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 , the Caputo derivative of fractional order α is defined by
where , and denotes the integer part of α.
The RiemannLiouville fractional derivative of order α for a continuous function f is defined by
where the righthand side is pointwise defined on .
Let be an interval in ℝ and . The RiemannLiouville fractional order integral of a function is defined by
whenever the integral exists.
Suppose that E is a Banach space which is partially ordered by a cone , that is, if and only if . We denote the zero element of E by θ. A cone P is called normal if there exists a constant such that implies (see [24]). Also, we define the order interval for all [24]. We say that an operator is increasing whenever implies . Also, means that there exist and such that (see [24]). Finally, put for all . It is easy to see that is convex and for all . We recall the following in our results. Let E be a real Banach space and let P be a cone in E. Let be an interval and let τ and φ be two positivevalued functions such that for all and is a surjection. We say that an operator is τφconcave whenever for all and [13]. We say that A is φconcave whenever for all t[13]. We recall the following result.
Theorem 1.1 ([13])
LetEbe a Banach space, letPbe a normal cone inE, and letbe an increasing andτφconcave operator. Suppose that there existssuch that. Then there areandsuch thatand, the operator A has a unique fixed point, and forand the sequencewith, we have.
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 is the RiemannLiouville fractional derivative of order α. Let . Consider the Banach space of continuous functions on with the sup norm and set . Then P is a normal cone.
Lemma 2.1Let, , , and. Then the problemwith the boundary value conditionhas a solutionif and only ifis a solution of the fractional integral equation
where
Proof From and the boundary condition, it is easy to see that . By the definition of a fractional integral, we get
Hence,
This completes the proof. □
Now, we are ready to state and prove our first main result.
Theorem 2.2Letbe given and letτandφbe two functions onsuch thatfor all. Suppose thatis a surjection andis increasing inufor each fixedt, andfor alland. Assume that there exist, andsuch that
for all, whereis the green function defined in Lemma 2.1. Then the problem (2.1) with the boundary value condition (2.2) has a unique positive solution. Moreover, for the sequence, we havefor all.
Proof By using Lemma 2.1, the problem is equivalent to the integral equation
where
Define the operator by . Then u is a solution for the problem if and only if . It is easy to check that the operator A is increasing on P. On the other hand,
for all and . Thus, the operator A is τφconcave. Since
for all , we get . Now, by using Theorem 1.1, the operator A has a unique positive solution . This completes the proof. □
Here, we give the following example to illustrate Theorem 2.2.
Example 2.1 Let be given. Consider the periodic boundary value problem
where , g is continuous on and . Put
Then . Now, define , , , and also for all t. Then is a surjection and for all . For each , we have
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 is the Caputo fractional derivative of order α. Let be the Banach space of continuous functions on with the sup norm and
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, , , and. Then the problemwith the boundary value conditionsandhas a solutionif and only ifis a solution of the fractional integral equation, where
Theorem 2.4Letbe given and letτandφbe two positivevalued functions onsuch thatfor all. Suppose thatis a surjection andis increasing inufor each fixedt, wheneverandotherwise, and alsofor alland. Assume that there exist, andsuch that
for all, whereis the green function defined in Lemma 2.3. Then the problem (2.3) with the boundary value conditions (2.4) has a unique positive solution. Moreover, for the sequence, we havefor all.
Proof It is sufficient to define the operator by
Now, by using a similar proof of Theorem 2.2, one can show that for all and , and also the operator A is τφconcave. By using Theorem 1.1, the operator A has a unique positive solution . This completes the proof by using Lemma 2.3. □
Below we present an example to illustrate Theorem 2.4.
Example 2.2 Let . Consider the periodic boundary value problem
where g is a continuous function on with . Put , and
Then . Now, define , , , and . Then it is easy to see that is a surjection map and for . Also, we have
for all . Now, put , and also . Then we have
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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