• PDF / 301,922 Bytes
• 16 Pages / 595.28 x 793.7 pts Page_size
• 8 Downloads / 273 Views

DOWNLOAD

REPORT

---

RESEARCH

Open Access

Existence of mild solutions for fractional impulsive neutral evolution equations with nonlocal conditions Shengquan Liang1* and Rui Mei2 * Correspondence: [email protected] 1 Gansu Polytechnic College of Animal Husbandry and Engineering, Huangyangzhen, 733006, People’s Republic of China Full list of author information is available at the end of the article

Abstract In this paper, by using the fractional power of an operator and some fixed point theorems, we study the existence of mild solutions for the nonlocal problem of Caputo fractional impulsive neutral evolution equations in Banach spaces. In the end, an example is given to illustrate the applications of the abstract results. MSC: 34K45; 35F25 Keywords: fractional impulsive neutral evolution equation; compact and analytic semigroup; mild solutions; fixed point theorem

1 Introduction During the past two decades, fractional differential equations have been proved to be valuable tools in the modeling of many phenomena in various fields of engineering, physics, and economics, and hence they have gained considerable attention. Some basic theory for the initial value problem of fractional differential (or evolution) equations was discussed in [–]. But all these papers did not consider the effect of impulsive conditions in the equations. Recently, Wang et al. [] studied the existence of mild solutions for the fractional impulsive evolution equations ⎧ q ⎪ ⎨ D u(t) + Au(t) = f (t, u(t)), t ∈ J = [, a], t = tk , u(tk+ ) = u(tk– ) + yk , k = , , . . . , m, ⎪ ⎩ u() = u

()

in a Banach space X, where a >  is a constant, Dq denotes the Caputo fractional derivative of order q ∈ (, ), A : D(A) ⊂ X → X is a closed linear operator and –A generates a C -semigroup T(t) (t ≥ ) in X, f : J × X → X is continuous, yk , u are the elements of X,  = t < t < t < · · · < tm < tm+ = a, u(tk+ ) and u(tk– ) represent the right and left limits of u(t) at t = tk , respectively. By using some fixed point theorems of compact operator, they derive many existence and uniqueness results concerning the mild solutions for problem () under the different assumptions on the nonlinear term f . For more articles about the fractional impulsive evolution equations, we refer to [–] and the references therein. On the other hand, the fractional neutral differential equations have also been studied by many authors. Many methods of nonlinear analysis have been employed to research this ©2014 Liang and Mei; 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.

Liang and Mei Advances in Difference Equations 2014, 2014:101 http://www.advancesindifferenceequations.com/content/2014/1/101

Page 2 of 16

problem; see [, –]. But, as far as we know, papers considering the fractional impulsive neutral evolution equations are seldom. In th