## Abstract and Applied Analysis

- Abstr. Appl. Anal.
- Volume 2005, Number 2 (2005), 121-158.

### A new topological degree theory for densely defined quasibounded $({\widetilde{S}}_{+})$-perturbations of multivalued maximal monotone operators in reflexive Banach spaces

Athanassios G. Kartsatos and Igor V. Skrypnik

#### Abstract

Let $X$ be an infinite-dimensional real reflexive Banach space with dual space ${X}^{\ast}$ and $G\subset X$ open and bounded. Assume that $X$ and ${X}^{\ast}$ are locally uniformly convex. Let $T:X\supset D\left(T\right)\to {2}^{{X}^{\ast}}$ be maximal monotone and $C:X\supset D\left(C\right)\to {X}^{\ast}$ quasibounded and of type $\left({\tilde{S}}_{+}\right)$. Assume that $L\subset D\left(C\right)$, where $L$ is a dense subspace of $X$, and $0\in T\left(0\right)$. A new topological degree theory is introduced for the sum $T+C$. Browder's degree theory has thus been extended to densely defined perturbations of maximal monotone operators while results of Browder and Hess have been extended to various classes of single-valued densely defined generalized pseudomonotone perturbations $C$. Although the main results are of theoretical nature, possible applications of the new degree theory are given for several other theoretical problems in nonlinear functional analysis.

#### Article information

**Source**

Abstr. Appl. Anal., Volume 2005, Number 2 (2005), 121-158.

**Dates**

First available in Project Euclid: 17 May 2005

**Permanent link to this document**

https://projecteuclid.org/euclid.aaa/1116340205

**Digital Object Identifier**

doi:10.1155/AAA.2005.121

**Mathematical Reviews number (MathSciNet)**

MR2179439

**Zentralblatt MATH identifier**

1110.47049

#### Citation

Kartsatos, Athanassios G.; Skrypnik, Igor V. A new topological degree theory for densely defined quasibounded $({\widetilde{S}}_{+})$-perturbations of multivalued maximal monotone operators in reflexive Banach spaces. Abstr. Appl. Anal. 2005 (2005), no. 2, 121--158. doi:10.1155/AAA.2005.121. https://projecteuclid.org/euclid.aaa/1116340205