- Open Access
Existence of solutions for nonlinear singular fractional differential equations with fractional derivative condition
© Li; licensee Springer. 2014
Received: 9 September 2014
Accepted: 12 November 2014
Published: 25 November 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].
where , are real numbers, is the standard Riemann-Liouville fractional derivative, f satisfies the Caratheodory conditions on , f is positive, and is singular at .
where , are the Caputo fractional derivatives, is a continuous function, is a continuous function, and , (), is a real number.
where is a given function, , , and represents the standard Caputo fractional derivative of order q.
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 .
then is a Banach space.
Definition 2.1 
Definition 2.2 
exists almost everywhere on ( denotes the integer part of the real number α).
Lemma 2.1 
Lemma 2.2 
where , , and .
Lemma 2.3 
Lemma 2.4 
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.
where is defined by (2.1). The proof is complete. □
3 Existence and uniqueness results
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.
where B denotes the beta function.
Case 3. For and . The proof is similar to Case 2, here we just leave it out. This completes the proof. □
is continuous on .
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.
, with .
Therefore, as , i.e., is continuous.
So, is bounded.
Step 3. We will prove that is equicontinuous.
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:
for each and all .
Then the BVP (1.1) has a unique solution.
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.
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 .
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.
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.
Let , then all conditions in Theorem 3.2 are satisfied. Then the problem (4.2) has a solution.
The author is very grateful to the referees for their very helpful comments and suggestions, which greatly improved the presentation of this paper.
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