Existence, Uniqueness, and Ulam Stability of Conformable Integro-Dynamic Equations with Deviating Arguments on Time Scales
DOI:
https://doi.org/10.15377/2409-5761.2026.13.5Keywords:
Ulam–Hyers stability, Integro-dynamic equations, Conformable calculus on time scales, Nonlinear functional dynamic equations, Fixed point theory (Banach and Schauder).Abstract
In this paper, we investigate a class of nonlinear conformable integro-dynamic equations with deviating arguments on time scales. By transforming the considered problem into an equivalent integral equation, sufficient conditions for the existence and uniqueness of solutions are established using Schauder’s and Banach’s fixed point theorems. Furthermore, Hyers–Ulam and generalized Hyers–Ulam stability results are derived under suitable assumptions. The obtained results extend several existing studies on nonlinear integro-dynamic equations within the framework of conformable calculus on time scales.
AMS Subject Classifications: 34N05, 39A12, 34A08, 47H10.
References
[1] Hilger S. Analysis on measure chains-A unified approach to continuous and discrete calculus. Results Math. 1990; 18(1-2): 18-56. https://doi.org/10.1007/BF03323153
[2] Agarwal RP, Hazarika B, Tikare S, editors. Dynamic Equations on Time Scales and Applications. Boca Raton (FL): CRC Press; 2024. https://doi.org/10.1201/9781003467908
[3] Bohner M, Peterson AC. Dynamic Equations on Time Scales: An Introduction with Applications. Boston (MA): Birkhuser; 2001. https://doi.org/10.1007/978-1-4612-0201-1
[4] Bohner M, Peterson AC. Advances in Dynamic Equations on Time Scales. Boston (MA): Birkhuser; 2003. https://doi.org/10.1007/978-0-8176-8230-9
[5] Khuddush M, Prasad KR. Positive almost periodic solutions for a time scale model of fishery with time varying variable delays and harvesting term. J Appl Nonlinear Dyn. 2023; 12(2): 257-71. https://doi.org/10.5890/JAND.2023.06.005
[6] Prasad KR, Khuddush M. Existence and global exponential stability of positive almost periodic solutions for a time-scales model of hematopoiesis with multiple time-varying variable delays. Int J Differ Equ. 2019; 14(2): 149-67. https://doi.org/10.37622/IJDE/14.2.2019.149-167
[7] Bohner M, Hassan TS, Li T. Fite-Hille-Wintner-type oscillation criteria for second-order half-linear dynamic equations with deviating arguments. Indag Math (N S). 2018; 29(2): 548-60. https://doi.org/10.1016/j.indag.2017.10.006
[8] Bohner M, Li T, Saker SH, Candan T, Mert R. Kamenev-type criteria for nonlinear damped dynamic equations. Sci China Math. 2015; 58(7): 1445-52. https://doi.org/10.1007/s11425-015-4974-8
[9] Affan SE, Elabbasy EM, El-Matary BM, Hassan TS, Hassan AM. Second-order advanced dynamic equations on time scales: Oscillation analysis via monotonicity properties. AIMS Math. 2025; 10(2): 4473-91. https://doi.org/10.3934/math.2025206
[10] Georgiev SG, Raffoul YN, Tikare S. Oscillatory and nonoscillatory behavior of nonlinear delay dynamic equations on time scales. Electron Res Arch. 2026; 34(5): 3516-29. https://doi.org/10.3934/era.2026157
[11] Hassan TS, Mesmouli MB, Iambor LF, Glalah BA. Kneser-type oscillation criteria for half-linear third-order dynamic equations on time scales. Mathematics. 2026; 14(11): 1966. https://doi.org/10.3390/math14111966
[12] Sikorska-Nowak A. Integro-differential equations on time scales with Henstock-Kurzweil delta integrals. Discuss Math Differ Incl Control Optim. 2011; 31(1): 71-90. https://doi.org/10.7151/dmdico.1128
[13] Sui Y. Oscillation criteria for the second-order neutral advanced dynamic equations on time scales. Appl Math Lett. 2025; 166: 109543. https://doi.org/10.1016/j.aml.2025.109543
[14] Xing Y, Han M, Zheng G. Initial value problem for first-order integro-differential equation of Volterra type on time scales. Nonlinear Anal. 2005; 60(3): 429-42. https://doi.org/10.1016/S0362-546X(04)00369-4
[15] Liu G, Xiang X, Peng Y. Nonlinear integro-differential equations and optimal control problems on time scales. Comput Math Appl. 2011; 61(2): 155-69. https://doi.org/10.1016/j.camwa.2010.10.013
[16] Ahmed AA, Hazarika B. On abstract nonlinear integro-dynamic equations in time scale. arXiv preprint arXiv:2404.11616; 2024.
[17] Shen Y. The Ulam stability of first-order linear dynamic equations on time scales. Results Math. 2017; 72(4): 1881-95. https://doi.org/10.1007/s00025-017-0725-1
[18] Hamza A, Oraby M. Stability of abstract dynamic equations on time scales. Adv Differ Equ. 2012; 2012: 143. https://doi.org/10.1186/1687-1847-2012-143
[19] Mil'ce A. Asymptotically almost automorphic solutions for some integro-dynamic equations with nonlocal initial conditions on time scales. Dyn Contin Discrete Impuls Syst Ser A Math Anal. 2016; 23: 27-46.
[20] Bohner M, Scindia PS, Tikare S. Qualitative results for nonlinear integro-dynamic equations via integral inequalities. Qual Theory Dyn Syst. 2022; 21: 106. https://doi.org/10.1007/s12346-022-00636-4
[21] Khuddush M, Vali SK, Ramakrishna MV. Qualitative results for nonlinear iterative integro-dynamic equation with deviating arguments on time scales. J Anal. 2025; 33: 2367-88. https://doi.org/10.1007/s41478-025-00937-2
[22] Lauran M. Existence results for some differential equations with deviating argument. Filomat. 2011; 25: 21-31. https://doi.org/10.2298/FIL1102021L
[23] Anderson DR, Georgiev SG. Conformable Dynamic Equations on Time Scales. Boca Raton (FL): Chapman & Hall/CRC; 2021. https://doi.org/10.1201/9781003057406
[24] Bayour B, Hammoudi A, Torres DFM. A truly conformable calculus on time scales. Glob Stoch Anal. 2018; 5(1): 1-14.
[25] Khalil R, Al Horani M, Yousef A, Sababheh M. A new definition of fractional derivative. J Comput Appl Math. 2014; 264: 65-70. https://doi.org/10.1016/j.cam.2014.01.002
Downloads
Published
Issue
Section
License
Copyright (c) 2026 Mahammad Khuddush
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
All the published articles are licensed under the terms of the Creative Commons Attribution Non-Commercial License (CC BY-NC 4.0) which permits unrestricted, non-commercial use, distribution and reproduction in any medium, provided the work is properly cited.
