On Maximal Monotone Operators and the Sum Problem in General Banach Spaces

Abstract

Maximal monotone operators play an important role in non-linear modern analysis.
In this thesis, we focus on the most famous and significant open problem that is “Sumproblem” in monotone operator theory. First, we provide some powerful sufficient conditions for partial solutions of the Sum problem. The maximality of the sum of an ultramaximal monotone operator and a maximal monotone operator of type (D) with Br_ezis-Crandall-Pazy conditions in Grothendieck Banach spaces is established. Also, we work on Br_ezis-Haraux conditions in Grothendieck Banach spaces. Finally, a sufficient condition is provided for the convergence of a sequence of maximal monotone operators of type (D) in general Banach spaces. The representability of the lower limit of a sequence of maximal monotone operators of type (D) is discussed through their representative functions. Moreover, some conditions are given which guarantee the maximality of the lower limit of a sequence of maximal monotone operators of type (D).