- Research Article
- Open Access
Positive Solutions of m-Point Boundary Value Problems for Fractional Differential Equations
© Zhi-Wei Lv. 2011
- Received: 28 November 2010
- Accepted: 19 January 2011
- Published: 9 February 2011
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.
- Banach Space
- Fractional Order
- Fractional Derivative
- Fractional Type
- Fractional Differential Equation
with and denotes the th Pseudo-derivative of , denotes the Pseudo fractional differential operator of order , is a continuous real-valued function on , and is a vector-valued Pettis-integrable function.
New results on the problem will be obtained.
Recall the following well-known definition and lemma (for more details on cone theory, see ).
a nonempty convex closed set is called a cone if it satisfied the following two conditions:
implies , where denotes the zero element of .
a cone is said to be normal if there exists a constant such that implies .
where is the smallest integer greater than or equal to .
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 real-valued functions , such that
for implies .
In the following, we will prove our main results.
It follows from the condition that .
The proof is complete.
If , then function in Lemma 2.1 satisfies the following conditions:
(i) , for ,
(ii) , for s, ,
Thus, for .
So, for . This, together with for , yields for .
Observing the express of , , and , we see that holds.
The proof is complete.
Let condition be satisfied. Suppose that . Then, problem (1.4) has at least one positive solution.
which implies that .
It is obvious from .
As , the right-hand side of the above inequality tends to zero, so, is equicontinuous.
By the Arzelá-Ascoli theorem, we conclude that the operator is completely continuous. Thus, our conclusion follows from Schauder fixed point theorem, and the proof is complete.
By Theorem 2.4, we know that BVP (1.4) has at least one positive solution in .
BVP (1.4) has a positive solution in , which is minimal positive solution.
From and the proof of Theorem 2.4, it may be concluded that and .
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 .
Let be any positive solution of BVP (1.4) in . It is obvious that .
Taking limits as in (2.32), we get .
BVP (1.4) has a positive solution in , which is maximal positive solution.
Thus, and .
Let be any positive solution of BVP (1.4) in .
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., [3–17] and references therein), fractional differential equations in Banach spaces have also been investigated by many authors (cf. e.g., [1, 2, 18–26] and references therein). In our coming papers, we will present more results on fractional differential equations in Banach spaces.
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 .
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