Existence and uniqueness result for fractional dynamic equations on metric-like spaces

Existence and uniqueness result for fractional dynamic equations on metric-like spaces

Authors

  • ˙ Inci M. Erhan, Najeh Redjel

Keywords:

time scale, Caputo fractional dynamic equation, fixed point, metric-like

Abstract

The problem of existence and uniqueness of solutions of initial value problems associated with a nonlinear fractional dynamic equation of Caputo type on an arbitrary time scale of order α > 0 is stated as a fixed point problem on a metric-like space. The initial conditions are assummed to be homogeneous. A theorem on the existence and uniqueness of a solution of the problem is stated and proved. Examples on two different time scales verifying the theoretical findings are presented and numerical computation of several initial terms of the iterative sequence of approximations is included.

References

A Amini-Harandi, Metric-like spaces, partial metric spaces and fixed points, Fixed Point Theory ans Applications, 2012 (2012) Art. 204.

R.P. Agarwal, U. Aksoy, E. Karapınar and ¨ ˙I. M. Erhan, F-contraction mappings on metric-like space in connection with integral equations on time scales, RACSAM, 114(3), (2020) Art. 147.

A. Ahmadkhanlu and M. Jahanshani, On the existence and uniqueness of solution of initial value problem for fractional order differential equations on time scales, Bulletin of the Iranian Mathematical Society, 38 (2012) 241–252.

G. A. Anastassiou, Principles of delta fractional calculus on time scales and inequalities, Mathematical and Computer Modelling, 52 (2010) 556–566.

A.Benkerrouche, M. S. Souid, E. Karapınar, A. Hakem On the boundary value problems of Hadamard fractional differential equations of variable order, Mathematical Methods in the Applied Sciences, 46(3) (2023) 3187–3203.

N. Benkhettou, A.M.C.Brito da Cruz, D. F.M.Torres, A fractional calculus on arbitrary time scales: Fractional differenti ation and fractional integration, Signal Processing, 107 (2015) 230–237.

M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications , Birkh¨auser, Boston, 2001.

M. Bohner and R. P. Agarwal, Basic Calculus on Time Scales and some of its Applications, Result. Math., 35 (1999) 3–22.

M. Bohner and S. Georgiev, Multivariable Dynamic Calculus on Time Scales, Springer, 2016.

R. Diaz and E. Pariguan. On hypergeometric functions and Pochhammer k-symbol., https://doi.org/10.48550/arXiv.math/0405596.

˙Inc. M. Erhan, On the existence and uniqueness of solutions o fractional dynamic equations on time scales, J. Nonlinear and Convex Analysis, 23(8) (2022) 1545-1558.

S. Georgiev, ˙I. M. Erhan, Series solution method for Cauchy problems with fractional ∆-derivative on time scales, Fractional Differential Calculus, 9 (2019) 243–261.

S. Georgiev, Fractional Dynamic Calculus and Fractional Dynamic Equations on Time Scales, Springer, 2017.

N. Hussain, J. R. Roshan, V. Oarvanch and ZKadelburg, Fixed Points of Contractive Mapping in b-Metric-like spaces, The Scientific World Journal, Vol 2014, Article ID 471827, 15 pages.

J. E. Lazreg, N. Benkhettou, M. Benchohra, E. Karapınar, Neutral functional sequential differential equations with Caputo fractional derivative on time scales, Fixed Point Theory and Algorithms for Sciences and Engineering, 1 (2022) 1–16.

Y. Li, S. Bai, D. O’Regan, Monotone iterative positive solutions for a fractional differential system with coupled Hadamard type fractional integral conditions, Journal of Applied Analysis and Computation, 13(3) (2023) 1556–1580.

D. Mozyrska, D. F. M. Torres, M. Wyrwas, Solutions of systems with the Caputo–Fabrizio fractional delta derivative on time scales, Nonlinear Analysis: Hybrid Systems, 32 (2019) 168–176.

R. Sevinik-Adıg¨uzel, E. Karapınar and ˙I. M. Erhan, A Solution to Nonlinear Volterra Integro-Dynamic Equations via Fixed Point Theory, Filomat, 33(16) (2019), 5331–5343.

R. Sevinik-Adıg¨uzel, U. Aksoy, E. Karapınar and ¨ ˙I. M. Erhan, On the solution of a boundary value problem associated with a fractional differential equation, Math. Methods in Appl.Sciences, Early Access, DOI: 10.1002/mma.6652.

R. Sevinik-Adıg¨uzel, U. Aksoy, E. Karapınar and ¨ ˙I. M. Erhan, Uniqueness of solution for higher-order nonlinear fractional differential equations with multi-point and integral boundary conditions , RACSAM, 115 (2021) Art.115, DOI10.1007/s13398-021-01095-3.

R. Sevinik-Adıg¨uzel, U. Aksoy, E. Karapınar and ¨ ˙I. M. Erhan, On the solutions of fractional differential equations via Geraghty type hybrid contractions, Applied and Computational Mathematics, 20 (2021) 313–333.

N. Shobkolaei, S. Sedghi, J. R. Roshan, and N. Hussain,Suzuki type fixed point results in metric-like spaces, Journal od Function Spaces and Applications, (2013), Article ID 143686, 9 pages.

X. Wu, T. Sun, New oscillation criteria for a class of higher order neutral functional dynamic equations on time scales, Journal of Applied Analysis and Computation, 13(2) (2023) 734–757.

Downloads

Published

2023-12-30

Issue

Section

Articles

How to Cite

Existence and uniqueness result for fractional dynamic equations on metric-like spaces. (2023). Advances in the Theory of Nonlinear Analysis and Its Application, 7(5), 81-93. https://doi.org/10.17762/atnaa.v7.i5.327