Abstract
We study the existence and uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem
where is a real number, is the RiemannLiouville fractional derivative and is continuous, (f is singular at ). Our approach is based on a coupled fixed point theorem on ordered metric spaces. An example is given to illustrate our main result.
MSC: 34A08, 34B16, 47H10.
Keywords:
singular fractional differential equation; positive solution; coupled fixed point; coupled lower and upper solution; ordered metric space1 Introduction
Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, fluid flows, electrical networks, viscoelasticity, aerodynamics, and many other branches of science. For details, see [111].
In the last few decades, fractionalorder models are found to be more adequate than integerorder models for some real world problems. Recently, there have been some papers dealing with the existence and multiplicity of solutions (or positive solutions) of nonlinear initial fractional differential equations by the use of techniques of nonlinear analysis (fixed point theorems, LeraySchauder theory, etc.); see [2,4,5,11].
Recently, there have been many exciting developments in the field of fixed point theory on partially ordered metric spaces. The first result in this direction was given by Turinici [12]. In [13], Ran and Reurings extended the Banach contraction principle in partially ordered sets with some applications to matrix equations. The obtained result by Ran and Reurings was further extended and refined by many authors; see [1419].
Very recently, Shurong Sun et al.[20] discussed the existence and uniqueness of a positive solution to the singular nonlinear fractional differential equation boundary value problem
where is a real number, is the RiemannLiouville fractional derivative and is continuous, (f is singular at ), is nondecreasing for all .
Motivated by the above mentioned work, in this paper we investigate the existence and uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem
where is a real number, is the RiemannLiouville fractional derivative and is continuous, (f is singular at ), for all is nondecreasing with respect to the first component, and it is decreasing with respect to the second component. Our approach is based on a recent coupled fixed point theorem on ordered metric spaces established by Harjani et al.[17]. We end the paper with an example that illustrates our main result.
2 Preliminaries
In this section, we recall some basic definitions and properties from fractional calculus theory. For more details about fractional calculus, we refer the readers to [1,3,10].
Definition 2.1 The RiemannLiouville fractional derivative of order of a continuous function is given by
where , denotes the integer part of number α, provided that the right side is pointwise defined on .
Definition 2.2 The RiemannLiouville fractional integral of order of a function is given by
provided that the right side is pointwise defined on .
From the definition of the RiemannLiouville derivative, we can obtain the following statement.
Lemma 2.1 (see [10])
Let. If we assume, then the fractional differential equation
has, , as unique solutions, whereNis the smallest integer greater than or equal toα.
Lemma 2.2 (see [10])
Assume thatwith a fractional derivative of orderthat belongs to. Then
for some, , whereNis the smallest integer greater than or equal toα.
The Green function of fractional differential equation boundary value problem is given by
Lemma 2.3 (see [10])
Letand. The unique solution to
is
where
Hereis called the Green function of boundary value problem (3)(4).
The following properties of the Green function will be used later.
Lemma 2.4 (see [10])
The following properties hold:
Let be a partially ordered set endowed with a metric d such that is complete metric space. Let be a given mapping.
Definition 2.3 We say that is directed if for every there exists such that and .
Definition 2.4 We say that is regular if the following conditions hold: () = if is a nondecreasing sequence in X such that , then for all n;; () = if is a decreasing sequence in X such that , then for all n..
Example 2.1 Let , , be the set of real continuous functions on . We endow X with the standard metric d given by
We define the partial order ⪯ on X by
Let . For , that is, for all , we have and . This implies that is directed. Now, let be a nondecreasing sequence in X such that as , for some . Then, for all , is a nondecreasing sequence of real numbers converging to . Thus we have for all n, that is, for all n. Similarly, if is a decreasing sequence in X such that as , for some , we get that for all n. Then we proved that is regular.
Definition 2.5 (see [15])
An element is called a coupled fixed point of F if and .
Definition 2.6 (see [15])
We say that F has the mixed monotone property if for all , , we have
Denote by Φ the set of functions satisfying: () = φ is continuous;; () = φ is nondecreasing;; () = ..
The following two lemmas are fundamental in the proofs of our main results.
Lemma 2.5 (see [17])
Letbe a partially ordered set and suppose that there exists a metricdonXsuch thatis a complete metric space. Letbe a mapping having the mixed monotone property onXsuch that
for allwithand, where. Suppose also thatis regular and there existsuch that
ThenFhas a coupled fixed point. Moreover, ifandare the sequences inXdefined by
then
Lemma 2.6 (see [17])
Adding to the hypotheses of Lemma 2.5 the conditionis regular, we obtain the uniqueness of the coupled fixed point. Moreover, we have the equality.
3 Main result
Let Banach space be endowed with the norm . We define the partial order ⪯ on E by
In Example 2.1, we proved that with the classic metric given by
satisfies the following properties: is directed and is regular.
where 0 denotes the zero function.
Definition 3.1 (see [15])
We say that is a coupled lower and upper solution to (1)(2) if
and
Our main result is the following.
Theorem 3.1Let, , is continuous, andis continuous on. Assume that there existssuch that forwith, and,
where, . Suppose also that (1)(2) has a coupled lower and upper solution. Then the boundary value problem (1)(2) has a unique positive solution. The sequencesanddefined by
Proof Suppose that u is a solution of boundary value problem (1)(2). Then
• Step 1. We shall prove that
Let . Let us prove that . We have
By the continuity of in , it is easy to check that . Now, let . We have to prove that
We distinguish three cases:
Case 1. . Since is continuous on , there exists a constant such that for all . We have
Using Lemma 2.3, we have
where denotes the beta function.
Now, we have
Case 3. and . The proof is similar to that of Case 2, so we omit it.
Thus we proved that is continuous on for all . Moreover, taking into account Lemma 2.4 and as for all , , our claim (10) is proved. Now the mapping
is well defined.
• Step 2. We shall prove that F has the mixed monotone property with respect to the partial order ⪯ given by (7).
Let such that and . From (8), we have
that is,
which gives us that
Then F has the mixed monotone property.
• Step 3. We shall prove that F satisfies the contractive condition (5) for some .
Let such that and . For all , using (8), we have
Thus we have
which implies that
Now, using the above inequality, (11) and the fact that , we get
Thus we proved that for all such that and , we have
• Step 4. Existence of such that and .
We take , the coupled lower and upper solution to (1)(2).
Now, from Lemmas 2.5 and 2.6, there exists a unique such that , that is is the unique positive solution to (1)(2). The convergence of the sequences and to follows immediately from (6). □
Now, we end this paper with the following example.
Example 3.1 Consider the boundary value problem
In this case, , for . Note that f is continuous on and . Let and . For all with , and , we have
On the other hand,
Consider now, the pair defined by and . Using Lemma 2.4(iv), one can show easily that is a coupled lower and upper solution to (12)(13).
Finally, applying Theorem 3.1, we deduce that (12)(13) has one and only one positive solution .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Acknowledgement
This work was supported by the Research Center, College of Science, King Saud University.
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