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This article is part of the series Professor Anthony To-Ming Lau's contributions to the development of Fixed Point Theory and Applications..

Open Access Research

Boundary value problem of a nonlinear Langevin equation with two different fractional orders and impulses

Guotao Wang1, Lihong Zhang2* and Guangxing Song2

Author Affiliations

1 School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi, 041004, P.R. China

2 Department of Mathematics, China University of Petroleum, Qingdao, Shandong, 266555, P.R. China

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Fixed Point Theory and Applications 2012, 2012:200  doi:10.1186/1687-1812-2012-200

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/200


Received:15 March 2012
Accepted:5 October 2012
Published:6 November 2012

© 2012 Wang et al.; 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 a new type of a Langevin equation involving two different fractional orders and impulses. Sufficient conditions are formulated for the existence and uniqueness of solutions of the given problems.

MSC: 34A08, 34B10, 34B37, 46N10.

Keywords:
nonlinear fractional Langevin equation; impulse; two different fractional orders; nonlocal conditions; uniqueness; fixed point theorem

1 Introduction

Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. These characteristics of the fractional derivatives make the fractional-order models more realistic and practical than the classical integer-order models. In fact, fractional differential equations appear naturally in a number of fields such as physics, geophysics, polymer rheology, regular variation in thermodynamics, biophysics, blood flow phenomena, aerodynamics, electro-dynamics of complex medium, viscoelasticity, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, fitting of experimental data, nonlinear oscillation of earthquake, the fluid-dynamic traffic model, etc. For more details and applications, we refer the reader to the books [1-3]. For some recent development on the topic, see [4-15] and the references therein.

It is well known that a Langevin equation is widely used to describe the evolution of physical phenomena in fluctuating environments [16-18]. However, for the systems in complex media, an integer-order Langevin equation does not provide the correct description of the dynamics. One of the possible generalizations of a Langevin equation is to replace the integer-order derivative by a fractional-order derivative in it. This gives rise to a fractional Langevin equation, see [19-22] and the references therein.

In 2008, Lim, Li and Teo [23] firstly introduced a new type of a Langevin equation with two different fractional orders. The solution to this new version of a fractional Langevin equation gives a fractional Gaussian process parametrized by two indices, which provides a more flexible model for fractal processes as compared with the usual one characterized by a single index. In 2009, Lim and Teo [24] discussed the fractional oscillator process with two indices. In 2010, by using the contraction mapping principle and Krasnoselskii’s fixed point theorem, Ahmad and Nieto [25] studied a Langevin equation involving two fractional orders with Dirichlet boundary conditions. Recently, the existence of solutions for a three-point boundary value problem of a Langevin equation with two different fractional orders has also been studied in [26].

Motivated by the above-mentioned works, in this paper, we consider the following nonlinear Langevin equation with two different fractional orders and impulses in a Banach space E:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M1">View MathML</a>

(1.1)

with one of the following three boundary conditions:

(1.2)

(1.3)

(1.4)

where CD is the Caputo fractional derivative, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M5">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M6">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M7">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M8">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M9">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M10">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M11">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M12">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M13">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M14">View MathML</a> denote the right and the left limits of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M15">View MathML</a> at <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M16">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M17">View MathML</a>), respectively. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M18">View MathML</a> has a similar meaning for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M19">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M20">View MathML</a>. Evidently, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M21">View MathML</a> is a Banach space endowed with the sup-norm <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M22">View MathML</a>.

Nonlocal conditions were initiated by Byszewski [27] when he proved the existence and uniqueness of mild and classical solutions of nonlocal Cauchy problems. Many authors since then have considered the existence and multiplicity of solutions (or positive solutions) of nonlocal problems. The recent results on nonlocal problems of fractional differential equations can be found in [29-40]. As remarked by Byszewski [28], the nonlocal condition can be more useful than the standard initial (boundary) condition to describe some physical phenomena. For example, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M23">View MathML</a> may be given by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M24">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M25">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M26">View MathML</a>, are given constants and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M27">View MathML</a>.

Impulsive differential equations, which provide a natural description of observed evolution processes, are regarded as important mathematical tools for the better understanding of several real world problems in applied sciences. The theory of impulsive differential equations of integer order has found its extensive applications in realistic mathematical modeling of a wide variety of practical situations and has emerged as an important area of investigation. The impulsive differential equations of fractional order have also attracted a considerable attention and a variety of results can be found in the papers [41-51].

To the best knowledge of the authors, no paper has considered nonlinear Langevin equations involving two different fractional orders and impulses, i.e., problems (2.1), (3.1) and (3.2). This paper fills this gap in the literature.

This paper is organized as follows. In Section 2, we present some preliminary results. Consequently, problem (2.1) is reduced to an equivalent integral equation. Then, by using the fixed point theory, we study the existence and uniqueness of a Dirichlet boundary value problem for nonlinear Langevin equations involving two different fractional orders and impulses. In Section 3, we indicate some generalizations to nonlocal Dirichlet boundary value problems. The last section is devoted to an example illustrating the applicability of the imposed conditions. These results can be considered as a contribution to this emerging field.

2 Dirichlet boundary value problem

In this section, we consider the following Dirichlet boundary value problem:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M28">View MathML</a>

(2.1)

For the sake of convenience, we introduce the following notations:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M29">View MathML</a>

Definition 2.1 A function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M30">View MathML</a> with its Caputo derivative of fractional order existing on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M31">View MathML</a> is a solution of (2.1) if it satisfies (2.1).

Lemma 2.1[1]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M32">View MathML</a>, then the fractional differential equation

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M33">View MathML</a>

has a solution

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M34">View MathML</a>

Lemma 2.2[1]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M32">View MathML</a>, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M36">View MathML</a>

for some<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M37">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M38">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M39">View MathML</a>.

2.1 Existence result

Lemma 2.3For any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M40">View MathML</a>, a functionuis a solution of the following Dirichlet boundary value problem:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M41">View MathML</a>

(2.2)

if and only ifuis a solution of the fractional integral equation

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M42">View MathML</a>

(2.3)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M43">View MathML</a>

(2.4)

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M44">View MathML</a>, by (2.2), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M45">View MathML</a>

(2.5)

We may apply Lemma 2.2 to reduce the equation (2.5) to an equivalent integral equation

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M46">View MathML</a>

for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M47">View MathML</a>.

Thus,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M48">View MathML</a>

(2.6)

for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M47">View MathML</a>.

Similarly, by Lemma 2.2, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M50">View MathML</a>

(2.7)

for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M51">View MathML</a>. Combining with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M52">View MathML</a>, we get that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M53">View MathML</a>.

Substituting the value of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M54">View MathML</a> in (2.7), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M55">View MathML</a>

(2.8)

for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M51">View MathML</a>.

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M57">View MathML</a>, then

for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M59">View MathML</a>.

Thus, we have

In view of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M61">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M62">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M63">View MathML</a>

Hence,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M64">View MathML</a>

(2.9)

By a similar process, we can get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M65">View MathML</a>

(2.10)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M66">View MathML</a>

(2.11)

By the same method, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M67">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M68">View MathML</a>

(2.12)

By (2.12) and the condition <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M69">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M70">View MathML</a>

(2.13)

Substituting the value of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M71">View MathML</a> in (2.8) and (2.12) and letting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M72">View MathML</a>, we can get (2.3). Conversely, assume that u is a solution of the impulsive fractional integral equation (2.3). Then by a direct computation, it follows that the solution given by (2.3) satisfies (2.2). This completes the proof. □

2.2 Nonlinear problem

Define the constant:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M73">View MathML</a>

(2.14)

Theorem 2.1Assume that

(H1) There exist constants<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M74">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M75">View MathML</a>) such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M76">View MathML</a>

Then problem (2.1) has a unique solution provided<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M77">View MathML</a>, where Λ is given by (2.14).

Proof Define the operator <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M78">View MathML</a> as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M79">View MathML</a>

(2.15)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M80">View MathML</a>

Then the equation (2.1) has a solution if and only if the operator T has a fixed point.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M81">View MathML</a>. By (2.15), we have

Using the condition (H1), by computation, we can get

Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M84">View MathML</a>.

As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M85">View MathML</a>, therefore, A is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle. □

3 Nonlocal Dirichlet boundary value problems

In this section, we consider the following nonlocal Dirichlet boundary value problems:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M86">View MathML</a>

(3.1)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M87">View MathML</a>

(3.2)

For the forthcoming analysis, we need the following assumptions:

(H2) There exists a constant <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M88">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M89">View MathML</a>.

(H3) There exists a constant <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M90">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M91">View MathML</a>.

Theorem 3.1Assume (H1), (H2) hold if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M92">View MathML</a>, then problem (3.1) has a unique solution, where Λ is given by (2.14).

Theorem 3.2Assume (H1), (H3) hold if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M93">View MathML</a>, then problem (3.2) has a unique solution, where Λ is given by (2.14).

The proofs of Theorem 3.2 and Theorem 3.1 are similar. Here we only prove Theorem 3.1.

Proof We transform the problem (3.1) into a fixed point problem. Consider the operator <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M94">View MathML</a> as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M95">View MathML</a>

(3.3)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M96">View MathML</a>

The rest of the proof is almost the same as that of Theorem 2.1, so we omit it. □

4 Example

The following example is a direct application of our main result.

Example 4.1 Consider the following Dirichlet boundary value problem of a nonlinear Langevin equation with two different fractional orders and impulses:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M97">View MathML</a>

(4.1)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M98">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M99">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M100">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M101">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M102">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M103">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M104">View MathML</a>.

Obviously, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M105">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M106">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M107">View MathML</a>. Further,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M108">View MathML</a>

Therefore, by Theorem 2.1, we can get that the above equation (4.1) has a unique solution on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/200/mathml/M109">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors completed the paper together. All authors read and approved the final manuscript.

Acknowledgements

We would like to express our gratitude to the anonymous reviewers and editors for their valuable comments and suggestions which have improved the quality of the present paper. The research was supported by the Natural Science Foundation for Young Scientists of Shanxi Province (2012021002-3), China.

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