Journal of Function Spaces
Volume 2020 (2020), Article ID 5623589, 10 pages
https://doi.org/10.1155/2020/5623589

Research Article

Positive Solutions for a Weakly Singular Hadamard-Type Fractional Differential Equation with Changing-Sign Nonlinearity

Xinguang Zhang,1,2 Lixin Yu,¹ Jiqiang Jiang,³ Yonghong Wu,² and Yujun Cui⁴

¹School of Mathematical and Informational Sciences, Yantai University, Yantai, 264005 Shandong, China
²Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia
³School of Mathematical Sciences, Qufu Normal University, Qufu, 273165 Shandong, China
⁴Department of Mathematics, Shandong University of Science and Technology, Qingdao, 266590 Shandong, China

Correspondence should be addressed to Xinguang Zhang; zxg123242@163.com and Lixin Yu; lixinyu970@gmail.com

Received 30 April 2020; Accepted 20 June 2020; Published 1 August 2020

Academic Editor: Dumitru Motreanu

Copyright © 2020 Xinguang Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we focus on the existence of positive solutions for a class of weakly singular Hadamard-type fractional mixed periodic boundary value problems with a changing-sign singular perturbation. By using nonlinear analysis methods combining with some numerical techniques, we further discuss the effect of the perturbed term for the existence of solutions of the problem under the positive, negative, and changing-sign cases. The interesting points are that the nonlinearity can be singular at the second and third variables and be changing-sign.

1. Introduction

In this paper, we study the existence of positive solutions for the following Hadamard-type fractional differential equation with mixed periodic boundary conditions:

where , is the Hadamard fractional derivative of order , and is a continuous function with singularity at the second and third space variables; is a variable sign function.

Nonlocal characteristics are the most important property of the fractional differential operator; because of this, the fractional differential equations can describe many viscoelasticities and memory phenomena of natural science. For example, some viscoelastic materials, such as the silicone gel with the property of weak frequency dependency, often involve a complicated strain-stress relationship; if let and be the stress and the strain, respectively, then the stress decays after a shear jump is governed by the following fractional order viscoelasticity Kelvin-Voigt equation [1, 2]:

where , are constants and are Riemann-Liouville fractional derivatives. In practice, the study of the qualitative properties of solutions for the corresponding fractional models such as existence, uniqueness, multiplicity, and stability is necessary to analyze and control the model under consideration [3–21]. In [3], Zhang et al. considered a singular fractional differential equation with signed measure

where are the standard Riemann-Liouville derivatives, is denoted by a Riemann-Stieltjes integral and is a function of bounded variation and can be a signed measure; the nonlinearity may be singular at both and . By using the spectral analysis of the relevant linear operator and Gelfand’s formula combining the calculation of a fixed point index of the nonlinear operator, some sufficient conditions for the existence of positive solutions were established. Recently, by using the fixed point index theory, Wang [10] established the existence and multiplicity of positive solutions for a class of singular fractional differential equations with nonlocal boundary value conditions, where the nonlinearity may be singular at some time and space variables.

In the recent years, to improve and develop the fractional calculus, there are several kinds of fractional derivatives and integral operators with different kernels such as Hadamard, Erdelyi-Kober, Caputo-Fabrizio derivatives, Hilfer derivatives, and integrals to be given to enrich the application of the fractional calculus such as the Rubella disease and human liver model [22–25]. In particular, the Hadamard derivative is a nonlocal fractional derivative with singular logarithmic kernel. So the study of Hadamard fractional differential equations is relatively difficult; see [26–32].

In this paper, we are interested in the existence of solutions for the Hadamard-type fractional differential equation (1) which involves a singular perturbed term ; we will further discuss the effect of the perturbed term for the existence of solutions when is positive, negative, and changing-sign. Our main tools rely on nonlinear analysis methods as well as some numerical techniques. Thus, in order to make our work be more self-contained, a brief overview for these methods and techniques should be necessary. In recent work [33–41], some fixed point theorems were employed to study the qualitative properties of solutions for various types of differential equations. For obtaining numerical and analytical results, many authors [42–59] also developed iterative techniques to solve some nonlinear problems with practical applications. In addition, variational methods [60–78] and upper and lower solution methods [2, 25, 79, 80] also offered wonderful tools for dealing with various nonlinear ordinary and partial differential equations arising from natural science fields. These analysis and techniques not only improved and perfected the relative theoretical framework of differential equations but also gave some new understand for the corresponding natural phenomena.

Our work has some new features. Firstly, the equation contains a Hadamard-type fractional derivative which has a singular logarithmic kernel. Secondly, the nonlinearity involves a perturbed term which can be positive, negative, or changing-sign. Thirdly, the nonlinearity is allowed to have weakly singular at space variables, which is a class of interesting natural phenomena. In the end, the effect of the perturbed term for the existence of solutions of the equation is discussed, and the criteria on the existence of positive solutions are established for all cases of the perturbed term including positive, negative, and changing-sign cases. The rest of this paper is organized as follows. In Section 2, we firstly introduce the concept of the Hadamard fractional integral and differential operators and then give the logarithmic kernel and Green function of periodic boundary value problem and their properties. Our main results are summarized in Section 3 which includes three theorems for three different situations. An example is given in Section 4.

2. Basic Definitions and Preliminaries

Before starting our main results, we firstly recall the definition of the Hadamard-type fractional integrals and derivatives; for detail, see [81].

Let , , and be a finite or infinite interval of . The -order left Hadamard fractional integral is defined by

and the left Hadamard fractional derivative is defined by

In what follows, we firstly consider the linear auxiliary problem

It follows from [26] that the problem (6) has a unique solution:

where

is Green’s function of (6). Now, let , then the Hadamard-type fractional differential equation (1) reduces to the following second order integrodifferential equation:

In order to obtain the solution of the second order integrodifferential equation (9), let us consider the linear periodic boundary value problem

Clearly, by Fredholm’s alternative, the nonhomogeneous equation has a unique -periodic solution:

where is the Green function of line equation (10) subject to periodic boundary conditions .

Thus, it follows from (10) that the equation (9) is equivalent to the following integral equation:

As a result, in order to obtain the solution of equation (1), it is sufficient to find the fixed point of the following operator:

Lemma 1 (see [26]). Green’s function has the following properties:

(i)(ii)For all , the following inequality holds:

In order to obtain the properties of the Green function , define the best Sobolev constants as

where is the gamma function. Let and denote means that for all and for in a subset of positive measure. Define

Then, the following lemma is a direct consequence of Theorem 2.1 and Corollary 2.3 in [82]:

Lemma 2. Let , assume and . If then for all .

3. Main Results

In this section, we firstly give our work space which equips the maximum norm . Let

then is a real Banach space and is a normal cone of with normal constant 1. For convenience, we denote a set of functions as

Now, let us list the hypotheses to be used in the rest the paper.

(F0)

(F1) There exist a constant and two functions with on any subinterval of , such that

Suppose is the unique solution of the following equation:

then from (10), can be written as

Similar to (21), we denote as

Let

By Lemma 2 and (F0)–(F1), we know that .

Now we shall divide into three cases to discuss its influence for the solution of equation (1).

3.1. Positive Case

In this case, we have the following result:

Theorem 3. Assume that (F0)–(F1) hold. If , then the Hadamard-type fractional differential equation (1) has at least one positive solution.

Proof. Let and be fixed positive constants; we introduce a closed convex set of cone :

For any , it follows from Lemma 1 that

With the help of (25), for any , from (19), one gets

It follows from (26) and (F1) that

Thus, (27), (28), Lemma 1, (F1), and the Arzela-Ascoli theorem guarantee that the operator is completely continuous.

In the following, we shall choose a suitable such that maps the closed convex set into itself. To do this, we only need to choose such that

In fact, take it follows from and that there exists a sufficiently large such that

which imply that (29) holds and .

Thus, from Schauder’s fixed point theorem, has a fixed point , and hence, the Hadamard-type fractional differential equation (1) has at least one positive solution:

3.2. Negative Case

For this case, we have the following existence result:then the Hadamard-type fractional differential equation (1) has at least one positive solution.

Theorem 4. Assume that (F0)–(F1) hold. If , and

Proof. Firstly, from (26) and F1, we have

Thus, by (33) and (34), in order to guarantee , it is sufficient to choose such that

Let us fix ; to ensure (35) holds, we only need to choose some such that

Obviously, the function in has a minimum at

Taking , since we have

and from (32), we also have

which implies that (36) holds if . Consequently, we have .

Thus, from Schauder’s fixed point theorem, has a fixed point , and hence, the Hadamard-type fractional differential equation (1) has at least one positive solution:

Remark 5. Note that the right side of inequality (32) is negative because of the weak force condition , so if , the inequality (32) is always valid. Thus, if , we can omit the assumption (32).

3.3. Changing-Sign Case for

In order to establish the existence result under the case where is changing-sign, we need the following lemma.

Lemma 6. If and the equation

has a unique solution which satisfies

Proof. Let

then

On the other hand, since , one has

Therefore,

Moreover,

Thus, it follows from (44) to (47) that equation (41) has a unique positive solution .

Theorem 7. Assume that (F0)–(F1) hold. Let be the unique solution of equation (41), if and satisfy

then the Hadamard-type fractional differential equation (1) has at least one positive solution.

Proof. In this case, following the strategy and notations of (33) and (34), we have

Thus, to ensure , it is sufficient to choose such that

To do this, we fix Clearly, if satisfies

then the inequalities of (50) hold.
Let

notice that

and ; consequently, there exists such that

which implies that solves the equation

i.e., is the unique solution of equation (41).
On the other hand, since

the function gets the minimum at , i.e.,

Taking , then the assumption (48) implies that (52) holds.
Thus, we only need to prove that the inequality is also satisfied. Notice that solves (41), we have

It follows from that

and then

Thus, it follows from

that

That is, under the assumptions of Theorem 7, (51) and (52) all hold. Thus, according to Schauder's fixed point theorem, the Hadamard-type fractional differential equation (1) has at least one positive solution

Remark 8. In this case, the nonlinearity of equation (1) may be singular at and ; moreover, can be changing-sign function, which is allowed to be singular at some .

4. Examples

In this section, we give some examples with positive, negative, and changing-sign perturbations to demonstrate the application of our main results.

We consider the following Hadamard-type fractional differential equation with different perturbations:

Example 9. Consider the case of equation (65) with positive force term .

Conclusion. The Hadamard-type fractional differential equation (65) has at least one positive solution.

Proof. Take , and

then

It is easy to check that the Green function of the equation with periodic boundary conditions is

Thus, , and (F1) holds.
Now, let us compute and . Note that

By simple computations, one has

Now, take and let . Clearly, satisfies the inequalities

thus, according to Theorem 3, the Hadamard-type fractional differential equation (65) has at least one positive solution.

Example 10. We consider the case of Hadamard-type fractional differential equation (65) with negative force term .

Conclusion. The Hadamard-type fractional differential equation (65) has at least one positive solution.

Proof. We still take , , then as Example 9, we have , and (F1) holds.
Now, let us compute and . Note that

By simple computations, one has

So , and

thus, according to Theorem 4, the Hadamard-type fractional differential equation (65) has at least one positive solution.

Example 11. Consider the case of Hadamard-type fractional differential equation (65) with changing-sign force term

Conclusion. The Hadamard-type fractional differential equation (65) has at least one positive solution.

Proof. It follows from Example 9 that , and (F1) holds.
Now, let us compute and . Note that

By simple computations, we have

Consequently, equation (41) reduces to

which has a unique solution . Thus, one has

which implies that (48) holds.
According to Theorem 7, the Hadamard-type fractional differential equation (65) has at least one positive solution.

5. Conclusion

The force effect from outside in many complex processes often leads to the system governed equations possessing perturbations. In this work, we establish several criteria on the existence of positive solutions for a Hadamard-type fractional differential equation with singularity in all of the cases where the perturbed term is positive, negative, and changing-sign. The main advantage of our assumption is that it provides an effective bound for the perturbed term . This classification is valid and reasonable, and it is also easier to get the solution of the target equation by simple calculation.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Authors’ Contributions

The study was carried out in collaboration among all authors. All authors read and approved the final manuscript.

Acknowledgments

The authors are supported financially by the National Natural Science Foundation of China (11871302 and 11571296).

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