Abstract
We consider the existence of at least one positive solution of the problem , , under the circumstances that , , where , is the Riemann-Liouville fractional derivative, and represents a nonlinear nonlocal boundary condition. By imposing some relatively mild structural conditions on f, , , and φ, one positive solution to the problem is ensured. Our results generalize the existing results and an example is given as well.
MSC: 34A08, 34B18.
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1 Introduction
In this paper we consider the existence of at least one positive solution of the fractional differential equation
subject to the boundary conditions
here is some measurable set, , is the Riemann-Liouville fractional derivative and φ is a linear functional having the form
where the integral appearing in (3) is taken in the Lebesgue-Stieltjes sense, θ is a function of bounded variation.
Let us review briefly some recent results on such problems in order to see our problem (1)-(2) in a more appropriate context.
So far, in view of their various applications in science and engineering, such as fluid mechanics, control system, viscoelasticity, porous media, edge detection, optical systems, electromagnetism and so forth, see [1]–[15], fractional differential equations have attracted great attention of mathematicians.
There are a great number of works on the existence of solutions of various classes of ordinary differential equations and fractional differential equations; readers may refer to [16]–[32].
Some of them discussed two-point boundary value problems. For example, Bai and Lü [6] studied the following two-point boundary value problem of fractional differential equations:
where , is the standard Riemann-Liouville fractional derivative. By means of Guo-Krasnosel’skiǐ’s fixed point theorem and the Leggett-Williams fixed point theorem they obtained the existence of positive solutions.
Some authors discussed multi-point boundary value problems, for instance, by using fixed point index theory, the Krein-Rutman theorem and some other methods, Jiang [7] studied the eigenvalue interval of the multi-point boundary value problem
where , is the Caputo derivative, , .
There are also results on fractional boundary value problem with integral boundary conditions, let us refer to Vong [8]. He investigated positive solutions of the nonlocal boundary value problem for a class of singular fractional differential equations with an integral boundary condition,
where , and μ is a function of bounded variation.
To proceed, Goodrich [9] considered the existence of at least one positive solution of the ordinary differential equation
in which the boundary condition is more general.
Motivated by the above works, we decided to consider the problem (1)-(2). As to the novel contributions of this work, we hold in the first place that the problem discussed in [9] is an ordinary differential equation, while we take a look into the fractional differential equation under the same boundary conditions. Secondly, the boundary conditions are more flexible and general than often. Let us take the condition into consideration, where , , are defined in the sequel. If , the conditions are the standard Dirichlet boundary conditions. Readers might refer to Bai and Lü [6]. As far as we are concerned, varies among many sorts of functionals. If (where is defined in the sequel) or , our conditions reduce to integral boundary conditions, while if , we have multi-point boundary conditions. Thirdly, compared to Goodrich [9], we make an adjustment to the Green function and define a function instead of a constant which affects the defined cone.
This paper is organized as follows. In Section 2, we review some preliminaries and lemmas. In Section 3, a theorem and five corollaries about the existence of at least one positive solution of problem (1)-(2) are obtained. Lastly, we give an example to illustrate the obtained theorem.
2 Preliminaries and lemmas
For the convenience of the readers, we give some background materials from fractional calculus theory to facilitate the analysis of the boundary value problem (1)-(2).
Definition 2.1
([2])
The Riemann-Liouville fractional integral of order of a function is given by
provided the right side is pointwise defined on .
Definition 2.2
([2])
The Riemann-Liouville typed fractional derivative of order α () of a continuous function is given by
where , denotes the integer part of number α, provided that the right side is pointwise defined on .
Lemma 2.1
([2])
Let. If we assume, then the fractional differential equation
has, , , as a unique solution, where n is the smallest integer greater than or equal to α.
Lemma 2.2
([2])
Letwith a fractional derivative of order α () that belongs to. Then
where n is the smallest integer greater than or equal to α.
Remark 2.1
([2])
The Riemann-Liouville type fractional derivative and integral of order α () have the following properties:
Lemma 2.3
Letand. Then the fractional differential equation boundary value problem
has a unique solution,
where
Proof
We may apply Lemma 2.2 to reduce (1) to an equivalent integral equation,
Consequently, the general solution of (1) is
By (2), we have
Therefore, the unique solution of problem (1) and (2) is
The proof is complete. □
Lemma 2.4
([6])
Letbe an arbitrary but fixed interval. Then the functiondefined by (8) satisfies the following conditions:
-
(1)
, for ;
-
(2)
there exists a positive function such that
(9)
Lemma 2.5
([4])
Let B be a Banach space, and letbe a cone. Assume, are open and bounded subsets of B with, , and letbe a completely continuous operator such that
-
(i)
, , and , ; or
-
(ii)
, , and , .
Then T has a fixed point in.
In order to get the main results, we first need some structure on , , φ, and f appearing in problem (1)-(2).
Let B be the Banach space on equipped with the usual supremum norm . Then define the cone by
where .
Define the operator by
Here we come to the nine significant assumptions.
(H1): Let and be real-valued continuous functions.
(H2): The functional can be written in the form
where satisfy , and , are continuous linear functionals.
(H3): There is a constant such that the functional φ in (11) satisfies the inequality
for all . Furthermore, there is a constant such that the functional in (11) satisfies whenever .
(H4): For each given , there are and whenever and we have
(H5): There exists a function satisfying the growth condition
for some , having the property that for each given , there is such that
for all , whenever .
(H6): Assume that the nonlinearity splits in the sense that , for continuous functions and such that a is not identically zero on any subinterval of .
(H7): Suppose .
(H8): Suppose .
(H9): For each both
and
hold, where (17) holds for each .
3 Main results
In this section we state and prove the existence theorem of problem (1)-(2).
Lemma 3.1
Let T be the operator defined in (10). Assume conditions (H1)-(H9) hold. Then, and the operator T is completely continuous.
Proof
Now we divide the proof into two steps; in the first step we prove that , then in the next, the conclusion that the operator T is completely continuous is treated.
Step 1. Here we are going to show that . In fact, since , , a, and g are all nonnegative, it is easy to find that whenever , it follows that , for each , where we use the fact , following .
On the other hand, provided we get
where .
Consequently, for each we deduce that
where the inequality follows from assumption (H9). Thus, , therefore as desired we conclude .
Step 2. In this part we turn to the proof that in the sequel is bounded as well as equicontinuous with the help of the Arzelà-Ascoli theorem.
With the continuity of , , a, and g, it is easy to find T is continuous.
Let be bounded, namely, there exists a number , such that for each we have . By the continuity of , , and φ, we find , are bounded, so there exist constants and such that and . is a measurable set, take , then
Hence is bounded.
For each , , , and assuming there is a , such that , for each put
where , we find , so that is equicontinuous.
Indeed,
By the continuity of , , and φ, we find , are bounded, so there exist constants and such that and . is a measurable set, take , then
Next we discuss the following three cases.
Case 1. .
Case 2. .
Case 3. .
Hence .
From the Arzelà-Ascoli theorem, T is completely continuous. The proof is complete. □
With Lemma 3.1 in hand, we are now ready to present the first existence theorem for problem (1) and (2).
Theorem 3.1
Assume that conditions (H1)-(H9) hold. Suppose that
and that. Then problem (1)-(2) has at least one positive solution.
Proof
Begin by selecting a number such that
From (H7), there exists a number such that whenever . Then set
for we find
from the definition of norm , we get , and so, T is a cone expansion on .
From condition (27), we choose a positive number ε small enough, so that we may assume
Since , it follows that , if , then , and then by condition (H4), we have
For , we may select such that . By the definition of the cone, we have
If , then
Hence, by condition (H5) we get
Provided that
Next we are going to discuss these two cases: g is bounded and unbounded on , respectively.
Now if g is bounded on , then there exists sufficiently large such that
Indeed, we might assume without loss of generality that
where ε is selected sufficiently small such that both (32) and (35) hold. We define a number
Set
Then for each we find that
whence T is a cone compression on .
On the other hand, assume g is unbounded on . From condition (H8) there is a number such that whenever , where is picked with
Since g is unbounded on , we can find a number satisfying
such that for any .
Take
Then for each we find that
whence T is a cone compression on .
Therefore, in either case, define , we find whenever . From Lemma 2.5, we claim that problem (1)-(2) has at least one positive solution; the proof is complete. □
Next we are going to give some corollaries since admits a wide variety of functionals. First we assume , then, respectively, that , and , where is not Lebesgue null. In addition, we know is also well defined and if the problem as well as the boundary conditions are similar to [6].
Corollary 3.1
Assume that conditions (H1), (H6)-(H9) hold, then the problem with Dirichlet conditions
has at least one positive solution.
Corollary 3.2
Assume that conditions (H1)-(H9) hold. Suppose, in addition,
andis not Lebesgue null, and, then the problem with integral conditions
has at least one positive solution.
Corollary 3.3
Assume that conditions (H1)-(H9) hold. Suppose, in addition,
andis not Lebesgue null, and, then the problem with multi-point conditions
has at least one positive solution.
Corollary 3.4
Assume that conditions (H1)-(H9) hold. Suppose, in addition,
andis not Lebesgue null, and the total variation of θ oversatisfies, then the problem with the Lebesgue-Stieltjes integral conditions
has at least one positive solution.
Specially, take , for any we have
then
thus we have the following corollary.
Corollary 3.5
Assume that conditions (H1)-(H4) and (H6)-(H9) hold. Suppose, in addition,
then the problem
has at least one positive solution.
4 Example
In this part we give an example of Theorem 3.1.
Define in the first place by
where
Then define , by
and
It is clear that
and moreover
so conditions (H4) and (H5) hold, and we see , , and .
Now we consider the boundary value problem
where is a given function with conditions (H7) and (H8) satisfied. Here is chosen such that and .
What is more, for each
and
Then we find that and , so condition (H3) is met as well. Finally, after straightforward numerical calculations, condition (H9) can also be achieved, since
As a consequence, each of conditions (H1)-(H9) is satisfied. From Theorem 3.1, problem (55) has at least one positive solution.
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Acknowledgements
The authors sincerely thank the reviewers for their valuable suggestions and useful comments, which have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (61374074), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2013AL003), also supported by Graduate Innovation Foundation of University of Jinan (YCX13013).
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Feng, W., Sun, S., Li, X. et al. Positive solutions to fractional boundary value problems with nonlinear boundary conditions. Bound Value Probl 2014, 225 (2014). https://doi.org/10.1186/s13661-014-0225-0
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DOI: https://doi.org/10.1186/s13661-014-0225-0