\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 151, pp. 1--21.\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} \title[\hfilneg EJDE-2017/151\hfil Monotone operator inclusions] {Nontrivial solutions of inclusions involving perturbed maximal monotone operators} \author[D. R. Adhikari \hfil EJDE-2017/151\hfilneg] {Dhruba R. Adhikari} \address{Dhruba R. Adhikari \newline Department of Mathematics, Kennesaw State University, Georgia 30060, USA} \email{dadhikar@kennesaw.edu} \dedicatory{Communicated by Pavel Drabek} \thanks{Submitted June 11, 2016. Published June 25, 2017.} \subjclass[2010]{47H14, 47H05, 47H11} \keywords{Strong quasiboundedness; Browder and Skrypnik degree theories; \hfill\break\indent maximal monotone operator; bounded demicontinuous operator of type $(S_+)$} \begin{abstract} Let $X$ be a real reflexive Banach space and $X^*$ its dual space. Let $L: X\supset D(L)\to X^*$ be a densely defined linear maximal monotone operator, and $T:X\supset D(T)\to 2^{X^*}$, $0\in D(T)$ and $0\in T(0)$, be strongly quasibounded maximal monotone and positively homogeneous of degree 1. Also, let $C:X\supset D(C)\to X^*$ be bounded, demicontinuous and of type $(S_+)$ w.r.t. to $D(L)$. The existence of nonzero solutions of $Lx+Tx+Cx\ni0$ is established in the set $G_1\setminus G_2$, where $G_2\subset G_1$ with $\overline G_2\subset G_1$, $G_1, G_2$ are open sets in $X$, $0\in G_2$, and $G_1$ is bounded. In the special case when $L=0$, a mapping $G:\overline G_1\to X^*$ of class $(P)$ introduced by Hu and Papageorgiou is also incorporated and the existence of nonzero solutions of $Tx+ Cx+ Gx\ni 0$, where $T$ is only maximal monotone and positively homogeneous of degree $\alpha\in (0, 1]$, is obtained. Applications to elliptic partial differential equations involving $p$-Laplacian with $p \in (1, 2]$ and time-dependent parabolic partial differential equations on cylindrical domains are presented. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction and preliminaries} Let $X$ be a real reflexive Banach space with its dual space $X^*$. The norms of $X, X^*$ will be denoted by $\|\cdot\|_X$ and $\|\cdot\|_{X^*}$, respectively. We denote by $\langle x^*,x\rangle$ the value of the functional $x^*\in X^*$ at $x\in X$. The symbols $\partial D, \overset\circ D, \overline D$, denote the strong boundary, interior and closure of the set $D$, respectively. The symbol $B_Y(0,R)$ denotes the open ball of radius $R$ with center at $0$ in a Banach space $Y$. If $\{x_n\}$ is a sequence 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 $Y$. The operator $T$ is said to be ``compact'' if it maps bounded subsets of $D(T)$ onto relatively compact subsets in $Y$. It is said to be ``demicontinuous'' if it is strong-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 $X^*$ and satisfies $$ J(\alpha x) = \alpha J(x), \quad (\alpha, x)\in \mathbb{R}_+\times X. $$ 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 if and only if $R(T+\lambda J) = X^*$ for all $\lambda\in (0,\infty)$. If $T$ is maximal monotone, then the operator $T_t\equiv(T^{-1}+tJ^{-1})^{-1}:X\to X^*$ called the Yosida approximant is bounded, demicontinuous, maximal monotone and such that $T_tx\rightharpoonup T^{\{0\}}x$ as $t\to 0^+$ for every $x\in D(T)$, where $T^{\{0\}}x$ denotes the element $y^*\in Tx$ of minimum norm, i.e., $\|T^{\{0\}}x\|=\inf\{\|y^*\|: y^*\in Tx\}$. In our setting, this infimum is always attained and $D(T^{\{0\}})=D(T)$. Also, $T_tx\in TJ_tx$, where $J_t \equiv I-tJ^{-1}T_t:X\to X$ and satisfies $\lim_{t\to 0}J_tx=x$ for all $x\in\overline{\operatorname{co}D(T)}$, where $\operatorname{co}A$ denotes the convex hull of the set $A$. In addition, $x\in D(T)$ and $t_0 > 0$ imply $\lim_{t\to t_0}T_tx=T_{t_0}x$. The operators $T_t,J_t$ were introduced by Br\'ezis, Crandall and Pazy in \cite{BCP}. For their basic properties, we refer the reader to \cite{BCP} as well as Pascali and Sburlan \cite[pp. 128-130]{PS}. We need the following lemmas about maximal monotone operators. \begin{lemma}[{\cite[p. 915]{Zeidler1}}] \label{lemA} Let $T:X\supset D(T)\to 2^{X^*}$ be maximal monotone. Then the following are true: \begin{itemize} \item[(i)] $\{x_n\}\subset D(T)$, $x_n\to x_0$ and $Tx_n\owns y_n\rightharpoonup y_0$ imply $x_0\in D(T)$ and $y_0\in Tx_0$. \item[(ii)] $\{x_n\}\subset D(T)$, $x_n\rightharpoonup x_0$ and $Tx_n\owns y_n\to y_0$ imply $x_0\in D(T)$ and $y_0\in Tx_0$. \end{itemize} \end{lemma} The next lemma is essentially due to Br\'ezis, Crandall and Pazy \cite{BCP}, and its proof can be found in \cite{AK}. \begin{lemma}\label{L1} Assume that the operators $T:X\supset D(T)\to 2^{X^*}$ and $S:X\supset D(S)\to 2^{X^*}$ are maximal monotone, with $0\in D(T)\cap D(S)$ and $0\in S(0)\cap T(0)$. Assume, further, that $T+S$ is maximal monotone and that there is a sequence $\{t_n\}\subset (0,\infty)$ such that $t_n\downarrow 0$, and a sequence $\{x_n\}\subset D(S)$ such that $x_n\rightharpoonup x_0\in X$ and $T_{t_n}x_n+w^*_n\rightharpoonup y_0^*\in X^*$, where $w^*_n\in Sx_n$. Then the following are true. \begin{itemize} \item[(i)] The inequality \begin{equation} \label{L12} \lim_{n\to\infty}\langle T_{t_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 T_{t_n}x_n+w^*_n,x_n-x_0\rangle = 0, \end{equation} then $x_0\in D(T+S)$ and $y_0^*\in (T+S)x_0$. \end{itemize} \end{lemma} \begin{definition} \rm An operator $T:X \supset D(T) \to 2^{X^*}$ is said to be ``strongly quasibounded" if for every $S>0$ there exists $K(S)>0$ such that $$ \|x\| \le S,\quad \text{and}\quad \langle x^*, x \rangle \le S,\quad \text{for some }x^*\in Tx, $$ imply $\|x^*\| \le K(S)$. \end{definition} Browder and Hess have shown in \cite{BrowderHess1972} that a monotone operator $T$ with $0\in{\mathaccent"7017 D}(T)$ is strongly quasibounded. The proof of the following lemma, which is due to Browder and Hess \cite{BrowderHess1972}, can also be found in \cite[Lemma D]{Kartsatos2008}. \begin{lemma}\label{L2} Let $T:X \supset D(T) \to 2^{X^*}$ be a strongly quasibounded maximal monotone operator such that $0 \in T(0)$. Let $\{t_n\} \subset (0, \infty)$ and $\{u_n\} \subset X$ be such that $$ \|u_n\| \le S,\quad \langle T_{t_n}u_n, u_n \rangle \le S,\quad\text{for all } n, $$ where $S$ is a positive constant. Then there exists a number $K=K(S)>0$ such that $\|T_{t_n}u_n\| \le K$ for all $n$. \end{lemma} \begin{definition}\label{D31} \rm An operator $G: X\supset D(G)\to 2^{X^*}$ is said to belong to class $(P)$ if it maps bounded sets to relatively compact sets, for every $x\in D(G)$, $G(x)$ is closed and convex subsets of $X^*$ and $G(\cdot)$ is upper-semicontinuous (usc), i.e., for every closed set $F\subset X^*$, the set $G^-(F) = \{x\in D(G): G(x) \cap F \ne \emptyset\}$ is closed in $X$. \end{definition} An important fact about a compact-set valued upper-semicontinuous operator $G$ is that it is closed. Furthermore, for every sequence $\{[x_n, y_n]\}\subset Gr(G)$ such that $x_n\to x\in D(G)$, the sequence $\{y_n\}$ has a cluster point in $G(x)$. \begin{definition}\label{def1} \rm Let $L:X\supset D(L)\to X^*$ be a densely defined linear maximal monotone operator and $C: X\supset D(C)\to X^*$ be bounded and demicontinuous. We say that $C: X\supset D(C)\to X^*$ is of type $(S_+)$ w.r.t. to $D(L)$ if for every sequence $\{x_n\}\subset D(L)\cap D(C)$ with $x_n\rightharpoonup x_0$ in $X$, $Lx_n\rightharpoonup Lx_0$ 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 $X$. In this case, if $L=0$, then $C$ is of class $(S_+)$. \end{definition} \begin{definition}\label{def2} \rm The family $C(t):X\supset D\to X^*, t\in[0,1]$, of operators is said to be a ``homotopy of type $(S_+)$ w.r.t. $D(L)$" if for any sequences $\{x_n\}\subset D(L)\cap D$ with $x_n\rightharpoonup x_0$ in $X$ and $Lx_n\rightharpoonup Lx_0$ in $X^*$, $\{t_n\}\subset[0,1]$ 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 $X^*$. In this case, if $L=0$, then $C(t)$ is a homotopy of type $(S_+)$. A homotopy of type $(S_+)$ w.r.t. $D(L)$ is ``bounded" if the set $$ \{C(t)x: t\in[0,1], x\in D\} $$ is bounded. \end{definition} Let $G$ be an open and bounded subset of $X$. Let $L:X\supset D(L)\to X^*$ be densely defined linear maximal monotone, $T:X\supset D(T)\to 2^{X^*}$ maximal monotone, and $C(s) :X\supset \overline{G}\to X^*$, $s\in[0,1]$, a bounded homotopy of type $(S_+)$ w.r.t. $D(L)$. Since the graph $Gr(L)$ of $L$ is closed in $X\times X^*$, the space $Y=D(L)$ associated with the graph norm $$ \|x\|_Y = \|x\|_X + \|Lx\|_{X^*}, \quad x\in Y, $$ becomes a real reflexive Banach space. We may now 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. Note that since $j:Y\to X$ is continuous, we have $D(j^*)=X^*$, which 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, and $$ \overline{j^{-1}(G)} \subset j^{-1}(\overline G), \quad \partial (j^{-1}(G))\subset j^{-1}(\partial G). $$ We define $M:Y\to Y^*$ by $$ ( Mx, y) = \langle Ly, J^{-1}(Lx)\rangle ,\quad x, y\in D(L). $$ Here, the duality pair $(\cdot, \cdot )$ 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 $J^{-1}(Lx)\in D(L^*)$ and \begin{gather} Mx = j^*\circ L^*\circ J^{-1}( Lx),\label{M2}\\ ( 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 $My\in j^*(X^*)$. We now define $\hat L: Y\to Y^*$ and $\hat C(s): j^{-1}(\overline G)\to Y^*$ by $$ \hat L = j^*\circ L \circ j \quad\text{and}\quad \hat C(s) = j^*\circ C(s)\circ j $$ respectively, and for every $t >0$, we also define $\hat T_t:Y\to Y^*$ by $$ \hat T_{t} = j^*\circ T_{t}\circ j, $$ where $T_{t}$ is the Yosida approximant of $T$. Kartsatos and the author developed a new degree theory in \cite{AK2008} for the triplet $L+T+C$, where $L$ is densely defined linear maximal monotone, $T$ is (possibly nonlinear) maximal monotone and strongly quasibounded, and $C$ is bounded, demicontinuous and of type $(S_+)$ w.r.t. the set $D(L)$. This degree theory extends the degree theory of Berkovits and Mustonen \cite{BM} who considered the case $T=0$. As in \cite{BM}, the construction of the degree mapping in \cite{AK2008} uses the graph norm topology of the space $Y=D(L)$ and is based on the Skrypnik degree and its invariance under homotopies of type $(S_+)$. In fact, it is shown that the mapping \begin{equation} H(t,x):=\hat L+\hat T_t+\hat C+tMx,\quad (t,x)\in(0,\infty)\times j^{-1}(\overline G),\label{L16} \end{equation} has the Skrypnik degree, ${\rm d}_{\rm S}(H(t,\cdot),\widetilde G,0)$, under the usual boundary condition on the boundary of an open and bounded set $\widetilde G\subset Y$, which remains fixed for all sufficiently small $t\in(0,\infty)$. Then the degree is defined by \begin{equation} d(L+T+C,G,0) = \lim_{t\downarrow 0}{\rm d}_{\rm S}(\hat L+\hat T_t+\hat C+tM,\widetilde G,0),\label{L17} \end{equation} where $G$ is an open bounded subset of $X$ related to $\widetilde G$. The operator $C$ above satisfies the $(S_+)$-condition w.r.t.~$Y=D(L)$ and $T$ is strongly quasibounded and maximal monotone with $0\in T(0)$. In order to show that the degree ${\rm d}_{\rm S}$ is fixed as above, it can be shown, in addition, that the family of mappings $f^t := H(t,\cdot)$ is a homotopy of class $(S_+)$ in the sense of Browder \cite[Definition 3, p. 69]{BR1983} on every interval $[t_1,t_2]\subset(0,t_0]$, where $t_0$ is an appropriate fixed positive number. The approach discussed here is that of Berkovits and Mustonen in \cite{BM} and Addou and Memri in \cite{AM}. In Section 2, we establish the existence of nonzero solutions of the inclusion $Lx+Tx+Cx\ni 0$, where $L$, $C$ are as above and $T$ is a strongly quasibounded maximal monotone operator and positively homogeneous of degree 1. This result is in the spirit of similar results in \cite{AK} for operators of the form $T+C$, where $T$ is single-valued maximal monotone, $0=T(0)$, and $C$ bounded demicontinuous and of type $(S_+)$. Mild and natural boundary conditions are considered in order to establish the result by utilizing the graph norm topology on $D(L)$ and relevant topological degree theory. The theory is applicable to parabolic partial differential equations in divergence form on cylindrical domains. In Section 3, the existence of nonzero solutions of $Tx+Cx+ Gx\ni 0$ is established by utilizing the topological degree theories developed by Browder \cite{BrowderHess1972} and Skrypnik \cite{Skrypnik1994}. In this case, $T$ is only maximal monotone with $0\in T(0)$ and positively homogeneous of degree $\alpha\in (0, 1]$, and $C$ is bounded demicontinuous of type of $(S_+)$. This result extends and generalizes a similar result in \cite{AK} for $\alpha =1$ and $G = 0$ and has applications to elliptic boundary value problems involving $p$-Laplacian. For additional facts and various topological degree theories related to the subject of this paper, the reader is referred to Kartsatos and the author \cite{AK1}, Kartsatos and Lin \cite{KartsatosLin2003}, and Kartsatos and Skrypnik \cite{KartsatosSkrypnik2005b,KartsatosSkrypnik}. For information on various concepts and ideas of Nonlinear Analysis used herein, the reader is referred to Barbu \cite{BA}, Browder \cite{Browder1976}, Pascali and Sburlan \cite{PS}, Simons \cite{Simons}, Skrypnik \cite{Skrypnik1986,Skrypnik1994}, and Zeidler \cite{Zeidler1}. The following lemma from \cite{AK2016} about the boundedness of the solutions of a homotopy equation will be needed in the sequel. \begin{lemma}\label{L3} Let $G\subset X$ be open and bounded. Assume the following: \begin{itemize} \item[(A1)] $L:X\supset D(L)\to X^*$ is linear, maximal monotone with $D(L)$ dense in $X$; \item[(A2)] $T:X\supset D(T)\to 2^{X^*}$ is strongly quasibounded, maximal monotone with $0\in T(0)$; \item[(A3)] $C(t):X\supset\overline G\to X^*$ is a bounded homotopy of type $(S_+)$ w.r.t. $D(L)$. \end{itemize} Then, for a continuous curve $f(s), 0\le s \le 1$, in $X^*$, the set $$ K=\big\{x\in j^{-1}(\overline G): \hat L + \hat T_t+\hat C(s)+tMx = j^*f(s), \text{ for some }t>0,\; s\in[0,1]\big\} $$ is bounded in $Y$. Thus, there exists $R>0$ such that $K\subset B_Y(R)$, where $ B_Y(R))$ is the open ball of $Y$ of radius $R$. \end{lemma} Lemma \ref{L4} below taken from Kartsatos and Skrypnik \cite{KartsatosSkrypnik2005a} will be used in the proof of Theorem~\ref{Th1}. \begin{lemma}\label{L4} Let $T:X\supset D(T)\to 2^{X^*}$ be maximal monotone and such that $0\in D(T)$ and $0\in T(0)$. Then the mapping $(t,x)\to T_tx$ is continuous on the set $(0,\infty)\times X$. \end{lemma} \begin{definition}\label{D32} \rm An operator $T: X\supset D(T)\to 2^{X^*}$ is said to be positively homogeneous of degree $\alpha >0$ if, for a fixed $\alpha >0$, $x\in D(T)$ implies $tx\in D(T)$ for all $t\in\mathbb{R}_+$ and $T(tx) = t^\alpha Tx$. \end{definition} The following lemma, which plays an important role in the existence theorems of Section 2 and Section 3, shows in particular that the Yosida approximants of a positively homogeneous maximal monotone operator of degree $\alpha$ are also positively homogeneous only when $\alpha = 1$. \begin{lemma}\label{L5} Let $T:X\supset D(T)\to 2^{X^*}$ is maximal monotone and positively homogeneous of degree $\alpha>0$. Then, for each $t>0$, the Yosida approximant $T_t$ satisfies \begin{equation}\label{138} T_t(sx) = s^\alpha T_{ts^{\alpha-1}}(x)\quad \text{for all } (s, x)\in (0, +\infty)\times X. \end{equation} \end{lemma} \begin{proof} Let $$ y= T_t(sx) =(T^{-1} + t J^{-1}) ^{-1}(sx), $$ for $t, s>0$, $x\in X$. The homogeneity of the duality mapping $J$ implies \begin{align*} y \in T(-t J^{-1} y + sx) &= T\Big(s\Big(-\frac{t}{s} J^{-1} y + x\Big)\Big)\\ &= s^{\alpha}T\Big(-\frac{t}{s} J^{-1} y + x\Big)\\ &= s^{\alpha}T\Big(-\frac{t}{s^{1-\alpha}} J^{-1}\big(\frac{y}{s^\alpha}\big) + x\Big). \end{align*} This is equivalent to $$ x\in T^{-1}\Big(\frac{y}{s^\alpha}\Big) + ts^{\alpha-1} J^{-1}\Big(\frac{y}{s^\alpha}\Big) $$ and $$ y = s^\alpha (T^{-1} + ts^{\alpha-1} J^{-1})^{-1}x = s^\alpha T_{ts^{\alpha-1}}(x). $$ \end{proof} \section{Nonzero solutions of $Lx+Tx+Cx\ni 0$} Guo and Lakshmikantham have shown in \cite{GL} the following result for compact operators defined on a cone in a Banach space. The operator $T$ satisfies non-contractive and non-expansive type of conditions only on the boundary of the subsets $G_1, G_2$ of $X$ for the existence of a nonzero fixed of $T$. \begin{theorem} \label{thmA} Let $X$ be a Banach space and $K$ a positive cone in $X$ which induces a partial ordering $``\le"$ in $X$. Let $G_1$, $G_2\subset X$ be open, $0\in G_2$, $\overline{G_2}\subset G_1$, $G_1$ bounded, and $T:K\cap \overline G_1\to K$ compact with $T(0)=0$. Suppose that one of the following two conditions holds. \begin{enumerate} \item $Tx \not\ge x$ for $x\in K\cap\partial G_1$, and $Tx \not\le x$ for $x\in K\cap\partial G_2$; \item $Tx \not\le x$ for $x\in K\cap\partial G_1$, and $Tx \not\ge x$ for $x\in K\cap\partial G_2$. \end{enumerate} Then there exists a fixed point of $T$ in $K\cap(G_1\setminus G_2)$. \end{theorem} By imposing certain conditions only on the boundary of sets $G_1, G_2$, the author and Kartsatos \cite{AK} established the existence of nonzero solutions of $Tx+Cx=0$, where $T$ is positively homogeneous of degree 1 and single-valued maximal monotone, and $C$ is a bounded demicontinuous of type $(S_+)$. The following result is obtained in the spirit of \cite[Theorem 6, p.1246]{AK} in the context of the Berkovits-Mustonen theory in \cite{BM}. \begin{theorem}\label{Th1} 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 $T:X\supset D(T) \to 2^{X^*}$ strongly quasibounded, maximal monotone and positively homogeneous of degree $1$. Also, let $C:\overline{G_1}\to X^*$ be bounded, demicontinuous and of type $(S_+)$ w.r.t. to $D(L)$. Moreover, assume the following: \begin{itemize} \item[(H1)] there exists $v^*\in X^*\setminus\{0\}$ such that $Lx+Tx+Cx\not\ni \lambda v^*$ for all $(\lambda,x)\in\mathbb{R}_+\times( D(L)\cap D(T)\cap \partial G_1)$, and \item[(H2)] $Lx+Tx+Cx+ \lambda Jx \not\ni 0$ for all $(\lambda,x)\in~\mathbb{R}_+\times( D(L)\cap D(T)\cap \partial G_2)$. \end{itemize} Then the inclusion $Lx+Tx+Cx\ni 0$ has a solution $x\in D(L)\cap D(T)\cap(G_1\setminus G_2)$. \end{theorem} \begin{proof} To solve the inclusion \begin{equation}\label{T1} Lx+Tx+Cx\ni 0, \quad x\in\overline{G_1}, \end{equation} let us consider the associated equation \begin{equation}\label{T2} \hat Lx+\hat T_tx+\hat Cx +t Mx=0, \quad t\in (0, +\infty),\; x\in j^{-1}(\overline{G_1}). \end{equation} One can show as in \cite{AK2008} that there exists $R>0$ such that the open ball $B_Y(0, R) = \{y\in Y: \|y\|_Y < R\}$ contains all solutions of \eqref{T2}. We shall prove that \eqref{T2} has a solution $x_t\in j^{-1}(G_1\setminus G_2)$ for all sufficiently small $t$. We first claim that there exist $\tau_0>0$ , $t_0>0$ such that \begin{equation}\label{T3} \hat Lx+\hat T_tx+\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 $\tau_n\to \infty$, $t_n\downarrow 0$ and \begin{equation}\label{T4} \hat Lx_n+\hat T_{t_n}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 obtain the boundedness of $\{M(x_n)\}$. It is clear that $\hat Cx_n$ is bounded in $Y^*$, and therefore \eqref{T4} implies that $\|\hat Lx_n + \hat T_{t_n}x_n\|_{Y^*} \to \infty$. Define $$ \alpha_n = \frac{1}{\|\hat Lx_n + \hat T_{t_n}x_n\|_{Y^*}} \quad\text{and}\quad u_n=\alpha_n x_n. $$ It is obvious that $u_n\to 0$ in $Y$. Since $T$ is positively homogeneous of degree 1, $T_t$ is also positively homogeneous of degree 1 by Lemma~\ref{L5}. From \eqref{T4}, we obtain \begin{equation}\label{T5} (\hat L +\hat T_{t_n})(\alpha_n x_n)+\alpha_n\hat Cx_n+t_n\alpha_n Mx_n =\tau_n\alpha_n j^*v^*. \end{equation} Since $\|(\hat L +\hat T_{t_n})(\alpha_n x_n)\|_{Y^*} = 1$, \eqref{T5} implies $$ \tau_n\alpha_n \to \frac{1}{\|j^*v^*\|_{Y^*}}, $$ and therefore $$ (\hat L +\hat T_{t_n})(u_n )=(\hat L +\hat T_{t_n})(\alpha_n x_n )\to y_0, $$ where $$ y_0 = \frac{j^*v_*}{\|j^*v^*\|_{Y^*}}. $$ Since $u_n\to 0$, we have $$ \lim_{n\to\infty} \langle(\hat L+\hat T_{t_n})u_n, u_n\rangle = \langle y_0, 0\rangle =0. $$ Since $\hat L, \hat T_{t_n}$, and $\hat L+\hat T_{t_n}$ are maximal monotone, by Lemma \ref{L1}, (ii), we have $$ y_0= (\hat L+\hat T)(0) =0, $$ 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 T_t 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 $(S_+)$ w.r.t. $D(L)$. 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 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 $Y$, $T_{t_n}x_n\rightharpoonup w^*$ in $X^*$ and $Cx_n\rightharpoonup c^*$ and \begin{equation}\label{T7} \hat Lx_n +\hat T_{t_n} x_n+\hat Cx_n+t_n Mx_n =s_n\tau_0 j^*v^*. \end{equation} Here, the boundedness of $\{T_{t_n}\}$ follows as in Step I of \cite[Prop.~1]{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 $x_n\in j^{-1}(\partial G_1) = \partial G_1\cap Y \subset \partial G_1$. From \eqref{T7} we obtain \begin{equation}\label{T8} \langle Lx_n+ T_{t_n} x_n+ Cx_n+ t_nL^* J^{-1}(Lx_n), x_n-x_0\rangle =s_n\tau_0\langle v^*, x_n-x_0\rangle . \end{equation} If we assume \begin{equation} \label{T9} \limsup_{n\to\infty}\langle Cx_n, x_n - x_0\rangle >0, \end{equation} we easily get a contradiction using a standard argument in relation to Lemma~\ref{L1}, (i). This is because $L+T$ is maximal monotone because $T$ is strongly quasibounded (cf. Pascali and Sburlan \cite[Proposition, p. 142]{PS}). Consequently, \begin{equation}\label{T10} \limsup_{n\to\infty}\langle Cx_n, x_n - x_0\rangle \le 0. \end{equation} Since $C$ is demicontinuous and of type $(S_+)$ w.r.t. $D(L)$, we obtain $x_n\to x_0$ and $Cx_n\rightharpoonup c^*=Cx_0$. From \eqref{T8}, we obtain $$ \lim_{n\to\infty}\langle Lx_n + T_{t_n}x_n, x_n - x_0\rangle = 0. $$ Using Lemma~\ref{L1}, (ii), we obtain $x_0\in D(T)$ and $w^*\in Tx_0$. Then, in view of \eqref{T8}, it follows that $$ \langle Lx_0 + w^*+ Cx_0 - s_0\tau_0 v^*, u\rangle = 0 $$ for all $u\in Y$. Since $Y$ is dense in $X$, we have $$ Lx_0 + Tx_0+ Cx_0 \ni s_0\tau_0 v^*, $$ which contradicts the hypothesis (H1) 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 $s\in [0, 1]$. The mapping $H(s, x)$ is an admissible homotopy for the Skrypnik's degree. The Skyrpnik's degree, ${\rm d}_{\rm S} (H(s, \cdot), G_R^1(Y), 0)$, is well-defined and remains constant for all $s\in [0, 1]$. Also, the degree, ${\rm d}(L+T+C, G_1, 0)$, developed in \cite{AK2008} is defined as $$ {\rm d}(L+T+C-\tau_0v^*, G_1, 0)= \lim_{t\to0+}{\rm d}_{\rm S} (H(1, \cdot), G_R^1(Y), 0). $$ By shrinking $t_0$ further if necessary, we have $$ {\rm d}(L+T+C-\tau_0v^*, G_1, 0)= {\rm d}_{\rm S} (H(1, \cdot), G_R^1(Y), 0), \quad \text{for all } t\in (0, t_0]. $$ Suppose, if possible, that $$ {\rm d}_{\rm S} (H(1, \cdot), G_R^1(Y), 0)\ne 0$$ for some $t_1\in (0, t_0]$. Then there exists $x_0\in G_R^1(Y)$ such that $$\hat Lx +\hat T_{t_1} x+\hat Cx+t_1 Mx = \tau_0 j^*v^*.$$ This contradicts the choice of $\tau_0$ as stated in \eqref{T3}. Since $${\rm d}_{\rm S} (H(0, \cdot), G_R^1(Y), 0)= {\rm d}_S (H(1, \cdot), G_R^1(Y), 0),$$ we have \begin{equation}\label{D1} {\rm d}_S (\hat L+ \hat T_t + \hat C + tM, 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 $$ \widetilde{H}(s, x)= s(\hat Lx +\hat T_t x+\hat Cx) +t Mx+ (1-s)\hat Jx, \quad s\in [0, 1], \; x\in j^{-1}(\overline {G_2}). $$ As in \cite[Step III, p.29]{AK2016}, it can be shown that there exists $t_0>0$ (choose it even smaller than the one used previously if necessary) such that all the solutions $$ \widetilde{H} (s, x) = 0, \; t\in (0, t_0], \;s\in [0, 1] $$ are bounded in $Y$. We enlarge the previous number $R>0$ if necessary so that all solutions of $\widetilde{H}(s, x) = 0$ as above are contained in $B_Y(0, R)$ in $Y$. We first show that there exists $t_1\in (0, t_0]$ 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, $G_R^2(Y) := j^{-1}(G_2)\cap B_Y(0, R)$. Suppose that the contrary is true. Then there must exist sequences $\{t_n\}\subset (0, t_0]$, $\{s_n\}\subset [0, 1]$, $\{x_n\}\subset\partial G_R^2(Y) $ such that \begin{equation}\label{T11} s_n(\hat Lx_n +\hat T_{t_n} x_n+\hat Cx_n)+t_n Mx_n+ (1-s_n)\hat Jx_n = 0. \end{equation} We may assume that $t_n\downarrow 0$, $s_n\to s_0$, $x_n\rightharpoonup x_0$ in $X$ and $Lx_n \rightharpoonup Lx_0$ in $X^*$. As in the previous part, we can show that $x_n\in\partial G_2 \cap Y \subset \partial G_2$. If $s_n = 0$ for some $n$, then we obtain $t_n Mx_n + \hat Jx_n = 0$. Since $M$ is monotone for such $x_n$'s by \eqref{M2}, \eqref{M2b}, and $\hat J$ is strictly monotone, we obtain $x_n = 0$ which is a contradiction to $0\in G_2$. We may now assume that $s_n\in (0, 1]$. Suppose $s_0 =0$. Dividing both sides of \eqref{T11}, we obtain \begin{equation} \label{T12} \hat Lx_n +\hat T_{t_n} x_n+\hat Cx_n+\frac{t_n}{s_n} Mx_n = -\frac{1-s_n}{s_n}\hat Jx_n , \end{equation} which implies $$ \langle Cx_n, x_n \rangle \le -\frac{(1-s_n)}{s_n}\|x_n\|_X^2. $$ Since $x_n\in\partial G_2$, the sequence $\{\|x_n\|_X\}$ is bounded away from zero. This leads to a contradiction to the boundedness of $\{\langle Cx_n, x_n\rangle\}$ because ${(1-s_n)}/{s_n}\to \infty$. Assume that $s_0= 1$. Now, by Lemma~\ref{L2}, the strong quasiboundedness of $T$ implies that the sequence $\{T_{t_n}x_n\}$ is bounded, and so we may assume that $T_{t_n}x_n\rightharpoonup w^*$ for some $w^*\in X^*$. From \eqref{T11}, we obtain \begin{equation}\label{T13} \lim_{n\to\infty}\langle Lx_n+ T_{t_n}x_n + Cx_n, x_n-x_0\rangle =0. \end{equation} If \eqref{T9} is true, we obtain a contradiction to (i) of Lemma~\ref{L1}. Therefore \eqref{T10} must hold true. With \eqref{T13}, this implies $x_n\to x_0\in\partial G_2$, and therefore $x_0\in D(T)$ and $Lx_0+Tx_0+Cx_0\ni 0$. This is a contradiction to hypothesis (H2) for $\lambda =0$. For the remaining case $s_0\in (0, 1)$, one can see that \eqref{T12} is replaced with \begin{equation}\label{T14} \limsup_{n\to\infty}\langle Lx_n+ T_{t_n}x_n + Cx_n, x_n-x_0\rangle \le 0. \end{equation} We may assume that $T_{t_n}x_n\rightharpoonup w^* (\text{some})\in X^*$. By using the monotonicity of $L$, $T_{t_n}$, the continuity of $T_t$ from Lemma~\ref{L4} and a standard argument, we obtain $x_n\to x_0\in\partial G_2$, and hence \eqref{T12} implies $$ \langle Lx_0 + w^*+Cx_0+\frac{1-s_0}{s_0}Jx_0, u\rangle = 0 $$ for all $u\in Y$. By the density of $Y$ in $X$, we obtain $$ Lx_0 + Tx_0+Cx_0+\frac{1-s_0}{s_0}Jx_0\ni 0,$$ which contradicts hypothesis (H2). At this time, 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 T_t x+\hat Cx)+t Mx + (1-s)\hat Jx =0, \quad s\in [0, 1], \; x\in \overline {G_R^2(Y)}. \end{equation} It is already shown 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, $\hat L +\hat T_t +\hat C+tM$ and $tM+ \hat J$. We also note here that $tM + \hat J$ is strictly monotone. Therefore $\widetilde{H}(s, x)$ is an admissible homotopy for the Skrypnik'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}_{\rm S}(\widetilde{H} (0, \cdot), G_R^2(Y), 0). \end{equation} This implies \begin{equation}\label{D2} {\rm d}_S(\hat L+\hat T_t +\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 $$ {\rm d}_S(\hat L+\hat T_t +\hat C+t M, G_R^1(Y), 0) \ne {\rm d}_S(\hat L+\hat T_t +\hat C+t M, G_R^2(Y), 0) $$ for all $t\in (0,t_0]$. By the excision property of the Skrypnik'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 $$ \hat Lx+\hat T_t x+\hat Cx+t Mx=0. $$ We now pick a sequence $\{t_n\}\subset (0, t_0]$ such that $t_n\downarrow 0$, and denote the corresponding solution $x_t$ by $x_n$, i.e. $$ \hat Lx_n+\hat T_{t_n}x_n +\hat Cx_n+t_n Mx_n=0. $$ Since $Y$ is reflexive, we have $x_n\rightharpoonup x_0\in Y$ by passing to a subsequence. This implies $x_n\to x_0$ in $X$ and $Lx_n \rightharpoonup Lx_0$ in $X^*$. By the strong quasiboundedness of $T$, we may assume that $T_{t_n}x_n\rightharpoonup w^*\in X^*$. If \eqref{T9} holds, then we obtain a contradiction by Lemma~\ref{L1}, (i). Then \eqref{T10} must be valid. Since $C$ is of type $(S_+)$ w.r.t. $D(L)$, we obtain $x_n\to x_0\in\overline{G_R^1(Y)\setminus G_R^2(Y)}$, and by Lemma \ref{lemA}, we have $x_0\in D(T)$ and $ Lx_0+ w^*+ Cx_0 = 0$, and therefore $Lx_0 +Tx_0+Cx_0 \ni 0$. It remains to show that $x_0\in G_1\setminus G_2$. Since \[ G_R^1(Y)\setminus G_R^2(Y) =(G_1\setminus G_2) \cap Y\cap B_Y(0, R) \subset G_1\setminus G_2, \] we have $x_n\in G_1\setminus G_2$ for all $n$, and so $$ x_0\in\overline{G_1\setminus G_2} \subset (G_1\setminus G_2) \cup \partial (G_1\setminus G_2)\subset (G_1\setminus G_2) \cup \partial G_1 \cup \partial G_2 $$ By hypotheses (H1) and (H2), $x_0\not\in \partial G_1\cup\partial G_2$. Thus, $x_0\in D(L)\cap D(T)\cap (G_1\setminus G_2)$. \end{proof} \section{Nonzero solutions of $Tx+Cx+Gx\ni 0$} Hu and Papageorgiou \cite{HP} generalized the degree theory of Browder \cite{Browder1983} to the mappings of the form $T+C+G$, where $T$ is maximal monotone with $0\in T(0)$, $C$ bounded demicontinuous of type $(S_+)$ and $G$ belongs to class $(P)$. In this section, with an application of Browder and Skrypnik degree theories, the existence of nonzero solutions of the inclusion $Tx+Cx+Gx\ni 0$ is established with an additional condition of positive homogeneity of degree $\alpha\in (0, 1]$ on $T$. The result extends and generalizes a similar result by Kartsatos and the author in \cite[ Theorem 6, p.1246, for $\alpha =1$ and $G=0$]{AK} to a multivalued $T$ with $\alpha \in (0, 1]$ and $G\ne 0$. This result is new for $\alpha\in (0, 1)$ and applies to partial differential equations involving $p$-Laplacian with $p\in (1, 2]$. In what follows, the norms in $X$ and $X^*$ are both denoted by $\|\cdot\|$ and will be understood from the context of their use. \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 $T:X\supset D(T)\to 2^{X^*}$ be maximal monotone, and positively homogeneous of degree $\alpha\in(0, 1]$, $C:\overline{ G}_1\to X^*$ bounded, demicontinuous and of type $(S_+)$, and $G:\overline{G}_1\to 2^{X^*}$ of class $(P)$. Moreover, assume the following: \begin{itemize} \item[(H3)] There exists $v^*_0\in X^*\setminus\{0\}$ such that $Tx+Cx +Gx\not\ni \lambda v^*_0$ for every $(\lambda,x)\in\mathbb{R}_+\times(D(T)\cap\partial G_1)$; \item[(H4)] $Tx+Cx+ Gx+\lambda Jx \not\ni 0$ for every $(\lambda,x)\in \mathbb{R}_+\times(D(T)\cap\partial G_2)$. \end{itemize} Then the inclusion $Tx+Cx+Gx\ni 0$ has a nonzero solution $x\in D(T)\cap(G_1\setminus G_2)$. \end{theorem} \begin{proof} We consider the inclusion $$Tx + Cx + Gx \ni 0$$ and then the associated approximate equation \begin{equation} T_tx + Cx + g_\epsilon x = 0. \label{5} \end{equation} Here, $\epsilon >0$ and $g_\epsilon:\overline{G_1}\to X^*$ is an approximate continuous Cellina-selection (cf. \cite{HP}, \cite[Lemma 6, p. 236]{Aubin1984}) satisfying $$ g_\epsilon x\in G(B_\epsilon(x)\cap \overline{G_1})+B_\epsilon(0) $$ for all $x\in \overline{G_1}$ and $g_\epsilon(\overline{G_1})\subset\overline{\operatorname{conv}} G(\overline{G_1})$. We show that equation \eqref{5} has a solution $x_{t, \epsilon}$ in $G_1\setminus G_2$ for all sufficiently small $t$ and $\epsilon$. To this end, we first show that there exist $\tau_0>0$, $t_0>0$ and $\epsilon_0>0$ such that the equation \begin{equation}\label{6*} T_tx + Cx+ g_\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 $\tau_n\to\infty$, $t_n\downarrow 0$, $\epsilon_n\downarrow 0$ and \begin{equation}\label{7} T_{t_n}x_n + Cx_n +g_{\epsilon_n}x_n = \tau_n v_0^*. \end{equation} We may assume that $g_{\epsilon_n}x_n\to g^*\in X^*$ in view of the properties of $G$. Then $\|T_{t_n}x_n\|\to \infty$ as $\|\tau_n v_0^*\|\to\infty$ and $\{Cx_n\}$ is bounded. Thus, from \eqref{7}, we obtain \begin{equation}\label{8} \frac{T_{t_n}x_n}{\|T_{t_n}x_n\|} + \frac{Cx_n}{\|T_{t_n}x_n\|} + \frac{g_{\epsilon_n}x_n}{\|T_{t_n}x_n\|}= \frac{\tau_n}{\|T_{t_n}x_n\|}v_0^*, \end{equation} In view of \eqref{138}, we obtain \begin{equation} \label{144} \frac{T_{t_n}x_n}{\|T_{t_n}x_n\|} = T_{{t_n}{\lambda_n}}\Big(\frac{x_n}{\|T_{t_n}x_n\|^{1/\alpha}}\Big), \end{equation} where $$ \lambda_n =\|T_{t_n}x_n\|^{{(\alpha-1)}/\alpha}. $$ It clear that $\lambda_n\to 0$ for $\alpha\in(0, 1)$ and $\lambda_n = 1$ for $\alpha = 1$. Then \eqref{8} implies $$ 1-\big\|\frac{Cx_n}{\|T_{t_n}x_n\|} + \frac{g_{\epsilon_n}x_n}{\|T_{t_n}x_n\|}\big\| \le \frac{\tau_n\|v_0^*\|}{\|T_{t_n}x_n\|} \le 1 +\big\|\frac{Cx_n}{\|T_{t_n}x_n\|} + \frac{g_{\epsilon_n}x_n}{\|T_{t_n}x_n\|}\big\|. $$ Thus, \begin{equation}\label{10} \frac{\tau_n\|v_0^*\|}{\|T_{t_n}x_n\|}\to 1 \quad\text{and} \quad \frac{\tau_n}{\|T_{t_n}x_n\|}\to\frac{1}{\|v_0^*\|} \quad\text{as } n\to\infty. \end{equation} Let $$ u_n = \frac{x_n}{\|T_{t_n}x_n\|^{1/\alpha}}. $$ We have $u_n\to 0$. By \eqref{8}, \eqref{144} and \eqref{10}, we obtain $T_{t_n\lambda_n}u_n\to h$ with $$ h = \frac{v_0^*}{\|v_0^*\|}. $$ Therefore $$ \lim_{n\to\infty}\langle T_{t_n\lambda_n}u_n, u_n\rangle = \langle h, 0\rangle = 0. $$ Since $t_n\lambda_n\to 0$, by (ii) of Lemma~\ref{L1} with $S=0$ we obtain, $0\in D(T)$ and $h= T(0)$. Since $T(0) = 0$, this is a contradiction to $\|h\| = 1$. We now consider the homotopy mapping \begin{equation}\label{13} H_1(s,x, t, \epsilon) = T_tx+Cx+g_\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. For every $s\in[0,1]$ the operator $x\mapsto Cx- s\tau_0v_0^*$ is demicontinuous and bounded on $\overline{G_1}$. In order to see that it is of type $(S_+)$, assume that $\{x_n\}\subset\overline{G_1}$ satisfies $x_n\rightharpoonup x_0\in X$ and $$\limsup_{n\to\infty}\langle Cx_n- s\tau_0 v_0^*, x_n -x_0\rangle \le 0.$$ Then $$\limsup_{n\to\infty}\langle Cx_n, x_n -x_0\rangle \le 0,$$ which by the $(S_+)$-property of $C$, implies $x_n\to x_0\in\overline{G_1}$. Before we consider the Skrypnik degree of this homotopy on the set $G_1$, we show that the equation $H_1(s, x, t, \epsilon) = 0$ has no solution on the boundary of $G_1$ for all sufficiently small $t\in(0, t_0]$, $\epsilon\in(0, \epsilon_0]$ and all $s\in[0,1]$. To this end, assume the contrary and let $\{x_n\}\subset\partial G_1$, $\{t_n\}\subset(0, t_0]$, $\{s_n\}\subset[0,1]$ and $\{\epsilon_n\}\subset(0, \epsilon_0]$ such that $t_n\downarrow 0$, $s_n\to s_0$ for some $s_0\in[0,1]$, $\epsilon_n\downarrow 0$ and $$ T_{t_n}x_n + Cx_n+ g_{\epsilon_n}x_n = s_n \tau_0 v_0^*. $$ We may assume that $x_n\rightharpoonup x_0\in X$. Since $\{Cx_n\}$ is bounded, we may assume that $Cx_n\rightharpoonup y^*_0\in X^*$ and $g_{\epsilon_n}x_n\to g^*$. Then we have $T_{t_n}x_n\rightharpoonup - y_0^* -g^*+s_0\tau_0v_0^*$. From $$ \langle T_{t_n}x_n, x_n - x_0\rangle + \langle Cx_n, x_n-x_0\rangle = \langle g_{\epsilon_n}x_n+s_n\tau_0v_0^*, x_n -x_0\rangle, $$ we obtain \begin{equation}\label{14} \lim_{n\to\infty}[\langle T_{t_n}x_n, x_n - x_0\rangle+ \langle Cx_n, x_n-x_0\rangle] = 0. \end{equation} Let us assume that \begin{equation}\label{15} \limsup_{n\to\infty}\langle Cx_n, x_n- x_0\rangle >0. \end{equation} Then there exists a subsequence of $\{x_n\}$, which we still denote by $\{x_n\}$, such that \begin{equation}\label{16} \lim_{n\to\infty}\langle Cx_n, x_n-x_0\rangle = q, \end{equation} for some constant $q>0$. By \eqref{14} and \eqref{16}, we obtain $$ \lim_{n\to\infty}\langle T_{t_n}x_n, x_n-x_0\rangle= -q < 0. $$ Applying (i) of Lemma~\ref{L1} with $S =0$, we obtain a contradiction. Therefore \eqref{15} is false and we now only have $$ \limsup_{n\to\infty}\langle Cx_n, x_n - x_0\rangle \le 0. $$ Since $C$ is of type $(S_+)$, we have $x_n\to x_0\in\partial G_1$. Since $C$ is also demicontinuous, $Cx_n\rightharpoonup Cx_0$. This implies $$ T_{t_n}x_n \rightharpoonup - Cx_0- g^*+ s_0 \tau_0v_0^*. $$ Applying (ii) of Lemma~\ref{L1} with $S =0$, we obtain $x_0\in D(T)\cap\partial G_1$ and $$ Tx_0+Cx_0+Gx_0\ni s_0\tau_0v_0^*, $$ which is a contradiction to our hypothesis (H3). Thus, we may now choose $t_0$ and $\epsilon_0$ further so that we also have that $H_1(s, x, t, \epsilon) = 0$ has no solution $x\in\partial G_1$ for all $t\in(0,t_0]$, $\epsilon\in(0, \epsilon_0]$ and all $s\in[0,1]$. It is clear that the mapping $H_1(s, x, t, \epsilon)$ is an admissible homotopy for Skrypnik's degree and the Skrypnik degree ${\rm d}_{\rm S}(H_1(s,\cdot, t, \epsilon), G_1, 0)$ is well-defined and is constant for all $s\in[0,1]$ and for all $t\in(0,t_0]$, $\epsilon\in(0, \epsilon_0]$. Consequently, the Browder's degree generalized by Hu and Papageorgiou \cite{HP}, ${\rm d}_{\rm HP}$ , is well-defined and satisfies \begin{equation}\label{17} {\rm d}_{\rm HP}(T+C+G-\tau_0v_0^*, G_1, 0) = {\rm d}_{\rm S}(T_t+C+g_\epsilon- \tau_0v_0^*, G_1, 0) \end{equation} for $t\in(0,t_0],\epsilon\in(0, \epsilon_0]$. Assume that $$ {\rm d}_{\rm S}(H_1(1, \cdot,t_1,\epsilon_1), G_1, 0)\ne 0, $$ for some sufficiently small $t_1\in(0, t_0]$ and $\epsilon_1\in(0, \epsilon_0]$. Then, the equation $$ T_{t_1}x +Cx +g_{\epsilon_1} x = \tau_0v_0^* $$ has a solution in the set $G_1$. However, this contradicts our choice of the number $\tau_0$ in \eqref{6*}. Consequently, $$ {\rm d}_{\rm S}(T_t+C+g_\epsilon, G_1, 0) = {\rm d}_{\rm S}(H_1(0, \cdot,t_1,\epsilon_1), G_1, 0)= 0, \quad t\in(0, t_0],\; \epsilon\in(0, \epsilon_0]. $$ We next consider the homotopy mapping \begin{equation}\label{18} H_2(s, x, t,\epsilon) = s(T_tx+Cx +g_\epsilon x)+(1-s)Jx, \quad (s, x)\in[0,1]\times\overline{G_2}. \end{equation} We first show that there exist $t_1\in(0, t_0]$, $\epsilon_1\in(0, \epsilon_0]$ such that the equation $H_2(s, x, t,\epsilon)= 0$ has no solution on $\partial G_2$ for any $s\in[0,1]$, any $t\in(0, t_1]$ and any $\epsilon\in(0, \epsilon_1]$. Let us assume the contrary. Then there exist sequences $t_n\in(0, t_0]$, $\epsilon_n\in(0, \epsilon_1]$, $s_n\in[0,1]$, and $x_n\in\partial G_2$ such that $t_n\downarrow 0$, $\epsilon_n\downarrow 0$, $s_n\to s_0\in[0,1]$, $x_n\rightharpoonup x_0\in X$, $Cx_n\rightharpoonup y_0^*\in X^*$, $g_{\epsilon_n}x_n\to g^*\in X^*$, $Jx_n\rightharpoonup z_0^*\in X^*$, and \begin{equation}\label{19} s_n(T_{t_n}x_n +Cx_n +g_{\epsilon_n}x_n) + (1- s_n) Jx_n = 0. \end{equation} $s_n = 0$ is impossible because $J(0) = 0$ and $J$ is injective, we may assume that $s_n> 0 $, for all $n$. If $s_n \to 0$, \begin{equation}\label{20} \langle T_{t_n}x_n + Cx_n, x_n\rangle = -\Big(\frac{1}{s_n} -1\Big)\langle Jx_n , x_n\rangle -\langle g_{\epsilon_n}x_n, x_n\rangle \to - \infty \end{equation} because $\{\|x_n\|\}$ is bounded below away from zero. Since $\langle T_{t_n}x_n , x_n \rangle \ge 0$ and $\{\langle Cx_n, x_n\rangle\}$ is bounded, we see that \eqref{20} is impossible. Thus $s_0\in(0, 1]$ and \eqref{19} implies that $$ T_{t_n}x_n\rightharpoonup -y_0^*-g^*-\Big(\frac1{s_0}-1\Big)z_0^*. $$ Also, from \eqref{19}, \begin{equation} \label{2100} \begin{aligned} &\langle T_{t_n}x_n + Cx_n, x_n - x_0\rangle \\ &=-\Big(\frac1{s_n} -1\Big) \langle g_{\epsilon_n}x_n+Jx_n, x_n - x_0\rangle \\ &=-\Big(\frac1{s_n} -1\Big)\big[\langle Jx_n- Jx_0, x_n - x_0\rangle +\langle g_{\epsilon_n}x_n+Jx_0, x_n- x_0\rangle\big]\\ &\le -\Big(\frac1{s_n} -1\Big)\langle g_{\epsilon_n}x_n+Jx_0, x_n- x_0\rangle, \end{aligned} \end{equation} by the monotonicity of the duality mapping $J$. Since $s_0\in(0,1]$ and $x_n\rightharpoonup x_0$, we see from \eqref{2100} that $$ \limsup_{n\to\infty}\{q_n := \langle T_{t_n}x_n+Cx_n,x_n-x_0\rangle\} \le 0. $$ Let \begin{equation}\label{22} \limsup_{n\to\infty}\langle Cx_n,x_n-x_0\rangle > 0. \end{equation} Then, for some subsequence of $\{n\}$ denoted by $\{n\}$ again, we have \begin{equation}\label{23} \lim_{n\to\infty}\langle Cx_n,x_n-x_0\rangle = q > 0. \end{equation} From $$ \langle T_{t_n}x_n,x_n-x_0\rangle =q_n-\langle Cx_n,x_n-x_0\rangle, $$ we see that $$ \limsup_{n\to\infty}\langle T_{t_n}x_n,x_n-x_0\rangle \le \limsup_{n\to\infty}q_n+\lim_{n\to\infty}[-\langle Cx_n,x_n-x_0\rangle] \le -q < 0. $$ This implies $$ \limsup_{n\to\infty}\langle T_{t_n}x_n,x_n-x_0\rangle < 0. $$ Using (i) of Lemma~\ref{L1}, we conclude that \eqref{22} is impossible, and therefore \eqref{22} holds with ``$\le$" in place of ``$>$''. Since $C$ is of type $(S_+)$, we have $x_n\to x_0\in\partial G_2$. This implies $Cx_n\rightharpoonup Cx_0,Jx_n\to Jx_0$ and $$ T_{t_n}x_n\rightharpoonup -Cx_0-g^*-\Big(\frac{1}{s_0}-1\Big)Jx_0.$$ Since $x_n\to x_0$, we have $$\lim_{n\to\infty}\langle T_{t_n}x_n, x_n-x_0\rangle = 0. $$ Using $ii$ of Lemma~\ref{L1}, we have $x_0\in D(T)$ and $$ -Cx_0-g^*-\Big(\frac{1}{s_0}-1\Big)Jx_0\in Tx_0. $$ By a property of the selection $g_{\epsilon_n}x_n$ (cf. \cite[p. 238]{HP}), we have $g^*\in G(x_0)$. This implies $$ Tx_0 + Cx_0 + Gx_0 +\Big(\frac{1}{s_0} -1\Big)Jx_0\ni 0. $$ We arrived at a contradiction to our hypothesis (H4) because $x_0\in D(T)\cap \partial G_2$. 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 is now clear that the mapping $H_2(s, x, t,\epsilon)$ is an admissible homotopy for Skrypnik's degree and so the Skrypnik degree ${\rm d}_{\rm S}(H_2(s, \cdot, t, \epsilon), G_2, 0)$ 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 Skrypnik degree, for all $t\in(0, t_0]$, $\epsilon\in(0,\epsilon_0]$, we have \begin{align*} {\rm d}_{\rm S}(H_2(1, \cdot, t, \epsilon), G_2, 0) &= {\rm d}_{\rm S}(T_t+C+g_\epsilon, G_2, 0)\\ &= {\rm d}_{\rm S}(H_2(0, \cdot, t, \epsilon), G_2, 0)\\ &= {\rm d}_{\rm S}(J, G_2, 0) = 1. \end{align*} Thus, for all $t\in(0, t_0]$, $\epsilon\in(0,\epsilon_0]$, we have $$ {\rm d}_{\rm S}(T_t+C+g_\epsilon, G_1, 0) \ne {\rm d}_{\rm S}(T_t+C+g_\epsilon, G_2,0). $$ From the excision property of the Skrypnik degree, which is an easy consequence of its finite-dimensional approximations, we obtain a solution $x_{t, \epsilon}\in G_1\setminus G_2$ of $T_tx +Cx+g_\epsilon x = 0$ for every $t\in(0, t_0]$ and every $\epsilon\in(0, \epsilon_0]$. We let $t_n\in(0, t_0]$ and $\epsilon_n\in(0, \epsilon_0]$ be such that $t_n\downarrow 0$, $\epsilon_n\downarrow 0$ and let $x_n\in G_1\setminus G_2$ be the corresponding solutions of $T_tx +Cx+g_\epsilon x = 0$. We have $$ T_{t_n}x_n +Cx_n+ g_{\epsilon_n}x_n = 0. $$ We may assume that $x_n\rightharpoonup x_0$ and $g_{\epsilon_n}x_n\to g^*\in X^*$. We have $$ \langle T_{t_n}x_n , x_n -x_0\rangle = -\langle Cx_n + g_{\epsilon_n}x_n, x_n- x_0\rangle. $$ If $$ \limsup_{n\to\infty}\langle Cx_n+g_{\epsilon_n}x_n, x_n- x_0\rangle >0, $$ then we obtain a contradiction from (i) of Lemma~\ref{L1}. Consequently, $$ \limsup_{n\to\infty}\langle Cx_n+g_{\epsilon_n}x_n, x_n- x_0\rangle \le 0, $$ 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 $T_{t_n}x_n \rightharpoonup -Cx_0-g^*$. Using this in (ii) of Lemma~\ref{lemA}, we obtain $x_0\in D(T)$ and $-Cx_0-g^*\in Tx_0$. By a property of the selection $g_{\epsilon_n}x_n$ (cf. \cite[p. 238]{HP}), we have $g^*\in G(x_0)$ and therefore $Tx_0+Cx_0+Gx_0\ni 0$ by Lemma~\ref{lemA}. 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 conditions (H3) and (H4), we have $x_0\notin \partial G_1\cup\partial G_2$. Thus, $x_0\in D(T)\cap(G_1\setminus G_2)$ and the proof is complete. \end{proof} \section{Applications} \noindent\textbf{Application 1.} 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 $$ \xi(u)= (D^\alpha u)_{|\alpha|\le m} ,\quad \eta(u)= (D^\alpha u)_{|\alpha|\le m-1} , \quad \zeta(u)= (D^\alpha u)_{|\alpha|= m}, $$ where $$ D^\alpha u =\prod_{i=1}^N\Big(\frac{\partial}{\partial x_i}\Big)^{\alpha_i}. $$ Also, let $q= p/(p-1)$. We now consider the partial differential operator in divergence form $$ (Au)(x) = \sum_{|\alpha|\le m}(-1)^{|\alpha|} D^\alpha A_\alpha(x, u(x), \dots, D^mu(x)),\quad x\in\Omega. $$ The coefficients $A_\alpha :\Omega\times\mathbb{R}^{N_0}\to \mathbb{R}$ are assumed to be Carath\'eodory functions, 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 consider the following conditions: \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 $x\in \Omega$, $\eta\in\mathbb{R}^{N_1}$, $\zeta_1, \zeta_2\in\mathbb{R}^{N_2}$ 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 $x\in \Omega$, $\xi_1, \xi_2\in\mathbb{R}^{N_0}$. \item[(H8)] There exist $c_2>0$, $\kappa_2\in L^1(\Omega)$ such that $$ \sum_{|\alpha|\le m}A_\alpha(x,\xi)\xi_\alpha \ge c_2|\xi|^p-\kappa_2(x),\quad x\in\Omega,\; \xi\in\mathbb{R}^{N_0}. $$ \end{itemize} If an operator $T: W_0^{m,p}(\Omega)\to W^{-m, q}(\Omega)$ is given by \begin{equation} \label{176} \langle Tu, 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 conditions (H5), (H7) imply that it is bounded, continuous and monotone (cf. e.g. Kittila \cite[pp. 25-26]{Kittila}, Pascali and Sburlan \cite[pp. 274-275]{PS}). Since $T$ is continuous, it is maximal monotone. Similarly, condition (H5), with $A$ replaced by $B$, implies that the operator \begin{equation} \label{177} \langle Cu, v\rangle = \int_\Omega\sum_{|\alpha|\le m} B_\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 $B$ in place of $A$ everywhere, imply that the operator $C$ is of type $(S_+)$ (cf. Kittila \cite[ p. 27]{Kittila}). We also consider a multifunction $H:\Omega\times \mathbb{R}^{N_1}\to 2^{\mathbf{R}}$ such that \begin{itemize} \item [(H9)] $H(x, r) = [\varphi(x, r), \psi(x, r)]$ is measurable in $x$ and u.s.c. in $r$, where $\varphi, \psi :\Omega\times \mathbb{R}^{N_1}\to \mathbf R$ are measurable functions; \item[(H10)] $|H(x, r)| = \max[|\varphi(x, r)|, |\psi(x, r)|] \le a(x) + c_2|r| $ a.e. on $\Omega\times\mathbb{R}^{N_1}$ and $a(\cdot) \in L^q(\Omega)$, $c_2>0$. \end{itemize} Define $G:W_0^{m,p}\to 2^{W^{-m, q}(\Omega)}$ by \begin{align*} Gu= \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 $G$ is u.s.c and compact with closed and convex values (cf. \cite[p. 254]{HP}), and therefore is of class $(P)$. We now state the following theorem as an application of Theorem~\ref{Th2}. \begin{theorem}\label{Th3} Assume that the operators $T$, $C$ and $G$ defined as above with $T(0) = 0$, $C(0)=0$. Assume, further, that the rest of the conditions of Theorem~\ref{Th2} are satisfied for two balls $G_1 = B_r(0)$ and $G_2= B_q(0)$, where $01$ ($p>2$ for $p$-Laplacian operator $A$ in Theorem~\ref{Th3}) needs further work. \smallskip \noindent\textbf{Application 2.} Let $\Omega$ be a bounded open set in $\mathbb{R}^N$ with smooth boundary, $m\ge 1$ an integer, and $T>0$. Set $Q= \Omega\times [0, a]$. We consider the differential operator \begin{equation}\label{IV1} \begin{split} &\frac{\partial u(x,t)}{\partial t} +\sum_{|\alpha|\le m}(-1)^{|\alpha|}D^\alpha A_\alpha(x,t,u(x,t), Du(x,t),\dots ,D^mu(x,t))\\ &+\sum_{|\alpha| \le m}(-1)^{|\alpha|}D^\alpha B_\alpha(x,t,u(x,t), Du(x,t),\dots ,D^mu(x,t))\\ \end{split} \end{equation} in $Q$. The coefficients $A_\alpha=A_\alpha (x,t,\xi)$, are defined for $(x,t)\in Q$, $\xi=\{\xi_\gamma$, $|\gamma|\le m\}=(\eta,\zeta)\in\mathbb{R}^{N_0}$ with $\eta=\{\eta_\gamma, |\gamma|\le m-1\}\in\mathbb{R}^{N_1}$, $\zeta=\{\zeta_\gamma, |\gamma|=m\}\in\mathbb{R}^{N_2}$, and $N_1+N_2 = N_0$. We assume that each coefficient $A_\alpha(x,t,\xi)$ satisfies the usual Carath\'eodory conditions. We consider the following conditions. \begin{itemize} \item[(H11)] (Continuity) For some $p\ge 2$, $c_1>0$, $g\in L^q(Q)$ 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 $(x,t)\in Q$, $\xi=(\eta,\zeta)\in\mathbb{R}^{N_0}$, $|a|\le m$. \item[(H12)] (Monotonicity) $$ \sum_{|\alpha| \le m}(A_\alpha(x,t,\xi_1)- A_\alpha(x,t,\xi_2))(\xi_{1_\gamma}-\xi_{2_\gamma}) \ge 0, \quad (x,t)\in Q,\; \xi_1,\xi_2\in\mathbb{R}^{N_0}. $$ \item[(H13)] (Leray-Lions) \[ \sum_{|\alpha| = m}(A_\alpha(x,t,\eta,\zeta)- A_\alpha(x,t,\eta,\zeta^*))(\zeta_\gamma-\zeta^*_\gamma) > 0, \] for $(x,t)\in Q$, $\eta\in\mathbb{R}^{N_1}$, $\zeta,\zeta^*\in\mathbb{R}^{N_2}$. \item[(H14)] (Coercivity) There exist $c_0>0$ and $h\in L^1(Q)$ such that $$ \sum_{|a|\le m}A_\alpha(x,t,\xi) \ge c_0|\xi|^p-h(x,t),\quad (x,t)\in Q,\; \xi\in\mathbb{R}^{N_0}. $$ \end{itemize} Under the condition (H11), the second term of \eqref{IV1} generates a continuous bounded operator $ T:X\to X^*$, where $X=L^p(0,a;V), X^*=L^q(0,a;V^*)$, and $V=W_0^{m,p}(\Omega)$. It is defined by $$ \langle Tu,v\rangle=\sum_{|\alpha|\le m}\int_QA_\alpha(x,t,u,Du,\dots ,D^mu)D^\alpha v, \quad u,v\in X. $$ This operator is also maximal monotone under the condition (H12). Under (H11), (H13) and (H14) (with ``A" replaced by ``B" and the other necessary changes) the third term of \eqref{IV1} generates a continuous, bounded operator $C$ which satisfies the condition $(S_+)$ w.r.t. $D(L)$, where the operator $L$ is defined below. The operator $C$ is defined by $$ \langle Cu,v\rangle=\sum_{|\alpha| \le m}\int_QB_\alpha(x,t,u,Du,\dots ,D^mu)D^\alpha v, \quad u,v\in X. $$ 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 $$ \langle Lu,v\rangle = \int_0^a\langle u'(t),v(t)\rangle dt,\quad u\in D(L),\; v\in X. $$ The symbol $u'(t)$ above is the generalized derivative of $u(t)$, i.e. $$ \int_0^a \langle u'(t), \varphi(t)\rangle \,dt =-\int_0^a \langle \varphi'(t), u(t) \rangle \,dt,\quad \varphi\in C_0^\infty(0,a; X). $$ One can verify, as in Zeidler \cite{Zeidler1}, that $L$ is a linear densely defined maximal monotone operator. Let $K$ be an unbounded closed convex proper subset of $ X$ with $0\in\overset\circ K$. Let $\varphi_K:X\to\mathbb{R}_+\cup\{\infty\}$ be defined by \begin{equation*} \varphi_K(x)= \begin{cases} 0 &\text{if $x\in K,$}\\ \infty &\text{otherwise.} \end{cases} \end{equation*} The function $\varphi_K$ is proper convex and lower semicontinuous on $X$, and $x^*\in\partial\varphi_K(x)$, for $x\in K$, if and only if $$ \langle x^*,y-x\rangle \le 0,\quad\text{for all } y\in K. $$ Also, \begin{gather*} D(\partial\varphi_K)=K\quad\text{and}\quad 0\in\partial\varphi_K(x),\quad x\in K,\\ \partial\varphi_K(x)=\{0\},\quad x\in {\mathaccent"7017 K}. \end{gather*} The operator $\partial\varphi_K:X\supset K\to 2^{X^*}$ is maximal monotone with $0\in\overset{\circ} {D}(\partial\varphi_K)$ and $0\in\partial\varphi_K(0)$. It is thus strongly quasibounded. For these facts see, e.g., Kenmochi \cite{Kenmochi1974}. In addition, the sum $\partial\varphi_K+ T$ is a multivalued strongly quasibounded maximal monotone operator from $K$ to $2^{X^*}$. As an application of Theorem~\ref{Th1}, we state the following theorem. \begin{theorem} \label{thm4} Assume that the operators $L,T,C$ are as above with $A_\alpha$ satisfying (H11), (H12) and $ T(0)=0$, $C(0) =0$ , and $B_\alpha$ satisfying (H11), (H13) and (H14) with the necessary notational changes. Assume, further, that the rest of the conditions of Theorem~\ref{Th1} are satisfied for two balls $G_1 = B_r(0)$ and $G_2= B_q(0)$, in $X = L^p(0, a, V)$, where $0