Variational Problem and Maximal Monotone Operators

Abstract

Differential problems and differential inclusions are fundamental topics in applied mathematics, given their close connection to the theory of maximal monotone operators and their pivotal role in the analysis of numerous math- ematical models in various fields. This framework includes the problem of finding the set of zeros for the sum of maximal monotone operators, which forms the basis for a lot of applications in non-linear analysis, optimization, and some theories related of fixed point. This thesis aims to study the problem of finding a set of zeros for the sum of a family of maximal monotone operators, with a focus on develop- ing efficient algorithms and achieving their convergence properties. First, we reviewed the most important algorithms known in the scientific litera- ture for dealing with the case of a single maximal monotone operator, then the case of a sum of maximal monotone operators, presenting the most no- table modifications and improvements proposed by researchers to accelerate convergence and enhance numerical stability. Next, we addressed the funda- mental problem of finding the zeros of a finite family of maximal monotone operators, dividing it into several special cases. In the context of summing two maximal monotone operators, we proposed an improved method for find- ing solutions based on the Yosida approximation, with a theoretical analysis of the method’s properties and a proof of its convergence. We also presented a new algorithm for dealing with the case of The sum of a pair of maximal monotone operators corresponding to a finite family of nonexpansive oper- ators, Demonstrating strong convergence of their Sequences generated by iteration under appropriate conditions. At a later stage, we generalized the study to the case of the sum of a finite family of maximal monotone operators, and proposed a general algo- rithm based on a radial function to compute the zero set of the problem, with a detailed theoretical analysis of its properties and proof of the appropriate convergence results. At the end of the thesis, we addressed the problem of the difference between two maximal periodic effects and proposed a new method for finding their solutions, studying their convergence properties and high- lighting their contribution to expanding the range of possible applications of the theory of maximal monotone operators. The results obtained in this thesis contribute to enriching the literature on algorithms for solving comprehensiveness problems associated with maximal monotone operators and open new horizons for their applications in the fields of nonlinear analysis and mathematical optimization.