A class of nonlinear fractional multipoint boundary value problems at resonance is considered in this article. The existence results are obtained by the method of the coincidence degree theory of Mawhin. An example is given to illustrate the results.
Keywords:coincidence degree; fractional differential equation; resonance; multipoint boundary conditions
The subject of fractional calculus has gained considerable popularity during the past decades, due mainly to its frequent appearance in a variety of different areas such as physics, aerodynamics, polymer rheology, etc. (see [1-3]). Many methods have been introduced for solving fractional differential equations (FDEs for short in the remaining), such as the Laplace transform method, the iteration method, the Fourier transform method, etc. (see ).
Recently, there have been many works related to the existence of solutions for multipoint boundary value problems (BVPs for short in the remaining) at nonresonance of FDEs (see [5-11]). Motivated by the above articles and recent studies on FDEs (see [12-19]), we consider the existence of solutions for a nonlinear fractional multipoint BVPs at resonance in this article.
In , Zhang and Bai considered the following fractional three-point boundary value problems at resonance:
where is a natural number; is a real number; and are the standard Riemann-Liouville derivative and integral respectively; is continuous; ; are given constants such that . In their article, they made the operator and got . In , Bai discussed fractional m-point boundary value problems at resonance with the case of .
In 2010, Bai and Jiang studied the fractional differential equation of boundary value problems at resonance with the case of respectively (see [18,19]), and we can see that they obtained the results by the assumption that a specific algebraic expression is not equal to zero; for example,
is referred to as a condition in . We will show that the assumption like above is not necessary.
In this article, we will use the coincidence degree theory to study the existence of solutions for a nonlinear FDEs at resonance which is given by
with boundary conditions
We will always suppose that the following conditions hold:
The rest of this article is organized as follows: In Section 2, we give some definitions, lemmas and notations. In Section 3, we establish theorems of existence result for BVPs (1.1)-(1.2). In Section 4, we give an example to illustrate our result.
Definition 2.3 ()
Lemma 2.4 ()
Lemma 2.5 ()
Now, we briefly recall some notations and an abstract existence result, which can be found in . Let YZ be real Banach spaces, be a Fredholm map of index zero, and be continuous projectors such that
Lemma 2.6 ()
Lemma 2.7 ()
Lemma 2.8 ()
3 Main results
Then, let us make some assumptions which will be used throughout the article.
Lemma 3.2If condition (C) holds andLis defined by (2.1), then
Then in view of condition (C), (1.2) and Lemma 2.4, x satisfies
It is equal to
similarly, we can derive that
Furthermore, Q is a continuous linear projector.
However, the determinant of coefficients is as follows
but the determinant of coefficients is as follows
For any , in view of the definition of operators Kp and L, we have . On the other hand, if , we have , . Therefore, by Lemma 2.5 and definitions of operators and L, we know that , which implies that . By the definition of , we have
It follows from Lemma 2.4 that
Then, we have
Proof In order to prove N is L-compact, we only need to prove that is bounded and is compact. Since the function f satisfies Carathéodory conditions and , for each , there exists a such that, for a.e. and every , we have . By the definition of operators Q and on the interval , it is easy to get that and are bounded. Thus, there exists a constant with each , such that .
By (3.7)-(3.9) and Lemma 2.4, we have that
Furthermore, we have
By (H1) and the definition of N, we have
which yield that
Furthermore, from the previous inequalities, we know that
By the first parts of (H2) and (H3), similar to the proof of Lemma 3.7, then
Now with Lemmas 3.2-3.8 in hands, we can begin to prove our main result - Theorem 3.1.
Finally, we will prove that (iii) of Lemma 2.6 is satisfied. We let I as the identity operator in the Banach space Y and , according to Lemma 3.8 (or Remark 3.9) we know that for all , . By the homotopic property of degree, we have
so (iii) of Lemma 2.6 is satisfied.
According to Theorem 3.1, we have the following corollary.
Corollary 3.10Suppose that (H1) is replaced by the following condition,
and the others in Theorem 3.1 are not changed, then BVPs (1.1)-(1.2) have at least one solution.
4 An example
Thus, we have
The authors declare that they have no competing interests.
NX designed all the steps of proof in this research and also wrote the article. WBL suggested many good ideas in this article. LSX helped to draft the first manuscript and gave an example to illustrate our result. All authors read and approved the final manuscript.
The authors would like to acknowledge the anonymous referee for many helpful comments and valuable suggestions on this article. This work is sponsored by Fundamental Research Funds for the Central Universities (2012LWB44).
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