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D. Baleanu, H. Mohammadi, Sh. Rezapour, "Positive Solutions of an Initial Value Problem for Nonlinear Fractional Differential Equations", Abstract and Applied Analysis, vol. 2012, Article ID 837437, 7 pages, 2012. https://doi.org/10.1155/2012/837437
Positive Solutions of an Initial Value Problem for Nonlinear Fractional Differential Equations
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.
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 . Also, the Riemann-Liouville fractional derivative of order is defined by , where and the right side is pointwise defined on (). The formula of Laplace transform for the Riemann-Liouville derivative is defined by when the limiting values are finite and . This formula simplifies to . Also, two-parametric Mittag-Leffler function is defined by for and . Analytic properties and asymptotical expansion of this function are given in . 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 )
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 . We need the following fixed point theorems for obtaining our results.
Lemma 1.1 (see ). 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 ). 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 ). 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.
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.
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.
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