SOME NEW STABILITY RESULTS OF VOLTERRA INTEGRAL EQUATIONS ON TIME SCALES

Year 2022, Volume: 4 Issue: 2, 44 - 54, 30.10.2022

Zeynep Kalkan , Aynur Şahin

https://doi.org/10.47087/mjm.1145159

Cited By: 2

Abstract

In this paper, we generalize two types of Volterra integral equations given on time scales and examine their Hyers-Ulam and Hyers-Ulam-Rassias stabilities. We also prove these stability results for the non-homogeneous nonlinear Volterra integral equation on time scales and provide an example to support these results. Moreover, we show that the general Volterra type integral equation given on time scales has the Hyers-Ulam-Rassias stability. Our results extend and improve some recent developments announced in the current literature.

Keywords

Hyers-Ulam stability, Hyers-Ulam-Rassias stability, Volterra integral equations, time scales

References

• S. Abbas and M. Benchohra,Ulam-Hyers stability for the Darboux problem for partial fractional differential and integro-differential equations via Picard operators . Results Math. 65, (2014), 67-79.
• S. D. Akgöl, Asymptotic equivalence of impulsive dynamic equations on time scales, Hacet. J. Math. Stat., (2022), 1-15.
• E. H. Alaa and A. G. Ghallab, Stability of a Volterra integral equation on time scales, (2017), arxiv:1701.01217v1 [math.DS].
• S. Andras and A. R. Meszaros, Ulam-Hyers stability of dynamic equations on time scales via Picard operators, Appl. Math. Comput. 219 no. 9, (2013), 4853-4864.
• J. H. Bae and K. W. Jun, On the generalized Hyers-Ulam-Rassias stability of a quadratic functional equation, Bull. Korean Math. Soc. 38 no. 2, (2001), 325-336.
• M. Bohner and A. Peterson, Dynamic Equations on Time Scales. An Introduction with Applications, Birkhäuser Boston, MA, 2001.
• M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser Boston, MA, 2003.
• A. Bielecki, Une remarque sur la méthode de Banach–Cacciopoli–Tikhonov dans la théorie des équations différentielles ordinaires, Bull. Polish Acad. Sci. Cl. III 4, (1956), 261–264.
• L. Cadariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4 no. 1, Article 4, (2003).
• M. Gachpazan and O. Baghani, Hyers-Ulam stability of nonlinear integral equation, Fixed Point Theory Appl., Article ID 927640, 6 pages, (2010).
• S. G. Georgiev and İ. M. Erhan, Adomian polynomials method for dynamic equations on time scales, Adv. Theory Nonlinear Anal. Appl., 5, no. 3, (2021), 300-315.
• S. Hilger, Analysis on Measure chain-A unified approach to continuous and discrete calculus, Results Math. 18, (1990 ), 18-56.
• L. Hua, Y. Li and J. Feng, On Hyers-Ulam stability of dynamic integral equation on time scales, Math. Aeterna 4, no. 6, (2014), 559-571.
• J. Huang, S. M. Jung and Y. Li, On Hyers-Ulam stability of nonlinear differential equations, Bull. Korean Math. Soc. 52, no. 2, (2015), 685-697.
• D. H. Hyers, On the stability of linear functional equation, Proc. Nat. Acad. Sci. USA, 27, no.4, (1941), 222-224.
• S. M. Jung, A fixed point approach to the stability of a Volterra integral equation, Fixed Point Theory Appl., Article ID 57064, 9 pages, (2007).
• T. M. Rassias, On the stability of linear mapping in Banach spaces, Proc. Am. Math. Soc. 72, (1978), 297-300.
• A. Reinfelds and S. Christian, Hyers-Ulam stability of Volterra type integral equations on time scales, Adv. Dyn. Syst. Appl. 15, no. 1, (2020), 39-48.
• I. A. Rus, Ulam stability of ordinary differential equations, Studia. Univ. Babe¸s-Bolyai, Math. 54, no. 4, (2009), 125-133.
• A. Şahin, H. Arısoy and Z. Kalkan, On the stability of two functional equations arising in mathematical biology and theory of learning, Creat. Math. Inform. 28 no. 1,(2019), 91-95.
• B. Sözbir, S. Altundağ and M. Başarır, On the (delta,f)-Lacunary statistical convergence of the functions, Maltepe J. Math. 1, no. 2, (2020), 1-8.
• B. Sözbir, S. Altundağ and M. Başarır, On the Δ_(Λ^2)^f-statistical convergence on product time scale, Univ. J. Math. Appl., 3, no. 4, (2020), 138-143.
• C. C. Tisdell and A. Zaidi, Basic qualitative and quantitative results for solutions to nonlinear dynamic equations on time scales with an application to economic modelling, Nonlinear Anal. 68, no. 11, (2008), 3504–3524.
• N. Tok and M. Başarır, On the λ_h^α -statistical convergence of the functions defined on the time scale, Proc. Inter. Math. Sci., 1, no. 1, (2019), 1-10.
• S. M. Ulam, A Collection of the Mathematical Problems, Interscience, New York, 1960.
• N. A. Yaseen, Hyers-Ulam-Rassias stability for Volterra integral equations on time scales, Journal of the ACS, 8, (2014), 33-44.