Abstract
We prove that there are unique positive solutions for a new kind of fractional differential equation with a negatively perturbed term boundary value problem. Our methods rely on an iterative algorithm which requires constructing an iterative scheme to approximate the solution. This allows us to calculate the estimation of the convergence rate and the approximation error.
1. Introduction
In this work, we present the uniqueness of a positive solution for the following boundary value problem (BVP for short): where is the standard fractional derivative of order satisfying , and may be singular at , .
Fractional calculus differential equations are an important branch of differential equations. In recent years, it has attracted the interest of many researchers and has become a hot-button issue [1–14]. Compared with the integer order, it has a wider range of applications as it can be used to describespecific problems more precisely, such as the problem in complex analysis, polymer rheology, physical chemistry, electrical networks, and many other branches of science, For specific applications, see [15, 16, 20–28]. In [3], the authors study the following BVP:
They obtained the existence of multiple positive solutions by means of the Gou-Krasnoselskii fixed point theorem. In [4], the authors also study the same BVP. By constructing a special -positive operator and using its properties, they obtained a unique solution for BVP (2).
BVP (1) is more general than the problem in papers [3] and [4] in four aspects. First, the nonlinear term has two space variables and can be singular with respect to the second space variable. Second, the method we used is different from papers [3] and [4]. Through constructing an iterative process which can be from any initial value, only by the iterative algorithm, we can prove that it converges uniformly to the unique positive solution. Third, we calculate the estimation of the convergence rate and the approximation error. Finally, we compare with it [3] and [4]. We lower the conditional constraint on the nonlinear term since we do not need the monotone of the nonlinear term. So the result of this paper is most general, not only did it weaken the restrictions but it also strengthened the conclusions in [3] and [4]. Also, we can show that the main result in [4] is a corollary of our work.
The rest of our presentation is as follows. In Section 2, we recall some definitions and lemmas. In Section 3, we establish the result of the uniqueness of the positive solutions to BVP (1).
Finally, in Section 4, an illustrative example is also presented.
2. Preliminaries
We first list some definitions and lemmas which will be used later.
Definition 1 (see [5]). Let , and the Riemann-Liouvill standard fractional integral and the Riemann-Liouvill standard fractional derivative of order of a function are given by where and denotes the integer part of the real number and provides that the right side integral is pointwise defined in .
Lemma 2 (see [3]). Suppose that and , then the boundary value problem is given by where
Lemma 3 (see [3]). The function obtained in Lemma 2 is continuous on and satisfies the following properties: (1)for all , (2)for all
In the rest of this paper, we assume that the following conditions hold:
H1. is increasing with respect to , decreasing with respect to , and for any , , there exists a constant such that
H2. are Lebesgue integrable, and does not vanish identically on any subinterval of .
H3. The third condition is as follows:
The basic space used in this paper is . Define a set in as follows:
Denote . Evidently, , i.e., is not empty.
Let the operator be defined by
3. Main Results
In this section, we will prove the existence of the positive solution to BVP (1) and calculate the approximation error and the convergence rate.
Theorem 4. Assume that H1, H2, and H3 hold. Then, BVP (1) has a unique positive solution satisfying where is a constant that belongs to .
Proof. The solution of BVP (1) coincides with the fixed point of operator . So our goal is to show that operator has a unique fixed point in . We divide our proof in four steps.
Step 1. We verify that operator is well defined. From H1, H2, and H3, for any and , we know that So, operator is well defined.
Step 2. We verify the properties of .
Let and
Consequently, we can prove that there exists constants such that
Therefore, the operator is well defined. It follows from H1 that is nondecreasing with respect to and nonincreasing with respect to .
Step 3. We will establish the existence of a positive solution to BVP (1). Since , from the above steps, we have . According to the definition of , there exists a constant such that Taking where is a fixed number satisfying In fact, . From (19), we get and . Moreover, and Combining with is nondecreasing with respect to and nonincreasing with respect to , so we have On the other hand, for any nature number , denote by and we have Therefore, for any nature number and , we obtain So, there exists such that uniformly on . By the same argument, we can also prove that uniformly on . Since is continuous, we can take the limits in and we get . Therefore, is a positive solution of BVP (1). Owing to , for any , there exists a constant such that holds.
Step 4. We further show its uniqueness. Let be another positive solution of BVP (1), then for any , there exists a constant such that Taking defined in (21) as being small enough such that . So In view of , by means of the nondecreasing , we obtain Taking limits to the above inequality, we get . Therefore, the solutions of BVP (1) are unique which completes the proof of Theorem 4.
Now, we are in a position to construct the successive sequence which converges to the unique solution.
Theorem 5. Suppose conditions H1, H2, and H3 are satisfied. Then, for any initial value , the successive sequence uniformly converges to the unique positive solution where the error estimation is the same order infinitesimal of , where and determined by .
Proof. According to Theorem 4, notice that the positive solution is unique; for any , there exists a constant , such that We shall adopt the similar argument as in the proof of Theorem 4; take to be a fixed number , thus Let then Taking limits to inequality (37), by the same method of the proof to (26), we can show that uniformly converges to the unique positive solution of BVP (1), and which means the error estimation is the same order infinitesimal of , where and determined by .
The proof is completed.
4. Example
Let us illustrate the main results with an example.
Example. Take .
We consider the following BVP:
For any , take , we can prove that
Combined with the expression of , , and , it is easy to see that H1 and H2 are holding. In addition,
So all of the assumptions in Theorem 4 are satisfied. Then, BVP (39) has a unique positive solution . For any initial value , we can construct the successive iterative sequence as follows which uniformly converges to the unique positive solution on . The error estimation is the same order infinitesimal of , i.e., where is a constant and determined by the initial value . In addition, for any , there exists a constantthat satisfies
5. Conclusions
We obtain the uniqueness of the positive solutions for a new kind of fractional differential equation with a negatively perturbed term. Through constructing an iterative process , which can be from any initial value , only by the iterative algorithm can we prove that it converges uniformly to the unique positive solution where the error estimation is the same order infinitesimal of .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there is no conflict of interests regarding the publication of this paper.
Authors’ Contributions
All authors typed, read, and approved the final manuscript.
Acknowledgments
This project is supported by the Natural Science Foundation of Shandong Province (Grant Nos. ZR2017LA002 and ZR2019MA022) and the Doctoral Research Foundation of Weifang University (Grant No. 2017BS02).