# Topological Degree and Variational Inequality Theories for Pseudomonotone Perturbations of Maximal Monotone Operators

2013

Dissertation

Ph.D.

## Degree Granting Department

Mathematics and Statistics

## Major Professor

Athanassios G. Kartsatos

## Keywords

Existence of zeros, Homotopy invariance, Local solvability, multivalued (S+) operators, Nonlinear problems, Quasimonotone

## Abstract

Let X be a real reflexive locally uniformly convex

Banach space with locally uniformly convex dual space X*

. Let G be a

bounded open subset of X. Let T:X⊃ D(T)⇒ 2X*

be maximal

monotone and S: X ⇒ 2X*

be bounded

pseudomonotone and such that 0 notin cl((T+S)(D(T)∩partG)). Chapter 1 gives general introduction and mathematical prerequisites. In

Chapter 2 we develop a homotopy invariance and uniqueness results for the degree theory constructed by Zhang and Chen for multivalued (S+) perturbations of

maximal monotone operators. Chapter 3 is devoted to the construction of a new topological degree

theory for the sum T+S with the degree mapping d(T+S,G,0) defined by

d(T+S,G,0)=limepsilondarr

0+

dS+(T+S+ J,G,0),

where dS+ is the degree for bounded (S+)-perturbations of maximal

monotone operators. The uniqueness and homotopy invariance result of

this degree mapping are also included herein. As applications of the theory, we give associated mapping theorems as well as degree theoretic

proofs of known results by Figueiredo, Kenmochi and Le.

In chapter 4, we consider T:X D(T)⇒ 2X*

to be maximal monotone and S:D(S)=K⇒ 2X*

at least pseudomonotone, where K is a nonempty, closed

and convex subset of X with 0isinKordm. Let Phi:X⇒ ( infin, infin] be a

proper, convex and lower-semicontinuous function. Let f*

isin X*

be fixed. New

results are given concerning the solvability of perturbed variational inequalities

for operators of the type T+S associated with the function f. The associated

range results for nonlinear operators are also given, as well as extensions and/or

improvements of known results by Kenmochi, Le, Browder, Browder and Hess,

Figueiredo, Zhou, and others.

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