Abstract
By employing the monotone iterative method, this paper not only establishes the existence of the minimal and maximal positive solutions for multipoint fractional boundary value problem on an unbounded domain, but also develops two computable explicit monotone iterative sequences for approximating the two positive solutions. An example is given for the illustration of the main result.
1. Introduction
The fractional calculus has been recognized as an effective modeling methodology for describing hereditary properties of various materials and processes widely. For a lot of applications, we refer the reader to the books [1–5]. For some new development on the topic, see [6–17] and the references therein.
Recently, there has been a significant development on boundary value problems for fractional differential equations on infinite intervals; see papers [18–26], in which authors are devoted to investigating the existence of solutions and positive solutions by employing some fixed point theorems, Leray-Schauder nonlinear alternative theorem, or fixed point index theory.
By using Schauder’s fixed point theorem combined with the diagonalization method, Arara et al. [18] studied the existence of the bounded solution of the following problem on infinite intervals: where , , , and is the Caputo fractional derivative of order .
In [19], Zhao and Ge investigated the existence of positive solutions for the following fractional boundary value problem by employing the Leray-Schauder nonlinear alternative theorem: where , , , , and is the standard Riemann-Liouville fractional derivative.
Liang and Zhang [20] were concerned with the following nonlinear fractional differential equations with multipoint fractional boundary conditions on an unbounded domain: where , , denotes the Riemann-Liouville fractional derivative, , and , , satisfy . By using the fixed point index theory, authors gave sufficient conditions for the existence of multiple positive solutions to the above multi-point fractional boundary value problem.
However, very interesting and important question is “If we know the existence of the solution, how can we find it?” This question motivates us to reconsider problem (3). In this paper, we not only establish the existence of two positive solutions for problem (3), but also develop two computable explicit monotone iterative sequences for approximating the minimal and maximal positive solutions of (3), which is indeed an important and useful contribution to the existing literature on the topic. In addition, to start our work, we employ the monotone iterative method, which is different from the ones used in [18–26]. Let us state that this method was widely used for nonlinear problem; see, for instance, [27–38].
2. Preliminaries and Several Lemmas
In this section, we present some useful definitions and related theorems.
Definition 1 (see [2]). The Riemann-Liouville fractional derivative of order for a continuous function is defined by provided the right-hand side is pointwise defined on and is the integer part of .
Definition 2 (see [2]). The Riemann-Liouville fractional integral of order for a function is defined as provided that such integral exists.
Lemma 3 (see [20]). Let . For , the fractional boundary value problem has a unique solution where with
Lemma 4 (see [20]). For , then Green’s function has the following properties:(1)(2)where
For the forthcoming analysis, we will use a Banach space: equipped with the norm
Define a cone by and an operator as follows:
Observe that multi-point fractional boundary value problem (3) has a solution if and only if the integral operator has a fixed point.
3. Main Results
In this section, we shall construct two explicit monotone iterative sequences which converge to the minimal and maximal positive solutions of (3).
Theorem 5. Assume that the following conditions hold:(H1), on any subinterval of , and when is bounded, is bounded on ;(H2) does not identically vanish on any subinterval of and ;(H3) is nondecreasing for any , and there exists a constant , such that for .
Then the multi-point fractional boundary value problem (3) has the minimal and maximal positive solutions , in , which can be obtained by the following two explicit monotone iterative sequences:
Moreover,
Proof. By a similar process used in [20], it is easy to show that is completely continuous.
Now denote ; then we have . In fact, let ; then by and (12), we have
That is, .
Denote that , and , for all . Since and , then and . So, we have
By condition , for and , we have
This proves that is a nondecreasing operator.
So, we have
By the induction, define , . Then the sequence and satisfies the following relation:
In view of the complete continuity of the operator and , then is relative compact. That is, has a convergent subsequence and there exists a such that as . This, together with (23), holds .
Since is continuous and , then we have . That is, is a fixed point of the operator .
Denote that , , and , for all . Since and , then and . By , we have
Since is nondecreasing, then we have
By the induction, define , . Then the sequence and satisfies the following relation:
With an analysis exactly parallel to the proving process of , we have that there exists a such that .
Since is continuous and , we have . That is, is a fixed point of the operator .
Now, we are in a position to show that and are the maximal and minimal positive solutions of (3) in .
Let be any solution of (3). That is . Noting that is nondecreasing and , then we have , for all .
Similarly, we can obtain
Since and , it follows from (23)(27) that
Since , for all , then is not a solution of problem (3). Thus, by (28), we know that and are the maximal and minimal positive solutions of (3) in , which can be obtained by the corresponding iterative sequences in (17).
This completes the proof.
4. Example
Example 1. Take , , , , and . Consider the following boundary value problem:
where and
Now, we show that is bounded on when is bounded. Since
Then we have . So condition holds.
In view of , condition holds.
By a simple computation, we have that and . Taking , it follows that
Hence, condition holds. Thus all conditions of Theorem 5 are satisfied. Therefore, the fractional boundary value problem (29) has the minimal and maximal positive solutions in , which can be obtained by two explicit monotone iterative sequences.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to express their gratitude to the anonymous reviewers and editors for their valuable comments and suggestions which improved the quality of the original paper. This work is supported by the NNSF of China (no. 61373174) and the Natural Science Foundation for Young Scientists of Shanxi Province, China (no. 2012021002-3).