International Conference on Applications of Mathematics to Nonlinear Sciences. Electron. J. Diff. Eqns., Conference 24 (2017), pp. 1-10.

Existence results for multivalued operators of monotone type in reflexive Banach spaces

Dhruba R. Adhikari

Let $X$ be a real reflexive Banach space and $X^*$ its dual space. Let $T:X\supset D(T) \to 2^{X^*}$ be an operator of class $\mathcal A_G(S_+)$, where $G\subset X$. A result concerning the existence of pathwise connected sets in the range of T is established, and as a consequence, an open mapping theorem is proved. In addition, for certain operators T of class $\mathcal B_G(S_+)$, the existence of nonzero solutions of $0\in Tx$ in $G_1\setminus G_2$, where $G_1, G_2 \subset X$ satisfy $0\in G_2$ and $\overline{G_2}\subset G_1$, is established. The Skrypnik's topological degree theory is used, utilizing approximating schemes for operators of classes $\mathcal A_G(S_+)$ and $\mathcal B_G(S_+)$, along with the methodology of a recent invariance of domain result by Kartsatos and the author.

Published November 15, 2017.
Math Subject Classifications: 47H14, 47H05, 47H11.
Key Words: Browder and Skrypnik degree theory; invariance of domain; nonzero solutions; bounded demicontinuous operator of type (S+).

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Dhruba R. Adhikari
Department of Mathematics
Kennesaw State University
Georgia 30060, USA

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