## Positive solutions for impulsive fractional differential equations with generalizedperiodic boundary value conditions

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RESEARCH

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Positive solutions for impulsive fractional differential equations with generalized periodic boundary value conditions Kaihong Zhao* and Ping Gong * Correspondence: [email protected] Department of Applied Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650093, China

Abstract By constructing Green’s function, we give the natural formulae of solutions for the following nonlinear impulsive fractional diﬀerential equation with generalized periodic boundary value conditions: ⎧c q ⎨ Dt u(t) = f (t, u(t)), u(tk ) = Ik (u(tk )), ⎩ au(0) – bu(1) = 0,

t ∈ J = J\{t1 , . . . , tm }, J = [0, 1], u (tk ) = Jk (u(tk )), k = 1, . . . , m, au (0) – bu (1) = 0, q

where 1 < q < 2 is a real number, c Dt is the standard Caputo diﬀerentiation. We present the properties of Green’s function. Some suﬃcient conditions for the existence of single or multiple positive solutions of the above nonlinear fractional diﬀerential equation are established. Our analysis relies on a nonlinear alternative of the Schauder and Guo-Krasnosel’skii ﬁxed point theorem on cones. As applications, some interesting examples are provided to illustrate the main results. MSC: 34B10; 34B15; 34B37 Keywords: impulsive fractional diﬀerential equation; positive solutions; boundary value problems; ﬁxed point theorem

1 Introduction In recent years, the fractional order diﬀerential equation has aroused great attention due to both the further development of fractional order calculus theory and the important applications for the theory of fractional order calculus in the ﬁelds of science and engineering such as physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, Bode’s analysis of feedback ampliﬁers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, ﬁtting of experimental data, and so forth. Many papers and books on fractional calculus diﬀerential equation have appeared recently. One can see [–] and the references therein. In order to describe the dynamics of populations subject to abrupt changes as well as other phenomena such as harvesting, diseases, and so on, some authors have used an impulsive diﬀerential system to describe these kinds of phenomena since the last century. For the basic theory on impulsive diﬀerential equations, the reader can refer to the monographs of Bainov and Simeonov [], Lakshmikantham et al. [] and Benchohra et al. []. ©2014 Zhao and Gong; 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.