Electron. J. Differential Equations, Vol. 2017 (2017), No. 151, pp. 1-21.

Nontrivial solutions of inclusions involving perturbed maximal monotone operators

Dhruba R. Adhikari

Let X be a real reflexive Banach space and $X^*$ its dual space. Let $L: X\supset D(L)\to X^*$ be a densely defined linear maximal monotone operator, and $T:X\supset D(T)\to 2^{X^*}$, $0\in D(T)$ and $0\in T(0)$, be strongly quasibounded maximal monotone and positively homogeneous of degree 1. Also, let $C:X\supset D(C)\to X^*$ be bounded, demicontinuous and of type $(S_+)$ w.r.t. to D(L). The existence of nonzero solutions of $Lx+Tx+Cx\ni0$ is established in the set $G_1\setminus G_2$, where $G_2\subset G_1$ with $\overline G_2\subset G_1$, $G_1, G_2$ are open sets in X, $0\in G_2$, and $G_1$ is bounded. In the special case when L=0, a mapping $G:\overline G_1\to X^*$ of class (P) introduced by Hu and Papageorgiou is also incorporated and the existence of nonzero solutions of $Tx+ Cx+ Gx\ni 0$, where T is only maximal monotone and positively homogeneous of degree $\alpha\in (0, 1]$, is obtained. Applications to elliptic partial differential equations involving p-Laplacian with $p \in (1, 2]$ and time-dependent parabolic partial differential equations on cylindrical domains are presented.

Submitted June 11, 2016. Published June 25, 2017.
Math Subject Classifications: 47H14, 47H05, 47H11.
Key Words: Strong quasiboundedness; Browder and Skrypnik degree theories; maximal monotone operator; bounded demicontinuous operator of type $(S_+)$.

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Dhruba R. Adhikari
Department of Mathematics
Kennesaw State University
Georgia 30060, USA
email: dadhikar@kennesaw.edu

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