Existence and Uniqueness of Positive Solution for a Boundary Value Problem of Fractional Order
We are concerned with the existence and uniqueness of positive solutions for the following nonlinear fractional boundary value problem: , , , , where denotes the standard Riemann-Liouville fractional derivative. Our analysis relies on a fixed point theorem in partially ordered sets. Some examples are also given to illustrate the results.
Differential equations of fractional order occur more frequently in different research and engineering areas such as physics, chemistry, economics, and control of dynamical. Indeed, we can find numerous applications in viscoelasticity, electrochemistry control, porous media, and electromagnetism. (see, e.g., [1–7]).
On the other hand, some basic theory for the initial value problems of fractional differential equations involving the Riemann-Liouville differential operator has been discussed by Lakshmikantham and Vatsala , Lakshmikantham , El-Sayed and El-Maghrabi , Bai , Bai and Ge , Bai and Lü , Zhang , and Kempfle et al. [19, 20].
In  the authors studied the following two-point boundary value problem of fractional order: and they proved the existence of positive solutions by means of the Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem.
In  the author investigated the existence of solutions of Since boundary values are nonzero, the Riemann-Liouville fractional derivative is not suitable and the author used the Caputo fractional derivative .
Motivated by these works, in this paper we discuss the existence and uniqueness of positive solutions for the following nonlinear boundary value problem of fractional order:
This problem was studied in , where the authors use lower and upper solution method and the Schauder fixed point theorem which cannot ensure the uniqueness of the solution. The practical relevance of appears in problems related with other areas as physics and economics which can be modeled by these fractional boundary values problems. Particularly, these problems appear in the Hamiltonian formulation for the lagrangians depending on fractional derivatives of coordinates when the systems are nonconservative (see, e.g., ).
Our main interest in this paper is to give an alternative answer to the main results of the paper .
The main tool used in our study is a fixed point theorem in partially ordered sets which gives us uniqueness of the solution.
2. Preliminaries and Previous Results
For the convenience of the reader, we present here some definitions, lemmas and basic results that will be used in the proofs of our theorems.
Definition 2.1. The Riemann-Liouville fractional integral of order of a function is given by provided that the right-hand side is pointwise defined on and where denotes the gamma function.
Lemma 2.3. Let and . Then fractional differential equation has for some () and as unique solution.
Lemma 2.4. Assume that with a fractional derivative of order that belongs to . Then for some () and .
Lemma 2.5. Given and , the unique nonnegative solution for is where
In the sequel, we present the fixed-point theorems which we will use later. These results appear in .
Theorem 2.6. Let be a partially ordered set and suppose that there exists a metric in such that is a complete metric space. Assume that satisfies the following condition Let be a nondecreasing mapping such that where is a continuous and nondecreasing function such that is positive in , and . If there exists with then has a fixed point.
In our considerations we will work in the Banach space with the standard norm .
Notice that this space can be equipped with a partial order given by In  it is proved that with the classical metric given by satisfies condition (2.9) of Theorem 2.6. Moreover, for , as the function , satisfies condition (2.11).
Finally, by we denote the class of functions continuous, nondecreasing, positive in and .
By we denote the class of functions continuous, nondecreasing, satisfying that , where denotes the identity mapping on .
3. Main Result
The main result of the paper is the following.
Theorem 3.1. Problem (1.3) has a unique positive solution if the following conditions are satisfied. (H1) is continuous and nondecreasing with respect to the second argument.(H2)There exists such that . (H3)There exists such that, for with and , where .
Lemma 3.2. , and is a continuous function on .
Proof. The continuity of is easily checked. In order to prove the nonnegativness of , for , it is obvious that In the case of with , we have As , we have and, consequently, Taking into account that the function with and is decreasing we have The last inequality and (3.3) give us with . Finally, notice that , and this finishes the proof.
Lemma 3.3. One has
Proof. Since and if we put , then, as we deduce that is strictly increasing and, consequently,
In the sequel, we give the proof of Theorem 3.1.
Proof of Theorem 3.1. Consider the cone
Obviously, is a closed set of , and, thus, is a complete metric space with the distance given by . can be equipped with a partial order defined by
Using a similar argument to that in , it can be proved that satisfies condition (2.9) of Theorem 2.6. Moreover, as for the function , satisfies condition (2.11).
Now, we consider the operator defined on and given by By (H1) and Lemma 3.2, applies into itself.
In the sequel we check that satisfies the assumptions of Theorem 2.6.
Firstly, we prove that is a nondecreasing operator. In fact, by (H1), for with and , we have Now, we prove that satisfies the contractive condition appearing in Theorem 2.6.
In fact, for and , taking into account assumption (H3), we get As and, thus, is nondecreasing and by Lemma 3.3, from the last inequality we obtain Using the fact that (assumption (H3)), we have Put , As , this means that . The last inequality gives us This proves that satisfies the contractive condition of Theorem 2.6.
Finally, as (Lemma 3.2) and (assumption (H1)), we have where 0 denotes the zero function.
Now, Theorem 2.6 shows that problem (1.3) has at least one nonnegative solution. As satisfies condition (2.11), we obtain the uniqueness of the solution.
In what follows, we will prove that this solution is positive (this means that , for ).
Finally, we will prove that the zero function is not the solution for problem (1.3). In fact, in contrary case, the zero function is a fixed point of and, thus, we have The nonnegative character of the functions and and the last expression give us This and the fact that a.e. () for any because is given by a polynomial implies Taking into account assumption (H2), for certain . By the continuity of we can find a set with and , where is the Lebesgue measure, such that for . This contradicts (3.21).
This proves that the zero function is not the solution for problem (1.3). Now, we will prove that the solution is positive.
In the contrary case, we find such that . As the solution is a fixed point of the operator , this means that Since and, thus, and by the fact that is nondecreasing in the second variable and , we can get and this inequality implies
Using a similar reasoning to the one above used we obtain a contradiction.
Therefore, , for .
This finishes the proof.
Remark 3.4. In Theorem 3.1, condition (H2) seems to be a strong condition in order to obtain a positive solution for problem (1.3), but when there is uniqueness of solution one will see that this condition is a very adjusted one. More precisely, under the assumption that problem (1.3) has a unique nonnegative solution one has
In fact, if for certain the argument used in the proof of Theorem 3.1 give us that is a positive solution.
For the other implication, suppose that for any . Under this assumption, our problem (1.3) admits as solutions the function and the zero function and this contradicts the hypothesis about uniqueness of solution to problem (1.3). Therefore, for certain .
Remark 3.5. Notice that the assumptions in Theorem 3.1 are invariant by additive perturbations. More precisely, if for any and satisfies (H1) and (H3) of Theorem 3.1, then , with a nondecreasing continuous function with for certain , satisfies (H1), (H2), and (H3) of Theorem 3.1 and the following nonlinear boundary value problem of fractional order: has a unique positive solution (by Theorem 3.1).
In the sequel we present an example where the results can be applied.
Example 3.6. Consider the fractional boundary value problem
In this case, for . Obviously, is a continuous function and for . As for , is nondecreasing with respect to the second variable.
Besides, for and , we have A straightforward calculation gives us that satisfies that .
Moreover, in this case , and we have Finally, Theorem 3.1 proves the existence and uniqueness of a positive solution for problem (3.27).
4. A Final Remark
Theorem 4.1 (see [21, Theorem 3.1]). Problem (1.3) has a positive solution if the following conditions are satisfied: is nondecreasing relative to , for , where , and there exists a positive constant such that
Example 4.2. Consider the fractional boundary value problem
with and .
In this case, , for . Obviously, is continuous and nondecreasing with respect to the second variable since .
Besides, if and , we have In this case, and it is easily seen that belongs to .
Moreover, in this case, and, as , we have As for any , Theorem 3.1 gives us the existence and uniqueness of positive solution for problem (4.2).
On the other hand, we will show that with and does not satisfy () of Theorem 4.1. In fact, suppose that there exists such that This implies that Notice that , and, consequently, taking limit as in the last inequality, we get this is false because and the function is decreasing when . Therefore, problem (4.2) can be covered by Theorem 3.1 and it cannot be studied by Theorem 4.1.
Our main contribution in this paper is to prove under certain assumptions the existence and the uniqueness of positive solution for problem (1.3) which was treated in . In  the question of uniqueness of solution was not considered. Moreover, we present an example which can be covered by the results of this paper and cannot be treated by the ones obtained in .
This paper was partially supported by Ministerio de Educación y Ciencia Project MTM 2007/65706.
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