- Research Article
- Open Access
Employing of Some Basic Theory for Class of Fractional Differential Equations
© A. Babakhani and D. Baleanu 2011
- Received: 19 September 2010
- Accepted: 15 November 2010
- Published: 29 December 2010
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)).
- Fractional Order
- Fractional Derivative
- Global Existence
- Fractional Differential Equation
- Initial Value Problem
where and . Recall that is the Banach space of continuous functions from the interval into endowed with uniform norm.
where . We denote by and by . Also and refer to and , respectively.
Let and . Then
can be found in [19, page 53] and (ii) is an immediate consequence of (1.6) and (i).
We now first discuss a fundamental result relative to the fractional inequalities.
Let , , and
one of the foregoing inequalities being strict. Moreover, if is nondecreasing in for each and , then one has .
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.
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.
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.
so that and will be observing in the proof of this theorem.
This proves that is a solution of the IVP (1.2), and the proof is complete.
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.
exists on . Clearly . The uniform continuity of gives argument as before (as in Theorem 3.1), that is a solution of IVP (1.2).
where , , and .
Using Theorem 2.1, we get on as . Therefore, the proof is complete.
We need the following comparison result before we proceed further.
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.
with exist in .
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.
- 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
- Hilfe R (Ed): Applications of Fractional Calculus in Physics. World Scientific, River Edge, NJ, USA; 2000:viii+463.Google Scholar
- 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
- Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.MATHGoogle Scholar
- Podlubny I: Geometric and physical interpretation of fractional integration and fractional differentiation. Fractional Calculus & Applied Analysis 2002,5(4):367-386.MathSciNetMATHGoogle Scholar
- 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
- Lakshmikantham V, Leela S, Vasundhara J: Theory of Fractional Dynamic Systems. Cambridge Academic, Cambridge, UK; 2009.MATHGoogle Scholar
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives. Gordon and Breach Science, Yverdon, Switzerland; 1993:xxxvi+976.MATHGoogle Scholar
- 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
- 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
- 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
- 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
- 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
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.