Advances in Mathematical Physics
Volume 2017 (2017), Article ID 6971743, 6 pages
https://doi.org/10.1155/2017/6971743

Research Article

The Asymptotic Behaviours of a Class of Neutral Delay Fractional-Order Pantograph Differential Equations

Baojun Miao and Xuechen Li

School of Mathematics and Statistics, Xuchang University, Xuchang 461000, China

Correspondence should be addressed to Baojun Miao; bjmiao2005@126.com

Received 1 April 2017; Revised 8 July 2017; Accepted 3 August 2017; Published 22 October 2017

Academic Editor: Antonio Scarfone

Copyright © 2017 Baojun Miao and Xuechen Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

By using fractional calculus and the summation by parts formula in this paper, the asymptotic behaviours of solutions of nonlinear neutral fractional delay pantograph equations with continuous arguments are investigated. The asymptotic estimates of solutions for the equation are obtained, which may imply asymptotic stability of solutions. In the end, a particular case is provided to illustrate the main result and the speed of the convergence of the obtained solutions.

1. Introduction

Throughout the past decade, the study of the theory of fractional differential equations has gained a lot of great interests and attracted growing attention. Fractional differential equation has been used successfully in many applied fields such as physics, quantum mechanics, field theory, chemistry, optimal control, and engineering. Hence, the existence and uniqueness of fractional differential equations have been extensively researched by many experts and scholars [1–7]. It is well known that the pantograph equation arises in quite different fields of pure and applied mathematics and have been investigated extensively [8–10]. Recently, due to its importance in many applied fields and playing an extremely important role in explaining many different phenomena, it is also interesting to study the existence and asymptotic behaviours of the generalized and multipantograph equations. Yüzbasi et al. [8] investigated the numerical solution of generalized pantograph equation with a linear functional argument by virtue of introducing a collocation method based on the Bessel polynomials for the approximate solution of the pantograph equations. Rahimkhani et al. [11] obtained numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet. Yu [12] used variational iteration method to solve the multipantograph delay equation, and sufficient conditions were given to assure the convergence of the method. Balachandran and Kiruthika [13] studied the existence of solutions of abstract fractional pantograph equations by using the fractional calculus and fixed point theorems. Peics [14] researched the asymptotic behaviour of solutions of difference equations with continuous argument and obtained asymptotic estimates. Doha et al. [15] using the Jacobi rational-Gauss method reduced solving the generalized pantograph equation to a system of algebraic equations and obtained reasonable numerical results. Motivated and inspired by thoughts and methods of the above recent papers, based on the difference pantograph equations [14], we establish the following neutral delay fractional pantograph equations with continuous arguments of the form

where , and are all continuous real function and are assumed to satisfy the following two hypotheses throughout this paper:

and .

There exists continuous real function which satisfies the following: when , , where and ; when , , which implies .

The aim of the paper is to research the asymptotic behaviour of solutions for fractional delay equations with continuous arguments (1), give asymptotic estimates for the speed of convergence of solutions, and apply our results to the particular cases. The paper is organized as follows. In Section 2, we present some notations and definitions of fractional calculus needed in the rest of the paper. Section 3 is devoted to the main results of the paper by using fractional calculus and the summation by parts formula. In Section 4, we obtain the speed of the convergence of the solutions for the initial value problem (1) by using Theorem 3 to deal with special case.

2. Preliminaries

For convenience of the reader, we need to introduce some notations and properties of fractional calculus which will be used in the proof of our results. Let be the set of natural numbers, and let be the set of real numbers and .

Use the notations

for ,   and arbitrary real function . The difference operator is defined by , where the function is defined for . The difference operator is defined by , where the function is defined for .

Let be a positive real number and set

for . Then is an increasing sequence such that

Fix a point such that , define the natural number such that and , and the set is defined by

A solution of (1) is a function which is defined for and satisfies (1) for . For a given real bounded continuous function on , (1) has a unique solution satisfying the initial condition .

Definition 1 (see [16]). The Riemann-Liouville fractional integral operator of order , of a function , is defined by

where is the Euler Gamma function.

Definition 2 (see [16]). The Riemann-Liouville fractional derivative of order , ,  , is defined by

where the function has absolutely continuous derivatives up to order .

3. Main Results

In this section, let , .

Theorem 3. Assume that the conditions and hold; there exists a real continuous function bounded on the initial interval such that

and there exists a real number such that

where

Assume that is a solution of the initial value problem (1); then

Proof. It is easy to see that (1) is equivalent to the following integral equation:

that is,

Let ; then the function will satisfy the equation

Let and ; then (12) is equivalent to

Summing up both sides of the above equality from to , we get

So from (16), we obtain

Let

Since for and , by utilizing the summation by parts formula, we can get

From the above inequality, we get

Further, we can obtain

Consequently,

that is, .

Remark 4. In Theorem 3, suppose that the function is monotone increasing; then the sequence has only positive members and the assumption

implies the existence of a real number that satisfies condition (9). If the function is monotone decreasing, then condition (9) is satisfied with .

Remark 5. In Theorem 3, if is positive continuous real function, then the solution of the initial value problem (1) is nonnegative for .

Theorem 6. If the conditions and hold, there exists a real continuous function bounded on the initial interval such that

and there exists a real number such that

where

Assume that is a solution of the positive initial value problem (1); then

Proof. It is similar to the proof of Theorem 3. It is seen that (1) is equivalent to the following integral equation:

By utilizing the hypothesis (24), it is easy to see that the function is positive, by the virtue of Remark 5, and it implies that the function is also positive. So we can get the following estimate:

Define for ,

Because of for and , by means of the summation by parts formula, we can draw the conclusion

From the above inequality, it follows that

Hence, ; that is, .

Remark 7. In Theorem 6, if the function is monotone decreasing, then the sequence defined by (26) is all negative and the assumption implies the existence of a real number which satisfies condition (25). If the function is monotone increasing, then condition (25) is satisfied with .

Theorem 8. Assume that conditions and hold; there exists a real continuous function bounded on the initial interval such that

and there exist real numbers and such that conditions (9) and (25) hold for all positive integer . Suppose that is a solution of the positive initial value problem (1); then

Proof. According to the consequence of Theorems 3 and 6, we can easily obtain the proof.

4. Particular Case

In this section, we use Theorem 3 to deal with special case. By seeking an appropriate function , we can also obtain the speed of the convergence of the solutions for the initial value problem (1).

Corollary 9. Let satisfy for real numbers . Suppose that conditions and hold; there exist real numbers and such that , ,

Let be a solution of the initial value problem (1); then

where , , and

Proof. By using relations and , we can obtain

Let . Because of , we have

Consequently, condition (8) is effective. Furthermore, we obtain

It follows that Theorem 3 and Remark 4 imply the conclusion.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The present investigation is supported by the Natural Science Foundation of Henan Province (Grant no. 142300410103).

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