- Open Access
On positive solutions for a class of singular nonlinear fractional differential equations
© Jleli and Samet; licensee Springer 2012
- Received: 29 March 2012
- Accepted: 18 May 2012
- Published: 12 July 2012
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 Riemann-Liouville 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.
- singular fractional differential equation
- positive solution
- coupled fixed point
- coupled lower and upper solution
- ordered metric space
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 [1–11].
In the last few decades, fractional-order models are found to be more adequate than integer-order 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, Leray-Schauder 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 . In , 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 [14–19].
where is a real number, is the Riemann-Liouville fractional derivative and is continuous, (f is singular at ), is nondecreasing for all .
where is a real number, is the Riemann-Liouville 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. . We end the paper with an example that illustrates our main result.
where , denotes the integer part of number α, provided that the right side is pointwise defined on .
provided that the right side is pointwise defined on .
From the definition of the Riemann-Liouville derivative, we can obtain the following statement.
Lemma 2.1 (see )
has, , as unique solutions, where N is the smallest integer greater than or equal to α.
Lemma 2.2 (see )
for some, , where N is the smallest integer greater than or equal to α.
The Green function of fractional differential equation boundary value problem is given by
Lemma 2.3 (see )
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 )
, , where.
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..
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 )
An element is called a coupled fixed point of F if and .
Definition 2.6 (see )
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 )
Then F has a coupled fixed point. Moreover, ifandare the sequences in X defined by
Lemma 2.6 (see )
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.
satisfies the following properties: is directed and is regular.
where 0 denotes the zero function.
Definition 3.1 (see )
Our main result is the following.
converge uniformly to.
where denotes the beta function.
Case 3. and . The proof is similar to that of Case 2, so we omit it.
is well defined.
Step 2. We shall prove that F has the mixed monotone property with respect to the partial order ⪯ given by (7).
Then F has the mixed monotone property.
Step 3. We shall prove that F satisfies the contractive condition (5) for some .
where and .
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.
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 .
This work was supported by the Research Center, College of Science, King Saud University.
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