Positive Solutions for BVP of Fractional Differential Equation with Integral Boundary Conditions

Abstract

In this paper, we consider a class of boundary value problems of nonlinear fractional differential equation with integral boundary conditions. By applying the monotone iterative method and some inequalities associated with Green’s function, we obtain the existence of minimal and maximal positive solutions and establish two iterative sequences for approximating the solutions to the above problem. It is worth mentioning that these iterative sequences start off with zero function or linear function, which is useful and feasible for computational purpose. An example is also included to illustrate the main result of this paper.

1. Introduction

Fractional calculus has widespread applications in many fields of science and engineering, for example, viscoelasticity, continuum mechanics, bioengineering, rheology, electrical networks, control theory of dynamical systems, and optics and signal processing [1, 2].

In the past decades, the existence of solutions or positive solutions for boundary value problems (BVPs for short) of nonlinear fractional differential equations attracted considerable attention from many authors, see [3–19] and the references therein.

Recently, the monotone iterative method has been applied to study BVPs of nonlinear fractional differential equations. For example, in [20], Cui et al. discussed the BVPwhere and denotes the standard Riemann–Liouville fractional derivative of order . The authors obtained the existence of maximal and minimal solutions and the uniqueness result for BVP (1). In 2014, Sun and Zhao [21] investigated the following BVP with integral boundary conditions:where and is the standard Riemann–Liouville fractional derivative of order . By means of the monotone iterative method, they proved the existence of a positive solution and established an iterative sequence for approximating the solution to BVP (2). For relevant results, one can refer to [22–25].

Motivated by the aforementioned works, in this paper, we consider the following BVP of nonlinear fractional differential equation with integral boundary conditions:where and denote the standard Caputo fractional derivatives of order and order , respectively. Throughout this paper, we always assume that and are nonnegative constants satisfying , and and are continuous.

The main tool used is the following theorem [26].

Theorem 1. Let be a normal cone of a Banach space and . Suppose that  is completely continuous  is monotone increasing on   is a lower solution of , that is,   is an upper solution of , that is, Then, the iterative sequencessatisfyand converge to, respectively, and which are fixed points of .

2. Preliminaries

First, we present the definitions of Riemann–Liouville fractional integral and fractional derivative and Caputo fractional derivative on a finite interval of the real line, which may be found in [1].

In this section, we always assume that and denotes the integer part of .

Definition 1. The Riemann–Liouville fractional integral of order on is defined by

Definition 2. The Riemann–Liouville fractional derivative of order on is defined bywhere .

Definition 3. Let be the Riemann–Liouville fractional derivative of order . Then, the Caputo fractional derivative of order on is defined bywhere

Lemma 1 (see [2]). Let . Then, the equation is satisfied for .

Lemma 2 (see [1]). Let be given by (9). Then, the following relations hold:(1).(2).For convenience, we denote

Lemma 3. Let . Then, for any , the BVPhas a unique solutionHere,where

Proof. In view of the equation in (11), Theorem 3.24 [1], and , we haveBy (15), Lemma 1, and Lemma 2, we obtainIt follows from (15) and (16) and the boundary conditions in (11) thatwhich together with (15) shows thatFrom (18), we getand so,which together with (18) implies thatIn what follows, we let

Lemma 4. satisfies the following properties:(1).(2).

Proof. Since (1) is obvious, we only need to prove that (2) holds.
First, it is clear that for .
Now, we verify that for . In fact, if , thenand if , thenBy the definition of and the condition , we may obtain the following remark.

Remark 1. and for .
In the remainder of this paper, we always assume that the following condition is fulfilled:Now, we define

Lemma 5. has the following property:

Proof. In view of the definition of and Lemma 4, it is obvious.

3. Main Results

For convenience, we let

Theorem 2. Assume that for and the following condition is satisfied:Then, BVP (3) possesses a minimal positive solution and a maximal positive solution .

Proof. Let be equipped with the normThen, is a normal cone in Banach space . Note that this induces an order relation “” in by defining if and only if .
Now, we define and byThen, it is easy to know that is completely continuous, and fixed points of are nonnegative solutions of BVP (3).

Step 1. We assert that is monotone increasing on .
Let and . Then, for , which together with implies thatThis shows that .

Step 2. We prove that is a lower solution of .
For any , we havewhich indicates that .

Step 3. We show that is an upper solution of .
In view of Lemma 5 and , we getwhich implies that .

Step 4. We claim that BVP (3) possesses a minimal positive solution and a maximal positive solution .
In fact, if we construct sequences and as in the following,then it follows from Theorem 1 thatand and converge to, respectively, and , which are nonnegative solutions of BVP (3).
In view of Lemma 5, Remark 1, the definitions of and , and the assumption for , we getand so,which shows that and are positive solutions of the BVP (3).
Moreover, if is a positive solution of the BVP (3), then it follows from the fact is monotone increasing on that . This indicates that and are minimal and maximal positive solutions of the BVP (3), respectively.

Example 1. Consider the following BVP:Since , a simple calculation shows thatMoreover, in view of and for , we getwhich indicates that is fulfilled. At the same time, we also obtain thatNow, if we let , then it is easy to know that is continuous, for , and (C2) is fulfilled. Therefore, it follows from Theorem 2 that BVP (39) possesses a minimal positive solution and a maximal positive solution . In addition, the two iterative schemes are as follows:

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant 11661049).