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Positive Solutions of mPoint Boundary Value Problems for Fractional Differential Equations
Advances in Difference Equations volumeÂ 2011, ArticleÂ number:Â 571804 (2011)
Abstract
We discuss the existence of minimal and maximal positive solutions for fractional differential equations with multipoint boundary value conditions, and new results are given. An example is also given to illustrate the abstract results.
1. Introduction
Recently, [1] discussed the existence of positive solutions for the following boundary value problem of fractional order differential equation
where is the standard RiemannLiouville fractional derivative of order , , , , , and satisfies CarathÃ©odorytype conditions. Moreover, [2] considered the following nonlinear point boundary value problem of fractional type:
where takes values in a reflexive Banach space ,
with and denotes the th Pseudoderivative of , denotes the Pseudo fractional differential operator of order , is a continuous realvalued function on , and is a vectorvalued Pettisintegrable function.
In this paper, we consider the existence of minimal and maximal positive solutions for the following multiplepoint boundary value problem:
where is the standard RiemannLiouville fractional derivative,
, is continuous, , , , , , and
New results on the problem will be obtained.
Recall the following wellknown definition and lemma (for more details on cone theory, see [3]).
Definition 1.1.
Let be a real Banach space. Then,

(a)
a nonempty convex closed set is called a cone if it satisfied the following two conditions:

(i)
implies ,

(ii)
implies , where denotes the zero element of .

(b)
a cone is said to be normal if there exists a constant such that implies .
Lemma 1.2.
Assume that with a fractional derivative of order that belongs to . Then,
where is the smallest integer greater than or equal to .
2. Main Results
Let and . Then, is the Banach space endowed with the norm and is normal cone.
We list the following assumptions to be used in this paper.
there exist two nonnegative realvalued functions , such that
for implies .
In the following, we will prove our main results.
Lemma 2.1.
Let . Then, the fractional differential equation
has a unique solution which is given by
where
in which
where
Proof.
Using Lemma 1.2, we have
It follows from the condition that .
Thus,
This, together with the relation , yields
From the boundary value condition , we deduce that
Thus,
The proof is complete.
Lemma 2.2.
If , then function in Lemma 2.1 satisfies the following conditions:
(i)â€‰â€‰, for ,
(ii)â€‰â€‰, for s,,
where
in which
Proof.
When , we have
Thus, for .
Furthermore, we conclude that
So, for . This, together with for , yields for .
Observing the express of , , and , we see that holds.
The proof is complete.
Remark 2.3.
From the express of and , we see that
Thus,
Now, we define an operator by
Theorem 2.4.
Let condition be satisfied. Suppose that . Then, problem (1.4) has at least one positive solution.
Proof.
Let , where
Step 1.
, for any
which implies that .
Step 2.
is continuous.
It is obvious from .
Step 3.
is equicontinuous.
From (2.11) and (2.18), for anyâ€‰â€‰, , , we conclude that
As , the righthand side of the above inequality tends to zero, so, is equicontinuous.
By the ArzelÃ¡Ascoli theorem, we conclude that the operator is completely continuous. Thus, our conclusion follows from Schauder fixed point theorem, and the proof is complete.
Theorem 2.5.
Besides the hypotheses of Theorem 2.4, we suppose that holds. Then, BVP (1.4) has minimal positive solution in and maximal positive solution in ; Moreover, , as uniformly on , where
Proof.
By Theorem 2.4, we know that BVP (1.4) has at least one positive solution in .
Step 1.
BVP (1.4) has a positive solution in , which is minimal positive solution.
From (2.18) and (2.22), one can see that
This, together with , yields that
From and the proof of Theorem 2.4, it may be concluded that and .
Let
Thus,
By the complete community of , we know that is relatively compact. So, there exists a and a subsequence
such that converges to uniformly on . Since is normal and is nondecreasing, it is easily seen that the entire sequence converges to uniformly on . being closed convex set in and imply that .
From
and , we see that
By (2.30), (2.22), and Lebesgue's dominated convergence theorem, we get
Let be any positive solution of BVP (1.4) in . It is obvious that .
Thus,
Taking limits as in (2.32), we get .
Step 2.
BVP (1.4) has a positive solution in , which is maximal positive solution.
Let
It is obvious that
Thus, and .
By (2.18), (2.23), and , we have
This, together with , yields that
Using a proof similar to that of Step 1, we can show that
Let be any positive solution of BVP (1.4) in .
Obviously,
This, together with , implies
Taking limits as in (2.39), we obtain .
The proof is complete.
On the other hand, we note that in these years, going with the significant developments of various differential equations in abstract spaces (cf., e.g., [3â€“17] and references therein), fractional differential equations in Banach spaces have also been investigated by many authors (cf. e.g., [1, 2, 18â€“26] and references therein). In our coming papers, we will present more results on fractional differential equations in Banach spaces.
3. An Example
Example 3.1.
Consider the following boundary value problem
where , , , , ,
, . By computation, we deduce that
From Remark 2.3, we get
Therefore,
On the one hand, it is obvious that . Thus, is satisfied.
For , we see that , which implies that holds.
Hence, by Theorem 2.5, BVP (3.1) has minimal and maximal positive solutions in .
Furthermore, we can conclude that
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Lv, ZW. Positive Solutions of mPoint Boundary Value Problems for Fractional Differential Equations. Adv Differ Equ 2011, 571804 (2011). https://doi.org/10.1155/2011/571804
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DOI: https://doi.org/10.1155/2011/571804
Keywords
 Banach Space
 Fractional Order
 Fractional Derivative
 Fractional Type
 Fractional Differential Equation