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Open Access Research

Periodic BVPs for fractional order impulsive evolution equations

Xiulan Yu1 and JinRong Wang23*

Author Affiliations

1 College of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, Shanxi, 030031, P.R. China

2 Department of Mathematics, Guizhou University, Guiyang, Guizhou, 550025, P.R. China

3 School of Mathematics and Computer Science, Guizhou Normal College, Guiyang, Guizhou, 550018, P.R. China

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Boundary Value Problems 2014, 2014:35  doi:10.1186/1687-2770-2014-35

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/35


Received:26 November 2013
Accepted:16 January 2014
Published:7 February 2014

© 2014 Yu and Wang; 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.

Abstract

In this paper, we study periodic BVPs for fractional order impulsive evolution equations. The existence and boundedness of piecewise continuous mild solutions and design parameter drift for periodic motion of linear problems are presented. Furthermore, existence results of piecewise continuous mild solutions for semilinear impulsive periodic problems are showed. Finally, an example is given to illustrate the results.

MSC: 34B05, 34G10, 47D06.

Keywords:
fractional order; impulsive evolution equations; periodic BVPs

1 Introduction

In order to describe dynamics of populations subject to abrupt changes as well as other evolution processes such as harvesting, diseases, and so forth, many researchers have used impulsive differential systems to describe the model since the last century. For a wide-ranging bibliography and exposition on this important object see for instance the monographs of [1-4] and the papers [5-12].

Fractional differential equations appear naturally in fields such as viscoelasticity, electrical circuits, nonlinear oscillation of earthquakes etc. In particular, impulsive fractional evolution equations are used to describe many practical dynamical systems in many evolutionary processes models. Recently, Wang et al.[13] discussed Cauchy problems and nonlocal problems for impulsive fractional evolution equations involving the Caputo fractional derivative. However, periodic boundary value problems (BVPs for short) for impulsive fractional evolution equations have not been studied extensively.

In this paper we study the periodic BVPs for impulsive fractional evolution equations. Firstly, we discuss periodic BVPs for impulsive fractional evolution equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M1">View MathML</a>

(1)

in Banach space X, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M2">View MathML</a> is the Caputo fractional derivative of order q with the lower limit zero, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M3">View MathML</a> is the generator of a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M4">View MathML</a>-semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M5">View MathML</a> on a Banach space X, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M6">View MathML</a> is continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M8">View MathML</a> are the elements of X, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M9">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M11">View MathML</a> represent respectively the right and left limits of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M12">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M13">View MathML</a>.

Secondly, we design parameter drift for the above periodic motion. We study the following impulsive periodic BVPs with parameter perturbations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M14">View MathML</a>

(2)

where p is a given function and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M15">View MathML</a> is a small parameter perturbation that may be caused by some adaptive control algorithms or parameter drift.

Finally, we consider semilinear impulsive periodic problems:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M16">View MathML</a>

(3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M17">View MathML</a> is continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M18">View MathML</a> is continuous.

The rest of this paper is organized as follows. In Section 2, the existence and boundedness of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M19">View MathML</a> are given. In Section 3, the existence and boundedness of PC-mild solutions and the design parameter drift for such a periodic motion are presented. In Section 4, existence results of PC-mild solutions for impulsive periodic problems are showed. Finally, an example is presented to illustrate the theory.

2 Existence and boundedness of operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M19">View MathML</a>

Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M21">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M22">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M23">View MathML</a>. We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M24">View MathML</a> the Banach space of all continuous functions from J into X with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M25">View MathML</a>. We also introduce the set of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M26">View MathML</a> = {<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M27">View MathML</a> is continuous at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M28">View MathML</a>, and x is continuous from the left and has right-hand limits at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M29">View MathML</a>}. Endowed with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M30">View MathML</a>

it is easy to see <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M31">View MathML</a> is a Banach space.

For measurable functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M32">View MathML</a>, define the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M34">View MathML</a>. We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M35">View MathML</a> the Banach space of all Lebesgue measurable functions l with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M36">View MathML</a>.

Definition 2.1 ([14])

The fractional integral of order γ with the lower limit zero for a function f is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M37">View MathML</a>

provided the right side is point-wise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M38">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M39">View MathML</a> is the gamma function.

Definition 2.2 ([14])

The Riemann-Liouville derivative of order γ with the lower limit zero for a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M40">View MathML</a> can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M41">View MathML</a>

Definition 2.3 ([14])

The Caputo derivative of order γ for a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M40">View MathML</a> can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M43">View MathML</a>

Remark 2.4 If f is an abstract function with values in X, then the integrals which appear in Definitions 2.1 and 2.2 are taken in Bochner’s sense.

As in our previous work [13], by a PC-mild solution of (1) we mean the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M44">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M45">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M46">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M47">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M48">View MathML</a>

here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M49">View MathML</a> is a probability density function defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M50">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M51">View MathML</a>

Lemma 2.5 (see Lemma 2.9 [15])

The operatorhas the following properties:

(i) For any fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M54">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M55">View MathML</a>are linear and bounded operators, i.e., for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M57">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M58">View MathML</a>.

(ii) Both<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M59">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M60">View MathML</a>are strongly continuous.

(iii) For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M54">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M55">View MathML</a>are compact operators if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M64">View MathML</a>is compact.

Suppose that here the bounded operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M65">View MathML</a> exists given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M66">View MathML</a>

(4)

We present sufficient conditions for the existence and boundedness of the operator B.

Lemma 2.6 (see Theorem 3.3 and Remark 3.4 [16])

The operatorBdefined in (4) exists and is bounded, if one of the following three conditions holds:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M64">View MathML</a>is compact for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M61">View MathML</a>and the homogeneous linear nonlocal problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M69">View MathML</a>

has no non-trivialPC-mild solutions.

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M70">View MathML</a>, then the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M71">View MathML</a>is invertible and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M72">View MathML</a>.

(iii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M73">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M74">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M75">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M76">View MathML</a>and the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M71">View MathML</a>is invertible and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M78">View MathML</a>.

3 Linear impulsive periodic problems and robustness

In this section, we consider the existence of PC-mild solutions of (1) and of design parameter drift for (2).

We list the following assumptions.

(HA): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M79">View MathML</a> is the infinitesimal generator of a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M4">View MathML</a>-semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M81">View MathML</a>.

(HF): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M82">View MathML</a> is strongly measurable for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M83">View MathML</a> and there exist a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M84">View MathML</a> and a real-valued function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M85">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M86">View MathML</a>, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M83">View MathML</a>.

(HB): The operator B defined in (4) exists and is bounded.

We first give an existence theorem of PC-mild solutions of (1).

Theorem 3.1Assume that (HA), (HF), and (HB) are satisfied. Then (1) has aPC-mild solution given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M88">View MathML</a>

(5)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M89">View MathML</a>

(6)

Further, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M90">View MathML</a>

(7)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M91">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M92">View MathML</a>.

Proof We consider an impulsive Cauchy problem,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M93">View MathML</a>

(8)

It follows from the expression of the initial value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M94">View MathML</a> that the mild solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M95">View MathML</a> of (8) corresponding to the initial value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M96">View MathML</a> must be the PC-mild solution of (1).

For the estimation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M95">View MathML</a>, by Lemma 2.5,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M98">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M91">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M92">View MathML</a>. The desired results are obtained. □

Remark 3.2 In Theorem 3.1, we replace (HB) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M5">View MathML</a>; it is a compact semigroup and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M102">View MathML</a>

has no non-trivial mild solutions. Then one can use the Fredholm alternative theorem to derive that the operator equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M103">View MathML</a> has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M104">View MathML</a>. Thus, the PC-mild solution of (1) is unique.

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M105">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M106">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M107">View MathML</a>. It can be seen that endowed with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M108">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M109">View MathML</a> is a Banach space. Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M110">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M111">View MathML</a>

and χ is a nonnegative function.

We introduce the assumption (HP):

(HP1): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M112">View MathML</a> is measurable in t.

(HP2): There exists a nonnegative function ϖ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M113">View MathML</a> and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M83">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M115">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M116">View MathML</a>, and we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M117">View MathML</a>

(HP3): There exists a nonnegative function χ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M118">View MathML</a>, and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M83">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M120">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M116">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M122">View MathML</a>

By a PC-mild solution of (2), we mean the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M44">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M124">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M46">View MathML</a>.

The following result shows that given a periodic motion we can design periodic motion controllers that are robust with respect to parameter drift.

Theorem 3.3Let (HA), (HF), (HB), and (HP) hold. Then there is a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M126">View MathML</a>such that, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M127">View MathML</a>, (2) has aPC-mild solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M128">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M129">View MathML</a>

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M130">View MathML</a>uniformly on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M83">View MathML</a>where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M132">View MathML</a>is the mild solution of (1).

Proof By (HB), one can choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M133">View MathML</a>

to be fixed. Consider the map on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M135">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M136">View MathML</a>

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M137">View MathML</a>.

By the assumption (HP), we can choose a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M126">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M139">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M140">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M141">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M142">View MathML</a>, one can verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M143">View MathML</a>

(9)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M144">View MathML</a>

This implies that is a contraction mapping on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M146">View MathML</a>. Then, has a unique fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M148">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M149">View MathML</a>

(10)

which is just the PC-mild solution of (2).

From the expressions (9) and (10), one can get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M150">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M130">View MathML</a> uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M83">View MathML</a>. □

4 Semilinear impulsive periodic problems

We impose the following assumptions.

(HF1): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M153">View MathML</a> is continuous and there exist a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M84">View MathML</a> and a real-valued function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M155">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M156">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M157">View MathML</a>.

(HF2): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M153">View MathML</a> is continuous and maps a bounded set into a bounded set.

(HF3): For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M159">View MathML</a>, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M160">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M161">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M162">View MathML</a>

(HI1): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M18">View MathML</a> is continuous and there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M164">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M165">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M157">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M167">View MathML</a>.

(HI2): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M18">View MathML</a> is continuous and there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M169">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M170">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M167">View MathML</a>.

Theorem 4.1Let (HA), (HB), (HI1), and (HF1) be satisfied. Then (3) has a uniquePC-mild solution onJprovided that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M173">View MathML</a>

(11)

Proof Consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M174">View MathML</a>

(12)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M175">View MathML</a>

(13)

and we define an operator Q on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M26">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M177">View MathML</a>

(14)

Clearly, Q is well defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M26">View MathML</a> due to our assumptions.

Then, we only need to show that Q is a contraction on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M26">View MathML</a>.

In general, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M180">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M167">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M182">View MathML</a>

Hence, the condition (11) allows us to conclude, in view of the Banach contraction mapping principle, that Q has a unique fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M44">View MathML</a>, which is just the unique PC-mild solution of (3). □

Theorem 4.2Suppose that (HA), (HB), (HI2), and (HF2) and (HF3) are satisfied. Then for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M184">View MathML</a>, (3) has at least aPC-mild solution onJ.

Proof Consider the mapping

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M185">View MathML</a>

by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M186">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M187">View MathML</a>

(15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M188">View MathML</a> is defined in (13) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M189">View MathML</a>

(16)

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M180">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M191">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M192">View MathML</a>

Thus, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M193">View MathML</a>.

Just like the proof in our previous work [13], one can prove that Q is a continuous mapping from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M194">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M194">View MathML</a> and it is a compact operator. Now, Schauder’s fixed point theorem implies that Q has a fixed point, which gives rise to a PC-mild solution. □

5 Example

We consider impulsive fractional differential equations with periodic boundary conditions,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M196">View MathML</a>

(17)

in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M197">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M198">View MathML</a> will be chosen later.

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M199">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M200">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M201">View MathML</a>. Then A is the infinitesimal generator of a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M4">View MathML</a>-semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M203">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M204">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M205">View MathML</a> is also compact and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M206">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M53">View MathML</a>. By the Fredholm alternative theorem, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M208">View MathML</a> exists and is bounded where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M209">View MathML</a> is defined in Section 2.

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M210">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M211">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M213">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M214">View MathML</a> is a continuous function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M215">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M216">View MathML</a>. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M217">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M213">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M220">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M221">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M222">View MathML</a>.

Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M223">View MathML</a>

Thus, one can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M224">View MathML</a> such that (11) holds. Therefore, (17) has a unique PC-mild solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/35/mathml/M225">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

This work was carried out in collaboration between all authors. JRW raised these interesting problems in this research. JRW and XLY proved the theorems, interpreted the results and wrote the article. All authors defined the research theme, read and approved the manuscript.

Acknowledgements

This work is partially supported by Key Support Subject (Applied Mathematics), Key Project on the Reforms of Teaching Contents, Course System of Guizhou Normal College and Doctor Project of Guizhou Normal College (13BS010) and Guizhou Province Education Planning Project (2013A062).

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