Impulsive fractional dynamic equation with non-local initial condition on time scales

Bikash Gogoi; Utpal Kumar Saha; Bipan Hazarika

doi:10.5269/bspm.65039

Bikash Gogoi National Institute of Technology Arunachal Pradesh https://orcid.org/0000-0001-9911-5999
Utpal Kumar Saha National Institute of Technology Arunachal Pradesh https://orcid.org/0000-0002-0143-1625
Bipan Hazarika Gauhati University https://orcid.org/0000-0002-0644-0600

Résumé

In this manuscript we investigate the existence and uniqueness of an im-pulsive fractional dynamic equation on time scales involving non-local initial condition with help of Caputo nabla derivative. The existency is based on the Scheafer’s fixed point theorem along with the Arzela-Ascoli theorem and Banach contraction theorem. The comparison of the Caputo nabla derivative and Riemann-Liouvile nabla derivative of fractional order are also discussed in the context of time scale.

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Bibliographies de l'auteur

Bikash Gogoi, National Institute of Technology Arunachal Pradesh

Department of Basic and Applied Science

Utpal Kumar Saha, National Institute of Technology Arunachal Pradesh

Department of Basic and Applied Science

Bipan Hazarika, Gauhati University

Department of Mathematics

Références

R. P. Agarwal, M. Bohner, D. O’Regan, A. Peterson, Dynamic equations on time scales: a survey, J. Comput. Appl. Math. 141 (2002) 1–26).

R. Knapik, Impulsive differential equations with non local conditions, Morehead Electronic Journal of Applicable Mathematics, Issue 2 - MATH-2002-03).

A. V. Letnikov, Theory of differentiation of arbitrary order, Mat. Sb. 3(1868) 1–68 (In Russian).

A. Ali, I. Mahariq, K. Shah, T. Abdeljawad, B. Al-Sheikh, Stability analysis of initial value problem of pantograph-type implicit fractional differential equations with impulsive conditions, Ali et al. Advances in Difference Equations (2021) 2021:55 https://doi.org/10.1186/s13662-021-03218-x.

C. Kou, J. Liu, Y. Ye, Existence and Uniqueness of Solutions for the Cauchy-Type Problems of Fractional Differential Equations, Discrete Dynamics in Nature and Society, 2010, Article ID 142175, https://doi.org/10.1155/2010/142175.

N. Benkhettou, A. Hammoudi, D. F. M. Torres, Existence and uniqueness of solution for a fractional Riemann-Liouville initial value problem on time scales, J. King Saud University - Science, 28(1)(2016) 87–92.

N. H. Du, N. C. Liem, C. J. Chyan, S. W. Lin, Lyapunov Stability of Quasilinear Implicit Dynamic Equations on Time Scale, J. Inequa. Appl. Article number: 979705 (2011) 27 pages doi:10.1155/2011/979705.

M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser Boston, (2004).

M. Bohner, A. Peterson, Dynamic Equations On Time Scales: An Introduction with Application, Birkhäuster, Boston, MA, (2001).

K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A WileyInterscience Publ. (1993).

L. Debnath, Recent application of fractional calculus to science and engineering, Int. J. Math. Math. Sci. Vol. 2003 Article ID 753601, https://doi.org/10.1155/S0161171203301486.

K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.

K. Dishlieva, Impulsive Differential Equations and Applications, J Applied Computat Mathemat 2012, 1:6 DOI: 10.4172/2168-9679.1000e117).

E. R. Duke, Solving Higher Order Dynamic Equation on Time Scales First Order System, (2006). Theses, Dissertations). and Capstones. Paper 577.

G. S. Guseinov, Integration on Time Scale, J. Math. Anal. Appl. 285(2003) 107–127.

S. Hilger, Analysis on measure chains, a unified approach to continuous and discrete calculus, Results Math. 18(1990) 18–56.

S. Hilger, Differential and difference calculus, unified, Nonlinear Anal. 30(5)(1997) 2683–2694.

V. Kumar, M. Malik, Existence, Uniqueness and Stability of nonlinear implicit fractional dynamical equation with impulsive condition on time scales, Nonauton. Dyn. Syst. 2019; 6:65-80.

V. Kumar, M. Malik, Existence and stability of fractional integro differential equation with non-instantaneous integrable impulses and periodic boundary condition on time scales, J. King Saud University-science, 31(2019) 1311-1317.

I. Podlubny, Fractional Differential Equation, Academic Press, New York, (1999).

I. Stamova, G. Stamov, Applied Impulsive Mathematical Models, CMS Books in Mathematics, DOI 10.1007/978-3- 319-28061-5_4.

S. S. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Switzerland, 1993.

S. Abbas, Dynamic equation on time scale with almost periodic coefficients, Nonautonomous Dynamical Systems, 2020.

S. Tikare, Nonlocal Initial Value Problems For First Order Dynamic Equations on Time Scale, Appl. Math. E-Notes, 21(2021) 410–420.

S. Tikare, M. Bohner, B. Hazarika, R. P. Agarwal, Dynamic Local and Nonlocal Initial Value Problems in Banach Spaces, Rend. Circ. Mat. Palermo, II. Ser. (2021). https://doi.org/10.1007/s12215-021-00674-y.

S. Tikare, C. C. Tisdell, Nonlinear dynamic equations on time scales with impulses and nonlocal conditions, J. Class. Anal. vol 16, number 2(2020), 125-140.

D. Sytnyk, R. Melnik, Mathematical models with nonlocal initial conditions: Am exemplification from quantum mechanics, Math. Comput. Appl. 2021, 26, 73. https://doi.org/10.3390/mca26040073.

B. Telli, M. S. Souid, L1 -solutions of the initial value problems for implicit differential equations with Hadamard fractional derivative, J. Appl. Anal. 2021; https://doi.org/10.1515/jaa-2021-2048.

B. Gogoi, U.K. Saha, B. Hazarika, D.F.M. Torres, H. Ahmad, Nabla Fractional Derivative and Fractional Integral on Time Scales. Axioms 2021, 10, 317. https://doi.org/10.3390/axioms10040317).

J. Zhu, L. Wu, Fractional Cauchy problem with Caputo nabla derivative on time scales, Abst. Appl. Anal. 23(2015) 486–054.

J. Dong, Y. Feng, J. Jiang, A Note on Implicit Fractional Differential Equations, Mathemtica Aeterna, 7(3)(2017) 261–267.

Z. Zhu, Y. Zhu, Fractional Cauchy problem with Riemann-Liouville fractional delta derivative on time scales, Abst. Appl. Anal. 19(2013) 401–596.

Z. Gao, L. Yang, G. Liu, Existence and Uniqueness of Solutions to Impulsive Fractional Integro-Differential Equations with Nonlocal Conditions, Applied Mathematics, 2013, 4, 859-863, http://dx.doi.org/10.4236/am.2013.46118.