Existence and Uniqueness of Solutions to Impulsive Fractional Integro-Differential Equations with Nonlocal Conditions ()

Zhenghui Gao, Liu Yang, Gang Liu

Department of Mathematics and Computational Science, Hengyang Normal University, Hengyang, China.

**DOI: **10.4236/am.2013.46118
PDF
HTML XML
5,859
Downloads
11,213
Views
Citations

Department of Mathematics and Computational Science, Hengyang Normal University, Hengyang, China.

In this article, by using Schaefer fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions for a class of impulsive integro-differential equations with nonlocal conditions involving the Caputo fractional derivative.

Keywords

Caputo Fractional Derivative; Impulses; Nonlocal Conditions; Existence; Uniqueness; Fixed Point

Share and Cite:

Gao, Z. , Yang, L. and Liu, G. (2013) Existence and Uniqueness of Solutions to Impulsive Fractional Integro-Differential Equations with Nonlocal Conditions. *Applied Mathematics*, **4**, 859-863. doi: 10.4236/am.2013.46118.

1. Introduction

Fractional differential equations appear naturally in a number of fields such as physics, engineering, biophysics, blood flow phenomena, aerodynamics, electron-analytical chemistry, biology, control theory, etc., An excellent account in the study of fractional differential equations can be found in [1-11] and references therein. Undergoing abrupt changes at certain moment of times like earthquake, harvesting, shock etc, these perturbations can be well-approximated as instantaneous change of state or impulses. Furthermore, these processes are modeled by impulsive differential equations. In 1960, Milman and Myshkis introduced impulsive differential equations in their papers [12]. Based on their work, several monographs have been published by many authors like Semoilenko and Perestyuk [13], Lak-shmikantham et al. [14], Bainov and Semoinov [15,16], Bainov and Covachev [17] and Benchohra et al. [18]. Impulsive fractional differential equations represent a real framework for mathematical modelling to real world problems. Significant progress has been made in the theory of impulsive fractional differential equations [19-21].

We consider a class of impulsive fractional integrodifferential equations with nonlocal conditions of the form

(1.1)

(1.2)

(1.3)

Where is the Caputo fractional derivative, the function is continuous and the function is continuous,

and represent the right and left limits of at, and is a continuous function,.

Nonlocal conditions were initiated by Byszewski [22] who proved the existence and uniqueness of mild and classical solutions of nonlocal Cauchy problems. As remarked by Byszewski [23,24], the nonlocal condition can be more useful than the standard initial condition to describe some physical phenomena. For example, may be given by

where are given constants and .

In this article, our aim is to show sufficient conditions for the existence and uniqueness of solutions of solutions to impulsive fractional integro-differential equations with nonlocal conditions.

2. Preliminaries

In this section, we introduce some notations, definitions and preliminary facts which are used throughout this paper. By we denote the Banach space of all continuous functions from into with the norm

Definition 2.1 [5,8]: The fractional (arbitrary) order integral of the function of order is defined by

where is the gamma function, when

Definition 2.2 [5,8]: For a function given on the interval, Riemann-Liouville fractional-order derivative of order of, is defined by

here and denotes the integer part of

, when.

Definition 2.3 [14]: For a function given on the interval, the Caputo fractional-order derivative of order of, is defined by

where.

Lemma 2.4 [25]: (Schaefer’s fixed point theorem). Let be a Banach space and be a completely continuous operator. If the set

is bounded, then has at least a fixed point in X.

3. Existence of Solutions

Consider the set of functions

Definition 3.1: A function whose -derivative exists on is said to be a solution of (1.1)-(1.3), if satisfies the equation

on and satisfies the conditions

where.

To prove the existence of solutions to (1.1)-(1.3), we need the following auxiliary lemmas.

Lemma 3.2: Let, then the equation

has solutions

Lemma 3.3: Let, then

for some.

As a consequence of Lemma 3.2 and Lemma 3.3, we have the following result Lemma 3.4: Let, and let be continuous. A function is a solution of the fractional integral equation

(3.1)

if and only if is a solution of the fractional nonlocal BVP

(3.2)

(3.3)

(3.4)

Proof Assume satisfies (3.2)-(3.4).

If then.

Lemma 3.3 implies

If, by Lemma 3.3, it follows that

If, then from Lemma 3.3 we get

If, then again from we have (3.1).

Conversely, assume that satisfies the impulsive fractional integral equation (3.1). If, then and using the fact that is the left inverse of, we get.

If and using the fact that, where is a constant, we conclude that

Also, we can easily show that

Theorem: Assume that:

(H_{1}) There exists a constant such that

for each and each;

(H_{2}) There exists a constant such that

, for each and;

(H_{3}) There exists a constant such that

, for each, then the problem

(1.1)-(1.3) has at least one solution on.

Proof Consider the operator

defined by

Clearly, the fixed points of the operator are solution of the problem (1.1)-(1.3).

We shall use Schaefer’s fixed point theorem to prove that has a fixed point. The proof will be given in several steps.

Step 1: is continuous.

Let be a sequence such that in. Then for each

Since is continuous function, we have

as.

For each,

Since and are continuous functions, we have as.

Therefore, is continuous.

Step 2: maps bounded sets into bounded sets in.

Indeed, it is enough to show that for any, there exists a positive constant such that for each

, we have

. By (H_{1}), (H_{2}) and (H_{3}), for each, we have

For, we have

Let

then

Step 3: maps bounded sets into equicontinuous sets of.

Let, be a bounded set of as in Step 2, and let. For

, we have

For, we have

As, the right-hand side of the above inequality tends to zero. As a consequence of Steps 1 to 3 together with the Arzel’a-Ascoli theorem, we can conclude that is completely continuous.

As a consequence of Lemma 2.4 (Schaefer’s fixed point theorem), we deduce that has a fixed point which is a solution of the problem (1.1)-(1.3).

4. Acknowledgements

This work was supported by the natural science foundation of Hunan Province (13JJ6068, 12JJ9001), Hunan provincial science and technology department of science and tech-neology project (2012SK3117), Science foundation of Hengyang normal university of China (No. 12B35) and Construct program of the key discipline in Hunan Province.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | [1] J. A. Tenreiro Machado, V. Kiryakova and F. Mainardi, “Recent History of Fractional Calculus,” Communications in Nonlinear Science and Numerical Simulation, Vol. 16, No. 3, 2011, pp. 1140-1153. |

[2] | A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, “Theory and Applications of Fractional Differential Equations,” North-Holland Mathematics Studies, Vol. 204, 2006. doi:10.1016/S0304-0208(06)80001-0 |

[3] | K. Diethelm, “The Analysis of Fractional Differential Equations,” Springer-Verlag, Berlin, Heidelberg, 2010. doi:10.1007/978-3-642-14574-2 |

[4] | K. S. Miller and B. Ross, “An Introduction to the Fractional Calculus and Fractional Differential Equations,” John Wiley, New York, 1993. |

[5] | I. Podlubny, “Fractional Differential Equations,” Academic Press, San Diego, New York, London, 1999. |

[6] | S. G. Samko, A. A. Kilbas and O. I. Marichev, “Fractional Integral and Derivatives,” Gordon and Breach Science Publisher, London, 1993. |

[7] | J. Sabatier, O. P. Agrawal and J. A. Tenreiro Machado, “Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering,” Springer, Berlin, 2007. doi:10.1007/978-1-4020-6042-7 |

[8] | V. Lakshmikantham, S. Leela and J. Vasundhara Devi, “Theory of Fractional Dynamic Systems,” Cambridge Academic Publishers, Cambridge, 2009. |

[9] | S. Zhang, “Positive Solutions for Boundary-Value Problems of Nonlinear Fractional Differential Equations,” Electronic Journal of Qualitative Theory of Differential Equations, Vol. 36, 2006, pp. 1-12. |

[10] | B. Ahmad and J. J. Nieto, “Existence of Solution for Non-Local Boundary Value Problems of Higher-Order Nonlinear Fractional Differential Equations,” Abstract and Applied Analysis, Vol. 2009, 2009, pp. 1-9. doi:10.1155/2009/494720 |

[11] | A. A. Kilbas and S. A. Marzan, “Nonlinear Differential Equations with the Caputo Fractional Derivative in the Space of Continuously Differentiable Functions,” Differential Equations, Vol. 41, No. 1, 2005, pp. 84-89. doi:10.1007/s10625-005-0137-y |

[12] | V. D. Milman and A. D. Myshkis, “On the Stability of Motion in the Presence of Impulses (Russian),” Siberial Mathematical Journal, Vol. 1, No. 2, 1960, pp. 233-237. |

[13] | A. M. Samoilenko and N. A. Perestyuk, “Differential Equations with Impulses,” Viska Scola, Kiev, 1987 (in Russian). |

[14] | V. Lakshmikantham, D. D. Baino and P. S. Simeonov, “Theory of Impulsive Differential Equations,” World Scientific Publishing Corporation, Singapore City, 1989. doi:10.1142/0906 |

[15] | D. D. Baino and P. S. Simeonov, “Systems with Impulsive Effects,” Horwood, Chichister, 1989. |

[16] | D. D. Baino and P. S. Simeonov, “Impulsive Differential Equations: Periodic Solutions and Its applications,” Longman Scientific and Technical Group, England, 1993. |

[17] | D. D. Baino and V. C. Covachev, “Impulsive Differential Equations with a Small Perturbations,” World Scientific, New Jersey, 1994. doi:10.1142/2058 |

[18] | M. Benchohra, J. Henderson and S. K. Ntonyas, “Impulsive Differential Equations and Inclusions,” Hindawi Publishing Corporation, New York, 2006. doi:10.1155/9789775945501 |

[19] | R. P. Agarwal, M. Benchohra and B. A. Salimani, “Existence Results for Differential Equations with Fractional Order and Impulses,” Memoir on Differential Equations and Mathematical Physics, Vol. 44, 2008, pp. 1-21. |

[20] | M. Benchohra and B. A. Salimani, “Existence and Uniqueness of Solutions to Impulsive Fractional Differential Equations,” Electronic Journal of Differential Equations, Vol. 2009, No. 10, 2009, pp. 1-11. |

[21] | M. Fecken, Y. Zhong and J. Wang, “On the Concept and existence of Solutions for Impulsive Fractional Differential Equations,” Communications in Non-Linear Science and numerical Simulation, Vol. 17, No. 7, 2012, pp. 3050-3060. doi:10.1016/j.cnsns.2011.11.017 |

[22] | L. Byszewski and V. Lakshmikantham, “Theorem about the Existence and Uniqueness of a Solution of a Nonlocal Abstract Cauchy Problem in a Banach Space,” Journal of Applied Analysis, Vol. 40, 1991, pp. 11-19. doi:10.1080/00036819008839989 |

[23] | L. Byszewski, “Theorems about Existence and Uniqueness of Solutions of a Semilinear Evolution Nonlocal Cauchy Problem,” Journal of Mathematical Analysis and Applications, Vol. 162, No. 2, 1991, pp. 494-505. doi:10.1016/0022-247X(91)90164-U |

[24] | L. Byszewski, “Existence and Uniqueness of Mild and Classical Solutions of Semilinear Functional-Differential Evolution Nonlocal Cauchy Problem,” Selected Problems of Mathematics, 50th Anniversary Cracow University of Technology, No. 6, Cracow University of Technology, Krakow, 1995, pp. 25-33. |

[25] | J. X. Sun, “Nonlinear Functional Analysis and Its Application,” Science Press, Beijing, 2008. |

Journals Menu

Contact us

+1 323-425-8868 | |

customer@scirp.org | |

+86 18163351462(WhatsApp) | |

1655362766 | |

Paper Publishing WeChat |

Copyright © 2024 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.