ABSTRACT
This paper is a survey of recent development of the theory of Contractors and Contractor Directions, which is basically designed for solving equations in abstract spaces, and of its applications to classical and modern analysis. The theory which is presented in the research monograph: Contractors and Contractor Directions - Theory and Applications, Marcel Dekker, 1977 (304 pages), is characterized by its generality, unifying property, capacity for applications, potential for further development, and flexibility concerning adaptation of methods related to some other branches of mathematics. Since, the publication of the monograph, both the theory and its applications, have undergone intensive development. General existence and surjectivity theorems have been obtained for new classes of equations: with closed operators as well as with operators without continuity conditions. The most surprising result is that the theory of contractor directions provides a tool for solving general quasilinear evolution equations of “hyperbolic” type in nonreflexive Banach spaces whereas no other method is available. Kato’s theory applies to reflexive Banach spaces.
A new concept has been introduced: generalized contractor directions. Among other developments: self-accelerating iterative methods of contractor directions, the diagonal method of contractor directions which is 4an iterative method of solving equations in locally convex metric spaces with application to optimization problems in such spaces. More progress is expected along with applications: to nonlinear integral equations, to nonlinear evolutions equations and partial differential equations of hyperbolic type, to eigenvalue problems and bifurcation theory, to nonlinear programming and general control theory. The conclusion of the program will be the second volume of the monograph mentioned above.
