  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 $\bar G_2\subset G_1$, $0\in
G_2$, 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 $(S_+)$ w.r.t.
$D(L)$, and $T:\bar G_1\to 2^{X^*}$ 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.

Functional Analysis