zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $C\left[a,b\right]$. 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 $\left(p\right)$ with values in $\left[0,1\right)$ 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.

MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47N10 Applications of operator theory in optimization, convex analysis, programming, economics 47N70 Applications of operator theory in systems theory, circuits, and control theory 49L20 Dynamic programming method (infinite-dimensional problems)