INITIAL VALUE PROBLEM FOR A CLASS OF SEMI-LINEAR FRACTIONAL ITERATIVE DIFFERENTIAL EQUATIONS

Mi Zhou; Lu Zhang

doi:10.11948/20230353

2024 Volume 14 Issue 5

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2024 > 14(5): 2733-2749

Next Article Previous Article

Mi Zhou, Lu Zhang. INITIAL VALUE PROBLEM FOR A CLASS OF SEMI-LINEAR FRACTIONAL ITERATIVE DIFFERENTIAL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(5): 2733-2749. doi: 10.11948/20230353

Citation:

Mi Zhou, Lu Zhang. INITIAL VALUE PROBLEM FOR A CLASS OF SEMI-LINEAR FRACTIONAL ITERATIVE DIFFERENTIAL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(5): 2733-2749. doi: 10.11948/20230353

INITIAL VALUE PROBLEM FOR A CLASS OF SEMI-LINEAR FRACTIONAL ITERATIVE DIFFERENTIAL EQUATIONS

Mi Zhou^1, ,,
Lu Zhang^2,

1.
School of Mathematics and Physics, University of South China, Hengyang, Hunan 421001, China
2.
Faculty of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, China

Author Bio: Email: lzhang@xtu.edu.cn(L. Zhang)
Corresponding author: Email: zhoum aa@sina.com(M. Zhou)

Abstract

Abstract

An initial value problem of a class of semi-linear fractional order iterative differential equations is researched in this paper. The existence of solution is acquired in respect of Banach space $C(I, I)$ and $C_{K, q}(I, I)$ for fractional order iterative differential equations. Nevertheless, because the operator is Hölder continuous rather than Lipschitz continuous, uniqueness results can not be obtained. Additionally, a change of solution to $[k, \beta]$ for the $k\in I$ will arise from a small perturbation of the initial value. Our analysis is on the basis of the properties of Mittag-Leffler function and Schauder's fixed point theorem. Lastly, some examples are provided to demonstrate our results.
- Fractional order iterative differential equations /
- Mittag-Leffler function /
- initial value problem /
- existence
MSC: 26A33, 33E12, 34A12

FullText(HTML)

References

[1]	J. Alzabut, M. Khuddush, A. G. M. Selvam and D. Vignesh, Second order iterative dynamic boundary value problems with mixed derivative operators with applications, Qual. Theor. Dyn. Syst., 2023, 22(1), 1–32. doi: 10.1007/s12346-022-00693-9 CrossRef Google Scholar
[2]	J. H. Barrett, Differential equations of non-integer order, Canad J. Math., 1954, 6, 529–541. doi: 10.4153/CJM-1954-058-2 CrossRef Google Scholar
[3]	A. Bouakkaz, Positive periodic solutions for a class of first-order iterative differential equations with an application to a hematopoiesis model, Carpathian J. Math., 2022, 38(2), 347–355. doi: 10.37193/CJM.2022.02.07 CrossRef Google Scholar
[4]	A. Buica, Existence and continuous dependence of solutions of some functional-differential equations, Semin. Fixed Point Theory, 1995, 3(1), 1–14. Google Scholar
[5]	A. Chidouh, A. Guezane-Lakoud and R. Bebbouchi, Positive solutions of the fractional relaxation equation using lower and upper solutions, Vietnam J. Math., 2016, 44, 739–748. doi: 10.1007/s10013-016-0192-0 CrossRef Google Scholar
[6]	F. Damag and A. Kilicman, Sufficient conditions on existence of solution for nonlinear fractional iterative integral equation, J. Nonlinear Sci. Appl., 2017, 10, 368–376. doi: 10.22436/jnsa.010.02.03 CrossRef Google Scholar
[7]	J. H. Deng and J. R. Wang, Existence and approximation of solutions of fractional order iterative differential equations, Cent. Eur. J. of Phys., 2013, 11(10), 1377–1386. Google Scholar
[8]	K. Diethelm and A. D. Freed, On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, in: F. Keil, W. Mackens, H. Voss, J. Werther (Eds), Science Computing in Chemical Engineering Ⅱ-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, 217–224. Springer-Verlag, Heidelberg, 1999. Google Scholar
[9]	A. Granas and J. Dugundji, Fixed Point Theory, Spring, New York, 2003. Google Scholar
[10]	A. Guerfi and A. Ardjouni, Existence, uniqueness, continuous dependence and Ulam stability of mild solutions for an iterative fractional differential equation, Cubo (Temuco), 2022, 24(1), 83–94. doi: 10.4067/S0719-06462022000100083 CrossRef Google Scholar
[11]	A. A. Hamoud and K. P. Ghadle, Some new results on nonlinear fractional iterative Volterra-Fredholm integro differential equations, Twms J. Appl. Eng. Math., 2022, 12, 1283–1294. Google Scholar
[12]	A. Hamoud, N. Mohammed and K. Ghadle, Existence and uniqueness results for Volterra-Fredholm integro differential equations, Adv. Theory Nonlinear Anal. Appl., 2020, 4(4), 361–372. Google Scholar
[13]	A. A. Hamoud, N. M. Mohammed and K. P. Ghadle, Existence, uniqueness and stability results for nonlocal fractional nonlinear Volterra-Fredholm integro differential equations, Discontinuity, Nonlinearity, and Complexity, 2022, 11(2), 343–352. Google Scholar
[14]	J. W. He and Y. Zhou, Hölder regularity for non-autonomous fractional evolution equations, Fract. Calc. Appl. Anal., 2022, 25(2), 924–961. Google Scholar
[15]	J. W. He, Y. Zhou, L. Peng and B. Ahmad, On well-posedness of semilinear Rayleigh-Stokes problem with fractional derivative on $\mathbb{R}^N$, Adv. Nonlinear Anal., 2022, 11(1), 580–597. $\mathbb{R}^N$" target="_blank">Google Scholar
[16]	A. S. Hegazi, E. Ahmed and A. E. Matou, On chaos control and synchronization of the commensurate fractional order Liu system, Lect, Commun. Nonlinear Sci., 2013, 18(5), 1193–1202. doi: 10.1016/j.cnsns.2012.09.026 CrossRef Google Scholar
[17]	R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. Google Scholar
[18]	R. W. Ibrahim, Existence of deviating fractional differential equation, Cubo Math. J., 2012, 14(3), 129–142. doi: 10.4067/S0719-06462012000300009 CrossRef Google Scholar
[19]	R. W. Ibrahim, Existence of iterative Cauchy fractional differential equation, J. Math., 2013. DOI: 10.1155/2013/838230. CrossRef Google Scholar
[20]	R. W. Ibrahim, A. Kilicman and F. H. Damag, Existence and uniqueness for a class of iterative fractional differential equations, Adv. Difference Equ., 2015, 1, 1–13. Google Scholar
[21]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier, Amsterdam, 2006. Google Scholar
[22]	A. Kiliman and F. H. M. Damag, Some solution of the fractional iterative integro-differential equations, Malays. J. Math. Sci., 2018, 12(1), 121–141. Google Scholar
[23]	C. Li, R. Wu and R Ma, Existence of solutions for Caputo fractional iterative equations under several boundary value conditions, AIMS Math., 2023, 8(1), 317–339. doi: 10.3934/math.2023015 CrossRef Google Scholar
[24]	F. Liu and K. Burrage, Novel techniques in parameter estimation for fractional dynamical models arising from biological systems, Comput. Math. Appl., 2011, 62(3), 822–833. doi: 10.1016/j.camwa.2011.03.002 CrossRef Google Scholar
[25]	F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Soliton Fract., 1996, 7(9), 1461–1477. doi: 10.1016/0960-0779(95)00125-5 CrossRef Google Scholar
[26]	B. Mansouri, A. Ardjouni and A. Djoudi, Periodicity and continuous dependence in iterative differential equations, Rend Circ. Mat. Palerm., 2019, 69(2), 1–16. Google Scholar
[27]	R. Nigmatullin, T. Omay and D. Baleanu, On fractional filtering versus conventional filtering in economics, Commun. Nonlinear Sci. Numer. Simul., 2010, 15(4), 979–986. doi: 10.1016/j.cnsns.2009.05.027 CrossRef Google Scholar
[28]	I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. Google Scholar
[29]	K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 2011, 382(1), 426–447. doi: 10.1016/j.jmaa.2011.04.058 CrossRef Google Scholar
[30]	V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Nonlinear Physical Science, Spring, Heidelberg, 2011. Google Scholar
[31]	S. Unhaley and S. Kendre, On existence and uniqueness results for iterative fractional integro-differential equation with deviating arguments, Appl. Math. E-Notes, 2019, 19, 116–127. Google Scholar
[32]	J. R. Wang and J. H. Deng, Fractional order differential equations with iterations of linear modification of the argument, Adv. Difference Equ., 2013. DOI: 10.1186/1687-1847-2013-329. CrossRef Google Scholar
[33]	J. R. Wang, M. Fečkan and Y. Zhou, Presentation of solutions of impulsive fractional Langevin equations and existence results, Eur. Phys. J. -Spec. Top., 2013, 222(8), 1857–1874. doi: 10.1140/epjst/e2013-01969-9 CrossRef Google Scholar
[34]	J. H. Wang, M. Feckan M and Y. Zhou, Fractional order iterative functional differential equations with parameter, Appl. Math. Model., 2013, 37(8), 6055–6067. doi: 10.1016/j.apm.2012.12.011 CrossRef Google Scholar
[35]	B. J. West, Colloquium: Fractional calculus view of complexity: A tutorial, Rev. Mod. Phys., 2014, 86(4), 1169–1184. doi: 10.1103/RevModPhys.86.1169 CrossRef Google Scholar
[36]	D. Yang and W. Zhang, Solutions of equivariance for iterative differential equations, Appl. Math. Lett., 2004, 17(7), 759–765. doi: 10.1016/j.aml.2004.06.002 CrossRef Google Scholar
[37]	S. Zhang and X. Su, The existence of a solution for a fractional differential equation with nonlinear boundary conditions considered using upper and lower solutions in reverse order, Comput. Math. Appl., 2011, 62(3), 1269–1274. doi: 10.1016/j.camwa.2011.03.008 CrossRef Google Scholar
[38]	M. Zhou, Well-posedness of nonlinear fractional quadratic iterative differential equations, J. Anal., 2023, 31(2), 881–897. doi: 10.1007/s41478-022-00484-0 CrossRef Google Scholar
[39]	Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. Google Scholar