Abstract
In this paper, we study the existence of a class of higher-order singular semipositone fractional differential systems with coupled integral boundary conditions and parameters. By using the properties of the Green’s function and the Guo-Krasnosel’skii fixed point theorem, we obtain some existence results of positive solutions under some conditions concerning the nonlinear functions. The method of this paper is a unified method for establishing the existence of positive solutions for a large number of nonlinear differential equations with coupled boundary conditions. In the end, examples are given to demonstrate the validity of our main results.
MSC:34B16, 34B18.
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1 Introduction
Coupled boundary conditions arise in the study of reaction-diffusion equations, Sturm-Liouville problems, mathematical biology and so on; see [1–4]. Leung [5] studied the following reaction-diffusion system for prey-predator interaction:
subject to the coupled boundary conditions
where , a, r, , are positive constants, have Hölder continuous partial derivatives up to second order in compact sets, η is a unit outward normal at ∂ Ω and p and q have Hölder continuous first derivatives in compact subsets of . The functions , respectively represent the density of prey and predator at time and at position . Similar coupled boundary conditions are also studied in [6] for a biochemical system.
Existence theory for boundary value problems of ordinary differential equations is well studied. However, differential equations with fractional order are a generalization of the ordinary differential equations to non-integer order. This generalization is not a mere mathematical curiosity but rather has interesting applications in many areas of science and engineering such as electrochemistry, control, porous media, electromagnetism, etc. There has been a significant development in the study of fractional differential equations in recent years; see, for example, [7–13]. Wang et al. [14] researched a coupled system of nonlinear fractional differential equations
where , , , are continuous functions, , are also two standard Riemann-Liouville fractional derivatives. By using the Banach fixed point theorem and nonlinear differentiation of Leray-Schauder type, the existence and uniqueness of positive solutions are obtained.
In [15], Yang considered the positive solutions to boundary values problem for a coupled system of nonlinear fractional differential equations as follows:
where , are continuous, are nonnegative and integrable functions, are continuous, and , are standard Riemann-Liouville fractional derivatives. By applying the Banach fixed point theorem, nonlinear differentiation of Leray-Schauder type and the fixed point theorems of cone expansion and compression of norm type, sufficient conditions for the existence and nonexistence of positive solutions to a general class of integral boundary value problems for a coupled system of fractional differential equations are obtained.
Inspired by the above mentioned work and wide applications of coupled boundary conditions in various fields of sciences and engineering, in this paper, we research the existence result to a class of singular semipositone fractional differential systems with coupled integral boundary conditions of the type
where is a parameter, , , is the standard Riemann-Liouville derivative. is a constant, is right continuous on , left continuous at , and nondecreasing on , , denotes the Riemann-Stieltjes integrals of x with respect to , is a continuous function and may be singular at for . By a positive solution of system (1.1), we mean that , satisfies (1.1) and , for all .
To the best knowledge of the authors, there is seldom earlier literature studying fractional differential system with coupled integral boundary conditions like system (1.1), especially when () may be sign-changing, and may be singular at and . Motivated by the results mentioned above, this paper attempts to fill part of this gap in the literature.
2 Preliminaries and lemmas
For convenience of the reader, we present some necessary definitions about fractional calculus theory.
Let and let u be piecewise continuous on and integrable on any finite subinterval of . Then, for , we call
the Riemann-Liouville fractional integral of u of order α.
The Riemann-Liouville fractional derivative of order , , is defined as
where ℕ denotes the natural number set, the function is n times continuously differentiable on .
Let , if the fractional derivatives and are continuous on , then
where , n is the smallest integer greater than or equal to α.
Lemma 2.2 Assume that the following condition () holds.
()
Let (), then the system with the coupled boundary conditions
has a unique integral representation
where
and
Proof System (2.1) is equivalent to the system of integral equations
Integrating (2.4) and (2.5) with respect to and respectively, we have
Therefore, we can get
Note that
Thus, system (2.6) and (2.7) has a unique solution for and . By Cramer’s rule and simple calculations, it follows that
Substituting (2.8) and (2.9) into (2.4) and (2.5), we have
So (2.2) holds. The proof is completed. □
Lemma 2.3 For , the functions and () defined as (2.3) satisfy
where
Proof By [[18], Lemma 3.2], for any , we have
So, by (2.3) and (2.14), we have
By a similar proof as (2.15) and (2.16), we also obtain
then we know that (2.10) holds.
By [[18], Lemma 3.2], for any , we also have
So, by (2.3) and (2.17), we have
By a similar proof as (2.18) and (2.19), we also obtain
then we know that (2.11) holds.
On the other hand, by (2.3) and (2.14), we also have
So, we can get that (2.12) holds. By a similar proof as (2.20) and (2.21), we also obtain
which implies that (2.13) holds. The proof is completed. □
Remark 2.1 From Lemma 2.3, for , we have
where , ϱ, ρ are defined as Lemma 2.3, .
In the rest of the paper, we always suppose that the following assumption holds:
() is continuous and satisfies
where are continuous and may be singular at , is continuous and
Lemma 2.4 Assume that (), () hold. Then the system with the coupled boundary conditions
has a unique solution
which satisfies
Proof It follows from Lemmas 2.2, 2.3 and conditions (), () that (2.22) and (2.23) hold. The proof is completed. □
Next we consider the following singular nonlinear system:
where , is defined as (2.22), ().
Lemma 2.5 If is a solution of system (2.24) with , for any , then is a positive solution of system (1.1).
Proof In fact, if is a positive solution of system (2.24) such that , for any , then from system (2.24) and the definition of , we have
Let , , , then
which implies that
Thus, system (2.25) becomes
Then, by (2.26), is a positive solution of system (1.1). The proof is completed. □
Let , then X is a Banach space with the norm
Let
where ω is defined as Remark 2.1. It is easy to see that K is a positive cone in X. Under the above conditions (), (), for any , we can define an integral operator by
We know that is a positive solutions of system (1.1) if and only if is a fixed point of T in K.
Lemma 2.6 Assume that (), () hold. Then is a completely continuous operator.
Proof By a routine discussion, we know that is well defined, so we only prove . For any , , by Remark 2.1, we have
On the other hand,
Then we have
i.e.,
In the same way as (2.29) and (2.30), we can prove that
Therefore, we have .
According to the Ascoli-Arzela theorem, we can easily get that is completely continuous. The proof is completed. □
In order to obtain the existence of the positive solutions of system (1.1), we will use the following cone compression and expansion fixed point theorem.
Lemma 2.7 [19]
Let P be a positive cone in a Banach space E, and are bounded open sets in E, , , is a completely continuous operator. If the following conditions are satisfied:
or
then A has at least one fixed point in .
3 Main results
Theorem 3.1 Assume that (), () hold and that for any fixed , the following conditions are satisfied:
() There exists a constant
such that
()
where ω is defined as Remark 2.1, ρ is defined as Lemma 2.3,
Then system (1.1) has at least one positive solution . Moreover, satisfies , , for some positive constant .
Proof Let . For any , , by the definition of , we know that
So, for any , by condition () and Lemma 2.3, we have
Similarly as (3.1), for any , by condition (), we also have
Consequently, we have
On the other hand, by the first inequality in (), there exists such that , and also there exists such that
Choose . Let . For any , by the definition of and (2.23), we have
Thus, for any , by (3.3)-(3.5), we have
Hence, for any , by (3.6) and Lemma 2.3, we conclude that
Consequently,
Obviously, by the second inequality in (), (3.7) is still valid.
It follows from the above discussion, (3.2), (3.7), Lemmas 2.6 and 2.7, that for any fixed , T has a fixed point and . Since , we have
Let , , , then we have
By Lemma 2.5, we know that for any fixed , system (1.1) has at least one positive solution ; moreover, satisfies , , . The proof is completed. □
Remark 3.1 From the proof of Theorem 3.1, we know that the conclusion of Theorem 3.1 is valid if condition () is replaced by
where
Theorem 3.2 Assume that (), () hold and that for any fixed , the following conditions are satisfied:
() There exists a constant
such that
()
where , (), are defined in Theorem 3.1. Then system (1.1) has at least one positive solution . Moreover, satisfies , , for some positive constant .
The proof of Theorem 3.2 is similar to that of Theorem 3.1, and so we omit it.
Remark 3.2 The conclusion of Theorem 3.2 is valid if inequality (3.8) in condition () is replaced by
where is defined in Remark 3.1.
Theorem 3.3 Assume that (), () hold and that the following is satisfied:
()
Then there exist , such that system (1.1) has at least one positive solution provided , . Moreover, satisfies , , for some positive constant .
Proof Choose . Let
where ω is defined as Remark 2.1, ρ is defined as Lemma 2.3, ().
Let . For any , , by the definition of , we know that
So, for , , by Lemma 2.3, we have
Similarly as (3.9), for any , by condition (), we also have
Consequently, we have
On the other hand, by the first inequality in (), choose such that
where ω is defined as Remark 2.1, ϱ is defined as Lemma 2.3. Then there exists such that
Let
For any , by (2.23), we have
Thus, for any , by (3.11)-(3.13), we have
Hence, for any , by (3.14) and Lemma 2.3, we have
Consequently,
Obviously, by the second inequality in (), (3.15) is still valid.
It follows from the above discussion, (3.10), (3.15), Lemmas 2.6 and 2.7, that for any , , T has a fixed point and . Since , we have
Let , , , then we have
By Lemma 2.5 we know that for any , , system (1.1) has at least one positive solution ; moreover, satisfies , , . The proof is completed. □
Remark 3.3 From the proof of Theorem 3.3, we know that the conclusion of Theorem 3.3 is valid if condition () is replaced by
Theorem 3.4 Assume that (), () hold and that the following condition is satisfied:
()
and
where , . Then there exist such that system (1.1) has at least one positive solution provided , . Moreover, satisfies , , for some positive constant .
Proof It follows from
of (), there exists such that
Select
In proving the theorem, we assume
and
where . For any , by (2.23), we have
Thus, for any , by (3.16)-(3.18), we have
Hence, for any , by (3.19) and Lemma 2.3, we have
Consequently,
Obviously, by the inequality
in (), (3.20) is still valid.
On the other hand, choose such that
Then, for the above , by the first inequality in (), there exists such that for any , we have
Then we have
where . Select
Assume
For any , by (3.21) and Lemma 2.3, we have
Similarly as (3.22), for any , by (3.21) and Lemma 2.3, we also have
Consequently, we have
It follows from the above discussion, (3.20), (3.23), Lemmas 2.6 and 2.7, that for any , , T has a fixed point and . Since , by the same method as Theorem 3.3, we know that for any , , system (1.1) has at least one positive solution . Moreover, satisfies , , . The proof is completed. □
Remark 3.4 From the proof of Theorem 3.4, we know that the conclusion of Theorem 3.4 is valid if the second equality of condition () is replaced by
Remark 3.5 From the proof of Theorem 3.4, we know that the conclusion of Theorem 3.4 is valid if the second equality of condition () is replaced by
where ∧ is defined in Theorem 3.4.
Similarly as Remark 3.4, the conclusion of Theorem 3.4 is also valid if the second equality of condition () is replaced by
4 Example
Consider the fractional differential system
where () is a parameter, , , , , . Then we have
So, condition () holds.
Next, in order to demonstrate the application of our main results obtained in Section 3, we choose two different sets of functions () such that satisfies the conditions of Theorems 3.3 and 3.4.
Case 1. Let , , . Take , , , , , then
By a direct calculation, we have
So condition () holds.
In addition, choose , we know
so condition () of Theorem 3.3 is satisfied.
Therefore, by Theorem 3.3, we obtain that system (4.1) has at least one positive solution provided () is small enough.
Case 2. Let , , . Take , , , , , , then
By a direct calculation, we have
So condition () holds.
In addition, choose , we know
and
so condition () of Theorem 3.4 is satisfied.
Therefore, by Theorem 3.4, we obtain that system (4.1) has at least one positive solution provided () is sufficiently large.
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Acknowledgements
The first and second authors were supported financially by the National Natural Science Foundation of China (11371221), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001) and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province. The third author was supported financially by the Australia Research Council through an ARC Discovery Project Grant.
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The study was carried out in collaboration between all authors. YW completed the main part of this paper and gave two examples; LSL and YHW corrected the main theorems and polished the manuscript. All authors read and approved the final manuscript.
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Wang, Y., Liu, L. & Wu, Y. Positive solutions for a class of higher-order singular semipositone fractional differential systems with coupled integral boundary conditions and parameters. Adv Differ Equ 2014, 268 (2014). https://doi.org/10.1186/1687-1847-2014-268
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DOI: https://doi.org/10.1186/1687-1847-2014-268