Existence of Positive Solutions for Multiterm Fractional Differential Equations of Finite Delay with Polynomial Coefficients
Existence of positive solutions has been studied by A. Babakhani and V. Daftardar-Gejji (2003) in case of multiterm nonautonomous fractional differential equations with constant coefficients. In the present paper we discuss existence of positive solutions in case of multiterm fractional differential equations of finite delay with polynomial coefficients.
In last 30 years, the theory of ordinary differential equations of fractional order has become a new important branch (see, e.g., [1–5] and the references therein). Numerous applications of such equations have been presented [3–10]. Existence of positive solution of fractional ordinary differential equations has been well investigated for fractional functional differential equations [1, 6, 11–14]. Ye et al.  have addressed the question of existence of positive solutions for the nonlinear fractional functional differential equation by using the sub- and supersolution method, where , is the standard Riemann-Liouville fractional derivative, and is continuous, as usual, is the space of continuous function from to , , equipped with the sup norm: and denotes the function in defined by They require that the nonlinearity is increasing in for each .
As a pursuit of this in the present paper, we deal with the existence of positive solutions in the case of multiterm differential equations with polynomial coefficients of the fractional type: where and is the standard Riemann-Liouville fractional derivative, , , and is a given continuous function, .
Let be a real Banach space with a cone . introduces a partial order in in the following manner :
Definition 2.1 (see ). For the order interval is defined as
Definition 2.2 (see ). A cone is called normal, if there exists a positive constant such that and implies , where denotes the zero element of .
Definition 2.4 (see [16, 17]). The left-sided Riemann-Liouville fractional derivative of a function is defined as
where , . We denote by and by . If the fractional derivative is integrable, then [16, page 71]
If is continuous on , then and (2.5) reduces to
Proposition 2.5. Let be continuous on , and let be a nonnegative integer, then where
The proof of the above proposition can be found in [17, page 53].
Corollary 2.6. Let , and , , , then
Theorem 2.7 (see ). Let be a Banach space with closed and convex. Assume that is a relatively open subset of with and is a continuous and compact map. Then either(1) has a fixed point in , or(2)there exists and with .
3. Existence of Positive Solution
In this section, we discuss the existence of positive solutions for (1.4). Using (2.5), (2.6), and Corollary 2.6, (1.4) is equivalent to the integral equation where Let be the function defined by then , for each with , we denote by the function define by We can decompose as , , which implies , for . Therefore, (3.1) is equivalent to the integral equation where is defined (3.2). Set and let be the seminorm in defined by and is a Banach space with norm . Let be a cone of , and Define the operator by
Theorem 3.1. Suppose that the following conditions hold:(1)there exist such that , for , , and , ,(2), where Then (1.4) has at least a positive solution , satisfying , where
Proof. We will show that the operator is continuous and completely continuous.Step 1. The operator is continuous in view of the continuity of .Step 2. maps bounded sets into bounded sets in .
Let be bounded; that is, there exists a positive constant such that , for all . For each , we have where is defined in (3.9). It follows that Hence is bounded.
Step 3. maps bounded sets into equicontinuous sets of .
We will show that is equicontinuous. For each , and , then for given , choose where , , , and . If , Hence is equicontiuous. The Arzela-Ascoli theorem implies that is compact and is continuous and completely continuous.
Step 4. We now show that there exists an open set with for and . Let be any solution of , , where is given by (3.8); since is continuous and completely continuous, we have So Now, by (3.10) and (3.17), we know that any solution of (3.8) satisfies ; let Therefore, Theorem 2.7 guarantees that (3.1) has at least a positive solution . Hence, (1.4) has at least a positive solution , satisfying and the proof is complete.
Note that we can complete the above mentioned procedure by using only the continuity of without condition (), but with our procedure and details of condition () in Theorem 3.1 answers all the questions exist in the following remark.
Remark 3.2. When is continuous on , , (i.e., is singular at ) in (1.4). Suppose , such that is a continuous function on , then is continuous on by Lemma in [12, page 613]. We also obtain results about the existence to (1.4) by using a nonlinear alternative of Leray-Schauder type. The proof is similar to that of Theorem 3.1 as long as we let(1), for , , and , , (2), then (1.4) has at least a positive solution , satisfying , where
4. Unique Existence of Solution
In this section, we will give uniqueness of positive solution to (1.4).
Proof. Let . Then we obtain where is given in (3.8). Hence, In view of Banach fixed point theorem has unique fixed point in , which is the unique positive solution of (2.7) and (1.4) has a unique positive solution in .
Remark 4.2. When , then condition (i) reduces to the Lipschitz condition.
Example 4.3. Let and , . Consider the equation Then (4.3) is equivalent to the integral equation, Here , , then , , , , then , , and , then , . Hence In view of (2.8) and that , and we obtain If , , , in the above equation satisfy the conditions required in Theorem 4.1, the iterated sequence is for where , , is the unique solution, which may not be positive, where is Mittag-Leffler function.
The authors thank the referee’s efforts for their remarks and Professor Ferhan Merdivenci Atici, Western Kentucky University, USA, for her regular contact.
A. M. A. El-Sayed, “Nonlinear differential equations of orbitrary order,” Nonlinear Analysis: Theory, Methods & Applications, vol. 32, pp. 181–186, 1998.View at: Google Scholar
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006.View at: MathSciNet
S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, Switzerland, 1993.View at: MathSciNet
M. Benchohra, J. Henderson, S. K. Ntouyas, and A. Ouahab, “Existence results for fractional order functional differential equations with infinite delay,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1340–1350, 2008.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
C. Zhong, X. Fan, and W. Chen, Nonlinear Functional Analysis and Its Application, Lanzhou University Press, Lanzhou, China, 1998.
I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, London, UK, 1999.View at: MathSciNet
K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993.View at: MathSciNet