## Recent Advances in Function Spaces and its Applications in Fractional Differential Equations 2020

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Xinguang Zhang, Lixin Yu, Jiqiang Jiang, Yonghong Wu, Yujun Cui, "Positive Solutions for a Weakly Singular Hadamard-Type Fractional Differential Equation with Changing-Sign Nonlinearity", *Journal of Function Spaces*, vol. 2020, Article ID 5623589, 10 pages, 2020. https://doi.org/10.1155/2020/5623589

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

**Academic Editor:**Dumitru Motreanu

#### 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).

#### References

- H. Schiessel, R. Metzler, A. Blumen, and T. F. Nonnenmacher, “Generalized viscoelastic models: their fractional equations with solutions,”
*Journal of Physics A: Mathematical and General*, vol. 28, no. 23, pp. 6567–6584, 1995. View at: Publisher Site | Google Scholar - J. He, X. Zhang, L. Liu, Y. Wu, and Y. Cui, “A singular fractional Kelvin-Voigt model involving a nonlinear operator and their convergence properties,”
*Boundary Value Problems*, vol. 2019, no. 1, Article ID 112, 2019. View at: Publisher Site | Google Scholar - X. Zhang, L. Liu, Y. Wu, and B. Wiwatanapataphee, “The spectral analysis for a singular fractional differential equation with a signed measure,”
*Appl. Math. Comput.*, vol. 257, pp. 252–263, 2015. View at: Publisher Site | Google Scholar - J. Mao, Z. Zhao, and C. Wang, “The exact iterative solution of fractional differential equation with nonlocal boundary value conditions,”
*Journal of Function Spaces*, vol. 2018, Article ID 8346398, 6 pages, 2018. View at: Publisher Site | Google Scholar - M. Li and J. Wang, “Exploring delayed Mittag-Leffler type matrix functions to study finite time stability of fractional delay differential equations,”
*Appl. Math. Comput.*, vol. 324, pp. 254–265, 2018. View at: Publisher Site | Google Scholar - Y. Cui and Y. Zou, “Monotone iterative method for differential systems with coupled integral boundary value problems,”
*Boundary Value Problems*, vol. 2013, no. 1, Article ID 245, 2013. View at: Publisher Site | Google Scholar - Q. Feng and F. Meng, “Traveling wave solutions for fractional partial differential equations arising in mathematical physics by an improved fractional Jacobi elliptic equation method,”
*Mathematical Methods in the Applied Sciences*, vol. 40, no. 10, pp. 3676–3686, 2017. View at: Publisher Site | Google Scholar - M. Li and J. Wang, “Representation of solution of a Riemann-Liouville fractional differential equation with pure delay,”
*Appl. Math. Lett.*, vol. 85, pp. 118–124, 2018. View at: Publisher Site | Google Scholar - F. Wang and Y. Yang, “Quasi-synchronization for fractional-order delayed dynamical networks with heterogeneous nodes,”
*Appl. Math. Comput.*, vol. 339, pp. 1–14, 2018. View at: Publisher Site | Google Scholar - X. Zhang, L. Liu, and Y. Wu, “The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium,”
*Applied Mathematics Letters*, vol. 37, pp. 26–33, 2014. View at: Publisher Site | Google Scholar - Y. Wang and L. Liu, “Positive solutions for a class of fractional infinite-point boundary value problems,”
*Boundary Value Problems*, vol. 2018, no. 1, Article ID 118, 2018. View at: Publisher Site | Google Scholar - Y. Wang, “Positive solutions for a class of two-term fractional differential equations with multipoint boundary value conditions,”
*Advances in Difference Equations*, vol. 2019, no. 1, Article ID 304, 2019. View at: Publisher Site | Google Scholar - X. Hao, H. Wang, L. Liu, and Y. Cui, “Positive solutions for a system of nonlinear fractional nonlocal boundary value problems with parameters and p-Laplacian operator,”
*Boundary Value Problems*, vol. 2017, no. 1, Article ID 182, 2017. View at: Publisher Site | Google Scholar - Y. Wang, “Existence and multiplicity of positive solutions for a class of singular fractional nonlocal boundary value problems,”
*Boundary Value Problems*, vol. 2019, no. 1, Article ID 92, 2019. View at: Publisher Site | Google Scholar - X. Zhang, L. Liu, and Y. Wu, “Multiple positive solutions of a singular fractional differential equation with negatively perturbed term,”
*Mathematical and Computer Modelling*, vol. 55, no. 3-4, pp. 1263–1274, 2012. View at: Publisher Site | Google Scholar - Y. Wang and L. Liu, “Positive solutions for a class of fractional 3-point boundary value problems at resonance,”
*Advances in Difference Equations*, vol. 2017, no. 1, Article ID 7, 2017. View at: Publisher Site | Google Scholar - Y. Wang and L. Liu, “Necessary and sufficient condition for the existence of positive solution to singular fractional differential equations,”
*Advances in Difference Equations*, vol. 2015, no. 1, Article ID 207, 2015. View at: Publisher Site | Google Scholar - X. Zhang, L. Liu, B. Wiwatanapataphee, and Y. Wu, “The eigenvalue for a class of singular
*p*-Laplacian fractional differential equations involving the Riemann–Stieltjes integral boundary condition,”*Applied Mathematics and Computation*, vol. 235, pp. 412–422, 2014. View at: Publisher Site | Google Scholar - J. Wu, X. Zhang, L. Liu, and Y. Wu, “Positive solutions of higher-order nonlinear fractional differential equations with changing-sign measure,”
*Advances in Difference Equations*, vol. 2012, no. 1, Article ID 71, 2012. View at: Publisher Site | Google Scholar - J. Jiang, L. Liu, and Y. Wu, “Positive solutions to singular fractional differential system with coupled boundary conditions,”
*Commun. Nonlinear Sci. Numer. Simul.*, vol. 18, no. 11, pp. 3061–3074, 2013. View at: Publisher Site | Google Scholar - Y. Wang, “Necessary conditions for the existence of positive solutions to fractional boundary value problems at resonance,”
*Appl. Math. Lett.*, vol. 97, pp. 34–40, 2019. View at: Publisher Site | Google Scholar - D. Baleanu, A. Jajarmi, H. Mohammadi, and S. Rezapour, “A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative,”
*Chaos, Solitons & Fractals*, vol. 134, article 109705, 2020. View at: Publisher Site | Google Scholar - D. Baleanu, S. Rezapour, and Z. Saberpour, “On fractional integro-differential inclusions via the extended fractional Caputo-Fabrizio derivation,”
*Boundary Value Problems*, vol. 2019, no. 1, Article ID 79, 2019. View at: Publisher Site | Google Scholar - D. Baleanu, H. Mohammadi, and S. Rezapour, “A mathematical theoretical study of a particular system of Caputo-Fabrizio fractional differential equations for the Rubella disease model,”
*Advances in Difference Equations*, vol. 2020, no. 1, Article ID 184, 2020. View at: Publisher Site | Google Scholar - K. Liu, M. Feckan, D. O'Regan, and J. Wang, “Hyers-Ulam stability and existence of solutions for differential equations with Caputo-Fabrizio fractional derivative,”
*Mathematics*, vol. 7, no. 4, p. 333, 2019. View at: Publisher Site | Google Scholar - Y. Ding, J. Jiang, D. O’Regan, and J. Xu, “Positive solutions for a system of Hadamard-type fractional differential equations with semipositone nonlinearities,”
*Complexity*, vol. 2020, Article ID 9742418, 14 pages, 2020. View at: Publisher Site | Google Scholar - X. Wu, J. Wang, and J. Zhang, “Hermite-Hadamard-type inequalities for convex functions via the fractional integrals with exponential kernel,”
*Mathematics*, vol. 7, no. 9, p. 845, 2019. View at: Publisher Site | Google Scholar - P. Yang, J. Wang, and Y. Zhou, “Representation of solution for a linear fractional delay differential equation of Hadamard type,”
*Advances in Difference Equations*, vol. 2019, no. 1, 2019. View at: Publisher Site | Google Scholar - J. Jiang, D. O’Regan, J. Xu, and Y. Cui, “Positive solutions for a Hadamard fractional p-Laplacian three-point boundary value problem,”
*Mathematics*, vol. 7, no. 5, p. 439, 2019. View at: Publisher Site | Google Scholar - K. Liu, J. R. Wang, and D. O’Regan, “On the Hermite-Hadamard type inequality for
*ψ*-Riemann–Liouville fractional integrals via convex functions,”*Journal of Inequalities and Applications*, vol. 2019, no. 1, Article ID 27, 2019. View at: Publisher Site | Google Scholar - J. Mao, Z. Zhao, and C. Wang, “The unique positive solution for singular Hadamard fractional boundary value problems,”
*Journal of Function Spaces*, vol. 2019, Article ID 5923490, 6 pages, 2019. View at: Publisher Site | Google Scholar - J. Jiang, D. O’Regan, J. Xu, and Z. Fu, “Positive solutions for a system of nonlinear Hadamard fractional differential equations involving coupled integral boundary conditions,”
*Journal of Inequalities and Applications*, vol. 2019, no. 1, Article ID 204, 2019. View at: Publisher Site | Google Scholar - J. Jiang, L. Liu, and Y. Wu, “Symmetric positive solutions to singular system with multi-point coupled boundary conditions,”
*Appl. Math. Comput.*, vol. 220, pp. 536–548, 2013. View at: Publisher Site | Google Scholar - J. He, X. Zhang, L. Liu, and Y. Wu, “Existence and nonexistence of radial solutions of the Dirichlet problem for a class of general k-Hessian equations,”
*Nonlinear Analysis: Modelling and Control*, vol. 23, no. 4, pp. 475–492, 2018. View at: Publisher Site | Google Scholar - X. Zhang, J. Jiang, Y. Wu, and Y. Cui, “Existence and asymptotic properties of solutions for a nonlinear Schrodinger elliptic equation from geophysical fluid flows,”
*Applied Mathematics Letters*, vol. 90, pp. 229–237, 2019. View at: Publisher Site | Google Scholar - J. Wang, A. Zada, and H. Waheed, “Stability analysis of a coupled system of nonlinear implicit fractional anti-periodic boundary value problem,”
*Mathematical Methods in the Applied Sciences*, vol. 42, no. 18, pp. 6706–6732, 2019. View at: Publisher Site | Google Scholar - Z. You, M. Feckan, and J. Wang, “Relative controllability of fractional delay differential equations via delayed perturbation of Mittag-Leffler functions,”
*Journal of Computational and Applied Mathematics*, vol. 378, article 112939, 2020. View at: Publisher Site | Google Scholar - P. Yang, J. Wang, and M. Feckan, “Periodic nonautonomous differential equations with noninstantaneous impulsive effects,”
*Mathematical Methods in the Applied Sciences*, vol. 42, no. 10, pp. 3700–3720, 2019. View at: Publisher Site | Google Scholar - F. Sun, L. Liu, X. Zhang, and Y. Wu, “Spectral analysis for a singular differential system with integral boundary conditions,”
*Mediterranean Journal of Mathematics*, vol. 13, no. 6, pp. 4763–4782, 2016. View at: Publisher Site | Google Scholar - L. Liu, F. Sun, X. Zhang, and Y. Wu, “Bifurcation analysis for a singular differential system with two parameters via to topological degree theory,”
*Nonlinear Analysis: Modelling and Control*, vol. 22, no. 1, pp. 31–50, 2017. View at: Publisher Site | Google Scholar - X. Zhang, L. Liu, Y. Wu, and B. Wiwatanapataphee, “Nontrivial solutions for a fractional advection dispersion equation in anomalous diffusion,”
*Applied Mathematics Letters*, vol. 66, pp. 1–8, 2017. View at: Publisher Site | Google Scholar - J. Wu, X. Zhang, L. Liu, Y. Wu, and B. Wiwatanapataphee, “Iterative algorithm and estimation of solution for a fractional order differential equation,”
*Boundary Value Problems*, vol. 2016, no. 116, 2016. View at: Publisher Site | Google Scholar - X. Zhang, L. Liu, Y. Wu, and Y. Cui, “The existence and nonexistence of entire large solutions for a quasilinear Schrodinger elliptic system by dual approach,”
*Journal of Mathematical Analysis and Applications*, vol. 464, no. 2, pp. 1089–1106, 2018. View at: Publisher Site | Google Scholar - S. Liu, J. Wang, D. Shen, and D. O'Regan, “Iterative learning control for noninstantaneous impulsive fractional-order systems with varying trial lengths,”
*International Journal of Robust and Nonlinear Control*, vol. 28, no. 18, pp. 6202–6238, 2018. View at: Publisher Site | Google Scholar - T. Ren, H. Xiao, Z. Zhou et al., “The iterative scheme and the convergence analysis of unique solution for a singular fractional differential equation from the eco-economic complex system’s co-evolution process,”
*Complexity*, vol. 2019, Article ID 9278056, 15 pages, 2019. View at: Publisher Site | Google Scholar - Y. Sun, L. Liu, and Y. Wu, “The existence and uniqueness of positive monotone solutions for a class of nonlinear Schrodinger equations on infinite domains,”
*Journal of Computational and Applied Mathematics*, vol. 321, pp. 478–486, 2017. View at: Publisher Site | Google Scholar - H. Che, H. Chen, and M. Li, “A new simultaneous iterative method with a parameter for solving the extended split equality problem and the extended split equality fixed point problem,”
*Numerical Algorithms*, vol. 79, no. 4, pp. 1231–1256, 2018. View at: Publisher Site | Google Scholar - X. Zhang, C. Mao, L. Liu, and Y. Wu, “Exact iterative solution for an abstract fractional dynamic system model for bioprocess,”
*Qualitative Theory of Dynamical Systems*, vol. 16, no. 1, pp. 205–222, 2017. View at: Publisher Site | Google Scholar - X. Zhang, L. Liu, and Y. Wu, “The entire large solutions for a quasilinear Schrodinger elliptic equation by the dual approach,”
*Applied Mathematics Letters*, vol. 55, pp. 1–9, 2016. View at: Publisher Site | Google Scholar - S. Liu, J. R. Wang, D. Shen, and D. O’Regan, “Iterative learning control for differential inclusions of parabolic type with noninstantaneous impulses,”
*Applied Mathematics and Computation*, vol. 350, pp. 48–59, 2019. View at: Publisher Site | Google Scholar - X. Zhang, L. Liu, Y. Wu, and Y. Cui, “Entire blow-up solutions for a quasilinear p-Laplacian Schrodinger equation with a non-square diffusion term,”
*Applied Mathematics Letters*, vol. 74, pp. 85–93, 2017. View at: Publisher Site | Google Scholar - T. Ren, S. Li, X. Zhang, and L. Liu, “Maximum and minimum solutions for a nonlocal p-Laplacian fractional differential system from eco-economical processes,”
*Boundary Value Problems*, vol. 2017, no. 1, 2017. View at: Publisher Site | Google Scholar - X. Zhang, J. Xu, J. Jiang, Y. Wu, and Y. Cui, “The convergence analysis and uniqueness of blow-up solutions for a Dirichlet problem of the general k-Hessian equations,”
*Applied Mathematics Letters*, vol. 102, article 106124, 2020. View at: Publisher Site | Google Scholar - L. Guo, L. Liu, and Y. Wu, “Iterative unique positive solutions for singular p-Laplacian fractional differential equation system with several parameters,”
*Nonlinear Analysis: Modelling and Control*, vol. 23, no. 2, pp. 182–203, 2018. View at: Publisher Site | Google Scholar - X. Zhang, L. Liu, Y. Wu, and Y. Cui, “A sufficient and necessary condition of existence of blow-up radial solutions for a -Hessian equation with a nonlinear operator,”
*Nonlinear Analysis: Modelling and Control*, vol. 25, pp. 126–143, 2020. View at: Google Scholar - X. Zhang, L. Liu, Y. Wu, and L. Caccetta, “Entire large solutions for a class of Schrödinger systems with a nonlinear random operator,”
*Journal of Mathematical Analysis and Applications*, vol. 423, no. 2, pp. 1650–1659, 2015. View at: Publisher Site | Google Scholar - J. Wu, X. Zhang, L. Liu, and Y. Wu, “Twin iterative solutions for a fractional differential turbulent flow model,”
*Boundary Value Problems*, vol. 2016, no. 98, 2016. View at: Publisher Site | Google Scholar - J. Wu, X. Zhang, L. Liu, Y. Wu, and Y. Cui, “Convergence analysis of iterative scheme and error estimation of positive solution for a fractional differential equation,”
*Mathematical Modelling and Analysis*, vol. 23, no. 4, pp. 611–626, 2018. View at: Publisher Site | Google Scholar - J. Wu, X. Zhang, L. Liu, Y. Wu, and Y. Cui, “The convergence analysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular decreasing nonlinearity,”
*Boundary Value Problems*, vol. 2018, no. 82, 2018. View at: Publisher Site | Google Scholar - J. Liu and Z. Zhao, “Multiple solutions for impulsive problems with non-autonomous perturbations,”
*Applied Mathematics Letters*, vol. 64, pp. 143–149, 2017. View at: Publisher Site | Google Scholar - A. Mao, R. Jing, S. Luan, J. Chu, and Y. Kong, “Some nonlocal elliptic problem involving positive parameter,”
*Topological Methods in Nonlinear Analysis*, vol. 42, pp. 207–220, 2013. View at: Google Scholar - J. Liu and Z. Zhao, “An application of variational methods to second-order impulsive differential equation with derivative dependence,”
*Electronic Journal of Differential Equations*, vol. 2014, no. 62, 2014. View at: Google Scholar - A. Mao and W. Wang, “Nontrivial solutions of nonlocal fourth order elliptic equation of Kirchhoff type in R3,”
*Journal of Mathematical Analysis and Applications*, vol. 459, no. 1, pp. 556–563, 2018. View at: Publisher Site | Google Scholar - X. Zhang, L. Liu, and Y. Wu, “Variational structure and multiple solutions for a fractional advection-dispersion equation,”
*Computers Mathematics with Application*, vol. 68, no. 12, pp. 1794–1805, 2014. View at: Publisher Site | Google Scholar - M. Shao and A. Mao, “Multiplicity of solutions to Schrodinger-Poisson system with concave-convex nonlinearities,”
*Applied Mathematics Letters*, vol. 83, pp. 212–218, 2018. View at: Publisher Site | Google Scholar - B. Zhu, L. Liu, and Y. Wu, “Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay,”
*Applied Mathematics Letters*, vol. 61, pp. 73–79, 2016. View at: Publisher Site | Google Scholar - B. Zhu, L. Liu, and Y. Wu, “Local and global existence of mild solutions for a class of semilinear fractional integro-differential equations,”
*Fractional Calculus and Applied Analysis*, vol. 20, no. 6, 2017. View at: Publisher Site | Google Scholar - X. He, A. Qian, and W. Zou, “Existence and concentration of positive solutions for quasi-linear Schrodinger equations with critical growth,”
*Nonlinearity*, vol. 26, pp. 3137–3168, 2013. View at: Google Scholar - X. Zhang, L. Liu, Y. Wu, and Y. Cui, “Existence of infinitely solutions for a modified nonlinear Schrödinger equation via dual approach,”
*Electronic Journal of Differential Equations*, vol. 147, pp. 1–15, 2018. View at: Google Scholar - A. Mao, Y. Zhu, and S. Luan, “Existence of solutions of elliptic boundary value problems with mixed type nonlinearities,”
*Boundary Value Problems*, vol. 2012, no. 1, Article ID 97, 2012. View at: Publisher Site | Google Scholar - X. Zhang, L. Liu, Y. Wu, and Y. Cui, “New result on the critical exponent for solution of an ordinary fractional differential problem,”
*Journal of Function Space*, vol. 2017, article 3976469, pp. 1–4, 2017. View at: Publisher Site | Google Scholar - F. Sun, L. Liu, and Y. Wu, “Finite time blow-up for a class of parabolic or pseudo-parabolic equations,”
*Computers Mathematics with Application*, vol. 75, no. 10, pp. 3685–3701, 2018. View at: Publisher Site | Google Scholar - J. Sun and T. Wu, “Steep potential well may help Kirchhoff type equations to generate multiple solutions,”
*Nonlinear Analysis*, vol. 190, p. 111609, 2020. View at: Publisher Site | Google Scholar - J. Sun and T.-f. Wu, “Bound state nodal solutions for the non-autonomous Schrodinger-Poisson system in R3,”
*Journal of Differential Equations*, vol. 268, no. 11, pp. 7121–7163, 2020. View at: Publisher Site | Google Scholar - J. Liu and Z. Zhao, “Existence of positive solutions to a singular boundary-value problem using variational methods,”
*Electronic Journal of Differential Equations*, vol. 2014, p. 135, 2014. View at: Google Scholar - F. Sun, L. Liu, and Y. Wu, “Finite time blow-up for a thin-film equation with initial data at arbitrary energy level,”
*Journal of Mathematical Analysis and Applications*, vol. 458, no. 1, pp. 9–20, 2018. View at: Publisher Site | Google Scholar - A. Mao and X. Zhu, “Existence and multiplicity results for Kirchhoff problems,”
*Mediterranean Journal of Mathematics*, vol. 14, no. 2, p. 58, 2017. View at: Publisher Site | Google Scholar - X. Zhang, J. Jiang, Y. Wu, and Y. Cui, “The existence and nonexistence of entire large solutions for a quasilinear Schrodinger elliptic system by dual approach,”
*Applied Mathematics Letters*, vol. 100, p. 106018, 2020. View at: Publisher Site | Google Scholar - F. Yan, M. Zuo, and X. Hao, “Positive solution for a fractional singular boundary value problem with
*p*-Laplacian operator,”*Boundary Value Problems*, vol. 2018, no. 1, Article ID 51, 2018. View at: Publisher Site | Google Scholar - J. He, X. Zhang, L. Liu, Y. Wu, and Y. Cui, “Existence and asymptotic analysis of positive solutions for a singular fractional differential equation with nonlocal boundary conditions,”
*Boundary Value Problems*, vol. 2018, no. 1, Article ID 189, 2018. View at: Publisher Site | Google Scholar - A. Kilbas, H. Srivastava, and J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, Elsevier, Boston, MA, USA, 2006. - P. J. Torres, “Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem,”
*Journal of Differential Equations*, vol. 190, no. 2, pp. 643–662, 2003. View at: Publisher Site | Google Scholar

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