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Detection of glaucoma using retinal fundus images: A comprehensive review

  • Received: 12 December 2020 Accepted: 01 February 2021 Published: 02 March 2021
  • Content-based image analysis and computer vision techniques are used in various health-care systems to detect the diseases. The abnormalities in a human eye are detected through fundus images captured through a fundus camera. Among eye diseases, glaucoma is considered as the second leading case that can result in neurodegeneration illness. The inappropriate intraocular pressure within the human eye is reported as the main cause of this disease. There are no symptoms of glaucoma at earlier stages and if the disease remains unrectified then it can lead to complete blindness. The early diagnosis of glaucoma can prevent permanent loss of vision. Manual examination of human eye is a possible solution however it is dependant on human efforts. The automatic detection of glaucoma by using a combination of image processing, artificial intelligence and computer vision can help to prevent and detect this disease. In this review article, we aim to present a comprehensive review about the various types of glaucoma, causes of glaucoma, the details about the possible treatment, details about the publicly available image benchmarks, performance metrics, and various approaches based on digital image processing, computer vision, and deep learning. The review article presents a detailed study of various published research models that aim to detect glaucoma from low-level feature extraction to recent trends based on deep learning. The pros and cons of each approach are discussed in detail and tabular representations are used to summarize the results of each category. We report our findings and provide possible future research directions to detect glaucoma in conclusion.

    Citation: Amsa Shabbir, Aqsa Rasheed, Huma Shehraz, Aliya Saleem, Bushra Zafar, Muhammad Sajid, Nouman Ali, Saadat Hanif Dar, Tehmina Shehryar. Detection of glaucoma using retinal fundus images: A comprehensive review[J]. Mathematical Biosciences and Engineering, 2021, 18(3): 2033-2076. doi: 10.3934/mbe.2021106

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  • Content-based image analysis and computer vision techniques are used in various health-care systems to detect the diseases. The abnormalities in a human eye are detected through fundus images captured through a fundus camera. Among eye diseases, glaucoma is considered as the second leading case that can result in neurodegeneration illness. The inappropriate intraocular pressure within the human eye is reported as the main cause of this disease. There are no symptoms of glaucoma at earlier stages and if the disease remains unrectified then it can lead to complete blindness. The early diagnosis of glaucoma can prevent permanent loss of vision. Manual examination of human eye is a possible solution however it is dependant on human efforts. The automatic detection of glaucoma by using a combination of image processing, artificial intelligence and computer vision can help to prevent and detect this disease. In this review article, we aim to present a comprehensive review about the various types of glaucoma, causes of glaucoma, the details about the possible treatment, details about the publicly available image benchmarks, performance metrics, and various approaches based on digital image processing, computer vision, and deep learning. The review article presents a detailed study of various published research models that aim to detect glaucoma from low-level feature extraction to recent trends based on deep learning. The pros and cons of each approach are discussed in detail and tabular representations are used to summarize the results of each category. We report our findings and provide possible future research directions to detect glaucoma in conclusion.



    During the past few decades, singular differential equations have been widely investigated by many scholars. Singular differential equations appear in many problems of applications such as the Kepler system describing the motion of planets around stars in celestial mechanics [12], nonlinear elasticity [10] and Brillouin focusing systems [2]. We refer to the classical monograph [31] for more information about the application of singular differential equations in science. Owing to the extensive applications in many branches of science and industry, singular differential equations have gradually become one of the most active research topics in the theory of ordinary differential equations. Up to this time, some necessary work has been done by scholars, including Torres [31,33], Mawhin [17], O'Regan [29], Ambrosetti [1], Fonda [12,13], Chu [5,8] and Zhang [36,37], etc.

    In the current literature on singular differential equations, the problem on the existence and multiplicity of periodic solutions is one of the hot topics. Lazer and Solimini [25] first applied topological degree theory to study the periodic solutions of singular differential equations, the results also reveal that there are essential differences between repulsive singularity and attractive singularity. In order to avoid the collision between periodic orbit and singularity in the case of repulsive singularity, a strong force condition was first introduced by Gordon [16]. After that, various variational methods and topological methods based on topological degree theory have been widely used, including the method of upper and lower solutions [9,18,19,20], fixed point theorems [7], continuation theory of coincidence degree [21,24,26], and nonlinear Leray-schauder alternative principle [5,6,8]. From present literature, the existence periodic solutions are convenient to prove if the singular term satisfies the strong force condition. It is worth noting that Torres obtained existence results of periodic solutions in the case of a weak singularity condition of the singular term, see the reference [32] for details. Until now, the work on the existence of periodic solutions with weak conditions is much less than the work with strong force conditions, see [22,27,32].

    Because of singular Rayleigh equations are widely applied in many fields, such as engineering technique, physics and mechanics fields [14,30]. Singular Rayleigh equations usually have multiple regulations and local periodic vibration phenomena. Hence, periodic solutions of singular Rayleigh equation becomes one key issue of singular Rayleigh equations. However, most of the results in the references [3,4,15,23,34,35] are concerned about one solution, while fewer works are concerned about multiple periodic solutions. Therefore, it is valuable to investigate the existence of multiple periodic solutions for singular Rayleigh equations in both theory and practice.

    Motivated by the above literature, the main purpose of this paper is to verify the existence and multiplicity of periodic solutions of the following singular Rayleigh equation

    x+f(x)+g(x)=e(t), (1.1)

    where fC(R,R), eC(R/TZ,R), and gC((0,+),R) may be singular at the origin. We discuss both repulsive and attractive singularity with some weak conditions for the term g. It is said that Eq (1.1) has a repulsive singularity at the origin if

    limx0+g(x)=

    and has an attractive singularity at the origin if

    limx0+g(x)=+.

    The proof of the main results in this study is based on Mawhin's coincidence degree and the method of upper and lower solutions. Compared to the existing results about periodic problems of singular Rayleigh equations, the novelties lie in two aspects: (1) the singular term g has a weaker force condition; (2) the existence of arbitrarily many periodic solutions are concerned.

    The rest of this paper is organized as follows. Some preliminary results are presented in Section 2. The main results will be presented and proved in Section 3. Finally, in Section 4, some examples and numerical solutions are expressed to illustrate the application of our results.

    In this section, we first recall some basic results on the continuation theorem of coincidence degree theory [28].

    Let X and Y be two real Banach spaces. A linear operator

    L:Dom(L)XY

    is called a Fredholm operator of index zero if

    (i) ImL is a closed subset of Y,

    (ii)  dimKerL=codimImL<.

    If L is a Fredholm operator of index zero, then there exist continuous projectors

    P:XX,Q:YY

    such that

    ImP=KerL,ImL=KerQ=Im(IQ).

    It follows that

    L|DomLKerP:(IP)XImL

    is invertible and its inverse is denoted by KP.

    If Ω is a bounded open subset of X, the continuous operator

    N:ΩXY

    is said to be L-compact in ˉΩ if

    (iii) KP(IQ)N(ˉΩ) is a relative compact set of X,

    (iv) QN(ˉΩ) is a bounded set of Y.

    Lemma 2.1. [28] Let Ω be an open and bounded set of X, L:D(L)XY be a Fredholm operator of index zero and the continuous operator N:ˉΩXY be L-compact on ˉΩ. In addition, if the following conditions hold:

    (A1) LxλNx,(x,λ)Ω×(0,1),

    (A2) QNx0,xKerLΩ,

    (A3) deg{JQN,ΩKerL,0}0,

    where J:ImQKerL is an homeomorphism map. Then Lx=Nx has at least one solution in ˉΩ.

    In order to apply Lemma 2.1 to Eq (1.1), let X=C1T, Y=CT, where

    C1T={x|xC1(R,R),x(t+T)=x(t)},
    CT={x|xC(R,R),x(t+T)=x(t)}.

    Define

    x=max{x,x},x=maxt[0,T]|x(t)|.

    Clearly, X, Y are two Banach spaces with norms and . Meanwhile, let

    L:DomL={x:xC2(R,R)C1T}XY,Lx=x.

    Then

    KerL=R,ImL={x:xY,T0x(t)dt=0},

    hence, L is a Fredholm operator of index zero. Define the projects P and Q by

    P:XX,[Px](t)=x(0)=x(T),
    Q:YY,[Qx](t)=1TT0x(t)dt.

    Obviously,

    ImP=KerL,KerQ=ImL.

    Let Lp=L|DomLKerP, then Lp is invertible and its inverse is denoted by Kp:ImLDomL,

    [Kpx](t)=tTT0(Ts)x(s)ds+t0(ts)x(s)ds,

    Let N:XY, such that

    [Nx](t)=[f(x(t))+g(x(t))]+e(t).

    It is easy to show that QN and KP(IQ)N are continuous by the Lebesgue convergence theorem. By Arzela-Ascoli theorem, we get that QN(¯Ω) and Kp(IQ)N(¯Ω) are compact for any open bounded set Ω in X. Therefore, N is L-compact on ¯Ω.

    For the sake of convenience, we denote

    mint[0,T]e(t)=e,maxt[0,T]e(t)=e,ω=T1q(eqc)1p1.

    Moreover, we list the following condition

    (H0) There exist two constants c>0 and p1, such that

    f(x)xc|x|p,(t,x)R2.

    Obviously, we have f(0)=0.

    Lemma 3.1. Assume that x is a T-periodic solution of Eq (1.1). Then the following inequalities hold

    g(x(s1))e(s1)e,g(x(t1))e(t1)e,

    where s1 and t1 be the maximum point and the minimum point of x(t) on [0,T].

    Proof. Obviously,

    x(s1)=x(t1)=0,x(s1)0andx(t1)0.

    Combining these with Eq (1.1), we get

    g(x(s1))e(s1)0,g(x(t1))e(t1)0.

    Then we have

    g(x(s1))e(s1)e,g(x(t1))e(t1)e.

    Theorem 3.2. Assume that (H0) holds and Eq (1.1) has a repulsive singularity at the origin. Suppose further that

    (H1) There exist only two positive constants ξ1 and η1 with η1>ξ1>ω, such that

    g(ξ1)=e,g(η1)=e.

    Then Eq (1.1) has a positive T-periodic solution x1 satisfies

    ξ1ωx1(t)η1+ω,mintRx1(t)η1,tR. (3.1)

    Proof. Since Eq (1.1) can be written as an operator equation Lx=Nx, so we consider an auxiliary equation Lx=λNx,

    x(t)+λ[f(x(t)+g(x(t))]=λe(t),λ(0,1). (3.2)

    Suppose that xX is a periodic solution of the above equation. Multiplying both sides of Eq (3.2) by x(t) and integrating on the interval [0,T], then we have

    T0f(x(t))x(t)dt=T0e(t)x(t)dt.

    By using the Hölder's inequality, it follows from (H0) and the above equality that

    cxppT0f(x(t))x(t)dt=T0e(t)x(t)dteqxp,

    where 1q+1p=1. Then we can obtain from the above inequality that

    xp(eqc)1p1. (3.3)

    By Lemma 3.1 and (H1), we can deduce that

    x(s1)ξ1,x(t1)η1. (3.4)

    Then, by (3.3) and (3.4), we have

    |x(t)|=|x(t1)+tt1x(s)ds||x(t1)|+T0|x(s)|ds|x(t1)|+(T0ds)1q(T0|x(s)|pds)1p|x(t1)|+T1qxPη1+T1q(eqc)1p1=η1+ω (3.5)

    and

    |x(t)|=|x(s1)+ts1x(s)ds||x(s1)|T0|x(s)|ds|x(s1)|(T0ds)1q(T0|x(s)|pds)1p|x(s1)|T1qxPξ1T1q(eqc)1p1=ξ1ω. (3.6)

    Combining with the above two inequalities, we get

    ξ1ωx(t)η1+ω,t[0,T]. (3.7)

    By (H0) and the continuity of f, it is immediate to see that

    f(x)0,ifx0andf(x)0,ifx<0.

    Therefore, let us define two sets

    I1={t[0,T]|x(t)0}

    and

    I2={t[0,T]|x(t)<0}.

    Integrating the Eq (3.2) over the sets I1, I2, we get

    I1f(x(t))dt+I1g(x(t))dt=I1e(t)dt

    and

    I2f(x(t))dt+I2g(x(t))dt=I2e(t)dt,

    which imply that

    T0|f(x(t))|dtT0|g(x(t))|dt+T0|e(t)|dt. (3.8)

    Then by (3.2), (3.7) and (3.8), we obtain

    |x(t)|=|tt1x(s)ds|T0|x(t)|dtλ(T0|f(x(t))|dt+T0|g(x(t))|dt+T0|e(t)|dt)<2(T0|g(x(t))|dt+T0|e(t)|dt)2T(gω+¯|e|):=M1, (3.9)

    where

    gω=maxξ1ωx(t)η1+ω|g(x)|

    and ¯|e| is the mean value of |e(t)| on the interval [0,T].

    Obviously, ξ1, η1 and M1 are positive constants independent of λ. Take three positive constants h1, h2 and ˜M1 with

    h1<ξ1ω<η1+ω<h2,˜M1>M1 (3.10)

    and let

    Ω1={x:xX,h1<x(t)<h2,|x(t)|<˜M1,t[0,T]}.

    Obviously, Ω1 is an open bounded set of X. By the definition of N, we know that N is L-compact on the ˉΩ1. By (3.7), (3.9) and (3.10), we get that

    xΩ1DomL,LxλNx,λ(0,1).

    Hence, the condition (A1) in Lemma 2.1 is satisfied.

    Next, we verify that the condition (A2) of Lemma 2.1 is satisfied. Clearly, if xΩ1KerL=Ω1R, we have QNx0. If it does not hold, then there exists xΩ1R, such that QNx=0, and x(t)ζ is a constant. That is

    1TT0[g(ζ)+e(t)]dt=0,

    i.e.,

    g(ζ)¯e=0.

    This implies that

    eg(ζ)e,

    which together with (H1) yield

    ζ[ξ1,η1].

    This contradicts x=ζΩ1R. Thus,

    QNx0,xΩ1KerL. (3.11)

    Finally, we prove that the condition (A3) of Lemma 2.1 is also satisfied. Define

    H(μ,x)=μx+(1μ)JQN(x),

    where

    J=I:ImLKerL,Jx=x.

    Then, by (3.11), we notice that

    xH(μ,x)0,(μ,x)[0,1]×Ω1KerL.

    Therefore, we have

    deg{JQNx,Ω1KerL,0}=deg{H(0,x),Ω1KerL,0}=deg{H(1,x),Ω1KerL,0}0.

    To sum up the above discussion, we have proven that all of the conditions of Lemma 2.1 are satisfied. Therefore, Eq (1.1) has a T-periodic solution x1 in Ω1. Moreover, by (3.4) and (3.7), we get that (3.1) holds.

    Theorem 3.3. Assume that (H0) holds and Eq (1.1) has a repulsive singularity at the origin. Suppose further that

    (H2) There exist only four positive constants ξ1, ξ2, η1, η2 with ξ2>η2>η1>ξ1>ω, such that

    g(ξ1)=e=g(ξ2),g(η1)=e=g(η2).

    Then Eq (1.1) has two positive T-periodic solutions x1 and x2, which satisfy

    ξ1ωx1(t)η1+ω,mint[0,T]x1(t)η1,t[0,T] (3.12)

    and

    η2x2(t)ξ2,t[0,T]. (3.13)

    Proof. From Lemma 3.1 and (H2), we obtain

    ξ1x(s1)ξ2

    and

    x(t1)η1orx(t1)η2.

    Therefore, we have either

    η2x(t)ξ2,t[0,T] (3.14)

    or

    ξ1x(s1)ξ2,x(t1)η1. (3.15)

    (1) If (3.14) holds, notice that

    α(t)=η2

    is a constant lower solution of Eq (1.1) and

    β(t)=ξ2

    is a constant upper solution of Eq (1.1). Then by the method of upper and lower solutions (see [11,31]), we know that Eq (1.1) has a positive T-periodic solution x2 such that (3.13) holds.

    (2) If (3.15) holds, by (3.3), analysis similar to that in (3.5) and (3.6), we have

    ξ1ωx(t)η1+ω,t[0,T]. (3.16)

    By using similar arguments of Theorem 3.2, it follows from (3.16) that there exists a constant M2>0 such that

    |x|<M2.

    Clearly, ξ1, η1, M2 are all independent of λ. Take three constants u1,u2, and ˜M2 with

    0<u1<ξ1ω<η1+ω<u2,˜M2>M2

    and set

    Ω2={x:xX,u1<x(t)<u2,|x(t)|<˜M2,t[0,T],mint[0,T]x(t)η1}.

    The remainder can be proved in the same way as in the proof of Theorem 3.2. Then, Eq (1.1) has a positive T-periodic solution x1 in Ω2 such that (3.12) holds.

    To sum up the above discussion, we plainly conclude that Eq (1.1) has at least two positive T-periodic solutions.

    Remark 1. In Theorem 3.3, if η1=η2, we can only get that Eq (1.1) has at least one positive T-periodic solution.

    Proof. If η1=η2, by Lemma 3.1 and (H2), we can only get

    ξ1x(s1)ξ2.

    As in the proof of Theorem 3.3, we can prove that Eq (1.1) has a positive T-periodic solutions x1 such that

    ξ1ωx1(t)ξ2,maxt[0,T]x1(t)ξ1,forallt[0,T]. (3.17)

    Moreover, by the method of upper and lower solutions (see [11,31]), we can also get that Eq (1.1) has a positive T-periodic solution x2 such that

    η1=η2x2(t)ξ2,t[0,T]. (3.18)

    But, by (3.17) and (3.18), we are not sure that x1 is different from x2.

    Therefore, we just can assert that Eq (1.1) has at least one positive T-periodic solution.

    Furthermore, Theorem 3.3 can be generalized to arbitrarily many periodic solutions.

    Theorem 3.4. Assume that (H0) holds and Eq (1.1) has a repulsive singularity at the origin. Suppose further that

    (Hn) There exist only 2n positive constants ξ1, ξ2, ξn, η1, η2, ηn, with

    ξn>ηn>ηn1>ξn1>η1>ξ1>ω,ifniseven,ηn>ξn>ξn1>ηn1>η1>ξ1>ω,ifnisodd (3.19)

    and

    η2i1<ξ2i+1ω,i=1,2[n2], (3.20)

    where [] stands for the integer part, such that

    g(ξ1)=g(ξ2)==g(ξn)=e,
    g(η1)=g(η2)==g(ηn)=e.

    Then Eq (1.1) has at least n different positive T-periodic solutions.

    Proof. The case n=1 and n=2, one can see Theorem 3.2 and Theorem 3.3.

    Let us define the following sets

    B2i1={x|xX,ξ2i1ωx(t)η2i1+ω,maxtRx(t)ξ2i1mintRx(t)η2i1},i=1,2[n+12],B2i={x|xX,η2ix(t)ξ2i,},i=1[n2].

    By (3.19) and (3.20), notice that BiBj=, for ij, i,j=1,2n.

    For the case n=3, by Lemma 3.1 and (H3), we have

    ξ1x(s1)ξ2orx(s1)ξ3

    and

    x(t1)η1orη2x(t1)η3.

    Then

    ξ1x(s1)ξ2andx(t1)η1 (3.21)

    or

    η2x(t)ξ2 (3.22)

    or

    x(s1)ξ3andη2x(t1)η3. (3.23)

    By (3.21) and (3.22), as in the proof of Theorem 3.3, we can prove that Eq (1.1) has two different positive T-periodic solutions x1 and x2 with x1B1 and x2B2. By (3.23), analysis similar to that in the proof of Theorem 3.2 shows that Eq (1.1) has a positive T-periodic solution x3 belonging to B3. Then, by the facts, we get that Eq (1.1) has at least 3 different positive T-periodic solutions.

    Similar arguments apply to the case n>3, we can prove that Eq (1.1) has n different positive T-periodic solutions x1, x2, xn with xiBi, i=1,2,n.

    The proof is completed.

    Theorem 3.5. Assume that (H0) holds and Eq (1.1) has an attractive singularity at the origin. Suppose further that

    (C1) There exist only two positive constants ξ1, η1 with η1>ξ1, such that

    g(ξ1)=e,g(η1)=e.

    Then Eq (1.1) has at least one positive T-periodic solution.

    Proof. Obviously,

    α(t)=ξ1

    is a constant lower solution of Eq (1.1) and

    β(t)=η1

    is a constant upper solution of Eq (1.1). Then by the method of upper and lower solutions (see [11,31]), we know that Eq (1.1) has a positive T-periodic solution x such that α(t)x(t)β(t) for every t.

    Theorem 3.6. Assume that (H0) holds and Eq (1.1) has an attractive singularity at the origin. Suppose further that

    (C2) There exist only four positive constants ξ1<η1<η2<ξ2, such that

    g(ξ1)=e=g(ξ2),g(η1)=e=g(η2).

    Then Eq (1.1) has at least two positive T-periodic solutions.

    Proof. The proof of Theorem 3.6 works almost exactly as the proof Theorem 3.3. It is easy to get that

    x(s1)η1orx(s1)η2

    and

    ξ1x(t1)ξ2,

    which together with Lemma 3.1 yield that

    ξ1x(t)η1,t[0,T] (3.24)

    or

    ξ1x(t1)ξ2,x(s1)η2. (3.25)

    (1) If (3.24) holds, by the method of upper and lower solutions align (see [11,31]), we get that Eq (1.1) has at least one positive T-periodic solution x such that

    ξ1x(t)η1,t[0,T].

    (2) If (3.25) holds, repeating the proof of Theorem 3.2, we can construct an open bounded set

    Ω3={x:xX,r1<x(t)<r2,|x(t)|<˜M3,t[0,T]},

    such that Eq (1.1) has at least one positive T-periodic solutions in Ω3.

    To sum up the above discussion, we have proved that Eq (1.1) has at least two positive T-periodic solutions.

    Similar as in the proof of Theorem 3.4, we can generalize Theorem 3.6 to arbitrarily many periodic solutions.

    Theorem 3.7. Assume that (H0) holds and Eq (1.1) has an attractive singularity at the origin. Suppose further that

    (Cn) There exist only 2n positive constants ξ1, ξ2, ξn, η1, η2, ηn with

    ξn>ηn>ηn1>ξn1>η1>ξ1>0,ifniseven,ηn>ξn>ξn1>ηn1>η1>ξ1>0,ifnisodd

    and

    ξ2i<η2i+2ω,i=1,2[n22]

    such that

    g(ξ1)=g(ξ2)==g(ξn)=e,
    g(η1)=g(η2)==g(ηn)=e.

    Then Eq (1.1) has at least n positive T-periodic solutions.

    In this section, some examples and numerical solutions are given to illustrate the application of our results.

    Example 1. Consider the following equation:

    x+13.2x+3.3x4x2=3.8sin(πt)+2. (4.1)

    Conclusion: Eq (4.1) has at least one positive 2-periodic solution.

    Proof. Corresponding to Eq (1.1), we have

    f(x)=13.2x,g(x)=3.3x4x2,e(t)=3.8sin(πt)+2.

    Obviously, e=5.8, e=1.8. It is easy to see that there exist only two positive constants ξ10.912, η12.047 such that

    g(ξ1)=e=1.8,g(η1)=e=5.8.

    Moreover, it is easy to check that ξ1>ω. Then, by Theorem 3.2, we get that Eq (4.1) has at least one positive 2-periodic solution. Applying Matlab software, we obtain numerical periodic solution of Eq (4.1), which is shown in Figure 1.

    Figure 1.  The periodic solution of Eq (4.1) as x(0)=1.22,x(0)=0,t[0,20].

    Example 2. Consider the following equation:

    x+83x6.4x1.7x2=17.521+sin2(t). (4.2)

    Conclusion: Eq (4.2) has at least two positive π-periodic solutions.

    Proof. Corresponding to Eq (1.1), we have

    f(x)=83x,g(x)=6.4x1.7x2,e(t)=17.521+sin2(t).

    Obviously, e=8.76, e=17.52. It is easy to check that exist only four positive constants ξ10.3323, η10.5805, η21.177, ξ22.7 such that

    g(ξ1)=g(ξ2)=e=17.52,g(η1)=g(η2)=e=8.76.

    Moreover, it is easy to check that ξ1>ω. Then, by Theorem 3.3, we get that Eq (4.2) has at least two positive π-periodic solutions. We obtain two numerical periodic solutions of Eq (4.2), which are shown in Figures 2 and 3, respectively.

    Figure 2.  The periodic solution of Eq (4.2) as x(0)=0.413,x(0)=0,t[0,30].
    Figure 3.  The periodic solution of Eq (4.2) as x(0)=1.86,x(0)=0,t[0,20].

    Example 3. Consider the following equation:

    x+95(x)3+5.6x3x2=3sin(πt)+2. (4.3)

    Conclusion: Eq (4.3) has at least one positive 2-periodic solution.

    Proof. Corresponding to Eq (1.1), we have

    f(x)=95(x)3,g(x)=5.6x3x2,e(t)=3sin(πt)+2.

    Obviously, e=5, e=1. It is easy to see that there exist only two positive constants ξ10.757, η11.24 such that

    g(ξ1)=e=1,g(η1)=e=5.

    Moreover, it is easy to check that ξ1>ω. Then, by Theorem 3.2, we get that Eq (4.3) has at least one positive 2-periodic solution. Applying Matlab software, we obtain numerical periodic solution of Eq (4.3), which is shown in Figure 4.

    Figure 4.  The periodic solution of Eq. (4.3) as x(0)=0.835,x(0)=0,t[0,20].

    In this paper, we study the existence and multiplicity of positive periodic solutions of the singular Rayleigh differential equation (1.1). Based on the continuation theorem of coincidence degree theory and the method of upper and lower solutions, we construct some subsets Bk, k=1,2,,n of C1T with BiBj=, for ij, i,j=1,2,,n, such that the equation (1.1) has a positive T-periodic solution in each set Bk, k=1,2,,n. That is, the equation (1.1) has at least n distinct positive T-periodic solutions. We discuss both the repulsive singular case and the attractive singular case, and the singular term has a weaker force condition than the literatures about strong force condition. Some results in the literature are generalized and improved. It should be pointed out that it is the first time to study the existence of arbitrarily many periodic solutions of singular Rayleigh equations. In addition, some typical numerical examples and the corresponding simulations have been presented at the end of this paper to illustrate our theoretical analysis.

    We would like to express our great thanks to the referees for their valuable suggestions. Zaitao Liang was supported by the Natural Science Foundation of Anhui Province (No. 1908085QA02), the National Natural Science Foundation of China (No.11901004) and the Key Program of Scientific Research Fund for Young Teachers of AUST (No. QN2018109). Hui Wei was supported by the Postdoctoral Science Foundation of Anhui Province (No. 2019B318) and the National Natural Science Foundation of China (No. 11601007).

    The authors declare that they have no competing interests concerning the publication of this article.



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