Nonlinear Problems: Analytical and Computational Approach with ApplicationsView this Special Issue
Positive and Nondecreasing Solutions to an m-Point Boundary Value Problem for Nonlinear Fractional Differential Equation
We are concerned with the existence and uniqueness of a positive and nondecreasing solution for the following nonlinear fractional m-point boundary value problem: , where denotes the standard Riemann-Liouville fractional derivative, is a continuous function, for , and . Our analysis relies on a fixed point theorem in partially ordered sets. Some examples are also presented to illustrate the main results.
Many papers and books on fractional differential equations have appeared recently. Most of them are devoted to the solvability of the linear fractional equation in terms of a special function (see, e.g., [1, 2]) and to problems of analyticity in the complex domain . Moreover, Delbosco and Rodino  considered the existence of a solution for the nonlinear fractional differential equation where , , is a given continuous function in . They obtained existence results by using the Schauder fixed point theorem and the Banach contraction principle.
Recently, El-Shahed  considered the following nonlinear fractional boundary value problem: They used the Krasnoselskii’s fixed point theorem on cone expansion and compression to show the existence and nonexistence of positive solutions for the above fractional boundary value problem.
In , Liang and Zhang considered the following nonlinear fractional boundary value problem: and by means of lower and upper solution method and fixed point theorems, they obtained some results on the existence of positive solutions to the above boundary value problem.
The question of uniqueness of the solution is not treated in .
Recently, in  Caballero et al. studied the fractional boundary value problem appearing in  by using a fixed point theorem in partially ordered sets, and the authors obtained uniqueness of the solution.
In this paper we discuss the existence and uniqueness of a positive and nondecreasing solution for the following -point nonlinear boundary value problem of fractional order: where for , , and, moreover, .
Recently, this problem has been studied in [8, 9]. In  the author studies the existence and multiplicity of positive solutions for problem (1.4), and he uses Krasnoselskii and Leggett-Williams fixed point theorems. The question of uniqueness and monotonicity of the solution is not treated. In , the authors investigate the existence and uniqueness of positive and nondecreasing solutions for problem (1.4), and the main tools in this paper are a fixed point theorem in partially ordered sets and the lower and upper solution method.
Our study is based on a different fixed point theorem in partially ordered sets than the one used in .
2. Preliminaries and Previous Results
For the convenience of the reader, we present here some definitions, lemmas, and results that will be used in the proofs of our main results.
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 Euler gamma function given by
Definition 2.2. The Riemann-Liouville fractional derivative of order of a function is defined by where and denote the integer part of .
Lemma 2.3. Let and . Then the fractional differential equation has as unique solutions.
Lemma 2.4. Assume that with a fractional derivative of order that belongs to . Then for some and .
Lemma 2.5. Given , then the unique solution of is where Green’s function is given by taking where and denotes the characteristic function of the set for .
Lemma 2.7. Under the assumption for and where , the Green’s function appearing in Lemma 2.5 satisfies .
Remark 2.8. It is easily checked that is a continuous function on .
The following lemmas appear in .
Lemma 2.9. The function appearing in Lemma 2.5 is strictly increasing in the first variable.
Lemma 2.10. The function appearing in Lemma 2.5 satisfies
For convenience, we will denote by the constant In the sequel we present the fixed point theorem we will use later and which appears in .
Firstly, we need to introduce the following class of functions. By we denote the class of those functions satisfying implies .
Theorem 2.11 (see ). Let be a partially ordered set, and suppose that there exists a metric in such that is a complete metric space. Let be a nondecreasing mapping such that there exists an element with .
Suppose that there exists such that for with .
Assume that either is continuous or is such that Besides, if then has a unique fixed point.
In our considerations, we will work in the space , with the standard distance given by for .
Notice that this space can be equipped with a partial order given by for and .
3. Main Result
Our starting point in this section is to present the class of functions which we use later. By we will denote the class of those functions satisfying the following conditions:(C1) is nondecreasing;(C2)For any , ;(C3), where is the class of functions appearing in Section 2.
Examples of functions in are with and .
In what follows, we formulate our main result.
Theorem 3.1. Suppose that the following assumptions are satisfied:(a) is continuous;(b) is nondecreasing with respect to the second variable for each ;(c)there exists and such that
for with and .
Then problem (1.4) has a unique nonnegative and nondecreasing solution.
Proof. Consider the cone .
Notice that, as is a closed set of , is a complete metric space with the distance given by satisfying conditions (2.14) and (2.15) of Theorem 2.11.
Now, for we define the operator by where is Green’s function defined in Section 2.
By Lemma 2.7, Remark 2.8, and (a), it is clear that applies the cone into itself.
Now, we will check that assumptions in Theorem 2.11 are satisfied.
Firstly, the operator is nondecreasing.
In fact, by assumption (b), for we have On the other hand, for and and taking into account our hypotheses, we can obtain Since is not nondecreasing and taking into account Lemma 2.10 and assumption (c), we get Thus, for and , where .
Obviously, the last inequality is satisfied for .
Thus condition (2.13) in Theorem 2.11 holds with . Moreover, since and are nonnegative functions, Finally, Theorem 2.11 tells us that problem (1.4) has a unique nonnegative solution .
In what follows, we will prove that the unique nonnegative solution for problem (1.4) is nondecreasing.
In fact, by Lemmas 2.5 and 2.9 it is easily seen that is strictly increasing and, since is a fixed point of the operator , we have These facts and the nonnegative character of give us that is nondecreasing.
Now, we present a sufficient condition for the existence and uniqueness of a positive and strictly increasing solution for problem (1.4) (positive solution means a solution satisfying for . The proof of this fact is similar to the proof of Theorem 3.6 of . We present it for completeness.
Theorem 3.2. Under assumptions of Theorem 3.1 and adding the following assumption (d) for certain , one obtains existence and uniqueness of a positive and strictly increasing solution for problem (1.4).
Proof. Consider the nonnegative solution for Problem (1.4) whose existence is guaranteed by Theorem 3.1.
Notice that satisfies Firstly, we will prove that for .
In fact, in contrary case we can find such that . Consequently, As , and is nondecreasing with respect to the second variable (assumption (b)), from the last expression we can get and, thus, This fact and the nonnegative character of the functions and imply By Lemma 2.9, and since and for any , we have On the other hand, assumption (d) gives us that for certain and, thus, .
This fact and the continuity of imply the existence of a set with and , where is the Lebesgue measure, such that for any .
This contradicts (3.14).
Therefore, for .
In the sequel, we will show that is strictly increasing.
In fact, since , we have Now, we take with .
We can consider two cases.
Case 1 (). Suppose that .
Using a similar argument similar o the one used in the proof of the positive character of , we obtain a contradiction.Case 2 (). Suppose that .
In this case, we have
Since (Lemma 2.3), we get
Again, the same reasoning that we use earlier gives us a contradiction.
Therefore, . This finished the proof.
Remark 3.3. In Theorem 3.2, the condition for certain seems to be a strong condition in order to obtain a positive solution for problem (1.4), but when the solution is unique, we will see that this condition is very adjusted one. In fact, suppose that problem (1.4) has a unique nonnegative solution , then we have
Indeed, if for any , then it is easily seen that the zero function is a solution for problem (1.4) and the uniqueness of solution gives us .
The reverse implication is obvious.
Remark 3.4. Notice that assumptions in Theorem 3.1 are invariant by nonnegative and continuous perturbations. More precisely, if for any and satisfies conditions (a), (b), and (c) of Theorem 3.1, then , where continuously and , satisfies assumptions of Theorem 3.2, and this means that the boundary value problem with for , and has a unique positive and strictly increasing solution.
In the sequel, we present an example which illustrates our results.
Example 3.5. Consider the following boundary value problem:
In this case, , , , , and . Besides, , and
It is easily seen that satisfies condition (a) of Theorem 3.1.
Since for , satisfies (b) of Theorem 3.1.
Moreover, for and , we have where . It is easily proved that belongs to the class . Since for , Theorem 3.2 states that problem (3.20) has a unique positive and strictly increasing solution for .
4. Some Remark
Theorem 4.1. Problem (1.4) has a unique positive and strictly increasing with solution if the following conditions are satisfied:(1) is continuous and nondecreasing with respect to the second variable and for with ( denotes the Lebesgue measure),(2)there exists such that for with and
The main tool used by the authors in  for the proof of Theorem 4.1 is a fixed point theorem in partially ordered sets which appears in . This fixed point theorem uses the following class of functions .
By we denote the class of functions continuously such that if , then the following conditions are satisfied:(1), and it is nondecreasing;(2);(3) is positive in .
In what follows we will prove that the classes of functions (see Section 3) and are not comparable.
Example 4.2. This example appears in .
Consider the function defined by It is easily proved that .
On the other hand, it is easily seen that is not increasing, and, consequently, .
Example 4.3. Consider the function given by .
It is easily proved that .
On the other hand, since and when , we have that .
Theorem 4.4. Problem (1.4) has a positive solution if the following conditions are satisfied:( is nondecreasing relative to , for , and there exists a positive constant such that for , where
Example 4.5. Consider the fractional boundary value problem In this case, , and . Besides, .
Moreover, for and , we have where . It is easily checked that .
On the other hand, we will show that condition appearing in Theorem 4.4 is not satisfied.
In fact, suppose that there exists such that for any .
Since with and , we have Taking limit when in the last expression, we get , which is false, since and the function is decreasing when .
In , the author proves the following result.
Theorem 4.6. Assume that is continuous, for , and with .In addition, suppose that one of the following two conditions holds:,.Then Problem (1.4) has at least one positive solution.
In what follows we present an example which can be treated by our results and it cannot be studied by Theorem 4.6.
Example 4.7. Consider Problem (3.20) which appears in Example 3.5.
We prove that this example can be treated by Theorem 3.2 and we obtained a unique positive and strictly increasing solution for this problem when .
On the other hand, in this case, since , we have Since , Theorem 4.6 cannot be used in this case.
Example 4.8. Consider the following boundary value problem In this case, , , and . Besides, .
It is easily seen that satisfies conditions (a) and (b) of Theorem 3.1.
Moreover, in  it is proved that if , Using this fact, for and , we have where .
In  it is proved that .
Our main contribution is that for we obtain uniqueness and strictly increasing character for the solution of Problem (4.11).
This paper was partially supported by the Ministerio de Educación y Ciencia, Project MTM 2007/65706.
K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives. Theory and applications, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993.
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 B.V., Amsterdam, The Netherlands, 2006.