Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces
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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.
KeywordsBanach Space Fractional Order Fractional Derivative Fractional Calculus Normal Cone
The field of fractional differential equations has been subjected to an intensive development of the theory and the applications (see, for example, [1, 2, 3, 4, 5, 6] 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 Leray-Schauder theorem, stability, etc. (see, for example, [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] 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, [20, 21, 22, 23] 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 non-local 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.
where , and denotes the integer part of α.
where the right-hand side is pointwise defined on .
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 ). Also, we define the order interval for all . We say that an operator is increasing whenever implies . Also, means that there exist and such that (see ). 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 positive-valued functions such that for all and is a surjection. We say that an operator is τ-φ-concave whenever for all and . We say that A is φ-concave whenever for all t . We recall the following result.
Theorem 1.1 ()
Let E be a Banach space, let P be a normal cone in E, and let be an increasing and τ-φ-concave operator. Suppose that there exists such that . Then there are and such that and , the operator A has a unique fixed point , and for and the sequence with , we have .
2 Main results
on partially ordered Banach spaces with two types of boundary conditions and two types of fractional derivatives, Riemann-Liouville and Caputo.
2.1 Existence results for the fractional differential equation with the Riemann-Liouville fractional derivative
where is the Riemann-Liouville 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.
This completes the proof. □
Now, we are ready to state and prove our first main result.
for all , where is 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 have for all .
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.
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
It is known that P is a normal cone. Similar to the proof of Lemma 2.1, we can prove the following result.
for all , where is 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 have for all .
Now, by using a similar proof of Theorem 2.2, one can show that for all and , and also the operator A is τ-φ-concave. 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.
Thus, by using Theorem 2.4, the problem has a unique solution in .
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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