Existence of Solutions of a Discrete Fourth-Order Boundary Value Problem

Abstract

Let 𝑎,𝑏 be two integers with 𝑏−𝑎≥5 and let 𝕋₂={𝑎+2,𝑎+3,…,𝑏−2}. We show the existence of solutions for nonlinear fourth-order discrete boundary value problem Δ⁴𝑢(𝑡−2)=𝑓(𝑡,𝑢(𝑡), Δ²𝑢(𝑡−1)), 𝑡∈𝕋₂, 𝑢(𝑎+1)=𝑢(𝑏−1)=Δ²𝑢(𝑎)=Δ²𝑢(𝑏−2)=0 under a nonresonance condition involving two-parameter linear eigenvalue problem. We also study the existence and multiplicity of solutions of nonlinear perturbation of a resonant linear problem.

1. Introduction

The deformations of an elastic beam whose both ends are simply supported are described by a fourth-order two-point boundary value problem 𝑦^′′′′+𝑔(𝑥)𝑦=𝑒(𝑥),0<𝑥<1,𝑦(0)=𝑦(1)=𝑦^′′(0)=𝑦^′′(1)=0 (1.1) See studies by Aftabizadeh [1] and Gupta in [2]. The existence of solutions of nonlinear boundary value problems of fourth-order differential equations has been studied by many authors; see [1–12] and the references therein. For example, Aftabizadeh [1] proved an existence theorem for nonlinear boundary value problems 𝑦^′′′′=𝑓(𝑥,𝑦,𝑦^′′),0<𝑥<1,𝑦(0)=𝑦₀,𝑦^′′(0)=𝑦₁,𝑦(1)=𝑦₀,𝑦^′′(1)=𝑦₁, (1.2) under several conditions that 𝑓 is a bounded function. Yang [3] obtained existence results of (1.2) under the following assumption.(A) There are constants 𝑎,𝑏,𝑐≥0 with 𝑎/𝜋⁴+𝑏/𝜋²<1 such that |𝑓(𝑥,𝑦,𝑢)|≤𝑎|𝑦|+𝑏|𝑢|+𝑐. (1.3)

Del Pino and Manásevich [4] extended Yang's result and proved the following.

Theorem A. Assume that the pair (𝛼,𝛽) satisfies 𝛼(𝑘𝜋)⁴+𝛽(𝑘𝜋)²≠1 (1.4) for all 𝑘∈ℕ and that there are positive constants 𝑎,𝑏, and 𝑐 such that 𝑎max_𝑘∈ℕ{1|𝑘⁴𝜋⁴−𝛼−𝛽𝑘²𝜋²|}+𝑏max_𝑘∈ℕ{𝜆_𝑘|𝑘⁴𝜋⁴−𝛼−𝛽𝑘²𝜋²|}<1,|𝑓(𝑡,𝑢,𝑣)−(𝛼𝑢−𝛽𝑣)|≤𝑎|𝑢|+𝑏|𝑣|+𝑐 (1.5) for all 𝑥∈(0,1), 𝑢,𝑣∈ℝ, then (1.2) possesses at least one solution.

Of course, the natural question is whether or not the similar existence can be established for the corresponding discrete analog of (1.2) of the form Δ⁴𝑢(𝑡−2)=𝑓(𝑡,𝑢(𝑡),Δ²𝑢(𝑡−1)),𝑡∈𝕋₂,𝑢(𝑎+1)=𝑟₁,𝑢(𝑏−1)=𝑟₂,Δ²𝑢(𝑎)=𝑟₃,Δ²𝑢(𝑏−2)=𝑟₄, (1.6)(1.7) where 𝕋₂={𝑎+2,𝑎+3,…,𝑏−2}, 𝑟_𝑖∈ℝ for 𝑖∈{1,2,3,4}.

The purpose of this paper is to show that the answer is yes. To this end, we state and prove a spectrum result of two-parameter linear eigenvalue problem Δ⁴𝑢(𝑡−2)+𝛽Δ²𝑢(𝑡−1)−𝛼𝑢(𝑡)=0,𝑡∈𝕋₂,𝑢(𝑎+1)=𝑢(𝑏−1)=Δ²𝑢(𝑎)=Δ²𝑢(𝑏−2)=0. (1.8)(1.9) This result is a slightly generalized version of Shi and Wang [13, Theorem 2.1]. In Section 3, we use Leray-Schauder principle to study the existence of solutions of (1.6), (1.7) under some nonresonant conditions involving the spectrum of (1.8), (1.9). Section 4 is considered with some perturbations of resonant linear problems. We established some a priori bounds and used these together with bifurcation arguments to prove the existence and multiplicity of solutions.

Finally, we note that the existence of solutions of second-order discrete boundary value problems has also received much attention; see studies by Agarwal and Wong in [14], Henderson in [15], and the references therein. However, relatively little is known about the existence of solutions of fourth-order discrete boundary value problems. The likely reason may be that the structure of spectrum of the corresponding linear eigenvalue problem is not very clear. To our best knowledge, only He and Yu [16] as well as Zhang et al. [17] dealt with the discrete problem of the form Δ⁴𝑢(𝑡−2)=𝑓(𝑡,𝑢(𝑡)),𝑡∈𝕋₂,𝑢(𝑎)=𝑢(𝑏)=Δ²𝑢(𝑎)=Δ²𝑢(𝑏−2)=0. (1.10)(1.11) As we will see in Section 2, (1.9) has more advantage than (1.11) in the study of the spectrum of two-parameter linear eigenvalue problems.

2. Spectrum of Two-Parameter Linear Eigenvalue Problem

Let 𝑎,𝑏 be two integers with 𝑏−𝑎≥5. Recall 𝕋₂={𝑎+2,𝑎+3,…,𝑏−2}. (2.1) Let 𝕋₀={𝑎,𝑎+1,…,𝑏},𝕋₁={𝑎+1,𝑎+2,…,𝑏−1},Λ={1,2,…,𝑏−𝑎−3}. (2.2) Let 𝑋 be the Banach space 𝑋={𝑢∣𝑢∶𝕋₀→ℝ,𝑢(𝑎+1)=𝑢(𝑏−1)=Δ²𝑢(𝑎)=Δ²𝑢(𝑏−2)=0} (2.3) under the norm ‖𝑢‖_𝑋∶=max{|𝑢(𝑗)|∣𝑗∈𝕋₂}. (2.4) Let 𝑌 be the Banach space 𝑌={𝑦∣𝑦∶𝕋₁→ℝ,𝑦(𝑎+1)=𝑦(𝑏−1)=0} (2.5) equipped with the norm ‖𝑦‖_𝑌∶=max{|𝑦(𝑗)|∣𝑗∈𝕋₂}. (2.6) Let 𝑍 be the Banach space 𝑍={𝑧∣𝑧∶𝕋₂→ℝ} (2.7) equipped with the norm ‖𝑧‖_𝑍∶=max{|𝑧(𝑗)|∣𝑗∈𝕋₂}. (2.8)

Remark 2.1. For any 𝑧∈𝑍 with 𝑧={𝑧(𝑎+2),𝑧(𝑎+3),…,𝑧(𝑏−2)}, (2.9) it determines a unique element 𝑦∈𝑌 by 𝑦={0,𝑧(𝑎+2),𝑧(𝑎+3),…,𝑧(𝑏−2),0} (2.10) and a unique element 𝑥∈𝑋 by 𝑥={−𝑧(𝑎+2),0,𝑧(𝑎+2),𝑧(𝑎+3),…,𝑧(𝑏−2),0,−𝑧(𝑏−2)}. (2.11) Hence, the Banach spaces 𝑋, 𝑌, and 𝑍 are homomorphic with each other. Denote the natural homomorphism from 𝑍 to 𝑋 by 𝑗.

Now, we define a linear operator 𝐿∶𝑋→𝑍 by 𝐿𝑢(𝑡)∶=Δ⁴𝑢(𝑡−2)+𝛽Δ²𝑢(𝑡−1)−𝛼𝑢(𝑡),𝑡∈𝕋₂. (2.12) For 𝑘∈Λ, let 𝜆_𝑘 be the 𝑘th-eigenvalue of the second-order linear eigenvalue problem Δ²𝑢(𝑡−1)+𝜆𝑢(𝑡)=0,𝑡∈𝕋₂,𝑢(𝑎+1)=𝑢(𝑏−1)=0. (2.13) It is well known that 𝜆_𝑘 is simple, and the corresponding eigenfunction 𝜓_𝑘(𝑡)∶=sin𝑘𝜋(𝑡−𝑎−1)𝑏−𝑎−2,𝑡∈𝕋₁,𝑘∈Λ. (2.14) See the study by Kelly and Peterson in [18].

The following result is considered with the spectrum of two-parameter eigenvalue problem: Δ⁴𝑢(𝑡−2)+𝛽Δ²𝑢(𝑡−1)−𝛼𝑢(𝑡)=0,𝑡∈𝕋₂,𝑢(𝑎+1)=𝑢(𝑏−1)=Δ²𝑢(𝑎)=Δ²𝑢(𝑏−2)=0. (2.15)(2.16) It is a slightly generalized version of Shi and Wang [13, Theorem 2.1].

Proposition 2.2. (𝛼,𝛽) is an eigenvalue pair of (2.15), (2.16) if and only if 𝛼𝜆²_𝑘+𝛽𝜆_𝑘=1 (2.17) for some 𝑘∈Λ.

Proof. Let 𝑟₁,𝑟₂∈ℂ such that 𝑟₁+𝑟₂=𝛽,𝑟₁𝑟₂=−𝛼. (2.18) Define two second-order difference operators 𝐿₁,𝐿₂∶𝑌→𝑌 by (𝐿₁𝑦)(𝑡)∶=Δ²𝑦(𝑡−1)+𝑟₁𝑦(𝑡),𝑡∈𝕋₂,(𝐿₁𝑦)(𝑎+1)=0,(𝐿₁𝑦)(𝑏−1)=0,(𝐿₂𝑦)(𝑡)∶=Δ²𝑦(𝑡−1)+𝑟₂𝑦(𝑡),𝑡∈𝕋₂,(𝐿₂𝑦)(𝑎+1)=0,(𝐿₂𝑦)(𝑏−1)=0. (2.19) Then, for 𝑦∈𝑌 and 𝑡∈𝕋₁, 𝐿₂∘𝐿₁𝑦(𝑡)=𝐿₂(Δ²𝑦(𝑡−1)+𝑟₁𝑦(𝑡))=Δ²(Δ²𝑦(𝑡−2)+𝑟₁𝑦(𝑡−1))+𝑟₂(Δ²𝑦(𝑡−1)+𝑟₁𝑦(𝑡))=Δ⁴𝑦(𝑡−2)+(𝑟₁+𝑟₂)Δ²𝑦(𝑡−1)+𝑟₁𝑟₂𝑦(𝑡)=Δ⁴𝑦(𝑡−2)+𝛽Δ²𝑦(𝑡−1)−𝛼𝑦(𝑡)=(𝐿|_𝑌)𝑦(𝑡). (2.20) We claim that if (2.15), (2.16) possess a nontrivial solution 𝑦, then either 𝑟₁=𝜆_𝑘 or 𝑟₂=𝜆_𝑘 for some 𝑘∈Λ. In either case, sin(𝑘𝜋(𝑡−𝑎−1)/(𝑏−𝑎−2)),(𝑡∈𝕋₁), is a nontrivial solution of (2.15), (2.16).
In fact, if 𝑟₂≠𝜆_𝑘 for all 𝑘∈Λ, then (2.20) implies that (𝐿₁𝑦)(𝑡)=0,𝑡∈𝕋₁. (2.21) This is Δ²𝑦(𝑡−1)+𝑟₁𝑦(𝑡)=0,𝑡∈𝕋₂,𝑦(𝑎+1)=𝑦(𝑏−1)=0. (2.22) Thus, 𝑟₁=𝜆_𝑘 for some 𝑘∈Λ, and 𝑦(𝑡)=sin𝑘𝜋(𝑡−𝑎−1)𝑏−𝑎−2,𝑡∈𝕋₁. (2.23)
If 𝑟₂=𝜆_𝑘 for some 𝑘∈Λ, then (2.20) implies that 𝐿₁𝑦(𝑡)=𝛾𝜓_𝑘(𝑡),𝑡∈𝕋₁, (2.24) for some 𝛾∈ℝ⧵{0}. This is Δ²𝑦(𝑡−1)+𝑟₁𝑦(𝑡)=𝛾𝜓_𝑘(𝑡),𝑡∈𝕋₂,𝑦(𝑎+1)=𝑦(𝑏−1)=0. (2.25)(2.26) Since 𝛾≠0, it follows that 𝑟₁≠𝜆_𝑘,𝑘∈Λ. (2.27) This implies that (2.25), (2.26) have a unique solution 𝑦(𝑡)∶=(𝐿₁)⁻¹𝛾𝜓_𝑘(𝑡),𝑡∈𝕋₁. (2.28)
We show that 𝑟₁=𝜆_𝑘+𝛾,𝑦(𝑡)=𝜓_𝑘(𝑡). (2.29)
In fact, from (2.25) we have ^𝑏−2_𝑡=𝑎+2𝛾𝜓²_𝑘(𝑡)=^𝑏−2_𝑡=𝑎+2(Δ²𝑦(𝑡−1)+𝑟₁𝑦(𝑡))𝜓_𝑘=(𝑡)_{𝑏−2𝑡=𝑎+2}(Δ²𝜓_𝑘(𝑡−1)+𝑟₁𝜓_𝑘=(𝑡))𝑦(𝑡)_{𝑏−2𝑡=𝑎+2}(𝑟₁−𝜆_𝑘)𝜓_𝑘=(𝑡)𝑦(𝑡)_{𝑏−2𝑡=𝑎+2}(𝑟₁−𝜆_𝑘)𝜓²_𝑘(𝑡), (2.30) which implies that 𝛾=𝑟₁−𝜆_𝑘, and, subsequently, 𝑦(𝑡)=𝜓_𝑘(𝑡).
Therefore, the claim is true.
Now, (2.17) follows by substituting this solution into (2.15), (2.16). Reciprocally, if (2.17) holds, then, clearly, sin(𝑘𝜋(𝑡−𝑎−1)/(𝑏−𝑎−2)), (𝑡∈𝕋₁) is a nontrivial solution of (2.15), (2.16).

Remark 2.3. From the proof of Proposition 2.2, we see that if (2.15) subjects to (1.9), then we can factor 𝐿|_𝑌 as follows: 𝐿|_𝑌=𝐿₂∘𝐿₁. (2.31) However, this cannot be done if (2.15) subjects to (1.11). So, (1.9) has more advantage than (1.11) in the study of the spectrum of two-parameter linear eigenvalue problems.

Next, for 𝑗∈ℕ, let us set 𝐿_𝑗𝛼={(𝛼,𝛽)∣𝜆²_𝑗+𝛽𝜆_𝑗=1}. (2.32) In view of the Proposition 2.2, we call 𝐿_𝑗 an eigenline of (2.15), (2.16). We note that an eigenvalue pair (𝛼,𝛽) can belong to at most two eigenlines. If (𝛼,𝛽) belongs to just one 𝐿_𝑗, then the corresponding eigenspace is spanned by sin(𝑘𝜋(𝑡−𝑎−1)/(𝑏−𝑎−2)). If (𝛼,𝛽) belongs to 𝐿_𝑗∩𝐿_𝑘(𝑗≠𝑘), then the corresponding eigenspace is spanned by sin(𝑗𝜋(𝑡−𝑎−1)/(𝑏−𝑎−2)) and sin(𝑘𝜋(𝑡−𝑎−1)/(𝑏−𝑎−2)).

Suppose that the pair (𝛼,𝛽) is not an eigenvalue pair of (2.15), (2.16), that is, 𝛼𝜆²_𝑘+𝛽𝜆_𝑘≠1 (2.33) for all 𝑘∈Λ, and that ℎ∈𝑍: ℎ(𝑡)=^{𝑏−𝑎−3}_𝑘=1ℎ_𝑘sin𝑘𝜋(𝑡−𝑎−1)𝑏−𝑎−2,𝑡∈𝕋₂. (2.34) From the Fredholm Alternative, it follows that the boundary value problem Δ⁴𝑢(𝑡−2)+𝛽Δ²𝑢(𝑡−1)−𝛼𝑢(𝑡)=ℎ(𝑡),𝑡∈𝕋₂,𝑢(𝑎+1)=𝑢(𝑏−1)=Δ²𝑢(𝑎)=Δ²𝑢(𝑏−2)=0 (2.35) has a unique solution for each ℎ∈𝑍. Moreover, this solution admits a Fourier series expansion of the form 𝑢(𝑡)=^{𝑏−𝑎−3}_𝑘=1ℎ_𝑘sin(𝑘𝜋(𝑡−𝑎−1)/(𝑏−𝑎−2))(𝜆²_𝑘−𝛼−𝛽𝜆_𝑘),𝑡∈𝕋₂,𝑢(𝑎+1)∶=0,𝑢(𝑏−1)∶=0,𝑢(𝑎)∶=−𝑢(𝑎+2),𝑢(𝑏)∶=−𝑢(𝑏−2). (2.36) Also, we have Δ²𝑢(𝑡−1)=−^{𝑏−𝑎−3}_𝑘=1𝜆_𝑘ℎ_𝑘sin(𝑘𝜋(𝑡−𝑎−1)/(𝑏−𝑎−2))(𝜆²_𝑘−𝛼−𝛽𝜆_𝑘),𝑡∈𝕋₂,Δ²𝑢(𝑎)∶=0,Δ²𝑢(𝑏−2)∶=0. (2.37)

From (2.36) and (2.37), we can easily see that the operators 𝐴∶𝑍→𝑍, 𝐵∶𝑍→𝑍 defined by 𝐴(ℎ)(𝑡)=𝑢(𝑡),𝐵(ℎ)(𝑡)=Δ²𝑢(𝑡−1),𝑡∈𝕋₂, (2.38) are compact linear operators. In (2.38), 𝑢 is the solution of (2.35), (2.16) corresponding to ℎ∈𝑍. The norms of 𝐴 and 𝐵 are, respectively, given by ‖𝐴‖_𝑍→𝑍=max_𝑘∈Λ{1|𝜆²_𝑘−𝛼−𝛽𝜆_𝑘|},‖𝐵‖_𝑍→𝑍=max_𝑘∈Λ{𝜆_𝑘|𝜆²_𝑘−𝛼−𝛽𝜆_𝑘|}. (2.39)

Finally, as an immediate consequence of Proposition 2.2, we have the following.

Proposition 2.4. Let 𝛾 and 𝛿 be two constants with (𝛾,𝛿)∈[0,∞)×[0,∞) and 𝛾+𝛿>0. Then the generalized eigenvalues of problem Δ⁴𝑢(𝑡−2)+𝜇(𝛿Δ²𝑢(𝑡−1)−𝛾𝑢(𝑡))=0,𝑡∈𝕋₂,𝑢(𝑎+1)=𝑢(𝑏−1)=Δ²𝑢(𝑎)=Δ²𝑢(𝑏−2)=0 (2.40) are given by 𝜇₁(𝛾,𝛿)<𝜇₂(𝛾,𝛿)<⋯<𝜇_{𝑏−𝑎−3}(𝛾,𝛿), (2.41) where 𝜇_𝑘𝜆(𝛾,𝛿)=^2𝑘𝛾+𝛿𝜆_𝑘,𝑘∈Λ. (2.42) The generalized eigenfunction corresponding to 𝜇_𝑘(𝛾,𝛿) is 𝜓_𝑘(𝑡)=sin𝑘𝜋(𝑡−𝑎−1)𝑏−𝑎−2,𝑡∈𝕋₁. (2.43)

3. Existence Results for Nonresonant Problems

Theorem 3.1. Assume that the pair (𝛼,𝛽) satisfies 𝛼𝜆²_𝑘+𝛽𝜆_𝑘≠1 (3.1) for all 𝑘∈Λ and that there are positive constants 𝑎^∗,𝑏^∗, and 𝑐^∗ such that 𝑎^∗max_𝑘∈Λ{1|𝜆²_𝑘−𝛼−𝛽𝜆_𝑘|}+𝑏^∗max_𝑘∈Λ{𝜆_𝑘|𝜆²_𝑘−𝛼−𝛽𝜆_𝑘|}<1,|𝑓(𝑡,𝑢,𝑣)−(𝛼𝑢−𝛽𝑣)|≤𝑎^∗|𝑢|+𝑏^∗|𝑣|+𝑐^∗ (3.2)(3.3) for all 𝑡∈𝕋₂, 𝑢,𝑣∈ℝ, then (1.6), (1.7) possess at least one solution.

Remark 3.2. It is not difficult to see that (3.1), (3.2) imply that 𝑎^∗|𝜆²_𝑘−𝛼−𝛽𝜆_𝑘|+𝑏^∗𝜆_𝑘|𝜆²_𝑘−𝛼−𝛽𝜆_𝑘|<1 (3.4) for 𝑘∈Λ. It turns out that (3.4) is equivalent to the fact that the square [𝛼−𝑎^∗,𝛼+𝑎^∗]×[𝛽−𝑏^∗,𝛽+𝑏^∗] does not intersect any of the eigenlines 𝐿_𝑗 of (2.15), (2.16). From this point of view, (3.1), (3.2) can be thought of as a two-parameter nonresonance condition relative to the eigenlines 𝐿_𝑗.

Proof of Theorem 3.1. It is easy to check that the problem Δ⁴𝑢(𝑡−2)=0,𝑡∈𝕋₂,𝑢(𝑎+1)=𝑟₁,𝑢(𝑏−1)=𝑟₂,Δ²𝑢(𝑎)=𝑟₃,Δ²𝑢(𝑏−2)=𝑟₄ (3.5) has a unique solution 𝑙(𝑡). Set 𝑦(𝑡)∶=𝑢(𝑡)−𝑙(𝑡),𝑡∈𝕋₀. (3.6) Then (1.6),(1.7) can be rewritten as Δ⁴𝑦(𝑡−2)=𝑓(𝑡,𝑦(𝑡)+𝑙(𝑡),Δ²(𝑦(𝑡−1)+𝑙(𝑡−1))),𝑡∈𝕋₂,𝑦(𝑎+1)=𝑦(𝑏−1)=Δ²𝑦(𝑎)=Δ²𝑦(𝑏−2)=0. (3.7) Since |𝑓(𝑡,𝑦(𝑡)+𝑙(𝑡),Δ²(𝑦(𝑡−1)+𝑙(𝑡−1)))−[𝛼(𝑦(𝑡)+𝑙(𝑡))−𝛽(Δ²(𝑦(𝑡−1)+𝑙(𝑡−1)))]|≤𝑎^∗|𝑦|+𝑏^∗|Δ²𝑦(𝑡−1)|+𝑐^∗∗ (3.8) with 𝑐^∗∗=𝑎^∗max_𝑡∈𝕋0|𝑙(𝑡)|+𝑏^∗max_𝑡∈𝕋1|Δ²𝑙(𝑡−1)|+𝑐^∗, (3.9) it follows that (3.2) and (3.3) still hold except that 𝑐^∗ is replaced by 𝑐^∗∗. So, we may suppose that 𝑟₁=𝑟₂=𝑟₃=𝑟₄=0 in (1.7).
Let us define 𝑇∶𝑍×𝑍→𝑍×𝑍 by 𝑇(𝑢,𝑣)=(𝐴(𝑓(⋅,𝑢,𝑣)−(𝛼𝑢−𝛽𝑣)),𝐵(𝑓(⋅,𝑢,𝑣)−(𝛼𝑢−𝛽𝑣))), (3.10) where 𝐴 and 𝐵 are the operators defined in (2.38). The growth condition (3.3) together with the compactness 𝐴 and 𝐵 implies that 𝑇 is a completely continuous operator. By Remark 2.1, the problem Δ⁴𝑢(𝑡−2)=𝑓(𝑡,𝑢(𝑡),Δ²𝑢(𝑡−1)),𝑡∈𝕋₂,𝑢(𝑎+1)=𝑢(𝑏−1)=Δ²𝑢(𝑎)=Δ²𝑢(𝑏−2)=0 (3.11) is equivalent to the fixed point problem in 𝑍×𝑍: (𝑢,𝑣)=𝑇(𝑢,𝑣). (3.12) We will study this fixed point problem by means of the well-known Leray-Schauder principle [18]. To do this, we show that there is a uniform bound independent of 𝜆∈[0,1] for the solutions of the equation (𝑢,𝑣)=𝜆𝑇(𝑢,𝑣). (3.13) Thus, let (𝑢,𝑣) be a solution of (3.13). From the definition of 𝑇 and (3.3), we obtain the result that ‖𝑢‖_𝑍≤‖𝐴‖_𝑍→𝑍{𝑎‖𝑢‖_𝑍+𝑏‖𝑣‖_𝑍+𝑐},‖𝑣‖_𝑍≤‖𝐵‖_𝑍→𝑍{𝑎‖𝑢‖_𝑍+𝑏‖𝑣‖_𝑍+𝑐}. (3.14)(3.15) Combining (3.14) and (3.15) and using (3.2) and (2.38), we obtain the existence of a constant 𝑀=𝑀(𝑎^∗,𝑏^∗,𝑐^∗,‖𝐴‖_𝑍→𝑍,‖𝐵‖_𝑍→𝑍) such that ‖𝑢‖_𝑍+‖𝑣‖_𝑍≤𝑀. (3.16) By the Leray-Schauder principle [19], we conclude the existence of at least one solution of (3.12), and the theorem follows.

4. Existence and Multiplicity Results for Perturbations of Resonant Linear Problems

In this section, we consider the perturbations of resonant linear problems of the form Δ⁴𝑢(𝑡−2)+(𝜇_𝑘+𝜇)(𝛿Δ²𝑢(𝑡−1)−𝛾𝑢(𝑡))=𝑔(𝑡,𝑢(𝑡))+ℎ(𝑡),𝑡∈𝕋₂,(𝑢(𝑎+1)=𝑢(𝑏−1)=Δ²𝑢(𝑎)=Δ²𝑢(𝑏−2)=0, (4.1)_𝜇)(4.1) where (𝛾,𝛿)∈[0,∞)×[0,∞) with 𝛾+𝛿>0, 𝜇_𝑘=𝜇_𝑘(𝛾,𝛿), and 𝑔 and ℎ satisfy the following.(H1) (Sublinear growth condition) 𝑔∶𝕋₂×ℝ→ℝ is continuous, and there exist 𝛼∈[0,1), 𝐶₁,𝐶₂∈(0,∞) such that |𝑔(𝑡,𝑠)|≤𝐶₁|𝑠|^𝛼+𝐶₂,𝑠∈ℝ,𝑡∈𝕋₂ (4.2)(H2) There exists 𝛽>0 such that 𝑠𝑔(𝑡,𝑠)>0,for𝑡∈𝕋₂,|𝑠|>𝛽 (4.3)(H3)ℎ∶𝕋₂→ℝ satisfies ^𝑏−2_𝑡=𝑎+2ℎ(𝑡)𝜓_𝑘(𝑡)=0. (4.4)

We will establish some a priori bounds and use these together with Leray-Schauder continuation and bifurcation arguments to reduce results which say that there are multiple solutions of (4.1)_𝜇, (4.1) for 𝜇 on one side of zero and guarantee the existence of at least one solution for 𝜇=0 and 𝜇 on the other side of zero. To wit, we have the following.

Theorem 4.1. Let (H1), (H2), and (H3) hold. Then there exist 𝜇₋<0<𝜇₊ such that (4.1)_𝜇, (4.1) have(1)at least one solution if 𝜇∈[0,𝜇₊],(2)at least three solutions if 𝜇∈[𝜇₋,0).

We have the following “dual” theorem if (H2) is replaced by the assumption(H2^′) that there exists 𝛽>0 such that 𝑠𝑔(𝑡,𝑠)<0,for𝑡∈𝕋₂,|𝑠|>𝛽. (4.5)

Theorem 4.2. Let (H1), (H2′), and (H3) hold. Then there exist 𝜇₋<0<𝜇₊ such that (4.1)_𝜇, (4.1) have(1)at least one solution if 𝜇∈[𝜇₋,0],(2)at least three solutions if 𝜇∈(0,𝜇₊].

Define ℒ∶𝑋→𝑍 by (ℒ𝑦)(𝑡)=Δ⁴𝑢(𝑡−2)+𝜇_𝑘(𝛿Δ²𝑢(𝑡−1)−𝛾𝑢(𝑡)),𝑡∈𝕋₂. (4.6) Define 𝐹∶𝑍→𝑍 by (𝐹𝑢)(𝑡)=𝑔(𝑡,𝑢(𝑡)),𝑡∈𝕋₂. (4.7) It is easy to check that 𝐹∶𝑍→𝑍 is continuous. Obviously (4.1)_𝜇, (4.1) are equivalent to ℒ𝑢+𝜇(𝛿Δ²𝑢(𝑡−1)−𝛾𝑢(𝑡))=𝐹(𝑢)+ℎ.( (4.7)_𝜇)

Define an operator 𝑃∶𝑋→𝑋 by (𝑃𝑥)(𝑡)=𝜓_𝑘(𝑡)^𝑏−2_𝑠=𝑎+2𝑥(𝑠)𝜓_𝑘(𝑠),𝑡∈𝕋₀, (4.8) where 𝜓_𝑘(𝑎)∶=−𝜓_𝑘(𝑎+2),𝜓_𝑘(𝑏)∶=−𝜓_𝑘(𝑏−2). (4.9)

It is easy to show the following.

Lemma 4.3. 𝑃 is a projection and Im(𝑃)=Ker(ℒ).

Define an operator 𝐸∶𝑍→𝑍 by (𝐸𝑦)(𝑡)=𝑦(𝑡)−𝜓_𝑘(𝑡)^𝑏−2_𝑠=𝑎+2𝑦(𝑠)𝜓_𝑘(𝑠),𝑡∈𝕋₂. (4.10) Obviously, we have the following.

Lemma 4.4. 𝐸 is a projection and Im(𝐸)=Im(ℒ).

It is clear that 𝑋=𝑋_𝑃⊕𝑋_𝐼−𝑃,𝑍=𝑍_𝐼−𝐸⊕𝑍_𝐸, (4.11) where 𝐼 represents the identity operator and 𝑋_𝑃,𝑋_𝐼−𝑃,𝑍_𝐼−𝐸, and 𝑍_𝐸 are the images of 𝑃,𝐼−𝑃,𝐼−𝐸, and 𝐸, respectively.

It is obvious that the restriction of ℒ to 𝑋_𝐼−𝑃 is a bijection from 𝑋_𝐼−𝑃 onto 𝑍_𝐸, the image of ℒ. We define 𝑀∶𝑍_𝐸→𝑋_𝐼−𝑃 by 𝑀∶=(ℒ|_{𝑋𝐼−𝑃})⁻¹. (4.12) Since ker(ℒ)=span{𝜓_𝑘}, we see that each 𝑥∈𝑋 can be uniquely decomposed into 𝑥=𝜌𝜓_𝑘+𝑣 (4.13) for some 𝜌∈ℝ, and 𝑣∈𝑋_𝐼−𝑃. For 𝑧∈𝑍, we also have the decomposition 𝑧=𝜏𝜓_𝑘+ℎ, (4.14) with 𝜏∈ℝ and ℎ∈𝑍_𝐸.

Lemma 4.5. Equations (4.1)_𝜇, (4.1) are equivalent to the system ℒ𝑣+𝜇(𝛿Δ²𝑣(𝑡−1)−𝛾𝑣(𝑡))=𝐸𝐹(𝜌𝜓_𝑘+𝑣)+ℎ,(𝜇^𝑏−2_𝑠=𝑎+2(𝛿Δ²𝜓_𝑘(𝑠−1)−𝛾𝜓_𝑘(𝑠))𝜓_𝑘(𝑠)=^𝑏−2_𝑠=𝑎+2𝜓_𝑘(𝑠)𝑓(𝑠,𝜌𝜓_𝑘(𝑠)+𝑣(𝑠)). (4.9)_𝜇)(4.15)

Lemma 4.6. Let (H1) and (H2) hold. Then there exists 𝑅₀ such that any solution 𝑦 of (4.1)_𝜇, (4.1) satisfies ‖𝑦‖_𝑋<𝑅₀ (4.16) as long as 10≤𝜇≤𝛿∶=2‖𝑀𝐽‖_{𝑋𝐼−𝑃→𝑋𝐼−𝑃}, (4.17) where 𝐽∶𝑋→𝑍 is defined by (𝐽𝑥)(𝑡)∶=𝛿Δ²𝑥(𝑠−1)−𝛾𝑥(𝑠),𝑡∈𝕋₂. (4.18)

Proof. Obviously (ℒ+𝜇𝐽)|_{𝑋𝐼−𝑃}∶𝑋_𝐼−𝑃→𝑍_𝐸 is invertible for |𝜇|≤𝛿. Moreover, by (4.17), ‖(ℒ+𝜇𝐽)|_𝑋⁻¹_𝐼−𝑃‖_{𝑍𝐸→𝑋𝐼−𝑃}=‖[(ℒ|_{𝑋𝐼−𝑃})(𝐼+𝜇𝑀𝐽)]⁻¹‖_{𝑍𝐸→𝑋𝐼−𝑃}=‖[(𝐼+𝜇𝑀𝐽)]⁻¹‖_{𝑋𝐼−𝑃→𝑋𝐼−𝑃}‖𝑀‖_{𝑍𝐸→𝑋𝐼−𝑃}≤2‖𝑀‖_{𝑍𝐸→𝑋𝐼−𝑃}. (4.19) Let 𝑦=𝜌𝜓_𝑘+𝑣 be any solution of (4.1)_𝜇, (4.1). Then we have that, if 𝜌≠0, ‖𝑣‖_𝑋=‖(ℒ+𝜇𝐽)|_𝑋⁻¹_𝐼−𝑃𝐸(ℎ−𝐹(𝜌𝜓_𝑘+𝑣))‖_𝑋≤‖(ℒ+𝜇𝐽)|_𝑋⁻¹_𝐼−𝑃‖_{𝑍𝐸→𝑋𝐼−𝑃}‖𝐸‖_{𝑍→𝑍𝐸}[‖ℎ‖_𝑍+𝐶₁(|𝜌|‖𝜓_𝑘‖_𝑍+‖𝑣‖_𝑍)^𝛼+𝐶₂]≤2‖𝑀‖_{𝑍𝐸→𝑋𝐼−𝑃}‖𝐸‖_{𝑍→𝑍𝐸}[‖ℎ‖_𝑍+𝐶₁(|𝜌|‖𝜓_𝑘‖_𝑍+‖𝑣‖_𝑍)^𝛼+𝐶₂]≤2‖𝑀‖_{𝑍𝐸→𝑋𝐼−𝑃}‖𝐸‖_{𝑍→𝑍𝐸}[‖ℎ‖_𝑍+𝐶₁(|𝜌|‖𝜓_𝑘‖_𝑋+‖𝑣‖_𝑋)^𝛼+𝐶₂]=2‖𝑀‖_{𝑍𝐸→𝑋𝐼−𝑃}‖𝐸‖_{𝑍→𝑍𝐸}[‖ℎ‖_𝑍+𝐶₁(|𝜌|‖𝜓_𝑘‖_𝑋)^𝛼(1+‖𝑣‖_𝑋|𝜌|‖𝜓_𝑘‖_𝑋)^𝛼+𝐶₂]≤2‖𝑀‖_{𝑍𝐸→𝑋𝐼−𝑃}‖𝐸‖_{𝑍→𝑍𝐸}[‖ℎ‖_𝑍+𝐶₁(|𝜌|‖𝜓_𝑘‖_𝑋)^𝛼(1+𝛼‖𝑣‖_𝑋|𝜌|‖𝜓_𝑘‖_𝑋)+𝐶₂]=2‖𝑀‖_{𝑍𝐸→𝑋𝐼−𝑃}‖𝐸‖_{𝑍→𝑍𝐸}×[‖ℎ‖_𝑍+𝐶₁(|𝜌|‖𝜓_𝑘‖_𝑋)^𝛼𝛼(1+(|𝜌|‖𝜓_𝑘‖_𝑋)^1−𝛼‖𝑣‖_𝑋(|𝜌|‖𝜓_𝑘‖_𝑋)^𝛼)+𝐶₂], (4.20) and hence ‖𝑣‖_𝑋(|𝜌|‖𝜓_𝑘‖_𝑋)^𝛼≤𝐶₃(|𝜌|‖𝜓_𝑘‖_𝑋)^𝛼+𝐶₄+𝛼𝐶₄(|𝜌|‖𝜓_𝑘‖_𝑋)^1−𝛼‖𝑣‖_𝑋(|𝜌|‖𝜓_𝑘‖_𝑋)^𝛼, (4.21) where 𝐶₃=‖𝑀‖_{𝑍𝐸→𝑋𝐼−𝑃}‖𝐸‖_{𝑍→𝑍𝐸}(‖ℎ‖_𝑍+𝐶₂),𝐶₄=2𝐶₁‖𝑀‖_{𝑍𝐸→𝑋𝐼−𝑃}‖𝐸‖_{𝑍→𝑍𝐸}. (4.22) If |𝜌|≥(2𝛼𝐶₄)^1/(1−𝛼)‖𝜓_𝑘‖_𝑋=∶𝐶, (4.23) then we have ‖𝑣‖_𝑋(|𝜌|‖𝜓_𝑘‖_𝑋)^𝛼≤2𝐶₃(𝐶‖𝜓_𝑘‖_𝑋)^𝛼+2𝐶₄=∶𝐶^∗. (4.24)
If we assume that the conclusion of the lemma is false, we obtain a sequence {𝜂_𝑛} with 0≤𝜂_𝑛≤𝛿 and 𝜂_𝑛→0, and a sequence of corresponding solutions {𝑦_𝑛=𝜌_𝑛𝜓_𝑘+𝑣_𝑛} of (4.1)_𝜂𝑛, (4.1) such that ‖𝑦_𝑛‖_𝑋→+∞. From (4.24), we conclude that it is necessary that |𝜌_𝑛|→+∞. We may assume that 𝜌_𝑛→+∞,𝜌_𝑛≥𝐶∀𝑛∈ℕ (4.25) since the other case can be treated by the same way. Thus (4.24) yields that ‖𝑣_𝑛‖_𝑋∶≤𝐶|𝜌_𝑛|^𝛼 (4.26) with 𝐶∶=𝐶^∗‖𝜓_𝑘‖^𝛼_𝑋.
Now from (4.15), we get that 𝜌_𝑛𝜂_𝑛^𝑏−2_𝑠=𝑎+2(𝛿Δ²𝜓_𝑘(𝑠−1)−𝛾𝜓_𝑘(𝑠))𝜓_𝑘(𝑠)=^𝑏−2_𝑠=𝑎+2𝜓_𝑘(𝑠)𝑓(𝑠,𝜌_𝑛𝜓_𝑘(𝑠)+𝑣_𝑛(𝑠)). (4.27) By (4.17) and (4.27), it follows that ^𝑏−2_𝑠=𝑎+2𝜓_𝑘(𝑠)𝑓(𝑠,𝜌_𝑛𝜓_𝑘(𝑠)+𝑣_𝑛(𝑠))≤0. (4.28)
Let 𝐴⁺={𝑡∣𝑡∈{𝑎+2,…,𝑏−2},𝜓_𝑘𝐴(𝑡)>0},₋={𝑡∣𝑡∈{𝑎+2,…,𝑏−2},𝜓_𝑘(𝑡)<0}. (4.29) It is easy to see that 𝐴⁺∪𝐴⁻≠∅,min{|𝜓_𝑘(𝑡)|∣𝑡∈𝐴⁺∪𝐴⁻}>0. (4.30) Combining (4.30) and (4.26), we conclude that there exists a positive constant Γ such that, for 𝑛∈ℕ, ‖𝑣_𝑛‖_𝑋≤Γ(|𝜌_𝑛|min{|𝜓_𝑘(𝑡)|∣𝑡∈𝐴⁺∪𝐴⁻})^𝛼, (4.31) which implies that lim_{𝜌𝑛→+∞}min{𝜌_𝑛𝜓_𝑘(𝑡)+𝑣_𝑛(𝑡)∣𝑡∈𝐴⁺}=+∞,lim_{𝜌𝑛→+∞}min{𝜌_𝑛𝜓_𝑘(𝑡)+𝑣_𝑛(𝑡)∣𝑡∈𝐴⁻}=−∞. (4.32) Applying (4.32), (4.30), and (H2), we conclude that ^𝑏−2_𝑠=𝑎+2𝜓_𝑘(𝑠)𝑓(𝑠,𝜌_𝑛𝜓_𝑘(𝑠)+𝑣_𝑛(𝑠))=_𝑠∈𝐴+𝜓_𝑘(𝑠)𝑓(𝑠,𝜌_𝑛𝜓_𝑘(𝑠)+𝑣_𝑛(𝑠))+_{𝑠∈𝐴−}𝜓_𝑘(𝑠)𝑓(𝑠,𝜌_𝑛𝜓_𝑘(𝑠)+𝑣_𝑛(𝑠))>0, (4.33) which contracts (4.28).

Using the similar arguments, we may establish the following lemma.

Lemma 4.7. Let (H1) and (H2′) hold. Then there exists 𝑅₀ such that any solution 𝑦 of (4.1)_𝜇, (4.1) satisfies ‖𝑦‖_𝑋<𝑅₀ (4.34) as long as −𝛿≤𝜇≤0, (4.35) where 𝛿 is given in (4.17).

Lemma 4.8. Let (H1) and (H2) hold. Then there exists 𝑅₁∶𝑅₁≥𝑅₀ such that, for 0≤𝜇≤𝛿 and 𝑅≥𝑅₁, one has deg(𝑗∘(ℒ+𝜇𝐽−𝐹−ℎ),𝐵(𝑅),0)=deg(𝑗∘(ℒ+𝛿𝐽),𝐵(𝑅),0)=±1, (4.36) where 𝑗∶𝑍→𝑋 is the natural homomorphism, 𝐵(𝑅)={𝑢∈𝑋‖𝑢‖_𝑋<𝑅}, and “deg” denotes Leray-Schauder degree when 𝜇≠0 and coincidence degree when 𝜇=0 (see the study by Gaines and Mawhin in [20]). Therefore (4.7)_𝜇 has a solution in 𝐵(𝑅) for 𝜇∈[0,𝛿].

Proof. By Lemma 4.6 and the definition of ℒ, the degree deg(𝑗∘(ℒ+𝜇𝐽−𝐹−ℎ),𝐵(𝑅),0) (4.37) is well defined for 𝜇∈[0,𝛿] and is a constant with respect to 𝜇.
Now if (𝜇,𝑦)∈[0,1]×𝑋 is a solution of ℒ𝑦+𝛿(𝛿Δ²𝑦(𝑠−1)−𝛾𝑦(𝑠))−𝜇(𝐹(𝑦)+ℎ)=0, (4.38) then we have ‖𝑦‖_𝑋=𝜇‖(ℒ+𝛿𝐽)⁻¹(ℎ+𝐹(𝑦))‖_𝑋≤‖(ℒ+𝛿𝐽)⁻¹‖_{𝑍𝐸→𝑋𝐼−𝑃}(‖ℎ‖_𝑍+𝐶₁‖𝑦‖^𝛼_𝑋+𝐶₂). (4.39) Hence there exists 𝑅^′₀>0 such that ‖𝑦‖_𝑋<𝑅^′₀. Thus if 𝑅₁=max{𝑅^′₀,𝑅₀}, then we have, whenever 𝑅>𝑅₁, that deg(𝑗∘(ℒ+𝛿𝐽−𝐹−ℎ),𝐵(𝑅),0)=deg(𝑗∘(ℒ+𝛿𝐽),𝐵(𝑅),0)=±1, (4.40) which completes the proof.

By a similar manner we may establish the following.

Lemma 4.9. Let (H1) and (H2^′) hold. Then there exists 𝑅₁∶𝑅₁≥𝑅₀ such that, for 0≥𝜇≥−𝛿 and 𝑅≥𝑅₁, one has deg(𝑗∘(ℒ+𝜇𝐽−𝐹−ℎ),𝐵(𝑅),0)=deg(𝑗∘(ℒ−𝛿𝐽),𝐵(𝑅),0)=±1. (4.41) Therefore (4.7)_𝜇 has a solution in 𝐵(𝑅) for 𝜇∈[−𝛿,0].

Lemma 4.10. Let (H1) and (H2) hold. Then there exists 𝛿₁>0 such that, for −𝛿₁<𝜇<0, one has deg(𝑗∘(ℒ+𝜇𝐽−𝐹−ℎ),𝐵(𝑅₁),0)=deg(𝑗∘(ℒ+𝛿₁𝐽),𝐵(𝑅₁),0)=±1. (4.42)

Proof. Let 𝜏₀=inf_{𝑥∈𝜕𝐵(𝑅1)∩𝑋}‖𝑗∘(ℒ𝑥−𝐹(𝑥)−ℎ)‖_𝑋. (4.43) Then it is not difficult to check that 𝜏₀>0. Hence if we take 𝛿₁ so small that 𝛿₁𝑅₁<𝜏₀, then for 𝜇∈[−𝛿₁,𝛿₁], deg(𝑗∘(ℒ+𝜇𝐽−𝐹−ℎ),𝐵(𝑅₁),0)=deg(𝑗∘(ℒ−𝐹−ℎ),𝐵(𝑅₁),0)=±1. (4.44)

Lemma 4.11. Let (H1) and (H2^′) hold. Then there exists 𝛿₁>0 such that, for 0≤𝜇≤𝛿₁, one has deg(𝑗∘(ℒ+𝜇𝐽−𝐹−ℎ),𝐵(𝑅),0)=deg(𝑗∘(ℒ−𝛿₁𝐽),𝐵(𝑅),0)=±1. (4.45)

Proof of Theorem 4.1. By the study of Massabò and Pejsachowicz in [21, Theorem 1.1], (4.7)_𝜇 has a continuum 𝒞_∗={(𝜇,𝑦_𝜇)} of solutions with ‖𝑦_𝜇‖_𝑋<𝑅₁ and 𝜇∈[−𝛿₁,𝛿]. On the other hand, since 𝐹 is ℒ-completely continuous and satisfies (H1) and since 𝜇=0 is a simple eigenvalue, it follows from the study by Rabinowitz in [22, Theorem 1.6] that 𝜇=0 is a bifurcation point from infinity for (4.7)_𝜇. Moreover, there exist two continua 𝒞^±_∞={(𝜇,𝑦_𝜇)}⊂ℝ×𝑋 (4.46) of solutions of (4.7)_𝜇, bifurcating from infinity at 𝜇=0, that is, there exists 𝜖₀∈(0,1/𝑅₁), such that for all 𝜖∶0<𝜖≤𝜖₀ there exist two continua 𝒞⁺_𝜖 and 𝒞⁻_𝜖 with 𝒞⁺_𝜖⊂𝒞⁺_∞,𝒞⁻_𝜖⊂𝒞⁻_∞,𝒞^±_𝜖⊂{(𝜇,𝑦_𝜇)∶‖𝑦_𝜇‖_𝑋≥1𝜖,|𝜇|<𝜖}=∶𝑈_𝜖(0,∞), (4.47) and 𝒞^±_𝜖 connects (0,∞) to 𝜕𝑈_𝜖(0,∞). Notice that (𝜇,𝑦)∈𝒞⁺_𝜖 implies that 𝑦(𝑎+2)>0, and (𝜇,𝑦)∈𝒞⁻_𝜖 implies that 𝑦(𝑎+2)<0. So, 𝒞⁺_𝜖∩𝒞⁻_𝜖=∅. (4.48) Now, Lemma 4.6 implies that 𝒞^±_𝜖⊆{(𝜇,𝑦_𝜇)∣‖𝑦_𝜇‖_𝑋≥1𝜖,−𝜖<𝜇<0}. (4.49) This completes the proof.

Proof of Theorem 4.2. Using similar arguments, we may get the desired results.

Acknowledgments

The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC (no. 10671158), NWNU-KJCXGC-03-47, the Spring-sun program (no. Z2004-1-62033), SRFDP (no. 20060736001), and the SRF for ROCS, SEM (2006 [311]).