Abstract and Applied Analysis

Volume 2012 (2012), Article ID 837437, 7 pages

doi:10.1155/2012/837437

Research Article

Positive Solutions of an Initial Value Problem for Nonlinear Fractional Differential Equations

D. Baleanu,1,2 H. Mohammadi,³ and Sh. Rezapour³

¹Department of Mathematics, Cankaya University, Ogretmenler Cad. 14 06530, Balgat, Ankara, Turkey

²Institute of Space Sciences, Magurele, Bucharest, Romania

³Department of Mathematics, Azarbaijan University of Shahid Madani, Tabriz, Iran

Received 23 January 2012; Accepted 20 March 2012

Academic Editor: Juan J. Trujillo

Copyright © 2012 D. 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 investigate the existence and multiplicity of positive solutions for the nonlinear fractional differential equation initial value problem , where is the standard Riemann-Liouville differentiation and is continuous. By using some fixed-point results on cones, some existence and multiplicity results of positive solutions are obtained.

1. Introduction

Fractional differential equations have been subjected to an intense debate during the last few years (see, e.g., [1–5] and the references therein). This trend is due to the intensive development of the theory of fractional calculus itself and by the applications of such constructions in various sciences such as physics, mechanics, chemistry, and engineering [5–15]. The fractional differential equations started to be used extensively in studying the dynamical systems possessing memory effect. Comprehensive treatment of the fractional equations techniques such as Laplace and Fourier transform method, method of Green function, Mellin transform, and some numerical techniques are given in [5, 7, 9] and the references therein. In classical approach, linear initial fractional differential equations are solved by special functions [9, 16]. In some papers, for nonlinear problems, techniques of functional analysis such as fixed point theory, the Banach contraction principle, and Leray-Schauder theory are applied for solving such kind of the problems (see, e.g., [17–19] and the references therein). The existence of nonlinear fractional differential equations of one time fractional derivative is considered in [6, 7, 9, 20]. Also, the existence and multiplicity of positive solutions to nonlinear Dirichlet problem

where is continuous and is the Riemann-Liouville differentiation, have been reviewed by some authors (see e.g., [18–21] and the references therein).

In this paper, by using some fixed-point results, we investigate the existence and multiplicity of positive solutions for the nonlinear fractional differential equation initial value problem

where , is the standard Riemann-Liouville differentiation, and is continuous. Now, we present some necessary notions. The Riemann-Liouville fractional integral of order is defined by [20]. Also, the Riemann-Liouville fractional derivative of order is defined by , where and the right side is pointwise defined on ([20]). The formula of Laplace transform for the Riemann-Liouville derivative is defined by

when the limiting values are finite and . This formula simplifies to [21]. Also, two-parametric Mittag-Leffler function is defined by for and [21]. Analytic properties and asymptotical expansion of this function are given in [9]. For example, if ,, and is a real constant, then , whenever and . Also, by using the formula for integration of the Mittag-Leffler function term by term, we have (see [9])

Let be a cone in a Banach space . The map is said to be a nonnegative continuous concave functional whenever is continuous and for all and [20]. We need the following fixed point theorems for obtaining our results.

Lemma 1.1 (see [22]). Let be a Banach space, a cone in , and two bounded open balls of centered at the origin with . Suppose that is a completely continuous operator such that either

(i) and , or(ii), and ,

holds. Then A has a fixed point in .

Lemma 1.2 (see [23]). Let be a cone in a real Banach space , , , and positive real numbers, a nonnegative concave functional on such that for all and

Suppose that is completely continuous and there exist constants such that

(), and for some we have ,() for all with ,() for all with .

Then A has at least three fixed points , , and such that , , with .

Note that the condition implies whenever .

2. Main Results

As we know, there is an integral form of the solution for the following equation:

Suppose that the functions and are continuous on . Then is a solution for (2.1), where and is the two-parameter function of the Mittag-Leffler type (see [9]). Now, we give an equivalent solution for (2.1). In fact, if we apply the Laplace transform to (2.1), then by using a calculation and finding the inverse Laplace transform we get that is an equivalent solution for (2.1). In this way, note that

where . But, we have

Since , we get and so

Now, we establish some results on existence and multiplicity of positive solutions for the problem (2.1). Let be endowed via the order if and only if for all . Consider the cone and the nonnegative continuous concave functional . Now, we give our first result.

Lemma 2.1. Define by , where and is the two-parameter function of the Mittag-Leffler type. Then is completely continuous.

Proof. Since the mappings and are nonnegative and continuous, it is easy to see that is continuous. Now, we show that is a relatively compact operator. This implies that is completely continuous. Let be a bounded subset. Then there exists a positive constant such that for all . Put . Then, for each , we have

where and . Thus, is uniformly bounded. Now, we show that is equicontinuous. Let and . Thus,

Now, by using the formula for integration of the Mittag-Leffler function term by term given in ( ), we obtain that

Thus, by using the formula , we obtain a common factor . This implies that small changes of cause small changes of . that is, is equicontinuous. Now by using the Arzela-Ascoli theorem, we get that is a relatively compact operator.

Theorem 2.2. Suppose that in the problem (1.2) there exists a positive real number such that

() for all ,() for all with .

Then the problem (1.2) has a positive solution such that .

Example 2.3. Consider the nonlinear fractional differential equation initial value problem

Put and . Since for all and for all , by using Theorem 2.2 we get that this problem has a positive solution we get that this problem has a positive solution with .

Proof. First, let us to consider the operator , where . By using Lemma 2.1, is completely continuous and note that is a solution of the problem (1.2) if and only if . Let and we have for all . By using the assumption , we have

and so . Also, for we have for all . By using the assumption we have

This completes the proof.

Theorem 2.4. Suppose that in the problem (2.1) there exist positive real numbers such that

() for all ,() for all , where

() for all .

Then the problem (2.1) has at least there positive solutions , , and such that , and .

Proof. Define . Then, for all . Note that, the assumption () implies that for all . Thus,

Hence, is a operator on . Also, note that the assumption () implies that for all . Thus, the condition () in Lemma 1.2 holds. It is sufficient that we show that the condition () in Lemma 1.2 holds. Put for all . It is easy to see that and . Thus, and so for all and . But, the assumption () implies that for all and so

Thus, for all . This shows that the condition () in Lemma 1.2 holds. This completes the proof.

Acknowledgments

Research of the second and third authors was supported by Azarbaijan University of Shahid Madani. Also, the authors express their gratitude to the referees for their helpful suggestions which improved final version of this paper.

References

R. P. Agarwal, D. O'Regan Donal, and S. Staněk, “Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 371, no. 1, pp. 57–68, 2010.

K. Balachandran, S. Kiruthika, and J. J. Trujillo, “Remark on the existence results for fractional impulsive integrodifferential equations in Banach spaces,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 6, pp. 2244–2247, 2012.

F. Chen, J. J. Nieto, and Y. Zhou, “Global attractivity for nonlinear fractional differential equations,” Nonlinear Analysis. Real World Applications, vol. 13, no. 1, pp. 287–298, 2012.

D. Băleanu, R. P. Agarwal, O. G. Mustafa, and M. Coşulschi, “Asymptotic integration of some nonlinear differential equations with fractional time derivative,” Journal of Physics A, vol. 44, no. 5, Article ID 055203, 2011.

D. Băleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, Complexity, Nonlinearity and Chaos, World Scientific, 2012.

A. A. Kilbas, O. I. Marichev, and S. G. Samko, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, Switzerland, 1993.

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Application of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier, 2006.

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.

I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.

R. Gorenflo and F. Mainardi, “Fractional calculus: integral and differential equations of fractional order,” in Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996), vol. 378 of CISM Courses and Lectures, pp. 223–276, Springer, Vienna, Austria, 1997.

R. Gorenflo and F. Mainardi, “Fractional relaxation of distributed order,” in Complexus Mundi, pp. 33–42, World Scientific, Hackensack, NJ, USA, 2006.

F. Mainardi, A. Mura, G. Pagnini, and R. Gorenflo, “Sub-diffusion equations of fractional order and their fundamental solutions,” in Proceedings of the International Symposium on Mathematical Methods in Engineering, J. A. Tenreiro-Machado and D. Baleanu, Eds., pp. 23–55, Springer, Ankara, Turkey, 2006.

F. Mainardi, Y. Luchko, and G. Pagnini, “The fundamental solution of the space-time fractional diffusion equation,” Fractional Calculus & Applied Analysis, vol. 4, no. 2, pp. 153–192, 2001.

A. V. Chechkin, R. Gorenflo, I. M. Sokolov, and V. Yu. Gonchar, “Distributed order time fractional diffusion equation,” Fractional Calculus & Applied Analysis, vol. 6, no. 3, pp. 259–279, 2003.

A. Kochubei, “Distributed order calculus and equations of ultraslow diffusion,” Journal of Mathematical Analysis and Applications, vol. 340, no. 1, pp. 252–281, 2008.

K. S. Miller, “Fractional differential equations,” Journal of Fractional Calculus, vol. 3, pp. 49–57, 1993.

V. Daftardar-Gejji and A. Babakhani, “Analysis of a system of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 293, no. 2, pp. 511–522, 2004.

S. Q. Zhang, “The existence of a positive solution for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 252, no. 2, pp. 804–812, 2000.

S. Q. Zhang, “Existence of positive solution for some class of nonlinear fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 278, no. 1, pp. 136–148, 2003.

Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005.

M. Stojanović, “Existence-uniqueness result for a nonlinear n-term fractional equation,” Journal of Mathematical Analysis and Applications, vol. 353, no. 1, pp. 244–255, 2009.

M. A. Krasnoselski, Positive Solutions of Operator Equations, P. Noordhoff, Groningen, The Netherlands, 1964.

R. W. Leggett and L. R. Williams, “Multiple positive fixed points of nonlinear operators on ordered Banach spaces,” Indiana University Mathematics Journal, vol. 28, no. 4, pp. 673–688, 1979.