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Fixed points for generalized contractions and applications to control theory. (English) Zbl 1144.54031

The authors give a valuable contribution to the nonlinear functional analysis of fixed point theorems. Results of that kind allow a variety of applications in various areas of applied mathematics such as continuous optimization, game theory, differential and integral equations, dynamical systems, numerical analysis, computational statistics, and control theory, which is indeed the main field of application in this article. Actually, the examples considered are on optimal control of ordinary differential equations; the control is done by dynamic programming. Furthermore, the integral equations are defined on the space C[a,b]. The authors introduce the notions weak/strong topological contraction and p-contraction and use this generalized concept to obtain new fixed point theorems. The latter are for self-mappings from a topological resp. metric space into itself being a topological contraction resp. metric p-contraction, respectively; they yield results on existence, uniqueness of a fixed point and elements on how to localize and iteratively – along an “orbit” – find that solution. The theory is mainly for metric spaces with special regard to compact metric spaces. The central idea of a p-contraction lies in the replacement of a contraction constant by a state dependent function (p) with values in [0,1) and not being larger than a constant that is less than 1 at all values of the operator.

For the future, these investigations could be extended and applied on further modern problems from, e.g., science, technology, the finance sector or operations research, which would be interesting and valuable.

54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47N10Applications of operator theory in optimization, convex analysis, programming, economics
47N70Applications of operator theory in systems theory, circuits, and control theory
49L20Dynamic programming method (infinite-dimensional problems)