Topological degree for quasibounded multivalued (S̃)₊-perturbations of maximal monotone operators

ABSTRACT

Let X be an infinite dimensional real reflexive Banach space with dual space $X^{\ast }$ and $G\subset X$ open and bounded. Let $T:X\supset D\left(T\right)\to 2^{X^{\ast }}$ be a maximal monotone operator with $0\in D\left(T\right)$ and $0\in T\left(0\right)$, and let $C:X\supset D\left(C\right)\to 2^{X^{\ast }}$ be densely defined strongly quasibounded and of type $\left({\tilde{S}}_{+}\right)$. A new topological degree theory is introduced for the sum T+C with a degree mapping $\mathrm{d}(T+C,G,0)$ defined eventually in terms of the Ma degree for multivalued compact operators. Unlike single-valued operators considered by Kartsatos and Skrypnik, the operator C here is multivalued so that the multivalued generalized pseudomonotone operators considered by Browder and Hess include such C and even T+C. Consequently, the main existence results of Browder and Hess are obtained via the new degree theory and some of their existence results are extended. An application of the theory to elliptic partial differential inclusions in divergence form is included.

Keywords:

• Topological degree theories
• Ma degree
• maximal monotone operators
• densely defined strongly quasibounded perturbations of type $\left({\tilde{S}}_{+}\right)$

2010 Mathematics Subject Classifications:

• 47H11
• 47H14
• 47H07

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Dhruba R. Adhikari http://orcid.org/0000-0001-5871-694X

Additional information

Funding

The author is thankful to the College of Science and Mathematics at Kennesaw State University for supporting this research through the 2018 Research Stimulus Program.