On degree theory for non-monotone type fractional order delay differential equations

Kamal Shah; Muhammad Sher; Asad Ali; Thabet Abdeljawad; Kamal Shah; Muhammad Sher; Asad Ali; Thabet Abdeljawad

doi:10.3934/math.2022526

AIMS Mathematics

2022, Volume 7, Issue 5: 9479-9492. doi: 10.3934/math.2022526

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On degree theory for non-monotone type fractional order delay differential equations

1.
Department of Mathematics and Sciences, Prince Sultan University, P. O. Box.11586, Riyadh, Saudi Arabia
2.
Department of Mathematics, University of Malakand, Chakdara Dir(L), P. O. Box. 18000, Khyber Pakhtunkhwa, Pakistan
3.
Department of Mathematics, Hazara University, Mansehra, P. O. Box. 21300, Khyber Pakhtunkhwa, Pakistan
4.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan

Received: 31 December 2021 Revised: 27 February 2022 Accepted: 04 March 2022 Published: 14 March 2022
MSC : 26A33, 34A08

In this paper, we establish a qualitative theory for implicit fractional order differential equations (IFODEs) with nonlocal initial condition (NIC) with delay term. Because area related to investigate existence and uniqueness of solution is important field in recent times. Also researchers are using existence theory to derive some prior results about a dynamical problem weather it exists or not in reality. In literature, we have different tools to study qualitative nature of a problem. On the same line the exact solution of every problem is difficult to determined. Therefore, we use technique of numerical analysis to approximate the solutions, where stability analysis is an important aspect. Therefore, we use a tool from non-linear analysis known as topological degree theory to develop sufficient conditions for existence and uniqueness of solution to the considered problem. Further, we also develop sufficient conditions for Hyers- Ulam type stability for the considered problem. To justify our results, we also give an illustrative example.
- arbitrary order differential equations,
- existence theory,
- topological degree theory
Citation: Kamal Shah, Muhammad Sher, Asad Ali, Thabet Abdeljawad. On degree theory for non-monotone type fractional order delay differential equations[J]. AIMS Mathematics, 2022, 7(5): 9479-9492. doi: 10.3934/math.2022526

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Abstract

In this paper, we establish a qualitative theory for implicit fractional order differential equations (IFODEs) with nonlocal initial condition (NIC) with delay term. Because area related to investigate existence and uniqueness of solution is important field in recent times. Also researchers are using existence theory to derive some prior results about a dynamical problem weather it exists or not in reality. In literature, we have different tools to study qualitative nature of a problem. On the same line the exact solution of every problem is difficult to determined. Therefore, we use technique of numerical analysis to approximate the solutions, where stability analysis is an important aspect. Therefore, we use a tool from non-linear analysis known as topological degree theory to develop sufficient conditions for existence and uniqueness of solution to the considered problem. Further, we also develop sufficient conditions for Hyers- Ulam type stability for the considered problem. To justify our results, we also give an illustrative example.

References

[1]	K. B. Oldham, Fractional differential equations in electrochemistry, Adv. Eng. Soft., 41 (2010), 9–12. https://doi.org/10.1016/j.advengsoft.2008.12.012 doi: 10.1016/j.advengsoft.2008.12.012
[2]	I. Podlubny, Fractional differential equations, Elsevier, 1998.
[3]	R. Hilfer, Applications of fractional calculus in physics, Singapore: World Scientific, 2000. https://doi.org/10.1142/3779
[4]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006.
[5]	K. Kavitha, V. Vijayakumar, A. Shukla, K. S. Nisar, R. Udhayakumar, Results on approximate controllability of Sobolev-type fractional neutral differential inclusions of Clarke sub-differential type, Chaos Soliton. Fract., 151 (2021), 111264. https://doi.org/10.1016/j.chaos.2021.111264 doi: 10.1016/j.chaos.2021.111264
[6]	M. M. Raja, V. Vijayakumar, R. Udhayakumar, Y. Zhou, A new approach on the approximate controllability of fractional differential evolution equations of order $1 < r < 2$ in Hilbert spaces, Chaos Soliton. Fract., 141 (2020), 110310. https://doi.org/10.1016/j.chaos.2020.110310 doi: 10.1016/j.chaos.2020.110310
[7]	K. S. Nisar, V. Vijayakumar, Results concerning to approximate controllability of non-densely defined Sobolev-type Hilfer fractional neutral delay differential system, Math. Method. Appl. Sci., 44 (2021), 13615–13632. https://doi.org/10.1002/mma.7647 doi: 10.1002/mma.7647
[8]	F. A. Rihan, Q. M. Al-Mdallal, H. J. AlSakaji, A. Hashish, A fractional-order epidemic model with time-delay and nonlinear incidence rate, Chaos Soliton. Fract., 126 (2019), 97–105. https://doi.org/10.1016/j.chaos.2019.05.039 doi: 10.1016/j.chaos.2019.05.039
[9]	R. Chinnathambi, F. A. Rihan, Stability of fractional-order prey–predator system with time-delay and Monod–Haldane functional response, Nonlinear Dyn., 92 (2018), 1637–1648. https://doi.org/10.1007/s11071-018-4151-z doi: 10.1007/s11071-018-4151-z
[10]	K. Kavitha, V. Vijayakumar, R. Udhayakumar, N. Sakthivel, K. S. Nisar, A note on approximate controllability of the Hilfer fractional neutral differential inclusions with infinite delay, Math. Method. Appl. Sci., 44 (2021), 4428–4447. https://doi.org/10.1002/mma.7040 doi: 10.1002/mma.7040
[11]	M. M. Raja, V. Vijayakumar, R. Udhayakumar, A new approach on approximate controllability of fractional evolution inclusions of order $1 < r < 2$ with infinite delay, Chaos Soliton. Fract., 141 (2020), 110343. https://doi.org/10.1016/j.chaos.2020.110343 doi: 10.1016/j.chaos.2020.110343
[12]	L. Wang, X.-B. Shu, Y. Cheng, R. Cui, Existence of periodic solutions of second-order nonlinear random impulsive differential equations via topological degree theory, Res. Appl. Math., 12 (2021), 100215. https://doi.org/10.1016/j.rinam.2021.100215 doi: 10.1016/j.rinam.2021.100215
[13]	T. Abdeljawad, M. A. Hajji, Q. M. Al-Mdallal, F. Jarad, Analysis of some generalized ABC–fractional logistic models, Alex. Eng. J., 59 (2020), 2141–2148. https://doi.org/10.1016/j.aej.2020.01.030 doi: 10.1016/j.aej.2020.01.030
[14]	A. Babakhani, Q. Al-Mdallal, On the existence of positive solutions for a non-autonomous fractional differential equation with integral boundary conditions, Comput. Methods Differ. Equ., 9 (2021), 36–51. https://doi.org/10.22034/CMDE.2020.29444.1420 doi: 10.22034/CMDE.2020.29444.1420
[15]	K. Shah, M. Sher, T. Abdeljawad, Study of evolution problem under Mittag-Leffler type fractional order derivative, Alex. Eng. J., 59 (2020), 3945–3951. https://doi.org/10.1016/j.aej.2020.06.050 doi: 10.1016/j.aej.2020.06.050
[16]	F. A. Rihan, H. J. Alsakaji, C. Rajivganthi, Stochastic SIRC epidemic model with time-delay for COVID-19, Adv. Differ. Equ., 2020 (2020), 502. https://doi.org/10.1186/s13662-020-02964-8 doi: 10.1186/s13662-020-02964-8
[17]	K. I. Isife, Positive solutions of a class of nonlinear boundary value fractional differential equations, Journal of Fractional Calculus and Nonlinear Systems, 2 (2021), 12–30.
[18]	J. R. Ockendon, A. B. Tayler, The dynamics of a current collection system for an electric locomotive, Proc. R. Soc. Lond. A, 322 (1971), 447–468. https://doi.org/10.1098/rspa.1971.0078 doi: 10.1098/rspa.1971.0078
[19]	M. Sher, K. Shah, M. Feckan, R. A. Khan, Qualitative analysis of multi-terms fractional order delay differential equations via the topological degree theory, Mathematics, 8 (2020), 218. https://doi.org/10.3390/math8020218 doi: 10.3390/math8020218
[20]	K. Balachandran, S. Kiruthika, J. J. Trujillo, Existence of solutions of nonlinear fractional pantograph equations, Acta Math. Sci., 33 (2013), 712–720. https://doi.org/10.1016/S0252-9602(13)60032-6 doi: 10.1016/S0252-9602(13)60032-6
[21]	D. Vivek, K. Kanagarajan, S. Harikrishnan, Existence and uniqueness results for pantograph equations with generalized fractional derivative, Journal Nonlinear Analysis and Application, 2 (2017), 105–112. https://doi.org/10.5899/2017/jnaa-00370 doi: 10.5899/2017/jnaa-00370
[22]	D. Vivek, K. Kanagarajan, S. Sivasundaram, Theory and analysis of nonlinear neutral pantograph equations via Hilfer fractional derivative, Nonlinear Studies, 24 (2017), 699–712.
[23]	F. Isaia, On a nonlinear integral equation without compactness, Acta Math. Univ. Comen., 75 (2006), 233–240.
[24]	Y. Guo, M. Chen, X.-B. Shu, F. Xu, The existence and Hyers-Ulam stability of solution for almost periodical fractional stochastic differential equation with fBm, Stoch. Anal. Appl., 39 (2021), 643–666. https://doi.org/10.1080/07362994.2020.1824677 doi: 10.1080/07362994.2020.1824677
[25]	Y. Guo, X.-B. 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 $1 < \beta < 2$, Bound. Value Probl., 2019 (2019), 59. https://doi.org/10.1186/s13661-019-1172-6 doi: 10.1186/s13661-019-1172-6
[26]	A. Khan, H. M. Alshehri, T. Abdeljawad, Q. M. Al-Mdallal, H. Khan, Stability analysis of fractional nabla difference COVID-19 model, Results Phys., 22 (2021), 103888. https://doi.org/10.1016/j.rinp.2021.103888 doi: 10.1016/j.rinp.2021.103888
[27]	J. Mawhin, Topological degree and boundary value problems for nonlinear differential equations, In: Topological methods for ordinary differential equations, Berlin, Heidelberg: Springer, 1993, 74–142. https://doi.org/10.1007/BFb0085076
[28]	O. Aberth, Computation of topological degree using interval arithmetic, and applications, Math. Comp., 62 (1994), 171–178. https://doi.org/10.1090/S0025-5718-1994-1203731-4 doi: 10.1090/S0025-5718-1994-1203731-4
[29]	L. Wang, X.-B. Shu, Y. Cheng, R. Cui, Existence of periodic solutions of second-order nonlinear random impulsive differential equations via topological degree theory, Results in Applied Mathematics, 12 (2021), 100215. https://doi.org/10.1016/j.rinam.2021.100215 doi: 10.1016/j.rinam.2021.100215
[30]	Y. Zhou, J. Wang, L. Zhang, Basic theory of fractional differential equations, 2 Eds., Singapore: World Scientific, 2016. https://doi.org/10.1142/10238
[31]	S. M. Ullam, Problems in modern mathematics (Chapter VI), Science Editors, New York: Wiley, 1940.
[32]	D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U. S. A., 27 (1941), 222–224. https://doi.org/10.1073/pnas.27.4.222 doi: 10.1073/pnas.27.4.222
[33]	T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300. https://doi.org/10.1090/S0002-9939-1978-0507327-1 doi: 10.1090/S0002-9939-1978-0507327-1
[34]	J. Wang, Y. Zhou, M. Feckan, Nonlinear impulsive problems for fractional differential equations and Ulam stability, Comput. Math. Appl., 64 (2012), 3389–3405. https://doi.org/10.1016/j.camwa.2012.02.021 doi: 10.1016/j.camwa.2012.02.021

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