Existence and stability analysis of solutions for fractional differential equations with delay

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DOI:

https://doi.org/10.26637/mjm11S/003

Abstract

In this manuscript, we establish the existence, uniqueness, and stability of solutions for fractional differential equations with delay. Then we solve the version of the problem with constant delay. We utilize the Bielecki Norm and the Ulam-Hyers stability for our results.

Keywords:

Existence, Stability, Fractional differential equations, Delay

Mathematics Subject Classification:

26A33
  • Mesfin Etefa NEERLab Laboratory, Department of Mathematics, Bowie State University, 14000 Jericho Park Rd.,Bowie, MD 20715, USA.
  • Gaston N'Guerekata NEERLab Laboratory, Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, M.D. 21251, USA.
  • Pages: 43-52
  • Date Published: 01-10-2023
  • Vol. 11 No. S (2023): Malaya Journal of Matematik (MJM): Special Issue Dedicated to Professor Gaston M. N'Guérékata’s 70th Birthday

D.D. B AINOV AND P.S. S IMEONOV , Stability Theory of Differential Equations with Impulse Effects: Theory

and Applications, Ellis Horwood, Chichester, UK, 1989.

M. B ENCHOHRA , J. H ENDERSON , S.K. N TOUYAS , Impulsive Differential Equations and Inclusions, vol. 2,

Hindawi Publishing Corporation, New York, 2006.

M. B ENCHOHRA , B. A. S LIMANI , Existence and Uniqueness of Solutions to Impulsive Fractional Differential Equations, Electronic Journal of Differential Equations, Vol. 2009(2009), No. 10, pp. 1–11.

L. B YSZEWSKI , Theorems about existence and uniqueness of solutions of a semilinear evolution nonlocal

Cauchy problem, J. Math. Anal. Appl., 162(1991), 494–505. DOI: https://doi.org/10.1016/0022-247X(91)90164-U

L. B YSZEWSKI , Existenceanduniquenessofmildandclassicalsolutionsofsemilinearfunctional-differential

evolution nonlocal Cauchy problem, Selected problems of mathematics, 25–33, 50th Anniv. Cracow Univ.

Technol. Anniv. Issue, 6, Cracow Univ. Technol., Krakow, 1995.

C. C ORDUNEANU , Principles of Differential and Integral Equations, Allyn and Bacon, Boston, 1971.

K. D ENG , Exponential decay of semilinear parabolic equations with nonlocal conditions, J. Math. Anal.

Appl., 179 (1993), 630–637. DOI: https://doi.org/10.1006/jmaa.1993.1373

F. D EVELIA AND O. D UMANA , Existence and stability analysis of solution for fractional delay differential

equations, Filomat, 37(6)(2023), 1869–1878. DOI: https://doi.org/10.2298/FIL2306869D

M. M. E TEFA , G. M. N’G U ´ ER ´ EKATA AND M. B ENCHOHRA , Existence and Uniqueness of Solutions

to Impulsive Fractional Differential Equations via the Deformable Derivative, Appl. Anal., 2021.

https://doi.org/10.1080/00036811.2021.1979224. DOI: https://doi.org/10.1080/00036811.2021.1979224

M. M. E TEFA AND G. M. N’G U ´ ER ´ EKATA , Existence Results for Fractional Order Functional Differential

Equations with Infinite Delay in the Sense of the Deformable Derivative, Analele Universit˘ at ¸ii Oradea,

Fasc. Matematica, Tom XXIX(2)(2022), 111–121.

K. E ZZINBI AND J. L IU , Nondensely defined evolution equations with nonlocal conditions, Math. Comput.Modelling, 36(2002), 1027–1038.

A. G RANAS AND J. D UGUNDJI , Fixed Point Theory, Springer-Verlag, New York, 2003. DOI: https://doi.org/10.1007/978-0-387-21593-8

U.N. K ATUGAMPOLA , A new fractional derivative with classical properties, arxiv: 1410.6535v2, (2010), pp.

-8 Modelling 36 (2002) 1027–1038. DOI: https://doi.org/10.1016/S0895-7177(02)00256-X

V. L AKSHMIKANTHAM , D.D. B AINOV AND P.S. S IMEONOV , Theory of Impulsive Differential Equations, World Scientific, Singapore, London, 1989.

A.M. M ATHAI AND H.J. H AUBOLD , An Introduction to Fractional Calculus, Mathematics Research

Developments, Nova Science Publishers, New York, 2017.

A. M ERAJ AND D.N. P ANDEY , Existence and Uniqueness of mild solution and approximate controllability

of fractional evolution equations with deformable derivative, J. Nonlinear Evol. Equ. and Appl.,

(7)(2019), 85–100.

M.M EBRAT AND G.M. N’G U ´ ER ´ EKATA , A Cauchy Problem for Some Fractional Differential Equation via

Deformable Derivatives, J. Nonlinear Evol. Equ. and Appl., 2020(4), pp. 1–9.

M. M EBRAT AND G.M. N’G U ´ ER ´ EKATA , An Existence Result for some Fractional- Integro Differential

Equations in Banach Spaces via the Deformable Derivative, Journal of Mathematical Extension, Accepted

for publication, May 2021.

G.M. N’G U ´ ER ´ EKATA , A Cauchy problem for some fractional abstract differential equation with non local conditions, Nonlinear Anal., 70(2009), 1873–1876. DOI: https://doi.org/10.1016/j.na.2008.02.087

G.M. N’G U ´ ER ´ EKATA , Existence and uniqueness of an integral solution to some Cauchy problem with

nonlocal conditions, Differential and Difference Equations and Applications, Vol. 843–849, Hindawi Publ.

Corp., New York, 2006.

I. A. R US , Picard operators and applications, Sci. Math. Jpn., 58(2003), 191–219.

I. A. R US , Fixed points, upper and lower fixed points: abstract Gronwall lemmas, Carpathian J. Math.,

(2004), 125–134.

F. Z ULFEQARR , A. U JLAYAN AND P. A HUJA , A new fractional derivative and its fractional integral with some applications, arXiv : 1705.00962v1, (2017), pp. 1–11. New York, 1993.

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Published

01-10-2023

How to Cite

Etefa, M., and G. . N’Guerekata. “Existence and Stability Analysis of Solutions for Fractional Differential Equations With Delay”. Malaya Journal of Matematik, vol. 11, no. S, Oct. 2023, pp. 43-52, doi:10.26637/mjm11S/003.