A Qualitative Study on Fractional Logistic Integro-differential Equations in an Arbitrary Time Scale
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Authors: N. SARKAR, M. SEN, D. SAHA AND B. HAZARIKA
DOI: 10.46793/KgJMat2603.403S
Abstract:
This manuscript deals with the investigation related to uniqueness and existence of solution of fractional order nonlinear pantograph integro-differential equation in arbitrary time scale. The fractional derivatives are defined in Riemann-Liouville sense, the primary tools are taken as Banach contraction principle and Schauder’s fixed point theory to establish the theoretical outcomes. Finally, we give examples to show the efficiency of our results.
Keywords:
Fractional integro-differential equation, logistic equation, time scale calculus, fixed point theory.
References:
[1] S. Abbas, M. Benchohra and G. M. N’Guerekata, Topics in Fractional Differential Equations, Developments in Mathematics 27, Springer, New York, 2012.
[2] A. Aghajani, Y. Jalilian and J. J. Trujillo, On the existence of solutions of fractional integro-differential equations, Fract. Calc. Appl. Anal. 15(1) (2012), 44–69.
[3] B. Ahmad and S. Sivasundaram, Some existence results for fractional integro-differential equations with nonlinear conditions, Commun. Appl. Anal. 12(2) (2008), 107–112.
[4] A. Ahmadkhanlu and M. Jahanshahi, On the existence and uniqueness of solution of initial value problem for fractional order differential equations on time scales, Bull. Iranian Math. Soc. 38(1) (2012), 241–252.
[5] A. Alkhazzan, J. Wang, C. Tunç, X. Ding, Z. Yuan and Y. Nie, On existence and continuity results of solution for multi-time scale fractional stochastic differential equation, Qual. Theory Dyn. Syst. 22 (2023), Article ID 49. https://doi.org/10.1007/s12346-023-00750-x
[6] M. Andersen, Properties of some density-dependent integrodifference equation population models, Math. Biosci. 104(1) (1991), 135–157.
[7] K. Balachandran, S. Kiruthika and J. J. Trujillo, Existence results for fractional impulsive integrodifferential equations in Banach spaces, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), 1970–1977.
[8] D. Baleanu, S. Etemad and Sh. Rezapour, On a fractional hybrid integro-differential equation with mixed hybrid integral boundary value conditions by using three operators, Alexandria Engineering Journal 59(5) (2020), 3019–3027.
[9] N. Balkani, R. H. Haghi and S. Rezapour, Approximate solutions for a fractional q-integro-difference equation, Journal of Mathematical Extension 13(3) (2019), 109–120.
[10] N. Benkhettou, A. Hammoudi and D. F. M. Torres, Existence and uniqueness of solution for a fractional Riemann-Liouville initial value problem on time scales, Journal of King Saud University 28(1) (2016), 87–92.
[11] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Application, Bikhauster, Boston, MA, 2001.
[12] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Bikhauser Boston, 2004.
[13] H. V. S. Chauhan, B. Singh, C. Tunç and O. Tunç, On the existence of solutions of non-linear 2D Volterra integral equations in a Banach Space, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116 (2022), 11 pages. https://doi.org/10.1007/s13398-022-01246-0
[14] J. M. Cushing, Integro-Differential Equations and Delay Models in Population Dynamics, Springer, Berlin, Heidelberg, 2013.
[15] A. M. A. El-Sayed, A. E. M. El-Mesiry and H. A. A. El-Saka, On fractional-order logistic equation, Appl. Math. Lett. 20(7) (2007), 817–823.
[16] M. Fazly and M. Hesaaraki, Periodic solutions for predator-prey systems with Beddington-DeAngelis functional response on time scales, Nonlinear Anal. Real. 9(3) (2008), 1224–1235.
[17] J. Gao, Q. Wang and L. Zhang, Existence and stability of almost-periodic solutions for cellular neural networks with time-varying delays in leakage terms on time scales, Appl. Math. Comput. 237 (2014), 639–649.
[18] E. Girejko and D. F. M. Torres, The existence of solutions for dynamic inclusions on time scales via duality, Appl. Math. Lett. 25(11) (2012), 1632–1637.
[19] D. Gupta, M. Sen, N. Sarkar and B. Miah, A qualitative investigation on Caputo fractional neutral VF integro differential equation and its uniform stability, J. Anal. (2023). https://doi.org/10.1007/s41478-023-00604-4
[20] S. Hilger, Analysis on measure chains-a unified approach to continuous and discrete calculus, Result. Math. 18 (1990), 18–56.
[21] S. Hilger, Differential and difference calculus-unified, Nonlinear Anal. 30(5) (1997), 2683–2694.
[22] S. B. Hsu and X. Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal. 40(2) (2008), 776–789.
[23] A. Iserles and Y. Liu, On pantograph inregro-differential equations, J. Integral Equ. Appl. 6(2) (1994), 213–237.
[24] M. M. Khader, N. H. Sweilam and B. N. Kharrat, Numerical simulation for solving fractional Riccati and logistic differential equations as a difference equation, Appl. Appl. Math. 15(1) (2020), 655–663.
[25] N. A. Khan, A. Ara and M. Jamil, Approximations of the nonlinear Volterra’s population model by an efficient numerical method, Math. Methods Appl. Sci. 34(14) (2011), 1733–1738.
[26] N. A. Khan, A. Mahmood, N. A. Khan and A. Ara, Analytical study of nonlinear fractional-order integro-differential equations: Revisit Volterra’s population model, Int. J. Diff. Equ. (2012), Article ID 845945. https://doi.org/10.1155/2012/845945
[27] Y. Li, L. Zhao and P. Liu, Existence and exponential stability of periodic solution of high-order Hopfield neural network with delays on time scales, Discrete Dyn. Nat. Soc. (2009), Article ID 573534. https://doi.org/10.1155/2009/573534
[28] E. Messina and A. Vecchio, Stability analysis of linear Volterra equations on time scales under bounded perturbations, Appl. Math. Lett. 59 (2016), 6–11.
[29] Y. Muroya, E. Ishiwata and H. Brunner, On the attainable order of collocation methods for pantograph integro-differential equations, J. Comput. Appl. Math. 152(1–2) (2003), 347–366.
[30] D. Saha, M. Sen, N. Sarkar and S. Saha, Existence of a solution in the Holder space for a nonlinear functional integral equation, Armenian J. Math. 12(7) (2020), 1–8.
[31] N. Sarkar and M. Sen, An investigation on existence and uniqueness of solution for integro differential equation with fractional order, J. Physics: Conf. Series 1849(1) (2021), Article ID 012011. https://doi:10.1088/1742-6596/1849/1/012011
[32] A. Souahi, A. Guezane-Lakoud and R. Khaldi, On some existence and uniqueness results for a class of equations of order 0 < α ≤ 1 on arbitrary time scales, Int. J. Differ. Equ. 8 (2016), Article ID 7327319. https://doi:10.1155/2016/7327319
[33] O. Tunç and C. Tunç, Solution estimates to Caputo proportional fractional derivative delay integro-differential equations, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117 (2023), Aricle ID 12. https://doi.org/10.1007/s13398-022-01345-y
[34] C. Wang, Existence and exponential stability of piecewise mean-square almost periodic solutions for impulsive stochastic Nicholson’s blowflies model on time scales, Appl. Math. Comput. 248 (2014), 101–112.
[35] Q. Wang and Z. Liu, Existence and stability of positive almost periodic solutions for a competitive system on time scales, Math. Comput. Simulat. 138 (2017), 65–77.
[36] H. Xu, Analytical approximations for a population growth model with fractional order, Commun. Nonlinear Sci. Numer. Simul. 14(5) (2009), 1978–1983.
[37] A. Zammit-Mangion and C. K. Wikle, Deep integro-difference equation models for spation-temporal forecasting, Spatial Statistics 37 (2020), Article ID 1004008. https://doi.org/10.1016/j.spasta.2020.100408
[38] Z. Zhang, D. Hao and D. Zhou, Global asymptotic stability by complex-valued inequalities for complex-valued neural networks with delays on period time scales, Neurocomputing 219(5) (2017), 494–501.
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