The Exact Iterative Solution of Fractional Differential Equation with Nonlocal Boundary Value Conditions

Abstract

We deal with a singular nonlocal fractional differential equation with Riemann-Stieltjes integral conditions. The exact iterative solution is established under the iterative technique. The iterative sequences have been proved to converge uniformly to the exact solution, and estimation of the approximation error and the convergence rate have been derived. An example is also given to demonstrate the results.

Dedicated to our advisors

1. Introduction

Fractional differential equations arise in many engineering and scientific disciplines; see [1–5]. Much attention has been paid to study fractional differential equations both with initial and boundary conditions; see, for example, [6, 7]. In [8, 9], they focused on sign-changing solution for some fractional differential equations. In [10], they get the existence of solutions for impulsive fractional differential equations. In [11–13], they get the existence and multiplicity of nontrivial solutions for a class of fractional differential equations. The mainly techniques authors need are fixed point theory, variational method, and global bifurcation techniques.

Also, ordinary differential equations and partial differential equations involving nonlocal boundary conditions have been studied extensively in recent years, see [14–22], including integral boundary conditions and multipoint boundary conditions.

In [23], authors obtained results on the uniqueness of positive solution for problem where is a real number. Under the assumption that where , and is the first eigenvalue of the corresponding linear operator.

Motivated by the above works, we study the following nonlocal boundary value problems:where denotes the left-handed Riemann-Liouville derivative of order q and is a real number. denotes a Stieltjes integral with a suitable function of bounded variation. Different from [23] and other works, we only use the iterative methods to obtain the existence and uniqueness of positive solution. Moreover, the estimation of the approximation error and the convergence rate have also been derived.

For clarity in presentation, we also list below some assumptions to be used later in the paper.

is continuous, and for , is increasing with respect to and there exists a constant such that, for ,It is easy to see that if , then

,

2. Preliminaries

For the convenience of the reader, we present here some necessary definitions from fractional calculus theory. These definitions and properties can be found in the recent monograph [23].

Definition 1. The Riemann-Liouville fractional integral of order of a function is given by provided that the right-hand side is pointwise defined on

Definition 2. The Riemann-Liouville fractional derivative of order of a continuous function is given by where , , provided that the right-hand side is pointwise defined on In particular,

Lemma 3 (see [13]). Assume that hold. Let Then boundary value problem has the unique solution given by the following formula: where

One can prove that have the following properties.

Lemma 4. Note that is the Green function of problem (8).

Lemma 5 (see [12]). For , one has where

Lemma 6. where is a constant and is nonnegative for any

Proof. We have the estimation where and Thus, (13) holds.

3. The Main Results

Throughout this paper, we will work in the space , which is a Banach space if it is endowed with the norm for any

Define the set in as follows:

there exists positive constants such that

And define the operator .Evidently Therefore, is not empty.

Theorem 7. Assume that - hold. And Then BVP (3) has at least one positive solution , and there exist constants satisfying

Proof. It is clear that is a solution of (3) if and only if is a fixed point of .
Claim 1. The operator is nondecreasing.
In fact, for , it is obvious that , , and for For any , we have that, for , and where and are positive constants satisfying Thus, it follows that there are constants such that, for , Therefore, for any , , i.e., is the operator From (16), it is easy to see that is nondecreasing for Hence, Claim 1 holds.
Claim 2. We take . Let and be fixed numbers satisfyingand assume thatThenand there exists such thatuniformly on
In fact, since Therefore, From (24), we have and
On the other hand, and since and is nondecreasing, by induction, (26) holds.
Let , and then It follows from (4) that And for any natural number , Thus, for any natural number and , we havewhich implies that there exists such that (27) holds and Claim 2 holds.
Letting in and noting the fact that is continuous, we obtain , which is a positive solution of BVP (3). The proof of Theorem 7 is now complete.

Theorem 8. Assume that - hold. Then
(i) BVP (3) has unique positive solution , and there exist constants with such that (ii) For any initial value , there exists a sequence that uniformly converges to the unique positive solution , and one has the error estimationwhere is a constant with and determined by

Proof. Let be defined in (24) and (25).
(i) It follows from Theorem 7 that BVP (3) has a positive solution , which implies that there exist constants and with such that satisfies (18). Let be another positive solution of BVP (3); then from Theorem 7 we have that there exist constants and with such that Let defined in (23) be small enough such that and defined in (23) be large enough such that Then Note that and is nondecreasing; we haveLetting in (36), we obtain that Hence, the positive solution of BVP (3) is unique.
(ii) From (i), we know that the positive solution to BVP (3) is unique. For any , there exist constants and with such that Similar to (i), we can let and defined by (23) satisfy and Then Let Note that is nondecreasing; we haveLetting in (39), it follows that uniformly converges to the unique positive solution for BVP (3), where At the same time, (33) follows from (31). Thus, the proof of the theorem is complete.

4. An Example

where Analysis 1. Let and then for any , we take and have Then holds.

In addition, we have Then and hold.

And

Hence all conditions of Theorem 7 are satisfied, and consequently we have the following corollary.

Corollary 9. Problem (41) has unique positive solution . For any initial value , the successive iterative sequence generated by uniformly converges to the unique positive solution on . One has the error estimation where is a constant with and determined by the initial value And there are constants with such that

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the Natural Science Foundation of China (11571197), the Science Foundation of Qufu Normal University of China (XJ201112), Jining University Youth Research Foundation (2015QNK102), a project of Shandong Province Higher Educational Science and Technology Program (J17KB143), Shandong Education Science 13th Five-Year Plan Project (BYK2017003).