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Existence of solutions for nonlinear singular fractional differential equations with fractional derivative condition
Advances in Difference Equations volume 2014, Article number: 292 (2014)
In this paper, we investigate a boundary value problem for singular fractional differential equations with a fractional derivative condition. The existence and uniqueness of solutions are obtained by means of the fixed point theorem. Some examples are presented to illustrate our main results.
Differential equations of fractional order have recently been addressed by many researchers of various fields of science and engineering such as physics, chemistry, biology, economics, control theory, and biophysics; see [1, 2]. On the other hand, fractional differential equations also serve as an excellent tool for the description of memory and hereditary properties of various materials and processes. With these advantages, the model of fractional order become more and more practical and realistic than the classical of integer order, such effects in the latter are not taken into account. As a result, the subject of fractional differential equations is gaining much attention and importance.
Recently, much attention has been focused on the study of the existence and uniqueness of solutions for boundary value problem of fractional differential equations with nonlocal boundary conditions by the use of techniques of nonlinear analysis (fixed point theorems, Leray-Schauder theory, the upper and lower solution method, etc.); see [3–17].
In , Agarwal et al. investigated the existence of solutions for the singular fractional boundary value problems
where , are real numbers, is the standard Riemann-Liouville fractional derivative, f satisfies the Caratheodory conditions on , f is positive, and is singular at .
In , Yan et al. studied the existence and uniqueness of solutions for a class of fractional differential equations with integral boundary conditions
where , are the Caputo fractional derivatives, is a continuous function, is a continuous function, and , (), is a real number.
In , Guezane-Lakoud and Bensebaa discussed the existence and uniqueness of solutions for a fractional boundary value problem with a fractional derivative condition,
where is a given function, , , and represents the standard Caputo fractional derivative of order q.
Motivated by all the works above, this paper deals with the existence and uniqueness of solutions for the singular fractional boundary value problem with a fractional derivative condition,
where , , is continuous, may be singular at , is the standard Caputo derivative.
The paper is organized as follows. In Section 2, we shall introduce some definitions and lemmas to prove our main results. In Section 3, we establish some criteria for the existence for the boundary value problem (1.1) by using the Banach fixed point theorem and the Schauder fixed point theorem. Finally, we present two examples to illustrate our main results.
2 Preliminaries and lemmas
In this section, we present definitions and some fundamental facts from fractional calculus which can be found in .
Let , , endowed with the norm
then is a Banach space.
Definition 2.1 
If and , then the Riemann-Liouville fractional integral is defined as
Definition 2.2 
Let , . If , then the Caputo fractional derivative of order α defined by
exists almost everywhere on ( denotes the integer part of the real number α).
Lemma 2.1 
Let and . Then the following relations hold:
Lemma 2.2 
For , , the homogeneous fractional differential equation has a solution
where , , and .
Lemma 2.3 
Let , and . Then
Lemma 2.4 
Let , and . Then for all we have
Lemma 2.5 (Schauder fixed point theorem)
Let be a complete metric space, let U be a closed convex subset of E, and let be a mapping such that the set is relatively compact in E. Then A has at least one fixed point.
Lemma 2.6 For and , , the unique solution of
is given by
Proof By Lemma 2.2, we get
for some , . So, we have
From the conditions , we obtain . Hence,
The condition implies that
Therefore, can be written as
where is defined by (2.1). The proof is complete. □
3 Existence and uniqueness results
Define the operator by
Lemma 3.1 Let , , is continuous, and . Suppose that is continuous on . Then the function is continuous on .
Proof By the continuity of and . It is easy to know that . Now we separate the process into three cases.
Case 1. For and . Because of the continuity of , there exists a constant such that , , then
where B denotes the beta function.
Case 2. For and , then
Case 3. For and . The proof is similar to Case 2, here we just leave it out. This completes the proof. □
Lemma 3.2 Let , , is continuous, and . Suppose that is continuous on . Then
is continuous on .
Proof From we obtain and . Hence, there exist two constants and such that , , for . Since is continuous on , we have
Observing that , are continuous on , we can show is continuous on by using the same method as in Lemma 3.1. The proof is completed. □
Lemma 3.3 Let , , is continuous, and . Assume that is continuous on . Then the operator is completely continuous.
Proof For , , by Lemma 3.1 and Lemma 3.2, we have . Now we separate the proof into three steps.
Step 1. Proof of is continuous.
Let and . If and , then . By the continuity of , we know that is uniformly continuous on . Thus for , there exists (), such that
, with .
It follows from (3.2) that
On the other hand, by (3.1), we get
Therefore, as , i.e., is continuous.
Step 2. Let be bounded, then there exists a positive constant b such that , . Since is continuous on , we see that there exists a positive constant L such that
Thus, by (3.3) and (3.4), we have
So, is bounded.
Step 3. We will prove that is equicontinuous.
For all , and we have
As , the right-hand sides of the inequalities (3.5) and (3.6) tend to 0, consequently , i.e., is equicontinuous.
By means of the Arzela-Ascoli theorem, we conclude that T is completely continuous. □
Now we are in the position to establish the main results.
Theorem 3.1 Assume that:
(H1) There exist two constants and such that
for each and all .
Then the BVP (1.1) has a unique solution.
Proof We shall use the Banach fixed point theorem. For this, we need to verify that T is a contraction. Let , then from (H1) and (3.3)-(3.4) we obtain
Taking (3.7) and (3.8) into account, we acquire ; then it is a contraction. As a consequence of the Banach fixed point theorem, we deduce that T has a fixed point which is the unique solution of the BVP (1.1). The proof is complete. □
Next, we will use the Schauder fixed point theorem to prove our result.
For the sake of convenience, we set
Theorem 3.2 Assume that , , is continuous, and , is continuous on . Then the BVP (1.1) has a solution.
Proof Let . First, we prove that .
In fact, for each , we have
Hence, we can conclude that
From Lemma 3.1 and Lemma 3.2, we know that , . Consequently, . From Lemma 3.3, we find that is completely continuous. By Lemma 2.5, we deduce that the problem (1.1) has a solution. This completes the proof. □
We illustrate our work with two examples.
Example 4.1 Consider the following fractional boundary value problem:
where , . Simple calculus gives
So, , then by Theorem 3.1, the problem (4.1) has a unique solution.
Remark 4.1 If , then . However, by Theorem 3.2, the problem (4.1) still has a solution.
Example 4.2 Let us consider the fractional boundary value problem
Let , then all conditions in Theorem 3.2 are satisfied. Then the problem (4.2) has a solution.
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The author is very grateful to the referees for their very helpful comments and suggestions, which greatly improved the presentation of this paper.
The author declares to have no competing interests.
The author read and approved the final manuscript.
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Cite this article
Li, R. Existence of solutions for nonlinear singular fractional differential equations with fractional derivative condition. Adv Differ Equ 2014, 292 (2014). https://doi.org/10.1186/1687-1847-2014-292
- boundary value problem
- singular fractional differential equation
- Caputo fractional derivative
- fixed point theory