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Detecting phase transitions in collective behavior using manifold's curvature

  • Received: 23 September 2015 Revised: 19 July 2016 Published: 01 April 2017
  • MSC : Primary: 53C15, 53C21; Secondary: 58D15

  • If a given behavior of a multi-agent system restricts the phase variable to an invariant manifold, then we define a phase transition as a change of physical characteristics such as speed, coordination, and structure. We define such a phase transition as splitting an underlying manifold into two sub-manifolds with distinct dimensionalities around the singularity where the phase transition physically exists. Here, we propose a method of detecting phase transitions and splitting the manifold into phase transitions free sub-manifolds. Therein, we firstly utilize a relationship between curvature and singular value ratio of points sampled in a curve, and then extend the assertion into higher-dimensions using the shape operator. Secondly, we attest that the same phase transition can also be approximated by singular value ratios computed locally over the data in a neighborhood on the manifold. We validate the Phase Transition Detection (PTD) method using one particle simulation and three real world examples.

    Citation: Kelum Gajamannage, Erik M. Bollt. Detecting phase transitions in collective behavior using manifold's curvature[J]. Mathematical Biosciences and Engineering, 2017, 14(2): 437-453. doi: 10.3934/mbe.2017027

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  • If a given behavior of a multi-agent system restricts the phase variable to an invariant manifold, then we define a phase transition as a change of physical characteristics such as speed, coordination, and structure. We define such a phase transition as splitting an underlying manifold into two sub-manifolds with distinct dimensionalities around the singularity where the phase transition physically exists. Here, we propose a method of detecting phase transitions and splitting the manifold into phase transitions free sub-manifolds. Therein, we firstly utilize a relationship between curvature and singular value ratio of points sampled in a curve, and then extend the assertion into higher-dimensions using the shape operator. Secondly, we attest that the same phase transition can also be approximated by singular value ratios computed locally over the data in a neighborhood on the manifold. We validate the Phase Transition Detection (PTD) method using one particle simulation and three real world examples.


    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 M_3 > 0 so that \mathop {\lim }\limits_{k \to \infty } {\left\| {{v_k}} \right\|_\infty } = \mathop {\lim }\limits_{k \to \infty } \frac{{{{\left\| {{u_k}} \right\|}_\infty }}}{{{{\left\| {{u_k}} \right\|}}}} \ge {M_3} \ge 0. Therefore, v\neq 0 . Let {\Omega _1} = \left\{ {t \in [0, T]:v \ne 0} \right\} and {\Omega _2} = [0, T]\backslash {\Omega _1}. According to the condition ({H_7}) , there exists M_4 > 0 so that F(t, u) \ge 0, \; \forall t \in [0, T] and \left| u \right| \ge {M_4}. Combining with the condition ({H_5}) , there exist M_5, M_6 > 0 so that F(t, u) \ge - {M_5}{u^p} - {M_6}, \; \forall t \in [0, T], \; u \in \mathbb{R}. According to the Fatou lemma, one has \mathop {\liminf}\limits_{k \to \infty } \int_{{\Omega _2}} {\frac{{F(t, {u_k})}}{{\left\| {{u_k}} \right\|^\theta }}} dt > - \infty. Combining with the condition ({H_7}) , for t\in [0, T] , one has

    \begin{equation*} \mathop {\liminf}\limits_{k \to \infty } \int_0^T {\frac{{F(t, {u_k})}}{{\left\| {{u_k}} \right\|^\theta }}dt} = \mathop {\lim \inf }\limits_{k \to \infty } \int_{{\Omega _1}} {\frac{{F(t, {u_k})}}{{{{\left| {{u_k}} \right|}^\theta }}}{{\left| {{v_k}} \right|}^\theta }dt + } \mathop {\lim \inf }\limits_{k \to \infty } \int_{{\Omega _2}} {\frac{{F(t, {u_k})}}{{{{\left| {{u_k}} \right|}^\theta }}}{{\left| {{v_k}} \right|}^\theta }dt} \to \infty . \end{equation*}

    This contradicts (3.7). So, the sequence {\left\{ {{u_k}} \right\}_{k \in \mathbb{N}}} is bounded. Assume that {\left\{ {{u_k}} \right\}_{k \in \mathbb{N}}} possesses one subsequence, still recorded as {\left\{ {{u_k}} \right\}_{k \in \mathbb{N}}} ; there exists u \in E_0^{\alpha, p} so that {u_k} \rightharpoonup u in E_0^{\alpha, p} , so {u_k} \rightarrow u in C([0, T], \mathbb{R}) . For the last step, we show that u\neq 0 . According to the condition ({H_5}) , for \forall \varepsilon > 0 , there exists \delta > 0 so that

    \begin{equation} f(t, u)u \le \varepsilon {\left| u \right|^p}, \;\forall (t, u) \in [0, T] \times [ - \delta , \delta ]. \end{equation} (3.9)

    Suppose that {\left\| {{u_k}} \right\|_\infty } \le \delta ; for {u_k} \in \mathcal{N} , by ({H_2}) and (3.9), we obtain

    \begin{equation*} \begin{aligned} \Lambda_\infty ^{ - p}\left\| {{u_k}} \right\|_\infty ^p & \le \left\| {{u_k}} \right\|^p = \int_0^T {f(t, {u_k}(t)){u_k}(t)} dt - \sum\limits_{j = 1}^m {{I_j}({u_k}({t_j})){u_k}({t_j})} \hfill \\ & \le \varepsilon T\left\| {{u_k}} \right\|_\infty ^p - \sum\limits_{j = 1}^m {{a_j}} {\left\| {{u_k}} \right\|_\infty } - \sum\limits_{j = 1}^m {{d_j}} \left\| {{u_k}} \right\|_\infty ^{{\gamma_j} + 1}. \hfill \end{aligned} \end{equation*}

    There is one contradiction in the above formula, so the hypothesis is not valid, namely, {\left\| u \right\|_\infty } = \mathop {\lim }\limits_{k \to \infty } {\left\| {{u_k}} \right\|_\infty } \ge \delta > 0, so u\neq0 . According to Lemma 11, there is one unique y > 0 so that yu\in \mathcal{N} . Because \varphi is weakly lower semi-continuous,

    \begin{equation} m \le \varphi (yu) \le \mathop {\underline {\lim } }\limits_{k \to \infty } \varphi (y{u_k}) \le \mathop {\lim }\limits_{k \to \infty } \varphi (y{u_k}). \end{equation} (3.10)

    For \forall u_k \in \mathcal{N} , by (3.4) and (3.5), we get that {y_k} = 1 is one global maximum point of g , so \varphi (y{u_k}) \le \varphi ({u_k}) . Combined with (3.10), one has m \le \varphi (yu) \le \mathop {\lim }\limits_{k \to \infty } \varphi ({u_k}) = m. Therefore, m is obtained at yu\in \mathcal{N} .

    The proof process of Theorem 2 is given below.

    Proof of Theorem 2. By Lemmas 11 and 12, we know that there exists u\in \mathcal{N} so that \varphi(u) = m = {\inf _\mathcal{N}}\varphi > 0 , i.e., u is the non-zero critical point of \varphi \left| {_\mathcal{N}} \right. . By Lemma 10, one has \varphi'(u) = 0 ; thus, u is the non-trivial ground-state solution of Problem (1.1).

    Lemma 13. Let f\in C([0, T]\times \mathbb{R}, \mathbb{R}) I_j\in C (\mathbb{R}, \mathbb{R}) . Assume that the conditions ({H_0}) and ({H_8}) ({H_{15}}) hold. Then, \varphi satisfies the {{\text{(PS)}}_c} condition.

    Proof. Assume that there is the sequence {\left\{ {{u_n}} \right\}_{n \in \mathbb{N}}} \subset E_0^{\alpha, p} so that \varphi \left({{u_n}} \right) \to c and \varphi'\left({{u_n}} \right) \to 0 (n \rightarrow \infty) ; then, there is {c_1} > 0 so that, for n \in \mathbb{N} , we have

    \begin{equation} \left| {{\varphi}\left( {{u_n}} \right)} \right| \le {c_1}, \;{\left\| {{\varphi}^\prime \left( {{u_n}} \right)} \right\|_{{{\left( {E_0 ^{\alpha , p}} \right)}^*}}} \le {c_1}, \end{equation} (3.11)

    where {\left({E_0 ^{\alpha, p}} \right)^*} is the conjugate space of E_0 ^{\alpha, p} . Next, let us verify that {\left\{ {{u_n}} \right\}_{n \in \mathbb{N}}} is bounded in E_0 ^{\alpha, p} . If not, we assume that {\left\| {{u_n}} \right\|} \to + \infty as (n \rightarrow \infty) . Let {v_n} = \frac{{{u_n}}}{{{{\left\| {{u_n}} \right\|}}}} ; then, {\left\| {{v_n}} \right\|} = 1 . Since E_0 ^{\alpha, p} is one reflexive Banach space, there is one subsequence of \{{v_n}\} (still denoted as \left\{ {{v_n}} \right\} ), so that {v_n} \rightharpoonup v ( n \rightarrow \infty ) in E_0 ^{\alpha, p} ; then, {v_n} \rightarrow v in C([0, T], \mathbb{R}) . By ( {H_{11}} ) and ( {H_{12}} ), we get

    \begin{equation} | {f_2}(t, u)\cdot u| \le {b_1}(t)|u{|^{{r}}} , \;\;|{F_2}\left( {t, u} \right)| \le \frac{1}{r} {b_1}(t)|u{|^{{r}}}. \end{equation} (3.12)

    Two cases are discussed below.

    \bf{Case} 1: v \ne 0 . Let \Omega = \left\{ {t \in [0, T]|\left| {v\left(t \right)} \right| > 0} \right\} ; then, {\text{meas}}(\Omega) > 0 . Because {\left\| {{u_n}} \right\|} \to + \infty (n \to \infty) and \left| {{u_n}\left(t \right)} \right| = \left| {{v_n}\left(t \right)} \right| \cdot {\left\| {{u_n}} \right\|} , so for t \in {\Omega } , one has \left| {{u_n}\left(t \right)} \right| \to + \infty (n \to \infty) . On the one side, by (2.6), (2.8), (3.11), (3.12) and ( {H_{13}} ), one has

    \begin{equation*} \begin{aligned} &\int_0^T {{F_1}(t, {u_n})} dt = \frac{1}{p}\left\| {{u_n}} \right\|^p +\sum\limits_{j = 1}^m {\int_0^{{u_n}({t_j})} {{I_j}(t)} dt} -\int_0^T {{F_2}(t, {u_n})} dt - \varphi({u_n}) \hfill \\ & \le \frac{1}{p}\left\| {{u_n}} \right\|^p + \sum\limits_{j = 1}^m {{a_j}{\Lambda _\infty }\left\| {{u_n}} \right\|} + \sum\limits_{j = 1}^m {{d_j}\Lambda _\infty ^{{\gamma _j} + 1}{{\left\| {{u_n}} \right\|}^{{\gamma _j} + 1}}} + \frac{T}{r}\Lambda _\infty ^r{\left\| {{b_1}} \right\|_\infty }{\left\| {{u_n}} \right\|^r} + {c_1}. \hfill \end{aligned} \end{equation*}

    Since \gamma_j\in [0, \theta-1), \; \theta > p > r > 1 ,

    \begin{equation} {\lim _{n \to \infty }}\int_0^T {\frac{{{F_1}(t, {u_n})}}{{\left\| {{u_n}} \right\|^\theta }}} dt \le o(1), \;n \rightarrow \infty. \end{equation} (3.13)

    On the other side, Fatou's lemma combines with the properties of {\Omega} and ( {H_{10}} ), so we get

    \begin{equation*} \mathop {{\text{lim}}}\limits_{n \to \infty } \int_0^T {\frac{{{F_1}\left( {t, {u_n}} \right)}}{{\left\| {{u_n}} \right\|^\theta}}dt} \ge \mathop {{\text{lim}}}\limits_{n \to \infty } \int_{{\Omega}} {\frac{{{F_1}\left( {t, {u_n}} \right)}}{{\left\| {{u_n}} \right\|^\theta}}dt} = \mathop {{\text{lim}}}\limits_{n \to \infty } \int_{{\Omega}} {\frac{{{F_1}\left(t, {{u_n}} \right)}}{{{{\left| {{u_n}\left( t \right)} \right|}^\theta}}}{{\left| {{v_n}\left( t \right)} \right|}^\theta}dt} = + \infty . \end{equation*}

    This contradicts (3.13).

    \bf{Case} 2: v \equiv 0 . From ( {H_8} ), for \forall \varepsilon > 0 , there is {L_0} > 0 , so that \left| {{f_1}(t, u)} \right| \le \varepsilon {\left| u \right|^{p - 1}}, \; \left| u \right| \le {L_0}. So, for \left| u \right| \le {L_0} , there is {\varepsilon _0} > 0 so that \left| {u{f_1}(t, u) - \theta {F_1}(t, u)} \right| \le {\varepsilon _0}\left({1 + \theta } \right){u^p}. For (t, u)\in [0, T]\times [{L_0}, L] , there is {c_2} > 0 so that \left| {u{f_1}(t, u) - \theta {F_1}(t, u)} \right| \le {c_2}. Combined with the condition ( {H_9} ), one has

    \begin{equation} u{f_1}(t, u) - \theta {F_1}(t, u) \ge - {\varepsilon _0}\left( {1 + \theta } \right){u^p} - {c_2}, \;\forall \left( {t, u} \right) \in [0, T] \times \mathbb{R}. \end{equation} (3.14)

    By ( {H_{14}} ), we obtain that there exists {c_3} > 0 , such that

    \begin{equation} \theta \sum\limits_{j = 1}^m {\int_0^{{u_n}({t_j})} {{I_j}(t)} dt} - \sum\limits_{j = 1}^m {{I_j}({u_n}({t_j})){u_n}({t_j})} \ge - {c_3}. \end{equation} (3.15)

    By (2.6), (2.8), (2.9), (3.11), (3.12), (3.14) and (3.15), we get that there exists {c_4} > 0 such that

    \begin{equation*} \begin{aligned} & o\left( 1 \right) = \frac{{\theta {c_1} + {c_1}\left\| {{u_n}} \right\|}}{{{{\left\| {{u_n}} \right\|}^p}}} \ge \frac{{\theta \varphi \left( {{u_n}} \right) - \langle {{\varphi ^\prime }\left( {{u_n}} \right), {u_n}} \rangle }}{{{{\left\| {{u_n}} \right\|}^p}}} \\ & = \left( {\frac{\theta }{p} - 1} \right) + \frac{1}{{{{\left\| {{u_n}} \right\|}^p}}}\left[ {\theta \sum\limits_{j = 1}^m {\int_0^{{u_n}({t_j})} {{I_j}(t)} dt} - \sum\limits_{j = 1}^m {{I_j}({u_n}({t_j})){u_n}({t_j})} } \right] \\ & + \frac{1}{{{{\left\| {{u_n}} \right\|}^p}}}\int_0^T {\left[ {{u_n}{f_1}\left( {t, {u_n}} \right) - \theta {F_1}\left( {t, {u_n}} \right)} \right]dt}+ \frac{1}{{{{\left\| {{u_n}} \right\|}^p}}}\int_0^T {\left[ {{u_n}{f_2}\left( {t, {u_n}} \right) - \theta {F_2}\left( {t, {u_n}} \right)} \right]dt} \\ & \ge \left( {\frac{\theta }{p} - 1} \right) + \frac{1}{{{{\left\| {{u_n}} \right\|}^p}}}\int_0^T {\left[ { - {\varepsilon _0}\left( {1 + \theta } \right)u_n^p - {c_2}} \right]dt} - \frac{1}{{{{\left\| {{u_n}} \right\|}^p}}}\left( {\frac{\theta }{r} + 1} \right)\int_0^T {{b_1}(t)|{u_n}{|^r}dt} - \frac{1}{{{{\left\| {{u_n}} \right\|}^p}}}{c_3} \\ & \ge \left( {\frac{\theta }{p} - 1} \right) - {\varepsilon _0}\left( {1 + \theta } \right)\int_0^T {\frac{{|{u_n}{|^p}}}{{{{\left\| {{u_n}} \right\|}^p}}}dt} - \frac{{T{c_2}}}{{{{\left\| {{u_n}} \right\|}^p}}} - \frac{1}{{{{\left\| {{u_n}} \right\|}^p}}}\left( {\frac{\theta }{r} + 1} \right){\left\| {{b_1}} \right\|_{{L^1}}}\left\| {{u_n}} \right\|_\infty ^r - \frac{1}{{{{\left\| {{u_n}} \right\|}^p}}}{c_3} \\ & \ge \left( {\frac{\theta }{p} - 1} \right) - {\varepsilon _0}\left( {1 + \theta } \right)\int_0^T {|{v_n}{|^p}dt} - \frac{{T{c_2}}}{{{{\left\| {{u_n}} \right\|}^p}}} - \left( {\frac{\theta }{r} + 1} \right){\left\| {{b_1}} \right\|_{{L^1}}}\Lambda _\infty ^r{\left\| {{u_n}} \right\|^{r - p}} \ge \left( {\frac{\theta }{p} - 1} \right), \;n \to \infty . \end{aligned} \end{equation*}

    It is a contradiction. Thus, {\left\{ {{u_n}} \right\}_{n \in \mathbb{N}}} is bounded in E_0^{\alpha, p} . Assume that the sequence {\left\{ {{u_n}} \right\}_{n \in \mathbb{N}}} possesses one subsequence, still recorded as {\left\{ {{u_n}} \right\}_{n \in \mathbb{N}}} ; there exists u \in E_0 ^{\alpha, p} so that {u_n} \rightharpoonup u in E_0 ^{\alpha, p} ; then, {u_n} \rightarrow u in C([0, T], \mathbb{R}) . Therefore,

    \begin{equation} \left\{ \begin{array}{l} \langle {{\varphi^\prime }\left( {{u_n}} \right) - {\varphi^\prime }\left( u \right), {u_n} - u} \rangle \to 0, \;n \to \infty , \hfill \\ \int_0^T {\left[ {f\left( {t, {u_n}\left( t \right)} \right) - f\left( {t, u\left( t \right)} \right)} \right]\left[ {{u_n}\left( t \right) - u\left( t \right)} \right]dt} \to 0, \;n \to \infty , \hfill \\ \sum\limits_{j = 1}^m {\left( {{I_j}({u_n}({t_j})) - {I_j}(u({t_j}))} \right)\left( {{u_n}({t_j}) - u({t_j})} \right)} \to 0, \; n \to \infty , \hfill \\ \int_0^T {a(t)\left( {{\phi _p}({u_n}(t)) - {\phi _p}(u(t))} \right)\left( {{u_n}(t) - u(t)} \right)} dt \to 0, \; n \to \infty .\hfill \end{array} \right. \end{equation} (3.16)

    Through (2.9), we can get

    \begin{equation} \begin{aligned} & \langle {{\varphi ^\prime }\left( {{u_n}} \right) - {\varphi ^\prime }\left( u \right), {u_n} - u} \rangle = \int_0^T {\left( {{\phi _p}{(_0}D_t^\alpha {u_n}(t)) - {\phi _p}{(_0}D_t^\alpha u(t))} \right)\left( {_0D_t^\alpha {u_n}(t) - {}_0D_t^\alpha u(t)} \right)} dt \\ & + \int_0^T {a(t)\left( {{\phi _p}({u_n}(t)) - {\phi _p}(u(t))} \right)\left( {{u_n}(t) - u(t)} \right)} dt + \sum\limits_{j = 1}^m {\left( {{I_j}({u_n}({t_j})) - {I_j}(u({t_j}))} \right)\left( {{u_n}({t_j}) - u({t_j})} \right)} \\ & - \int_0^T {\left[ {f\left( {t, {u_n}\left( t \right)} \right) - f\left( {t, u\left( t \right)} \right)} \right]\left[ {{u_n}\left( t \right) - u\left( t \right)} \right]dt} . \end{aligned} \end{equation} (3.17)

    From [23], we obtain

    \begin{equation} \begin{aligned} & \int_{\text{0}}^T {\left( {{\phi _p}{(_0}D_t^\alpha {u_n}(t)) - {\phi _p}{(_0}D_t^\alpha u(t))} \right)\left( {_0D_t^\alpha {u_n}(t) - {}_0D_t^\alpha u(t)} \right)dt} \\ & \ge \left\{ \begin{array}{l} c\int_{\text{0}}^T {{{\left| {_0D_t^\alpha {u_n}(t) - {}_0D_t^\alpha u(t)} \right|}^p}dt} , \;p \ge 2, \\ c\int_{\text{0}}^T {\frac{{{{\left| {_0D_t^\alpha {u_n}(t) - {}_0D_t^\alpha u(t)} \right|}^2}}}{{{{\left( {\left| {_0D_t^\alpha {u_n}(t)} \right| + \left| {_0D_t^\alpha u(t)} \right|} \right)}^{2 - p}}}}dt} , \;1 < p < 2. \end{array} \right. \end{aligned} \end{equation} (3.18)

    If p\geq 2 , by (3.16)–(3.18), one has {\left\| {{u_n} - u} \right\|} \to 0 (n \rightarrow \infty) . If 1 < p < 2 , by the H \ddot{{\text{o}}} lder inequality, one has \int_{\text{0}}^T {{{\left| {_0D_t^\alpha {u_n}(t) - {}_0D_t^\alpha u(t)} \right|}^p}dt} \le c{\left({\int_{\text{0}}^T {\frac{{{{\left| {_0D_t^\alpha {u_n}(t) - {}_0D_t^\alpha u(t)} \right|}^2}}}{{{{\left({\left| {_0D_t^\alpha {u_n}(t)} \right| + \left| {_0D_t^\alpha u(t)} \right|} \right)}^{2 - p}}}}dt} } \right)^{\frac{p}{2}}}{\left({\left\| {{u_n}} \right\| + \left\| u \right\|} \right)^{\frac{{p\left({2 - p} \right)}}{2}}}. Thus,

    \begin{equation} \begin{aligned} & \int_{\text{0}}^T {\left( {{\phi _p}{(_0}D_t^\alpha {u_n}(t)) - {\phi _p}{(_0}D_t^\alpha u(t))} \right)\left( {_0D_t^\alpha {u_n}(t) - {}_0D_t^\alpha u(t)} \right)dt} \\ & \ge \frac{c}{{{{\left( {\left\| {{u_n}} \right\| + \left\| u \right\|} \right)}^{2 - p}}}}{\left( {\int_{\text{0}}^T {{{\left| {_0D_t^\alpha {u_n}(t) - {}_0D_t^\alpha u(t)} \right|}^p}dt} } \right)^{\frac{2}{p}}}. \end{aligned} \end{equation} (3.19)

    By (3.16), (3.17) and (3.19), one has {\left\| {{u_n} - u} \right\|} \to 0 (n \rightarrow \infty) . Hence, \varphi satisfies the {{\text{(PS)}}_c} condition.

    The proof of Theorem 3.

    \bf{Step} \; 1. Clearly, \varphi(0) = 0 . Lemma 13 implies that \varphi \in {C^1}\left({E_0 ^{\alpha, p}, \mathbb{R}} \right) satisfies the {{\text{(PS)}}_c} condition.

    \bf{Step} \; 2. For \forall {\varepsilon_1} > 0 , we know from ( H_8 ) that there is \delta > 0 so that

    \begin{equation} {F_1}(t, u) \le {\varepsilon_1} {\left| u \right|^p}, \;\forall t \in [0, T], \;\left| u \right| \le \delta . \end{equation} (3.20)

    For \forall u \in {E_0 ^{\alpha, p}} , by (2.5), (2.6), (2.8), (3.12) and ( {H_{15}} ), we get

    \begin{equation} \begin{aligned} \varphi (u)& = \frac{1}{p}{\left\| u \right\|^p} + \sum\limits_{j = 1}^m {\int_0^{u({t_j})} {{I_j}(t)} dt} - \int_0^T {F(t, u(t))} dt \ge \frac{1}{p}{\left\| u \right\|^p} - \int_0^T {F\left( {t, u\left( t \right)} \right)dt} \\ & \ge \frac{1}{p}{\left\| u \right\|^p} - {\varepsilon _1}\int_0^T {{{\left| u \right|}^p}dt} - \frac{1}{r}\int_0^T {{b_1}(t){{\left| u \right|}^r}dt} \ge \frac{1}{p}{\left\| u \right\|^p} - {\varepsilon _1}\Lambda _p^p{\left\| u \right\|^p} - \frac{1}{r}{\left\| {{b_1}} \right\|_{{L^1}}}\left\| u \right\|_\infty ^r \\ & \ge \left( {\frac{1}{p} - {\varepsilon _1}\Lambda _p^p - \frac{{\Lambda _\infty ^r}}{r}{{\left\| {{b_1}} \right\|}_{{L^1}}}{{\left\| u \right\|}^{r - p}}} \right){\left\| u \right\|^p}. \end{aligned} \end{equation} (3.21)

    Choose {\varepsilon _1} = \frac{1}{{2p\Lambda _p^p}} ; one has \varphi (u) \ge \left({\frac{1}{{2p}} - \frac{{\Lambda _\infty ^r}}{r}{{\left\| {{b_1}} \right\|}_{{L^1}}}{{\left\| u \right\|}^{r - p}}} \right){\left\| u \right\|^p}. Let \rho = {\left({\frac{r}{{4p\Lambda _\infty ^r{{\left\| {{b_1}} \right\|}_{{L^1}}}}}} \right)^{\frac{1}{{r - p}}}} and \eta = \frac{1}{{4p}}{\rho ^p} ; then, for u \in {\partial {B_\rho }} , we obtain \varphi(u) \ge \eta > 0 .

    \bf{Step} \; 3. From ( {H_{10}} ), for \left| u \right| \ge {L_1} , there exist {\varepsilon_2}, \; {\varepsilon_3} > 0 such that

    \begin{equation} {F_1}(t, u) \ge {\varepsilon _2}{\left| u \right|^\theta } - {\varepsilon _3}. \end{equation} (3.22)

    By ( {H_8} ), for \left| u \right| \leq {L_1} , there exist {\varepsilon_4}, \; {\varepsilon_5} > 0 such that

    \begin{equation} {F_1}(t, u) \ge - {\varepsilon _4}{u^p} - {\varepsilon _5}. \end{equation} (3.23)

    From (3.22) and (3.23), we obtain that there exist {\varepsilon_6}, \; {\varepsilon_7} > 0 so that

    \begin{equation} {F_1}(t, u) \ge {\varepsilon _2}{\left| u \right|^\theta } - {\varepsilon _6}{u^p} - {\varepsilon _7}, \;\forall t \in [0, T], \;u \in \mathbb{R}, \end{equation} (3.24)

    where {\varepsilon _6} = {\varepsilon _2}L_1^{\theta - p} + {\varepsilon _4} . For \forall u \in E_0 ^{\alpha, p}\backslash \{0\} and \xi \in {\mathbb{R}^ + } , by ( {H_{13}} ), (2.5), (2.6), (2.8), (3.24) and the H \ddot{{\text{o}}} lder inequality, we have

    \begin{equation*} \begin{aligned} & \varphi \left( {\xi u} \right) \le \frac{{{\xi ^p}}}{p}{\left\| u \right\|^p} + \sum\limits_{j = 1}^m {{a_j}\xi {\Lambda _\infty }\left\| u \right\|} + \sum\limits_{j = 1}^m {{d_j}{\xi ^{{\gamma _j} + 1}}\Lambda _\infty ^{{\gamma _j} + 1}{{\left\| u \right\|}^{{\gamma _j} + 1}}} - {\varepsilon _2}{\xi ^\theta }\int_0^T {{{\left| u \right|}^\theta }dt} + {\varepsilon _6}\Lambda _p^p{\xi ^p}{\left\| u \right\|^p} + {\varepsilon _7}T \\ & \le \left( {\frac{1}{p} + {\varepsilon _6}\Lambda _p^p} \right){\xi ^p}{\left\| u \right\|^p} + \sum\limits_{j = 1}^m {{a_j}\xi {\Lambda _\infty }\left\| u \right\|} + \sum\limits_{j = 1}^m {{d_j}{\xi ^{{\gamma _j} + 1}}\Lambda _\infty ^{{\gamma _j} + 1}{{\left\| u \right\|}^{{\gamma _j} + 1}}} - {\varepsilon _2}{\xi ^\theta }{\left( {{T^{\frac{{p - \theta }}{\theta }}}\int_0^T {{{\left| {u\left( t \right)} \right|}^p}dt} } \right)^{\frac{\theta }{p}}} + {\varepsilon _7}T \\ & \le \left( {\frac{1}{p} + {\varepsilon _6}\Lambda _p^p} \right){\xi ^p}{\left\| u \right\|^p} + \sum\limits_{j = 1}^m {{a_j}\xi {\Lambda _\infty }\left\| u \right\|} + \sum\limits_{j = 1}^m {{d_j}{\xi ^{{\gamma _j} + 1}}\Lambda _\infty ^{{\gamma _j} + 1}{{\left\| u \right\|}^{{\gamma _j} + 1}}} - {\varepsilon _2}{\xi ^\theta }{T^{\frac{{p - \theta }}{p}}}\left\| u \right\|_{{L^p}}^\theta + {\varepsilon _7}T. \end{aligned} \end{equation*}

    Since \theta > p > 1 and {\gamma _j} + 1 \in \left[{1, \theta } \right) , the above inequality indicates that \varphi({\xi_0} u)\rightarrow -\infty when {\xi_0} is large enough. Let e = {\xi_0}u ; one has \varphi(e) < 0 . Thus, the condition (ii) in Lemma 8 holds. Lemma 8 implies that \varphi possesses one critical value c ^{(1)} \ge \eta > 0 . The specific form is c ^{(1)} = \mathop {\inf }\limits_{g \in \Gamma } \mathop {\max }\limits_{s \in [0, 1]} \varphi\left({g\left(s \right)} \right), where \Gamma = \left\{ {g \in C\left({[0, 1], E_0 ^{\alpha, p}} \right):g(0) = 0, {\mkern 1mu} g(1) = e} \right\}. Hence, there is 0 \ne u ^{(1)} \in E_0 ^{\alpha, p} so that

    \begin{equation} \varphi\left( {u ^{(1)}} \right) = c^{(1)}\geq \eta > 0, \;\varphi^\prime \left( {u ^{(1)}} \right) = 0. \end{equation} (3.25)

    \bf{Step} \; 4 . Equation (3.21) implies that \varphi is bounded below in \overline {{B_\rho }} . Choose \sigma \in E_0^{\alpha, p} so that \sigma(t)\neq 0 in [0, T] . For \forall \; l \in (0, + \infty) , by (2.6), (2.8), ( {H_{10}} ), ( {H_{11}} ) and ( {H_{13}} ), we have

    \begin{equation} \begin{aligned} \varphi \left( {l\sigma } \right) & \le \frac{{{l^p}}}{p}{\left\| \sigma \right\|^p} + \sum\limits_{j = 1}^m {{a_j}l{\Lambda _\infty }\left\| \sigma \right\|} + \sum\limits_{j = 1}^m {{d_j}{l^{{\gamma _j} + 1}}\Lambda _\infty ^{{\gamma _j} + 1}{{\left\| \sigma \right\|}^{{\gamma _j} + 1}}} - \int_0^T {{F_2}\left( {t, l\sigma \left( t \right)} \right)dt} \\ & \le \frac{{{l^p}}}{p}{\left\| \sigma \right\|^p} + \sum\limits_{j = 1}^m {{a_j}l{\Lambda _\infty }\left\| \sigma \right\|} + \sum\limits_{j = 1}^m {{d_j}{l^{{\gamma _j} + 1}}\Lambda _\infty ^{{\gamma _j} + 1}{{\left\| \sigma \right\|}^{{\gamma _j} + 1}}} - {l^r}\int_0^T {b\left( t \right){{\left| {\sigma \left( t \right)} \right|}^r}dt} . \end{aligned} \end{equation} (3.26)

    Thus, from 1 < r < p and \gamma_j\in [0, \theta-1) , we know that for a small enough {l_0} satisfying {\left\| {{l_0}\sigma } \right\|} \le \rho , one has \varphi \left({{l_0}\sigma } \right) < 0 . Let u = {{l_0}\sigma } ; we have that c^{(2)} = \inf \varphi\left(u \right) < 0, \quad {\left\| u \right\|} \le \rho. Ekeland's variational principle shows that there is one minimization sequence {\left\{ {{v_k}} \right\}_{k \in \mathbb{N}}} \subset \overline {{B_\rho }} so that \varphi\left({{v_k}} \right) \to c^{(2)} and \varphi^\prime \left({{v_k}} \right) \to 0, \; k\rightarrow \infty, i.e., {\left\{ {{v_k}} \right\}_{k \in \mathbb{N}}} is one {{\text{(PS)}}_c} sequence. Lemma 13 shows that \varphi satisfies the {{\text{(PS)}}_c} condition. Thus, c^{(2)} < 0 is another critical value of \varphi . So, there exists 0 \ne u^{(2)} \in E_0^{\alpha, p} so that \varphi \left({u^{(2)}} \right) = c^{(2)} < 0, \; {\left\| {u^{(2)}} \right\|} < \rho.

    Proof. The functionals \Phi:E_0^{\alpha, p} \to \mathbb{R} and \Psi:E_0^{\alpha, p} \to \mathbb{R} are defined as follows:

    \Phi \left( u \right) = \frac{1}{p}{\left\| u \right\|^p}, \Psi \left( u \right) = \int_0^T {F(t, u(t))} dt - \frac{\mu }{\lambda }\sum\limits_{j = 1}^n {{J_j}(u({t_j}))} ;

    then, \varphi(u) = \Phi \left(u \right)-\lambda \Psi \left(u \right) . We can calculate that

    \mathop {{\text{inf}}}\limits_{u \in E_0^{\alpha , p}} \Phi \left( u \right) = \Phi \left( 0 \right) = 0, \Psi \left( 0 \right) = \int_0^T {F(t, 0)} dt - \frac{\mu }{\lambda }\sum\limits_{j = 1}^n {{J_j}(0) = 0}.

    Furthermore, \Phi and \Psi are continuous Gâteaux differentiable and

    \begin{equation} \langle {\Phi '(u), v} \rangle = \int_0^T {{\phi _p}{{{(_0}D_t^\alpha u(t))}_0}D_t^\alpha v(t)dt} + \int_0^T {a(t){\phi _p}(u(t))v(t)} dt, \end{equation} (3.27)
    \begin{equation} \langle {\Psi '(u), v} \rangle = \int_0^T {f(t, u(t))v(t)} dt - \frac{\mu }{\lambda }\sum\limits_{j = 1}^n {{I_j}(u({t_j}))v({t_j})} , \forall u, v \in E_0^{\alpha , p}. \end{equation} (3.28)

    In addition, \Phi ':E_0^{\alpha, p} \to {\left({E_0^{\alpha, p}} \right)^*} is continuous. It is proved that \Psi ':E_0^{\alpha, p} \to {\left({E_0^{\alpha, p}} \right)^*} is a continuous compact operator. Suppose that \{ {u_n}\} \subset E_0^{\alpha, p} , {u_n}\rightharpoonup u , n\rightarrow \infty ; then, \left\{ {{u_n}} \right\} uniformly converges to u on C([0, T]) . Owing to f\in C([0, T]\times \mathbb{R}, \mathbb{R}) and I_j\in C (\mathbb{R}, \mathbb{R}) , we have that f(t, {u_n})\rightarrow f(t, u) and {{I_j}({u_n}({t_j}))} \rightarrow {{I_j}(u({t_j}))} , n\rightarrow \infty . Thus, {\Psi '({u_n})} \rightarrow {\Psi '(u)} as n\rightarrow \infty . Then, \Psi ' is strongly continuous. According to Proposition 26.2 in [24], \Psi ' is one compact operator. It is proved that \Phi is weakly semi-continuous. Suppose that \left\{ {{u_n}} \right\} \subset E_0^{\alpha, p}, \left\{ {{u_n}} \right\}\rightharpoonup u ; then, \left\{ {{u_n}} \right\} \rightarrow u on C([0, T]) , and \mathop {\lim \inf }\limits_{n \to \infty } \left\| {{u_n}} \right\| \ge \left\| u \right\| . So, \mathop {\lim \inf }\limits_{n \to \infty } \Phi \left({{u_n}} \right) = \mathop {\lim \inf }\limits_{n \to \infty } \left({\frac{1}{p}{{\left\| {{u_n}} \right\|}^p}} \right) \ge \frac{1}{p}{\left\| u \right\|^p} = \Phi \left(u \right). Thus, \Phi is weakly semi-continuous. Because \Phi \left(u \right) = \frac{1}{p}{\left\| u \right\|^p}\rightarrow +\infty and \left\| u \right\|\rightarrow + \infty, \Phi is coercive. By (3.27), we obtain

    \begin{equation*} \begin{aligned} & \langle {{\Phi ^\prime }\left( {{u}} \right) - {\Phi ^\prime }\left( v \right), u - v} \rangle = \int_0^T {\left( {{\phi _p}{(_0}D_t^\alpha u(t)) - {\phi _p}{(_0}D_t^\alpha v(t))} \right)\left( {_0D_t^\alpha u(t) - {}_0D_t^\alpha v(t)} \right)} dt \\ & + \int_0^T {a(t)\left( {{\phi _p}(u(t)) - {\phi _p}(v(t))} \right)\left( {u(t) - v(t)} \right)} dt , \;\forall u, v \in E_0^{\alpha , p}. \end{aligned} \end{equation*}

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

    \begin{equation} \begin{aligned} & \int_{\text{0}}^T {\left( {{\phi _p}{(_0}D_t^\alpha u(t)) - {\phi _p}{(_0}D_t^\alpha v(t))} \right)\left( {_0D_t^\alpha u(t) - {}_0D_t^\alpha v(t)} \right)dt} \\ & \ge \left\{ \begin{array}{l} c\int_{\text{0}}^T {{{\left| {_0D_t^\alpha u(t) - {}_0D_t^\alpha v(t)} \right|}^p}dt} , \;p \ge 2, \\ c\int_{\text{0}}^T {\frac{{{{\left| {_0D_t^\alpha u(t) - {}_0D_t^\alpha v(t)} \right|}^2}}}{{{{\left( {\left| {_0D_t^\alpha u(t)} \right| + \left| {_0D_t^\alpha v(t)} \right|} \right)}^{2 - p}}}}dt} , \;1 < p < 2. \end{array} \right. \end{aligned} \end{equation} (3.29)

    If p\geq 2 , then \langle {{\Phi ^\prime }\left(u \right) - {\Phi ^\prime }\left(v \right), u - v} \rangle \geq c{\left\| u-v \right\|^p}. Thus, {\Phi ^\prime } is uniformly monotonous. When 1 < p < 2 , the H \ddot{{\text{o}}} lder inequality implies

    \begin{equation*} \begin{aligned} & \int_{\text{0}}^T {{{\left| {_0D_t^\alpha u(t) - {}_0D_t^\alpha v(t)} \right|}^p}dt} \\ &\le {\left( {\int_0^T {\frac{{{{\left| {{}_0D_t^\alpha u(t) - {{\kern 1pt} _0}D_t^\alpha v(t)} \right|}^2}}}{{{{\left( {\left| {{}_0D_t^\alpha u(t)} \right| + {\kern 1pt} \left| {_0D_t^\alpha v(t)} \right|} \right)}^{2 - p}}}}dt} } \right)^{\frac{p}{2}}} {\text{ }}{\left( {\int_0^T {{{\left( {\left| {{}_0D_t^\alpha u(t)} \right| + {\kern 1pt} \left| {_0D_t^\alpha v(t)} \right|} \right)}^p}dt} } \right)^{\frac{{2 - p}}{2}}} \\ & \le c_{1}{\left( {\int_{\text{0}}^T {\frac{{{{\left| {_0D_t^\alpha u(t) - {}_0D_t^\alpha v(t)} \right|}^2}}}{{{{\left( {\left| {_0D_t^\alpha u(t)} \right| + \left| {_0D_t^\alpha v(t)} \right|} \right)}^{2 - p}}}}dt} } \right)^{\frac{p}{2}}}{\left( {{{\left\| u \right\|}^p} + {{\left\| v \right\|}^p}} \right)^{\frac{{2 - p}}{2}}}, \end{aligned} \end{equation*}

    where c_{1} = {2^{\frac{{\left({p - 1} \right)\left({2 - p} \right)}}{2}}} > 0 . Then,

    \begin{equation*} \int_0^T {\frac{{{{\left| {{}_0D_t^\alpha u(t) - {{\kern 1pt} _0}D_t^\alpha v(t)} \right|}^2}}}{{{{\left( {\left| {{}_0D_t^\alpha u(t)} \right| + {\kern 1pt} \left| {_0D_t^\alpha v(t)} \right|} \right)}^{2 - p}}}}dt} \ge \frac{c_{2}}{{{{\left( {\left\| {u} \right\| + \left\| v \right\|} \right)}^{2 - p}}}}{\left( {\int_{\text{0}}^T {{{\left| {_0D_t^\alpha u(t) - {}_0D_t^\alpha v(t)} \right|}^p}dt} } \right)^{\frac{2}{p}}}, \end{equation*}

    where c_{2} = \frac{1}{{c_1^{\frac{2}{p}}}}. Combined with (3.29), we can get

    \begin{equation} \int_{\text{0}}^T {\left( {{\phi _p}{(_0}D_t^\alpha u(t)) - {\phi _p}{(_0}D_t^\alpha v(t))} \right)\left( {_0D_t^\alpha u(t) - {}_0D_t^\alpha v(t)} \right)dt} \ge \frac{c}{{{{\left( {\left\| {u} \right\| + \left\| v \right\|} \right)}^{2 - p}}}}{\left( {\int_{\text{0}}^T {{{\left| {_0D_t^\alpha u(t) - {}_0D_t^\alpha v(t)} \right|}^p}dt} } \right)^{\frac{2}{p}}}. \end{equation} (3.30)

    Thus, \langle {{\Phi ^\prime }\left(u \right) - {\Phi ^\prime }\left(v \right), u - v} \rangle \ge \frac{{c{{\left\| {u - v} \right\|}^2}}}{{{{\left({\left\| u \right\| + \left\| v \right\|} \right)}^{2 - p}}}}. So, {\Phi ^\prime } is strictly monotonous. Theorem 26.A(d) in [24] implies that {\left({{\Phi ^\prime }} \right)^{ - 1}} exists and is continuous. If x \in E_0^{\alpha, p} satisfies \Phi \left(x \right) = \frac{1}{p}{\left\| x \right\|^p} \le r , then, by (2.6), we obtain \Phi \left(x \right) \ge \frac{1}{{p\Lambda _\infty ^p}}\left\| x \right\|_\infty ^p , and

    \begin{equation*} \left\{ {x \in E_0^{\alpha , p}:\Phi \left( x \right) \le r} \right\} \subseteq \left\{ {x:\frac{1}{{p\Lambda _\infty ^p}}\left\| x \right\|_\infty ^p \le r} \right\} = \left\{ {x:\left\| x \right\|_\infty ^p \le pr\Lambda _\infty ^p} \right\} = \left\{ {x:{{\left\| x \right\|}_\infty } \le {\Lambda _\infty }{{\left( {pr} \right)}^{\frac{1}{p}}}} \right\}. \end{equation*}

    Therefore, from \lambda > 0 and \mu\geq 0 , we have

    \begin{equation*} \begin{aligned} & {\text{sup}}\left\{ {\Psi \left( x \right):\Phi \left( x \right) \le r} \right\} = \sup \left\{ {\int_0^T {F(t, x(t))} dt - \frac{\mu }{\lambda }\sum\limits_{j = 1}^n {{J_j}(x({t_j}))} :\Phi \left( x \right) \le r} \right\} \\ & \le \int_0^T {\mathop {\max }\limits_{\left| x \right| \le {\Lambda _\infty }{{\left( {pr} \right)}^{\frac{1}{p}}}} F(t, x)} dt + \frac{\mu }{\lambda }\mathop {\max }\limits_{\left| x \right| \le {\Lambda _\infty }{{\left( {pr} \right)}^{\frac{1}{p}}}} \sum\limits_{j = 1}^n {\left( { - {J_j}(x)} \right)} . \end{aligned} \end{equation*}

    If \mathop {\max }\limits_{\left| x \right| \le {\Lambda _\infty }{{\left({pr} \right)}^{\frac{1}{p}}}} \sum\limits_{j = 1}^n {\left({ - {J_j}(x)} \right)} = 0 , by \lambda < {A_r} , we get

    \begin{equation} {\text{sup}}\left\{ {\Psi \left( x \right):\Phi \left( x \right) \le r} \right\} < \frac{r}{\lambda }. \end{equation} (3.31)

    If \mathop {\max }\limits_{\left| x \right| \le {\Lambda _\infty }{{\left({pr} \right)}^{\frac{1}{p}}}} \sum\limits_{j = 1}^n {\left({ - {J_j}(x)} \right)} > 0 , for \mu \in [0, \gamma) , \gamma = \min \left\{ {\frac{{r - \lambda \int_0^T {\mathop {\max }\limits_{\left| x \right| \le {\Lambda _\infty }{{\left({pr} \right)}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. } p}}}} F\left({t, x} \right)dt} }}{{\mathop {\max }\limits_{\left| x \right| \le {\Lambda _\infty }{{\left({pr} \right)}^{\frac{1}{p}}}} \sum\limits_{j = 1}^n {\left({ - {J_j}(x)} \right)} }}, \frac{{\lambda \int_0^T {F(t, \omega)} dt - \frac{1}{p}{{\left\| \omega \right\|}^p}}}{{\sum\limits_{j = 1}^n {{J_j}(\omega ({t_j}))} }}} \right\} , we have

    \begin{equation*} \begin{aligned} & {\text{sup}}\left\{ {\Psi \left( x \right):\Phi \left( x \right) \le r} \right\} \le \int_0^T {\mathop {\max }\limits_{\left| x \right| \le {\Lambda _\infty }{{\left( {pr} \right)}^{\frac{1}{p}}}} F(t, x)} dt + \frac{\mu }{\lambda }\mathop {\max }\limits_{\left| x \right| \le {\Lambda _\infty }{{\left( {pr} \right)}^{\frac{1}{p}}}} \sum\limits_{j = 1}^n {\left( { - {J_j}(x)} \right)} \\ & < \int_0^T {\mathop {\max }\limits_{\left| x \right| \le {\Lambda _\infty }{{\left( {pr} \right)}^{\frac{1}{p}}}} F(t, x)} dt + \frac{{\frac{{r - \lambda \int_0^T {\mathop {\max }\limits_{\left| x \right| \le {\Lambda _\infty }{{\left( {pr} \right)}^{\frac{1}{p}}}} F\left( {t, x} \right)dt} }}{{\mathop {\max }\limits_{\left| x \right| \le {\Lambda _\infty }{{\left( {pr} \right)}^{\frac{1}{p}}}} \sum\limits_{j = 1}^n {\left( { - {J_j}(x)} \right)} }}}}{\lambda } \times \mathop {\max }\limits_{\left| x \right| \le {\Lambda _\infty }{{\left( {pr} \right)}^{\frac{1}{p}}}} \sum\limits_{j = 1}^n {\left( { - {J_j}(x)} \right)} \\ & < \int_0^T {\mathop {\max }\limits_{\left| x \right| \le {\Lambda _\infty }{{\left( {pr} \right)}^{\frac{1}{p}}}} F(t, x)} dt + \frac{{r - \lambda \int_0^T {\mathop {\max }\limits_{\left| x \right| \le {\Lambda _\infty }{{\left( {pr} \right)}^{\frac{1}{p}}}} F\left( {t, x} \right)dt} }}{\lambda }\\ & = \int_0^T {\mathop {\max }\limits_{\left| x \right| \le {\Lambda _\infty }{{\left( {pr} \right)}^{\frac{1}{p}}}} F(t, x)} dt + \frac{r}{\lambda } - \int_0^T {\mathop {\max }\limits_{\left| x \right| \le {\Lambda _\infty }{{\left( {pr} \right)}^{\frac{1}{p}}}} F\left( {t, x} \right)dt} \\ & = \frac{r}{\lambda }. \end{aligned} \end{equation*}

    Thus, (3.31) is also true. On the other side, for \mu < \gamma , one has

    \begin{equation} \Psi \left( \omega \right) = \int_0^T {F(t, \omega (t))} dt - \frac{\mu }{\lambda }\sum\limits_{j = 1}^n {{J_j}(\omega ({t_j}))} > \frac{{\Phi \left( \omega \right)}}{\lambda }. \end{equation} (3.32)

    By combining (3.31) and (3.32), we obtain that \frac{{\Psi \left(\omega \right)}}{{\Phi \left(\omega \right)}} > \frac{1}{\lambda } > \frac{{{\text{sup}}\left\{ {\Psi \left(x \right):\Phi \left(x \right) \le r} \right\}}}{r}. This means that the condition (i) of Lemma 9 holds.

    Finally, for the third step, we show that, for any \lambda \in {\Lambda _r} = ({A_l}, {A_r}), the functional \Phi-\lambda\Psi is coercive. By (1.4), we obtain

    \begin{equation} \int_0^T {F(t, x\left( t \right))} dt \le L\int_0^T {\left( {1 + {{\left| {x(t)} \right|}^\beta }} \right)} dt \le LT + LT\left\| x \right\|_\infty ^\beta \le LT + LT\Lambda _\infty ^\beta {\left\| x \right\|^\beta }, \;x\in E_0^{\alpha , p}. \end{equation} (3.33)

    and

    \begin{equation} - {J_j}(x({t_j})) \le {L_j}\left( {1 + {{\left| {x({t_j})} \right|}^{{d_j}}}} \right) \le {L_j}\left( {1 + \left\| x \right\|_\infty ^{{d_j}}} \right) \le {L_j}\left( {1 + \Lambda _\infty ^{{d_j}}{{\left\| x \right\|}^{{d_j}}}} \right). \end{equation} (3.34)

    By (3.34), we get

    \begin{equation} \sum\limits_{j = 1}^n {\left( { - {J_j}(x({t_j}))} \right)} \le \sum\limits_{j = 1}^n {{L_j}\left( {1 + \Lambda _\infty ^{{d_j}}{{\left\| x \right\|}^{{d_j}}}} \right).} \end{equation} (3.35)

    If \frac{\mu }{\lambda } \geq 0 , for x \in E_0^{\alpha, p} , by (3.33) and (3.35), we have

    \begin{equation*} \Psi \left( x \right) \le LT + LT\Lambda _\infty ^\beta {\left\| x \right\|^\beta } + \frac{\mu }{\lambda }\sum\limits_{j = 1}^n {{L_j}\left( {1 + \Lambda _\infty ^{{d_j}}{{\left\| x \right\|}^{{d_j}}}} \right)} = LT + \frac{\mu }{\lambda }\sum\limits_{j = 1}^n {{L_j}} + LT\Lambda _\infty ^\beta {\left\| x \right\|^\beta } + \frac{\mu }{\lambda }\sum\limits_{j = 1}^n {{L_j}\Lambda _\infty ^{{d_j}}{{\left\| x \right\|}^{{d_j}}}} . \end{equation*}

    Thus, \Phi \left(x \right) - \lambda \Psi \left(x \right) \ge \frac{1}{p}{\left\| x \right\|^p} - \lambda \left({LT + \frac{\mu }{\lambda }\sum\limits_{j = 1}^n {{L_j}} + LT\Lambda _\infty ^\beta {{\left\| x \right\|}^\beta } + \frac{\mu }{\lambda }\sum\limits_{j = 1}^n {{L_j}\Lambda _\infty ^{{d_j}}{{\left\| x \right\|}^{{d_j}}}} } \right), \; \forall x \in E_0^{\alpha, p}. If 0 < \beta and {d_j} < p , then \mathop {\lim }\limits_{\left\| x \right\| \to + \infty } \left({\Phi \left(x \right) - \lambda \Psi \left(x \right)} \right) = + \infty, \; \lambda > 0. Thus, \Phi - \lambda \Psi is coercive. When \beta = p , \Phi \left(x \right) - \lambda \Psi \left(x \right) \ge \left({\frac{1}{p} - \lambda LT\Lambda _\infty ^p} \right){\left\| x \right\|^p}- \lambda \left({LT + \frac{\mu }{\lambda }\sum\limits_{j = 1}^n {{L_j}} + \frac{\mu }{\lambda }\sum\limits_{j = 1}^n {{L_j}\Lambda _\infty ^{{d_j}}{{\left\| x \right\|}^{{d_j}}}} } \right). Choose L < \frac{{\int_0^T {\mathop {\max }\limits_{\left| x \right| \le {\Lambda _\infty }{{\left({pr} \right)}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. } p}}}} F\left({t, x} \right)dt} }}{{prT\Lambda _\infty ^p}}. We have that \frac{1}{p} - \lambda LT\Lambda _\infty ^p > 0 , for all \lambda < {A_r} . If 0 < {d_j} < p , we have that \mathop {\lim }\limits_{\left\| x \right\| \to + \infty } \left({\Phi \left(x \right) - \lambda \Psi \left(x \right)} \right) = + \infty, for all \lambda < {A_r} . Obviously, the functional \Phi - \lambda \Psi is coercive. Lemma 9 shows that \varphi = \Phi-\lambda\Psi possesses at least three different critical points in E_0^{\alpha, p} .

    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 \lambda > 0 , \mu \in ({\gamma ^*}, 0] , one has

    \begin{equation*} \begin{aligned} {\text{sup}}\left\{ {\Psi \left( x \right):\Phi \left( x \right) \le r} \right\} & = \sup \left\{ {\int_0^T {F(t, x(t))} dt - \frac{\mu }{\lambda }\sum\limits_{j = 1}^n {{J_j}(x({t_j}))} :\Phi \left( x \right) \le r} \right\} \\ & \le \int_0^T {\mathop {\max }\limits_{\left| x \right| \le {\Lambda _\infty }{{\left( {pr} \right)}^{\frac{1}{p}}}} F(t, x)} dt - \frac{\mu }{\lambda }\mathop {\max }\limits_{\left| x \right| \le {\Lambda _\infty }{{\left( {pr} \right)}^{\frac{1}{p}}}} \sum\limits_{j = 1}^n { { {J_j}(x)} } . \end{aligned} \end{equation*}

    If \mathop {\max }\limits_{\left| x \right| \le {\Lambda _\infty }{{\left({pr} \right)}^{\frac{1}{p}}}} \sum\limits_{j = 1}^n { { {J_j}(x)} } = 0 , by \lambda < {A_r} , we obtain

    \begin{equation} {\text{sup}}\left\{ {\Psi \left( x \right):\Phi \left( x \right) \le r} \right\} < \frac{r}{\lambda }. \end{equation} (8.1)

    For \mu \in ({\gamma ^*}, 0] , if \mathop {\max }\limits_{\left| x \right| \le {\Lambda _\infty }{{\left({pr} \right)}^{\frac{1}{p}}}} \sum\limits_{j = 1}^n { { {J_j}(x)} } > 0 , then (A.1) is also true. On the other hand, for \mu \in ({\gamma ^*}, 0] , we have

    \begin{equation} \Psi \left( \omega \right) = \int_0^T {F(t, \omega (t))} dt - \frac{\mu }{\lambda }\sum\limits_{j = 1}^n {{J_j}(\omega ({t_j}))} > \frac{{\Phi \left( \omega \right)}}{\lambda }. \end{equation} (8.2)

    Combining (A.1) and (A.2), we get \frac{{\Psi \left(\omega \right)}}{{\Phi \left(\omega \right)}} > \frac{1}{\lambda } > \frac{{{\text{sup}}\left\{ {\Psi \left(x \right):\Phi \left(x \right) \le r} \right\}}}{r}, which shows that the condition (i) of Lemma 9 holds. Finally, we show that \Phi-\lambda\Psi is coercive for \forall \lambda \in {\Lambda _r} = ({A_l}, {A_r}) . For x\in E_0^{\alpha, p} , by (1.7), we get

    \begin{equation} \int_0^T {F(t, x\left( t \right))} dt \le LT + LT\Lambda _\infty ^\beta {\left\| x \right\|^\beta }, \;{J_j}(x({t_j})) \le {L_j}\left( {1 + \Lambda _\infty ^{{d_j}}{{\left\| x \right\|}^{{d_j}}}} \right). \end{equation} (8.3)

    So,

    \begin{equation} \sum\limits_{j = 1}^n { { {J_j}(x({t_j}))} } \le \sum\limits_{j = 1}^n {{L_j}\left( {1 + \Lambda _\infty ^{{d_j}}{{\left\| x \right\|}^{{d_j}}}} \right).} \end{equation} (8.4)

    For x \in E_0^{\alpha, p} , if - \frac{\mu }{\lambda } \ge 0 , then, by (A.3) and (A.4), we have

    \begin{equation*} \Psi \left( x \right) \le LT + LT\Lambda _\infty ^\beta {\left\| x \right\|^\beta } - \frac{\mu }{\lambda }\sum\limits_{j = 1}^n {{L_j}\left( {1 + \Lambda _\infty ^{{d_j}}{{\left\| x \right\|}^{{d_j}}}} \right)} = LT -\frac{\mu }{\lambda }\sum\limits_{j = 1}^n {{L_j}} + LT\Lambda _\infty ^\beta {\left\| x \right\|^\beta } - \frac{\mu }{\lambda }\sum\limits_{j = 1}^n {{L_j}\Lambda _\infty ^{{d_j}}{{\left\| x \right\|}^{{d_j}}}} . \end{equation*}

    Thus, for \forall x \in E_0^{\alpha, p} , we get

    \Phi \left( x \right) - \lambda \Psi \left( x \right) \ge \frac{1}{p}{\left\| x \right\|^p} - \lambda \left( {LT - \frac{\mu }{\lambda }\sum\limits_{j = 1}^n {{L_j}} + LT\Lambda _\infty ^\beta {{\left\| x \right\|}^\beta } - \frac{\mu }{\lambda }\sum\limits_{j = 1}^n {{L_j}\Lambda _\infty ^{{d_j}}{{\left\| x \right\|}^{{d_j}}}} } \right).

    If 0 < \beta and {d_j} < p , then \mathop {\lim }\limits_{\left\| x \right\| \to + \infty } \left({\Phi \left(x \right) - \lambda \Psi \left(x \right)} \right) = + \infty, \; \lambda > 0. Thus, \Phi - \lambda \Psi is coercive. When \beta = p , \Phi \left(x \right) - \lambda \Psi \left(x \right) \ge \left({\frac{1}{p} - \lambda LT\Lambda _\infty ^p} \right){\left\| x \right\|^p} - \lambda \left({LT- \frac{\mu }{\lambda }\sum\limits_{j = 1}^n {{L_j}} - \frac{\mu }{\lambda }\sum\limits_{j = 1}^n {{L_j}\Lambda _\infty ^{{d_j}}{{\left\| x \right\|}^{{d_j}}}} } \right). Choose L < \frac{{\int_0^T {\mathop {\max }\limits_{\left| x \right| \le {\Lambda _\infty }{{\left({pr} \right)}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. } p}}}} F\left({t, x} \right)dt} }}{{prT\Lambda _\infty ^p}}. We have that \frac{1}{p} - \lambda LT\Lambda _\infty ^p > 0 for \lambda < {A_r} . If 0 < {d_j} < p for all \lambda < {A_r} , one has \mathop {\lim }\limits_{\left\| x \right\| \to + \infty } \left({\Phi \left(x \right) - \lambda \Psi \left(x \right)} \right) = + \infty. Obviously, the functional \Phi - \lambda \Psi is coercive. Lemma 9 shows that \varphi = \Phi-\lambda\Psi possesses at least three different critical points in E_0^{\alpha, p} .

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