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Abstract and Applied Analysis
Volume 2010 (2010), Article ID 865139, 8 pages
http://dx.doi.org/10.1155/2010/865139
Research Article

Asymptotically Linear Solutions for Some Linear Fractional Differential Equations

1Department of Mathematics & Computer Science, Çankaya University, Ogretmenler Cad. 14 06530, Balgat, Ankara, Turkey
2Department of Mathematics & Computer Science, University of Craiova, DAL, Tudor Vladimirescu 26, 200534 Craiova, Romania
3Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA

Received 19 September 2010; Accepted 9 November 2010

Academic Editor: Paul Eloe

Copyright © 2010 Dumitru Baleanu et al. 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

We establish here that under some simple restrictions on the functional coefficient the fractional differential equation , has a solution expressible as for , where designates the Riemann-Liouville derivative of order and .

1. Introduction

Consider the ordinary differential equation where the function is continuous such that Here, the functions and are continuous, and there exists with

Then, given , , (1.1) has a solution , defined in a neighborhood of , which is expressible as for , as for and, finally, as for when . Such a solution is called asymptotically linear in the literature. In particular, these developments apply to the homogeneous linear differential equation .

A unifying technique of proof for such estimates can be read in [1] and is based on the next reformulation of the differential equation (1.1) for some large enough. For a different approach, the so-called Riccatian method, in the case of intermediate asymptotic (, ), see the technique from [2, 3].

The study of asymptotically linear solutions to linear and nonlinear ordinary differential equations is of importance in fluid mechanics, differential geometry (Jacobi fields, e.g., [4, page 239]), bidimensional gravity (the geodesics of the Euclidean planar spray being the asymptotically linear solutions ), and others.

In this note, we are interested in the existence of a fractional variant for the problem of asymptotically linear solutions which can be formulated as follows: are there any nontrivial fractional differential equations which have only asymptotically linear solutions and also their solution sets contain solutions (asymptotically linear) for all the prescribed values of numbers , , and ? To the best of our knowledge, this is an open problem in the theory of fractional differential equations.

Fractional differential equations have been of great interest during the last few years. This follows from the intensive development of the theory of fractional calculus [5, 6] followed by the applications of its methods in various sciences and engineering [7]. We can mention that the fractional differential equations are playing an important role in fluid dynamics, traffic model with fractional derivative, measurement of viscoelastic material properties, modeling of viscoplasticity, control theory, economy, nuclear magnetic resonance, mechanics, optics, signal processing, and so on. Basically, the fractional differential equations are used to investigate the dynamics of the complex systems; the models based on these derivatives have given superior results as those based on the classical derivatives, see [8, page 305], [911].

To introduce a fractional differential operator of order , there are three options. The first two consist of a mixed ordinary differential-Caputo fractional differential operator, namely, , and, respectively, a Riemann-Liouville fractional differential operator .

We recall that represents the Riemann-Liouville derivative of order a of some function , cf. [8, page 68], and stands for Euler's function Gamma. Remark as well that, in general, for a, , see [8, page 74]. To deal with iterations, Miller and Ross [12] coined the term sequential fractional differential operator of order for the quantity , cf. [8, pages 108, 122].

Also, the quantity has been called the Caputo derivative in physics, see [8, page 79], and it is often preferred due to its sound explanation of what the initial data signify.

The first variant of differential operator was used in [13] to study the existence of solutions of nonlinear fractional differential equations that obey the restrictions

The second variant of differential operator, see [14], was employed to prove that, for any real numbers , , the linear fractional differential equation possesses a solution with the asymptotic development

A recent application of the Caputo derivative can be found in [15].

All of these fractional differential operators are based upon the natural splitting of the second-order operator , namely, . Here, we shall introduce a different fractionalizing of which is based on the identities stemming from the integration technique in the Lie algebra , cf. [16, page 23].

In the following section, we give a positive (partial) answer to the preceding open question. In fact, we produce some simple conditions regarding the continuous function such that, given , the fractional differential equation (FDE) below possesses a solution with the asymptotic development when .

2. Asymptotically Linear Solutions

Let us start with a result regarding the case of intermediate asymptotic.

Proposition 2.1. Set the numbers , , and , , such that Assume also that is confined to Then, the FDE has a solution , with , which verifies the asymptotic formula when .

Proof. Introduce the complete metric space , where and the metric is given by the usual formula In particular, for all .
Introduce the function via the formulas Since , we deduce that can be continued backward to 0; so, its extension belongs to . Also, .
Define further the integral operator by the formula The estimate shows that is well defined by taking into account (2.1), (2.2).
Now, given , , we have and so .
The operator being a contraction, it has a unique fixed point . Since when , the proof is complete.

Theorem 2.2. Assume that (2.1) holds true and verifies the sharper restriction where . Then, the solution of FDE (2.3) from Proposition 2.1 has the asymptotic development when .

Proof. Notice that where is the Beta function, cf. [8, page 6].
Via (2.9), we have the estimate
By means of L’Hôpital’s rule, we conclude that (recall (2.5))
The proof is complete.

Our main contribution is given next.

Theorem 2.3. Set the numbers , , , with , and , , such that Assume also that satisfies the inequality Then the FDE (1.9) has a solution , with and , which has the asymptotic development

Proof. Keeping the notations from Proposition 2.1, introduce the change of variables and the integral operator with the formula
As before, we have the estimates for all , , .
Finally, for the fixed point of the operator , we have that
The proof is complete.

3. Conclusion

A particular case of Theorem 2.3 is when , , that is, when the solution of (1.9) reads as for . Notice from (2.14) that the behavior of the functional coefficient is confined to . However, there is no restriction with respect to the (eventual) zeros of . On the other hand, in the recent contribution [13, Section??3], we were forced to request that the functional coefficient of the FDE has a unique zero in . In conclusion, the fractional differential operators proposed in (1.9), (2.3) allow more freedom for the functional coefficient.

Acknowledgment

We are indebted to a referee for several insightful suggestions.

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