A class of nonlocal impulsive differential equations with conformable fractional derivative

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DOI:

https://doi.org/10.56754/0719-0646.2403.0439

Abstract

In this paper, we deal with the Duhamel formula, existence, uniqueness, and stability of mild solutions of a class of nonlocal impulsive differential equations with conformable fractional derivative. The main results are based on the semigroup theory combined with some fixed point theorems. We also give an example to illustrate the applicability of our abstract results.

Keywords

Functional-differential equations with fractional derivativesFunctional-differential equations with fractional derivatives , Groups and semigroups of linear operators , Nonlocal conditions , Impulsive conditions , Conformable fractional derivatives
  • Pages: 439–455
  • Date Published: 2022-12-21
  • Vol. 24 No. 3 (2022)

N. Abada, M. Benchohra and H. Hammouche, “Existence results for semilinear differential evolution equations with impulses and delay”, Cubo, vol. 12, pp. 1–17, 2010.

T. Abdeljawad, “On conformable fractional calculus”, J. Comput. Appl. Math., vol. 279, pp. 57–66, 2015.

B. Ahmad, B. Alghamdi, R. P Agarwal and A. Alsaedi, “Riemann-Liouville fractional integro- differential equations with fractional non-local multi-point boundary conditions”, Fractals, vol. 30, no. 1, 11 pages, 2022.

M. Al-Masaeed, E. Rabei and A. Al-Jamel, “Wkb approximation with conformable operator”, 2021, ArXiv:2111.01547.

M. Atraoui and M. Bouaouid, “On the existence of mild solutions for nonlocal differential equations of the second order with conformable fractional derivative”, Adv. Difference Equ., Paper No. 447, 11 pages, 2021.

M. Benchohra, J. Henderson and S. K. Ntouyas, Impulsive differential equations and inclusions, Comtemporary mathematics and its applications 2, New York: Hindawi Publishing Corporation, 2006.

S. A. Bhanotar and M. K. Kaabar, “Analytical solutions for the nonlinear partial differential equations using the conformable triple Laplace transform decomposition method”, Int. J. Differ. Equ., 18 pages, 2021.

T. T. Binh, N. H. Luc, D. O‘Regan and N. H. Can, “On an initial inverse problem for a diffusion equation with a conformable derivative”, Adv. Difference Equ., Paper No. 481, 24 pages, 2019.

G. Bonnano, R. Rodríguez-López and S. Tersian, “Existence of solutions to boundary value problem for impulsive fractional differential equations”, Fract. Calc. Appl. Anal., vol. 17, no. 3, pp. 717–744, 2014.

M. Bouaouid, “Mild solutions of a class of conformable fractional differential equations with nonlocal conditions”, Acta Math. Appl. Sin. Engl. Ser., vol. 38, pp.1–13, 2022.

M. Bouaouid, M. Atraoui, K. Hilal and S. Melliani, “Fractional differential equations with nonlocal-delay condition”, J. Adv. Math. Stud., vol. 11, no. 2, pp. 214–225, 2018.

M. Bouaouid, M. Hannabou and K. Hilal, “Nonlocal conformable-fractional differential equations with a measure of noncompactness in Banach spaces”, J. Math., Art. ID 5615080, 6 pages, 2020.

M. Bouaouid, K. Hilal and M. Hannabou, “Integral solutions of nondense impulsive conformable-fractional differential equations with nonlocal condition”, J. Appl. Anal., vol. 27, no. 2, pp. 187–197, 2021.

M. Bouaouid, K. Hilal and S. Melliani, “Nonlocal conformable fractional Cauchy problem with sectorial operator”, Indian J. Pure Appl. Math., vol. 50, no. 4, pp. 999–1010, 2019.

M. Bouaouid, K. Hilal and S. Melliani, “Nonlocal telegraph equation in frame of the conformable time-fractional derivative”, Adv. Math. Phys., 7 pages, 2019.

M. Bouaouid, K. Hilal and S. Melliani, “Sequential evolution conformable differential equations of second order with nonlocal condition”, Adv. Difference Equ., Paper No. 21, 13 pages, 2019.

M. Bouaouid, K. Hilal and S. Melliani, “Existence of mild solutions for conformable-fractional differential equations with non local conditions”, Rocky Mountain J. Math, vol. 50, no. 3, pp. 871–879, 2020.

L. Byszewski, “Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem”, J. Math. Anal. Appl., vol. 162, no. 2, pp. 494–505, 1991.

R. C. Cascaval, E. C. Eckstein, C. L. Frota and J. A. Goldstein, “Fractional telegraph equations”, J. Math. Anal. Appl., vol. 276, no. 1, pp. 145–159, 2002.

J. Chen, F. Liu and V. Anh, “Analytical solution for the time-fractional telegraph equation by the method of separating variables”, J. Math. Anal. Appl., vol. 338, no. 2, pp. 1364–1377, 2008.

K. Deng, “Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions”, J. Math. Anal. Appl., vol. 179, no. 2, pp. 630–637, 1993.

A. El-Ajou, “A modification to the conformable fractional calculus with some applications”, Alexandria Eng. J., vol. 59, no. 4, pp. 2239–2249, 2020.

H. Eltayeb, I. Bachar and M. Gad-Allah, “Solution of singular one-dimensional Boussinesq equation by using double conformable Laplace decomposition method”, Adv. Difference Equ., Paper No. 293, 19 pages, 2019.

H. Eltayeb and S. Mesloub, “A note on conformable double Laplace transform and singular conformable pseudoparabolic equations”, J. Funct. Spaces, Art. ID 8106494, 12 pages, 2020.

F. Gao and C. Chunmei, “Improvement on conformable fractional derivative and its applications in fractional differential equations”, J. Funct. Spaces, Art. ID 5852414, 10 pages, 2020.

Y. Giga, H. Mitake and S. Sato, “On the equivalence of viscosity solutions and distributional solutions for the time-fractional diffusion equation”. J. Differential Equations, vol. 316, pp. 364–386, 2022.

M. Hannabou, M. Bouaouid and K. Hilal, “Controllability of mild solution of nonlocal conformable fractional differential equations”, Adv. Math. Phys., Art. ID 3671909, 8 pages, 2022.

S. Injrou, R. Karroum and N. Deeb, “Various exact solutions for the conformable time- fractional generalized Fitzhugh-Nagumo equation with time-dependent coefficients”, Int. J. Differ. Equ., Art. ID 8888989, 11 pages, 2021.

M. Jneid and A. El Chakik, “Analytical solution for some systems of nonlinear conformable fractional differential equations”, Far East J. Math. Sci. (FJMS), vol. 109, no. 2, pp. 243–259, 2018.

R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, “A new definition of fractional derivative”, J. Comput. Appl. Math., vol. 264, pp. 65–70, 2014.

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies 204, Amsterdam: Elsevier, 2006.

V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of impulsive differential equations, Series in modern applied mathematics 6, Singapore: World Scientific, 1989.

J. Liang, J. H. Liu and T. J. Xiao, “Nonlocal impulsive problems for nonlinear differential equations in Banach spaces”, Math. Comput. Modelling, vol. 49, no. 3–4, pp. 798–804, 2009.

J. Liu and C. Zhang, “Existence and stability of almost periodic solutions to impulsive stochastic differential equations”, Cubo, vol. 15, no. 1, pp. 77–96, 2013.

K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, New York: John Wiley & Sons, Inc., 1993.

S. Momani, “Analytic and approximate solutions of the space and time-fractional telegraph equations”, Appl. Math. Comput., vol. 170, no. 2, pp. 1126–1134, 2005.

G. M. Mophou, “Existence and uniqueness of mild solutions to impulsive fractional differential equations”, Nonlinear Anal., vol. 72, no. 3–4, pp. 1604–1615, 2010.

K. B. Oldham and J. Spanier, The fractional calculus, Mathematics in Science and Engineering 111, New York: Academic Press, Inc., 1974.

W. E. Olmstead and C. A. Roberts, “The one-dimensional heat equation with a nonlocal initial condition”, Appl. Math. Lett., vol. 10, no. 3, pp. 89–94, 1997.

A. Pazy, Semigropus of linear operators and applications to partial differential equations, New York: Springer New York, 1983.

I. Podlubny, Fractional differential equations, Mathematics in Science and Engineering 198, San Diego: Academic Press, Inc., 1999.

E. Rabei, A. Al-Jamel and M. Al-Masaeed, “The solution of conformable Laguerre differential equation using conformable Laplace transform”, 2021, ArXiv:2112.01322.

S. G. Samko, A. A Kilbas and O. I Marichev, Fractional integrals and derivatives: theory and applications, Switzerland: Gordon & Breach Science Publishers, 1993.

T. N. Thach, N. H. Can, V. V. Tri, “Identifying the initial state for a parabolic diffusion from their time averages with fractional derivative”, Math. Meth. Appl. Sci, 16 pages, 2021.

T. N. Thach, , D. Kumar, N. H. Luc and N. D. Phuong, “On a semilinear fractional reaction-diffusion equation with nonlocal conditions”, Alexandria Eng. J., vol. 60, no. 6, pp. 5511–5520, 2021.

T. N. Thach, D. Kumar, N. H. Luc and N. H. Tuan, “Existence and regularity results for stochastic fractional pseudo-parabolic equations driven by white noise”, Discrete Contin. Dyn. Syst. Ser. S, vol. 15, no. 2, pp. 481–499, 2022.

T. N. Thach and N. H. Tuan, “Stochastic pseudo-parabolic equations with fractional derivative and fractional Brownian motion”, Stoch. Anal. Appl., vol. 40, no.2, pp. 328–351, 2022.

N. H. Tuan, N. D. Phuong and T. N. Thach, “New well-posedness results for stochastic delay Rayleigh-Stokes equations”, Discrete Contin. Dyn. Syst. Ser. B, vol. 28, no. 1, pp. 347–358, 2023.

N. H. Tuan, T. N. Thach, N. H Can and D. O‘Regan, “Regularization of a multidimensional diffusion equation with conformable time derivative and discrete data”, Math. Methods Appl. Sci., vol. 44, no. 4, pp. 2879–2891, 2021.

A. Zavalishchin, “Impulse dynamic systems and applications to mathematical economics”, Dynam. Systems Appl., vol. 3, no. 3, pp. 443–449, 1994.

Y. Zhou, V. Vijayakumar and R. Murugesu, “Controllability for fractional evolution inclusions without compactness”, Evol. Equ. Control Theory, vol. 4, no. 4, pp. 507–524, 2015.

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Published

2022-12-21

How to Cite

[1]
M. Bouaouid, A. Kajouni, K. Hilal, and S. Melliani, “A class of nonlocal impulsive differential equations with conformable fractional derivative”, CUBO, vol. 24, no. 3, pp. 439–455, Dec. 2022.

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