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This article is part of the series A Tribute to Professor Ivan Kiguradze.

Open Access Research

A note on stability of impulsive differential equations

YuMei Liao12 and JinRong Wang12*

Author Affiliations

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

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

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Boundary Value Problems 2014, 2014:67  doi:10.1186/1687-2770-2014-67

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/67


Received:5 December 2013
Accepted:5 March 2014
Published:24 March 2014

© 2014 Liao 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 credited.

Abstract

In this note, we study a new class of ordinary differential equations with non-instantaneous impulses. Both existence and generalized Ulam-Hyers-Rassias stability results are established. Finally, an example is given to illustrate our theoretical results.

Keywords:
impulsive differential equations; non-instantaneous impulses; stability

1 Introduction

Many evolution processes studied in applied sciences are represented by differential equations. However, the situation is quite different in many modeled phenomena which have a sudden change in their states such as population dynamics, biotechnology processes, chemistry, engineering, medicine and so on. One of the mathematical models about such processes can be formulated by the following impulsive differential equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M1">View MathML</a>

(1)

where the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M2">View MathML</a> and impulsive conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M4">View MathML</a>. We set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M6">View MathML</a>. The fixed time sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M7">View MathML</a> is increasing, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M8">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M10">View MathML</a> represent the right and left limits of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M11">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M12">View MathML</a>, respectively. Here, the impulsive conditions are the combination of the traditional initial value problems and the short-term perturbations whose duration can be negligible in comparison with the duration of such a process.

However, the above short-term perturbations could not show the dynamic change of evolution processes completely in pharmacotherapy. As we know, the introduction of the drugs in the bloodstream and the consequent absorption for the body are a gradual and continuous process. Thus, we have to use a new model to describe such an evolution process. In fact, the above situation has fallen in a new impulsive action, which starts at an arbitrary fixed point and keeps active on a finite time interval. To achieve this aim, Hernández and O’Regan [1] introduced a new class of abstract semilinear impulsive differential equations with non-instantaneous impulses. Then, the concept of mild solutions and existence results are presented. Next, Pierri et al.[2] continued the work and developed the results in [1] and obtained new existence results in a fractional power space.

In 1940, the famous stability of functional equations was firstly offered by Ulam at Wisconsin University and concerned approximate homomorphisms. Thereafter, Ulam’s stability problem [3] has attracted many famous researchers, one can refer to the interesting monographs of Hyers [4,5], Rassias [6], Jung [7], Cădariu [8], and an important survey of Brillouët-Belluot et al.[9] via the recent special issue on Ulam-type stability edited by Brzdȩk et al.[10]. For the recent Ulam’s stability concepts and results on ordinary differential equations (with impulses), one can see [11,12] and reference therein.

Motivated by [1,2,11,12], we introduce a new Ulam-type stability concept for the following semilinear differential equations with non-instantaneous impulses:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M13">View MathML</a>

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M14">View MathML</a> are pre-fixed numbers, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M15">View MathML</a> is continuous, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M16">View MathML</a> is continuous for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M17">View MathML</a>.

The novelty of our paper is considering a new type of equation (2), then presenting a generalized Ulam-Hyers-Rassias stability definition and finding reasonable conditions on equation (2) to show that equation (2) is generalized Ulam-Hyers-Rassias stable.

In Section 2, we introduce a new Ulam-type stability concept for equation (2) (see Definition 2.2). In Section 3, we mainly prove a generalized Ulam-Hyers-Rassias stability result for equation (2) on a compact interval. Finally, an example is given to illustrate our theoretical results.

2 Preliminaries

Throughout this paper, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M18">View MathML</a> be the Banach space of all continuous functions from J into ℝ with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M19">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M20">View MathML</a>. We introduce the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M21">View MathML</a> := {<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M23">View MathML</a>, and there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M26">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M27">View MathML</a>} with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M28">View MathML</a>. Meanwhile, we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M29">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M30">View MathML</a>. Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M31">View MathML</a> endowed with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M32">View MathML</a> is also a Banach space.

By virtue of the concept about the solutions in [1], we can introduce the following definition.

Definition 2.1 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M33">View MathML</a> is called a classical solution of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M34">View MathML</a>

(3)

if x satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M35">View MathML</a>

Next, we adopt the idea in [12] and introduce a new Ulam-type stability concept for equation (2). Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M36">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M37">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M38">View MathML</a>. We consider the following inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M39">View MathML</a>

(4)

Definition 2.2 Equation (2) is generalized Ulam-Hyers-Rassias stable with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M40">View MathML</a> if there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M41">View MathML</a> such that for each solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M42">View MathML</a> of inequality (4), there exists a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M33">View MathML</a> of equation (2) with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M44">View MathML</a>

Remark 2.3 Definition 2.2 has practical meaning in the following sense. Consider an evolution process with not sudden changes of states but acting on an interval, which can be modeled by equation (2). Assume that we can measure the state of the process at any time to get a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M45">View MathML</a>. Putting this <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M45">View MathML</a> into equation (2), in general, we do not expect to get a precise solution of equation (2). All what is required is to get a function which satisfies the suitable approximation inequality (4). Our result of Section 3 will guarantee that there is a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M47">View MathML</a> of inequality (4) close to the measured output <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M45">View MathML</a> and closeness is defined in the sense of generalized Ulam-Hyers-Rassias stability. This technique is quite useful in many applications such as numerical analysis, optimization, biology and economics, where it is quite difficult to find the exact solution.

Remark 2.4 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M42">View MathML</a> is a solution of inequality (4) if and only if there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M50">View MathML</a> and a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M17">View MathML</a> (which depend on y) such that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M54">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M56">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M59">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M17">View MathML</a>.

Remark 2.5 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M42">View MathML</a> is a solution of inequality (4), then y is a solution of the following integral inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M64">View MathML</a>

(5)

In fact, by Remark 2.4 we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M65">View MathML</a>

(6)

Clearly, the solution of equation (6) is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M66">View MathML</a>

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M59">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M69">View MathML</a>

Proceeding as above, we derive that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M70">View MathML</a>

In order to deal with Ulam-type stability, we need the following result (see Theorem 16.4, [13]).

Lemma 2.6Let the following inequality hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M71">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M72">View MathML</a>, ais nondecreasing and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M26">View MathML</a>.

Then, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M76">View MathML</a>, the following inequality is valid:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M77">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M78">View MathML</a>.

3 Main results

We introduce the following assumptions:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M79">View MathML</a>.

(H2) There exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M80">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M81">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M82">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M83">View MathML</a> is strongly measurable for the first variable and is continuous for the second variable. There exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M84">View MathML</a> and a nondecreasing function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M85">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M86">View MathML</a>

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M87">View MathML</a> and there are positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M17">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M90">View MathML</a>

(H4) There exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M91">View MathML</a> and a nondecreasing function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M38">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M93">View MathML</a>

Concerning the existence results for the solutions about problem (3), one can repeat the same procedure in Theorems 2.1 and 2.2 of Hernández and O’Regan [1] to derive the following results. So we omit the proof here.

Theorem 3.1Assume that (H1), (H2) and (H3) are satisfied. Then problem (3) has the unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M33">View MathML</a>provided that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M95">View MathML</a>

(7)

Theorem 3.2Assume that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M82">View MathML</a>) and (H3) are satisfied, the functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M97">View MathML</a>are bounded. Then problem (3) has at least one solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M33">View MathML</a>provided that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M99">View MathML</a>

Now, we discuss the stability of equation (2) by using the concept of generalized Ulam-Hyers-Rassias in the above section.

Theorem 3.3Assume that (H1), (H2), (H3) and (H4) are satisfied. Then equation (2) is generalized Ulam-Hyers-Rassias stable with respect to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M40">View MathML</a>provided that (7) holds.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M42">View MathML</a> be a solution of inequality (4). Denote by x the unique solution of the impulsive Cauchy problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M102">View MathML</a>

(8)

Then we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M103">View MathML</a>

Keeping in mind (5), for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M17">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M106">View MathML</a>

and for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M17">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M109">View MathML</a>

and for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M110">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M111">View MathML</a>

Hence, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M17">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M114">View MathML</a>

Thus, by Lemma 2.6, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M115">View MathML</a>

(9)

for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M17">View MathML</a>.

Further, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M17">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M120">View MathML</a>

which yields that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M121">View MathML</a>

(10)

Moreover, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M110">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M123">View MathML</a>

By Gronwall’s inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M124">View MathML</a>

(11)

Summarizing, we combine (9), (10) and (11) and derive that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M125">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M54">View MathML</a>, which implies that equation (2) is generalized Ulam-Hyers-Rassias stable with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M40">View MathML</a>. The proof is completed. □

4 Example

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M128">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M129">View MathML</a>. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M130">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M131">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M132">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M133">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M134">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M135">View MathML</a>. We set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M136">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M137">View MathML</a>.

Consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M138">View MathML</a>

(12)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M139">View MathML</a>

(13)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M140">View MathML</a> be a solution of inequality (13). Then there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M141">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M142">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M143">View MathML</a>

(14)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M144">View MathML</a>, integrating (14) from 0 to t, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M145">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M135">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M147">View MathML</a>

After checking the conditions in Theorem 3.1, we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M148">View MathML</a>

(15)

has a unique solution. Let us take the solution x of problem (15) given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M149">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M144">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M151">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M135">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M153">View MathML</a>

which yields that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M154">View MathML</a>

Summarizing, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M155">View MathML</a>

which yields that equation (12) is generalized Ulam-Hyers-Rassias stable with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/67/mathml/M156">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. YML and JRW proved the theorems, interpreted the results and wrote the article. All authors defined the research theme, read and approved the manuscript.

Acknowledgements

The authors thank the referees for their careful reading of the manuscript and insightful comments, which helped to improve the quality of the paper. We would also like to acknowledge the valuable comments and suggestions from the editors, which vastly contributed to improving the presentation of the paper. This work is supported by Project of Guizhou Normal College (12YB023), Doctor Project of Guizhou Normal College (13BS010), Guizhou Province Education Planning Project (2013A062), Key Project on the Reforms of Teaching Contents and Course System and Key Support Subject (Applied Mathematics) of Guizhou Normal College.

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