Abstract
In this research, we introduce the concept of \(E_\alpha \)-type stability for fractional integro-differential equations with uncertainty. We propose different types of fuzzy \(E_\alpha \) stabilities for some classes of fuzzy integro-differential equations of fractional order. Besides, we present some new findings on the existence and uniqueness of the solutions of fuzzy integro-differential equations of fractional order using the proposed new concept.
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Ulam, S.M.: A Collection of Mathematical Problems. Interscience, New York (1960)
Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. 27, 222–224 (1941)
Rassias, Th.M.: On the stability of linear mappings in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)
Hyers, D.H., Isac, G., Rassias, Th.M.: Stability of Functional Equations in Several Variables. Birkhäuser, Basel (1998)
Jung, S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor (2001)
Jung, S.-M.: Hyers–Ulam stability of linear differential equations of first order. Appl. Math. Lett. 17, 1135–1140 (2004)
Wang, G., Zhou, M., Sun, L.: Hyers–Ulam stability of linear differential equations of first order. Appl. Math. Lett. 21, 1024–1028 (2008)
Rezaei, H., Jung, S.-M., Rassias, Th.M.: Laplace transform and Hyers–Ulam stability of linear differential equations. J. Math. Anal. Appl. 403, 244–251 (2013)
Abdollahpour, M.R., Aghayari, R., Rassias, Th.M.: HyersUlam stability of associated Laguerre differential equations in a subclass of analytic functions. J. Math. Anal. Appl. 437, 605–612 (2016)
Huang, J., Li, Y.: Hyers-Ulam stability of linear functional differential equations. J. Math. Anal. Appl. 426, 1192–1200 (2015)
Kim, B., Jung, S.M.: Bessel’s differential equation and its Hyers–Ulam stability. J. Inequalities Appl. 2007(1), 1–8 (2007)
Kim, S.S., Cho, Y.J., Eshaghi Gordji, M.: On the generalized Hyers–Ulam–Rassias stability problem of radical functional equations. J. Inequalities Appl. 2012(1), 1–13 (2012)
Li, C.P., Zhang, F.R.: A survey on the stability of fractional differential equations. Eur. Phys. J. Spec. Top. 193, 27–47 (2011)
Li, Y., Chen, Y., Podlubny, I.: Mittag–Leffler stability of fractional order nonlinear dynamic systems. Automatica 45, 19651969 (2009)
Li, Y., Chen, Y., Podlubny, I.: Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittageffler stability. Comput. Math. Appl. 59, 1810–1821 (2010)
Wang, J., Lv, L., Zhou, Y.: Ulam stability and data dependence for fractional differential equations with Caputo derivative. Electron. J. Qual. Theory Differ. Equ. 63, 1–10 (2011)
Wang, J., Lv, L., Zhou, Y.: New concepts and results in stability of fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 17, 2530–2538 (2012)
Wang, J., Zhou, Y., Fečkan, M.: Nonlinear impulsive problems for fractional differential equations and Ulam stability. Comput. Math. Appl. 64, 3389–3405 (2012)
Wang, J., Li, X.: \(E_\alpha \)-Ulam type stability of fractional order ordinary differential equations. J. Appl. Math. Comput. 45, 449–459 (2014)
Agarwal, R.P., Lakshmikantham, V., Nieto, J.J.: On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal. 72, 2859–2862 (2010)
Allahviranloo, T., Gouyandeh, Z., Armand, A.: Fuzzy fractional differential equations under generalized fuzzy Caputo derivative. J. Intell. Fuzzy Syst. 26, 1481–90 (2014)
Allahviranloo, T., Salahshour, S., Abbasbandy, S.: Explicit solutions of fractional differential equations with uncertainty. Soft Comput. 16, 297–302 (2012)
Salahshour, S., Allahviranloo, T., Abbasbandy, S., Baleanu, D.: Existence and uniqueness results for fractional differential equations with uncertainty. Adv. Differ. Equ. 2012, 112 (2012)
Mazandarani, M., Vahidian Kamyad, A.: Modified fractional Euler method for solving fuzzy fractional initial value problem. Commun. Nonlinear Sci. Numer. Simul. 18, 12–21 (2013)
Mazandarani, M., Najariyan, M.: Type-2 fuzzy fractional derivatives. Commun. Nonlinear Sci. Numer. Simul. 19, 2354–2372 (2014)
Salahshour, S., Allahviranloo, T., Abbasbandy, S.: Solving fuzzy fractional differential equations by fuzzy Laplace transforms. Commun. Nonlinear. Sci. Numer. Simul. 17, 1372–1381 (2012)
Ahmadian, A., Chang, C.S., Salahshour, S.: Fuzzy approximate solutions to fractional differential equations under uncertainty: operational matrices approach. IEEE Trans. Fuzzy Syst. https://doi.org/10.1109/TFUZZ.2016.2554156
Ahmadian, A., Salahshour, S., Amirkhani, H., Baleanu, D., Yunus, R.: An efficient Tau method for numerical solution of a fuzzy fractional kinetic model and its application to oil palm frond as a promising source of xylose. J. Comput. Phys. 264, 562–564 (2015)
Malinowski, M.T.: Random fuzzy fractional integral equations-theoretical foundations. Fuzzy Sets Syst. (In press). https://doi.org/10.1016/j.fss.2014.09.019
Hoa, N.V.: Fuzzy fractional functional differential equations under Caputo gH-differentiability. Commun. Nonlinear Sci. Numer. Simul. 22, 1134–1157 (2015)
Hoa, N.V.: Fuzzy fractional functional integral and differential equations. Fuzzy Sets Syst. (In press). https://doi.org/10.1016/j.fss.2015.01.009
Mirmostafaee, A.K., Moslehian, M.: Fuzzy versions of Hyers–Ulam–Rassias theorem. Fuzzy Sets Syst. 159, 720–729 (2008)
Mirmostafaee, A.K., Mirzavaziri, M., Moslehian, M.S.: Fuzzy stability of the Jensen functional equation. Fuzzy Sets Syst. 159, 730–738 (2008)
Shen, Y.: On the Ulam stability of first order linear fuzzy differential equations under generalized differentiability. Fuzzy Sets Syst. (In press). https://doi.org/10.1016/j.fss.2015.01.002
Diamond, P., Kloeden, P.E.: Metric Spaces of Fuzzy Sets: Theory and Applications. World Scientific, Singapore (1994)
Bede, B., Rudas, I.J., Bencsik, A.L.: First order linear fuzzy differential equations under generalized differentiability. Inform. Sci. 177, 1648–1662 (2007)
Ye, H., Gao, J., Ding, Y.: A generalized Gronwall inequality and its application to a fractionaldifferential equation. J. Math. Anal. Appl. 328, 1075–1081 (2007)
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The authors acknowledge the financial support from Universiti Putra Malaysia under Putra-IPB grant: GP-IPB/2017/9542402.
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Ahmadian, A., Salahshour, S., Senu, N., Ismail, F. (2018). Some New Results on the Stability of Fractional Integro-Differential Equations Under Uncertainty. In: Ghazali, R., Deris, M., Nawi, N., Abawajy, J. (eds) Recent Advances on Soft Computing and Data Mining. SCDM 2018. Advances in Intelligent Systems and Computing, vol 700. Springer, Cham. https://doi.org/10.1007/978-3-319-72550-5_6
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DOI: https://doi.org/10.1007/978-3-319-72550-5_6
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