\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small International Conference on Applications of Mathematics to Nonlinear Sciences,\newline \emph{Electronic Journal of Differential Equations}, Conference 24 (2017), pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \setcounter{page}{1} \title[\hfilneg EJDE-2017/conf/24\hfil Existence results for operators of monotone type] {Existence results for multivalued operators of monotone type in reflexive Banach spaces} \author[D. R. Adhikari\hfil EJDE-2017/conf/24\hfilneg] {Dhruba R. Adhikari} \address{Dhruba R. Adhikari \newline Department of Mathematics, Kennesaw State University, Georgia 30060, USA} \email{dadhikar@kennesaw.edu} \thanks{Published November 15, 2017.} \subjclass[2010]{47H14, 47H05, 47H11} \keywords{Browder and Skrypnik degree theory; invariance of domain; \hfill\break\indent nonzero solutions; bounded demicontinuous operator of type $(S_+)$} \begin{abstract} Let $X$ be a real reflexive Banach space and $X^*$ its dual space. Let $T:X\supset D(T) \to 2^{X^*}$ be an operator of class $\mathcal A_G(S_+)$, where $G\subset X$. A result concerning the existence of pathwise connected sets in the range of $T$ is established, and as a consequence, an open mapping theorem is proved. In addition, for certain operators $T$ of class $\mathcal B_G(S_+)$, the existence of nonzero solutions of $0\in Tx$ in $G_1\setminus G_2$, where $G_1, G_2 \subset X$ satisfy $0\in G_2$ and $\overline{G_2}\subset G_1$, is established. The Skrypnik's topological degree theory is used, utilizing approximating schemes for operators of classes $\mathcal A_G(S_+)$ and $\mathcal B_G(S_+)$, along with the methodology of a recent invariance of domain result by Kartsatos and the author. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction and preliminaries} In what follows, $X$ is a real reflexive Banach space and $X^*$ its dual space. The norms of both $X$ and $ X^*$ will be denoted by $\|\cdot\|$ which will be understood from the context of its use. We denote by $\langle x^*,x\rangle$ the value of the functional $x^*\in X^*$ at $x\in X$. The symbol $\partial D$ and $\overline D$ denote the strong boundary and closure of the set $D$, respectively. The symbol $B(x_0,r)$ denotes the open ball of radius $r$ with center at $x_0$. For a sequence $\{x_n\}$ in $X$, we denote its strong convergence to $x_0$ in $X$ by $x_n\to x_0$ and its weak convergence to $x_0$ in $X$ by $x_n\rightharpoonup x_0$. An operator $T : X\supset D(T)\to Y$ is said to be ``bounded'' if it maps bounded subsets of the domain $D(T)$ onto bounded subsets of $Y$, where $Y$ is another Banach space. The value of $T$ at $x$ will be denoted by either $Tx$ or any other notation clearly understood from the context of its use. The operator $T$ is said to be ``compact'' if it maps bounded subsets of $D(T)$ onto relatively compact subsets of $Y$. It is said to be ``demicontinuous'' if it is strong-to-weak continuous on $D(T)$. The symbols $\mathbb{R}$ and $\mathbb{R}_+$ denote $ (-\infty, \infty)$ and $[0,\infty)$, respectively. The normalized duality mapping $J:X \supset D(J)\to 2^{X^*}$ is defined by $$ Jx = \{x^*\in X^* : \langle x^*, x\rangle = \|x\|^2, \; \|x^*\| = \|x\|\}, \quad x\in X. $$ The Hahn-Banach theorem ensures that $D(J) = X$, and therefore $J:X\to 2^{X^*}$ is a multivalued mapping defined on the whole space $X$. By a well-known renorming theorem due to Trojanski \cite{Trojanski}, one can always renorm the reflexive Banach space $X$ with an equivalent norm with respect to which both $X$ and $X^*$ become locally uniformly convex (therefore strictly convex). Henceforth, we assume that $X$ is a locally uniformly convex reflexive Banach space. With this setting, the normalized duality mapping $J$ is single-valued homeomorphism from $X$ onto $X^*$. For a multivalued operator $T$ from $X$ to $X^*$, we write $T:X\supset D(T) \to 2^{X^*}$, where $D(T)=\{x\in X:Tx\neq\emptyset\}$ is the effective domain of $T$. We denote by $Gr(T)$ the graph of $T$, i.e., $Gr(T)=\{(x,y):x\in D(T),~y\in Tx\}.$ An operator $T:X\supset D(T)\to 2^{X^*}$ is said to be ``monotone'' if for every $x,y\in D(T)$ and every $u\in Tx,~v\in Ty$ we have $$ \langle u - v,x-y\rangle \ge 0. $$ A monotone operator $T$ is said to be ``maximal monotone'' if $Gr(T)$ is maximal in $X\times X^*$, when $X\times X^*$ is partially ordered by the set inclusion. In our setting, a monotone operator $T$ is maximal monotone if and only if $R(T+\lambda J) = X^*$ for all $\lambda\in (0,\infty)$. \begin{definition}\label{D1}\rm An operator $C: X\supset D(C)\to X^*$ is said to be of type $(S_+)$ if for every sequence $\{x_n\}\subset D(C)$ with $x_n\rightharpoonup x_0$ in $X$ and $$ \limsup_{n\to\infty} \langle Cx_n, x_n- x_0\rangle \le 0, $$ we have $x_n\to x_0\in\overline {D(C)}$ in $X$. \end{definition} \begin{definition}\label{D2} \rm An operator $C: X\supset D(C)\to X^*$ is said to be pseudomonotone if for every sequence $\{x_n\}\subset D(C)$ with $x_n\rightharpoonup x_0$ in $X$ and $$ \limsup_{n\to\infty} \langle Cx_n, x_n- x_0\rangle \le 0, $$ we have $\lim_{n\to\infty} \langle Cx_n, x_n- x_0\rangle = 0$, and if $x\in D(C)$, then $ Cx_n\rightharpoonup Cx_0$ in $X$. \end{definition} \begin{definition}\label{D3} \rm The family $C(t):X\supset D\to X^*, t\in[0,1]$, of operators is said to be a homotopy of type $(S_+)$ if for any sequences $\{x_n\}\subset D$ with $x_n\rightharpoonup x_0$ in $X$ and $\{t_n\}\subset[0,1]$ with $t_n\to t_0$ and $$ \limsup_{n\to\infty} \langle C(t_n)x_n, x_n - x_0\rangle \le 0, $$ we have $x_n\to x_0$ in $X,~x_0\in D$ and $C(t_n)x_n \rightharpoonup C(t_0)x_0$ in $X^*$. A homotopy $C(t)$ of type $(S_+)$ is bounded if the set $$ \{C(t)x~|~t\in[0,1],~x\in D\} $$ is bounded. \end{definition} We next define the classes $\mathcal B_G(S_+)$ and $\mathcal A_G(S_+)$ of multivalued operators from $X$ to $X^*$. \begin{definition}\label{D4} \rm Let $G$ be an open subset of $X$. An operator $T: X\supset D(T)\to 2^{X^*}$ is of class $\mathcal B_G(S_+)$ if there exists a sequence $\{T_n\}$, called an approximating sequence of $T$, of bounded demicontinuous mappings of type $(S_+)$ from $\overline G$ to $X^*$ with the following conditions. \begin{itemize} \item[(A1)] For each $C>0$ there exists $K\ge 0$ such that $\langle T_nx, x\rangle \ge -K$ for all $x\in\overline G$ with $\|x\|\le C$ and for all $n\in \mathbb N$. \item[(A2)] Let $\{t_n\}\subset [0,1]$, $\{x_n\}\subset \overline G$ with $t_n\to 0$, and let $\{T_{m_n}\}$ be any subsequence of $\{T_n\}$. If $x_n\rightharpoonup x$ in $X$ and $t_n T_{m_n}x_n \to z$ in $X^*$, then $z= 0$. \item[(A3)] Let $\{x_n\}\subset \overline G$ and $\{T_{m_n}\}$ be any subsequence of $\{T_n\}$. If $x_n\rightharpoonup x$ in $X$, $T_{m_n}x_n\rightharpoonup w$ in $X^*$ and $$\limsup_{n\to\infty} \langle T_{m_n}x_n, x_n\rangle \le \langle w, x\rangle,$$ then $x_n\to x$ in $X$, $x\in D(T)$ and $w\in Tx$. \end{itemize} If the condition (A2) above is replaced by the following condition, the operator $T$ is said to be of class $\mathcal A_G(S_+)$. \begin{itemize} \item[$(\rm \tilde A2)$] Let $\{t_n\}\subset [0,1]$, $\{x_n\}\subset \overline G$ with $t_n\to 0$, and let $\{T_{m_n}\}$ be any subsequence of $\{T_n\}$. If $x_n\to x$ in $X$ and $t_n T_{m_n}x_n \rightharpoonup z$ in $X^*$, then $z= 0$. \end{itemize} \end{definition} \begin{definition}\label{D5} \rm Let $G$ be an open subset of $X$. An operator $T: X\supset D(T)\to 2^{X^*}$ is of class $\mathcal B_G(PM)$ (or $\mathcal A_G(PM)$) if there exists a sequence $\{T_n\}$, called an approximating sequence of $T$, of bounded pseudomonotone mappings from $\overline G$ to $X^*$ satisfying the conditions (A1), (A2) (or (A1), ($\tilde{A}2$)) and the following condition. \begin{itemize} \item[$(\rm A4)$] Let $\{x_n\}\subset \overline G$ and $\{T_{m_n}\}$ be any subsequence of $\{T_n\}$. If $x_n\rightharpoonup x$ in $X$, $T_{m_n}x_n\rightharpoonup w$ in $X^*$ and $$\limsup_{n\to\infty} \langle T_{m_n}x_n, x_n\rangle \le \langle w, x\rangle,$$ then $\langle T_{m_n}x_n, x_n\rangle \to \langle w, x\rangle$, and if $x\in \overline{G}$, then $x\in D(T)$ and $w\in Tx$. \end{itemize} \end{definition} \begin{remark} \rm If $G\subset X$ is open, then the following property holds true (cf. \cite[Lemma 2.2, p.9]{Kittila}). If $T\in\mathcal A_G(PM)$ and $A$ bounded demincontinuous of type $(S_+)$ on $G$, then $T+A\in\mathcal A_G(S_+)$. In particular, $T+J\in\mathcal A_G(S_+)$. \end{remark} The operators of class $\mathcal {A}_G(S_+)$ were introduced by Kittila in \cite{Kittila} and are multivalued generalizations of bounded demicontinuous operators of type $(S_+)$. Several examples of operators of type $\mathcal{A}_G(PM)$ are given in \cite[pp.36--43]{Kittila} in the context of elliptic equations with zeroth-order strongly nonlinear perturbations, higher-order elliptic equations with lower-order strongly nonlinear perturbations, and elliptic equations with highest-order strongly nonlinear perturbations. A topological degree theory was developed in \cite{Kittila} for such operators, and then the theory was applied to the study of strongly nonlinear elliptic partial differential equations in divergence form. Kittil\"a \cite[p.13]{Kittila} also showed that a densely defined maximal monotone operator $T:X\supset D(T)\to 2^{X^*}$, $0\in D(T)$, $0\in T0$ satisfies $T\in \mathcal A_X(PM)$. It can be seen that the operator $T+A$ is also of class $\mathcal B_G{(S_+)}$, where $A$ bounded demincontinuous of type $(S_+)$ on $X$. In the proof of the result given below, we only include the part that is different from the one for showing $T\in \mathcal A_X(PM)$ in \cite{Kittila}, and therefore $T\in\mathcal B_X(PM)$. \begin{theorem} \label{thm7} Let $T: X\supset D(T)\to 2^{X^*}$ be a maximal monotone operator with $0\in D(T)$, $0\in T0$ and $\overline{D(T)} = X$. Then $T\in \mathcal B_X(PM)$. \end{theorem} \begin{proof} The Yosida approximant $ T_n = (T^{-1} + \frac1n J^{-1})^{-1} : X\to X^*$, where $n$ is a positive integer, is single-valued maximal monotone and continuous operator with $T_n0 =0$. It is well-known that $T_nx \rightharpoonup T^0x$ on $D(T)$, where $T^0x$ is the unique element of $Tx$ having minimal norm, i.e. $\|T^0x\| = {\rm dist} (0, Tx)$. We only prove that $T$ satisfies the condition (A2). The other conditions follow from exactly the same arguments as in the proof of \cite[Theorem~2.1]{Kittila}. To verify the condition (A2), let $\{t_n\}\subset [0,1]$, $\{x_n\}\subset \overline G$ be such that $t_n\to 0$ and $x_n\rightharpoonup x_0$ in $X$, and let $\{T_{m_n}\}$ be any subsequence of $\{T_n\}$ such that $t_n T_{m_n}x_n \to z$ in $X^*$. Let $x\in D(T)$. Then $T_{m_n}x \rightharpoonup T^0x$ in $X^*$, and so $t_nT_{m_n}x\to 0$. Since $T_{m_n}$ is monotone, we have $$ \langle t_nT_{m_n}x_n-t_nT_{m_n}x, x_n - x\rangle \ge 0. $$ Letting $n\to \infty$ yields \begin{equation}\label{PM} \langle z, x_0 - x\rangle \ge 0\quad \text{for all } x\in D(T). \end{equation} Let $y\in X$. Since $\overline{D(T)} = X$, there exists a sequence $\{y_j\}\subset D(T)$ such that $y_j \to x_0-y$. Substituting $y_j$ for $x$ in \eqref{PM}, we get \begin{equation*} \langle z, x_0 - y_j\rangle \ge 0\quad \text{for all } j. \end{equation*} Letting $j\to\infty$ yields $\langle z, y\rangle$. Since $y\in X$ is arbitrary, we obtain $z=0$. This verifies the condition (A2). \end{proof} The first main result of this paper is the existence of nonzero solutions of $0\in Tx$, where $T\in\mathcal B_G(S_+)$. For additional facts related to the existence of nonzero solutions of nonlinear operator equations in Banach spaces, the reader is referred to Kartsatos and the author \cite{AK2016}, and Ding and Kartsatos \cite{Kartsatos95}. The second main result concerns an open mapping theorem for operators of class $\mathcal A_G(S_+)$, which extends the open mapping theorem of Park in \cite{Park} for bounded demicontinous operators of type $(S_+)$. A multivalued degree for operators in $\mathcal A_G(S_+)$ is developed by Kittila \cite{Kittila} via the Skyrpnik's degree (cf. \cite{Skrypnik1994}). In this paper, the methodologies in \cite{Kittila}, a recent paper of the author and Kartsatos \cite{AK2016}, and Kartsatos and Skrypnik \cite{KartsatosSkrypnik} as well as various properties of the Skrypnik's degree have been utilized. Open mapping theorems date back as far as Brouwer~\cite{Brouwer} for continuous injections in $\mathbb{R}^n$. Schauder~\cite{schauder} extended the Brouwer's open mapping theorem to infinite dimensional Banach spaces for operators of the form $I+C$ with $C$ compact. Tromba~\cite{Tromba} extended the Schauder's result to Fredholm maps of index zero. For other results concerning various continuity conditions on the main operators, the reader is referred to Berkovits~\cite{BE}, Deimling~\cite{Deimling}, Kartsatos~\cite{Kartsatos85}, Nagumo~\cite{Nagumo}, Petryshyn~\cite{Petryshyn70, Petryshyn93} (for $A$-proper mappings), Skrypnik~\cite[p.59]{Skrypnik1994} and the references therein. For the existence of pathwise connected sets in the ranges of certain operators, the reader is referred to \cite{Kartsatos95, Kartsatos2008} and the references therein. \section{Main Results} The first main result is the existence of nonzero solutions of the operator inclusion $0\in Tx$, where $T :X\supset D(T)\to 2^{X^*}$ is of the class $\mathcal B_G(S_+)$, $G\subset X$. \begin{theorem} \label{thm8} Assume that $G_1, G_2\subset X$ are open, bounded with $0\in G_2$ and $\overline{G_2}\subset G_1$. Let $T: X\supset D(T)\to 2^{X^*}$ be an operator of class $\mathcal B_{G_1}(S_+)$. Moreover, we assume the following conditions. \begin{itemize} \item[(H1)] There exists $v^*\in X^*$, $v^*\ne 0$, such that $\lambda v^*\not\in Tx$ for every $(\lambda, x)\in \mathbb{R}_+\times (D(T) \cap \partial G_1)$. \item[(H2)] For every $(\lambda, x)\in \mathbb{R}_+\times (D(T)\cap \partial G_2)$, we have $0\notin (T +\lambda J)x$. \end{itemize} Then there exists $x\in D(T)\cap (G_1\setminus G_2)$ such that $0\in Tx$. \end{theorem} \begin{proof} Since $T\in\mathcal B_{G_1}(S_+)$, there exists an approximating sequence $\{T_n\}$ in the sense of Definition~\ref{D4}, satisfying the conditions (A1)--(A3). Consider the approximate equation \begin{equation} \label{1} T_n x = 0. \end{equation} We first show that \eqref{1} has a solution $x_n\in G_1\setminus G_2$ for sufficiently large $n$. To this end, we first show that there exists $\tau_0 >0$ and $n_0$ such that the equation \begin{equation} \label{2} T_n x = \tau v^* \end{equation} has no solution in $G_1$ for every $\tau \ge \tau_0$ and for all $n\ge n_0$. Assuming the contrary implies the existence of $\{\tau_n\}\subset (0, \infty)$, $\{x_n\}\subset G_1$, and a subsequence of $\{T_n\}$ which we again denote by $\{T_n\}$, such that $\tau_n\to \infty$, $x_n \rightharpoonup x_0$, and \begin{equation} \label{3} T_n x_n = \tau_n v^*. \end{equation} Since $v^*\ne 0$, we have $\|T_nx_n\| \to \infty$, and therefore $$ \frac{T_n x_n}{\|T_nx_n\|} \to \frac{v^*}{\|v^*\|}. $$ Let $ t_n= 1/\|T_nx_n\|$ and $ h ={v^*}/{\|v^*\|}$. This implies that $t_n T_nx_n \to h$ and $t_n \to 0$. By the condition (A2), we get $h = 0$, which is a contradiction. Consider the homotopy \begin{equation} \label{4} H_n(s, x):= T_n x - s\tau_0 v^*, \quad (s, x)\in [0,1]\times\overline{G_1}, \end{equation} where $n\ge n_0$. We show that the equation $H_n(s, x) =0$ has no solution on $\partial G_1$ for sufficiently large $n$ and for all $s\in [0, 1]$. Assume the contrary and let $\{x_n\}\subset\partial G_1$ and $\{s_n\}\subset [0, 1]$ be such that $s_n\to s_0$, $x_n\rightharpoonup x_0$, and $$ T_n x_n = s_n\tau_0 v^*. $$ Since $T_n x_n \to s_0\tau_0v^*$, the condition (A3) yields $x_n\to x_0\in\partial G_1$, $x_0\in D(T)$ and $s_0\tau_0v^*\in Tx_0$. This contradicts the hypothesis (H1). By following an argument used in the proof of \cite[Theorem 1]{Adhikari}, we can show that $T_n-s\tau_0v^*$ is a bounded demicontinuous mapping of type $(S_+)$ for each $n$, and therefore $H_n(s, x)$ is an admissible homotopy for the Skrypnik's degree, ${\rm d} _{S_+}$. Suppose that ${\rm d}_{S_+} (H_{n_1}(1, \cdot), G_1, 0) \ne 0$ for a sufficiently large $n_1\ge n_0$, then the equation $$ T_{n_1} x = \tau_0 v^*$$ has a solution $x\in G_1$; however, this contradicts our choice of $\tau_0$. Consequently, \begin{equation} \label{5} {\rm d}_{S_+} (T_n, G_1, 0)={\rm d}_{S_+} (H_n(0, \cdot), G_1, 0) ={\rm d}_{S_+} (H_n(1, \cdot), G_1, 0) = 0 \end{equation} for all $n\ge n_0$. Consider the homotopy $$ \mathcal H_n(s, x) = s T_nx + (1-s) Jx, \quad (s, x) \in [0, 1]\times \overline{G_2}. $$ We show that there exists $n_1\ge n_0$ such that the equation $\mathcal H_n(s, x) =0$ has no solution on $\partial G_2$ for any $s\in [0, 1]$ and for any $n\ge n_1$. Let us assume the contrary and choose sequences $\{x_n\}\subset \partial G_2$, $\{s_n\}\subset [0, 1]$, and a subsequence of $\{T_n\}$ denoted again by itself, such that $x_n\rightharpoonup x_0$, $s_n \to s_0$, and \begin{equation}\label{7} s_n T_n x_n + (1-s_n) Jx_n = 0. \end{equation} Since $J0 = 0$ and $J$ is injective, we must have $s_n >0$ for all $n$. Also, if $s_n = 1$ for all large $n$, then $T_n x_n = 0$, and the condition (A3) yields $x_n\to x_0\in\partial G_2$, $x_0 \in D(T)$, and $0 \in Tx_0$. This is a contradiction to (H2). Suppose now that $s_n \to 0$. Then $$ \langle T_n x_n , x_n \rangle = -\big(\frac{1}{s_n}-1\big) \langle Jx_n , x_n\rangle \to -\infty .$$ This is a contradiction to the condition (A1) because the boundedness $\{x_n\}$ implies the existence of $K\ge 0$ such that $\langle T_n x_n , x_n \rangle \ge -K$ for all $n$. Thus, $s_0\in (0, 1]$ and \eqref{7} implies $$ T_n x_n \rightharpoonup -\big(\frac{1}{s_0}-1\big)j^*=: w, $$ where $j^*\in X^*$ satisfies $Jx_n \rightharpoonup j^*$. Since $J$ is monotone, we have \begin{align*} \langle T_n x_n , x_n -x_0\rangle &= -\big(\frac{1}{s_n}-1\big) \langle Jx_n , x_n-x_0\rangle \\ &= -\big(\frac{1}{s_n}-1\big) [\langle Jx_n-Jx_0 , x_n-x_0\rangle + \langle Jx_0, x_n -x_0\rangle]\\ &\leq -\big(\frac{1}{s_n}-1\big) \langle Jx_0, x_n -x_0\rangle. \end{align*} Since $X$ is reflexive, $\langle Jx_0, x_n -x_0\rangle \to 0$. This implies that $$ \limsup_{n\to\infty}\langle T_n x_n , x_n -x_0\rangle\le 0. $$ This along with $$ \limsup_{n\to\infty}\langle T_n x_n , x_n\rangle\ le \limsup_{n\to\infty}\langle T_n x_n , x_n -x_0\rangle + \limsup_{n\to\infty}\langle T_n x_n , x_0\rangle $$ implies $$ \limsup_{n\to\infty}\langle T_n x_n , x_n\rangle\le \langle w, x_0\rangle. $$ The condition (A3) yields $x_n \to x_0\in\partial G_2$, $x_0\in D(T)$ and $w\in Tx_0$. The continuity of $J$ implies $Jx_n \to Jx_0 = j^*$, so that $$ w= -\big(\frac{1}{s_0}-1\big) Jx_0 \in Tx_0, $$ i.e. $$ 0\in Tx_0 +\big(\frac{1}{s_0}-1\big) Jx_0, $$ which contradicts (H2). For the sake of convenience, we assume that $n_0$ is sufficiently large so that we make take $n_1= n_0$. Since an affine homotopy of bounded demicontinuous $(S_+)$ mappings is an admissible homotopy for the Skrynik's degree, ${\rm d}_{S_+}$, we have \begin{align*} {\rm d}_{S_+}(T_n, G_2, 0) &= {\rm d}_{S_+} (\mathcal H_n(1, \cdot), G_2, 0) \\ &= {\rm d}_{S_+} (\mathcal H_n(0, \cdot), G_2, 0)\\ &= {\rm d}_{S_+} (J, G_2, 0) = 1 \end{align*} for all $n\ge n_0$. Thus, for all $n\ge n_0$, we have $$ {\rm d}_{S_+}(T_n , G_1, 0) \ne {\rm d}_{S_+}(T_n , G_2, 0). $$ By the excision property of the Skrypnik's degree, for each $n\ge n_0$ there exists a solution $x_n\in G_1\setminus G_2$ of $T_nx_n = 0$. We may assume that $x_n \rightharpoonup x_0$ in $X$ and the condition (A3) implies that $x_n \to x_0\in\overline{G_1\setminus G_2}$, $x_0\in D(T)$ and $0\in Tx_0$. Note that $$ \overline{G_1\setminus G_2} = (G_1\setminus G_2) \cup \partial (G_1\setminus G_2)\subset (G_1\setminus G_2) \cup \partial G_1 \cup \partial G_2. $$ By the conditions (H2) and (H2), $x_0\notin\partial G_1 \cup \partial G_2$. Thus, $x_0\in D(T)\cap (G_1\setminus G_2)$. \end{proof} We proceed to prove a result about placing pathwise connected sets in the ranges of certain operators of class $\mathcal A_G(S_+)$. As a consequence, we obtain an open mapping theorem for such operators. \begin{proposition}\label{P1} Let $T: X\supset D(T)\to 2^{X^*}$ be of class ${\mathcal A}_G(S_+)$ with an approximating sequence $\{T_n\}$, where $G\subset X$ is open and bounded. Assume that $T+\epsilon J (\cdot- x_0)$ is injective on $G$ for each $\epsilon > 0$ and for every $x_0\in D(T)$. Moreover, assume that for each $x_0\in D(T)$, there exists a bounded $\phi_{x_0}:\mathbb{R}^+\to\mathbb{R}$ such that $\langle T_n x, x_0\rangle \le \phi_{x_0}(\|x\|)$ for all $x\in \partial G$ and for all large $n$. For a pathwise connected set $M\subset X^*$, assume that $T(D(T)\cap G)\cap M\ne \emptyset$ and $T(D(T)\cap \partial G)\cap M= \emptyset$. Then $M\subset T(D(T)\cap G)$. \end{proposition} \begin{proof} Let $y_0\in T(D(T)\cap G)\cap M$. Then there exists $x_0\in D(T)\cap G$ such that $y_0\in Tx_0$. Let $p\in M$. Take $f:[0, 1]\to M$ be a path in $M$ such that $f(0)= y_0$ and $f(1) = p$. We now claim that there exist $n_0\in\mathbb N$ such that \begin{equation}\label{10} T_n x+\frac1n J(x-x_0) = f(t) \end{equation} has no solution $x\in\partial G$ for any $t\in[0,1]$ and for all $n\ge n_0$. Assuming the contrary and without loosing the generality, let $\{x_n\}\subset \partial G$ with $x_n\rightharpoonup x$ and $\{t_n\}\subset [0,1]$ with $t_n\to t_0$ be such that $$ T_n x_n+\frac1n J(x_n-x_0) = f(t_n). $$ This implies that $T_nx_n\to f(t_0)$. Since $x_n\rightharpoonup x$, we have $$ \limsup_{n\to\infty}\langle T_nx_n, x_n\rangle \le \langle f(t_0), x\rangle, $$ and then by the condition (A3) of Definition~\ref{D1}, we have $x_n\to x$, $x\in D(T)$ and $f(t_0)\in~ Tx$. Since $f(t_0)\in M$ and $x\in D(T)\cap\partial G$, we have a contradiction to $T(D(T)\cap \partial G)\cap M = \emptyset$. Consider the homotopy equation \begin{equation}\label{12} H_n(x, t)\equiv T_n x+\frac1n J(x-x_0) -f(t)=0. \end{equation} We have already established that this equation has no solution on $\partial G$ for sufficiently large $n$ and for any $t\in[0,1]$, and therefore this is an admissible homotopy of type $(S_+)$. Consider the homotopy equation \begin{equation}\label{13} G_n(x, t)\equiv (1-t)\Big(T_n x+\frac1n J(x-x_0)-y_0\Big)+t J(x-x_0)=0. \end{equation} We show that \eqref{13} has no solution on $\partial G$ for any $t\in[0,1]$ and for all $n\ge n_0$. If not, let $\{x_n\}\subset \partial G$ with $x_n\rightharpoonup x$ and $\{t_n\}\subset [0,1]$ with $t_n \to t_0$ such that \begin{equation}\label{14} (1-t_n)\Big(T_n x_n+\frac1n J(x_n-x_0)-y_0\Big)+t_n J(x_n-x_0)=0. \end{equation} Since \eqref{10} has no solution on $\partial G$ for any $n\ge n_0$ and $t\in[0,1]$, we see that $t_n =0$ is impossible for all large $n$. Since $J$ is injective, $t_n = 1$ is also impossible. Suppose $t_0 =1$. The equation \eqref{14} implies \begin{equation}\label{15} (1-t_n)\langle T_n x_n, x_n -x_0\rangle +a_n \|x_n-x_0\|^2=(1-t_n)\langle y_0,x_n-x_0\rangle, \end{equation} where $$ a_n = \frac {1-t_n}{n} + t_n. $$ Since $G$ is bounded, there exists $C>0$ such that $\|x_n\| \le C$ for all $n$. By the condition $(A_1)$, there exists a $K>0$ such that $\langle T_n x_n, x_n \rangle \ge - K$ for all $n$. Also, by the hypothesis, there exists a bounded function $\phi_{x_0}:\mathbb{R}_+\to\mathbb{R}$ such that $\langle T_nx_n, x_0\rangle \le \phi(\|x_n\|)$ for all $n$. Then, in view of\eqref{15}, we have \begin{equation}\label{16} -(1-t_n)K -(1-t_n)\phi_{x_0}(\|x_n\|) +a_n \|x_n-x_0\|^2\le (1-t_n)\langle y_0,x_n-x_0\rangle. \end{equation} Since $t_n\to 1$, $a_n\to 1$ and $\phi_{x_0}$ is bounded, letting $n\to \infty$ in \eqref{16} yields $x_n\to x_0\in\partial G$, which is a contradiction. Next, we assume that $t_0\in [0, 1)$. If $t_0 = 0$, define $ \alpha_n = \frac{t_n}{1-t_n}.$ Then $\alpha_n \downarrow 0$ and \begin{equation}\label{17} T_nx_n +\big( \frac{1}{n}+\alpha_n\big) J(x_n -x_0) = y_0. \end{equation} This equation is like \eqref{12} for which we have already proved the impossibility of solutions on $\partial G$ with $f(t)\equiv y_0$. For the remaining case, $t_0\in (0, 1)$, we define $$ \beta_n = \frac{1}{n} +\frac{t_n}{1-t_n}. $$ Then $ \beta_n\to\beta_0:= \frac{t_0}{1-t_0} >0$. Then the equation becomes \begin{equation}\label{18} T_n x_n+\beta_n J(x_n-x_0)=y_0. \end{equation} If $$ \limsup_{n\to\infty}\langle T_n x_n, x_n -x\rangle >0, $$ then, by passing to a subsequence, let $$ q:= \lim_{n\to\infty}\langle T_n x_n, x_n -x\rangle >0. $$ In view of \eqref{18}, this yields $$ \limsup_{n\to\infty}\langle \beta_n J(x_n -x_0), (x_n-x_0)-(x-x_0)\rangle = -q <0. $$ Since $\beta_n\to \beta_0>0$ and $J$ is of type $(S_+)$, we obtain $x_n\to x\in\partial G $. From this and \eqref{18}, we get $T_nx_n \to w:=- \beta_0 J(x-x_0)+y_0.$ By the condition (A3), we obtain $x\in D(T) $ and $w\in Tx$, i.e. $y_0\in Tx + \beta_0 J(x-x_0)$. This leads to a contradiction to the injectivity of $T+\epsilon J (\cdot-x_0)$ because $x\ne x_0$. Thus, $H_n(x, t)$ and $G_n(x, t)$ are admissible homotopies for the Skrypnik's degree, ${\rm d}_{S_+}$, for the mappings of type $(S_+)$. By the invariance of the degree under these homotopies, we have \begin{align*} {\rm d}_{S_+}(T_n+\frac1n J(\cdot-x_0)-p, G, 0) &= {\rm d}_{S_+}(H_n(\cdot, 1), G, 0)\\ &= {\rm d}_{S_+}(H_n(\cdot, 0), G, 0)\\ &= {\rm d}_{S_+}(G_n(\cdot, 0), G, 0)\\ &= {\rm d}_{S_+}(G_n(\cdot, 1), G, 0) \\ &= {\rm d}_{S_+}(J(\cdot - x_0), G, 0) = 1. \end{align*} Here, the last equality follows by considering the $(S_+)$-homotopy $$ Q(x, t) = (1-t) J(x-x_0) + t Jx $$ with a continuous curve $y(t) = tJx_0$ so that \begin{align*} {\rm d}_{S_+} (J(\cdot - x_0), G, 0) &= {\rm d}_{S_+} (Q(\cdot, 0), G, 0)\\ & = {\rm d}_{S_+} (Q(\cdot, 1), G, Jx_0)\\ &= {\rm d}_{S_+} (J, G, Jx_0) = 1. \end{align*} Therefore, for every $n$, there exists $x_n \in G$ such that $$ H_n(x_n, 1) = 0, $$ i.e. $$ T_n x_n + \frac{1}{n} J(x_n-x_0) = p, $$ which implies $T_n x_n \to p$. By the condition (A3), we deduce that $x_n\to x\in \overline {G}$, $x\in D(T)$, and $p\in Tx$. Since $T(D(T)\cap \partial G) \cap M=\emptyset$, we can only have $x\in G$. Since $p$ was an arbitrary point in $M$, we obtain $M\subset T(D(T)\cap G)$. \end{proof} We use Proposition~\ref{P1} to prove the following open mapping theorem for operators of class $\mathcal A_G(S_+)$. \begin{theorem}[Open Mapping]\label{invariance of domain} Let $T: X\supset D(T)\to 2^{X^*}$ be of class ${\mathcal A}_G(S_+)$ with an approximating sequence $T_n$, where $G\subset X$ is open. Assume that $T+\epsilon J (\cdot- x_0)$ is locally injective on $G$ for each $\epsilon \ge 0$ and for every $x_0\in D(T)$. Moreover, assume that for each $x_0\in D(T)$ and for each $r>0$, there exists a bounded $\phi_{x_0}:\mathbb{R}^+\to\mathbb{R}$ such that $\langle T_n x, x_0\rangle \le \phi_{x_0}(\|x\|)$ for all $x\in \overline G\cap \partial B(x_0,r)$ and for all large $n$. Then $T(D(T)\cap G)$ is open. \end{theorem} \begin{proof} Let $y_0\in T(D(T)\cap G)$. Then there exists $x_0\in G$ such that $y_0\in Tx_0$. Since $T$ is locally injective on $G$, there is $r>0$ such that $T$ is injective on $\overline {B(x_0, r)}\cap D(T)$, where $\overline{B(x_0, r)}\subset G$. It is then clear that $y_0\notin T(D(T)\cap \partial B(x_0, r))$. We claim that there exists $\delta>0$ such that $B(y_0, \delta) \cap T(D(T)\cap \partial B(x_0, r))=\emptyset$. Assume the contrary, and let $y_n\in B(y_0, 1/n) \cap T(D(T)\cap \partial B(x_0, r))$. Then $y_n\to y_0$ and $y_n \in Tx_n$ with $x_n\in\partial B(x_0, r)$, and therefore the condition (A3) applies because $x_n\rightharpoonup x$ (up to subsequence) and $y_n\to y_0$. We get $x_n\to x\in \partial B(x_0, r)$, $x\in D(T)$ and $y_0\in Tx$. This contradicts $y_0\notin T(D(T)\cap \partial B(x_0, r))$. Since $B(y_0, \delta) \cap T(D(T)\cap \partial B(x_0, r))=\emptyset$, $y_0\in T(D(T)\cap B(x_0, r))$ and the ball $B(y_0, \delta)$ is pathwise connected, we can apply Proposition~\ref{P1} to obtain $B(y_0, \delta) \subset T(D(T)\cap B(x_0, r))$. Since $y_0$ was arbitrary, $T(D(T)\cap G)$ is open. \end{proof} It would be interesting to establish analogous results via degree theories for operators of the form $A+T$, where $A: X\supset D(A)\to 2^{X^*}$ is maximal monotone and $T$ is of class $\mathcal A_G(S_+)$. Similar results are also expected for the sum $L+A+T$ in the spirit of results in \cite{AK2016}, where $L$ is linear maximal monotone operator (densely defined) and $T$ is of class $\mathcal A_G(S_+)$ with respect to $D(L)$. 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