JFSJournal of Function Spaces2314-88882314-8896Hindawi10.1155/2020/56235895623589Research ArticlePositive Solutions for a Weakly Singular Hadamard-Type Fractional Differential Equation with Changing-Sign Nonlinearityhttps://orcid.org/0000-0001-9250-6823ZhangXinguang12https://orcid.org/0000-0001-8804-6692YuLixin1https://orcid.org/0000-0003-3198-4631JiangJiqiang3https://orcid.org/0000-0003-1028-1785WuYonghong2https://orcid.org/0000-0002-6688-4748CuiYujun4MotreanuDumitru1School of Mathematical and Informational SciencesYantai UniversityYantai264005 ShandongChinaytu.edu.cn2Department of Mathematics and StatisticsCurtin University of TechnologyPerthWA 6845Australiacurtin.edu.au3School of Mathematical SciencesQufu Normal UniversityQufu273165 ShandongChinaqfnu.edu.cn4Department of MathematicsShandong University of Science and TechnologyQingdao266590 ShandongChinasdust.edu.cn2020182020202030042020200620201820202020Copyright © 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.

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.

National Natural Science Foundation of China1157129611871302
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: (1)DtβztbtDtβzt=ft,zt,Dtβzt+χt,1<t<e,z1=z1=ze=0,Dtβz1=Dtβze,Dtβ+1z1=Dtβ+1ze,where 2<β3, Dtβ is the Hadamard fractional derivative of order α,β, bLp1,e and f:1,e×\0×00, is a continuous function with singularity at the second and third space variables; χLp1,e,,p1 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 δt and ρt 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]: (2)δt=ωταDtαρt+ωτβDtβρt,

where α>β>0, ω,τ are constants and Dtα,Dtβ 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 . In , Zhang et al. considered a singular fractional differential equation with signed measure (3)Dtαxt=ft,xt,Dtβxt,t0,1,Dtβx0=0,Dtβx1=01DtβxsdAs,where Dtα,Dtβ are the standard Riemann-Liouville derivatives, 01xsdAs is denoted by a Riemann-Stieltjes integral and 0<β1<α2,αβ>1,A is a function of bounded variation and dA can be a signed measure; the nonlinearity ft,x,y may be singular at both t=0,1 and x=y=0. 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  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 . 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 .

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 χ; we will further discuss the effect of the perturbed term for the existence of solutions when χ 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 , 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  also developed iterative techniques to solve some nonlinear problems with practical applications. In addition, variational methods  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 .

Let α,Reα>0, n=Reα, and a,b be a finite or infinite interval of +. The α-order left Hadamard fractional integral is defined by (4)Iaαxt=1Γαatlntsα1xssds,ta,b,and the α left Hadamard fractional derivative is defined by (5)Dtαxt=1Γnαtddtnatlntsnα1xssds,ta,b.

In what follows, we firstly consider the linear auxiliary problem (6)Dtβzt=xt,1<t<e,z1=z1=ze=0.

It follows from  that the problem (6) has a unique solution: (7)zt=1eHt,sxsdss,where (8)Ht,s=1Γβlntβ11lnsβ2lntlnsβ1,1ste,lntβ11lnsβ2,1tse,is Green’s function of (6). Now, let xt=Dtβzt, then the Hadamard-type fractional differential equation (1) reduces to the following second order integrodifferential equation: (9)xt+btxt=ft,1eHt,sxsdss,xt+χt,t1,e,x1=xe,x1=xe.

In order to obtain the solution of the second order integrodifferential equation (9), let us consider the linear periodic boundary value problem (10)x+btx=0,x1=xe,x1=xe.

Clearly, by Fredholm’s alternative, the nonhomogeneous equation x+btu=at has a unique e1-periodic solution: (11)xt=1eGt,sasds,where Gt,s is the Green function of line equation (10) subject to periodic boundary conditions x1=xe,x1=xe.

Thus, it follows from (10) that the equation (9) is equivalent to the following integral equation: (12)xt=1eGt,sfs,1eHs,τxτdττ,xs+χsds.

As a result, in order to obtain the solution of equation (1), it is sufficient to find the fixed point of the following operator: (13)Txt=1eGt,sfs,1eHs,τxτdττ,xs+χsds.

Lemma 1 (see [<xref ref-type="bibr" rid="B26">26</xref>]).

Green’s function H has the following properties:

HC1,e×1,e,+

For all t,s1,e, the following inequality holds:

(14)1Γβlntβ1lns1lnsβ2Ht,s1Γβlns1lnsβ2

In order to obtain the properties of the Green function G, define the best Sobolev constants as (15)Yζ=2πζe11+1/ζ22+ζ12/ζΓ1/ζΓ1/2+1/ζ2,if1ζ<,4e1,ifζ=,where Γ is the gamma function. Let bLp1,e and denote b0 means that bt0 for all t1,e and bt>0 for t in a subset of positive measure. Define (16)p=pp1,if1p<,p=1,ifp=.

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

Lemma 2.

Let 1p, assume bLp0,T and bt0. If bpY2p, then Gt,s>0 for all t,s1,e×1,e.

3. Main Results

In this section, we firstly give our work space E=C1,e which equips the maximum norm x=max1text. Let (17)P=xE:xt0,t1,e,then E,· is a real Banach space and P is a normal cone of E with normal constant 1. For convenience, we denote a set of functions as (18)Λ=bLp1,e: b0,bpY2p,for1p.

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

(F0) bΛ

(F1) There exist a constant λ0,1 and two functions h1,h2P with h1,h20 on any subinterval of 1,e, such that (19)h1tu+vλft,u,vh2tu+vλ,u,v0,×0,,t1,e.

Suppose μt is the unique solution of the following equation: (20)x+btx=χt,t1,e,x1=xe,x1=xe,then from (10), μt can be written as (21)μt=1eGt,sχsds.

Similar to (21), we denote κ1t,κ2t as (22)κ1t=1+1β1Γβλ1eGt,sh1sds,κ2t=1eGt,sh2sds.

Let (23)μ=inf1teμt,μ=sup1teμt,κ1=min1teκ1t,κ1=max1teκ1t,κ2=min1teκ2t,κ2=max1teκ2t.

By Lemma 2 and (F0)–(F1), we know that κi,κi>0,i=1,2.

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

3.1. Positive Case <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M87"><mml:mi>χ</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>

In this case, we have the following result:

Theorem 3.

Assume that (F0)–(F1) hold. If μ0, then the Hadamard-type fractional differential equation (1) has at least one positive solution.

Proof.

Let l and L be fixed positive constants; we introduce a closed convex set of cone P: (24)B=xP:lxtL,t1,e.

For any xB, it follows from Lemma 1 that (25)llnsβ1β1Γβ1eHs,τxτdττLβ1Γβ.

With the help of (25), for any s1,e, from (19), one gets (26)h1sLλ1+1/β1Γβλfs,1eHs,τxτdττ,xsh2slλ1+lnsβ1/β1Γβλh2slλ.

It follows from (26) and (F1) that (27)Txt=1eGt,sfs,1eHs,τxτdττ,xsds+μt1Lλ1+1/β1Γβλ1eGt,sh1sdsκ1Lλ,(28)Txt=1eGt,sfs,1eHs,τxτdττ,xsds+μt1lλ1eGt,sh2sds+μκ2lλ+μ.

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

In the following, we shall choose a suitable L>0 such that T maps the closed convex set B into itself. To do this, we only need to choose 0<l<L such that (29)κ1Lλl,κ2lλ+μL.

In fact, take l=1/L, it follows from κ1>0 and 0<λ<1 that there exists a sufficiently large L>1 such that (30)κ1L1λ1,Lλκ2+μL,which imply that (29) holds and l<L.

Thus, from Schauder’s fixed point theorem, T has a fixed point xB, and hence, the Hadamard-type fractional differential equation (1) has at least one positive solution: (31)zt=1eHt,sxsdss.

3.2. Negative Case <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M113"><mml:mi>χ</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced><mml:mo>≤</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>

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 μ0, and (32)μκ1λ2κ2λ1/1λ211λ2,

Proof.

Firstly, from (26) and (F1), we have (33)Txt=1eGt,sfs,1eHs,τxτdττ,xsds+μt1Lλ1+1/β1Γβλ1eGt,sh1sds+μκ1Lλ+μ,(34)Txt=1eGt,sfs,1eHs,τxτdττ,xsds+μt1lλ1eGt,sh2sdsκ2lλ.

Thus, by (33) and (34), in order to guarantee T:BB, it is sufficient to choose 0<l<L such that (35)κ1Lλ+μl,κ2lλL.

Let us fix L=κ2/lλ; to ensure (35) holds, we only need to choose some l>0 such that (36)0<l<κ21/1+λ,μlκ1κ2λlλ2=gl.

Obviously, the function gl in 0, has a minimum at (37)l~=λ2κ1κ2λ1/1λ2.

Taking l=l~, since κ1κ2,0<λ2<1, we have (38)0<l=l~=λ2κ1κ2λ1/1λ2<κ2κ2λ1/1λ2=κ21/1+λ,and from (32), we also have (39)μκ1λ2κ2λ1/1λ211λ2=gl~gl,which implies that (36) holds if l=l~,L=κ2/l~λ. Consequently, we have T:BB.

Thus, from Schauder’s fixed point theorem, T has a fixed point xB, and hence, the Hadamard-type fractional differential equation (1) has at least one positive solution: (40)zt=1eHt,sxsdss.

Remark 5.

Note that the right side of inequality (32) is negative because of the weak force condition 0<λ<1, so if μ=0, the inequality (32) is always valid. Thus, if μ=0, we can omit the assumption (32).

3.3. Changing-Sign Case for <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M140"><mml:mi>χ</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced></mml:math></inline-formula>

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

Lemma 6.

If 0<λ<1 and x0,, the equation (41)x1λ2κ2+μxλ1+λ=λ2κ2κ1,has a unique solution l~ which satisfies (42)0<l~<ελ2κ2κ11/2+2λ.

Proof.

Let (43)φx=λ2κ2κ1x1λ2κ2+μxλ1+λ,then (44)φ0=limx0φx=λ2κ2κ1>0.

On the other hand, since 0<λ<1,κ2κ1, one has (45)0<εκ21/1+λ.

Therefore, (46)φε=ε2+2λε1λ2κ2+μελ1+λ=ε1λ2ε1+λ2κ2+μελ1+λ<ε1λ2ε1+λ2κ21+λ<0.

Moreover, (47)φx=1λ2xλ2κ2+μxλ1+λμλ1+λx1λ2κ2+μxλλxλ1<0,x0,.

Thus, it follows from (44) to (47) that equation (41) has a unique positive solution l~0,ε.

Theorem 7.

Assume that (F0)–(F1) hold. Let l~ be the unique solution of equation (41), if μ0,μ0 and satisfy (48)μl~κ1l~2λκ2+μl~λλ,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 (49)Txt1Lλ1+1/β1Γβλ1eGt,sh1sds+μκ1Lλ+μ,Txt1lλ1eGt,sh2sds+μκ2lλ+μ.

Thus, to ensure T:BB, it is sufficient to choose 0<l<L such that (50)κ1Lλ+μl,κ2lλ+μL.

To do this, we fix L=κ2/lλ+μ. Clearly, if l satisfies (51)0<l<L=κ2lλ+μ,(52)μlκ1l2λκ2+μlλλ,then the inequalities of (50) hold.

Let (53)ψx=xκ1x2λκ2+μxλλ,notice that (54)ψx=1κ1λ2xλ21κ2+λ2μxλλλxλ2κ2+μxλλ1xλ1κ2+μxλ2λ=1κ1λ2xλ21κ2+μxλλ1μxλκ2+μxλ=1κ1κ2λ2xλ21κ2+μxλλ+1,and ψ0=,ψ=1; consequently, there exists l~ such that (55)ψl~=1κ1κ2λ2l~λ21κ2+μl~λλ+1=0,which implies that l~ solves the equation (56)x1λ2κ2+μxλ1+λ=λ2κ2κ1,

i.e., l~ is the unique solution of equation (41).

On the other hand, since (57)ψx=1λ2λ2κ1κ2xλ22κ2+μxλλ+1+1+λλ3μκ1κ2xλ2+λ2κ2+μxλλ+2>0,the function ψx gets the minimum at l~, i.e., (58)ψl~=minx0,ψx.

Taking l=l~, then the assumption (48) implies that (52) holds.

Thus, we only need to prove that the inequality L>l is also satisfied. Notice that 0<l=l~<ελ2κ2κ11/2+2λ solves (41), we have (59)l1λ2κ2+μlλ1+λ=λ2κ2κ1.

It follows from L=κ2/lλ+μ that (60)l1+λL1+λ=λ2κ2κ1,and then (61)L=1lλ2κ2κ11/1+λ.

Thus, it follows from (62)l<λ2κ2κ11/2+2λ,that (63)L=1lλ2κ2κ11/1+λ>λ2κ2κ11/2+2λ>l.

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 (64)zt=1eHt,sxsdss.

Remark 8.

In this case, the nonlinearity ft,u,v of equation (1) may be singular at u=0 and v=0; moreover, χt can be changing-sign Lp1,e function, which is allowed to be singular at some t1,e.

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: (65)Dt9/2zt+14Dt5/2zt=392zt+Dt5/2zt1/2+χt,1<t<e,z1=z1=ze=0,Dt5/2z1=Dt5/2ze,Dt9/2z1=Dt9/2ze.

Example 9.

Consider the case of equation (65) with positive force term χt=1.

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

Proof.

Take λ=1/2, h1t=19,h2t=20 and (66)ft,u,v=392u+v1/2,u,v0,×0,,t1,e,then (67)19u+v1/2ft,u,v20u+v1/2,u,v0,×0,,t1,e.

It is easy to check that the Green function of the equation xt+1/4xt=at with periodic boundary conditions x1=xe,x1=xe is (68)Gt,s=sine1t+s2sints2,1ste,sine1s+t2sinst2,1tse.

Thus, Gt,s>0,t,s1,e×1,e, and (F1) holds.

Now, let us compute μ,μ and κ1,κ2. Note that (69)μt=1eGt,sχsds=sin12e12+12cos12e1212cos12e12e=0.1935.

By simple computations, one has (70)μ=μ=0.1935,κ1=κ1=3.0004,κ2=κ2=3.8700.

Now, take L=15.3615 and let l=1/L=0.0651. Clearly, L satisfies the inequalities (71)3.0004L1/21,3.87L1/2+0.1935L,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 χt=3.

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

Proof.

We still take λ=1/2, h1t=19,h2t=20, then as Example 9, we have Gt,s>0,t,s1,e×1,e, and (F1) holds.

Now, let us compute μ,μ and κ1,κ2. Note that (72)μt=1eGt,sχsds=3sin12e12+12cos12e1212cos12e12e=0.5805.

By simple computations, one has (73)μ=μ=0.5805,κ1=κ1=3.0004,κ2=κ2=3.8700.

So μ=0.58050, and (74)μ=0.5805κ1λ2κ2λ1/1λ211λ2=0.8295,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 (75)χt=1,t1,2,10000729,t2,e.

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

Proof.

It follows from Example 9 that Gt,s>0,t,s1,e×1,e, and (F1) holds.

Now, let us compute μ,μ and κ1,κ2. Note that (76)μt=1eGt,sχsds=sine1212cose12sine22cos32t,1t<2,1000072912ecose12+cose12+sine12sin12cose+22t,2te,κ1t=1+1β1Γβλ1eGt,sh1sds=15.50591eGt,sds,κ2t=1eGt,sh2sds=201eGt,sds.

By simple computations, we have (77)μ=0.0798,μ=1,κ1=κ1=3.0004,κ2=κ2=3.8700.

Consequently, equation (41) reduces to (78)x3/43.8700+x1/23/2=2.9029,which has a unique solution l~=0.2200. Thus, one has (79)μ=0.0798l~κ1l~2λκ2+μl~λλ=0.0969,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 χ. 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).