The Leray-Schauder Approach for the Degree of Perturbed Maximal Monotone

Author

Ibrahimou Boubakari, University of South Florida

Graduation Year

2007

Document Type

Dissertation

Degree

Ph.D.

Degree Granting Department

Mathematics and Statistics

Major Professor

Athanassios Kartsatos, Ph.D.

Committee Member

Marcus McWaters, Ph.D.

Committee Member

Arunava Mukherjea, Ph.D.

Committee Member

Boris Shekhtman, Ph.D.

Committee Member

Yuncheng You, Ph.D.

Keywords

Demicontinuous, Eigenvalue, Invariance of domain, Quasimonotone, Strongly quasibounded, Yosida Approximant

Abstract

In this work, we demonstrate that the Leray-Schauder topological degree theory can be used for the development of a topological degree theory for maximal monotone perturbations of demicontinuous operators of type (S+) in separable reflexive Banach spaces. This is an extension of Berkovits’ degree development for operators as the perturbations above.

Berkovits has developed a topological degree for demicontinuous mappings of type (S₊), and has shown that the degree mapping is unique under the assumption that it satisfies certain general properties. He proved that if f is a bounded demicontinous mapping of type (S₊), G is an open bounded subset of X, and 0 ∈/ f(∂G), then there exists ε₀ > 0 such that for every ε ∈ (0, ε₀) we have 0 ∈/ (I+ (1/ε)QQ^∗ (f))(∂G). Here, Q is a compact linear injection from a Hilbert space H into X, such that Q(H) is dense in X, and Q^∗ its adjoint. The map I+ 1 εQQ^∗ (f) is a compact displacement of the identity, for which the Leray-Schauder degree is well defined. The Berkovits degree is obtained as the limit of this Leray-Schauder degree as ε tends to zero. We utilize a demicontinuous (S₊)-approximation of the form T_t + f, where T_t is the Yosida approximant of T. Namely, we show that if G is an open bounded set in X and 0 ∈/ (T + f)(∂G), then there exist ε₀ > 0, t₀ > 0, such that for every ε ∈ (0, ε₀), t ∈ (0, t₀), we have 0 ∈/ (I + (1/ε)QQ^∗ (T_t + f))(∂G). Our degree is the limit of the Leray-Schauder degree of the compact displacement of the identity I + (1/ε)QQ^∗ (T_t + f) as ε, t → 0. Various extension of the degree has been considered. Finally some properties and applications in invariance of domain, eigenvalue and surjectivity results have also been discussed.

Scholar Commons Citation

Boubakari, Ibrahimou, "The Leray-Schauder Approach for the Degree of Perturbed Maximal Monotone" (2007). USF Tampa Graduate Theses and Dissertations.
https://digitalcommons.usf.edu/etd/641