Existence and Ulam Stability of Solutions for Conformable Impulsive Differential Equations
Tóm tắt
In this article, we use mathematical induction to derive the representation of the solution of conformable impulsive linear differential equations with constant coefficients. We present the existence of solutions to impulsive nonlinear differential equations with constant coefficients under mild conditions on the nonlinear term. In addition, we consider the concepts of Ulam stability for this type of equation and give Ulam–Hyers and Ulam–Hyers–Rassias stability results. Finally, we give examples to verify our theoretical results.
Tài liệu tham khảo
Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015)
Khalil, R., Horani, M.Al, Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math 264, 65–70 (2014)
Abdeljawad, T., Horani, M.A.L., Khalil, R.: Conformable fractional semigroups of operators. J. Semigroup Theory Appl. 2015(7), 1–9 (2015)
Chung, W.: Fractional Newton mechanics with conformable fractional derivative. J. Comput. Appl. Math. 290, 150–158 (2015)
Abdeljawad, T., Al-Mdallal, Q.M., Jarad, F.: Fractional logistic models in the frame of fractional operators generated by conformable derivatives. Chaos Solitons Frac. 119, 94–101 (2019)
Bohner, M., Hatipoǧlu, V.F.: Dynamic cobweb models with conformable fractional derivatives. Nonlinear Anal. Hybrid Syst. 32, 157–167 (2019)
Al-Rifae, M., Abdeljawad, T.: Fundamental results of conformable Sturm-Liouville eigenvalue problems. Complexity 2017:3720471 (2017). https://doi.org/10.1155/2017/3720471
Horani, M.A.L., Hammad, M.A., Khalilb, R.: Variation of parameters for local fractional nonhomogenous linear-differential equations. J. Math. Comput. Sci. 16, 147–153 (2016)
Abdeljawad, T., Alzabut, J., Jarad, F.: A generalized Lyapunov-type inequality in the frame of conformable derivatives. Adv. Differ. Equ. 321, 1–10 (2017)
Pospíšil, M., Pospíšilová Škripková, L.: Sturms theorems for conformable fractional differential equations. Math. Commun. 21, 273–281 (2016)
Hammad, M.A., Khalil, R.: Abel’s formula and Wronskian for conformable fractional differential equations. Int. J. Differ. Equ. Appl. 13, 177–183 (2014)
Zheng, A., Feng, Y., Wang, W.: The Hyers–Ulam stability of the conformable fractional differential equation. Math. Aeterna 5, 485–492 (2015)
Iyiola, O.S., Nwaeze, E.R.: Some new results on the new conformable fractional calculus with application using D’Alambert approach. Progr. Fract. Differ. Appl. 2, 1–7 (2016)
Bayour, B., Torres, D.F.M.: Existence of solution to a local fractional nonlinear differential equation. J. Comput. Appl. Math. 312, 127–133 (2017)
Tariboon, J., Ntouyas, S.K.: Oscillation of impulsive conformable fractional differential equations. Open Math. 14, 497–508 (2016)
Li, M., Wang, J., O’Regan, D.: Existence and Ulam’s stability for conformable fractional differential equations with constant coefficients. Bull. Malay. Math. Sci. Soc. 40, 1791–1812 (2019)
Jaiswal, A., Bahuguna, D.: Semilinear conformable fractional differential equations in Banach spaces. Differ. Equ. Dyn. Syst. 27, 313–325 (2019)
Pospíšil, M.: Laplace transform, Gronwall inequality and delay differential equations for general conformable fractional derivative. Commun. Math. Anal. 22, 14–33 (2019)
Zhou, Y.: Attractivity for fractional differential equations in Banach space. Appl. Math. Lett. 75, 1–6 (2018)
Zhou, Y.: Attractivity for fractional evolution equations with almost sectorial operators. Fract. Calc. Appl. Anal. 21, 786–800 (2018)
Zhou, Y., Peng, L., Huang, Y.Q.: Duhamel’s formula for time-fractional Schrödinger equations. Math. Methods Appl. Sci. 41, 8345–8349 (2018)
Wei, W., Xiang, X., Peng, Y.: Nonlinear impulsive integro-differential equation of mixed type and optimal controls. Optimization 55, 141–156 (2006)