The article presents a systematic investigation of an extension of the developments concerning F-contraction mappings which were proposed in 2012 by Wardowski. We develop the notion of F-contractions to the case of non-linear (F, FH )-dynamic-iterative scheme for Branciari Ćirić type-contractions and prove multi-valued fixed point results in controlled-metric spaces. An approximation of the dynamic-iterative scheme instead of the conventional Picard sequence is determined. The paper also includes a tangible example and a graphical interpretation that displays the motivation for such investigations. The work is illustrated by providing an application of the proposed non-linear (F, FH )-dynamic-iterative scheme to the Liouville-Caputo fractional derivatives and fractional differential equations.
- (F, F )-dynamic-iterative scheme
- Liouville-Caputo fractional differential equation
- controlled-metric space
- fixed points