Abstract
By using fixed point results on cones, we study the existence and uniqueness of positive solutions for some nonlinear fractional differential equations via given boundary value problems. Examples are presented in order to illustrate the obtained results.
1 Introduction
The field of fractional differential equations has been subjected to an intensive development of the theory and the applications (see, for example, [16] and the references therein). It should be noted that most of papers and books on fractional calculus are devoted to the solvability of linear initial fractional differential equations on terms of special functions. There are some papers dealing with the existence of solutions of nonlinear initial value problems of fractional differential equations by using the techniques of nonlinear analysis such as fixed point results, the LeraySchauder theorem, stability, etc. (see, for example, [719] and the references therein). In fact, fractional differential equations arise in many engineering and scientific disciplines such as physics, chemistry, biology, economics, control theory, signal and image processing, biophysics, blood flow phenomena and aerodynamics (see, for example, [2023] and the references therein). The main advantage of using the fractional nonlinear differential equations is related to the fact that we can describe the dynamics of complex nonlocal systems with memory. In this line of taught, the equations involving various fractional orders are important from both theoretical and applied view points. We need the following notions.
For a continuous function
where
The RiemannLiouville fractional derivative of order α for a continuous function f is defined by
where the righthand side is pointwise defined on
Let
whenever the integral exists.
Suppose that E is a Banach space which is partially ordered by a cone
Theorem 1.1 ([13])
LetEbe a Banach space, letPbe a normal cone inE, and let
2 Main results
We study the existence and uniqueness of a solution for the fractional differential equation
on partially ordered Banach spaces with two types of boundary conditions and two types of fractional derivatives, RiemannLiouville and Caputo.
2.1 Existence results for the fractional differential equation with the RiemannLiouville fractional derivative
First, we study the existence and uniqueness of a positive solution for the fractional differential equation
where
Lemma 2.1Let
where
Proof From
Thus,
Since
Hence,
This completes the proof. □
Now, we are ready to state and prove our first main result.
Theorem 2.2Let
for all
Proof By using Lemma 2.1, the problem is equivalent to the integral equation
where
Define the operator
for all
for all
Here, we give the following example to illustrate Theorem 2.2.
Example 2.1 Let
where
Then
Now, put
and
Thus, by using Theorem 2.2, the problem has a unique solution in
2.2 Existence results for the fractional differential equation with the Caputo fractional derivative
Here, we study the existence and uniqueness of a positive solution for the fractional differential equation
where
It is known that P is a normal cone. Similar to the proof of Lemma 2.1, we can prove the following result.
Lemma 2.3Let
Theorem 2.4Let
for all
Proof It is sufficient to define the operator
Now, by using a similar proof of Theorem 2.2, one can show that
Below we present an example to illustrate Theorem 2.4.
Example 2.2 Let
where g is a continuous function on
Then
for all
and
Thus, by using Theorem 2.4, the problem has a unique solution in
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Authors contributed equally in writing this article. Authors read and approved the final version of the manuscript.
Acknowledgements
This work is partially supported by the Scientific and Technical Research Council of Turkey. Research of the third and forth authors was supported by Azarbaidjan Shahid Madani University. Also, the authors express their gratitude to the referees for their helpful suggestions which improved final version of this paper.
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