Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 63, pp. 1-25.

Title: Solvability of inclusions involving perturbations of positively  homogeneous maximal monotone operators
       
Authors: Dhruba R. Adhikari (Kennesaw State Univ., Marietta, GA, USA)
         Ashok Aryal (Minnesota State Univ. Moorhead, Moorhead, MN, USA)
         Ghanshyam Bhatt (Tennessee State Univ., Nashville, TN, USA)
         Ishwari J. Kunwar (Fort Valley State Univ., Fort Valley, GA, USA)
         Rajan Puri (Wake Forest Univ., Winston-Salem, NC, USA)
         Min Ranabhat (Univ. of Delaware,  Newark, DE, USA)

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.   

Submitted April 1, 2022. Published August 30, 2022.
Math Subject Classifications: 47H14, 47H05, 47H11.
Key Words: Topological degree theory; operators of type $(S_+)$;
           monotone operator;  duality mapping; Yosida approximant.