Loading [MathJax]/jax/output/SVG/jax.js
Review Special Issues

Development and application of DNA molecular probes

  • Received: 02 January 2017 Accepted: 15 February 2017 Published: 21 February 2017
  • The development of DNA probes started from 1950's for diagnostic purposes and it is still growing. DNA probes are applied in several fields such as food, medical, veterinary, environment and security, with the aim of prevention, diagnosis and treatment. The use of DNA probes permits microorganism identification, including pathogen detection, and their quantification when used in specific systems. Various techniques obtained success by the utilization of specific DNA probes, that allowed the obtainment of rapid and specific results. From PCR, qPCR and blotting techniques that were first used in well equipped laboratories to biosensors such as fiber optic, surface plasmon resonance (SPR), electrochemical, and quartz crystal microbalance (QCM) biosensors that use different transduction systems. This review describes i) the design and production of primers and probes, and their utilization from the traditional techniques to the new emerging techniques like biosensors used in current applications; ii) the possibility to use labelled-free probes and probes labelled with an enzyme/fluorophore, etc.; iii) the different sensitivity obtained by using specific systems; and iv) the advantage obtained by using biosensors.

    Citation: Priya Vizzini, Lucilla Iacumin, Giuseppe Comi, Marisa Manzano. Development and application of DNA molecular probes[J]. AIMS Bioengineering, 2017, 4(1): 113-132. doi: 10.3934/bioeng.2017.1.113

    Related Papers:

    [1] Fugeng Zeng, Yao Huang, Peng Shi . Initial boundary value problem for a class of p-Laplacian equations with logarithmic nonlinearity. Mathematical Biosciences and Engineering, 2021, 18(4): 3957-3976. doi: 10.3934/mbe.2021198
    [2] Tingting Xue, Xiaolin Fan, Hong Cao, Lina Fu . A periodic boundary value problem of fractional differential equation involving p(t)-Laplacian operator. Mathematical Biosciences and Engineering, 2023, 20(3): 4421-4436. doi: 10.3934/mbe.2023205
    [3] Peng Shi, Min Jiang, Fugeng Zeng, Yao Huang . Initial boundary value problem for fractional p-Laplacian Kirchhoff type equations with logarithmic nonlinearity. Mathematical Biosciences and Engineering, 2021, 18(3): 2832-2848. doi: 10.3934/mbe.2021144
    [4] Hongyun Peng, Kun Zhao . On a hyperbolic-parabolic chemotaxis system. Mathematical Biosciences and Engineering, 2023, 20(5): 7802-7827. doi: 10.3934/mbe.2023337
    [5] Guodong Li, Ying Zhang, Yajuan Guan, Wenjie Li . Stability analysis of multi-point boundary conditions for fractional differential equation with non-instantaneous integral impulse. Mathematical Biosciences and Engineering, 2023, 20(4): 7020-7041. doi: 10.3934/mbe.2023303
    [6] Ting-Hao Hsu, Tyler Meadows, LinWang, Gail S. K. Wolkowicz . Growth on two limiting essential resources in a self-cycling fermentor. Mathematical Biosciences and Engineering, 2019, 16(1): 78-100. doi: 10.3934/mbe.2019004
    [7] Xiaomei Bao, Canrong Tian . Turing patterns in a networked vegetation model. Mathematical Biosciences and Engineering, 2024, 21(11): 7601-7620. doi: 10.3934/mbe.2024334
    [8] A. Vinodkumar, T. Senthilkumar, S. Hariharan, J. Alzabut . Exponential stabilization of fixed and random time impulsive delay differential system with applications. Mathematical Biosciences and Engineering, 2021, 18(3): 2384-2400. doi: 10.3934/mbe.2021121
    [9] Chun Lu, Bing Li, Limei Zhou, Liwei Zhang . Survival analysis of an impulsive stochastic delay logistic model with Lévy jumps. Mathematical Biosciences and Engineering, 2019, 16(5): 3251-3271. doi: 10.3934/mbe.2019162
    [10] Mohsen Dlala, Sharifah Obaid Alrashidi . Rapid exponential stabilization of Lotka-McKendrick's equation via event-triggered impulsive control. Mathematical Biosciences and Engineering, 2021, 18(6): 9121-9131. doi: 10.3934/mbe.2021449
  • The development of DNA probes started from 1950's for diagnostic purposes and it is still growing. DNA probes are applied in several fields such as food, medical, veterinary, environment and security, with the aim of prevention, diagnosis and treatment. The use of DNA probes permits microorganism identification, including pathogen detection, and their quantification when used in specific systems. Various techniques obtained success by the utilization of specific DNA probes, that allowed the obtainment of rapid and specific results. From PCR, qPCR and blotting techniques that were first used in well equipped laboratories to biosensors such as fiber optic, surface plasmon resonance (SPR), electrochemical, and quartz crystal microbalance (QCM) biosensors that use different transduction systems. This review describes i) the design and production of primers and probes, and their utilization from the traditional techniques to the new emerging techniques like biosensors used in current applications; ii) the possibility to use labelled-free probes and probes labelled with an enzyme/fluorophore, etc.; iii) the different sensitivity obtained by using specific systems; and iv) the advantage obtained by using biosensors.


    The article is concerned with the solvability of Dirichlet problems of the fractional p-Laplacian equation with impulsive effects, as follows:

    {tDαTϕp(0Dαtu(t))+a(t)ϕp(u(t))=λf(t,u(t)),ttj,a.e.t[0,T],Δ(tDα1Tϕp(C0Dαtu))(tj)=μIj(u(tj)),j=1,2,,n,nN,u(0)=u(T)=0, (1.1)

    where C0Dαt is the left Caputo fractional derivative, 0Dαt and tDαT are the left and right Riemann-Liouville fractional derivatives respectively, α(1/p,1], p>1, ϕp(x)=|x|p2x (x0), ϕp(0)=0, λ>0, μR, a(t)C([0,T],R), fC([0,T]×R,R), T>0, 0=t0<t1<t2<<tn<tn+1=T, IjC(R,R), and

    Δ(tDα1Tϕp(C0Dαtu))(tj)=tDα1Tϕp(C0Dαtu)(t+j)tDα1Tϕp(C0Dαtu)(tj),
    tDα1Tϕp(C0Dαtu)(t+j)=limtt+jtDα1Tϕp(C0Dαtu)(t),tDα1Tϕp(C0Dαtu)(tj)=limttjtDα1Tϕp(C0Dαtu)(t).

    Fractional calculus has experienced a growing focus in recent decades because of its application to real-world problems. This kind of problem has attracted the attention of many scholars and produced a series of excellent works [1,2,3,4,5,6,7,8]. In particular, left and right fractional differential operators have been widely used in the study of physical phenomena of anomalous diffusion, specifically, fractional convection-diffusion equations [9,10]. Recently, the equations containing left and right fractional differential operators have become a new field in the theory of fractional differential equations. For example, the authors of [11] first put forward the following fractional convection-diffusion equation:

    {aD(p0Dβt+qtDβT)Du(t)+b(t)Du(t)+c(t)u(t)=f,  a.e. t[0,T],  0β<1,u(0)=u(T)=0.

    The authors gained the relevant conclusions about the solution of the above-mentioned problems by using the Lax-Milgram theorem. In [12], the authors discussed the following problem:

    {ddt(120Dβt(u(t))+12tDβT(u(t)))+F(t,u(t))=0,  a.e. t[0,T],  0β<1,u(0)=u(T)=0.

    By applying the minimization principle and mountain pass theorem, the existence results under the Ambrosetti-Rabinowitz condition were obtained. The following year, in [13], the authors made further research on the following issues:

    {tDαT(0Dαtu(t))=F(t,u(t)),  a.e. t[0,T],  12<α1,u(0)=u(T)=0.

    Use of impulsive differential equations is an effective method to describe the instantaneous change of the state of things, and it can reflect the changing law of things more deeply and accurately. It has practical significance and application value in many fields of science and technology, such as signal communication, economic regulation, aerospace technology, management science, engineering science, chaos theory, information science, life science and so on. Due to the application of impulsive differential equations to practical problems, more and more attention has been paid to them in recent years, and many scholars at home and abroad have studied such problems. For example, in [14,15], using the three critical points theorem, the authors discussed the impulse problems as follows:

    {tDαT(C0Dαtu(t))+a(t)u(t)=λf(t,u(t)),ttj,a.e.t[0,T],α(12,1],Δ(tDα1T(C0Dαtu))(tj)=μIj(u(tj)),j=1,2,,n,u(0)=u(T)=0,

    where λ,μ>0, IjC(R,R), aC([0,T]) and there exist a1 and a2 such that 0<a1a(t)a2. In addition,

    Δ(tDα1T(C0Dαtu))(tj)=tDα1T(C0Dαtu)(t+j)tDα1T(C0Dαtu)(tj),
    tDα1T(C0Dαtu)(t+j)=limtt+j(tDα1T(C0Dαtu)(t)),tDα1T(C0Dαtu)(tj)=limttj(tDα1T(C0Dαtu)(t)).

    The p-Laplacian equation originated from the nonlinear diffusion equation proposed by Leibenson in 1983, when he studied the problem of one-dimensional variable turbulence of gas passing through porous media:

    ut=x(umx|umx|μ1),m=n+1.

    When m>1, the above equation is the porous medium equation; When 0<m<1, the above equation is a fast diffusion equation; When m=1, the above equation is a heat equation; However, when m=1,μ1, such equations often appear in the study of non-Newtonian fluids. In view of the importance of such equations, the above equation has been abstracted into the p-Laplacian equation:

    (ϕp(u))=f(t,u),

    where ϕp(x)=|x|p2x (x0),ϕp(0)=0,p>1. When p=2, the p-Laplacian equation is reduced to a classical second-order differential equation. Ledesma and Nyamoradi [16] researched the impulse problem with a p-Laplacian operator as below.

    {tDαT(|0Dαtu(t)|p20Dαtu(t))+a(t)|u(t)|p2u(t)=f(t,u(t)),ttj,a.e.t[0,T],Δ(tI1αT(|0Dαtu(tj)|p20Dαtu(tj)))=Ij(u(tj)),j=1,2,,n,nN,u(0)=u(T)=0, (1.2)

    where α(1p,1], p>1, fC([0,T]×R,R), IjC(R,R) and

    Δ(tI1αT(|0Dαtu(tj)|p20Dαtu(tj)))=tI1αT(|0Dαtu(t+j)|p20Dαtu(t+j))tI1αT(|0Dαtu(tj)|p20Dαtu(tj)),tI1αT(|0Dαtu(t+j)|p20Dαtu(t+j))=limtt+jtI1αT(|0Dαtu(t)|p20Dαtu(t)),tI1αT(|0Dαtu(tj)|p20Dαtu(tj))=limttjtI1αT(|0Dαtu(t)|p20Dαtu(t)).

    By using the mountain pass theorem and the symmetric mountain pass theorem, the authors acquired the related results of Problem (1.2) under the Ambrosetti-Rabinowitz condition. If α=1 and a(t)=0, then Problem (1.2) is reduced to the p-Laplacian equation with impulsive effects, as follows:

    {(|u|p2u)=f(t,u(t)),ttj,a.e.t[0,1],u(1)=u(0)=0,u(t+j)=u(tj),j=1,2,,n,Δ|u(tj)|p2u(tj)=Ij(u(tj)),j=1,2,,n.

    This problem has been studied in [17] and [18]. The main methods used in the above literature are the critical point theory and the topological degree theory. To show the major conclusions of literature [16], the following assumptions are first introduced below:

    (F1) There are θ>p and r>0, so that 0<θF(t,ξ)ξf(t,ξ),t[0,T],|ξ|r;

    (F2) f(t,ξ)=o(|ξ|p1),ξ0, for t[0,T];

    (F3) For j, there are cj>0 and γj(p1,θ1) so that |Ij(ξ)|cj|ξ|γj;

    (F4) For u large enough, one has Ij(ξ)ξθu0Ij(ξ)dξ,j=1,2,,n.

    Theorem 1. ([16]). If the conditions (F1)(F4) hold, then the impulsive problem (1.2) possesses one weak solution.

    The research work of this paper is to further study the impulse problem (1.1) on the basis of the above work. To compare with Theorem 1, the supposed conditions and main results are given as below.

    (H0) a(t)C([0,T],R) satisfies essinft[0,T]a(t)>λ1, where λ1=infuEα,p0{0}T0|0Dαtu(t)|pdtT0|u(t)|pdt>0;

    (H1) For tR, j=1,2,,m,mN, Ij(t) satisfies t0Ij(s)ds0;

    (H2) There are aj,dj>0 and γj[0,p1) so that |Ij(t)|aj+dj|t|γj,tR;

    (H3) The map sIj(s)/Ij(s)|s|p1|s|p1 is strictly monotonically decreasing on R{0};

    (H4) The map sf(t,s)/|s|p1 is strictly monotonically increasing on R{0}, for t[0,T];

    (H5) f(t,u)=o(|u|p1)(|u|0), uniformly for t[0,T];

    (H6) There are M>0, L>0 and θ>p so that

    uf(t,u)θF(t,u)M|u|p,t[0,T],|u|L,

    where F(t,u)=u0f(t,s)ds;

    (H7) lim|u|F(t,u)|u|θ=, uniformly for t[0,T].

    Theorem 2. Let fC1([0,T]×R,R) and IjC1(R,R). Assume that the conditions (H0)(H7) hold. Then, Problem (1.1) with λ=μ=1 has at least one nontrivial ground-state solution.

    Remark 1. Obviously, the conditions (H6) and (H7) are weaker than (F1) of Theorem 1. In addition, for this kind of problem, the existence of solutions has been discussed in the past, while the ground-state solutions have been rarely studied. Therefore, our finding extends and enriches Theorem 1 in [16].

    Next, further research Problem (1.1) with the concave-convex nonlinearity. The function fC([0,T]×R,R) studied here satisfies the following conditions:

    f(t,u)=f1(t,u)+f2(t,u), (1.3)

    where f1(t,u) is p-suplinear as |u| and f2(t,u) denotes p-sublinear growth at infinity. Below, some supposed conditions are given on f1 and f2, as below:

    (H8) f1(t,u)=o(|u|p1)(|u|0), uniformly for t[0,T];

    (H9) There are M>0,L>0 and θ>p so that

    uf1(t,u)θF1(t,u)M|u|p,t[0,T],|u|L,

    where F1(t,u)=u0f1(t,s)ds;

    (H10) lim|u|F1(t,u)|u|θ=, uniformly for t[0,T];

    (H11) There are 1<r<p and bC([0,T],R+), R+=(0,), so that

    F2(t,u)b(t)|u|r,(t,u)[0,T]×R,

    where F2(t,u)=u0f2(t,s)ds;

    (H12) There is b1L1([0,T],R+) so that |f2(t,u)|b1(t)|u|r1,(t,u)[0,T]×R;

    (H13) There are aj,dj>0 and γj[0,θ1) so that |Ij(t)|aj+dj|t|γj,tR;

    (H14) For t large enough, Ij(t) satisfies θt0Ij(s)dsIj(t)t;

    (H15) For tR, Ij(t) satisfies t0Ij(s)ds0.

    Theorem 3. Assume that the conditions (H0) and (H8)(H15) hold. Then, the impulse problem (1.1) with λ=μ=1 possesses at least two non-trivial weak solutions.

    Remark 2. Obviously, the conditions (H9) and (H10) are weaker than (F1) of Theorem 1. And, the condition (H13) is weaker than the condition (F3) of Theorem 1. Further, the function f studied in Theorem 3 contains both p-suplinear and p-sublinear terms, which is more general. Thus, our finding extends Theorem 1 in [16].

    Finally, the existence results of the three solutions of the impulse problem (1.1) in the case of the parameter μ0 or μ<0 are considered respectively. We need the following supposed conditions.

    (H16) There are L,L1,,Ln>0, 0<βp, 0<dj<p and j=1,,n so that

    F(t,x)L(1+|x|β),Jj(x)Lj(1+|x|dj),(t,x)[0,T]×R, (1.4)

    where F(t,x)=x0f(t,s)ds and Jj(x)=x0Ij(t)dt;

    (H17) There are r>0, and ωEα,p0 so that 1pωp>r, T0F(t,ω(t))dt>0,nj=1Jj(ω(tj))>0 and

    Al:=1pωpT0F(t,ω(t))dt<Ar:=rT0max|x|Λ(pr)1/1ppF(t,x)dt. (1.5)

    Theorem 4. Assume that the conditions (H0) and (H16)(H17) hold. Then, for every λΛr=(Al,Ar), there is

    γ:=min{rλT0max|x|Λ(pr)1/1ppF(t,x)dtmax|x|Λ(pr)1pnj=1(Jj(x)),λT0F(t,ω)dt1pωpnj=1Jj(ω(tj))} (1.6)

    so that, for each μ[0,γ), the impulse problem (1.1) possesses at least three weak solutions.

    (H18) There are L,L1,,Ln>0, 0<βp, 0<dj<p and j=1,,n so that

    F(t,x)L(1+|x|β),Jj(x)Lj(1+|x|dj); (1.7)

    (H19) There are r>0 and ωEα,p0 so that 1pωp>r, T0F(t,ω(t))dt>0,nj=1Jj(ω(tj))<0 and (1.5) hold.

    Theorem 5. Assume that the conditions (H0) and (H18)-(H19) hold. Then, for every λΛr=(Al,Ar), there is

    γ:=max{λT0max|x|Λ(pr)1/1ppF(t,x)dtrmax|x|Λ(pr)1pnj=1Jj(x),λT0F(t,ω)dt1pωpnj=1Jj(ω(tj))}

    so that, for each μ(γ,0], the impulse problem (1.1) possesses at least three weak solutions.

    Remark 3. The assumptions (H16) and (H18) study both 0<β<p and β=p. When p=2, the assumptions (H16) and (H18) contain the condition 0<β<2 in [14,15]. In addition, this paper allows a(t) to have a negative lower bound, satisfying essinft[0,T]a(t)>λ1, where λ1=infuEα,p0{0}T0|0Dαtu(t)|pdtT0|u(t)|pdt>0, and a(t) in [14,15] has a positive lower bound satisfying 0<a1a(t)a2. Thus, our conclusions extend the existing results.

    This paper studies Dirichlet boundary-value problems of the fractional p-Laplacian equation with impulsive effects. By using the Nehari manifold method, the existence theorem of the ground-state solution of the above impulsive problem is given. At the same time, the p-suplinear condition required for the proof is weakened. This is the research motivation for this paper. There is no relevant research work on this result. In addition, the existence and multiplicity theorems of nontrivial weak solutions to the impulsive problem are given by means of a variational method. In the process of building the proof, the conditions of nonlinear functions with the concave-convex terms are weakened and the conditions of impulsive terms and variable coefficient terms are weakened. Our work extends and enriches the existing results in [14,15,16], which is the innovation of this paper.

    Here are some definitions and lemmas of fractional calculus. For details, see [19].

    Definition 1. ([19]). Let u be a function defined on [a,b]. The left and right Riemann-Liouville fractional derivatives of order 0γ<1 for a function u denoted by aDγtu(t) and tDγbu(t), respectively, are defined by

    aDγtu(t)=ddtaDγ1tu(t)=1Γ(1γ)ddt(ta(ts)γu(s)ds),
    tDγbu(t)=ddttDγ1bu(t)=1Γ(1γ)ddt(bt(st)γu(s)ds),

    where t[a,b].

    Definition 2. ([19]). Let 0<γ<1 and uAC([a,b]); then, the left and right Caputo fractional derivatives of order γ for a function u denoted by CaDγtu(t) and CtDγbu(t), respectively, exist almost everywhere on [a,b]. CaDγtu(t) and CtDγbu(t) are respectively represented by

    CaDγtu(t)=aDγ1tu(t)=1Γ(1γ)ta(ts)γu(s)ds,
    CtDγbu(t)=tDγ1bu(t)=1Γ(1γ)bt(st)γu(s)ds,

    where t[a,b].

    Definition 3. ([20]). Let 0<α1 and 1<p<. Define the fractional derivative space Eα,p as follows:

    Eα,p={uLp([0,T],R)|0DαtuLp([0,T],R)},

    with the norm

    uEα,p=(upLp+0DαtupLp)1p, (2.1)

    where uLp=(T0|u(t)|pdt)1/1pp is the norm of Lp([0,T],R). Eα,p0 is defined by closure of C0([0,T],R) with respect to the norm uEα,p.

    Proposition 1 ([19]). Let u be a function defined on [a,b]. If caDγtu(t), ctDγbu(t), aDγtu(t) and tDγbu(t) all exist, then

    caDγtu(t)=aDγtu(t)n1j=0uj(a)Γ(jγ+1)(ta)jγ,t[a,b],
    ctDγbu(t)=tDγbu(t)n1j=0uj(b)Γ(jγ+1)(bt)jγ,t[a,b],

    where nN and n1<γ<n.

    Remark 4. For any uEα,p0, according to Proposition 1, when 0<α<1 and the boundary conditions u(0)=u(T)=0 are satisfied, we can get c0Dαtu(t)=0Dαtu(t) and ctDαTu(t)=tDαTu(t),t[0,T].

    Lemma 1. ([20]). Let 0<α1 and 1<p<. The fractional derivative space Eα,p0 is a reflexive and separable Banach space.

    Lemma 2. ([13]). Let 0<α1 and 1<p<. If uEα,p0, then

    uLpTαΓ(α+1)0DαtuLp. (2.2)

    If α>1/1pp, then

    uC0DαtuLp, (2.3)

    where u=maxt[0,T]|u(t)| is the norm of C([0,T],R), C=Tα1pΓ(α)(αqq+1)1q>0 and q=pp1>1.

    Combined with (2.2), we think over Eα,p0 with the norm as below.

    uEα,p=(T0|0Dαtu(t)|pdt)1p=0DαtuLp,uEα,p0. (2.4)

    Lemma 3. ([13]). If 1/1pp<α1 and 1<p<, then Eα,p0 is compactly embedded in C([0,T],R).

    Lemma 4. ([13]). Let 1/1pp<α1 and 1<p<. If the sequence {uk} converges weakly to u in Eα,p0, i.e., uku, then uku in C([0,T],R), i.e., uku0,k.

    To investigate Problem (1.1), this article defines a new norm on the space Eα,p0, as follows:

    u=(T0|0Dαtu(t)|pdt+T0a(t)|u(t)|pdt)1p.

    Lemma 5. ([16]). If essinft[0,T]a(t)>λ1, where λ1=infuEα,p0{0}T0|0Dαtu(t)|pdtT0|u(t)|pdt>0, then u is equivalent to uEα,p, i.e., there are Λ1, Λ2>0, so that Λ1uEα,puΛ2uEα,p and uEα,p0, where uEα,p is defined in (2.4).

    Lemma 6. Let 0<α1 and 1<p<. For uEα,p0, by Lemmas 2 and 5 and (2.4), we have

    uLpTαΓ(α+1)uEα,pΛpu, (2.5)

    where Λp=TαΛ1Γ(α+1). If α>1/1pp, then

    uTα1pΓ(α)(αqq+1)1quEα,pΛu, (2.6)

    where Λ=Tα1pΛ1Γ(α)(αqq+1)1q,q=pp1>1.

    Lemma 7. ([19]). Let α>0, p1, q1 and 1/1pp+1/1qq<1+α, or p1, q1 and 1/1pp+1/1qq=1+α. Assume that the function uLp([a,b],R) and vLq([a,b],R); then,

    ba[aDαtu(t)]v(t)dt=bau(t)[tDαbv(t)]dt. (2.7)

    By multiplying the equation in Problem (1.1) by vEα,p0 and integrating on [0,T], one has

    T0tDαTϕp(0Dαtu(t))v(t)dt+T0a(t)ϕp(u(t))v(t)dtλT0f(t,u(t))v(t)dt=0.

    According to Lemma 7, we can get

    T0tDαTϕp(0Dαtu(t))v(t)dt=nj=0tj+1tjv(t)d[tDα1Tϕp(0Dαtu(t))]=nj=0tDα1Tϕp(0Dαtu(t))v(t)|tj+1tj+nj=0tj+1tjϕp(0Dαtu(t))0Dαtv(t)dt=nj=1[tDα1Tϕp(0Dαtu(t+j))v(tj)tDα1Tϕp(0Dαtu(tj))v(tj)]+T0ϕp(0Dαtu(t))0Dαtv(t)dt = μnj=1Ij(u(tj))v(tj)+T0ϕp(0Dαtu(t))0Dαtv(t)dt.

    Definition 4. Let uEα,p0 be one weak solution of the impulse problem (1.1), if

    T0ϕp(0Dαtu(t))0Dαtv(t)dt+T0a(t)ϕp(u(t))v(t)dt+μnj=1Ij(u(tj))v(tj)λT0f(t,u(t))v(t)dt=0

    holds for vEα,p0.

    Define a functional φ:Eα,p0R as below:

    φ(u)=1pup+μmj=1u(tj)0Ij(t)dtλT0F(t,u(t))dt, (2.8)

    where F(t,u)=u0f(t,s)ds. According to the continuity of the functions f and Ij, it is easy to prove that φC1(Eα,p0,R). In addition,

    φ(u),v=T0ϕp(0Dαtu(t))0Dαtv(t)dt+T0a(t)ϕp(u(t))v(t)dt+μnj=1Ij(u(tj))v(tj)λT0f(t,u(t))v(t)dt,u,vEα,p0. (2.9)

    Thus, the critical point of φ(u) corresponds to a weak solution of the impulse problem (1.1). The ground-state solution here refers to the minimum energy solution of the functional φ.

    Definition 5. ([21]). Let X be a real Banach space, φC1(X,R). For {un}nNX, {un}nN possesses one convergent subsequence if φ(un)c(n) and φ(un)0 (n). Then, φ(u) satisfies the (PS)c condition.

    Lemma 8. ([21]). Let X be a real Banach space and φC1(X,R) satisfy the (PS)c condition. Assume that φ(0)=0 and

    (i) there exist ρ, η>0 such that φ|Bρη>0;

    (ii) there exists an eX/B¯Bρ¯Bρ such that φ(e)0.

    Then, φ has one critical value cη. Moreover, c can be described as c=infgΓmaxs[0,1]φ(g(s)), where Γ={gC([0,1],X):g(0)=0,g(1)=e}.

    Lemma 9. ([22]). Let X be one reflexive real Banach space, Φ:XR be one sequentially weakly lower semi-continuous, coercive and continuously Gâteaux differentiable functional whose Gâteaux derivative admits one continuous inverse on X and Ψ:XR be one continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that infxXΦ(x)=Φ(0)=Ψ(0)=0. Suppose there are r>0 and ¯xX with r<Φ(¯x) so that

    (i) sup{Ψ(x):Φ(x)r}<rΨ(¯x)Φ(¯x),

    (ii) for each λΛr=(Φ(¯x)Ψ(¯x),rsup{Ψ(x):Φ(x)r}), the functional ΦλΨ is coercive.

    Then, for each λΛr, the functional ΦλΨ possesses at least three distinct critical points in X.

    Define N={uEα,p0{0}|G(u)=0}, where G(u)=φ(u),u=up+mj=1Ij(u(tj))u(tj)T0f(t,u(t))u(t)dt. Then, any non-zero critical point of φ must be on N. For j=1,2,,m and t[0,T], by (H3) and (H4), one has

    Ij(u(tj))u2(tj)<(p1)Ij(u(tj))u(tj),(p1)f(t,u(t))u(t)<f(t,u(t))uu2(t). (3.1)

    So, for uN, by (3.1), we get

    G(u),u=pup+mj=1(Ij(u(tj))u2(tj)+Ij(u(tj))u(tj))T0(f(t,u(t))uu2(t)+f(t,u(t))u(t))dt=mj=1(Ij(u(tj))u2(tj)(p1)Ij(u(tj))u(tj))+T0((p1)f(t,u(t))u(t)f(t,u(t))uu2(t))dt<0. (3.2)

    The formula indicates that N has one C1 structure, which is a Nehari manifold. Here are some necessary lemmas to verify Theorem 2.

    Lemma 10. Let the assumptions given in (H3) and (H4) be satisfied. Additionally, we assume that uN is one critical point of φ|N; then, φ(u)=0. In other words, N is one natural constraint on φ(u).

    Proof. If uN is one critical point of φ|N, there is one Lagrange multiplier λR such that φ(u)=λG(u). Therefore, φ(u),u=λG(u),u=0. Combining with (3.2), we know that λ=0, so φ(u)=0.

    To discuss the critical point of φ|N, let us examine the structure of N.

    Lemma 11. Let the assumptions given in (H0) and (H7) be satisfied. For uEα,p0{0}, there is one unique y=y(u)>0 so that yuN.

    Proof. The first step is to show that there are ρ,σ>0 such that

    φ(u)>0,uBρ(0){0},φ(u)σ,uBρ(0). (3.3)

    It is easy to know that 0 is one strict local minimizer of φ. By (H5), for ε>0, there is δ>0 so that F(t,u)ε|u|p,|u|δ. So, for uEα,p0,u=ρ, uΛu=δ, by (H1), one has

    φ(u)=1pup+mj=1u(tj)0Ij(t)dtT0F(t,u(t))dt1pupT0F(t,u(t))dt1pupεTΛpup.

    Select ε=12pTΛp; one has φ(u)12pup. Let ρ=δΛ and σ=δp2pΛp. Therefore, we can conclude that there are ρ,σ>0 so that, for uBρ{0}, one has φ(u)>0, and for uBρ, one has φ(u)σ.

    Second, we prove that φ(yu) as y. In fact, by (H7), there exist c1,c2>0 so that

    F(t,u)c1|u|θc2,(t,u)[0,T]×R.

    By (H2), we have that φ(yu)yppup+mj=1ajCyu+mj=1djyγj+1Cγj+1γj+1uγj+1c1yθuθLθ+Tc2. Because γj[0,p1),p>1,θ>p, φ(yu), as y. Let g(y):=φ(yu), where y>0. From the above proof, it can be seen that there exists at least one yu=y(u)>0 so that g(yu)=maxy0g(y)=maxy0φ(yu)=φ(yuu). Next, we show that, when y>0, g(y) possesses one unique critical point, which must be the global maximum point. In fact, if y is the critical point of g, then

    g(y)=φ(yu),u=yp1up+mj=1Ij(yu(tj))u(tj)T0f(t,yu(t))u(t)dt=0.

    By (3.1), we obtain

    g(y)=(p1)yp2up+mj=1Ij(yu(tj))u2(tj)T0f(t,yu(t))(yu)u2(t)dt=1y2mj=1(Ij(yu(tj))(yu(tj))2(p1)Ij(yu(tj))yu(tj))+1y2T0((p1)f(t,yu(t))yu(t)f(t,yu(t))yu(yu(t))2)dt<0. (3.4)

    Therefore, if y is one critical point of g, then it must be one strictly local maximum point, and the critical point is unique. In addition, according to

    g(y)=φ(yu),u=1yφ(yu),yu, (3.5)

    if yuN, then y is one critical point of g. Define m=infNφ. By (3.3), we have that minfBρφσ>0.

    Lemma 12. Assume that the conditions (H0) and (H7) hold; then, there is uN so that φ(u)=m.

    Proof. According to the continuity of Ij and f and Lemma 4, it is easy to verify that φ is weakly lower semi-continuous. Let {uk}N be the minimization sequence of φ that satisfies φ(uk)infNφ=m, so

    φ(uk)=m+o(1),G(uk)=0. (3.6)

    Now, we show that {uk} is bounded in Eα,p0. Otherwise, uk as k. For uEα,p0{0}, choose vk=ukuk; then, vk=1. Since Eα,p0 is one reflexive Banach space, there is one subsequence of {vk} (still denoted as {vk}) such that vkv in Eα,p0; then, vkv in C([0,T],R). On the one hand, combining (2.8) and (H2), one has

    T0F(t,uk)dt=1pukp+mj=1uk(tj)0Ij(t)dtφ(uk)1pukp+mj=1ajΛuk+mj=1djΛγj+1γj+1uγj+1+M1,

    where M1>0. Because γj[0,p1),p>1,θ>p, we have that

    T0F(t,uk)ukθdto(1),k. (3.7)

    On the other side, according to the continuity of f, there is M2>0 so that

    |uf(t,u)θF(t,u)|M2,|u|L,t[0,T].

    Combining the condition (H6), we have

    uf(t,u)θF(t,u)M|u|pM2,|u|R,t[0,T]. (3.8)

    Combining the conditions (H1) and (H2), we get

    m+o(1)=φ(uk)=1pukp+mj=1uk(tj)0Ij(t)dtT0F(t,uk(t))dt1pukp1θT0uk(t)f(t,uk(t))dtMθT0|u(t)|pdtM2Tθ(1p1θ)ukp1θmj=1Ij(uk(tj))uk(tj)MTθupM2Tθ(1p1θ)ukp1θmj=1ajuk1θmj=1djukγj+1MTθupM2Tθ.

    This means that there is M3>0 so that limkvk=limkukukM30. Therefore, v0. Let Ω1={t[0,T]:v0} and Ω2=[0,T]Ω1. According to the condition (H7), there exists M4>0 so that F(t,u)0,t[0,T] and |u|M4. Combining with the condition (H5), there exist M5,M6>0 so that F(t,u)M5upM6,t[0,T],uR. According to the Fatou lemma, one has lim infkΩ2F(t,uk)ukθdt>. Combining with the condition (H7), for t[0,T], one has

    lim infkT0F(t,uk)ukθdt=liminfkΩ1F(t,uk)|uk|θ|vk|θdt+liminfkΩ2F(t,uk)|uk|θ|vk|θdt.

    This contradicts (3.7). So, the sequence {uk}kN is bounded. Assume that {uk}kN possesses one subsequence, still recorded as {uk}kN; there exists uEα,p0 so that uku in Eα,p0, so uku in C([0,T],R). For the last step, we show that u0. According to the condition (H5), for ε>0, there exists δ>0 so that

    f(t,u)uε|u|p,(t,u)[0,T]×[δ,δ]. (3.9)

    Suppose that ukδ; for ukN, by (H2) and (3.9), we obtain

    Λpukpukp=T0f(t,uk(t))uk(t)dtmj=1Ij(uk(tj))uk(tj)εTukpmj=1ajukmj=1djukγj+1.

    There is one contradiction in the above formula, so the hypothesis is not valid, namely, u=limkukδ>0, so u0. According to Lemma 11, there is one unique y>0 so that yuN. Because φ is weakly lower semi-continuous,

    mφ(yu)lim_kφ(yuk)limkφ(yuk). (3.10)

    For ukN, by (3.4) and (3.5), we get that yk=1 is one global maximum point of g, so φ(yuk)φ(uk). Combined with (3.10), one has mφ(yu)limkφ(uk)=m. Therefore, m is obtained at yuN.

    The proof process of Theorem 2 is given below.

    Proof of Theorem 2. By Lemmas 11 and 12, we know that there exists uN so that φ(u)=m=infNφ>0, i.e., u is the non-zero critical point of φ|N. By Lemma 10, one has φ(u)=0; thus, u is the non-trivial ground-state solution of Problem (1.1).

    Lemma 13. Let fC([0,T]×R,R) IjC(R,R). Assume that the conditions (H0) and (H8)(H15) hold. Then, φ satisfies the (PS)c condition.

    Proof. Assume that there is the sequence {un}nNEα,p0 so that φ(un)c and φ(un)0 (n); then, there is c1>0 so that, for nN, we have

    |φ(un)|c1,φ(un)(Eα,p0)c1, (3.11)

    where (Eα,p0) is the conjugate space of Eα,p0. Next, let us verify that {un}nN is bounded in Eα,p0. If not, we assume that un+ as (n). Let vn=unun; then, vn=1. Since Eα,p0 is one reflexive Banach space, there is one subsequence of {vn} (still denoted as {vn}), so that vnv (n) in Eα,p0; then, vnv in C([0,T],R). By (H11) and (H12), we get

    |f2(t,u)u|b1(t)|u|r,|F2(t,u)|1rb1(t)|u|r. (3.12)

    Two cases are discussed below.

    Case 1: v0. Let Ω={t[0,T]||v(t)|>0}; then, meas(Ω)>0. Because un+ (n) and |un(t)|=|vn(t)|un, so for tΩ, one has |un(t)|+ (n). On the one side, by (2.6), (2.8), (3.11), (3.12) and (H13), one has

    T0F1(t,un)dt=1punp+mj=1un(tj)0Ij(t)dtT0F2(t,un)dtφ(un)1punp+mj=1ajΛun+mj=1djΛγj+1unγj+1+TrΛrb1unr+c1.

    Since γj[0,θ1),θ>p>r>1,

    limnT0F1(t,un)unθdto(1),n. (3.13)

    On the other side, Fatou's lemma combines with the properties of Ω and (H10), so we get

    limnT0F1(t,un)unθdtlimnΩF1(t,un)unθdt=limnΩF1(t,un)|un(t)|θ|vn(t)|θdt=+.

    This contradicts (3.13).

    Case 2: v0. From (H8), for ε>0, there is L0>0, so that |f1(t,u)|ε|u|p1,|u|L0. So, for |u|L0, there is ε0>0 so that |uf1(t,u)θF1(t,u)|ε0(1+θ)up. For (t,u)[0,T]×[L0,L], there is c2>0 so that |uf1(t,u)θF1(t,u)|c2. Combined with the condition (H9), one has

    uf1(t,u)θF1(t,u)ε0(1+θ)upc2,(t,u)[0,T]×R. (3.14)

    By (H14), we obtain that there exists c3>0, such that

    θmj=1un(tj)0Ij(t)dtmj=1Ij(un(tj))un(tj)c3. (3.15)

    By (2.6), (2.8), (2.9), (3.11), (3.12), (3.14) and (3.15), we get that there exists c4>0 such that

    o(1)=θc1+c1ununpθφ(un)φ(un),ununp=(θp1)+1unp[θmj=1un(tj)0Ij(t)dtmj=1Ij(un(tj))un(tj)]+1unpT0[unf1(t,un)θF1(t,un)]dt+1unpT0[unf2(t,un)θF2(t,un)]dt(θp1)+1unpT0[ε0(1+θ)upnc2]dt1unp(θr+1)T0b1(t)|un|rdt1unpc3(θp1)ε0(1+θ)T0|un|punpdtTc2unp1unp(θr+1)b1L1unr1unpc3(θp1)ε0(1+θ)T0|vn|pdtTc2unp(θr+1)b1L1Λrunrp(θp1),n.

    It is a contradiction. Thus, {un}nN is bounded in Eα,p0. Assume that the sequence {un}nN possesses one subsequence, still recorded as {un}nN; there exists uEα,p0 so that unu in Eα,p0; then, unu in C([0,T],R). Therefore,

    {φ(un)φ(u),unu0,n,T0[f(t,un(t))f(t,u(t))][un(t)u(t)]dt0,n,mj=1(Ij(un(tj))Ij(u(tj)))(un(tj)u(tj))0,n,T0a(t)(ϕp(un(t))ϕp(u(t)))(un(t)u(t))dt0,n. (3.16)

    Through (2.9), we can get

    φ(un)φ(u),unu=T0(ϕp(0Dαtun(t))ϕp(0Dαtu(t)))(0Dαtun(t)0Dαtu(t))dt+T0a(t)(ϕp(un(t))ϕp(u(t)))(un(t)u(t))dt+mj=1(Ij(un(tj))Ij(u(tj)))(un(tj)u(tj))T0[f(t,un(t))f(t,u(t))][un(t)u(t)]dt. (3.17)

    From [23], we obtain

    T0(ϕp(0Dαtun(t))ϕp(0Dαtu(t)))(0Dαtun(t)0Dαtu(t))dt{cT0|0Dαtun(t)0Dαtu(t)|pdt,p2,cT0|0Dαtun(t)0Dαtu(t)|2(|0Dαtun(t)|+|0Dαtu(t)|)2pdt,1<p<2. (3.18)

    If p2, by (3.16)–(3.18), one has unu0 (n). If 1<p<2, by the H¨older inequality, one has T0|0Dαtun(t)0Dαtu(t)|pdtc(T0|0Dαtun(t)0Dαtu(t)|2(|0Dαtun(t)|+|0Dαtu(t)|)2pdt)p2(un+u)p(2p)2. Thus,

    T0(ϕp(0Dαtun(t))ϕp(0Dαtu(t)))(0Dαtun(t)0Dαtu(t))dtc(un+u)2p(T0|0Dαtun(t)0Dαtu(t)|pdt)2p. (3.19)

    By (3.16), (3.17) and (3.19), one has unu0 (n). Hence, φ satisfies the (PS)c condition.

    The proof of Theorem 3.

    Step1. Clearly, φ(0)=0. Lemma 13 implies that φC1(Eα,p0,R) satisfies the (PS)c condition.

    Step2. For ε1>0, we know from (H8) that there is δ>0 so that

    F1(t,u)ε1|u|p,t[0,T],|u|δ. (3.20)

    For uEα,p0, by (2.5), (2.6), (2.8), (3.12) and (H15), we get

    φ(u)=1pup+mj=1u(tj)0Ij(t)dtT0F(t,u(t))dt1pupT0F(t,u(t))dt1pupε1T0|u|pdt1rT0b1(t)|u|rdt1pupε1Λppup1rb1L1ur(1pε1ΛppΛrrb1L1urp)up. (3.21)

    Choose ε1=12pΛpp; one has φ(u)(12pΛrrb1L1urp)up. Let ρ=(r4pΛrb1L1)1rp and η=14pρp; then, for uBρ, we obtain φ(u)η>0.

    Step3. From (H10), for |u|L1, there exist ε2,ε3>0 such that

    F1(t,u)ε2|u|θε3. (3.22)

    By (H8), for |u|L1, there exist ε4,ε5>0 such that

    F1(t,u)ε4upε5. (3.23)

    From (3.22) and (3.23), we obtain that there exist ε6,ε7>0 so that

    F1(t,u)ε2|u|θε6upε7,t[0,T],uR, (3.24)

    where ε6=ε2Lθp1+ε4. For uEα,p0{0} and ξR+, by (H13), (2.5), (2.6), (2.8), (3.24) and the H¨older inequality, we have

    φ(ξu)ξppup+mj=1ajξΛu+mj=1djξγj+1Λγj+1uγj+1ε2ξθT0|u|θdt+ε6Λppξpup+ε7T(1p+ε6Λpp)ξpup+mj=1ajξΛu+mj=1djξγj+1Λγj+1uγj+1ε2ξθ(TpθθT0|u(t)|pdt)θp+ε7T(1p+ε6Λpp)ξpup+mj=1ajξΛu+mj=1djξγj+1Λγj+1uγj+1ε2ξθTpθpuθLp+ε7T.

    Since θ>p>1 and γj+1[1,θ), the above inequality indicates that φ(ξ0u) when ξ0 is large enough. Let e=ξ0u; one has φ(e)<0. Thus, the condition (ii) in Lemma 8 holds. Lemma 8 implies that φ possesses one critical value c(1)η>0. The specific form is c(1)=infgΓmaxs[0,1]φ(g(s)), where Γ={gC([0,1],Eα,p0):g(0)=0,g(1)=e}. Hence, there is 0u(1)Eα,p0 so that

    φ(u(1))=c(1)η>0,φ(u(1))=0. (3.25)

    Step4. Equation (3.21) implies that φ is bounded below in ¯Bρ. Choose σEα,p0 so that σ(t)0 in [0,T]. For l(0,+), by (2.6), (2.8), (H10), (H11) and (H13), we have

    φ(lσ)lppσp+mj=1ajlΛσ+mj=1djlγj+1Λγj+1σγj+1T0F2(t,lσ(t))dtlppσp+mj=1ajlΛσ+mj=1djlγj+1Λγj+1σγj+1lrT0b(t)|σ(t)|rdt. (3.26)

    Thus, from 1<r<p and γj[0,θ1), we know that for a small enough l0 satisfying l0σρ, one has φ(l0σ)<0. Let u=l0σ; we have that c(2)=infφ(u)<0,uρ. Ekeland's variational principle shows that there is one minimization sequence {vk}kN¯Bρ so that φ(vk)c(2) and φ(vk)0,k, i.e., {vk}kN is one (PS)c sequence. Lemma 13 shows that φ satisfies the (PS)c condition. Thus, c(2)<0 is another critical value of φ. So, there exists 0u(2)Eα,p0 so that φ(u(2))=c(2)<0,u(2)<ρ.

    Proof. The functionals Φ:Eα,p0R and Ψ:Eα,p0R are defined as follows:

    Φ(u)=1pup,Ψ(u)=T0F(t,u(t))dtμλnj=1Jj(u(tj));

    then, φ(u)=Φ(u)λΨ(u). We can calculate that

    infuEα,p0Φ(u)=Φ(0)=0,Ψ(0)=T0F(t,0)dtμλnj=1Jj(0)=0.

    Furthermore, Φ and Ψ are continuous Gâteaux differentiable and

    Φ(u),v=T0ϕp(0Dαtu(t))0Dαtv(t)dt+T0a(t)ϕp(u(t))v(t)dt, (3.27)
    Ψ(u),v=T0f(t,u(t))v(t)dtμλnj=1Ij(u(tj))v(tj),u,vEα,p0. (3.28)

    In addition, Φ:Eα,p0(Eα,p0) is continuous. It is proved that Ψ:Eα,p0(Eα,p0) is a continuous compact operator. Suppose that {un}Eα,p0, unu, n; then, {un} uniformly converges to u on C([0,T]). Owing to fC([0,T]×R,R) and IjC(R,R), we have that f(t,un)f(t,u) and Ij(un(tj))Ij(u(tj)), n. Thus, Ψ(un)Ψ(u) as n. Then, Ψ is strongly continuous. According to Proposition 26.2 in [24], Ψ is one compact operator. It is proved that Φ is weakly semi-continuous. Suppose that {un}Eα,p0, {un}u; then, {un}u on C([0,T]), and liminfnunu. So, liminfnΦ(un)=liminfn(1punp)1pup=Φ(u). Thus, Φ is weakly semi-continuous. Because Φ(u)=1pup+ and u+, Φ is coercive. By (3.27), we obtain

    Φ(u)Φ(v),uv=T0(ϕp(0Dαtu(t))ϕp(0Dαtv(t)))(0Dαtu(t)0Dαtv(t))dt+T0a(t)(ϕp(u(t))ϕp(v(t)))(u(t)v(t))dt,u,vEα,p0.

    From [23], we know that there is c>0 so that

    T0(ϕp(0Dαtu(t))ϕp(0Dαtv(t)))(0Dαtu(t)0Dαtv(t))dt{cT0|0Dαtu(t)0Dαtv(t)|pdt,p2,cT0|0Dαtu(t)0Dαtv(t)|2(|0Dαtu(t)|+|0Dαtv(t)|)2pdt,1<p<2. (3.29)

    If p2, then Φ(u)Φ(v),uvcuvp. Thus, Φ is uniformly monotonous. When 1<p<2, the H¨older inequality implies

    T0|0Dαtu(t)0Dαtv(t)|pdt(T0|0Dαtu(t)0Dαtv(t)|2(|0Dαtu(t)|+|0Dαtv(t)|)2pdt)p2 (T0(|0Dαtu(t)|+|0Dαtv(t)|)pdt)2p2c1(T0|0Dαtu(t)0Dαtv(t)|2(|0Dαtu(t)|+|0Dαtv(t)|)2pdt)p2(up+vp)2p2,

    where c1=2(p1)(2p)2>0. Then,

    T0|0Dαtu(t)0Dαtv(t)|2(|0Dαtu(t)|+|0Dαtv(t)|)2pdtc2(u+v)2p(T0|0Dαtu(t)0Dαtv(t)|pdt)2p,

    where c2=1c2p1. Combined with (3.29), we can get

    T0(ϕp(0Dαtu(t))ϕp(0Dαtv(t)))(0Dαtu(t)0Dαtv(t))dtc(u+v)2p(T0|0Dαtu(t)0Dαtv(t)|pdt)2p. (3.30)

    Thus, Φ(u)Φ(v),uvcuv2(u+v)2p. So, Φ is strictly monotonous. Theorem 26.A(d) in [24] implies that (Φ)1 exists and is continuous. If xEα,p0 satisfies Φ(x)=1pxpr, then, by (2.6), we obtain Φ(x)1pΛpxp, and

    {xEα,p0:Φ(x)r}{x:1pΛpxpr}={x:xpprΛp}={x:xΛ(pr)1p}.

    Therefore, from λ>0 and μ0, we have

    sup{Ψ(x):Φ(x)r}=sup{T0F(t,x(t))dtμλnj=1Jj(x(tj)):Φ(x)r}T0max|x|Λ(pr)1pF(t,x)dt+μλmax|x|Λ(pr)1pnj=1(Jj(x)).

    If max|x|Λ(pr)1pnj=1(Jj(x))=0, by λ<Ar, we get

    sup{Ψ(x):Φ(x)r}<rλ. (3.31)

    If max|x|Λ(pr)1pnj=1(Jj(x))>0, for μ[0,γ), γ=min{rλT0max|x|Λ(pr)1/1ppF(t,x)dtmax|x|Λ(pr)1pnj=1(Jj(x)),λT0F(t,ω)dt1pωpnj=1Jj(ω(tj))}, we have

    sup{Ψ(x):Φ(x)r}T0max|x|Λ(pr)1pF(t,x)dt+μλmax|x|Λ(pr)1pnj=1(Jj(x))<T0max|x|Λ(pr)1pF(t,x)dt+rλT0max|x|Λ(pr)1pF(t,x)dtmax|x|Λ(pr)1pnj=1(Jj(x))λ×max|x|Λ(pr)1pnj=1(Jj(x))<T0max|x|Λ(pr)1pF(t,x)dt+rλT0max|x|Λ(pr)1pF(t,x)dtλ=T0max|x|Λ(pr)1pF(t,x)dt+rλT0max|x|Λ(pr)1pF(t,x)dt=rλ.

    Thus, (3.31) is also true. On the other side, for μ<γ, one has

    Ψ(ω)=T0F(t,ω(t))dtμλnj=1Jj(ω(tj))>Φ(ω)λ. (3.32)

    By combining (3.31) and (3.32), we obtain that Ψ(ω)Φ(ω)>1λ>sup{Ψ(x):Φ(x)r}r. This means that the condition (i) of Lemma 9 holds.

    Finally, for the third step, we show that, for any λΛr=(Al,Ar), the functional ΦλΨ is coercive. By (1.4), we obtain

    T0F(t,x(t))dtLT0(1+|x(t)|β)dtLT+LTxβLT+LTΛβxβ,xEα,p0. (3.33)

    and

    Jj(x(tj))Lj(1+|x(tj)|dj)Lj(1+xdj)Lj(1+Λdjxdj). (3.34)

    By (3.34), we get

    nj=1(Jj(x(tj)))nj=1Lj(1+Λdjxdj). (3.35)

    If μλ0, for xEα,p0, by (3.33) and (3.35), we have

    Ψ(x)LT+LTΛβxβ+μλnj=1Lj(1+Λdjxdj)=LT+μλnj=1Lj+LTΛβxβ+μλnj=1LjΛdjxdj.

    Thus, Φ(x)λΨ(x)1pxpλ(LT+μλnj=1Lj+LTΛβxβ+μλnj=1LjΛdjxdj),xEα,p0. If 0<β and dj<p, then limx+(Φ(x)λΨ(x))=+,λ>0. Thus, ΦλΨ is coercive. When β=p, Φ(x)λΨ(x)(1pλLTΛp)xpλ(LT+μλnj=1Lj+μλnj=1LjΛdjxdj). Choose L<T0max|x|Λ(pr)1/1ppF(t,x)dtprTΛp. We have that1pλLTΛp>0, for all λ<Ar. If 0<dj<p, we have thatlimx+(Φ(x)λΨ(x))=+, for all λ<Ar. Obviously, the functional ΦλΨ is coercive. Lemma 9 shows that φ=ΦλΨ possesses at least three different critical points in Eα,p0.

    This paper studies the solvability of Dirichlet boundary-value problems of the fractional p-Laplacian equation with impulsive effects. For this kind of problems, the existence of solutions has been discussed in the past, while the ground-state solutions have been rarely studied. By applying the Nehari manifold method, we have obtained the existence result of the ground-state solution (see Theorem 2). At the same time, by the mountain pass theorem and three critical points theorem, some new existence results on this problem were achieved (see Theorems 3–5). In particular, this paper weakens the commonly used p-suplinear and p-sublinear growth conditions, to a certain extent, and expands and enriches the results of [14,15,16]. This theory can provide a solid foundation for studying similar fractional impulsive differential equation problems. For example, one can consider the solvability of Sturm-Liouville boundary-value problems of fractional impulsive equations with the p-Laplacian operator. In addition, the proposed theory can also be used to study the existence of solutions to the periodic boundary-value problems of the fractional p-Laplacian equation with impulsive effects and their corresponding coupling systems.

    This research was funded by the Natural Science Foundation of Xinjiang Uygur Autonomous Region (Grant No. 2021D01A65, 2021D01B35), Natural Science Foundation of Colleges and Universities in Xinjiang Uygur Autonomous Region (Grant No. XJEDU2021Y048) and Doctoral Initiation Fund of Xinjiang Institute of Engineering (Grant No. 2020xgy012302).

    The authors declare that there is no conflict of interest.

    Proof of Theorem 5

    Proof. This is similar to the proof process of Theorem 4. Since λ>0, μ(γ,0], one has

    sup{Ψ(x):Φ(x)r}=sup{T0F(t,x(t))dtμλnj=1Jj(x(tj)):Φ(x)r}T0max|x|Λ(pr)1pF(t,x)dtμλmax|x|Λ(pr)1pnj=1Jj(x).

    If max|x|Λ(pr)1pnj=1Jj(x)=0, by λ<Ar, we obtain

    sup{Ψ(x):Φ(x)r}<rλ. (8.1)

    For μ(γ,0], if max|x|Λ(pr)1pnj=1Jj(x)>0, then (A.1) is also true. On the other hand, for μ(γ,0], we have

    Ψ(ω)=T0F(t,ω(t))dtμλnj=1Jj(ω(tj))>Φ(ω)λ. (8.2)

    Combining (A.1) and (A.2), we get Ψ(ω)Φ(ω)>1λ>sup{Ψ(x):Φ(x)r}r, which shows that the condition (i) of Lemma 9 holds. Finally, we show that ΦλΨ is coercive for λΛr=(Al,Ar). For xEα,p0, by (1.7), we get

    T0F(t,x(t))dtLT+LTΛβxβ,Jj(x(tj))Lj(1+Λdjxdj). (8.3)

    So,

    nj=1Jj(x(tj))nj=1Lj(1+Λdjxdj). (8.4)

    For xEα,p0, if μλ0, then, by (A.3) and (A.4), we have

    Ψ(x)LT+LTΛβxβμλnj=1Lj(1+Λdjxdj)=LTμλnj=1Lj+LTΛβxβμλnj=1LjΛdjxdj.

    Thus, for xEα,p0, we get

    Φ(x)λΨ(x)1pxpλ(LTμλnj=1Lj+LTΛβxβμλnj=1LjΛdjxdj).

    If 0<β and dj<p, then limx+(Φ(x)λΨ(x))=+,λ>0. Thus, ΦλΨ is coercive. When β=p, Φ(x)λΨ(x)(1pλLTΛp)xpλ(LTμλnj=1Ljμλnj=1LjΛdjxdj). Choose L<T0max|x|Λ(pr)1/1ppF(t,x)dtprTΛp. We have that 1pλLTΛp>0 for λ<Ar. If 0<dj<p for all λ<Ar, one has limx+(Φ(x)λΨ(x))=+. Obviously, the functional ΦλΨ is coercive. Lemma 9 shows that φ=ΦλΨ possesses at least three different critical points in Eα,p0.

    [1] Li S, Qiu W, Zhang X, et al. (2016) A high-performance DNA biosensor based on the assembly of gold nanoparticles on the terminal of hairpin-structured probe DNA. Sens Actuat B Chem 223: 861-867.
    [2] Gudnason H, Dufva M, Duong BD, et al. (2008) An inexpensive and simple method for thermally stable immobilization of DNA on an unmodified glass surface: UV linking of poly(T)10-poly(C)10-tagged DNA probes. Biotechniques 45: 261-271. doi: 10.2144/000112905
    [3] Sakata T, Kamahori M, and Miyahara Y. (2004) Immobilization of oligonucleotide probes on Si3N4 surface and its application to genetic field effect transistor. Mat Sci Eng C Bio S 24: 827-832. doi: 10.1016/j.msec.2004.08.042
    [4] Kimura N (2006) One-step immobilization of poly(dT)-modified DNA onto non-modified plastic substrates by UV irradiation for microarrays. Biochem Biophys Res Commun 347: 477-484. doi: 10.1016/j.bbrc.2006.06.130
    [5] Kimura N, Oda R, Inaki Y, et al. (2004) Attachment of oligonucleotide probes to poly carbodiimide-coated glass for microarray applications. Nucl Ac Res 32: e68. doi: 10.1093/nar/gnh057
    [6] Zagorodko O, Spadavecchia J, Yanguas SA, et al. (2014) Highly sensitive detection of DNA hybridization on commercialized graphene-coated surface plasmon resonance interfaces. Anal Chem 86: 11211-11216.
    [7] Rodriguez LD, Gonzalez GP, Gattuso A, et al. (2014) Reducing time in the analysis of listeria monocytogenes in meat, dairy and vegetable products. Int J Food Microbiol 184: 98-105. doi: 10.1016/j.ijfoodmicro.2014.03.006
    [8] Abdalhai MH, Fernandes AM, Bashari M, et al. (2014) Rapid and sensitive detection of foodborne pathogenic bacteria (staphylococcus aureus) using an electrochemical DNA genomic biosensor and its application in fresh beef. J Agric Food Chem 62: 12659-12667. doi: 10.1021/jf503914f
    [9] Khemthongcharoen N, Wonglumsom W, Suppat A, et al. (2015) Piezoresistive microcantilever-based DNA sensor for sensitive detection of pathogenic vibrio cholerae O1 in food sample. Biosens Bioelectron 63: 347-353. doi: 10.1016/j.bios.2014.07.068
    [10] Stuken A, Dittami SM, Eikrem W, et al. (2013) Novel hydrolysis-probe based qPCR assay to detect saxitoxin transcripts of dinoflagellates in environmental samples. Harmful Algae 28: 108-117. doi: 10.1016/j.hal.2013.06.003
    [11] Ryu H, Henson M, Elk M, et al. (2013) Development of quantitative PCR assays targeting the 16S rRNA genes of enterococcus spp. and their application to the identification of enterococcus species in environmental samples. Appl Environ Microbiol 79: 196-204.
    [12] Khatera M, de Escosura MA, and Merkoçia A. (2016) Biosensors for plant pathogen detection. Biosens Bioelectron, In press.
    [13] Huang H, Bai W, Dong C, et al. (2015) An ultrasensitive electrochemical DNA biosensor based on graphene/Au nanorod/polythionine for human papillomavirus DNA detection. Biosens Bioelectron 68: 442-446. doi: 10.1016/j.bios.2015.01.039
    [14] Znazen A, Sellam H, Elleuch E, et al. (2015) Comparison of two quantitative real time PCR assays for rickettsia detection in patients from tunisia. Plos Negl Trop Dis 9: e0003487. doi: 10.1371/journal.pntd.0003487
    [15] Essaidi LM, Lyon M, Mamin A, et al. (2016) A new real-time RT-qPCR assay for the detection, subtyping and quantification of human respiratory syncytial viruses positive- and negative-sense RNAs. J Virol Meth 235: 9-14.
    [16] Gokcea G, Erdem A, Ceylana C, et al. (2016) Voltammetric detection of sequence-selective DNA hybridization related to toxoplasma gondii in PCR amplicons. Talanta 149: 244-249. doi: 10.1016/j.talanta.2015.11.071
    [17] Knapp J, Millon L, Mouzo L, et al. (2014) Real time PCR to detect the environmental faecal contamination by echinococcus multilocularis from red fox stools. Vet Paras 201: 40-47. doi: 10.1016/j.vetpar.2013.12.023
    [18] Chena Q, Zhang L, Jiang F, et al. (2017) MnO2 microsphere absorbing Cy5-labeled single strand DNA probe serving as powerful biosensor for effective detection of mycoplasma ovipneumoniae. Sens Actuat B Chem 244: 1138-1144. doi: 10.1016/j.snb.2017.01.104
    [19] Zhu D, Yan Y, Lei P, et al. (2014) A novel electrochemical sensing strategy for rapid and ultrasensitive detection of salmonella by rolling circle amplification and DNA-AuNPs probe. Anal Chim Acta 846: 44-50. doi: 10.1016/j.aca.2014.07.024
    [20] Blagden T, Schneider W, Melcher U, et al. (2016) Adaptation and validation of e-probe diagnostic nucleic acid analysis for detection of escherichia coli O157:H7 in metagenomic data from complex food matrices. J Food Prot 79: 574-581. doi: 10.4315/0362-028X.JFP-15-440
    [21] Kashisha, Soni DK, Mishra SK, et al. (2015) Label-free impedimetric detection of listeria monocytogenes based on poly-5-carboxy indole modified ssDNA probe. J Biotechnol 200: 70-76. doi: 10.1016/j.jbiotec.2015.02.025
    [22] Vasavirama K (2013) Molecular probes and their application. Int J Lifesc Bt Pharm Res 2: 32-42.
    [23] Manzano M and Iacumin L. (2015) Molecular techniques in food microbiology, in current applications of biotechnology, EZ. Dundar M, Bruschi F, Deeni Y, et al., chapter 2, Erciyes, Turkey, 9-23.
    [24] Zhao X, Lin CW, Wang J, et al. (2014) Advances in rapid detection methods for foodborne pathogens. J Microbiol Biotechnol 24: 297-312. doi: 10.4014/jmb.1310.10013
    [25] Manzano M, Cocolin L, Cantoni C, et al. (1997) Detection and identification of listeria monocytogenes from milk and cheese by a single step PCR. Mol Biotechnol 7: 85-88. doi: 10.1007/BF02821546
    [26] Aznar R and Alarcòn B. (2003) PCR detection of listeria monocytogenes: a study of multiple factors affecting sensitivity. J Appl Microbiol 95: 958-966. doi: 10.1046/j.1365-2672.2003.02066.x
    [27] Malorny B, Hoorfar J, Hugas M, et al. (2003) Interlaboratory diagnostic accuracy of a salmonella specific PCR-based method. Int J Food Microbiol 89: 241-249. doi: 10.1016/S0168-1605(03)00154-5
    [28] Yamamoto Y (2002) PCR in diagnosis of infection: detection of bacteria in cerebrospinal fluids. Clin Diagn Lab Immunol 9: 508-514.
    [29] Al Dragy WA and Baqer AA. (2014) Detection of escherichia coli O157:H7 in human patients stool and food by using multiplex PCR assays targeting the rfbE and the eaea genes compared with detection by biochemical test and serological assay. JNUS 17: 124-131.
    [30] Juskowiak B (2011) Nucleic acid-based fluorescent probes and their analytical potential. Anal Bioanal Chem 399: 3157-3176. doi: 10.1007/s00216-010-4304-5
    [31] de Boer P, Rahaoui H, Leer RJ, et al. (2015) Real-time PCR detection of campylobacter spp.: a comparison to classic culturing and enrichment. Food Microbiol 51: 96-100.
    [32] Gianfranceschi MV, Rodriguez LD, Hernandez M, et al. (2014) European validation of a real-time PCR-based method for detection of listeria monocytogenes in soft cheese. Int J Food Microbiol 184: 128-133. doi: 10.1016/j.ijfoodmicro.2013.12.021
    [33] Gattuso A, Gianfranceschi MV, Sonnessa M, et al. (2014) Optimization of a real time PCR based method for the detection of listeria monocytogenes in pork meat. Int J Food Microbiol 184: 106-108. doi: 10.1016/j.ijfoodmicro.2014.04.015
    [34] Rodriguez LD, Pla M, Scortti M, et al. (2005) A novel real-time PCR for listeria monocytogenes that monitors analytical performance via an internal amplification control. Appl Environ Microbiol 71: 9008-9012. doi: 10.1128/AEM.71.12.9008-9012.2005
    [35] Brooks JP, McLaughlin MR, Adeli A, et al (2016) Cultivation and qPCR detection of pathogenic and adminntibiotic resistant bacterial establishment in naive broiler houses. J Environ Qual 45: 958-966. doi: 10.2134/jeq2015.09.0492
    [36] Vendrame M, Iacumin L, Manzano M, et al. (2013) Use of propidium monoazide for the enumeration of viable oenococcus oeni in must and wine by quantitative PCR. Food Microbiol 35: 49-57. doi: 10.1016/j.fm.2013.02.007
    [37] Minguzzi S, Terlizzi F, Lanzoni C, et al. (2016) A rapid protocol of crude RNA/DNA extraction for RT-qPCR detection and quantification of 'candidatus phytoplasma prunorum'. Plos One 11: e0146515. doi: 10.1371/journal.pone.0146515
    [38] Calgua B, Rodriguez MJ, Hundesa A, et al. (2013) New methods for the concentration of viruses from urban sewage using quantitative PCR. J Virol Meth 187: 215-221. doi: 10.1016/j.jviromet.2012.10.012
    [39] Gimenez A, Clemente CP, Calgua B, et al. (2009) Comparison of methods for concentrating human adenoviruses, polyomavirus JC and noroviruses in source waters and drinking water using quantitative PCR. J Virol Meth 158: 104-109. doi: 10.1016/j.jviromet.2009.02.004
    [40] de Keuckelaere A, Baert L, Duarte A, et al. (2013) Evaluation of viral concentration methods from irrigation and processing water. J Virol Meth 187: 294-303. doi: 10.1016/j.jviromet.2012.11.028
    [41] Faye O, Diallo D, Diallo M, et al. (2013) Quantitative real-time PCR detection of zika virus and evaluation with field-caught mosquitoes. Virol J 10: 1-8. doi: 10.1186/1743-422X-10-1
    [42] Keller G and Manak MM. (1989) DNA probes. Stokton, New York.
    [43] Komminoth P (1992) Digoxigenin as an alternative probe labeling for in situ hybridization. Diagn Mol Pathol 1: 142-150. doi: 10.1097/00019606-199206000-00008
    [44] Musiani M, Venturoli S, Gallinella G, et al. (2007) Qualitative PCR-ELISA protocol for the detection and typing of viral genomes. Nat Prot 2: 2502-2510. doi: 10.1038/nprot.2007.311
    [45] Sue MJ, Yeap SK, Omar AR, et al. (2014). Application of PCR-ELISA in molecular diagnosis. Biomed Res Int 653014: 1-6.
    [46] Laoboonchai A, Kawamoto F, Thanoosingha N, et al. (2001) PCR-based ELISA technique for malaria diagnosis of specimens from Thailand. Trop Med Int Health 6: 458-462. doi: 10.1046/j.1365-3156.2001.00736.x
    [47] Kobets T, Badalov´a J, Grekov I, et al. (2010) Leishmania parasite detection and quantification using PCR-ELISA. Nat Prot 5: 1074-1080. doi: 10.1038/nprot.2010.68
    [48] Ziyaeyan M, Sabahi F, Alborzi A, et al. (2008) Quantification of human cytomegalovirus DNA by a new capture hybrid polymerase chain reaction enzyme-linked immunosorbent assay in plasma and peripheral bloodmononuclear cells of bonemarrow transplant recipients. Exp Clin Transplant 6: 294-300.
    [49] Peterson AW, Heaton RI, and Georgiadis RM. (2001) The effect of surface probe density on DNA hybridization. Nucl Acid Res 29: 5163-5168. doi: 10.1093/nar/29.24.5163
    [50] Xing JM, Zhang S, Du Y, et al. (2009) Rapid detection of intestinal pathogens in fecal samples by an improved reverse dot blot method. World J Gastroenterol 15: 2537-2542. doi: 10.3748/wjg.15.2537
    [51] Cecchini F, Iacumin L, Fontanot M, et al. (2012) Identification of the unculturable bacteria candidatus arthromitus in the intestinal content of trouts using dot blot and southern blot techniques. Vet Microbiol 156: 384-394.
    [52] Chen GF, Zhang CY, Wang YY, et al. (2015) Application of reverse dot blot hybridization to simultaneous detection and identification of harmful algae. Environ Sci Pollut Res Int 22: 1-13. doi: 10.1007/s11356-014-3220-1
    [53] Nestorova GG, Adapa BS, Kopparthy VL, et al. (2016) Lab-on-a-chip thermoelectric DNA biosensor for label-free detection of nucleic acid sequences. Sens Actuat B Chem 225: 174-180. doi: 10.1016/j.snb.2015.11.032
    [54] Wijesuriya D, Breslin K, Anderson G, et al. (1994). Regeneration of immobilized antibodies on fiber optic probes. Biosens Bioelectron 9: 585-592. doi: 10.1016/0956-5663(94)80051-0
    [55] Ferguson JA, Steemers FJ, and Walt DR. (2000). High-density fiber-optic DNA random microsphere array. Anal Chem 72: 5618-5624. doi: 10.1021/ac0008284
    [56] Almadidy A, Watterson J, Piunno P, et al. (2003) A fibre-optic biosensor for detection of microbial contamination, Can J Chem 81: 339-349.
    [57] Cecchini F, Manzano M, Yohai MY, et al. (2012) Chemiluminescent DNA optical fibre sensor for brettanomyces bruxellensis detection. J Biotechnol 157: 25-30. doi: 10.1016/j.jbiotec.2011.10.004
    [58] Yin MJ, Wu C, Shao LY, et al. (2013) Label-free, disposable fiber-optic biosensors for DNA hybridization detection. Analyst 138: 1988-1994. doi: 10.1039/c3an36791f
    [59] Yoo SY, Kim DK, Park TJ, et al. (2010) Detection of the most common corneal dystrophies caused by BIGH3 gene point mutations using a multispot gold-capped nanoparticle array chip. Anal Chem 82: 1349-1357. doi: 10.1021/ac902410z
    [60] Endo T, Kerman K, Nagatani N, et al. (2005) Label-free detection of peptide nucleic Admincid-DNA hybridization using localized surface plasmon resonance based optical biosensor. Anal Chem 77: 6976-6984. doi: 10.1021/ac0513459
    [61] Cheng XR, Hau BY, Endo T, et al. (2014) Au nanoparticle-modified DNA sensor based on simultaneous electrochemical impedance spectroscopy and localized surface plasmon resonance. Biosens Bioelectron 53: 513-518. doi: 10.1016/j.bios.2013.10.003
    [62] Qu J, Wu L, Liu H, et al. (2015) A novel electrochemical biosensor based on DNA for rapid and selective detection of cadmium. Int J Electrochem Sci 10: 4020-4028.
    [63] Patel MK, Solanki PR, Seth S, et al. (2009) CtrA gene based electrochemical DNA sensor for detection of meningitis. Electrochem Commun 11: 969-973. doi: 10.1016/j.elecom.2009.02.037
    [64] Li F, Chen W, and Zhang S. (2008) Development of DNA electrochemical biosensor based on covalent immobilization of probe DNA by direct coupling of sol-gel and self-assembly technologies. Biosens Bioelectron 24: 787-792. doi: 10.1016/j.bios.2008.06.047
    [65] Li F, Chen W, Dong P, et al. (2009) A simple strategy of probe DNA immobilization by diazotization-coupling on self-assembled 4-aminothiophenol for DNA electrochemical biosensor. Biosens Bioelectron 24: 2160-2164. doi: 10.1016/j.bios.2008.11.017
    [66] Chen KI, Li BR, and Chen YT. (2011) Silicon nanowire field-effect transistor-based biosensors for biomedical diagnosis and cellular recording investigation. Nano Today 6: 131-154. doi: 10.1016/j.nantod.2011.02.001
    [67] Lin CH, Hung CH, Hsiao CY, et al. (2009) Poly-silicon nanowire field-effect transistor for ultrasensitive and label-free detection of pathogenic DNA. Biosens Bioelectron 24: 3019-3024. doi: 10.1016/j.bios.2009.03.014
    [68] Rahman SFA, Yusofa NA, Hashim U, et al. (2016) Enhanced sensing of dengue virus DNA detection using O2 plasma treated-silicon nanowire based electrical biosensor. Anal Chim Acta 942: 74-85. doi: 10.1016/j.aca.2016.09.009
    [69] Gao AR, Lu N, Dai PF, et al. (2011) Silicon-nanowire-based CMOS-compatible field-effect transistor nanosensors for ultrasensitive electrical detection of nucleic acids. Nano Lett 11: 3974-3978. doi: 10.1021/nl202303y
    [70] Zhang GJ and Ning Y. (2012) Silicon nanowire biosensor and its applications in disease diagnostics: a review. Anal Chim Acta 749:1-15. doi: 10.1016/j.aca.2012.08.035
    [71] Dell'Atti D, Zavaglia M, Tombelli S, et al. (2007) Development of combined DNA-based piezoelectric biosensors for the simultaneous detection and genotyping of high risk human papilloma virus strains. Clin Chim Acta 383: 140-146. doi: 10.1016/j.cca.2007.05.009
    [72] Hao RZ, Songc HB, Zuo GM, et al. (2011) DNA probe functionalized QCM biosensor based on gold nanoparticle amplification for bacillus anthracis detection. Biosens Bioelectron 26: 3398-3404. doi: 10.1016/j.bios.2011.01.010
    [73] Moa XT, Zhoub YP, Leia H, et al. (2002) Microbalance-DNA probe method for the detection of specific bacteria in water. Enz Microb Technol 30: 583-589. doi: 10.1016/S0141-0229(01)00484-7
    [74] Skládal P, Dos SRC, Yamanaka H, et al. (2004) Piezoelectric biosensors for real-time monitoring of hybridization and detection of hepatitis C virus. J Virol Meth 117: 145-151. doi: 10.1016/j.jviromet.2004.01.005
    [75] Vizzini P (2013) QCM technique optimization for detection of brettanomyces bruxellensis. personal communication. University of Udine, Udine, Italy.
    [76] Yo Y, Moreira BG, Behlke MA, et al. (2006) Design of LNA probes that improve mismatch discrimination. Nucl Ac Res 34: e60. doi: 10.1093/nar/gkl175
    [77] Johnson MP, Haupt LM, and Griffths LR. (2004) Locked nucleic acid (LNA) single nucleotide polymorphism (SNP) genotype analysis and validation using real-time PCR. Nucl Ac Res 32: e55. doi: 10.1093/nar/gnh046
    [78] Yoon JH, Nam JS, Kim KJ, et al. (2013) Simple and rapid discrimination of embB codon 306 mutations in mycobacterium tuberculosis clinical isolates by a real-time PCR assay using an LNA-TaqMan probe. J Microbiol Meth 92: 301-306. doi: 10.1016/j.mimet.2012.12.014
    [79] Priya NG, Pandey N, and Rajagopal R. (2012) LNA probes substantially improve the detection of bacterial endosymbionts in whole mount of insects by fluorescent in-situ hybridization. BMC Microbiol 24: 12-18.
    [80] Wang Q, Wang X, Zhang J, et al. (2012) LNA real-time PCR probe quantification of hepatitis B virus DNA. Exper Therap Medic 3: 503-508.
    [81] Nitecki SS, Teape N, Carney BF, et al. (2015) A duplex qPCR for the simultaneous detection of escherichia coli O157:H7 and listeria monocytogenes using LNA probes. Lett Appl Microbiol 61: 20-27. doi: 10.1111/lam.12427
    [82] Priya NG, Pandey N, and Rajagopal R. (2012) LNA probes substantially improve the detection of bacterial endosymbionts in whole mount of insects by fluorescent in-situ hybridization. BMC Microbiol 12: 1-9. doi: 10.1186/1471-2180-12-1
    [83] Rohdea A, Hammerla JA, Appela B, et al. (2017) Differential detection of pathogenic yersinia spp. by fluorescence in situ hybridization. Food Microbiol 62: 39-45.
    [84] Rohdea A, Hammerl JA, and Dahouk SA. (2016) Detection of foodborne bacterial zoonoses by fluorescence in situ hybridization. Food Control 69: 297-305. doi: 10.1016/j.foodcont.2016.05.008
    [85] Fontenete S, Guimarães N, Leite M, et al. (2013) Hybridization-based detection of helicobacter pylori at human body temperature using advanced locked nucleic acid (LNA) Probes. Plos One 8: e81230. doi: 10.1371/journal.pone.0081230
    [86] Santos RS, Guimarães N, Madureira P, et al. (2014) Peptide nucleic acids with a structurally biased backbone: effects of conformational constraints and stereochemistry. J Biotechnol 187: 16-24. doi: 10.1016/j.jbiotec.2014.06.023
    [87] Stone NRH, Gorton RL, Barker K, et al. (2013) Evaluation of PNA-fish yeast traffic light for rapid identification of yeast directly from positive blood Cultures and adminssessment of clinical impact. J Clin Microbiol 51: 1301-1302. doi: 10.1128/JCM.00028-13
    [88] Ahn JJ, Lee SY, Hong JY, et al. (2015) Application of fluorescence melting curve analysis for dual DNA detection using single peptide nucleic acid probe. Biotechnol Prog 31: 730-735. doi: 10.1002/btpr.2054
    [89] Vilaivan T (2015) Pyrrolidinyl PNA with α/β-dipeptide backbone: from development to applications. Acc Chem Res 48: 1645-1656. doi: 10.1021/acs.accounts.5b00080
    [90] Teengam P, Siangproh W, Tuantranont A, et al. (2017) Electrochemical paper-based peptide nucleic acid biosensor for detecting human papillomavirus. Anal Chim Acta 952: 32-40. doi: 10.1016/j.aca.2016.11.071
  • This article has been cited by:

    1. Huiping Zhang, Wangjin Yao, Three solutions for a three-point boundary value problem with instantaneous and non-instantaneous impulses, 2023, 8, 2473-6988, 21312, 10.3934/math.20231086
  • Reader Comments
  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(12276) PDF downloads(1494) Cited by(5)

Figures and Tables

Figures(7)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog