The Exitence And Algorithm Of Solution For Variational Inequality And Variational Inclusion

In the 1960s, Hartman and Stampacchia construct the variational inequality theory. It is well known that Variational inequality theory is powerful tools of the current mathematical technology. Variational inequalities have many important applications in operation research, the random problem of probability and statistics, computer science, system science, engineering technology, transportation, economics and management et. al. In the last 20 years of the twentieth century, they have been paid close attention by many scholars.In the paper, we show the real background of the problems we study and the main works that have been studied by many authors. Based on the results of research in this field in recent years, we discuss the following problems:First, the auxiliary variational principle technique is applied to solve a class of variational inequality. An innovative iterative algorithm to cumpute approximate solution is constructed. The convergence of iterative sequences generated by the algorithm is proved.Second, the existence of solution for a kind of variational inclusion is discussed. We establish the equivalence between the variational inclusion and the general resolvent equation. Give a three-step iterative algorithm for this variational inclusion and prove the convergence of this iterative algorithm.Third, we study the existence of solution for a class of set-valued variational inclusion with accretive type mappings in Banach spaces. Some equivalent conditions as the sequence of iterations strongly converges to the only solution for variation inclusion are obtained.