Generalized exponential function and initial value problem for conformable dynamic equations

Awais Younus; Khizra Bukhsh; Manar A. Alqudah; Thabet Abdeljawad; Awais Younus; Khizra Bukhsh; Manar A. Alqudah; Thabet Abdeljawad

doi:10.3934/math.2022670

AIMS Mathematics

2022, Volume 7, Issue 7: 12050-12076. doi: 10.3934/math.2022670

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Generalized exponential function and initial value problem for conformable dynamic equations

1.
Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan
2.
Department of Mathematics, Ghazi University, Dera Ghazi Khan, Pakistan
3.
Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah Bint Abdulrahman University, P. O. Box 84428, Riyadh 11671, Saudi Arabia
4.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh, Saudi Arabia
5.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan

Received: 22 January 2022 Revised: 25 March 2022 Accepted: 06 April 2022 Published: 21 April 2022
MSC : 34A08, 34N05

In this article, we define the generalized exponential function on arbitrary time scales in the conformable setting and develop its fundamental characteristics. We address the fundamental theory of a conformable fractional dynamic equation on time scales, subject to the local and non-local initial conditions. We generalized the Grönwall type inequalities in a conformable environment. The generalized exponential function and the Grönwall's inequalities are indispensable for the study of the qualitative aspects of the local initial value problem. We developed some criteria related to global existence, extension and boundedness, as well as stability of solutions.
- generalized exponential function,
- conformable dynamic equation,
- local initial value problem,
- nonlocal initial value problem,
- Grönwall's inequalities
Citation: Awais Younus, Khizra Bukhsh, Manar A. Alqudah, Thabet Abdeljawad. Generalized exponential function and initial value problem for conformable dynamic equations[J]. AIMS Mathematics, 2022, 7(7): 12050-12076. doi: 10.3934/math.2022670

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Abstract

In this article, we define the generalized exponential function on arbitrary time scales in the conformable setting and develop its fundamental characteristics. We address the fundamental theory of a conformable fractional dynamic equation on time scales, subject to the local and non-local initial conditions. We generalized the Grönwall type inequalities in a conformable environment. The generalized exponential function and the Grönwall's inequalities are indispensable for the study of the qualitative aspects of the local initial value problem. We developed some criteria related to global existence, extension and boundedness, as well as stability of solutions.

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