Ulam stabilities of nonlinear coupled system of fractional differential equations including generalized Caputo fractional derivative

Tamer Nabil; Tamer Nabil

doi:10.3934/math.2021301

AIMS Mathematics

2021, Volume 6, Issue 5: 5088-5105. doi: 10.3934/math.2021301

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Ulam stabilities of nonlinear coupled system of fractional differential equations including generalized Caputo fractional derivative

Tamer Nabil ^{1,2
,
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1.
King Khalid University, College of Science, Department of Mathematics, Abha, Saudi Arabia
2.
Suez Canal University, Faculty of Computers and Informatics, Department of Basic Science, Ismailia, Egypt

Received: 03 December 2020 Accepted: 01 February 2021 Published: 09 March 2021
MSC : 47H10, 26A33, 34B27, 39B82

In this paper, we establish the existence and uniqueness of solution for a nonlinear coupled system of implicit fractional differential equations including $ \psi $-Caputo fractional operator under nonlocal conditions. Schaefer's and Banach fixed-point theorems are applied to obtain the solvability results for the proposed system. Furthermore, we extend the results to investigate several types of Ulam stability for the proposed system by using classical tool of nonlinear analysis. Finally, an example is provided to illustrate the abstract results.
- fixed-point,
- $ \psi $-Caputo operator,
- coupled implicit system,
- Ulam stability,
- existence of solution
Citation: Tamer Nabil. Ulam stabilities of nonlinear coupled system of fractional differential equations including generalized Caputo fractional derivative[J]. AIMS Mathematics, 2021, 6(5): 5088-5105. doi: 10.3934/math.2021301

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Abstract

In this paper, we establish the existence and uniqueness of solution for a nonlinear coupled system of implicit fractional differential equations including $ \psi $-Caputo fractional operator under nonlocal conditions. Schaefer's and Banach fixed-point theorems are applied to obtain the solvability results for the proposed system. Furthermore, we extend the results to investigate several types of Ulam stability for the proposed system by using classical tool of nonlinear analysis. Finally, an example is provided to illustrate the abstract results.

References

[1]	M. F. Elettreby, A. A. Al-Raezah, T. Nabil, Fractional-order model of two-prey one-predator system, Math. Probl. Eng., $\bf{ 2017}$ (2017), 6714538.
[2]	D. N. Tien, Fractional stochastic differential equations with applications to finance, J. Math. Anal. Appl., $\bf{ 397}$ (2013), 338-348.
[3]	G. M. Bahaa, Optimal control problem for variable-order fractional differential systems with time delay involving Atangana-Baleanu derivatives, Chaos, Solitons Fractals, $\bf{ 122}$ (2019), 129-142.
[4]	G. Q. Zeng, J. Chen, Y. X. Dai, L. M. Li, C. W. Zheng, M. R. Chen, Design of fractional order PID controller for automatic regulator voltage system based on multi-objective extremal optimization, Neurocomputing, $\bf{ 160}$ (2015), 173-184.
[5]	F. Dong, Q. Ma, Single image blind deblurring based on the fractional-order differential, Comput. Math. Appl., $\bf{ 78}$ (2019), 1960-1977.
[6]	E. Hashemizadeh, A. Ebrahimzadeh, An efficient numerical scheme to solve fractional diffusion-wave and fractional Klein-Gordon equations in fluid mechanics, Physica A, $\bf{ 503}$ (2018), 1189-1203.
[7]	V. E. Tarasov, E. C. Aifantis, On fractional and fractal formulations of gradient linear and nonlinear elasticity, Acta Mech., $\bf{ 230}$ (2019), 2043-2070.
[8]	G. Ali, K. Shah, T. Abdeljawad, H. Khan, G. U. Rahman, A. Khan, On existence and stability results to a class of boundary value problems under Mittag-Leffler power law, Adv. Differ. Equations, 2020 (2020), 1-13. doi: 10.1186/s13662-019-2438-0
[9]	H. Khan, Z. A. Khan, H. Tajadodi, A. Khan, Existence and data-dependence theorems for fractional impulsive integro-differential system, Adv. Differ. Equations, 2020 (2020), 1-11. doi: 10.1186/s13662-019-2438-0
[10]	H. Khan, J. F. Gomez-Aguilar, T. Abdeljawad, A. Khan, Existence results and stability criteria for ABC-fuzzy-Volterra integro-differential equation, Fractals, 28 (2020), 1-9.
[11]	A. Shah, R. A. Khan, A. Khan, H. Khan, J. F. Gomez-Aguilar, Investigation of a system of nonlinear fractional order hybrid differential equations under usual boundary conditions for existence of solution, Math. Methods Appl. Sci., 44 (2021), 1628-1638. doi: 10.1002/mma.6865
[12]	E. Zeidler, Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems, New York: Springer, 1986.
[13]	S. M. Ulam, A collection of mathematical problems, New York: Interscience, 1960.
[14]	D. H. Hyers, On the stability of the linear functional equations, Proc. Natl. Acad. Sci. USA, $ \bf{ 27}$ (1941), 222-224.
[15]	T. M. Rassias, On the stability of linear mapping in Banach spaces, Proc. Amer. Math. Soc., $\bf{ 72}$ (1978), 297-300.
[16]	D. H. Hyers, G. Isac, T. M. Rassias, Stability of Functional Equations in Several Variables, New York: Springer, 1998.
[17]	T. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta. Appl. Math., $\bf{ 62}$ (2000), 23-130.
[18]	A. Khan, H. Khan, J. F. Gomez-Aguilar, T. Abdeljawad, Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel, Chaos, Solitons Fractals, $\bf{ 127}$ (2019), 422-427.
[19]	H. Khan, T. Abdeljawad, M. Aslam, R. Khan, A. Khan, Existence of positive solution and Hyers-Ulam stability for a nonlinear singular-delay-fractional differential equation, Adv. Differ. Equations, $\bf{ 2019}$ (2019), 104.
[20]	Y. Guo, X. Shu, Y. Li, F. Xu, The existence and Hyers-Ulam stability of solution for an impulsive Riemann-Liouville fractional neutral functional stochastic differential equation with infinite delay of order $1<\beta<2$, Boundary Value Probl., 2019 (2019), 59. Available from: https://doi.org/10.1186/s13661-019-1172-6.
[21]	D. X. Cuong, On the Hyers-Ulam stability of Riemann-Liouville multi-order fractional differential equations, Afr. Mat., $ \bf{ 30}$ (2019), 1041-1047.
[22]	Y. Basci, S. Ogreki, A. Misir, On Hyers-Ulam stability for fractional differential equations including the new Caputo-Fabrizio fractional derivative, Mediterr. J. Math., $\bf{ 16}$ (2019), 131.
[23]	A. Ali, B. Samet, K. Shah, R. Khan, Existence and stability of solution of a toppled systems of differential equations non-integer order, Bound. Valu Probl., $\bf{ 2017}$ (2017), 16. Available from: https://doi.org/10.1186/s13661-017-0749-1.
[24]	H. Khan, Y. Li, W. Chen, D. Baleanu, A. Khan, Existence theorems and Hyers-Ulam stability for a coupled system of fractional differential equations with p-Laplacian operator, Bound. Value Probl., $\bf{ 2017}$ (2017), 157. Available from: https://doi.org/10.1186/s13661-017-0878-6.
[25]	Z. Ali, A. Zada, K. Shah, Ulam stability results for the solutions of nonlinear implicit fractional order differential equations, Hacettepe J. Math. Stat., $\bf{ 48}$ (2019), 1092-1109.
[26]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Amsterdam: Elsevier, 2006.
[27]	R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul., $\bf{ 44}$ (2017), 460-481.
[28]	R. Almeida, A. B. Malinowska, M. T. Monterio, Fractional differential equations with a Caputo derivative with respect to a kernel functions and their applications, Math. Method. Appl. Sci., $\bf{ 41}$ (2018), 336-352.
[29]	M. S. Abdo, A. G. Ibrahim, S. K. Panchal, Nonlinear implicit fractional differential equations involving $\psi$-Caputo fractional derivative, Proc. Jangieon Math. Soc., $\bf{ 22}$ (2019), 387-400.
[30]	R. Almeida, Functional differential equations involving the $\psi$-Caputo fractional derivative, fractal and fractional, 4 (2020), 1-8.
[31]	D. B. Pachpatte, On some $\psi$-Caputo fractional Cebysev like inequalities for functions of two and three variables, AIMS Math., 5 (2020), 2244-2260. doi: 10.3934/math.2020148
[32]	C. Derbazi, Z. Baitiche, M. S. Abdo, T. Abdeljawad, Qualitative analysis of fractional relaxation equation and coupled system with $\psi$-Caputo fractional derivative in Banach spaces, AIMS Math., 6 (2021), 2486-2509. doi: 10.3934/math.2021151
[33]	T. Nabil, Existence results for nonlinear coupled system of integral equations of Urysohn Volterra-Chandrasekhar mixed type, Demonstratio Mathematica, 53 (2020), 236-248. doi: 10.1515/dema-2020-0017
[34]	T. Nabil, Solvability of Fractional differential inclusion with a generalized Caputo derivative, J. Funct. Spaces, 2020 (2020), 1-11.
[35]	T. Nabil, On nonlinear fractional neutral differential equation with the $\psi$-Caputo fractional derivative, J. Math. Appl., 43 (2020), 99.
[36]	A. Granas, J. Dugundji, Fixed Point Theory, New York: Springer-Verlag, 2003.
[37]	J. K. Hale, S. M. V. Lunel, Introduction to Functional Differential Equations, New York: Springer-Verlag, 1993.
[38]	I. A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math., $\bf{ 26}$ (2010), 103-107.

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