Electron. J. Differential Equations, Vol. 2022 (2022), No. 63, pp. 1-25.

Solvability of inclusions involving perturbations of positively homogeneous maximal monotone operators

Dhruba R. Adhikari, Ashok Aryal, Ghanshyam Bhatt, Ishwari J. Kunwar, Rajan Puri, Min Ranabhat

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.
DOI: https://doi.org/10.58997/ejde.2022.63

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Dhruba R. Adhikari
Department of Mathematics
Kennesaw State University
Marietta, GA 30060, USA
email: dadhikar@kennesaw.edu
Ashok Aryal
Mathematics Department
Minnesota State University Moorhead
Moorhead, MN 56563, USA
email: ashok.aryal@mnstate.edu
Ghanshyam Bhatt
Department of Mathematical Sciences
Tennessee State University
Nashville, TN 37209, USA
email: gbhatt@tnstate.edu
Ishwari J. Kunwar
Department of Mathematics and Computer Science
Fort Valley State University
Fort Valley, GA 31030, USA
email: kunwari@fvsu.edu
Rajan Puri
Department of Mathematics
Wake Forest University
Winston-Salem, NC 27109, USA
email: purir@wfu.edu
Min Ranabhat
Department of Mathematical Sciences
University of Delaware
EWG 315, Newark, DE 19716, USA
email: ranabhat@udel.edu

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