Boundary Value Problems for Nonlinear Implicit Caputo–Hadamard-Type Fractional Differential Equations with Impulses
Tóm tắt
In this paper, the authors establish the existence and uniqueness of solutions for a class of boundary value problem for nonlinear implicit fractional differential equations with impulses and Caputo–Hadamard type fractional derivatives. The stability of this problem is also studied. The arguments are based on the Banach contraction principle and Schaefer’s fixed point theorem. Two examples are presented to show the applicability of the results.
Tài liệu tham khảo
Abbas, S., Benchohra, M., N’Guérékata, G.M.: Topics in Fractional Differential Equations. Springer, New York (2012)
Abbas, S., Benchohra, M., N’Guérékata, G.M.: Advanced Fractional Differential and Integral Equations. Nova Science Publishers, New York (2015)
Bainov, D.D., Hristova, S.G.: Integral inequalities of Gronwall type for piecewise continuous functions. J. Appl. Math. Stoch. Anal. 10, 89–94 (1997)
Benchohra, M., Henderson, J., Ntouyas, S.K.: Impulsive Differential Equations and Inclusions, vol. 2. Hindawi Publishing Corporation, New York (2006)
Butzer, P.L., Kilbas, A.A., Trujillo, J.J.: Compositions of Hadamard-type fractional integration operators and the semigroup property. J. Math. Anal. Appl. 269, 387–400 (2002)
Butzer, P.L., Kilbas, A.A., Trujillo, J.J.: Fractional calculus in the Mellin setting and Hadamard-type fractional integrals. J. Math. Anal. Appl. 269, 1–27 (2002)
Egbunonu, P., Guay, M.: Identification of switched linear systems using subspace and integer programming techniques. Nonlinear Anal. Hybrid Syst. 1, 577–592 (2007)
Graef, J.R., Henderson, J., Ouahab, A.: Impulsive Differential Inclusions. A Fixed Point Approach, De Gruyter Series in Nonlinear Analysis and Applications, vol. 20, De Gruyter, Berlin (2013)
Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2003)
Hadamard, J.: Essai sur l’étude des fonctions données par leurs developpement de Taylor. J. Math. Pures Appl. Ser. 8, 101–186 (1892)
Jarad, F., Abdeljawad, T., Baleanu, D.: Caputo-type modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2012, 142 (2012)
Kilbas, A.A.: Hadamard-type fractional calculus. J. Korean Math. Soc. 38, 1191–1204 (2001)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204. Elsevier Science B.V, Amsterdam (2006)
Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. Worlds Scientific, Singapore (1989)
Lygeros, J., Godbole, D.N., Sastry, S.: A verified hybrid controller for automated vehicles. IEEE Trans. Autom. Control 43, 522–539 (1998)
Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993)
Pepyne, D.L., Cassandras, C.G.: Optimal control of hybrid systems in manufacturing. Proc. IEEE 7, 1108–1123 (2000)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Samoilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations. World Scientific, Singapore (1995)
Tariboon, J., Ntouyas, S.K., Thaiprayoon, C.: Nonlinear Langevin equation of Hadamard-Caputo type fractional derivatives with nonlocal fractional integral conditions. Adv. Math. Phys. 2014, 1–15 (2014). doi:10.1155/2014/372749
Vaadrager, F., Van Schuppen, J. (eds.): Hybrid Systems, Computation and Control. Lecture Notes in Computer Sciences, vol. 1569. Springer, New York (1999)
Varaiya, P.: Smart cars on smart roads: problems of control. IEEE Trans. Autom. Control 38, 195–207 (1993)
Wang, J., Feckan, M., Zhou, Y.: Ulam’s type stability of impulsive ordinary differential equations. J. Math. Anal. Appl. 395, 258–264 (2012)
Wei, W., Xiang, X., Peng, Y.: Nonlinear impulsive integro-differential equations of mixed type and optimal controls. Optimization 55, 141–156 (2006)
Yukunthorn, W., Ntouyas, S.K., Tariboon, J.: Nonlinear fractional Caputo–Langevin equation with nonlocal Riemann–Liouville fractional integral conditions. Adv. Differ. Equ. 315, 18 (2014)
Yukunthorn, W., Ahmad, B., Ntouyas, S.K., Tariboon, J.: On Caputo–Hadamard type fractional impulsive hybrid systems with nonlinear fractional integral conditions. Nonlinear Anal. Hybrid Syst. 19, 77–92 (2016)