Positive Solutions of m-Point Boundary Value Problems for Fractional Differential Equations

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

(1.1)

where is the standard Riemann-Liouville fractional derivative of order , , , , , and satisfies Carathéodory-type conditions. Moreover, [2] considered the following nonlinear -point boundary value problem of fractional type:

(1.2)

where takes values in a reflexive Banach space ,

(1.3)

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.

In this paper, we consider the existence of minimal and maximal positive solutions for the following multiple-point boundary value problem:

(1.4)

where is the standard Riemann-Liouville fractional derivative,

(1.5)

, is continuous, , , , , , and

(1.6)

New results on the problem will be obtained.

Recall the following well-known 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,

(1.7)

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 real-valued functions , such that

(2.1)

for implies .

In the following, we will prove our main results.

Lemma 2.1.

Let . Then, the fractional differential equation

(2.2)

has a unique solution which is given by

(2.3)

where

(2.4)

in which

(2.5)

where

(2.6)

Proof.

Using Lemma 1.2, we have

(2.7)

It follows from the condition that .

Thus,

(2.8)

This, together with the relation , yields

(2.9)

From the boundary value condition , we deduce that

(2.10)

Thus,

(2.11)

The proof is complete.

Lemma 2.2.

If , then function in Lemma 2.1 satisfies the following conditions:

(i)  , for ,

(ii)  , for s,,

where

(2.12)

in which

(2.13)

Proof.

When , we have

(2.14)

Thus, for .

Furthermore, we conclude that

(2.15)

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

(2.16)

Thus,

(2.17)

Now, we define an operator by

(2.18)

Theorem 2.4.

Let condition be satisfied. Suppose that . Then, problem (1.4) has at least one positive solution.

Proof.

Let , where

(2.19)

Step 1.

, for any

(2.20)

which implies that .

Step 2.

is continuous.

It is obvious from .

Step 3.

is equicontinuous.

From (2.11) and (2.18), for any  , , , we conclude that

(2.21)

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.

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

(2.22)

(2.23)

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

(2.24)

This, together with , yields that

(2.25)

From and the proof of Theorem 2.4, it may be concluded that and .

Let

(2.26)

Thus,

(2.27)

By the complete community of , we know that is relatively compact. So, there exists a and a subsequence

(2.28)

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

(2.29)

and , we see that

(2.30)

By (2.30), (2.22), and Lebesgue's dominated convergence theorem, we get

(2.31)

Let be any positive solution of BVP (1.4) in . It is obvious that .

Thus,

(2.32)

Taking limits as in (2.32), we get .

Step 2.

BVP (1.4) has a positive solution in , which is maximal positive solution.

Let

(2.33)

It is obvious that

(2.34)

Thus, and .

By (2.18), (2.23), and , we have

(2.35)

This, together with , yields that

(2.36)

Using a proof similar to that of Step 1, we can show that

(2.37)

Let be any positive solution of BVP (1.4) in .

Obviously,

(2.38)

This, together with , implies

(2.39)

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.

3. An Example

Example 3.1.

Consider the following boundary value problem

(3.1)

where , , , , ,

(3.2)

, . By computation, we deduce that

(3.3)

From Remark 2.3, we get

(3.4)

Therefore,

(3.5)

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

(3.6)