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

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.

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 Δ𝑝(𝑘−1)Δ𝑦(𝑘−1)+𝑞(𝑘)𝑦(𝑘)=0,𝑘=1,2,…,𝑛, (1.1) 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 Sengül [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 Sengül [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 following 2𝜈-order fractional difference boundary value problem _𝑇Δ^𝜈_𝑡−1𝑡Δ^𝜈_𝜈−1𝑥(𝑡)=𝜆𝑓(𝑡+𝜈−1,𝑥(𝑡+𝜈−1)),𝑡∈[0,𝑇]_ℕ0,𝑥(𝜈−2)=_𝑡Δ^𝜈_𝜈−1𝑥(𝑡)_𝑡=𝑇=0, (1.2)(1.3) where 𝜈∈(0,1), _𝑡Δ^𝜈_𝜈−1 and _𝑇Δ^𝜈_𝑡 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 on 𝑓, 𝜈, and 𝜆).

For convenience, throughout this paper, we arrange ∑^𝑚_𝑖=𝑗𝑥(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 ℕ_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 Δ_𝑎^−𝜈𝑓(𝑡)=1Γ(𝜈)^𝑡−𝜈_𝑠=𝑎(𝑡−𝑠−1)^(𝜈−1)𝑓(𝑠), (2.1) for 𝑡∈ℕ_𝑎−𝜈. We also define the 𝜈th fractional difference for 𝜈>0 by Δ^𝜈𝑓(𝑡)∶=Δ^𝑁Δ^𝜈−𝑁𝑓(𝑡), where 𝑡∈ℕ_{𝑎+𝑁−𝜈} and 𝑁∈ℕ is chosen so that 0≤𝑁−1<𝜈≤𝑁.

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 _𝑡Δ^𝜈_𝑎𝑓(𝑡)=Δ_𝑡Δ_𝑎^{−(1−𝜈)}𝑓(𝑡)=1Γ()Δ1−𝜈^{𝑡+𝜈−1}_𝑠=𝑎(𝑡−𝑠−1)^(−𝜈)𝑓(𝑠),𝑡≡𝑎−𝜈+1(mod1_𝑏),Δ^𝜈_𝑡𝑓(𝑡)=−Δ_𝑏Δ_𝑡^{−(1−𝜈)}𝑓(𝑡)=1Γ()()1−𝜈−Δ^𝑏_{𝑠=𝑡+1−𝜈}(𝑠−𝑡−1)^(−𝜈)𝑓(𝑠),𝑡≡𝑏+𝜈−1(mod1). (2.2)

Definition 2.3. For 𝐼∈𝐶¹(𝔼,ℝ), we say 𝐼 satisfies the Palais-Smale condition (henceforth denoted by (PS) condition) if any sequence {𝑥_𝑛}⊂𝔼 for which 𝐼(𝑥_𝑛) is bounded and 𝐼^′(𝑥_𝑛)→0 as 𝑛→+∞ possesses a convergent subsequence.

Lemma 2.4 (see [18]). A real symmetric matrix 𝐴 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). Let 𝔼 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]). Let 𝔼 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)=inf_𝔼𝐼(𝑥) (resp., 𝐼(𝑥₀)=sup_𝔼𝐼(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.
Let 𝔹_𝑟 denote the open ball in a real Banach space of radius 𝑟 about 0, and let 𝜕𝔹_𝑟 denote its boundary. Now some critical point theorems needed later can be stated.

Lemma 2.7 (mountain pass theorem [8]). Let 𝔼 be a real Banach space and 𝐼∈𝐶¹(𝔼,ℝ), satisfying (PS) condition. Suppose 𝐼(𝜃)=0 and(I₁) there are constants 𝜌,𝛼>0 such that 𝐼|_𝜕𝔹𝜌≥𝛼,(I₂) there is 𝑒∈𝔼⧵𝔹_𝜌 such that 𝐼(𝑒)≤0.
Then 𝐼 possesses a critical value 𝑐≥𝛼. Moreover 𝑐 can be characterized as 𝑐=inf_𝑔∈Γmax_{𝑢∈𝑔([0,1])}𝐼(𝑢), (2.3) where Γ=𝑔∈𝐶([0,1],𝔼)∣𝑔(0)=𝜃,𝑔(1)=𝑒. (2.4)

Lemma 2.8 (see [7]). Let 𝔼 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 constants 𝜌,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 subspace 𝕏₂ 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]). Let 𝔼 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 𝕊^𝑗−1 (the 𝑗−1 dimensional unit sphere) by an odd map and sup_𝕂𝐼<0. Then 𝐼 possesses at least 𝑗 distinct pairs of critical points.

3. Main Results

Firstly, we establish variational framework. Let Ω=𝑥=(𝑥(𝜈−1),𝑥(𝜈),…,𝑥(𝜈+𝑇−1))^†∣𝑥(𝜈+𝑖−1)∈ℝ,𝑖=0,1,…,𝑇 (3.1) be the 𝑇+1-dimensional Hilbert space with the usual inner product and the usual norm ⟨𝑥,𝑧⟩=^{𝑇+𝜈−1}_{𝑡=𝜈−1}𝑥(𝑡)𝑧(𝑡),‖𝑥‖=^{𝑇+𝜈−1}_{𝑡=𝜈−1}|𝑥(𝑡)|^21/2,𝑥,𝑧∈Ω. (3.2) For 𝑟>1, we recall the 𝑟-norm on Ω: ‖𝑥‖_𝑟=(∑^{𝑇+𝜈−1}_{𝑡=𝜈−1}|𝑥(𝑡)|^𝑟)^1/𝑟. We also recall the standard fact that there exist positive constants 𝑐_𝑟 and 𝑐_𝑟, such that 𝑐_𝑟||𝑥||≤‖𝑥‖_𝑟≤𝑐_𝑟‖𝑥‖,𝑥∈Ω. (3.3)

Define a functional on Ω by 𝐼(𝑥)=12^𝑇_𝑡=−1^𝑡Δ^𝜈_𝜈−1𝑥(𝑡)²−𝜆^𝑇_𝑡=−1𝐹(𝑡+𝜈−1,𝑥(𝑡+𝜈−1)) (3.4) for 𝑥=(𝑥(𝜈−1),𝑥(𝜈),…,𝑥(𝜈+𝑇−1))^†∈Ω, where 𝐹(𝑡+𝜈−1,𝑥(𝑡+𝜈−1))=₀^{𝑥(𝑡+𝜈−1)}𝑓()𝑥()𝑡+𝜈−1,𝑠𝑑𝑠,𝜈−2=0,_𝑡Δ^𝜈_𝜈−1𝑥(𝑡)_𝑡=𝑇=−𝜈Γ(1−𝜈)^𝑇+𝜈_{𝑠=𝜈−1}(𝑇−𝑠−1)^{(−𝜈−1)}𝑥(𝑠)=0. (3.5) Obviously, 𝐼(𝜃)=0. Let 𝔼=𝜒=(𝑥(𝜈−2),𝑥(𝜈−1),…,𝑥(𝜈+𝑇))^†∈ℝ^𝑇+3∣𝑥(𝜈−2)=0,_𝑡Δ^𝜈_𝜈−1𝑥(𝑡)_𝑡=𝑇=0. (3.6) 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, 𝜕𝐼(𝑥)()=𝜕𝑥𝜈−1^𝑇_𝑡=−1^𝑡Δ^𝜈_𝜈−1𝑥(𝑡𝜕)^𝑡Δ^𝜈_𝜈−1𝑥(𝑡)()((=𝜕𝑥𝜈−1−𝜆𝑓𝜈−1,𝑥𝜈−1))−𝜈Γ(1−𝜈)^𝑇_𝑡=−1𝜕∑^𝑡+𝜈_{𝑠=𝜈−1}(𝑡−𝑠−1)^{(−𝜈−1)}𝑥(𝑠)𝜕𝑥(𝜈−1)^𝑡Δ^𝜈_𝜈−1𝑥(𝑡)((=𝜈−𝜆𝑓𝜈−1,𝑥𝜈−1))₂Γ²(1−𝜈)^𝑇_𝑡=−1(𝑡−𝜈)^{(−𝜈−1)𝑡+𝜈}_{𝑠=𝜈−1}(𝑡−𝑠−1)^{(−𝜈−1)}𝑥(𝑠)((=𝜈−𝜆𝑓𝜈−1,𝑥𝜈−1))₂Γ²(1−𝜈)^𝑇_𝑠=−1(𝑠−𝜈)^{(−𝜈−1)𝑠+𝜈}_{𝑢=𝜈−1}(𝑠−𝑢−1)^{(−𝜈−1)}𝑥(𝑢)((=𝜈−𝜆𝑓𝜈−1,𝑥𝜈−1))₂Γ²(1−𝜈)^𝑇_{𝑠=𝑡−𝜈}(𝑠−𝑡−1)^{(−𝜈−1)𝑠+𝜈}_{𝑢=𝜈−1}(𝑠−𝑢−1)^{(−𝜈−1)}𝑥(𝑢)_{𝑡=𝜈−1}((=−𝜆𝑓𝜈−1,𝑥𝜈−1))−𝜈Γ(1−𝜈)^𝑇_{𝑠=𝑡−𝜈}(𝑠−𝑡−1)^{(−𝜈−1)}_𝑠Δ^𝜈_𝜈−1𝑥(𝑠)_{𝑡=𝜈−1}((=−𝜆𝑓𝜈−1,𝑥𝜈−1))−1Γ(1−𝜈)^𝑇_{𝑠=𝑡−𝜈}(𝑠−𝑡−1)^(−𝜈)−(𝑠−𝑡)^(−𝜈)_𝑠Δ^𝜈_𝜈−1𝑥(𝑠)_{𝑡=𝜈−1}((=1−𝜆𝑓𝜈−1,𝑥𝜈−1))Γ())1−𝜈(−Δ^𝑇_{𝑠=𝑡−𝜈}(𝑠−𝑡)^(−𝜈)_𝑠Δ^𝜈_𝜈−1𝑥(𝑠)_{𝑡=𝜈−1}((=−𝜆𝑓𝜈−1,𝑥𝜈−1))_𝑇Δ^𝜈_𝑡−1𝑡Δ^𝜈_𝜈−1𝑥(𝑡)_{𝑡=𝜈−1}((,(𝑥)−𝜆𝑓𝜈−1,𝑥𝜈−1))𝜕𝐼(𝜈)=𝜕𝑥^𝑇_𝑡=−1^𝑡Δ^𝜈_𝜈−1𝑥(𝑡)𝜕_𝑡Δ^𝜈_𝜈−1𝑥(𝑡)(𝜈)((𝜈=𝜕𝑥−𝜆𝑓𝜈,𝑥))−𝜈Γ(1−𝜈)^𝑇_𝑡=−1𝜕∑^𝑡+𝜈_{𝑠=𝜈−1}(𝑡−𝑠−1)^{(−𝜈−1)}𝑥(𝑠)𝜕𝑥(𝜈)^𝑡Δ^𝜈_𝜈−1𝑥(𝑡)((𝜈=𝜈−𝜆𝑓𝜈,𝑥))₂Γ²(1−𝜈)^𝑇_𝑡=0(𝑡−𝜈−1)^{(−𝜈−1)𝑡+𝜈}_{𝑠=𝜈−1}(𝑡−𝑠−1)^{(−𝜈−1)}𝑥(𝑠)((𝜈=𝜈−𝜆𝑓𝜈,𝑥))₂Γ²(1−𝜈)^𝑇_𝑠=0(𝑠−𝜈−1)^{(−𝜈−1)𝑠+𝜈}_{𝑢=𝜈−1}(𝑠−𝑢−1)^{(−𝜈−1)}𝑥(𝑢)((𝜈=𝜈−𝜆𝑓𝜈,𝑥))₂Γ²(1−𝜈)^𝑇_{𝑠=𝑡−𝜈}(𝑠−𝑡−1)^{(−𝜈−1)𝑠+𝜈}_{𝑢=𝜈−1}(𝑠−𝑢−1)^{(−𝜈−1)}𝑥(𝑢)_𝑡=𝜈((𝜈=−𝜆𝑓𝜈,𝑥))−𝜈Γ(1−𝜈)^𝑇_{𝑠=𝑡−𝜈}(𝑠−𝑡−1)^{(−𝜈−1)}_𝑠Δ^𝜈_𝜈−1𝑥(𝑠)_𝑡=𝜈((𝜈=−𝜆𝑓𝜈,𝑥))−1Γ(1−𝜈)^𝑇_{𝑠=𝑡−𝜈}(𝑠−𝑡−1)^(−𝜈)−(𝑠−𝑡)^(−𝜈)_𝑠Δ^𝜈_𝜈−1𝑥(𝑠)_𝑡=𝜈((𝜈=1−𝜆𝑓𝜈,𝑥))Γ())1−𝜈(−Δ^𝑇_{𝑠=𝑡−𝜈}(𝑠−𝑡)^(−𝜈)_𝑠Δ^𝜈_𝜈−1𝑥(𝑠)_𝑡=𝜈((𝜈=−𝜆𝑓𝜈,𝑥))_𝑇Δ^𝜈_𝑡−1𝑡Δ^𝜈_𝜈−1𝑥(𝑡)_𝑡=𝜈((𝜈,⋮(𝑥)−𝜆𝑓𝜈,𝑥))𝜕𝐼()=𝜕𝑥𝜈+𝑇−1^𝑇_𝑡=−1^𝑡Δ^𝜈_𝜈−1𝑥(𝑡)𝜕_𝑡Δ^𝜈_𝜈−1𝑥(𝑡)()((=𝜕𝑥𝜈+𝑇−1−𝜆𝑓𝜈+𝑇−1,𝑥𝜈+𝑇−1))−𝜈Γ(1−𝜈)^𝑇_𝑡=−1𝜕∑^𝑡+𝜈_{𝑠=𝜈−1}(𝑡−𝑠−1)^{(−𝜈−1)}𝑥(𝑠)𝜕𝑥(𝜈+𝑇−1)^𝑡Δ^𝜈_𝜈−1𝑥(𝑡)((=𝜈−𝜆𝑓𝜈+𝑇−1,𝑥𝜈+𝑇−1))₂Γ²(1−𝜈)^𝑇_{𝑡=𝑇−1}(𝑡−𝜈−𝑇)^{(−𝜈−1)𝑡+𝜈}_{𝑠=𝜈−1}(𝑡−𝑠−1)^{(−𝜈−1)}𝑥(𝑠)((=𝜈−𝜆𝑓𝜈+𝑇−1,𝑥𝜈+𝑇−1))₂Γ²(1−𝜈)^𝑇_{𝑠=𝑇−1}(𝑠−𝜈−𝑇)^{(−𝜈−1)𝑠+𝜈}_{𝑢=𝜈−1}(𝑠−𝑢−1)^{(−𝜈−1)}𝑥(𝑢)((=𝜈−𝜆𝑓𝜈+𝑇−1,𝑥𝜈+𝑇−1))₂Γ²(1−𝜈)^𝑇_{𝑠=𝑡−𝜈}(𝑠−𝑡−1)^{(−𝜈−1)𝑠+𝜈}_{𝑢=𝜈−1}(𝑠−𝑢−1)^{(−𝜈−1)}𝑥(𝑢)_{𝑡=𝜈+𝑇−1}((=−𝜆𝑓𝜈+𝑇−1,𝑥𝜈+𝑇−1))−𝜈Γ(1−𝜈)^𝑇_{𝑠=𝑡−𝜈}(𝑠−𝑡−1)^{(−𝜈−1)}_𝑠Δ^𝜈_𝜈−1𝑥(𝑠)_{𝑡=𝜈+𝑇−1}((=−𝜆𝑓𝜈+𝑇−1,𝑥𝜈+𝑇−1))−1Γ(1−𝜈)^𝑇_{𝑠=𝑡−𝜈}(𝑠−𝑡−1)^(−𝜈)−(𝑠−𝑡)^(−𝜈)_𝑠Δ^𝜈_𝜈−1𝑥(𝑠)_{𝑡=𝜈+𝑇−1}((=1−𝜆𝑓𝜈+𝑇−1,𝑥𝜈+𝑇−1))Γ())1−𝜈(−Δ^𝑇_{𝑠=𝑡−𝜈}(𝑠−𝑡)^(−𝜈)_𝑠Δ^𝜈_𝜈−1𝑥(𝑠)_{𝑡=𝜈+𝑇−1}((=−𝜆𝑓𝜈+𝑇−1,𝑥𝜈+𝑇−1))_𝑇Δ^𝜈_𝑡−1𝑡Δ^𝜈_𝜈−1𝑥(𝑡)_{𝑡=𝜈+𝑇−1}−𝜆𝑓(𝜈+𝑇−1,𝑥(𝜈+𝑇−1)). (3.7) 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 Definition 2.2 that, for 𝑡∈[−1,𝑇]_ℕ−1, _𝑡Δ^𝜈_𝜈−1𝑥(𝑡)1=ΔΓ(1−𝜈)^{𝑡−(1−𝜈)}_{𝑡=𝜈−1}(𝑡−𝑠−1)^(−𝜈)𝑥(𝑠). (3.8)

We let 𝑧()=1𝑡+𝜈−1Γ(1−𝜈)^{𝑡−(1−𝜈)}_{𝑠=𝜈−1}(𝑡−𝑠−1)^(−𝜈)𝑥(𝑠), (3.9) then 𝑧()𝑧()=1𝜈−2=0,𝜈−1Γ(1−𝜈)^{0−(1−𝜈)}_{𝑠=𝜈−1}(−𝑠−1)^(−𝜈)𝑥(𝑠)(),𝑧(𝜈)=1=𝑥𝜈−1Γ(1−𝜈)^{1−(1−𝜈)}_{𝑠=𝜈−1}(1−𝑠−1)^(−𝜈)𝑥(𝑠)=()𝑥()(𝜈),𝑧()=11−𝜈𝜈−1+𝑥𝜈+1Γ(1−𝜈)^{2−(1−𝜈)}_{𝑠=𝜈−1}(2−𝑠−1)^(−𝜈)𝑥(𝑠)=(2−𝜈)(1−𝜈)𝑥()+()𝑥(𝜈)(),⋮𝑧()=12!𝜈−11−𝜈+𝑥𝜈+1𝜈+𝑇−1Γ(1−𝜈)^{𝑇−(1−𝜈)}_{𝑠=𝜈−1}(𝑇−𝑠−1)^(−𝜈)𝑥(𝑠)=(𝑇−𝜈)(𝑇−1−𝜈)⋯(1−𝜈)(𝑇)!𝑥()+()⋯()𝜈−1𝑇−𝜈−1)(𝑇−2−𝜈1−𝜈()!𝑥(𝜈)()𝑥()(),𝑇−1+⋯+1−𝜈𝜈+𝑇−2+𝑥𝜈+𝑇−1 (3.10) 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))^†: ()𝐵=100⋯01−𝜈10⋯02−𝜈)(1−𝜈()⋯()2!1−𝜈1⋯0⋮⋮⋮⋮𝑇−𝜈)(𝑇−1−𝜈1−𝜈()⋯()𝑇!𝑇−1−𝜈)(𝑇−2−𝜈1−𝜈(𝑇−1)!⋯⋯1_{(𝑇+1)×(𝑇+1)}. (3.11)

By Lemma 2.4, (𝐵⁻¹)^†𝐵⁻¹ is a positive definite matrix. Let 𝜆_min and 𝜆_max denote, respectively, the minimum and the maximum eigenvalues of (𝐵⁻¹)^†𝐵⁻¹.

Since 𝑥=𝐵⁻¹𝑧, we may easily see that 𝜆_min‖𝑧‖²≤‖𝑥‖²=𝑧^†𝐵^−1†,𝐵⁻¹𝑧≤𝜆_max‖𝑧‖². (3.12)

Then ||𝑥||→∞ if and only if ||𝑧||→∞. Next, let 𝐴=2−10⋯00−12−1⋯000−12⋯00⋮⋮⋮⋮⋮000⋯2−1000⋯−11_{(𝑇+1)×(𝑇+1)}. (3.13)

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

For convenience, we list the following assumptions.(C₁) There exists 𝜇∈(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 constant 𝜇>2 such that liminf_|𝑥|→∞(𝐹(𝑡+𝜈−1,𝑥)/|𝑥|^𝜇)>0 for 𝑡∈[0,𝑇]_ℕ0.(C₃) There exists a constant 𝑑>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 lim_𝑥→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 constant 𝑝₂ such that liminf_𝑥→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 𝐹(𝑡+𝜈−1,𝑥)≤𝑎|𝑥|^𝜇+𝑏,𝑡∈[0,𝑇]_ℕ0,|𝑥|≥𝜍, (3.14) 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 𝐹(𝑡+𝜈−1,𝑥)≤𝑎|𝑥|^𝜇+𝑎^′,(𝑡,𝑠)∈[0,𝑇]_ℕ0×ℝ. (3.15)
Combining with (3.3)–(3.15), we have 𝐼(𝑥)=12^𝑇_𝑡=−1^𝑡Δ^𝜈_𝜈−1𝑥(𝑡)²−𝜆^𝑇_𝑡=−1𝐹((=1𝑡+𝜈−1,𝑥𝑡+𝜈−1))2^𝑇_𝑡=−1(Δ𝑧(𝑡+𝜈−1))²−𝜆^𝑇_𝑡=−1𝐹((=1𝑡+𝜈−1,𝑥𝑡+𝜈−1))2^𝑇−1_𝑡=−1(Δ𝑧(𝑡+𝜈−1))²−𝜆^𝑇_𝑡=0𝐹((≥1𝑡+𝜈−1,𝑥𝑡+𝜈−1))2𝜆₁‖𝑧‖²−𝜆^𝑇_𝑡=0𝐹((≥1𝑡+𝜈−1,𝑥𝑡+𝜈−1))2𝜆₁‖𝑧‖²−𝜆|𝑎|^𝑇_𝑡=0|𝑥(𝑡+𝜈−1)|^𝜇−𝑎^′𝜆()≥1𝑇+12𝜆₁‖𝑧‖²−𝜆|𝑎|𝑐_𝜇^𝜇‖𝑥‖^𝜇−𝑎^′𝜆()≥1𝑇+12𝜆₁‖𝑧‖²−𝜆|𝑎|𝑐_𝜇^𝜇𝜆^(1/2)𝜇_max‖𝑧‖^𝜇−𝑎^′𝜆(𝑇+1). (3.16)
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|𝜆_max), 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 𝐼(𝑥)≤12𝜆_𝑇+1‖𝑧‖²−𝜆^𝑇_𝑡=0𝐹(𝑡+𝜈−1,𝑥(𝑡+𝜈−1)). (3.17)
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 𝐹(𝑡+𝜈−1,𝑥)≥𝜍|𝑥|^𝜇−𝑐,(𝑡,𝑥)∈[0,𝑇]_ℕ0×ℝ, (3.18) 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 𝐼(𝑥)≤12𝜆_𝑇+1‖𝑧‖²−𝜆𝜍𝑐_𝜇^𝜇𝜆^𝜇/2_min‖𝑧‖^𝜇+𝑐𝜆(𝑇+1). (3.19) 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,𝜆₁/2𝑑𝜆_max), 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 𝐼∈𝐶¹(Ω,ℝ), so functional 𝐼 is weakly upper semicontinuous. By Lemma 2.6, there exists 𝑥₀∈Ω such that 𝐼(𝑥₀)=sup_Ω𝐼 and 𝐼^′(𝑥₀)=0. Set 𝑐₀=sup_Ω𝐼. Let {𝑥_𝑛}⊂Ω, such that there exists 𝑀>0 and |𝐼(𝑥_𝑛)|≤𝑀 for 𝑛∈ℕ. By (C₂) and (3.19), we may see that −𝑀≤𝐼𝑥_𝑛≤12𝜆_𝑇+1𝑧_𝑛²−𝜆^𝜇/2_min𝜆𝜍𝑐_𝜇^𝜇𝑧_𝑛^𝜇+𝑐𝜆(𝑇+1), (3.20) that is, ()≤1−𝑀−𝑐𝜆𝑇+12𝜆_𝑇+1𝑧_𝑛²−𝜆^𝜇/2_min𝜆𝜍𝑐_𝜇^𝜇𝑧_𝑛^𝜇. (3.21)
In view of 𝜇>2, we see that {𝑧_𝑛}⊂Ω is bounded, and hence {𝑥_𝑛}⊂Ω is bounded. Since Ω is finite dimensional, there is a subsequence of {𝑥_𝑛}, 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 𝐼(𝑥)=12^𝑇−1_𝑡=−1(Δ𝑧(𝑡+𝜈−1))²−𝜆^𝑇_𝑡=0𝐹((≥1𝑡+𝜈−1,𝑥𝑡+𝜈−1))2𝜆₁‖𝑧‖²−𝜆^𝑇_𝑡=0𝐹((≥1𝑡+𝜈−1,𝑥𝑡+𝜈−1))2𝜆₁−𝜆_max𝜆𝑑‖𝑧‖². (3.22)
For 𝜆∈(0,𝜆₁/2𝑑𝜆_max), we choose 𝜌=𝛿 and 𝛾=((1/2)𝜆₁−𝜆𝜆_max)𝜌². Then we have 𝐼|_𝜕𝐵𝜌≥𝛾>0, so that the condition (I₁) in Lemma 2.7 holds.
Since 𝐼(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 𝑐=inf_𝑔∈Γmax_{𝑢∈𝑔([0,1])}𝐼(𝑢), (3.23) 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 𝑐₀=𝑐=inf_𝑔∈Γmax_{𝑢∈𝑔([0,1])}𝐼(𝑢), (3.24) that is, 𝑐₀=max_{𝑢∈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, 𝑐=inf_𝑔∈Γmax_{𝑢∈𝑔([0,1])}𝐼(𝑢), (3.25) 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=max_{𝑢∈𝑔([0,1])}𝐼(𝑢) for each 𝑔∈Γ. By the definitions of Γ and Γ, we may choose 𝑔₀∈Γ and 𝑔₀∈Γ such that 𝑔₀([0,1])∩𝑔₀([0,1])={𝜃}. Therefore, we get the maximum of the functional 𝐼 on 𝑔₀([0,1])⧵{𝜃} and 𝑔₀([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 𝜆_𝑁<𝜆_𝑁+1. Then, for 𝜆∈(𝜆_𝑁/2𝑞₁𝜆_min,𝜆_𝑁+1/2𝑞₁𝜆_max), 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,…,𝜂_𝑇+1}. Let 𝕏?=?span{𝜂₁,𝜂₂,…,𝜂_𝑁},𝕎?=?span{𝜂_𝑁+1,𝜂_𝑁+2,…,𝜂_𝑇+1}, then Ω=𝕏⊕𝕎.
By (C₄), for 𝜀∈(0,𝑞₁), there exists 𝜌>0, such that 𝑞₁−𝜀𝑥²≤𝐹(𝑡+𝜈−1,𝑥)≤𝑞₁+𝜀𝑥²,|𝑥|≤𝜌,𝑡∈[0,𝑇]_ℕ0. (3.26)
So, for 𝑥∈𝕏 with 0<||𝑥||≤𝜌, such that ^𝑇−1_𝑡=−1(Δ𝑧(𝑡+𝜈−1))²=𝑧^′𝐴𝑧≤𝜆_𝑁‖𝑧‖²≤𝜆_𝑁‖𝑥‖²𝜆_{min𝑇𝑡=0},𝐹(𝑡+𝜈−1,𝑥(𝑡+𝜈−1))≥𝑞₁−𝜀^𝑇_𝑡=0(𝑥(𝑡+𝜈−1))²=𝑞₁−𝜀‖𝑥‖². (3.27)
Since 𝑧=𝐵𝑥∈𝕏, we have 𝐼(𝑥)=12^𝑇−1_𝑡=−1(Δ𝑧(𝑡+𝜈−1))²−𝜆^𝑇_𝑡=0𝐹((≤𝜆𝑡+𝜈−1,𝑥𝑡+𝜈−1))_𝑁2𝜆_min−𝜆𝑞₁−𝜀‖𝑥‖². (3.28)
Thus, for 𝜆>𝜆_𝑁/2(𝑞₁−𝜀)𝜆_min, we have 𝐼(𝑥)<0 for 𝑥∈𝕏 with 0<||𝑥||≤𝜌.
Similarly, for 𝑥∈𝕎 with 0<||𝑥||≤𝜌, 𝐼(𝑥)=12^𝑇−1_𝑡=−1(Δ𝑧(𝑡+𝜈−1))²−𝜆^𝑇_𝑡=0𝐹((≥𝜆𝑡+𝜈−1,𝑥𝑡+𝜈−1))^𝑁+12𝜆_max−𝜆𝑞₁+𝜀‖𝑥‖², (3.29) then for 𝜆<𝜆_𝑁+1/2(𝑞₁+𝜀)𝜆_max, we have 𝐼(𝑥)>0 for 𝑥∈𝕎 with 0<||𝑥||≤𝜌. So, by Lemma 2.5, for 𝜀∈(0,𝑞₁), if 𝜆∈(𝜆_𝑁/2(𝑞₁−𝜀)𝜆_min,𝜆_𝑁+1/2(𝑞₁+𝜀)𝜆_max), functional 𝐼 possesses at least three critical points. By the arbitrariness of 𝜀, we get for 𝜆∈(𝜆_𝑁/2𝑞₁𝜆_min,𝜆_𝑁+1/2𝑞₁𝜆_max), 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,𝜆_𝑁+1/2𝑑𝜆_max), 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 𝐹(𝑡+𝜈−1,𝑥)≤𝑑𝑥²for|𝑥|≤𝜌,𝑡∈[0,𝑇]_ℕ0. (3.30)
For 𝑁∈[1,𝑇+1]_ℕ1, if we choose 𝕏₁=span{𝜂_𝑁+1,𝜂_𝑁+2,…,𝜂_𝑇+1}, then codim𝕏₁=𝑁.
So for 𝑥∈𝕏₁ with ||𝑥||≤𝜌, since 𝑧=𝐵𝑥, we have 𝐼(𝑥)=12^𝑇−1_𝑡=−1(Δ𝑧(𝑡+𝜈−1))²−𝜆^𝑇_𝑡=0𝐹((≥𝜆𝑡+𝜈−1,𝑥𝑡+𝜈−1))^𝑁+12𝜆_max−𝜆𝑑‖𝑥‖². (3.31)
Thus, for 𝜆∈(0,𝜆_𝑁+1/2𝑑𝜆_max), 𝐼|_{𝕏1⋂𝜕𝔹𝜌}≥𝛽>0, where 𝛽=(𝜆_𝑁+1/2𝜆_max−𝜆𝑑)𝜌², (I₃) of Lemma 2.8 holds.
Next if we choose 𝕏₂=span{𝜂₁,𝜂₂,…,𝜂_𝑇+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, for 𝜆∈(0,𝜆_𝑁+1/2𝑑𝜆_max), 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,𝜆₁/2𝑑𝜆_max), 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]_ℕ1, when 𝜆∈(𝜆_𝑁/2𝑝₂𝜆_min,∞), 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,…,𝜂_𝑁} and set 𝕂=𝕏₁∩𝜕𝔹_𝜌, then 𝕂 is homeomorphic to 𝕊^𝑁−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∩𝜕𝔹_𝜌1, 𝐼(𝑥)=12^𝑇_𝑡=−1(Δ𝑧(𝑡+𝜈−1))²−𝜆^𝑇_𝑡=−1𝐹((≤1𝑡+𝜈−1,𝑥𝑡+𝜈−1))2𝜆_𝑁‖𝑧‖²−𝜆𝑝₂‖𝑥‖²=𝜆_𝑁2𝜆_min−𝜆𝑝₂𝜌²₁. (3.32)
For 𝜆∈(𝜆_𝑁/2𝑝₂𝜆_min𝑥,+∞), we have sup_𝕂1𝐼(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 _𝑇Δ^𝜈_𝑡−1𝑡Δ^𝜈_𝜈−1𝑥(𝑡)=𝜆4(𝑡+𝜈−1)𝑥³(())+()𝑥𝑡+𝜈−1)(sin𝑥𝑡+𝜈−1+2𝑡+𝜈−1⁴(𝑡+𝜈−1)cos(𝑥(𝑡+𝜈−1)),𝑡∈[0,𝑇]_ℕ0,𝑥(𝜈−2)=_𝑡Δ^𝜈_𝜈−1𝑥(𝑡)_𝑡=𝑇=0, (4.1) 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 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, (4.2) we see that (C₂) and (C₃) hold. Thus, by Theorem 3.4, when 𝜆∈(0,𝜆₁/2𝜆_max), problem (4.1) has at least two nontrivial solutions.

Example 4.2. Consider the problem _𝑇Δ^𝜈_𝑡−1𝑡Δ^𝜈_𝜈−1𝑥(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)=_𝑡Δ^𝜈_𝜈−1𝑥(𝑡)_𝑡=𝑇=0. (4.3) Suppose there exists 𝑁₀∈[0,𝑇]_ℕ0 such that 𝜆_max𝜆_𝑁0<𝜆_min𝜆_𝑁0+1. If we choose 𝜇=𝑞₁=𝑎=1 in (C₁) and (C₄), then for 𝑡∈[0,𝑇]_ℕ0, we have limsup_|𝑥|→∞𝐹(𝑡+𝜈−1,𝑥)|𝑥|^𝜇=limsup_|𝑥|→∞2sin²𝑥−𝑥²𝑒^{−(𝑡+𝜈−1)𝑥}|𝑥|^𝜇<1=𝑎,lim_𝑥→0𝐹(𝑡+𝜈−1,𝑥)𝑥²=1=𝑞₁>0, (4.4) and hence (C₁) and (C₄) are satisfied. So, in view of Theorem 3.5, for 𝜆∈(𝜆_𝑁0/2𝜆_min,𝜆_𝑁0+1/2𝜆_max), problem (4.3) has at least three solutions.

Example 4.3. Consider the problem _𝑇Δ^𝜈_𝑡−1𝑡Δ^𝜈_𝜈−1𝑥(𝑡)=𝜆4(𝑡+𝜈−1)𝑥³(())−()𝑥𝑡+𝜈−1)(cos𝑥𝑡+𝜈−1+2𝑡+𝜈−1⁴(𝑡+𝜈−1)sin(𝑥(𝑡+𝜈−1)),𝑡∈[0,𝑇]_ℕ0,𝑥(𝜈−2)=_𝑡Δ^𝜈_𝜈−1𝑥(𝑡)_𝑡=𝑇=0. (4.5) 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,𝜆_𝑁+1/2𝜆_max), problem (4.5) has at least 𝑇+1−𝑁 pairs of solutions.

Example 4.4. Finally, consider the problem _𝑇Δ^𝜈_𝑡−1𝑡Δ^𝜈_𝜈−1𝑥(𝑡1)=𝜆)𝑥(())𝑥([]𝑡+𝜈−1sin((𝑡+𝜈−1𝑡+𝜈−1))+𝑥𝑡+𝜈−1cos((𝑡+𝜈−1𝑡+𝜈−1)),𝑡∈0,𝑇_ℕ0,𝑥(𝜈−2)=_𝑡Δ^𝜈_𝜈−1𝑥(𝑡)_𝑡=𝑇=0, (4.6) 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∈(𝜆_𝑁/𝜆_min,+∞), problem (4.6) has at least 𝑁 pairs of solutions.

Acknowledgment

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