# Multiple Solutions for a Fractional Difference Boundary Value Problem via Variational Approach

Authors: Zuoshi Xie; Yuanfeng Jin; Chengmin Hou
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

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.


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## Body

## 1. Introduction

Variational methods for dealing with difference equations have appeared as early as 1985 in [1] 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., [2–11]).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 [12] explores some of the theories of a discrete conjugate fractional BVP. Similarly, in [13], 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 [16] 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 [16], 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 following2ν-order fractional difference boundary value problem
(1.2)TΔt-1ν(Δν-1νtx(t))=λf(t+ν-1,x(t+ν-1)),t∈[0,T]ℕ0,(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]ℕ0={0,1,2,…,T},  f(t+ν-1,·):[ν-1,T+ν-1]ℕν-1×ℝ→ℝ 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 onf, 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 integerβ, we let ℕβ={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.
Theνth 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 t∈ℕa-ν. We also define the νth fractional difference for ν>0 by Δνf(t):=ΔNΔν-Nf(t), where t∈ℕa+N-ν and N∈ℕ is chosen so that 0≤N-1<ν≤N.Definition 2.2.
Letf 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)Δ∑s=at+ν-1‍(t-s-1)(-ν)f(s),t≡a-ν+1(mod1),bΔtνf(t)=-ΔbΔt-(1-ν)f(t)=1Γ(1-ν)(-Δ)∑s=t+1-νb‍(s-t-1)(-ν)f(s),t≡b+ν-1(mod1).Definition 2.3.
ForI∈C1(𝔼,ℝ), 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 [18]).
A real symmetric matrixA is positive definite if there exists a real nonsingular matrix M such that A=M†M, where M† is the transpose.Lemma 2.5 (see [9]: linking theorem).
Let𝔼 be a real Banach space, and I∈C1(𝔼,ℝ) satisfies (PS) condition and is bounded from below. Suppose I has a local linking at the origin θ, namely, there is a decomposition 𝔼=𝕐⊕𝕎 and a positive number ρ such that k=dim𝕐<∞,I(y)<I(θ) for y∈𝕐 with 0<∥y∥≤ρ; I(y)≥I(θ) for y∈𝕎 with ∥y∥≤ρ. Then I has at least three critical points.Lemma 2.6 (see [6]).
Let𝔼 be a real reflexive Banach space, and let the functional I:𝔼→ℝ be weakly lower (upper) semicontinuous and coercive, that is, lim||x||→∞I(x)=∞ (resp., anticoercive, i.e., lim||x||→∞I(x)=-∞). Then there exists x0∈𝔼 such that I(x0)=inf𝔼I(x) (resp.,  I(x0)=sup𝔼I(x)). Moreover, if I∈C1(𝔼,ℝ), 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.
Let𝔹r 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 [8]).
Let𝔼 be a real Banach space and I∈C1(𝔼,ℝ), satisfying (PS) condition. Suppose I(θ)=0 and(I1)
there are constantsρ,α>0 such that I|∂𝔹ρ≥α,(I2)
there ise∈𝔼∖𝔹¯ρ such that I(e)≤0.
ThenI possesses a critical value c≥α. Moreover c can be characterized as
(2.3)c=infg∈Γmaxu∈g([0,1])I(u),
where
(2.4)Γ={g∈C([0,1],𝔼)∣g(0)=θ,g(1)=e}.Lemma 2.8 (see [7]).
Let𝔼 be a reflexive Banach space and I∈C1(𝔼,ℝ) with I(θ)=0. Suppose that I is an even functional satisfying (PS) condition and the following conditions:(I3)
there are constantsρ,α->0 and a closed linear subspace 𝕏1 of 𝔼 such that codim 𝕏1=l<∞ and I|∂𝔹ρ∩𝕏1≥α-,(I4)
there is a finite dimensional subspace𝕏2 of 𝔼 with dim𝕏2=m, m>l, such that I(x)→-∞ as ||x||→∞, x∈𝕏2. Then I possesses at least m-l distinct pairs of nontrivial critical points.Lemma 2.9 (the Clark theorem [8]).
Let𝔼 be a real Banach space, I∈C1(𝔼,ℝ) with I even, bounded from below, and satisfying (PS) condition. Suppose I(θ)=0, there is a set 𝕂⊂𝔼 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)∈ℝ,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,z∈Ω.
For r>1, we recall the r-norm on Ω: ∥x∥r=(∑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||≤∥x∥r≤c¯r∥x∥,x∈Ω.Define a functional onΩ by
(3.4)I(x)=12∑t=-1T‍(Δν-1νtx(t))2-λ∑t=-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))†∈ℝT+3∣x(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 x∈Ω, we always imply that x can be extended to χ∈𝔼 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=-1T‍∂∑s=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=-1T‍∂∑s=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=-1T‍∂∑s=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, x∈Ω such that I′(x)=0, of the functional I on Ω.Next, observe by Definition2.2 that, for t∈[-1,T]ℕ-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=(100⋯01-ν10⋯0(2-ν)(1-ν)2!1-ν1⋯0⋮⋮⋮⋮(T-ν)(T-1-ν)⋯(1-ν)T!(T-1-ν)(T-2-ν)⋯(1-ν)(T-1)!⋯⋯1)(T+1)×(T+1).By Lemma2.4, (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.Sincex=B-1z, we may easily see that
(3.12)λmin∥z∥2≤∥x∥2=〈z†(B-1)†,B-1z〉≤λmax∥z∥2.Then||x||→∞ if and only if ||z||→∞. Next, let
(3.13)A=(2-10⋯00-12-1⋯000-12⋯00⋮⋮⋮⋮⋮000⋯2-1000⋯-11)(T+1)×(T+1).By direct verifications, we may find thatA 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.(C1)
There existsμ∈(0,2) such that limsup|x|→∞(F(t+ν-1,x)/|x|μ)<a for t∈[0,T]ℕ0, where a is a constant.(C2)
There is a constantμ>2 such that liminf|x|→∞(F(t+ν-1,x)/|x|μ)>0 for t∈[0,T]ℕ0.(C3)
There exists a constantd>0 such that limsup|x|→0(F(t+ν-1,x)/|x|2)<d for t∈[0,T]ℕ0.(C4)
F(t+ν-1,x) satisfies limx→0(F(t+ν-1,x)/|x|2)=q1>0 for t∈[0,T]ℕ0, where q1 is a constant.(C5)
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]ℕ0, and x∈ℝ.(C6)
There is a positive constantp2 such that liminfx→0(F(t+ν-1,x)/|x|2)>p2 for t∈[0,T]ℕ0.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]ℕ0,|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]ℕ0×[-ς,ς], there exists a′>0 such that
(3.15)F(t+ν-1,x)≤a|x|μ+a′,(t,s)∈[0,T]ℕ0×ℝ.
Combining with (3.3)–(3.15), we have
(3.16)I(x)=12∑t=-1T‍(Δν-1νtx(t))2-λ∑t=-1T‍F(t+ν-1,x(t+ν-1))=12∑t=-1T‍(Δz(t+ν-1))2-λ∑t=-1T‍F(t+ν-1,x(t+ν-1))=12∑t=-1T-1‍(Δz(t+ν-1))2-λ∑t=0T‍F(t+ν-1,x(t+ν-1))≥12λ1∥z∥2-λ∑t=0T‍F(t+ν-1,x(t+ν-1))≥12λ1∥z∥2-λ|a|∑t=0T‍|x(t+ν-1)|μ-a′λ(T+1)≥12λ1∥z∥2-λ|a|(c-μ)μ∥x∥μ-a′λ(T+1)≥12λ1∥z∥2-λ|a|(c-μ)μλmax(1/2)μ∥z∥μ-a′λ(T+1).
So, in view of our assumptionμ∈(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.
Ifμ=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 Theorem3.1, we have
(3.17)I(x)≤12λT+1∥z∥2-λ∑t=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]ℕ0, so
(3.18)F(t+ν-1,x)≥ς|x|μ-c,(t,x)∈[0,T]ℕ0×ℝ,
where c>0. Since F(t+ν-1,x)-ς|x|μ is continuous on [0,T]ℕ0×[-ς′,ς′], through (3.17), we obtain
(3.19)I(x)≤12λT+1∥z∥2-λς(cμ)μλminμ/2∥z∥μ+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 Theorem3.3 that I(x)→-∞ as ||x||→∞. Clearly, Ω is a real reflexive finite dimensional Banach space and I∈C1(Ω,ℝ), so functional I is weakly upper semicontinuous. By Lemma 2.6, there exists x0∈Ω 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∈ℕ. By (C2) and (3.19), we may see that
(3.20)-M≤I(xn)≤12λT+1∥zn∥2-λminμ/2λς(cμ)μ∥zn∥μ+cλ(T+1),
that is,
(3.21)-M-cλ(T+1)≤12λT+1∥zn∥2-λminμ/2λς(cμ)μ∥zn∥μ.
In view ofμ>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]ℕ0. Thus, for x∈Ω with ||x||≤δ, we have
(3.22)I(x)=12∑t=-1T-1‍(Δz(t+ν-1))2-λ∑t=0T‍F(t+ν-1,x(t+ν-1))≥12λ1∥z∥2-λ∑t=0T‍F(t+ν-1,x(t+ν-1))≥(12λ1-λmaxλd)∥z∥2.
Forλ∈(0,λ1/2dλmax), we choose ρ=δ and γ=((1/2)λ1-λλmax)ρ2. Then we have I|∂Bρ≥γ>0, so that the condition (I1) in Lemma 2.7 holds.
SinceI(x)→-∞ as ||x||→∞, we can find e∈Ω 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=infg∈Γmaxu∈g([0,1])I(u),
where Γ={g∈C([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=infg∈Γmaxu∈g([0,1])I(u),
that is, c0=maxu∈g([0,1])I(u) for each g∈Γ.
Similarly, we can also choose-e∈Ω such that I(-e)<0. Applying Lemma 2.7 again, we obtain another critical value of the functional I,
(3.25)c¯=infg¯∈Γ¯maxu∈g¯([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=maxu∈g([0,1])I(u) for each g∈Γ¯. By the definitions of Γ and Γ¯, we may choose g0∈Γ and g¯0∈Γ¯ 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]ℕ1 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=𝕏⊕𝕎.
By (C4), for ε∈(0,q1), there exists ρ>0, such that
(3.26)(q1-ε)x2≤F(t+ν-1,x)≤(q1+ε)x2,|x|≤ρ,t∈[0,T]ℕ0.
So, forx∈𝕏 with 0<||x||≤ρ, such that
(3.27)∑t=-1T-1(Δz(t+ν-1))2=z'Az≤λN∥z∥2≤λN∥x∥2λmin,∑t=0TF(t+ν-1,x(t+ν-1))≥(q1-ε)∑t=0T(x(t+ν-1))2=(q1-ε)∥x∥2.
Sincez=Bx∈𝕏, we have
(3.28)I(x)=12∑t=-1T-1‍(Δz(t+ν-1))2-λ∑t=0T‍F(t+ν-1,x(t+ν-1))≤(λN2λmin-λ(q1-ε))∥x∥2.
Thus, forλ>λN/2(q1-ε)λmin, we have I(x)<0 for x∈𝕏 with 0<||x||≤ρ.
Similarly, forx∈𝕎 with 0<||x||≤ρ,
(3.29)I(x)=12∑t=-1T-1‍(Δz(t+ν-1))2-λ∑t=0T‍F(t+ν-1,x(t+ν-1))≥(λN+12λmax-λ(q1+ε))∥x∥2,
then for λ<λN+1/2(q1+ε)λmax, we have I(x)>0 for x∈𝕎 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]ℕ0, 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]ℕ0.
ForN∈[1,T+1]ℕ1, if we choose 𝕏1=span{ηN+1,ηN+2,…,ηT+1}, then codim𝕏1=N.
So forx∈𝕏1 with ||x||≤ρ, since z=Bx, we have
(3.31)I(x)=12∑t=-1T-1‍(Δz(t+ν-1))2-λ∑t=0T‍F(t+ν-1,x(t+ν-1))≥(λN+12λmax-λd)∥x∥2.
Thus, forλ∈(0,λN+1/2dλmax), I|𝕏1⋂‍∂𝔹ρ≥β>0, where β=(λN+1/2λmax-λd)ρ2, (I3) of Lemma 2.8 holds.
Next if we choose𝕏2=span{η1,η2,…,ηT+1}, then for x∈𝕏2, in view of (C2) and Theorem 3.3, we get I(x)→-∞ as ||x||→∞. (I4) of Lemma 2.8 is satisfied.
Therefore, forλ∈(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 Theorem3.6, 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 Theorem3.4, 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]ℕ1, 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]ℕ1, if we choose 𝕏1=span{η1,…,ηN} and set 𝕂=𝕏1∩∂𝔹ρ, 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]ℕ0. So for x∈𝕂1=𝕏1∩∂𝔹ρ1,
(3.32)I(x)=12∑t=-1T‍(Δz(t+ν-1))2-λ∑t=-1T‍F(t+ν-1,x(t+ν-1))≤12λN∥z∥2-λp2∥x∥2=(λN2λmin-λp2)ρ12.
Forλ∈(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 Theorem3.5, 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 Section3 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]ℕ0,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]ℕ0,limsup|x|→0F(t+ν-1,x)x2=limsup|x|→∞(t+ν-1)x2(sinx+2)=0,t∈[0,T]ℕ0,
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]ℕ0,x(ν-2)=[Δtν-1νx(t)]t=T=0.
Suppose there exists N0∈[0,T]ℕ0 such that λmaxλN0<λminλN0+1. If we choose μ=q1=a=1 in (C1) and (C4), then for t∈[0,T]ℕ0, 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,limx→0F(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]ℕ0,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]ℕ0 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]ℕ0,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]ℕ0 and λ∈(λN/λmin,+∞), problem (4.6) has at least N pairs of solutions.



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*Source: 143914-2012-12-17.xml*