The existence of solutions for nonlinear fractional multipoint boundary value problems at resonance
© Xu et al.; licensee Springer 2012
Received: 17 January 2012
Accepted: 18 May 2012
Published: 28 June 2012
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.
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.
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 .
is referred to as a condition in . We will show that the assumption like above is not necessary.
where ; ; ; ; with satisfying Carathéodory conditions; and are the standard Riemann-Liouville fractional derivative and fractional integral respectively.
BVPs (1.1)-(1.2) being at resonance means that the associated linear homogeneous equation with boundary conditions (1.2) has as a nontrivial solution, where , .
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.
where is the Gamma function, provided the right side is pointwise defined on .
where , provided the right side is pointwise defined on .
Definition 2.3 ()
for each , the mapping is Lebesgue measurable;
for almost every , the mapping is continuous on ;
for each , there exists a such that, for a.e. and every , we have .
Lemma 2.4 ()
Lemma 2.5 ()
It follows that is invertible. We denote the inverse by . If Ω is an open bounded subset of Y such that , the map will be called L-compact on Ω if is bounded and is compact.
Lemma 2.6 ()
for each ;
for each ;
whereis a projection such thatandis a any isomorphism.
In this article, we use the Banach space with the norm .
Lemma 2.7 ()
with the norm defined by.
Lemma 2.8 ()
Thus, BVP (1.1) can be written as for each .
3 Main results
Then, let us make some assumptions which will be used throughout the article.
Theorem 3.1 If conditions (C), (H 1)-(H 3) hold, then BVPs (1.1)-(1.2) have at least one solution provided that.
Combining with the condition (1.2), we get .
Lemma 3.3 If condition (C) holds, then there exist two constantsandwithsuch that.
Proof From , we obtain that for any nonnegative integer l, there exists such that . If else, we obtain that , , .
Since the determinant of coefficients is not equal to zero, we have that (), which is a contradiction to condition (C).
Similarly, we can deduce that the determinant of coefficients is not equal to zero, so we have that (), which is a contradiction to condition (C). Thus, there exists such that .
which is a contradiction to (3.2). Therefore, there exists two constants and with such that . □
Furthermore, Q is a continuous linear projector.
then we have (), i.e., . Thus, .
we can deduce that . Hence, . Furthermore, we get . Therefore, , which means that L is a Fredholm operator of index zero.
It is clear that .
By the definition of the norm in space Y, we get . □
Lemma 3.5 Assumeis an open bounded subset such that, and N is defined by (2.2), then N is L-compact on.
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 .
Since is uniformly continuous on and , so and are equicontinuous. By Lemma 2.8, we get that is completely continuous. □
Lemma 3.6 Suppose (H 1)-(H 3) hold, then the setis bounded.
Therefore, is bounded. □
Lemma 3.7 Suppose (H 2) and (H 3) hold, then the setis bounded.
Proof For any and , then and . By (H2), we get that , then we have . By (H3), we have that , thus . Therefore, is bounded. □
Lemma 3.8 If the first parts of (H 2) and (H 3) hold, then the setis bounded.
Therefore, is bounded.
If , we have .
If , we get that and , similar to the proof of Lemma 3.7, is bounded. If else, we have that and . It contradicts (3.10), thus is bounded. □
Remark 3.9 If the other parts of (H2) and (H3) hold, then the set is bounded.
Now with Lemmas 3.2-3.8 in hands, we can begin to prove our main result - Theorem 3.1.
for every ;
for every .
so (iii) of Lemma 2.6 is satisfied.
Consequently, by Lemma 2.6, the equation has at least one solution in . Namely, BVPs (1.1)-(1.2) have at least one solution in the space Y. □
According to Theorem 3.1, we have the following corollary.
Corollary 3.10 Suppose that (H 1) is replaced by the following condition,
(H4) there exist functionsand a constantsuch that for all, ,
and the others in Theorem 3.1 are not changed, then BVPs (1.1)-(1.2) have at least one solution.
4 An example
By Corollary 3.10, the BVP (4.1) has at least one solution in .
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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