On the existence of a mild solution for impulsive hybrid fractional differential equations
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On the existence of a mild solution for impulsive hybrid fractional differential equations Huanmin Ge and Jie Xin* * Correspondence: [email protected] School of Mathematics and Statistics Science, Ludong University, Yantai, 264025, China
Abstract This paper is motivated by some recent contributions on the existence of solution of impulsive fractional differential equations and the theory of fractional hybrid differential equations by Agarwal, Ahmad, Baleanu, Benchohra, Feˇckan, Nieto, Sun, Bai, Zhou, Zhang and Wang. Here, we derive new existence results of a mild solution of impulsive hybrid fractional differential equations. Finally, an example is given to illustrate the result. Keywords: Caputo fractional derivative; impulsive hybrid differential equations; existence
1 Introduction The study of differential equations of fractional order is motivated by the intensive development of the theory of fractional calculus itself (see [–]) and the application of fractional differential equations in the modeling of many physical phenomena. There have been many works on the theory of fractional calculus and applications of it. Fractional differential equations, including Riemann-Liouville fractional derivative or Caputo fractional derivative, have received more and more attention (see [–]). In recent years, hybrid differential equations have attracted much attention. The theory of hybrid differential equations has been developed, and we refer the readers to the articles [–]. The authors [] discussed the following fractional hybrid differential equations involving Riemann-Liouville differential operators:
L
q
u(t) D,t [ f (t,u(t)) ] = g(t, u(t)),
u() = ,
a.e. t ∈ J := [, T],
()
q
where L D,t is the Riemann-Liouville fractional derivative of order q ∈ (, ) with the lower limit zero, f ∈ C(J × R, R \ {}) and g ∈ C(J × R, R). They developed existence of solutions under mixed Lipschitz and Carathéodory conditions. Moreover, they have established some fundamental fractional differential inequalities and the comparison principle. Some recent papers have treated the problem of the existence of solutions for impulsive fractional differential equations. ©2014 Ge and Xin; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Ge and Xin Advances in Difference Equations 2014, 2014:211 http://www.advancesindifferenceequations.com/content/2014/1/211
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The authors [] considered the following basic impulsive Cauchy problems: ⎧ q c ⎪ ⎨ D,t u(t) = f (t, u(t)), t ∈ J := J \ {t , . . . , tm }, – u(tk ) = Ik (u(tk )), k = , , . . . , m, ⎪ ⎩ u() = u ,
()
q
where c D,t is the generalized Caputo fractional derivative of order q ∈ (, ) with the lower limit zero and Ik : R → R and tk satisfy = t < t < · · · < tm < tm+ = T, u(tk+ ) = limε→+
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