Certain dynamic iterative scheme families and multi-valued fixed point results

Amjad Ali; Muhammad Arshad; Eskandar Emeer; Hassen Aydi; Aiman Mukheimer; Kamal Abodayeh; Amjad Ali; Muhammad Arshad; Eskandar Emeer; Hassen Aydi; Aiman Mukheimer; Kamal Abodayeh

doi:10.3934/math.2022677

AIMS Mathematics

2022, Volume 7, Issue 7: 12177-12202. doi: 10.3934/math.2022677

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Research article

Certain dynamic iterative scheme families and multi-valued fixed point results

1.
Department of Mathematics and Statistics, International Islamic University, Islamabad, Pakistan
2.
Department of Mathematics, Taiz University, Taiz, Yemen
3.
Université de Sousse, Institut Supérieur d'Informatique et des Techniques de Communication, H. Sousse 4000, Tunisia
4.
China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
5.
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa
6.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia

Received: 01 March 2022 Revised: 30 March 2022 Accepted: 07 April 2022 Published: 22 April 2022
MSC : 46T99, 47H10, 54H25

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 $, $ F_{H} $)-dynamic-iterative scheme for Branciari Ćirić type-contractions and prove multi-valued fixed point results in controlled-metric spaces. An approximation of the dynamic-iterative scheme instead of the conventional Picard sequence is determined. The paper also includes a tangible example and a graphical interpretation that displays the motivation for such investigations. The work is illustrated by providing an application of the proposed non-linear ($ F $, $ F_{H} $)-dynamic-iterative scheme to the Liouville-Caputo fractional derivatives and fractional differential equations.
- fixed points,
- controlled-metric space,
- ($ F $, $ F_{H} $)-dynamic-iterative scheme,
- Liouville-Caputo fractional differential equation
Citation: Amjad Ali, Muhammad Arshad, Eskandar Emeer, Hassen Aydi, Aiman Mukheimer, Kamal Abodayeh. Certain dynamic iterative scheme families and multi-valued fixed point results[J]. AIMS Mathematics, 2022, 7(7): 12177-12202. doi: 10.3934/math.2022677

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Abstract

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 $, $ F_{H} $)-dynamic-iterative scheme for Branciari Ćirić type-contractions and prove multi-valued fixed point results in controlled-metric spaces. An approximation of the dynamic-iterative scheme instead of the conventional Picard sequence is determined. The paper also includes a tangible example and a graphical interpretation that displays the motivation for such investigations. The work is illustrated by providing an application of the proposed non-linear ($ F $, $ F_{H} $)-dynamic-iterative scheme to the Liouville-Caputo fractional derivatives and fractional differential equations.

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