SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS USING FIXED POINT RESULTS IN GENERALIZED METRIC SPACES OF PEROV'S TYPE

SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS USING FIXED POINT RESULTS IN GENERALIZED METRIC SPACES OF PEROV'S TYPE

N. Fabiano, Z. Kadelburg, N. Mirkov, S. Radenovic

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Abstract

In 1964, A. I. Perov generalized the Banach contraction principle intro- ducing, following the work of ̄D. Kurepa, a new approach to fixed point problems, by defining generalized metric spaces (also known as vector valued metric spaces), and pro- viding some actual results for the first time. Using the recent approach of coordinate representation for a generalized metric of Jachymski and Klima, we verify in this ar- ticle some natural properties of generalized metric spaces, already owned by standard metric spaces. Among other results, we show that the theorems of Nemytckii (1936) and Edelstein (1962) are valid in generalized metric spaces, as well. A new application to fractional differential equations is also presented. At the end we state a few open questions for young researchers.

Keywords

Fixed point; vector-valued metric, pseudometric; Perov type; F-contraction; fractional differential equation.