On a conformable fractional differential equations with maxima

Mohammed Derhab

doi:10.26637/mjm1201/007

Abstract

This work is concerned with the existence and uniqueness of solutions for a class of first order conformable fractional differential equations with maxima. We also give some examples illustrating the application of our
results.

Keywords:

Conformable fractional differential equations, maxima, upper and lower solutions, monotone iterative technique, uniqueness

Mathematics Subject Classification:

26A33

Authors Imprint References Funding Related Articles Downloads

Mohammed Derhab Dynamic Systems and Applications Laboratory, Department of Mathematics Faculty of Sciences, University Abou-Bekr Belkaid Tlemcen, B.P. 119 Tlemcen 13000, Algeria.

Pages: 85-103
Date Published: 01-01-2024
Vol. 12 No. 01 (2024): Malaya Journal of Matematik (MJM)

T. Abdeljawad, On conformable fractional calculus. J. Comput. Appl. Math., 279(2015), 57-66. DOI: https://doi.org/10.1016/j.cam.2014.10.016

E. A.-B. Abdel-Salam And M. I. Nouh, Conformable fractional polytropic gas spheres, New Astronomy, 76(2020), Article 101322, 8 pages. DOI: https://doi.org/10.1016/j.newast.2019.101322

D. Anderson, E. Camrud and D. Ulness, On the nature of the conformable derivative and its applications to physics, J. Fract. Calc. Appl., 10(2019), 92-135.

V. G. Angelov and D. D. Bainov, On the functional differential equations with "maximums", Appl. Anal., 16(1983), 187-194. DOI: https://doi.org/10.1080/00036818308839468

J.A.D. Appleby AND H. Wu, Exponential growth and Gaussian-like fluctuations of solutions of stochastic differential equations with maximum functionals, J. Phys.: Conf. Ser, 138(012002) (2008), 1-25. DOI: https://doi.org/10.1088/1742-6596/138/1/012002

V. Azhmyakov, A. Ahmed and E. I. Verriest, On the optimal control of systems evolving with state suprema, : Proceedings of the 2016 IEEE 55th Conference on Decision and Control, Las Vegas, USA, 2016, 3617-3623. DOI: https://doi.org/10.1109/CDC.2016.7798813

D. D. BAinov AND S. G. HRistova, Monotone-iterative techniques of Lakshmikantham for a boundary value problem for systems of differential equations with maxima, J. Math. Anal. Appl., 190(1995), 391-401. DOI: https://doi.org/10.1006/jmaa.1995.1083

D. D. Bainov and S. G. HRistova, Differential equations with maxima, Chapman & Hall/CRC Pure and Applied Mathematics, 2011. DOI: https://doi.org/10.1201/b10877

B. Bendouma, A. Cabada, And A Hammoudi, Existence results for conformable fractional problems with nonlinear functional boundary conditions, Malaya Journal of Matematik, 7(2019), 700-708. DOI: https://doi.org/10.26637/MJM0704/0013

B. Bendouma, A. Cabada, and A Hammoudi, Existence results for systems of conformable fractional differential equations, Arch. Math. (Brno)., 55(2019), 69-82. DOI: https://doi.org/10.5817/AM2019-2-69

M. Bohner And V. F. HatipoĞLu, Cobweb model with conformable fractional derivatives, Math Meth Appl Sci., 41(2018),1-8. DOI: https://doi.org/10.1002/mma.4846

T. A. Burton, Volterra Integral and Differential Equations, Second Edition. Elsevier, Amsterdam, (2005).

T. A. Burton, Lyapunov Theory for Integral Equations with Singular Kernels and Fractional Differential Equations, Publisher Amazon.com, (2012).

H. Chen, S. Meng and Y. Cui, Monotone iterative technique for conformable fractional differential equations with deviating arguments, Discrete Dyn. Nat. Soc., 2018(2018), Article ID 5827127, 9 pages.

S. Dashkovskiy, O. Kichmarenko and K. Sapozhnikova, Approximation of solutions to the optimal control problems for systems with maximum, J. Math. Sci. (N. Y.), 243(2019), 192-203. DOI: https://doi.org/10.1007/s10958-019-04535-z

M. Derhab, Existence of extremal solutions for a class of conformable fractional differential equations with deviating arguments and with nonlocal initial Condition, Comm. Appl. Nonlinear Anal., 29(2022), 65 - 84.

G. Fernández-Anaya, S. Quezada-García, M.A. Polo-Labarrios and L.A. Quezada-Téllez, Novel solution to the fractional neutron point kinetic equation using conformable derivatives, Annals of Nuclear Energy, 160(2021) 108407 DOI: https://doi.org/10.1016/j.anucene.2021.108407

K. P. HadeleR, On the theory of lateral inhibition, Kybernetik, 14(1974), 161-165. DOI: https://doi.org/10.1007/BF00288918

A. Halanay, Differential Equations Stability, Oscillations, Time Lags, Academic Press New-York and London, 1966.

J.K. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied mathematical sciences, Springer-Verlag New York Inc. 1993. DOI: https://doi.org/10.1007/978-1-4612-4342-7

I. Karafyllis and Z. P. Jiang, Stability and Stabilization of Nonlinear Systems, Communications and Control Engineering Series. Springer-Verlag London, Ltd., London, 2011. DOI: https://doi.org/10.1007/978-0-85729-513-2

S. M. Khaled, E. R. El-Zahar And A. Ebaid, Solution of Ambartsumian delay differential equation with conformable derivative, Mathematics, 7(2019), 10 pages. DOI: https://doi.org/10.3390/math7050425

R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative. J. Comput. Appl. Math., 264(2014), 65-70. DOI: https://doi.org/10.1016/j.cam.2014.01.002

H. Kiskinova, M. Petkovab and A. Zahariev, About the Cauchy problem for nonlinear system with conformable derivatives and variable delays, AIP Conference Proceedings, 2172, 050006(2019), 9 pages. DOI: https://doi.org/10.1063/1.5133525

D. Kumar, A.R. Seadawy and A.K Joardar, Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology, Chinese Journal of Physics, 56(2018), $75-85$. DOI: https://doi.org/10.1016/j.cjph.2017.11.020

V. Lakshmikantham and B.G. Zhang, Monotone iterative technique for delay differential equations, Appl. Anal., 22(1986), 227-233. DOI: https://doi.org/10.1080/00036818608839620

S. LiuA, H. WANG, X. Li AND H. Li, The extremal iteration solution to a coupled system of nonlinear conformable fractional differential equations, J. Nonlinear Sci. Appl., 10(2017), 5082-5089. DOI: https://doi.org/10.22436/jnsa.010.09.44

E. Liz and S. Trofimchuk, On a dynamical model for happiness, Math. Model. Nat. Phenom., 18(2023), 15 pages. DOI: https://doi.org/10.1051/mmnp/2023008

S. Meng And Y. Cui, The extremal solution to conformable fractional differential equations involving integral boundary condition, Mathematics 7(2019), 273-281. DOI: https://doi.org/10.3390/math7020186

E. P. Popov, Automatic Regulation and Control, Nauka, Moscow, 1966. (in Russian).

C. Thaiprayoon, S. K. Ntouyas and J. Tariboon, Monotone iterative technique for nonlinear impulsive conformable fractional differential equations with delay, Communications in Mathematics and Applications, 12(2021), $11-27$. DOI: https://doi.org/10.26713/cma.v12i1.587

M. T. Terekhin And V. V. Kiryushin, Nonzero solutions to a two-point boundary-value periodic problem for differential equations with maxima, Russian Math. (Iz. VUZ), 54(2010), 43-53. DOI: https://doi.org/10.3103/S1066369X1006006X

O. Trofymchuk, E. Liz and S. Trofimchuk, The peak-end rule and its dynamic realization through differential equations with maxima, Nonlinearity, 36(2023), 507-536. DOI: https://doi.org/10.1088/1361-6544/aca50d

W.-Z. Wu, L. ZENG, C. Liu, W. XiE AND M. Goh, A time power-based grey model with conformable fractional derivative and its applications, Chaos, Solitons and Fractals, 155(2022), 111657. DOI: https://doi.org/10.1016/j.chaos.2021.111657

S. Yang, L. WANG AND S. Zhang, Conformable derivative: Application to non-Darcian flow in low-permeability porous media, Appl. Math. Lett., 79(2018), 105-110. DOI: https://doi.org/10.1016/j.aml.2017.12.006

W. Zhong AND L. WANG, Basic theory of initial value problems of conformable fractional differential equations, Adv. Difference Equ., 321(2018), 14 pages. DOI: https://doi.org/10.1186/s13662-018-1778-5

W. ZhONG And L. WANG, Positive solutions of conformable fractional differential equations with integral boundary conditions, Bound. Value Probl., 137(2018), 12 pages. DOI: https://doi.org/10.1186/s13661-018-1056-1