Nontrivial solutions of inclusions involving perturbed maximal monotone operators
Date
2017-06-25
Authors
Adhikari, Dhruba R.
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
Let X be a real reflexive Banach space and X* its dual space. Let L : X ⊃ D(L) → X* be a densely defined linear maximal monotone operator, and T : X ⊃ D(T) → 2X*, 0 ∈ D(T) and 0 ∈ T(0), be strongly quasibounded maximal monotone and positively homogeneous of degree 1. Also, let C : X ⊃ D(C) → X* be bounded, demicontinuous and of type (S+) w.r.t. to D(L). The existence of nonzero solutions of Lx + Tx + Cx ∋ 0 is established in the set G1 \ G2, where G2 ⊂ G1 with Ḡ2 ⊂ G1, G1, G2 are open sets in X, 0 ∈ G2, and G1 is bounded. In the special case when L = 0, a mapping G : Ḡ1 → X* of class (P) introduced by Hu and Papageorgiou is also incorporated and the existence of nonzero solutions of Tx + Cx + Gx ∋ 0, where T is only maximal monotone and positively homogeneous of degree α ∈ (0, 1], is obtained. Applications to elliptic partial differential equations involving p-Laplacian with p ∈ (1, 2] and time-dependent parabolic partial differential equations on cylindrical domains are presented.
Description
Keywords
Strong quasiboundedness, Browder and Skrypnik degree theories, Maximal monotone operator, Bounded demicontinuous operator of type (S+)
Citation
Adhikari, D. R. (2017). Nontrivial solutions of inclusions involving perturbed maximal monotone operators. <i>Electronic Journal of Differential Equations, 2017</i>(151), pp. 1-21.
Rights
Attribution 4.0 International