On fractional Volterra integrodifferential equations with fractional integrable impulses
Abstract
We consider a class of nonlinear fractional Volterra integrodifferential equation with fractional integrable impulses and investigate the existence and uniqueness results in the Bielecki’s normed Banach spaces. Further, Bielecki-Ulam type stabilities have been demonstrated on a compact interval. A concrete example is provided to illustrate the outcomes we acquired.
Keyword : fractional Volterra integrodifferential equation, integrable impulses, Banach contraction principle, existence of solutions, Bielecki norm, Bielecki-Ulam type stability
How to Cite
Sutar, S. T., & Kucche, K. D. (2019). On fractional Volterra integrodifferential equations with fractional integrable impulses. Mathematical Modelling and Analysis, 24(3), 457-477. https://doi.org/10.3846/mma.2019.028
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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A. Zada and S. Ali. Stability analysis of multi-point boundary value problem for sequential fractional differential equations with noninstantaneous impulses. Int. J. Nonlinear Sci. Numer. Simul., 19(7):763–774, 2018. https://doi.org/10.1515/ijnsns-2018-0040
A. Zada, W. Ali and C. Park. Ulam’s type stability of higher order nonlinear delay differential equations via integral inequality of Gronwall-Bellman-Bihari’s type. Appl. Math. Comput., 350:60–65, 2019. https://doi.org/10.1016/j.amc.2019.01.014
A. Zada and S.O. Shah. Hyers-Ulam stability of first-order non-linear delay dfferential equations with fractional integrable impulses. Hacettepe J. Math. Stat., 47(5):1196–1205, 2018. https://doi.org/10.15672/HJMS.2017.496
A. Zada, S. Shaleena and T. Li. Stability analysis of higher order nonlinear dfferential equations in β-normed spaces. Math. Meth. App. Sci., 42(4):1151– 1166, 2019. https://doi.org/10.1002/mma.5419
A. Zada, M. Yar and T. Li. Existence and stability analysis of nonlinear sequential coupled system of Caputo fractional differential equations with integral boundary conditions. Ann. Univ. Paedagog. Crac. Stud. Math., 417:103–125, 2018. https://doi.org/10.2478/aupcsm-2018-0009
M. Benchora, J. Hendroson and S. Ntouyas. Impulsive differential equations and inclusions, volume 2. Contemporary mathematics and its applications, New York, NY, USA; Hindawi, 2006. https://doi.org/10.1155/9789775945501
N. Eghbali, V. Kalvandi and J.M. Rassias. A fixed point approach to the MittagLeffler-Hyers-Ulam stability of a fractional integral equation. Open Mathematics, 14:237–246, 2016. https://doi.org/10.1515/math-2016-0019
M. Frigon and D. O’Regan. Existence results for first order impulsive differential equations. J. Math. Anal. Appl., 193:96–113, 1995. https://doi.org/10.1006/jmaa.1995.1224
M. Frigon and D. O’Regan. Impulsive differential equations with variable time. Nonlinear Analysis: TMA, 26:9113–9122, 1996. https://doi.org/10.1016/0362-546X(95)00053-X
M. Frigon and D. O’Regan. First order impulsive initial and periodic problems with variable moments. J. Math. Anal. Appl., 233(2):730–739, 1999. https://doi.org/10.1006/jmaa.1999.6336
I. Podlubny. Fractional differential equations. Academic Press, San Diego, 1999.
A.P. Prudnikov, Y.A. Brychkov and O.I. Marichev. Integral and series, Elementary Functions, volume I. Nauka, Moscow, 1981.
A.M. Samoilenko and N.A. Perestyuk. Impulsive differential equations, volume 14. World Scientific series on nonlinear science, Singapore, 1995. https://doi.org/10.1142/9789812798664
S.M. Ulam. A collection of the mathematical problems. Interscience Publ., New York, 1960.
J. Wang and X. Li. eα-Ulam type stability of fractional order ordinary differential equations. J. Appl. Math. Comput., 45:449–459, 2014. https://doi.org/10.1007/s12190-013-0731-8
J. Wang, Z. Lin and Y. Zhou. On the stability of new impulsive ordinary differential equations. Topological Methods in Nonlinear Analysis, 46(1):303–314, 2015. https://doi.org/10.12775/TMNA.2015.048
J. Wang, L. Lv and Y. Zhou. Ulam stability and data dependence for fractional differential equations with Caputo derivative. Qualit. Th. Diff. Equat., 63:1–10, 2011. https://doi.org/10.14232/ejqtde.2011.1.63
J. Wang, L. Lv and Y. Zhou. New concepts and results in stability of fractional differential equations. Commun. Nonlinear Sci. Numer. Simulat., 17:2530–2538, 2012. https://doi.org/10.1016/j.cnsns.2011.09.030
J. Wang, A. Zada and W. Ali. Differential equations with variable delay in quasi-Banach spaces. Int. J. Nonlinear Sci. Num., 19(5):553–560, 2018. https://doi.org/10.1515/ijnsns-2017-0245
J. Wang and Y. Zhang. A class of nonlinear differential equations with fractional integrable impulses. Commun. Nonlinear Sci. Numer. Simulat., 19:3001–3010, 2014. https://doi.org/10.1016/j.cnsns.2014.01.016
J. Wang and Y. Zhang. Existence and stability of solutions to nonlinear impulsive differential equations in β-normed spaces. Electronic J. Differential Equations, 83:1–10, 2014.
J. Wang and Y. Zhang. Ulam-Hyers-Mittag-Leffler stability of fractionalorder delay differential equations. Optimization, 63(8):1181–1190, 2014. https://doi.org/10.1080/02331934.2014.906597
Wei Wei, Xuezhu Li and Xia Li. New stability results for fractional integral equation. Computers and Mathematics with Applications, 64:3468–3476, 2012. https://doi.org/10.1016/j.camwa.2012.02.057
X.Wang, M. Arif and A. Zada. β-Hyers-Ulam-Rassias stability of semilinear nonautonomous impulsive system. Symmetry, 11(2):231, 2019. https://doi.org/10.3390/sym11020231
A. Zada and S. Ali. Stability analysis of multi-point boundary value problem for sequential fractional differential equations with noninstantaneous impulses. Int. J. Nonlinear Sci. Numer. Simul., 19(7):763–774, 2018. https://doi.org/10.1515/ijnsns-2018-0040
A. Zada, W. Ali and C. Park. Ulam’s type stability of higher order nonlinear delay differential equations via integral inequality of Gronwall-Bellman-Bihari’s type. Appl. Math. Comput., 350:60–65, 2019. https://doi.org/10.1016/j.amc.2019.01.014
A. Zada and S.O. Shah. Hyers-Ulam stability of first-order non-linear delay dfferential equations with fractional integrable impulses. Hacettepe J. Math. Stat., 47(5):1196–1205, 2018. https://doi.org/10.15672/HJMS.2017.496
A. Zada, S. Shaleena and T. Li. Stability analysis of higher order nonlinear dfferential equations in β-normed spaces. Math. Meth. App. Sci., 42(4):1151– 1166, 2019. https://doi.org/10.1002/mma.5419
A. Zada, M. Yar and T. Li. Existence and stability analysis of nonlinear sequential coupled system of Caputo fractional differential equations with integral boundary conditions. Ann. Univ. Paedagog. Crac. Stud. Math., 417:103–125, 2018. https://doi.org/10.2478/aupcsm-2018-0009