Sperm morphology analysis (SMA) is a significant factor in diagnosing male infertility. Therefore, healthy sperm detection is of great significance in this process. However, the traditional manual microscopic sperm detection methods have the disadvantages of a long detection cycle, low detection accuracy in large orders, and very complex fertility prediction. Therefore, it is meaningful to apply computer image analysis technology to the field of fertility prediction. Computer image analysis can give high precision and high efficiency in detecting sperm cells. In this article, first, we analyze the existing sperm detection techniques in chronological order, from traditional image processing and machine learning to deep learning methods in segmentation and classification. Then, we analyze and summarize these existing methods and introduce some potential methods, including visual transformers. Finally, the future development direction and challenges of sperm cell detection are discussed. We have summarized 44 related technical papers from 2012 to the present. This review will help researchers have a more comprehensive understanding of the development process, research status, and future trends in the field of fertility prediction and provide a reference for researchers in other fields.
Citation: Muhammad Suleman, Muhammad Ilyas, M. Ikram Ullah Lali, Hafiz Tayyab Rauf, Seifedine Kadry. A review of different deep learning techniques for sperm fertility prediction[J]. AIMS Mathematics, 2023, 8(7): 16360-16416. doi: 10.3934/math.2023838
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Sperm morphology analysis (SMA) is a significant factor in diagnosing male infertility. Therefore, healthy sperm detection is of great significance in this process. However, the traditional manual microscopic sperm detection methods have the disadvantages of a long detection cycle, low detection accuracy in large orders, and very complex fertility prediction. Therefore, it is meaningful to apply computer image analysis technology to the field of fertility prediction. Computer image analysis can give high precision and high efficiency in detecting sperm cells. In this article, first, we analyze the existing sperm detection techniques in chronological order, from traditional image processing and machine learning to deep learning methods in segmentation and classification. Then, we analyze and summarize these existing methods and introduce some potential methods, including visual transformers. Finally, the future development direction and challenges of sperm cell detection are discussed. We have summarized 44 related technical papers from 2012 to the present. This review will help researchers have a more comprehensive understanding of the development process, research status, and future trends in the field of fertility prediction and provide a reference for researchers in other fields.
In 2005, Rodríguez [1] used the Lyapunov-Schmidt method and Brower fixed-point theorem to discuss the following discrete Sturm-Liouville boundary value problem
{Δ[p(t−1)Δy(t−1)]+q(t)y(t)+λy(t)=f(y(t)), t∈[a+1,b+1]Z,a11y(a)+a12Δy(a)=0, a21y(b+1)+a22Δy(b+1)=0, |
where λ is the eigenvalue of the corresponding linear problem and the nonlinearity f is bounded.
Furthermore, in 2007, Ma [2] studied the following discrete boundary value problem
{Δ[p(t−1)Δy(t−1)]+q(t)y(t)+λy(t)=f(t,y(t))+h(t), t∈[a+1,b+1]Z,a11y(a)+a12Δy(a)=0, a21y(b+1)+a22Δy(b+1)=0, |
where f is subject to the sublinear growth condition
|f(t,s)|≤A|s|α+B,s∈R |
for some 0≤α<1 and A,B∈(0,∞). Additional results to the existence of solutions to the related continuous and discrete problems on the nonresonance and the resonance can be found in [3,4,5,6,7,8,9,10,11,12,13] and the references therein. For example, Li and Shu [14] considered the existence of solutions to the continuous Sturm-Liouville problem with random impulses and boundary value problems using the Dhage's fixed-point theorem and considered the existence of upper and lower solutions to a second-order random impulsive differential equation in [15] using the monotonic iterative method.
Inspired by the above literature, we use the solution set connectivity theory of compact vector field [16] to consider the existence of solutions to discrete resonance problems
{−Δ[p(t−1)Δy(t−1)]+q(t)y(t)=λkr(t)y(t)+f(t,y(t))+γψk(t)+¯g(t), t∈[1,T]Z,(a0λk+b0)y(0)=(c0λk+d0)Δy(0),(a1λk+b1)y(T+1)=(c1λk+d1)∇y(T+1), | (1.1) |
where p:[0,T]Z→(0,∞), q:[1,T]Z→R, ¯g:[1,T]Z→R, r(t)>0, t∈[1,T]Z, (λk,ψk) is the eigenpair of the corresponding linear problem
{−Δ[p(t−1)Δy(t−1)]+q(t)y(t)=λr(t)y(t), t∈[1,T]Z,(a0λ+b0)y(0)=(c0λ+d0)Δy(0),(a1λ+b1)y(T+1)=(c1λ+d1)∇y(T+1). | (1.2) |
It is worth noting that the difference between the problem (1.1) and the above questions is the eigenvalue that not only appears in the equation but also in the boundary conditions, which causes us considerable difficulties. Furthermore, it should be noted that these problems also apply to a number of physical problems, including those involving heat conduction, vibrating strings, and so on. For instance, Fulton and Pruess [17] discussed a kind of heat conduction problem, which has the eigenparameter-dependent boundary conditions. However, to discuss this kind of problem, we should know the spectrum of the problem (1.2). Fortunately, in 2016, Gao and Ma [18] obtained the eigenvalue theory of problem (1.2) under the conditions listed as follows:
(A1) δ0:=a0d0−b0c0<0,c0≠0, d1−b1≠0,
(A2) δ1:=a1d1−b1c1>0,c1≠0, b0+d0≠0,
which laid a theoretical foundation for this paper.
Under the conditions (A1) and (A2), we assume the following conditions hold:
(H1) (Sublinear growth condition) f:[1,T]Z×R→R is continuous and there exist α∈[0,1) and A,B∈(0,∞), such that
|f(t,y)|≤A|y|α+B, |
(H2) (Symbol condition) There exists ω>0, such that
yf(t,y)>0,t∈[1,T]Zfor|y|>ω, | (1.3) |
or
yf(t,y)<0,t∈[1,T]Zfor|y|>ω, | (1.4) |
(H3) ¯g:[1,T]Z→R satisfies
T∑s=1¯g(s)ψk(s)=0, | (1.5) |
(H4) f:[1,T]Z×R→R is continuous and
lim|y|→∞f(t,y)=0 |
uniformly for t∈[1,T]Z.
The organization of this paper is as follows. In the second section, we construct a completely new inner product space. In the new inner product space, we discuss the basic self-adjointness of the corresponding linear operator and the properties of the eigenpair of (1.2). Finally, under the above properties, the Lyapunov-Schmidit method is used to decompose the inner product space and transform our problem to an equivalent system, that is to say, finding the solutions of (1.1) is equivalent to finding the solutions of this system. Under the sublinear condition and sign conditions on nonlinear terms, an existence result of solutions to the problem (1.1) is obtained using Schauder's fixed-point theorem and the connectivity theories of the solution set of compact vector fields. Based on the first result, the existence of two solutions to the problem (1.1) is also obtained in this section.
Definition 2.1. ([19]) A linear operator P from the linear space X to itself is called the projection operator, if P2=P.
Lemma 2.2. ([16]) Let C be a bounded closed convex set in Banach space E, T:[α,β]×C→C(α<β) be a continuous compact mapping, then the set
Sα,β={(ρ,x)∈[α,β]×C|T(ρ,x)=x} |
contains a connected branch connecting {α}×C and {β}×C.
Lemma 2.3. ([20])(Schauder) Let D be a bounded convex closed set in E, A:D→D is completely continuous, then A has a fixed point in D.
First, we construct the inner product space needed in this paper.
Let
Y:={u|u:[1,T]Z→R}, |
then Y is a Hilbert space under the following inner product
⟨y,z⟩Y=T∑t=1y(t)z(t) |
and its norm is ‖y‖Y:=√⟨y,y⟩Y.
Furthermore, consider the space H:=Y⊕R2. Define the inner product as follows:
⟨[y,α,β]⊤,[z,ζ,ρ]⊤⟩=⟨y,z⟩Y+p(0)|δ0|αζ+p(T)|δ1|βρ, |
which norm is defined as
‖y∗‖=⟨[y,α,β]⊤,[y,α,β]⊤⟩12, |
where ⊤ is transposition to a matrix.
Let
y0,0=b0y(0)−d0Δy(0), y0,1=a0y(0)−c0Δy(0) |
and
yT+1,0=b1y(T+1)−d1∇y(T+1), yT+1,1=a1y(T+1)−c1∇y(T+1). |
For y∗=[y,α,β]⊤, define an operator L:D→H as follows:
Ly∗=[−Δ[p(t−1)Δy(t−1)]+q(t)y(t)−y0,0−yT+1,0]:=[Ly−y0,0−yT+1,0], |
where D={[y,α,β]⊤:y∈Y, y0,1=α, yT+1,1=β}. Define S:D→H as follows:
Sy∗=S[yαβ]=[ryαβ]. |
Then, the problem (1.2) is equivalent to the eigenvalue problem as follows:
Ly∗=λSy∗, | (2.1) |
that is, if (λk,y) is the eigenpair of the problem (1.2), then (λk,y∗) is the eigenpair of the opertor L. Conversely, if (λk,y∗) is the eigenpair of the operator L, then (λk,y) is the eigenpair of the problem (1.2).
Eventually, we define A:D→H as follows:
Ay∗=F(t,y∗)+[γψk+¯g,0,0]⊤, |
where F(t,y∗)=F(t,[y,α,β]⊤)=[f(t,y),0,0]⊤. Obviously, the solution of the problem (1.1) is equivalent to the fixed point of the following operator
Ly∗=λkSy∗+Ay∗. | (2.2) |
It can be seen that there is a homomorphism mapping (λk,y)↔(λk,y∗) between the problem (1.1) and the operator Eq (2.2).
Next, we are committed to obtaining the orthogonality of the eigenfunction.
Lemma 2.4. Assume that (λ,y∗) and (μ,z∗) are eigenpairs of L, then
⟨y∗,Lz∗⟩−⟨Ly∗,z∗⟩=(μ−λ)⟨y∗,Sz∗⟩. |
Proof Let y∗=[y,α,β]⊤∈D, z∗=[z,ζ,ρ]⊤∈D, then
⟨y∗,Lz∗⟩=⟨[y,α,β]⊤,[Lz,−z0,0,−zT+1,0]⊤⟩=⟨y,Lz⟩Y+p(0)|δ0|α(−z0,0)+p(T)|δ1|β(−zT+1,0)=μ⟨y,rz⟩Y+p(0)|δ0|α(μζ)+p(T)|δ1|β(μρ)=μ⟨y∗,Sz∗⟩. | (2.3) |
Similarly, we have
⟨Ly∗,z∗⟩=⟨[Ly,−y0,0,−yT+1,0]⊤,[z,ζ,ρ]⊤⟩=⟨Ly,z⟩Y+p(0)|δ0|(−y0,0)ζ+p(T)|δ1|(−yT+1,0)ρ=λ⟨ry,z⟩Y+p(0)|δ0|λαζ+p(T)|δ1|λβρ=λ⟨y∗,Sz∗⟩. | (2.4) |
It can be seen from (2.3) and (2.4)
⟨y∗,Lz∗⟩−⟨Ly∗,z∗⟩=(μ−λ)⟨y∗,Sz∗⟩. |
Lemma 2.5. The operator L is the self-adjoint operator in H.
Proof For y∗=[y,α,β]⊤∈D,z∗=[z,ζ,ρ]⊤∈D, we just need to prove that ⟨y∗,Lz∗⟩=⟨Ly∗,z∗⟩. By the definition of inner product in H. we obtain
⟨y∗,Lz∗⟩=⟨y,Lz⟩Y+p(0)|δ0|α(−z0,0)+p(T)|δ1|β(−zT+1,0), |
and
⟨Ly∗,z∗⟩=⟨Ly,z⟩Y+p(0)|δ0|(−y0,0)ζ+p(T)|δ1|(−yT+1,0)ρ. |
Therefore,
⟨y∗,Lz∗⟩−⟨Ly∗,z∗⟩=⟨y,Lz⟩Y−⟨Ly,z⟩Y+p(0)|δ0|[α(−z0,0)−(−y0,0)ζ]+p(T)|δ1|[β(−zT+1,0)−(−yT+1,0)ρ], |
where
⟨y,Lz⟩Y=T∑t=1y(t)(−Δ[p(t−1)Δz(t−1)]+q(t)z(t))=T∑t=1y(t)p(t−1)Δz(t−1)−T∑t=1y(t)p(t)Δz(t)+T∑t=1q(t)y(t)z(t)=T−1∑t=0y(t+1)p(t)Δz(t)−T∑t=1y(t)p(t)Δz(t)+T∑t=1q(t)y(t)z(t)=T−1∑t=0p(t)Δy(t)Δz(t)+p(0)y(0)Δz(0)−p(T)y(T)Δz(T)+T∑t=1q(t)y(t)z(t) |
and
⟨Ly,z⟩Y=T−1∑t=0p(t)Δy(t)Δz(t)+p(0)Δy(0)z(0)−p(T)Δy(T)z(T)+T∑t=1q(t)y(t)z(t). |
Moreover, from
α(−z0,0)−(−y0,0)ζ=[a0y(0)−c0Δy(0)][d0Δz(0)−b0z(0)]−[d0Δy(0)−b0y(0)][a0z(0)−c0Δz(0)]=(a0d0−b0c0)[y(0)Δz(0)−Δy(0)z(0)] |
and
β(−zT+1,0)−(−yT+1,0)ρ=[a1y(T+1)−c1∇y(T+1)][−b1z(T+1)+d1∇z(T+1)]−[−b1y(T+1)+d1∇y(T+1)][a1z(T+1)−c1∇z(T+1)]=(a1d1−b1c1)[y(T+1)∇z(T+1)−∇y(T+1)z(T+1)], |
we have
⟨y∗,Lz∗⟩−⟨Ly∗,z∗⟩=p(0)|y(0)Δy(0)z(0)Δz(0)|−p(T)|y(T)Δy(T)z(T)Δz(T)|−p(0)|y(0)Δy(0)z(0)Δz(0)|+p(T)|y(T+1)∇y(T+1)z(T+1)∇z(T+1)|=0. |
In order to obtain the orthogonality of the eigenfunction, we define a weighted inner product related to the weighted function r(t) in H. First, we define the inner product in Y as ⟨y,z⟩r=T∑t=1r(t)y(t)z(t).
Similarly, the inner product associated with the weight function r(t) in the space H is defined as follows:
⟨[y,α,β]⊤,[z,ζ,ρ]⊤⟩r=⟨y,z⟩r+p(0)|δ0|αζ+p(T)|δ1|βρ. |
Lemma 2.6. (Orthogonality theorem) Assume that (A1) and (A2) hold. If (λ,y∗) and (μ,z∗) are two different eigenpairs corresponding to L, then y∗ and z∗ are orthogonal under the weight inner product related to the weight function r(t).
Proof Assume that (λ,y∗) and (μ,z∗) is the eigenpair of L, then it can be obtained from Lemmas 2.4 and 2.5
0=(μ−λ)⟨y∗,Sz∗⟩=(μ−λ)⟨y∗,z∗⟩r. |
Therefore, if λ≠μ, then ⟨y∗,z∗⟩r=0, which implies that y∗ and z∗ are orthogonal to the inner product defined by the weighted function r(t).
Lemma 2.7. ([18]) Suppose that (A1) and (A2) hold. Then (1.2) has at least T or at most T+2 simple eigenvalues.
In this paper, we consider that λk is a simple eigenvalue, that is, the eigenspace corresponding to each eigenvalue is one-dimensional. Let ψ∗k=[ψk,α,β]⊤∈D be the eigenfunction corresponding to λk, and assume that it satisfies
⟨ψ∗k,ψ∗k⟩=1. | (2.5) |
Denote by L:=L−λkS, then the operator (2.2) is transformed into
Ly∗=Ay∗. | (2.6) |
Define P:D→D by
(Px∗)(t)=ψ∗k(t)⟨ψ∗k(t),x∗(t)⟩. |
Lemma 2.8. P is a projection operator and Im(P)=Ker(L).
Proof Obviously, P is a linear operator, next, we need to prove P2=P.
(P2x∗)(t)=P(Px∗)(t)=ψ∗k(t)⟨ψ∗k(t),Px∗(t)⟩=ψ∗k(t)⟨ψ∗k(t),ψ∗k(t)⟨ψ∗k(t),x∗(t)⟩⟩=ψ∗k(t)⟨ψ∗k(t),x∗(t)⟩⟨ψ∗k(t),ψ∗k(t)⟩=ψ∗k(t)⟨ψ∗k(t),x∗(t)⟩=(Px∗)(t). |
It can be obtained from the Definition 2.1, P is a projection operator. In addition, Im(P)=span{ψ∗k}=Ker(L).
Define H:H→H by
H([yαβ])=[yαβ]−⟨[yαβ],ψ∗k⟩ψ∗k. |
Lemma 2.9. H is a projection operator and Im(H)=Im(L).
Proof Obviously, H is a linear operator, next, we need to prove that H2=H.
H2([yαβ])=H(H[yαβ])=H[yαβ]−⟨H[yαβ],ψ∗k⟩ψ∗k=[yαβ]−⟨[yαβ],ψ∗k⟩ψ∗k−⟨[yαβ]−⟨[yαβ],ψ∗k⟩ψ∗k,ψ∗k⟩ψ∗k=[yαβ]−2⟨[yαβ],ψ∗k⟩ψ∗k+⟨⟨[yαβ],ψ∗k⟩ψ∗k,ψ∗k⟩ψ∗k=[yαβ]−2⟨[yαβ],ψ∗k⟩ψ∗k+⟨[yαβ],ψ∗k⟩⟨ψ∗k,ψ∗k⟩ψ∗k=H([yαβ]). |
It can be obtained from Definition 2.1 that H is a projection operator. On the one hand, for any [y,α,β]⊤∈H, we have
⟨H[yαβ],ψ∗k⟩=⟨[yαβ]−⟨[yαβ],ψ∗k⟩ψ∗k,ψ∗k⟩=⟨[yαβ],ψ∗k⟩−⟨⟨[yαβ],ψ∗k⟩ψ∗k,ψ∗k⟩=0, |
thus, Im(H)⊂Im(L). On the other hand, for any y∗∈Im(L), we have
⟨y∗,ψ∗k⟩=0. |
In summary, Im(H)=Im(L).
Denote that I is a identical operator, then
D=Im(P)⊕Im(I−P),H=Im(H)⊕Im(I−H). |
The restriction of the operator L on L|Im(I−P) is a bijection from Im(I−P) to Im(H). Define M:Im(H)→Im(I−P) by
M:=(L|Im(I−P))−1. |
It can be seen from KerL=span{ψ∗k} that there is a unique decomposition for any y∗=[y,α,β]⊤∈D
y∗=ρψ∗k+x∗, |
where ρ∈R,x∗=[x,α,β]⊤∈Im(I−P).
Lemma 2.10. The operator Eq (2.6) is equivalent to the following system
x∗=MHA(ρψ∗k+x∗), | (2.7) |
T∑t=1ψk(t)f(t,ρψk(t)+x(t))=γ(p(0)|δ0|α2+p(T)|δ1|β2−1):=θ, | (2.8) |
where α=a0ψk(0)−c0Δψk(0),β=a1ψk(T+1)−c1∇ψk(T+1).
Proof (ⅰ) For any y∗=ρψ∗k+x∗, we have
Ly∗=Ay∗ ⟺H(L(ρψ∗k+x∗)−A(ρψ∗k+x∗))=0⟺Lx∗−HA(ρψ∗k+x∗)=0⟺x∗=MHA(ρψ∗k+x∗). |
(ⅱ) Since ⟨Ly∗,ψ∗k⟩=0, we have ⟨Ay∗,ψ∗k⟩=0. Therefore,
⟨f(t,y)+γψk+¯g,ψk⟩Y=T∑t=1f(t,ρψk(t)+x(t))ψk(t)+T∑t=1γψk(t)ψk(t)+T∑t=1¯g(t)ψk(t)=0. |
Combining (H3) with (2.5), we have
T∑t=1ψk(t)f(t,ρψk(t)+x(t))=γ(p(0)|δ0|α2+p(T)|δ1|β2−1)=θ, |
where α=a0ψk(0)−c0Δψk(0),β=a1ψk(T+1)−c1∇ψk(T+1).
Let
A+={t∈{1,2,⋯,T} s.t. ψk(t)>0}, |
A−={t∈{1,2,⋯,T} s.t. ψk(t)<0}. |
Obviously,
A+∪A−≠∅, min{|ψk(t)||t∈A+∪A−}>0. |
Lemma 3.1. Supposed that (H1) holds, then there exist constants M0 and M1, such that
‖x∗‖≤M1(|ρ|‖ψk‖Y)α, |
where (ρ,x∗) is the solution of (2.7) and satisfies |ρ|≥M0.
Proof Since
A(ρψ∗k+x∗)=F(t,ρψ∗k+x∗)+[γψk+¯g,0,0]⊤=[f(t,ρψk+x)+γψk+¯g,0,0]⊤, |
we have
‖x∗‖≤‖M‖Im(H)→Im(I−P)‖H‖H→Im(H)[‖¯g‖Y+γ‖ψk‖Y+A(|ρ|‖ψk‖Y+‖x‖Y)α+B]=‖M‖Im(H)→Im(I−P)‖H‖H→Im(H)[‖¯g‖Y+A(|ρ|‖ψk‖Y)α(1+‖x‖Y|ρ|‖ψk‖Y)α+B−θ]≤‖M‖Im(H)→Im(I−P)‖H‖H→Im(H)[‖¯g‖Y+A(|ρ|‖ψk‖Y)α(1+α‖x‖Y|ρ|‖ψk‖Y)+B−θ]=‖M‖Im(H)→Im(I−P)‖H‖H→Im(H)[‖¯g‖Y+A(|ρ|‖ψk‖Y)α(1+α(|ρ|‖ψk‖Y)1−α‖x‖Y(|ρ|‖ψk‖Y)α)+B−θ]. |
Denote that
D0=‖M‖Im(H)→Im(I−P)‖H‖H→Im(H)(‖¯g‖Y+B−θ),D1=A‖M‖Im(H)→Im(I−P)‖H‖H→Im(H). |
Furthermore, we have
‖x∗‖(|ρ|‖ψk‖Y)α≤D0(|ρ|‖ψk‖Y)α+D1+αD1(|ρ|‖ψk‖Y)1−α‖x‖Y(|ρ|‖ψk‖Y)α≤D0(|ρ|‖ψk‖Y)α+D1+αD1(|ρ|‖ψk‖Y)1−α‖x∗‖(|ρ|‖ψk‖Y)α. |
So, if we let
αD1(|ρ|‖ψk‖Y)1−α≤12, |
we have
|ρ|≥(2αD1)11−α‖ψk‖Y:=M0. |
Thus,
‖x∗‖(|ρ|‖ψk‖Y)α≤2D0(M0‖ψk‖Y)α+2D1:=M1. |
This implies that
‖x∗‖≤M1(|ρ|‖ψk‖Y)α. |
Lemma 3.2. Suppose that (H1) holds, then there exist constants M0 and Γ, such that
‖x∗‖≤Γ(|ρ|min{|ψk(t)||t∈A+∪A−})α, |
where (ρ,x∗) is the solution of (2.7) and satisfies |ρ|≥M0.
According to Lemma 3.2, choose constant ρ0, such that
ρ0>max{M0,Γ(|ρ0|min{|ψk(t)||t∈A+∪A−})α}. | (3.1) |
Let
K:={x∗∈Im(I−P)|x∗=MHA(ρψ∗k+x∗),|ρ|≤ρ0}. |
Then, for sufficiently large ρ≥ρ0, there is
ρψk(t)+x(t)≥ω, ∀t∈A+,x∗∈K, | (3.2) |
ρψk(t)+x(t)≤−ω, ∀t∈A−,x∗∈K, | (3.3) |
and for sufficiently small ρ≤−ρ0, there is
ρψk(t)+x(t)≤−ω, ∀t∈A+,x∗∈K, | (3.4) |
ρψk(t)+x(t)≥ω, ∀t∈A−,x∗∈K. | (3.5) |
Theorem 3.3. Suppose that (A1), (A2) and (H1)–(H3) hold, then there exists a non-empty bounded set Ω¯g⊂R, such that the problem (1.1) has a solution if and only if θ∈Ω¯g. Furthermore, Ω¯g contains θ=0 and has a non-empty interior.
Proof We prove only the case of (1.3) in (H2), and the case of (1.4) can be similarly proved.
From (1.3) and (3.2)–(3.5), it is not difficult to see that
f(t,ρψk(t)+x(t))>0, ∀t∈A+, x∗∈K, |
f(t,ρψk(t)+x(t))<0, ∀t∈A−, x∗∈K, |
for sufficiently large ρ≥ρ0 and for sufficiently small ρ≤−ρ0,
f(t,ρψk(t)+x(t))<0, ∀t∈A+, x∗∈K, |
f(t,ρψk(t)+x(t))>0, ∀t∈A−, x∗∈K. |
Therefore, if ρ≥ρ0 is sufficiently large,
ψk(t)f(t,ρψk(t)+x(t))>0, ∀t∈A+∪A−, x∗∈K, | (3.6) |
if ρ≤−ρ0 is sufficiently small,
ψk(t)f(t,ρψk(t)+x(t))<0, ∀t∈A+∪A−, x∗∈K. | (3.7) |
Let
C:={x∗∈Im(I−P)|‖x∗‖≤ρ0}. |
Define Tρ:Im(I−P)→Im(I−P) by
Tρ:=MHA(ρψ∗k+x∗). |
Obviously, Tρ is completely continuous. By (3.1), for x∗∈C and ρ∈[−ρ0,ρ0],
‖Tρx∗‖≤Γ(|ρ|min{|ψk(t)||t∈A+∪A−})α≤Γ(|ρ0|min{|ψk(t)||t∈A+∪A−})α≤ρ0, |
i.e.,
Tρ(C)⊆C. |
According to Schauder's fixed point theorem, Tρ has a fixed point on C, such that Tρx∗=x∗. It can be seen from Lemma 2.10 that the problem (1.1) is equivalent to the following system
Ψ(s,x∗)=θ, (s,x∗)∈S¯g, |
where
S¯g:={(ρ,x∗)∈R×Im(I−P)|x∗=MHA(ρψ∗k+x∗)}, |
Ψ(ρ,x∗):=T∑s=1ψk(s)f(s,ρψk(s)+x(s)). |
At this time, the Ω¯g in Theorem 3.3 can be given by Ω¯g=Ψ(S¯g). There exists a solution to the problem (1.1) for θ∈Ω¯g.
From (3.6), (3.7) and A+∪A−≠∅, we can deduce that for any x∗∈K
T∑s=1ψk(s)f(s,−ρ0ψk(s)+x(s))<0, T∑s=1ψk(s)f(s,ρ0ψk(s)+x(s))>0. |
Thus,
Ψ(−ρ0,x∗)<0<Ψ(ρ0,x∗), ∀x∗∈K. | (3.8) |
According to Lemma 2.2, S¯g⊂RׯBρ0 contains a connected branch ξ−ρ0,ρ0 connecting {−ρ0}×C and {ρ0}×C. Combined with (3.8), Ω¯g contains θ=0 and has a non-empty interior.
Theorem 3.4. Suppose that (A1), (A2), (H2)–(H4) hold. Ω¯g as shown in Theorem 3.3, then there exists a nonempty set Ω∗¯g⊂Ω¯g∖{0}, such that problem (1.1) has at least two solutions for θ∈Ω∗¯g.
Proof We prove only the case of (1.3), and the case of (1.4) can be similarly proved. Since the condition (H4) implies that (H1), using Theorem 3.3, we know that there exists ρ0>0, such that
Ψ(ρ0,x∗)>0, ∀x∗∈K. |
Let
δ:=min{Ψ(ρ0,x∗)|x∗∈K}, |
then δ>0.
Next, we prove that problem (1.1) has at least two solutions for any θ∈(0,δ).
Let
S¯g:={(ρ,x∗)∈R×Im(I−P)|x∗=MHA(ρψ∗k+x∗)}, |
¯K:={x∗∈Im(I−P)|(ρ,x∗)∈S¯g}. |
By (H4), there exists a constant A0 such that
‖x∗‖≤A0, ∀x∗∈K. |
Similar to the derivation of Theorem 3.3, there exists ρ∗>ρ0 such that the following results hold:
(ⅰ) For ρ≥ρ∗, there is
ψk(t)f(t,ρψk(t)+x(t))>0, ∀t∈A+∪A−, x∗∈¯K, | (3.9) |
(ⅱ) For ρ≤−ρ∗, there is
ψk(t)f(t,ρψk(t)+x(t))<0, ∀t∈A+∪A−, x∗∈¯K. | (3.10) |
Let
C∗:={x∗∈Im(I−P)|‖x∗‖≤A0}. |
According to (H4), (3.9) and (3.10), we have
lim|ρ|→∞T∑s=1ψk(s)f(s,ρψk(s)+x(s))=0 |
uniformly for x∗∈¯K, i.e.
lim|ρ|→∞Ψ(ρ,x∗)=0, x∗∈¯K. |
Therefore, there exists a constant l:l>ρ∗>ρ0>0 such that S¯g contains a connected branch between {−l}×C∗ and {l}×C∗, and
max{|Ψ(ρ,x∗)||ρ=±l, (ρ,x∗)∈ξ−l,l}≤max{|Ψ(ρ,x∗)||(ρ,x∗)∈{−l,l}ׯK}≤θ3. |
It can be seen from the connectivity of ξ−l,l that there exist (ρ1,x∗1) and (ρ2,x∗2) in ξ−l,l(⊂S¯g), such that
Ψ(ρ1,x∗1)=θ, Ψ(ρ2,x∗2)=θ, |
where ρ1∈(−l,ρ0),ρ2∈(ρ0,l). It can be proved that ρ1ψ∗k+x∗1 and ρ2ψ∗k+x∗2 are two different solutions of problem (1.1).
In this section, we give a concrete example of the application of our major results of Theorems 3.3 and 3.4. We choose T=3,a0,d0,b1,c1=0 and a1,d1,b0,c0=1, which implies that the interval becomes [1,3]Z and the conditions (A1),(A2) hold.
First, we consider the eigenpairs of the corresponding linear problem
{−Δ2y(t−1)=λy(t), t∈[1,3]Z,y(0)=λΔy(0), λy(4)=∇y(4). | (4.1) |
Define the equivalent matrix of (4.1) as follows,
Aλ=(λ−2+λ1+λ101λ−2101λ−2+11−λ) |
Consequently, Aλy=0 is equivalent to (4.1). Let |Aλ|=0, we have
λ1=−1.4657,λ2=0.1149,λ3=0.8274,λ4=2.0911,λ5=3.4324, |
which are the eigenvalues of (4.1). Next, we choose λ=λ1=−1.4657, then we obtain the corresponding eigenfunction
ψ1(t)={1,t=1,3.4657,t=2,3.46572−1,t=3. |
Example 4.1. Consider the following problem
{−Δ2y(t−1)=−1.4657y(t)+f(t,y(t))+ψ1(t)+¯g(t), t∈[1,3]Z,y(0)=−1.4657Δy(0), −1.4657y(4)=∇y(4), | (4.2) |
where
f(t,s)={ts3,s∈[−1,1],t5√s,s∈(−∞,−1)∪(1,+∞), |
and
¯g(t)={0,t=1,3.46572−1,t=2,−3.4657,t=3. |
Then, for f(t,y(t)), we have |f(t,y(t))|≤3|y(t)|13. If we choose ω=1, yf(t,y)>0 for |y(t)|>1. For ¯g(t), we have 3∑s=1¯g(s)ψ1(s)=0.
Therefore, the problem (4.2) satisfies the conditions (A1),(A2), (H1)–(H3), which implies that the problem (4.2) has at least one solution by Theorem 3.3.
Example 4.2. Consider the following problem
{−Δ2y(t−1)=−1.4657y(t)+f(t,y(t))+ψ1(t)+¯g(t), t∈[1,3]Z,y(0)=−1.4657Δy(0), −1.4657y(4)=∇y(4), | (4.3) |
where
f(t,s)=tse|s|, t∈[1,3]Z |
and
¯g(t)={0,t=1,1−3.46572,t=2,3.4657,t=3. |
Then, for f(t,y(t)), we always have yf(t,y)>0 for all y(t)>0 or y(t)<0, f is continuous and satisfies
lim|y|→∞f(t,y)=0. |
For ¯g(t), we have 3∑s=1¯g(s)ψ1(s)=0.
Therefore, the problem (4.3) satisfies the conditions (A1),(A2), (H2)–(H4), which implies that the problem (4.3) has at least two solutions by Theorem 3.4.
The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.
Supported by National Natural Science Foundation of China [Grant No. 11961060] and Natural Science Foundation of Qinghai Province(No.2024-ZJ-931).
The authors declare that there are no conflicts of interest.
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