INTEGRAL SOLUTIONS OF AN INITIAL VALUE PROBLEM OF FRACTIONAL DYNAMIC EQUATIONS ON TIME SCALE

Bikash Gogoi (), Utpal Kumar Saha (), Bipan Hazarika (), Hijaz Ahmad, Dilber Uzun Ozsahin, M. Adel () and Maged F. Alotaibi ()
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Bikash Gogoi: Department of Mathematics, Sibsagar University, Sivasagar, 785665 Assam, India
Utpal Kumar Saha: Department of Basic and Applied Science, National Institute of Technology Arunachal Pradesh, Jote, 791113 Arunachal Pradesh, India
Bipan Hazarika: Department of Mathematics, Gauhati University, Guwahati, 781014 Assam, India
Hijaz Ahmad: Operational Research Center in Healthcare, Near East University, Nicosia/TRNC, 99138 Mersin 10, Turkey5Department of Mathematics, College of Science, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul 02841, South Korea6Department of Technical Sciences, Western Caspian University, Baku 1001, Azerbaijan
Dilber Uzun Ozsahin: Department of Medical Diagnostic Imaging College of Health Science, University of Sharjah 27272, Sharjah, United Arab Emirates8Research Institute for Medical and Health Sciences, University of Sharjah 27272, Sharjah, United Arab Emirates9Operational Research Center in Healthcare, Near East University Nicosia/TRNC, 99138 Mersin 10, Turkey
M. Adel: 0Department of Mathematics, Faculty of Science, Islamic University of Madinah, Medina, Saudi Arabia
Maged F. Alotaibi: 1Department of Physics, College of Science, King Abdulaziz University, Jeddah, Saudi Arabia

FRACTALS (fractals), 2025, vol. 33, issue 03, 1-14

Abstract: The purpose of this paper is to investigate the existence of the integral solution of a fractional dynamic problem in the domain of time scale calculus. The discussion has been done with the help of Schauder’s fixed point theorem and the Kolmogorov compactness criterion in time scales. The new notion of the Caputo derivative in terms of nabla derivative (Caputo ∇-derivative) is developed with the help of the recently developed new notion of Riemann–Liouville nabla derivative (Riemann–Liouville ∇-derivative). Some properties of the Caputo ∇-derivative are also discussed. Later, examples have been given to justify the main findings of the paper.

Keywords: Nabla (∇)-Derivative; Riemann–Liouville ∇-Derivative; Riemann–Liouville ∇-Integral; Caputo ∇-Derivative; Kolmogorov Compactness Criterion; Fixed Point Theorem (FPT); Schauder’s FPT; Banach FPT (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1142/S0218348X24400620

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