Existence and Controllability for Second-Order Functional Integro-Differential Equations with Infinite Delay and Random Effects

Abstract

This article explores the existence and controllability results within the classes of second-order functional integro-differential equations with infinite delay. The study uses a Schauder fixed-point theorem that works in a stochastic domain to show that there are mild solutions to this problem. Additionally, rigorous proofs outlining the controllability of these problems despite the complexities introduced by infinite delay are provided. As an example of how the theory can be used in real life, an example is given to show that the established theory works for dealing with systems with infinite delay in a controlled environment.

Keywords

Integro-differential equation / Schauder fixed point / Mild solution / Infinite delay / Semigroup theory / Controllability / 45J05 / 34K30 / 47G20 / 34K20 / 93B05

References

[1] Ahmed NU. Semigroup Theory with Applications to Systems and Control, 1991, New York, Wiley

[2] Baghli S, Benchohra M. Existence results for semilinear neutral functional differential equations involving evolution operators in Frechet spaces. Georgian Math. J., 2010, 17: 423-436

[3] Balachandran, K.: Controllability of nonlinear systems with delays in both state and control variables. Kybernetika 22(4), 340–348 (1986). http://dml.cz/dmlcz/124373

[4] Balachandran K, Park DG, Anthoni SM. Existence of solutions of abstract nonlinear second order neutral functional integrodifferential equations. Comput. Math. Appl., 2005, 46: 1313-1324

[5] Dhage BC, Ntouyas SK. Existence and attractivity results for nonlinear first order random differential equations. Opuscula Math., 2010, 30: 411-429

[6] Dhakne MB, Kucche KD. Second order Volterra-Fredholm functional integrodifferential equations. Malaya J. Math., 2012, 1(1): 1-8

[7] Engl HW. A general stochastic fixed-point theorem for continuous random operators on stochastic domains. J. Math. Anal. Appl., 1978, 66: 220-231

[8] Gou, H., Li, Y.: A study on controllability of impulsive fractional evolution equations via resolvent operators. Boundary Value Problems. 2021, 25 (2021). https://doi.org/10.1186/s13661-021-01499-5

[9] Gunasekar, T., Madhumitha, S., Prakaash, A.S., Sakthi, R., Ganapathy, G., Suba, M.: Existence and approximate controllability of random impulsive neutral functional differential equation with finite delay. Commun. Appl. Nonlinear Anal. 32(1), 35–55 (2025)

[10] Gunasekar T, Raghavendran P, Nisar KS. Existence, uniqueness, and stability results of fractional Volterra-Fredholm integro-differential equations with state-dependent delay. Qualit. Theory Dyn. Syst., 2025, 24: 54

[11] Gunasekar T, Raghavendran P, Santra SS, Sajid M. Existence and controllability results for neutral fractional Volterra-Fredholm integro-differential equations. J. Math. Comput. Sci., 2024, 34(4): 361-380

[12] Hale J, Kato J. Phase space for retarded equations with infinite delay. Funkcial Ekvac, 1978, 21: 11-41

[13] Hino Y, Murakami S, Naito T. Functional Differential Equations with Unbounded Delay, 1991, Berlin, Springer

[14] Lupulescu V, Lungan C. Random integral equations on time scales. Opuscula Math., 2013, 33: 323-335

[15] Madhumitha S, Gunasekar T, Alsinai A, Raghavendran P, Qahtan RSA. Existence and controllability for second-order functional differential equations with infinite delay and random effects. Int. J. Differ. Equ., 2024, 2024: 1-9

[16] Pachpatte, B. G.: Integral and Finite Difference Inequalities and Applications. North-Holland Mathematics Studies, Vol. 205. Elsevier Science B.V., Amsterdam (2006)

[17] Pavlackova M, Taddei V. Mild solutions of second-order semilinear impulsive differential inclusions in Banach spaces. Mathematics, 2022, 10(4672

[18] Raghavendran P, Gunasekar T, Ahmad J, Emam W. A study on the existence, uniqueness, and stability of fractional neutral Volterra-Fredholm integro-differential equations with state-dependent delay. Fract. Fract., 2024, 9(120

[19] Raghavendran P, Gunasekar T, Gochhait S. Application of artificial neural networks for existence and controllability in impulsive fractional Volterra-Fredholm integro-differential equations. Appl. Math. Sci. Eng., 2024, 32(12436440

[20] Raghavendran P, Gunasekar T, Santra SS, Baleanu D, Majumder D. Analytical study of existence, uniqueness, and stability in impulsive neutral fractional Volterra-Fredholm equations. J. Math. Comput. Sci., 2024, 38: 313-329

[21] Shaikhet, L.: Stability of stochastic differential equations with distributed and state-dependent delays. J. Appl. Math. Comput. 4(4), 181–188 (2020). https://doi.org/10.26855/jamc.2020.12.009

[22] Shen, G., Sakthivel, R., Ren, Y., Li, M.Y.: Controllability and stability of fractional stochastic functional systems driven by Rosenblatt process. Collectanea Mathematica 71, 63–82 (2020). https://doi.org/10.1007/s13348-019-00248-3

[23] Underwood RG, Young DF. Null controllability of nonlinear functional differential equations. SIAM J. Control Optim., 1979, 17(6753-772

[24] Yang H, Han X. Existence and uniqueness of periodic solutions for a class of higher order differential equations. Mediterranean J. Math., 2023, 20(5282

RIGHTS & PERMISSIONS

Shanghai University

Just Accepted

This article has successfully passed peer review and final editorial review, and will soon enter typesetting, proofreading and other publishing processes. The currently displayed version is the accepted final manuscript. The officially published version will be updated with format, DOI and citation information upon launch. We recommend that you pay attention to subsequent journal notifications and preferentially cite the officially published version. Thank you for your support and cooperation.