Ulam stability of Volterra integral equations on time scales via fixed point approach

Hifza Siddiqui; Sanket Tikare

doi:10.5269/bspm.78989

Hifza Siddiqui Ramniranjan Jhunjhunwala College, Mumbai, India
Sanket Tikare Ramniranjan Jhunjhunwala College

Abstract

This article is about the Ulam stability of Volterra integral equations on time scales. We present the Hyers--Ulam and Hyers--Ulam--Rassias stability by employing fixed point alternative on complete generalized metric spaces. An Example is provided to illustrate the effectiveness and benefit of the proven results.

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References

Agarwal, R. P., Hazarika, B., Tikare S. (eds.), Dynamic Equations on Time Scales and Applications, CRC Press Taylor & Francis Group, (2024).

Alghamdi, M. A., Aljehani, A., Bohner, M., Hamza, A.E., Hyers–Ulam and Hyers–Ulam–Rassias stability of first-order linear dynamic equations, Publ. Inst. Math., 109(123), 83-93, (2021).

Alghamdi, M. A., Alharbi, M., Bohner, M., Hamza, A. E., Hyers–Ulam and Hyers–Ulam–Rassias stability of first-order nonlinear dynamic equations, Qual. Theory Dyn. Syst., 20(45), (2021).

Anderson, D. R., Gates, B., Heuer, D., Hyers-Ulam stability of second-order linear dynamic equations on time scales, Commun. Appl. Anal, 16(3), 281-292, (2012).

Anderson, D. R., Ointsuka, M., Hyers–Ulam stability of first-order homogeneous linear dynamic equations on time scales, Demonstr. Math., 51(1), 198–210, (2018).

Anderson, D. R., Ointsuka, M., Hyers–Ulam stability of second-order linear dynamic equations on time scales, Acta Math. Sin., 41B(5), 1809–1826, (2021).

Andras, S., Meszaros, A. R., Ulam–Hyers stability of dynamic equations on time scales via Picard operators, Appl. Math. Comput., 209, 4853–4864 (2013).

Berger, H., Henrich, S., Jessee, J., Mikels, B., Mullen, J., Meyer, C.K., Beresford, D., Application of time scales calculus to the growth and development in populations of Stomoxys calcitrans (Diptera: Muscidae), Int. J. Difference Equ., 8(2), 125–134, (2013).

Bohner, M., Heim, J., Liu, A., Solow models on time scales, Cubo, 15(1), 13-32, (2013).

Bohner, M., Heim, J., Liu, A., Qualitative analysis of a Solow model on time scales, J. Concr. Appl. Math., 13(3-4), 183-197, (2015).

Bohner, M., Georgiev, S.G., Multivariable Dynamic Calculus on Time Scales, Springer, 2016.

Bohner, M., Mesquita, J., Streipert, S., The Beverton–Hold model on isolated time scales, Math. Biosci. Eng., 19(11), 11693-11716, (2022).

Bohner, M., Peterson, A., Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser, Boston, (2001).

Bohner, M., Peterson, A. (eds.), Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, (2003).

Bohner, M., Tikare, S., Ulam stability for first-order nonlinear dynamic equations, Sarajevo J. Math., 18 (31)(1), 1-15, (2022).

Bohner, M., Warth, H., The Beverton–Holt dynamic equation, Appl. Anal., 86(8), 1007–1015, (2007).

Diaz, J. B., Margolis, B., A fixed point theorem of the alternative, for contarctions on a generalized complete metric space, Bull. Amer. math. Soc. 74, 305-309, (1968).

Georgiev, S. G., Dynamic Calculus and Equations on Time Scales, De Gruyter, 2023.

Girejko, E., Machado, L., Malinowska, A. B., Martins, N., On consensus in the Cucker–Smale type model on isolated time scales, Discrete Contin. Dyn. Syst. Ser. S, 11(1), 77-89, (2018).

Guzowska, M., Malinowska, A. B., Ammi, M. R. S., Calculus of variations on time scales: applications to economic models, Adv. Differ. Equ., 2015, 203, (2015).

Hamza, A. E., Ghallab, A. G., Stability of a Volterra integral equation on time scales, arxiv:1701.0121v1 [math. DS], (2017).

Hua, L., Li, Y., Feng, J., On Hyers-Ulam stability of dynamic integral equation on time scales, Math. AEterna, 4(6), 559-571, (2014).

Hyers, D. H., On the stability of linear functional equation, Proc. Nat. Acad. Sci. USA 27(4), 222–224, (1941).

Lobo, J. Z., Tikare, S., Ulam stability for integro-dynamic equations on time scales, J. Nonlinear Convex Anal., 25(9), 2173-2183, (2024).

Malinowska, A. B., Torres, D.F.M., Necessary and sufficient conditions for local Pareto optimality on time scales, J. Math. Sci, 161(6), 803-810, (2009).

Rassias, Th. M., On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72(2), 297–300, (1978).

Reinfelds, A., Christian, S., Hyers–Ulam stability of Volterra type integral equations on time scales, Adv. Dyn. Syst. Appl, 15(1), 39-48, (2020).

Scindia, P., Tikare, S., El-Deeb, A. A., Ulam stability of first-order nonlinear impulsive dynamic equations, Bound. Value Probl., 2023 (86), (2023).

Shah, S. O., Tikare, S., Osman, M., Ulam type stability results of nonlinear impulsive Volterra–Fredholm integro-dynamic adjoint equations on time scale, Math., 11, 4498, (2023).

Shah, S. O., Zada, A., The Ulam stability of non-linear Volterra integro-dynamic equations on time scales, Note Mat., 39(2), 57-69, (2019).

Shen, Y., The Ulam stability of first order linear dynamic equations on time scales, Results Math., 72, 1881–1895 (2017).

Ulam, S. M., A Collection of the Mathematical Problems, Interscience, New York, (1960).