AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 143914 10.1155/2012/143914 143914 Research Article Multiple Solutions for a Fractional Difference Boundary Value Problem via Variational Approach Xie Zuoshi 1 Jin Yuanfeng 2 Hou Chengmin 2 Liu Lishan 1 School of Economics and International Trade Zhejiang University of Finance and Economics Hangzhou Zhejiang 310018 China zufe.edu.cn 2 Department of Mathematics Yanbian University Yanji 133002 China ybu.edu.cn 2012 17 12 2012 2012 28 04 2012 05 11 2012 08 11 2012 2012 Copyright © 2012 Zuoshi Xie et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

By establishing the corresponding variational framework and using the mountain pass theorem, linking theorem, and Clark theorem in critical point theory, we give the existence of multiple solutions for a fractional difference boundary value problem with parameter. Under some suitable assumptions, we obtain some results which ensure the existence of a well precise interval of parameter for which the problem admits multiple solutions. Some examples are presented to illustrate the main results.

1. Introduction

Variational methods for dealing with difference equations have appeared as early as 1985 in  in which the positive definiteness of quadratic forms (which are functionals) is related to the existence of “nodes” of solutions (or positive solutions satisfying “conjugate” boundary conditions) of linear self-adjoint second-order difference equations of the form (1.1)Δ(p(k-1)Δy(k-1))+q(k)y(k)=0,k=1,2,,n, where p(k) is real and positive for k=0,1,,n and q(k) is real for k=1,2,,n. Later there are interests in solutions of nonlinear difference equations under various types of boundary or subsidiary conditions, and more sophisticated methods such as the mountain pass theorems are needed to handle the existence problem (see, e.g., ).

Recently, fractional differential and difference “operators” are found themselves in concrete applications, and hence attention has to be paid to associated fractional difference and differential equations under various boundary or side conditions. For example, a recent paper by Atici and Eloe  explores some of the theories of a discrete conjugate fractional BVP. Similarly, in , a discrete right-focal fractional BVP is analyzed. Other recent advances in the theory of the discrete fractional calculus may be found in [14, 15]. In particular, an interesting recent paper by Atici and Şengül  addressed the use of fractional difference equations in tumor growth modeling. Thus, it seems that there exists some promise in using fractional difference equations as mathematical models for describing physical problems in more accurate manners.

In order to handle the existence problem for fractional BVPs, various methods (among which are some standard fixed-point theorems) can be used. In this paper, however, we show that variational methods can also be applied. A good reason for picking such an approach is that, in Atici and Şengül , some basic fractional calculuses are developed and a simple variational problem is demonstrated, and hence advantage can be taken in obvious manners. We remark, however, that fractional difference operators can be approached in different manners and one by means of operator convolution rings can be found in the book by Cheng [17, Chapter 3] published in 2003.

More specifically, in this paper, we are interested in the existence of multiple solutions for the following 2ν-order fractional difference boundary value problem (1.2)  TΔt-1ν(Δν-1νtx(t))=λf(t+ν-1,x(t+ν-1)),t[0,T]0,(1.3)x(ν-2)=[Δν-1νx(t)t]t=T=0, where ν(0,1),   tΔν-1ν and   TΔtν are, respectively, left fractional difference and the right fractional difference operators (which will be explained in more detail later), t[0,T]0={0,1,2,,T}, f(t+ν-1,·):[ν-1,T+ν-1]ν-1× is continuous, and λ is a positive parameter.

By establishing the corresponding variational framework and using critical point theory, we will establish various existence results (which naturally depend on f, ν, and λ).

For convenience, throughout this paper, we arrange i=jmx(i)=0, for m<j.

2. Preliminaries

We first collect some basic lemmas for manipulating discrete fractional operators. These and other related results can be found in [14, 16].

First, for any integer β, we let β={β,β+1,β+2,}. We define t(ν):=Γ(t+1)/Γ(t+1-ν), for any t and ν for which the right-hand side is defined. We also appeal to the convention that, if t+1-ν is a pole of the Gamma function and t+1 is not a pole, then t(ν)=0.

Definition 2.1.

The νth fractional sum of f for ν>0 is defined by (2.1)Δa-νf(t)=1Γ(ν)s=at-ν(t-s-1)(ν-1)f(s), for ta-ν. We also define the νth fractional difference for ν>0 by Δνf(t):=ΔNΔν-Nf(t), where ta+N-ν and N is chosen so that 0N-1<νN.

Definition 2.2.

Let f be any real-valued function and ν(0,1). The left discrete fractional difference and the right discrete fractional difference operators are, respectively, defined as (2.2)  tΔaνf(t)=ΔtΔa-(1-ν)f(t)=1Γ(1-ν)Δs=at+ν-1(t-s-1)(-ν)f(s),ta-ν+1(mod1),  bΔtνf(t)=-ΔbΔt-(1-ν)f(t)=1Γ(1-ν)(-Δ)s=t+1-νb(s-t-1)(-ν)f(s),tb+ν-1(mod1).

Definition 2.3.

For IC1(𝔼,), we say I satisfies the Palais-Smale condition (henceforth denoted by (PS) condition) if any sequence {xn}𝔼 for which I(xn) is bounded and I(xn)0 as n+ possesses a convergent subsequence.

Lemma 2.4 (see [<xref ref-type="bibr" rid="B19">18</xref>]).

A real symmetric matrix A is positive definite if there exists a real nonsingular matrix M such that A=MM, where M is the transpose.

Lemma 2.5 (see [<xref ref-type="bibr" rid="B9">9</xref>]: linking theorem).

Let 𝔼 be a real Banach space, and IC1(𝔼,) satisfies (PS) condition and is bounded from below. Suppose I has a local linking at the origin θ, namely, there is a decomposition 𝔼=𝕐𝕎 and a positive number ρ such that k=dim𝕐<,I(y)<I(θ) for y𝕐 with 0<yρ; I(y)I(θ) for y𝕎 with yρ. Then I has at least three critical points.

Lemma 2.6 (see [<xref ref-type="bibr" rid="B6">6</xref>]).

Let 𝔼 be a real reflexive Banach space, and let the functional I:𝔼 be weakly lower (upper) semicontinuous and coercive, that is, lim||x||I(x)= (resp., anticoercive, i.e., lim||x||I(x)=-). Then there exists x0𝔼 such that I(x0)=inf𝔼I(x) (resp., I(x0)=sup𝔼I(x)). Moreover, if IC1(𝔼,), then x0 is a critical point of functional I.

Recall that, in the finite dimensional setting, it is well known that a coercive functional satisfies the (PS) condition.

Let 𝔹r denote the open ball in a real Banach space of radius r about 0, and let 𝔹r denote its boundary. Now some critical point theorems needed later can be stated.

Lemma 2.7 (mountain pass theorem [<xref ref-type="bibr" rid="B8">8</xref>]).

Let 𝔼 be a real Banach space and IC1(𝔼,), satisfying (PS) condition. Suppose I(θ)=0 and

there are constants ρ,α>0 such that I|𝔹ρα,

there is e𝔼𝔹¯ρ such that I(e)0.

Then I possesses a critical value cα. Moreover c can be characterized as (2.3)c=infgΓmaxug([0,1])I(u), where (2.4)Γ={gC([0,1],𝔼)g(0)=θ,g(1)=e}.

Lemma 2.8 (see [<xref ref-type="bibr" rid="B7">7</xref>]).

Let 𝔼 be a reflexive Banach space and IC1(𝔼,) with I(θ)=0. Suppose that I is an even functional satisfying (PS) condition and the following conditions:

there are constants ρ,α->0 and a closed linear subspace 𝕏1 of 𝔼 such that codim 𝕏1=l< and I|𝔹ρ𝕏1α-,

there is a finite dimensional subspace 𝕏2 of 𝔼 with dim𝕏2=m, m>l, such that I(x)- as ||x||, x𝕏2. Then I possesses at least m-l distinct pairs of nontrivial critical points.

Lemma 2.9 (the Clark theorem [<xref ref-type="bibr" rid="B8">8</xref>]).

Let 𝔼 be a real Banach space, IC1(𝔼,) with I even, bounded from below, and satisfying (PS) condition. Suppose I(θ)=0, there is a set 𝕂𝔼 such that 𝕂 is homeomorphic to 𝕊j-1 (the j-1 dimensional unit sphere) by an odd map and sup𝕂I<0. Then I possesses at least j distinct pairs of critical points.

3. Main Results

Firstly, we establish variational framework. Let (3.1)Ω={x=(x(ν-1),x(ν),,x(ν+T-1))x(ν+i-1),i=0,1,,T} be the T+1-dimensional Hilbert space with the usual inner product and the usual norm (3.2)x,z=t=ν-1T+ν-1x(t)z(t),x=(t=ν-1T+ν-1|x(t)|2)1/2,x,zΩ. For r>1, we recall the r-norm on Ω: xr=(t=ν-1T+ν-1|x(t)|r)1/r. We also recall the standard fact that there exist positive constants cr and c¯r, such that (3.3)cr||x||xrc¯rx,xΩ.

Define a functional on Ω by (3.4)I(x)=12t=-1T(Δν-1νtx(t))2-λt=-1TF(t+ν-1,x(t+ν-1)) for x=(x(ν-1),x(ν),,x(ν+T-1))Ω, where (3.5)F(t+ν-1,x(t+ν-1))=0x(t+ν-1)f(t+ν-1,s)ds,x(ν-2)=0,[Δν-1νtx(t)]t=T=-νΓ(1-ν)s=ν-1T+ν(T-s-1)(-ν-1)x(s)=0. Obviously, I(θ)=0. Let (3.6)𝔼={χ=(x(ν-2),x(ν-1),,x(ν+T))T+3x(ν-2)=0,[Δν-1νtx(t)]t=T=0}. Then by (1.3) it is easy to see that 𝔼 is isomorphic to Ω. In the following, when we say xΩ, we always imply that x can be extended to χ𝔼 if it is necessary. Now we claim that if x=(x(ν-1), x(ν),,x(ν+T-1))Ω is a critical point of I, then χ=(x(ν-2), x(ν-1),,x(ν+T))𝔼 is precisely a solution of BVP (1.2) and (1.3). Indeed, since I can be viewed as a continuously differentiable functional defined on the finite dimensional Hilbert space Ω, the Frechet derivative I(x) is zero if and only if I(x)/x(i)=0 for all i=ν-1,ν,,ν+T-1.

By computation, (3.7)I(x)x(ν-1)=t=-1T(Δν-1νtx(t))tΔν-1νx(t)x(ν-1)-λf(ν-1,x(ν-1))=-νΓ(1-ν)t=-1Ts=ν-1t+ν(t-s-1)(-ν-1)x(s)x(ν-1)(Δν-1νtx(t))-λf(ν-1,x(ν-1))=ν2Γ2(1-ν)t=-1T(t-ν)(-ν-1)s=ν-1t+ν(t-s-1)(-ν-1)x(s)-λf(ν-1,x(ν-1))=ν2Γ2(1-ν)s=-1T(s-ν)(-ν-1)u=ν-1s+ν(s-u-1)(-ν-1)x(u)-λf(ν-1,x(ν-1))=ν2Γ2(1-ν)[s=t-νT(s-t-1)(-ν-1)u=ν-1s+ν(s-u-1)(-ν-1)x(u)]t=ν-1-λf(ν-1,x(ν-1))=-νΓ(1-ν)[s=t-νT(s-t-1)(-ν-1)(Δν-1νs)x(s)]t=ν-1-λf(ν-1,x(ν-1))=-1Γ(1-ν)[s=t-νT((s-t-1)(-ν)-(s-t)(-ν))(Δν-1νs)x(s)]t=ν-1-λf(ν-1,x(ν-1))=1Γ(1-ν)[(-Δ)s=t-νT(s-t)(-ν)(Δν-1νs)x(s)]t=ν-1-λf(ν-1,x(ν-1))=[Δt-1νT(Δν-1νt)x(t)]t=ν-1-λf(ν-1,x(ν-1)),I(x)x(ν)=t=-1T(Δν-1νtx(t))tΔν-1νx(t)x(ν)-λf(ν,x(ν))=-νΓ(1-ν)t=-1Ts=ν-1t+ν(t-s-1)(-ν-1)x(s)x(ν)(Δν-1νtx(t))-λf(ν,x(ν))=ν2Γ2(1-ν)t=0T(t-ν-1)(-ν-1)s=ν-1t+ν(t-s-1)(-ν-1)x(s)-λf(ν,x(ν))=ν2Γ2(1-ν)s=0T(s-ν-1)(-ν-1)u=ν-1s+ν(s-u-1)(-ν-1)x(u)-λf(ν,x(ν))=ν2Γ2(1-ν)[s=t-νT(s-t-1)(-ν-1)u=ν-1s+ν(s-u-1)(-ν-1)x(u)]t=ν-λf(ν,x(ν))=-νΓ(1-ν)[s=t-νT(s-t-1)(-ν-1)(Δν-1νs)x(s)]t=ν-λf(ν,x(ν))=-1Γ(1-ν)[s=t-νT((s-t-1)(-ν)-(s-t)(-ν))(Δν-1νs)x(s)]t=ν-λf(ν,x(ν))=1Γ(1-ν)[(-Δ)s=t-νT(s-t)(-ν)(Δν-1νs)x(s)]t=ν-λf(ν,x(ν))=[Δt-1νT(Δν-1νt)x(t)]t=ν-λf(ν,x(ν)),I(x)x(ν+T-1)=t=-1T(Δν-1νtx(t))tΔν-1νx(t)x(ν+T-1)-λf(ν+T-1,x(ν+T-1))=-νΓ(1-ν)t=-1Ts=ν-1t+ν(t-s-1)(-ν-1)x(s)x(ν+T-1)(Δν-1νtx(t))-λf(ν+T-1,x(ν+T-1))=ν2Γ2(1-ν)t=T-1T(t-ν-T)(-ν-1)s=ν-1t+ν(t-s-1)(-ν-1)x(s)-λf(ν+T-1,x(ν+T-1))=ν2Γ2(1-ν)s=T-1T(s-ν-T)(-ν-1)u=ν-1s+ν(s-u-1)(-ν-1)x(u)-λf(ν+T-1,x(ν+T-1))=ν2Γ2(1-ν)[s=t-νT(s-t-1)(-ν-1)u=ν-1s+ν(s-u-1)(-ν-1)x(u)]t=ν+T-1-λf(ν+T-1,x(ν+T-1))=-νΓ(1-ν)[s=t-νT(s-t-1)(-ν-1)(Δν-1νs)x(s)]t=ν+T-1-λf(ν+T-1,x(ν+T-1))=-1Γ(1-ν)[s=t-νT((s-t-1)(-ν)-(s-t)(-ν))(Δν-1νs)x(s)]t=ν+T-1-λf(ν+T-1,x(ν+T-1))=1Γ(1-ν)[(-Δ)s=t-νT(s-t)(-ν)(Δν-1νs)x(s)]t=ν+T-1-λf(ν+T-1,x(ν+T-1))=[Δt-1νT(Δν-1νt)x(t)]t=ν+T-1-λf(ν+T-1,x(ν+T-1)). So to obtain the existence of solutions for problem (1.2) and (1.3), we just need to study the existence of critical points, that is, xΩ such that I(x)=0, of the functional I on Ω.

Next, observe by Definition 2.2 that, for t[-1,T]-1, (3.8)  tΔν-1νx(t)=Δ1Γ(1-ν)t=ν-1t-(1-ν)(t-s-1)(-ν)x(s).

We let (3.9)z(t+ν-1)=1Γ(1-ν)s=ν-1t-(1-ν)(t-s-1)(-ν)x(s), then (3.10)z(ν-2)=0,z(ν-1)=1Γ(1-ν)s=ν-10-(1-ν)(-s-1)(-ν)x(s)=x(ν-1),z(ν)=1Γ(1-ν)s=ν-11-(1-ν)(1-s-1)(-ν)x(s)=(1-ν)x(ν-1)+x(ν),z(ν+1)=1Γ(1-ν)s=ν-12-(1-ν)(2-s-1)(-ν)x(s)=(2-ν)(1-ν)2!x(ν-1)+(1-ν)x(ν)+x(ν+1),z(ν+T-1)=1Γ(1-ν)s=ν-1T-(1-ν)(T-s-1)(-ν)x(s)=(T-ν)(T-1-ν)(1-ν)(T)!x(ν-1)+(T-ν-1)(T-2-ν)(1-ν)(T-1)!x(ν)++(1-ν)x(ν+T-2)+x(ν+T-1), that is, z=Bx, where z=(z(ν-1), z(ν),,z(ν+T-1)), x=(x(ν-1), x(ν),,x(ν+T-1)): (3.11)B=(10001-ν100(2-ν)(1-ν)2!1-ν10(T-ν)(T-1-ν)(1-ν)T!(T-1-ν)(T-2-ν)(1-ν)(T-1)!1)(T+1)×(T+1).

By Lemma 2.4, (B-1)B-1 is a positive definite matrix. Let λmin and λmax denote, respectively, the minimum and the maximum eigenvalues of (B-1)B-1.

Since x=B-1z, we may easily see that (3.12)λminz2x2=z(B-1),B-1zλmaxz2.

Then ||x|| if and only if ||z||. Next, let (3.13)A=(2-1000-12-1000-12000002-1000-11)(T+1)×(T+1).

By direct verifications, we may find that A is a positive definite matrix. Let η1,η2,,ηT+1 be the orthonormal eigenvectors corresponding to the eigenvalues λ1,λ2,,λT+1 of A, where 0<λ1<λ2<<λT+1.

For convenience, we list the following assumptions.

There exists μ(0,2) such that limsup|x|(F(t+ν-1,x)/|x|μ)<a for t[0,T]0, where a is a constant.

There is a constant μ>2 such that liminf|x|(F(t+ν-1,x)/|x|μ)>0 for t[0,T]0.

There exists a constant d>0 such that limsup|x|0(F(t+ν-1,x)/|x|2)<d for t[0,T]0.

F(t+ν-1,x) satisfies limx0(F(t+ν-1,x)/|x|2)=q1>0 for t[0,T]0, where q1 is a constant.

f(t+ν-1,x) is odd with respect to x, that is, f(t+ν-1,-x)=-f(t+ν-1,x), for t[0,T]0, and x.

There is a positive constant p2 such that liminfx0(F(t+ν-1,x)/|x|2)>p2 for t[0,T]0.

Theorem 3.1.

If (C1) holds, then for all λ>0, BVP (1.2), (1.3) has at least one solution.

Proof.

By (C1), we obtain (3.14)F(t+ν-1,x)a|x|μ+b,t[0,T]0,|x|ς, where ς is some sufficiently large numbers and b>0. Thus, by the continuity of F(t+ν-1,x)-a|x|μ on [0,T]0×[-ς,ς], there exists a>0 such that (3.15)F(t+ν-1,x)a|x|μ+a,(t,s)[0,T]0×.

Combining with (3.3)–(3.15), we have (3.16)I(x)=12t=-1T(Δν-1νtx(t))2-λt=-1TF(t+ν-1,x(t+ν-1))=12t=-1T(Δz(t+ν-1))2-λt=-1TF(t+ν-1,x(t+ν-1))=12t=-1T-1(Δz(t+ν-1))2-λt=0TF(t+ν-1,x(t+ν-1))12λ1z2-λt=0TF(t+ν-1,x(t+ν-1))12λ1z2-λ|a|t=0T|x(t+ν-1)|μ-aλ(T+1)12λ1z2-λ|a|(c-μ)μxμ-aλ(T+1)12λ1z2-λ|a|(c-μ)μλmax(1/2)μzμ-aλ(T+1).

So, in view of our assumption μ(0,2), we see that, for λ>0,  I(x) as x, that is, I(x) is a coercive map. In view of Lemma 2.6, we know that there exists at least one x¯Ω such that I(x¯)=0; hence BVP (1.2), (1.3) has at least one solution. The proof is completed.

Remark 3.2.

If μ=2 and a>0, from the proof of Theorem 3.1, we can get that, for λ(0,λ1/2|a|λmax), our functional I is also coercive.

Theorem 3.3.

If (C2) holds, then for all λ>0, BVP (1.2), (1.3) has at least one solution.

Proof.

Similar to the proof Theorem 3.1, we have (3.17)I(x)12λT+1z2-λt=0TF(t+ν-1,x(t+ν-1)).

By (C2), there exists ς>0 and ς>0 such that F(t+ν-1,x)ς|x|μ for |x|>ς with t[0,T]0, so (3.18)F(t+ν-1,x)ς|x|μ-c,(t,x)[0,T]0×, where c>0. Since F(t+ν-1,x)-ς|x|μ is continuous on [0,T]0×[-ς,ς], through (3.17), we obtain (3.19)I(x)12λT+1z2-λς(cμ)μλminμ/2zμ+cλ(T+1). Thus I(x)- as ||x|| for μ>2. That is, I(x) is an anticoercive. In view of Lemma 2.6, we know that there exists at least one x¯Ω such that I(x¯)=0; hence BVP (1.2),  (1.3) has at least one solution. The proof is completed.

Theorem 3.4.

Assume (C2) and (C3) hold. Then, for λ(0,λ1/2dλmax), the BVP (1.2), (1.3) possesses at least two nontrivial solutions.

Proof.

First, we know from Theorem 3.3 that I(x)- as ||x||. Clearly, Ω is a real reflexive finite dimensional Banach space and IC1(Ω,), so functional I is weakly upper semicontinuous. By Lemma 2.6, there exists x0Ω such that I(x0)=supΩI and I(x0)=0. Set c0=supΩI. Let {xn}Ω, such that there exists M>0 and |I(xn)|M for n. By (C2) and (3.19), we may see that (3.20)-MI(xn)12λT+1zn2-λminμ/2λς(cμ)μznμ+cλ(T+1), that is, (3.21)-M-cλ(T+1)12λT+1zn2-λminμ/2λς(cμ)μznμ.

In view of μ>2, we see that {zn}Ω is bounded, and hence {xn}Ω is bounded. Since Ω is finite dimensional, there is a subsequence of {xn}, which is convergent in Ω. Therefore, the (PS) condition is verified.

By (C3), there exists δ>0,F(t+ν-1,x)dx2 for |x|δ, t[0,T]0. Thus, for xΩ with ||x||δ, we have (3.22)I(x)=12t=-1T-1(Δz(t+ν-1))2-λt=0TF(t+ν-1,x(t+ν-1))12λ1z2-λt=0TF(t+ν-1,x(t+ν-1))(12λ1-λmaxλd)z2.

For λ(0,λ1/2dλmax), we choose ρ=δ and γ=((1/2)λ1-λλmax)ρ2. Then we have I|Bργ>0, so that the condition (I1) in Lemma 2.7 holds.

Since I(x)- as ||x||, we can find eΩ with sufficiently large norm ||e|| such that I(e)<0. Hence (I2) in Lemma 2.7 is satisfied. Thus, functional I has one critical value (3.23)c=infgΓmaxug([0,1])I(u), where Γ={gC([0,1],Ω)g(0)=θ,g(1)=e}. If c0>c, the proof is completed. It suffices to consider the case c0=c. Then (3.24)c0=c=infgΓmaxug([0,1])I(u), that is, c0=maxug([0,1])I(u) for each gΓ.

Similarly, we can also choose -eΩ such that I(-e)<0. Applying Lemma 2.7 again, we obtain another critical value of the functional I, (3.25)c¯=infg¯Γ¯maxug¯([0,1])I(u), where Γ¯={g¯C([0,1],Ω)g¯(0)=θ,g¯(1)=-e}. If c0>c¯, then the proof is completed. It suffices to consider the case where c0=c¯. Then c0=maxug([0,1])I(u) for each gΓ¯. By the definitions of Γ and Γ¯, we may choose g0Γ and g¯0Γ¯ such that g0([0,1])g¯0([0,1])={θ}. Therefore, we get the maximum of the functional I on g0([0,1]){θ} and g¯0([0,1]){θ}, respectively, that is, we find two distinct nontrivial critical points of the functional I. Therefore, our BVP (1.2), (1.3) possesses at least two nontrivial solutions.

Theorem 3.5.

Assume that (C1) and (C4) hold and that there exists N[1,T+1]1 such that λN<λN+1. Then, for λ(λN/2q1λmin,λN+1/2q1λmax), BVP (1.2), (1.3) has at least three solutions.

Proof.

By (C1) and Theorem 3.1, we obtain lim||x||I(x)=, thus functional I is bounded from below. Similar to the proof of (PS) condition in Theorem 3.4, we can verify that functional I satisfies (PS) condition in our hypothesis. In order to apply linking theorem, we prove functional I is local linking at origin θ as follows. Clearly, Ω=span{η1,η2,,ηT+1}. Let 𝕏=span{η1,η2,,ηN},𝕎=span{ηN+1,ηN+2,,ηT+1}, then Ω=𝕏𝕎.

By (C4), for ε(0,q1), there exists ρ>0, such that (3.26)(q1-ε)x2F(t+ν-1,x)(q1+ε)x2,|x|ρ,t[0,T]0.

So, for x𝕏 with 0<||x||ρ, such that (3.27)t=-1T-1(Δz(t+ν-1))2=z'AzλNz2λNx2λmin,t=0TF(t+ν-1,x(t+ν-1))(q1-ε)t=0T(x(t+ν-1))2=(q1-ε)x2.

Since z=Bx𝕏, we have (3.28)I(x)=12t=-1T-1(Δz(t+ν-1))2-λt=0TF(t+ν-1,x(t+ν-1))(λN2λmin-λ(q1-ε))x2.

Thus, for λ>λN/2(q1-ε)λmin, we have I(x)<0 for x𝕏 with 0<||x||ρ.

Similarly, for x𝕎 with 0<||x||ρ, (3.29)I(x)=12t=-1T-1(Δz(t+ν-1))2-λt=0TF(t+ν-1,x(t+ν-1))(λN+12λmax-λ(q1+ε))x2, then for λ<λN+1/2(q1+ε)λmax, we have I(x)>0 for x𝕎 with 0<||x||ρ. So, by Lemma 2.5, for ε(0,q1), if λ(λN/2(q1-ε)λmin,λN+1/2(q1+ε)λmax), functional I possesses at least three critical points. By the arbitrariness of ε, we get for λ(λN/2q1λmin,λN+1/2q1λmax), the problem (1.2), (1.3) possesses at least three solutions.

Theorem 3.6.

Assume (C2), (C3), and (C5) hold. Then, for each N[0,T]0, if λ(0,λN+1/2dλmax), then BVP (1.2), (1.3) possesses at least T+1-N pairs of solutions.

Proof.

By (C5), functional I is even, and based on the proof of Theorem 3.4, we know that I satisfies (PS) condition. In order to obtain our result, we need to verify (I3) and (I4) of Lemma 2.8.

First, in view of (C3), there exists ρ>0 such that (3.30)F(t+ν-1,x)dx2for  |x|ρ,t[0,T]0.

For N[1,T+1]1, if we choose 𝕏1=span{ηN+1,ηN+2,,ηT+1}, then codim𝕏1=N.

So for x𝕏1 with ||x||ρ, since z=Bx, we have (3.31)I(x)=12t=-1T-1(Δz(t+ν-1))2-λt=0TF(t+ν-1,x(t+ν-1))(λN+12λmax-λd)x2.

Thus, for λ(0,λN+1/2dλmax), I|𝕏1𝔹ρβ>0, where β=(λN+1/2λmax-λd)ρ2, (I3) of Lemma 2.8 holds.

Next if we choose 𝕏2=span{η1,η2,,ηT+1}, then for x𝕏2, in view of (C2) and Theorem 3.3, we get I(x)- as ||x||. (I4) of Lemma 2.8 is satisfied.

Therefore, for λ(0,λN+1/2dλmax), functional I possesses at least T+1-N pair of critical points in Ω, and problem (1.2), (1.3) has at least T+1-N pairs of solutions.

Remark 3.7.

In Theorem 3.6, if we choose N=0, then for λ(0,λ1/2dλmax), the BVP (1.2), (1.3) possesses at least T+1 pairs of solutions.

Obviously, compared with Theorem 3.4, the even condition (C5) ensures that the problem (1.2), (1.3) possesses more solutions.

Theorem 3.8.

Suppose (C1), (C5), and (C6) hold. Then for every N[1,T+1]1, when λ(λN/2p2λmin,), problem (1.2), (1.3) possesses at least N pairs of nontrivial solutions.

Proof.

I ( x ) is an even functional on Ω by (C5). From (C1), we obtain I(x) as ||x||, so it is clear that I is bounded from below on Ω and satisfies the (PS) condition. For N[1,T+1]1, if we choose 𝕏1=span{η1,,ηN} and set 𝕂=𝕏1𝔹ρ, then 𝕂 is homeomorphic to 𝕊N-1 by an odd map. By (C6), there exists ρ1>0 such that F(t+ν-1,x)p2x2 for |x|ρ1, t[0,T]0. So for x𝕂1=𝕏1𝔹ρ1, (3.32)I(x)=12t=-1T(Δz(t+ν-1))2-λt=-1TF(t+ν-1,x(t+ν-1))12λNz2-λp2x2=(λN2λmin-λp2)ρ12.

For λ(λN/2p2λminx,+), we have sup𝕂1I(x)<0. Therefore, by Lemma 2.9, functional I has at least N pairs of nontrivial solutions.

Remark 3.9.

From Theorem 3.5, it is easy to see that, when f is odd about the second variable, we can obtain more solutions of the problem (1.2), (1.3), and the number of solutions depends on where λ lies.

4. Applications

In the final section, we apply the results developed in Section 3 to some examples.

Example 4.1.

Consider the following problem (4.1)Δt-1νT(Δν-1νtx(t))=λ4(t+ν-1)x3(t+ν-1)(sinx(t+ν-1)+2)+(t+ν-1)x4(t+ν-1)cos(x(t+ν-1)),t[0,T]0,x(ν-2)=[Δν-1νtx(t)]t=T=0, where f(t+ν-1,x)=4(t+ν-1)x3(sinx+2)+(t+ν-1)x4cosx. Choose μ=4 and d=1 in (C2) and (C3). Since (4.2)liminf|x|F(t+ν-1,x)|x|4=liminf|x|(t+ν-1)(sinx+2)=t+ν-1>0,t[0,T]0,limsup|x|0F(t+ν-1,x)x2=limsup|x|(t+ν-1)x2(sinx+2)=0,t[0,T]0, we see that (C2) and (C3) hold. Thus, by Theorem 3.4, when λ(0,λ1/2λmax), problem (4.1) has at least two nontrivial solutions.

Example 4.2.

Consider the problem (4.3)Δt-1νT(Δν-1νtx(t))=λ4sin(x(t+ν-1))cos(x(t+ν-1))-e-(t+ν-1)x(t+ν-1)×(2x(t+ν-1)-(t+ν-1)x2(t+ν-1))t[0,T]0,x(ν-2)=[Δtν-1νx(t)]t=T=0. Suppose there exists N0[0,T]0 such that λmaxλN0<λminλN0+1. If we choose μ=q1=a=1 in (C1) and (C4), then for t[0,T]0, we have (4.4)limsup|x|F(t+ν-1,x)|x|μ=limsup|x|2sin2x-x2e-(t+ν-1)x|x|μ<1=a,limx0F(t+ν-1,x)x2=1=q1>0, and hence (C1) and (C4) are satisfied. So, in view of Theorem 3.5, for λ(λN0/2λmin,λN0+1/2λmax), problem (4.3) has at least three solutions.

Example 4.3.

Consider the problem (4.5)Δt-1νT(Δν-1νtx(t))=λ4(t+ν-1)x3(t+ν-1)(cosx(t+ν-1)+2)-(t+ν-1)x4(t+ν-1)sin(x(t+ν-1)),t[0,T]0,x(ν-2)=[Δν-1νtx(t)]t=T=0. Condition (C5) is satisfied. If we choose μ=4 and d=1 in (C2) and (C3), then by some simple calculation, we may show that the hypotheses (C2) and (C3) are fulfilled. Therefore, by Theorem 3.6, for any N[0,T]0 and λ(0,λN+1/2λmax), problem (4.5) has at least T+1-N pairs of solutions.

Example 4.4.

Finally, consider the problem (4.6)Δt-1νT(Δν-1νtx(t))=λ1t+ν-1sin((t+ν-1)x(t+ν-1))+x(t+ν-1)cos((t+ν-1)x(t+ν-1)),t[0,T]0,x(ν-2)=[Δν-1νtx(t)]t=T=0, where f(t+ν-1,x)=(1/(t+ν-1))sin((t+ν-1)x)+xcos((t+ν-1)x). Let a=μ=1 and p2=1/2. Then it is easy to verify that (C1), (C5), and (C6) hold. Thus, by Theorem 3.8, for each N[0,T]0 and λ(λN/λmin,+), problem (4.6) has at least N pairs of solutions.

Acknowledgment

This Project was supported by the National Natural Science Foundation of China (11161049).

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