Solvability of inclusions involving perturbations of positively homogeneous maximal monotone operators
Date
2022-08-30
Authors
Adhikari, Dhruba R.
Aryal, Ashok
Bhatt, Ghanshyam
Kunwar, Ishwari
Puri, Rajan
Ranabhat, Min
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
Let X be a real reflexive Banach and X* be its dual space. Let G1 and G2 be open subsets of X* such that Ḡ2 ⊂ G1, 0 ∈ G2, and G1 is bounded. Let L : X ⊃ D(L) → X* be a densely defined linear maximal monotone operator, A : X ⊃ D(A) → 2X* be a maximal monotone and positively homogeneous operator of degree γ > 0, C : X ⊃ D(C) → X* be a bounded demicontinuous operator of type (S+) with respect to D(L), and T : Ḡ1 → 2X* 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 ∋ 0 in the set G1 \ G2. Secondly, we assume that A is bounded and establish the existence of nonzero solutions of Lx + Ax + Cx ∋ 0 in G1 \ G2. We remove the restrictions γ ∈ (0, 1] for Ax + Cx + Tx ∋ 0 and γ = 1 for Lx + Ax + Cx ∋ 0 from such existing results in the literature. We also present applications to elliptic and parabolic partial differential equations in general divergence from satisfying Dirichlet boundary conditions.
Description
Keywords
Topological degree theory, Operators of type (S+), Monotone operator, Duality mapping, Yosida approximant
Citation
Adhikari, D. R., Aryal, A., Bhatt, G., Kunwar, I. J., Puri, R., & Ranabhat, M. (2022). Solvability of inclusions involving perturbations of positively homogeneous maximal monotone operators. <i>Electronic Journal of Differential Equations, 2022</i>(63), pp. 1-25.
Rights
Attribution 4.0 International