class of parabolic
problems, \emph{Rend. Mat. Appl.,} \textbf{12} (1992), no. 3, 597--621.
\bibitem{BM2008} J.~Berkovits, M.~Miettunen;
On the uniqueness of the {B}rowder degree, \emph{Proc. Amer. Math. Soc.,} \textbf{136} (2008), no. 10, 3467--3476.
\bibitem{BK} I.~Boubakari, A. G. Kartsatos; The {L}eray-{S}chauder approach to the degree theory for
{$(S_+)$}-perturbations of maximal monotone operators in
separable reflexive {B}anach spaces, \emph{Nonlinear Anal.,} \textbf{70} (2009), no. 12, 4350--4368.
\bibitem{BCP} H.~Br\'ezis, M. G. Crandall, A.~Pazy;
Perturbations of nonlinear maximal monotone sets in {Banach} spaces,
\emph{Comm. Pure Appl. Math.,} \textbf{23} (1970), 123--144.
\bibitem{BR1983} F. E. Browder;
The degree of mapping and its generalizations, \emph{Contemp.
Math.,} \textbf{21} (1983), 15--40.
\bibitem{Browder1983} F. E. Browder;
Fixed point theory and nonlinear problems, \emph{Bull. Amer. Math.
Soc.,} \textbf{9} (1983), no. 1, 1--39.
\bibitem{Browder1976} F. E. Browder;
Nonlinear operators and nonlinear equations of evolution in
{Banach} spaces, nonlinear functional analysis, \emph{Proc. Sympos. Pure Appl. Math.,} \textbf{18} (1976), 1--308.
\bibitem{BrowderHess1972} F. E. Browder, P.~Hess;
Nonlinear mappings of monotone type in {Banach}
spaces, \emph{J. Functional Analysis,} \textbf{11} (1972), 251--294.
\bibitem{Cioranescu} I. Cioranescu; Geometry of {B}anach spaces, duality mappings and nonlinear problems,
\emph{Kluwer Acad. Publ.,} Dordrecht, 1990.
\bibitem{Hess} P. Hess; On nonlinear mappings of monotone type homotopic to odd operators, \emph{J. Functional Analysis,} \textbf{11} (1972), 138--167.
\bibitem{HP} S.~Hu, N.~S. Papageorgiou;
Generalizations of {Browder's} degree,
\emph{Trans. Amer. Math. Soc.,} \textbf{347} (1995), no.1, 233--259.
\bibitem{KartsatosLin2003} A. G. Kartsatos, J.~Lin;
Homotopy invariance of parameter-dependent
domains and perturbation theory for maximal monotone and m-accretive
operators in {Banach} spaces, \emph{Adv. Differential Equations,} \textbf{8} (2003), no.2,
129--160.
\bibitem{Kartsatos2008} A. G. Kartsatos, J.~Quarcoo;
A new topological degree theory for
densely defined {$(S_+)_L$}-perturbations of multivalued maximal monotone
operators in reflexive separable {Banach} spaces, \emph{Nonlinear Anal.,}
\textbf{69} (2008), no. 8, 2339--2354.
\bibitem{KartsatosSkrypnik} A. G. Kartsatos, I. V. Skrypnik;
Degree theories and invariance of
domain for perturbed maximal monotone operators in {Banach} spaces, \emph{Adv. Differential Equations,} \textbf{12} (2007), no. 11, 1275--1320.
\bibitem{KartsatosSkrypnik2005a} A. G. Kartsatos, I. V. Skrypnik;
A new topological degree theory for densely defined quasibounded
$(\widetilde S_+)$-perturbations of multivalued maximal monotone operators in
reflexive {Banach} spaces, \emph{Abstr. Appl. Anal.,} (2005), no. 2, 121--158.
\bibitem{KartsatosSkrypnik2005b} A. G. Kartsatos, I. V. Skrypnik;
On the eigenvalue problem for perturbed nonlinear maximal
monotone operators in reflexive {Banach} spaces, \emph{Trans. Amer. Math. Soc.,}
\textbf{358} (2006), no. 9, 3851--3881.
% \bibitem{Kenmochi1974} N. Kenmochi;
%Nonlinear operators of monotone type in reflexive {Banach}
% spaces and nonlinear perturbations, \emph{Hiroshima Math. J.,} \textbf{4} (1974),
% 229--263.
\bibitem{Kittila} A. Kittil\"a;
On the topological degree for a class of mappings of
monotone type and applications to strongly nonlinear elliptic problems, \emph{Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertations,} \textbf{91} (1994), 48pp.
\bibitem{PS} D.~Pascali, S.~Sburlan;
Nonlinear mappings of monotone type, \emph{Sijthoff
and Noordhoof}, Bucharest, 1978.
\bibitem{Rockafellar} R. Rockafellar, Local boundedness of nonlinear monotone operators,
\emph{Michigan Math. J.}, \textbf{16} (1969) 397--407.
\bibitem{Simons} S.~Simons;
Minimax and monotonicity, vol. 1693, \emph{Springer-Verlag}, Berlin, 1998.
\bibitem{Skrypnik1986} I. V. Skrypnik;
Nonlinear elliptic boundary value problems, \emph{BG Teubner}, 1986.
\bibitem{Skrypnik1994} I. V. Skrypnik;
Methods for analysis of nonlinear elliptic boundary value
problems, vol. 139, \emph{American Mathematical Society}, 1994.
\bibitem{Troyanski} S. L. Troyanski;
On locally uniformly convex and differentiable norms in
certain non-separable {Banach} spaces, \emph{Studia Math.}, \textbf{37} (1971),
173--180.
\bibitem{Zeidler1} E.~Zeidler;
Nonlinear functional analysis and its applications,
\textbf{II/B}, \emph{Springer-Verlag}, New York, 1990.
\bibitem{ZC} S.-S. Zhang, Y.-Q. Chen; Degree theory for multivalued $(S)$-type mappings and
fixed point theorems, \emph{Appl. Math. Mech.}, \textbf{11} (1990), no. 5, 441--454.
\end{thebibliography}
\end{document}