Open Access

Employing of Some Basic Theory for Class of Fractional Differential Equations

Advances in Difference Equations20102011:296353

https://doi.org/10.1155/2011/296353

Received: 19 September 2010

Accepted: 15 November 2010

Published: 29 December 2010

Abstract

Basic theory on a class of initial value problem of some fractional differential equation involving Riemann-Liouville differential operators is discussed by employing the classical approach from the work of Lakshmikantham and A. S. Vatsala (2008). The theory of inequalities, local existence, extremal solutions, comparison result and global existence of solutions are considered. Our work employed recent literature from the work of (Lakshmikantham and A. S. Vatsala, (2008)).

1. Introduction

Differential equations of fractional order have recently proven to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetism, and so forth [15]. There has been a significant development in the study of fractional differential equations and inclusions in recent years; see the monographs of Kilbas et al. [6], Lakshmikantham et al. [7], Podlubny [4], and the survey by Agarwal et al. [8]. For some recent contributions on fractional differential equations, see [920] and the references therein. Very recently in [10, 11, 21, 22], the author and other researchers studied the existence and uniqueness of solutions of some classes of fractional differential equations with delay. For more details on the geometric and physical interpretation for fractional derivatives of both the Caputo types see [5, 23]. Băleanu and Mustafa [16] allowed for immediate applications of a general comparison-type result from the recent literature [24]. Lakshmikantham and Vatsal have discussed basic theory of fractional differential equations for initial value problem fractional differential equations type [24]
(1.1)
This paper deals with the basic theory of initial value problem (IVP) for a generalized class of fractional order differential equation of the form
(1.2)

where and . Recall that is the Banach space of continuous functions from the interval into endowed with uniform norm.

We begin in this section with the recall of some definitions and results for fractional calculus which are used throughout this paper [4, 19, 20].

The left-sided Riemann-Liouville fractional integral of a function of order is defined as
(1.3)
and the left sided Riemann-Liouville fractional derivative operator of order is defined by
(1.4)

where . We denote by and by . Also and refer to and , respectively.

Assume that , if the fractional derivative is integrable, then [4, page 72]
(1.5)
If is continuous on then is integrable, [25], and (1.5) reduces to
(1.6)

Proposition 1.1.

Let and . Then

  1. (i)

    ,

     
  2. (ii)

    ,

     
where is a nonnegative integer, and
(1.7)
Proof.
  1. (i)

    can be found in [19, page 53] and (ii) is an immediate consequence of (1.6) and (i).

     

2. Strict and Nonstrict Inequalities

Since is assumed to be continuous, the initial value problem (1.2) is equivalent to the following Volterra fractional integral [10, 25]:
(2.1)

Note.

Let us consider the notation of for second term in (2.1) which is used in throughout of text and so that
(2.2)

We now first discuss a fundamental result relative to the fractional inequalities.

Theorem 2.1.

Let , , and

  1. (i)

    ,

     
  2. (ii)

    ,

     

one of the foregoing inequalities being strict. Moreover, if is nondecreasing in for each and , then one has .

Proof.

Suppose that for each , the conclusion is not true. Then, because of the continuity of the functions involved and it follows that there exists a such that and
(2.3)
Let us suppose that the inequality (ii) is strict. Then using the nondecreasing nature of and (2.3) we get
(2.4)

which is a contradiction in view of (2.3). Hence the conclusion of this theorem holds and the proof is complete.

The next result is for nonstrict inequalities, which require a one-sided Lipschitz type condition.

Theorem 2.2.

Assume that the conditions of Theorem 2.1 hold with nonstrict inequalities (i) and (ii). Suppose further that
(2.5)
whenever and . Then, , and implies
(2.6)

Proof.

Set , for small so that we have,
(2.7)
Now,
(2.8)
In view of (2.7), using one-sided Lipschitz condition (2.5) we see that
(2.9)
Now, since , we arrive at
(2.10)

in view of the condition . We now apply Theorem 2.1 to the inequalities (i), (2.9), and (2.10) to get , . Since is arbitrary, we conclude that (2.6) is true, and we are done.

3. Local Existence and Extremal Conditions

In this section, we will consider the local existence and the existence of extremal solutions for the IVP (1.2). We now first discuss Peano's type existence result.

Theorem 3.1.

Assume that , where , and let on . Then the IVP (1.2) has at least one solution on , where
(3.1)

so that and will be observing in the proof of this theorem.

Proof.

Let be a continuous function on such that . For , we define a function on and
(3.2)
on , where . We observe that
(3.3)
where
(3.4)
Note that, , as . Then , and hence we can consider in the last above inequality. Thus .
(3.5)
because of the choice of . If , we can employ (3.2) to extend as a continuous function on , such that holds. Continuing this process, we can define over so that is satisfied on . Furthermore, letting we see that
(3.6)
Notice that and , when . Let,
(3.7)
firstly we have
(3.8)
We, therefore, set , and get
(3.9)
Finally, inequality (3.6) from inequality (3.9) becomes
(3.10)
provided that
(3.11)
It then follows from (3.4) and (3.6) that the family forms an equicontinuous and uniformly bounded functions. Ascoli-Arzela theorem implies that the existence of a sequence such that as , and exists uniformly on . Since is uniformly continuous, we obtain that tends uniformly to as , and, hence, term by term integration of (3.2) with yields
(3.12)

This proves that is a solution of the IVP (1.2), and the proof is complete.

Theorem 3.2.

Under the hypothesis of Theorem 3.1, there exists extremal solution for the IVP (1.2) on the interval , provided is nondecreasing in for each , where will be observed in the proof of this theorem.

Proof.

We will prove the existence of maximal solution only, since the case of minimal solution is very similar. Let and consider the fractional differential equation with an initial condition
(3.13)
We observe that is defined and continuous on
(3.14)
and on . We then deduce from Theorem 3.1 that IVP (3.6) has a solution on the interval . Now for , we have
(3.15)
where was introduced by (2.2). In view of Theorem 2.1 to get . Consider the family of functions on . We have
(3.16)
where was denoted by (3.4), and
(3.17)
Showing that the family is uniformly bounded. Also, if then there exists a constant such that
(3.18)
Following the computation similar to (3.10) with suitable changes. This proves that the family is equicontinuous. Hence there exists a sequence with as and the uniform limit
(3.19)

exists on . Clearly . The uniform continuity of gives argument as before (as in Theorem 3.1), that is a solution of IVP (1.2).

Next, we show that is required maximal solution of (1.2), on . Let be a any solution of (1.2) on . Then we have
(3.20)

where , , and .

Using Theorem 2.1, we get on as . Therefore, the proof is complete.

4. Global Existence

We need the following comparison result before we proceed further.

Theorem 4.1.

Assume that , are continuous, and is nondecreasing with respect to the second argument such that
(4.1)
where is defined by (3.5). Let be the maximal solution of
(4.2)
existing on such that . Then we have
(4.3)

Proof.

In view of the definition of the maximal solution , it is enough to prove, to conclude (4.3), that
(4.4)
where is any solution of
(4.5)
Now, it follows from (4.4) that
(4.6)

where . Then applying Theorem 2.1, we get immediately (4.1) and since uniformly on each , the proof is complete.

We are now in position to prove global existence result.

Theorem 4.2.

Assume that continuous. Let there exists function continuous and nondecreasing with respect to the second argument such that
(4.7)
If the maximal solution of the initial value problem
(4.8)
exists in then all solutions of the initial value problem
(4.9)

with exist in .

Proof.

Let be any solution of IVP (4.8) such that , which exists on for , and the value of cannot be increased further. Set for . Then using the assumption (4.4), we get
(4.10)
Applying the comparison Theorem 4.1, we obtain
(4.11)
Since is assumed to exist on , it follows that
(4.12)
Now, let . Then employing the arguments similar to estimating (3.19) and using (4.7) and the bounded of we arrive at
(4.13)
Letting and using Cauchy criterion, it follows that exists. We define and consider the new IVP
(4.14)

By the assumed local existence, we find that can be continued beyond , contradicting our assumption. Hence, every solution of (4.9) exists on , and the proof is complete.

Authors’ Affiliations

(1)
Faculty of Basic Science, Babol University of Technology
(2)
Department of Mathematics and Computer Science, Cankaya University
(3)
Institute of Space Sciences

References

  1. Glockle WG, Nonnenmacher TF: A fractional calculus approach to self-similar protein dynamics. Biophysical Journal 1995,68(1):46-53. 10.1016/S0006-3495(95)80157-8View ArticleGoogle Scholar
  2. Hilfe R (Ed): Applications of Fractional Calculus in Physics. World Scientific, River Edge, NJ, USA; 2000:viii+463.Google Scholar
  3. Metzler R, Schick W, Kilian H-G, Nonnenmacher TF: Relaxation in filled polymers: a fractional calculus approach. The Journal of Chemical Physics 1995,103(16):7180-7186. 10.1063/1.470346View ArticleGoogle Scholar
  4. Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.MATHGoogle Scholar
  5. Podlubny I: Geometric and physical interpretation of fractional integration and fractional differentiation. Fractional Calculus & Applied Analysis 2002,5(4):367-386.MathSciNetMATHGoogle Scholar
  6. Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies. Volume 204. Elsevier Science B.V., Amsterdam, The Netherlands; 2006:xvi+523.Google Scholar
  7. Lakshmikantham V, Leela S, Vasundhara J: Theory of Fractional Dynamic Systems. Cambridge Academic, Cambridge, UK; 2009.MATHGoogle Scholar
  8. Agarwal RP, Benchohra M, Hamani S: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Applicandae Mathematicae 2010,109(3):973-1033. 10.1007/s10440-008-9356-6MathSciNetView ArticleMATHGoogle Scholar
  9. Daftardar-Gejji V, Babakhani A: Analysis of a system of fractional differential equations. Journal of Mathematical Analysis and Applications 2004,293(2):511-522. 10.1016/j.jmaa.2004.01.013MathSciNetView ArticleMATHGoogle Scholar
  10. Babakhani A, Enteghami E: Existence of positive solutions for multiterm fractional differential equations of finite delay with polynomial coefficients. Abstract and Applied Analysis 2009, 2009:-12.Google Scholar
  11. Babakhani A: Positive solutions for system of nonlinear fractional differential equations in two dimensions with delay. Abstract and Applied Analysis 2010, 2010:-16.Google Scholar
  12. Baleanu D, Golmankhaneh AK, Nigmatullin R, Golmankhaneh AK: Fractional Newtonian mechanics. Central European Journal of Physics 2010,8(1):120-125. 10.2478/s11534-009-0085-xMATHGoogle Scholar
  13. Baleanu D, Trujillo JI: A new method of finding the fractional Euler-Lagrange and Hamilton equations within Caputo fractional derivatives. Communications in Nonlinear Science and Numerical Simulation 2010,15(5):1111-1115. 10.1016/j.cnsns.2009.05.023MathSciNetView ArticleMATHGoogle Scholar
  14. Baleanu D: New applications of fractional variational principles. Reports on Mathematical Physics 2008,61(2):199-206. 10.1016/S0034-4877(08)80007-9MathSciNetView ArticleMATHGoogle Scholar
  15. Benchohra M, Hamani S, Ntouyas SK: Boundary value problems for differential equations with fractional order. Surveys in Mathematics and Its Applications 2008, 3: 1-12.MathSciNetMATHGoogle Scholar
  16. Băleanu D, Mustafa OG: On the global existence of solutions to a class of fractional differential equations. Computers & Mathematics with Applications 2010,59(5):1835-1841. 10.1016/j.camwa.2009.08.028MathSciNetView ArticleMATHGoogle Scholar
  17. Chang Y-K, Nieto JJ: Some new existence results for fractional differential inclusions with boundary conditions. Mathematical and Computer Modelling 2009,49(3-4):605-609. 10.1016/j.mcm.2008.03.014MathSciNetView ArticleMATHGoogle Scholar
  18. Ouahab A: Some results for fractional boundary value problem of differential inclusions. Nonlinear Analysis: Theory, Methods & Applications 2008,69(11):3877-3896. 10.1016/j.na.2007.10.021MathSciNetView ArticleMATHGoogle Scholar
  19. Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons, New York, NY, USA; 1993:xvi+366.MATHGoogle Scholar
  20. Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives. Gordon and Breach Science, Yverdon, Switzerland; 1993:xxxvi+976.MATHGoogle Scholar
  21. Belarbi A, Benchohra M, Ouahab A: Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces. Applicable Analysis 2006,85(12):1459-1470. 10.1080/00036810601066350MathSciNetView ArticleMATHGoogle Scholar
  22. Benchohra M, Henderson J, Ntouyas SK, Ouahab A: Existence results for fractional order functional differential equations with infinite delay. Journal of Mathematical Analysis and Applications 2008,338(2):1340-1350. 10.1016/j.jmaa.2007.06.021MathSciNetView ArticleMATHGoogle Scholar
  23. Heymans N, Podlubny I: Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheologica Acta 2006,45(5):765-771. 10.1007/s00397-005-0043-5View ArticleGoogle Scholar
  24. Lakshmikantham V, Vatsala AS: Basic theory of fractional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008,69(8):2677-2682. 10.1016/j.na.2007.08.042MathSciNetView ArticleMATHGoogle Scholar
  25. Babakhani A, Daftardar-Gejji V: Existence of positive solutions for multi-term non-autonomous fractional differential equations with polynomial coefficients. Electronic Journal of Differential Equations 2006, 2006: 129-12.MathSciNetMATHGoogle Scholar

Copyright

© A. Babakhani and D. Baleanu 2011

This article is published under license to BioMed Central Ltd. 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.