Necessary and Sufficient Condition for the Existence of Solutions to a Discrete Second-Order Boundary Value Problem

Abstract

This paper is concerned with the existence of solutions for the discrete second-order boundary value problem Δ²𝑢(𝑡−1)+𝜆₁𝑢(𝑡)+𝑔(Δ𝑢(𝑡))=𝑓(𝑡), 𝑡∈{1,2,…,𝑇}, 𝑢(0)=𝑢(𝑇+1)=0, where 𝑇>1 is an integer, 𝑓∶{1,…,𝑇}→ℝ, 𝑔∶ℝ→ℝ is bounded and continuous, and 𝜆₁ is the first eigenvalue of the eigenvalue problem Δ²𝑢(𝑡−1)+𝜆𝑢(𝑡)=0, 𝑡∈𝕋, 𝑢(0)=𝑢(𝑇+1)=0.

1. Introduction

Let 𝑔∶ℝ→ℝ and 𝑝∶[0,𝜋]→ℝ be continuous. The nonlinear two-point boundary value problem of ordinary differential equation𝑢^′′(𝑡)+𝑢(𝑡)+𝑔𝑢^′(𝑡)(),𝑢(0)(𝜋)=𝑝,𝑡∈0,𝜋=𝑢=0, (1.1) is very important in applications. Let us mention the problems arising in viscosity, nonlinear oscillations, electric circuits, and so forth. The term 𝑔(𝑢^′) may be regarded as a nonlinear damping term in resonance problems and its appears, for example, in Rayleigh’s equation (which is closely connected with a theory of oscillation of violin string), in oscillations of a simple pendulum under the action of viscous damping, in dry (Coulomb) friction (which occurs when the surfaces of two solids are contact and relative motion without lubrication), and in some cases of van der Pol oscillator, see [1–4] and the references therein.

Since the pioneer work of Landesman and Lazer [5], the problems of the type𝑢^′′(𝑡)(𝑡)(𝑢)(𝑡)(),𝑢(0)(𝜋)+𝑢+𝑔=𝑓,𝑡∈0,𝜋=𝑢=0, (1.2) (where 𝑔 is independent of 𝑢′) have been extensively studied in the past forty years, see Iannacci and Nkashama [6] and the references therein.

It has been remarked (see [7, 8]) that conditions of the Landesmen-Lazer type are not appropriated to yield the existence of solutions to (1.1). Thus, it is usually much more difficult to deal with (1.1) than to deal with (1.2), see Kannan et al. [7], Cañada and Drábek [8], Habets and Sanchez [9], Drábek et al. [10], and Del Toro and Roca [11].

In [8], Cañada and Drábek used the well-known Lyapunov-Schmidt method and the Schauder fixed point theorem to find a necessary and sufficient condition for the existence of solutions of (1.1). To wit, they proved

Theorem A (See [8, Theorem  3.1]). Let 𝑝∶[0,𝜋]→ℝ be continuous and let 𝑝(𝑡)=𝑠2𝜋sin𝑡+𝑝(𝑡),𝑠∈ℝ,^𝜋₀𝑝(𝑡)sin𝑡𝑑𝑡=0. (1.3)

Let 𝑔∶ℝ→ℝ be continuous and bounded with 𝑔(−∞)=𝑔(+∞) and 𝑔(𝜉)<𝑔(+∞) for 𝜉∈ℝ, where𝑔(−∞)∶=lim_{𝑠→−∞}𝑔(𝑠),𝑔(+∞)∶=lim_𝑠→+∞𝑔(𝑠). (1.4)

Then for any 𝑝∈𝐶[0,𝜋] with ∫^𝜋₀𝑝(𝑡)sin𝑡𝑑𝑡=0, there exists a real number 𝑔_𝑝<22/𝜋𝑔(+∞) such that (1.1) has at least one solution 𝑢∈𝐶²[0,𝜋] if and only if𝑠∈𝑔_𝑝,22𝜋𝑔(+∞). (1.5)

It is the purpose of this paper to establish the similar results for the discrete analogue of (1.1) of the formΔ²𝑢(𝑡−1)+𝜆₁𝑢(𝑡)((𝑡(𝑡)𝑢(0)()+𝑔Δ𝑢))=𝑓,𝑡∈𝕋,=𝑢𝑇+1=0, (1.6) where 𝑇>1 is an integer, 𝕋∶={1,…,𝑇}, 𝑔∶ℝ→ℝ is bounded and continuous, 𝑓∶𝕋→ℝ, 𝜆₁ is the first eigenvalue of the linear eigenvalue problemΔ²𝑢()(𝑡)𝑢(0)()𝑡−1+𝜆𝑢=0,𝑡∈𝕋,=𝑢𝑇+1=0. (1.7) Finally, it is worth remarking that the existence of solutions for nonlinear problemΔ²𝑢(𝑡−1)+𝜆₁𝑢(𝑡)(𝑢(𝑡(𝑡)𝑢(0)()+𝑔))=𝑓,𝑡∈𝕋,=𝑢𝑇+1=0, (1.8) which is a discrete analogue of (1.2), has been studied by Rodriguez [12] and Ma [13]. For other recent results on the existence of solutions of discrete problems, see [14–21] and the reference therein.

The rest of this paper is arranged as follows. In Section 2, we give some preliminaries and develop the methods of lower and upper solutions for the more generalized problems, that is, the case of the nonlinearity 𝑔=𝑔(𝑡,𝑢,Δ𝑢); in Section 3, we state our main result and provide the proof.

2. Preliminaries

Recall that 𝕋={1,2,…,𝑇}. Let 𝕋={0,1,…,𝑇+1}. Let 𝑋∶={𝑢∣𝑢∶𝕋→ℝ}, 𝑌∶={𝑢∣𝑢∶𝕋→ℝ} be equipped with the norm‖𝑢‖_𝑋=max_𝑘∈𝕋|𝑢(𝑘)|,‖𝑢‖_𝑌=max_𝑘∈𝕋|𝑢(𝑘)|, (2.1)

respectively. It is easy to see that (𝑋,‖⋅‖_𝑋) and (𝑌,‖⋅‖_𝑌) are Banach spaces.

Assume that 𝑔₀∶𝕋×ℝ²→ℝ is a continuous function, bounded by a constant 𝑀>0:𝑔₀𝑡,𝜂,𝜉≤𝑀 (2.2)

for 𝑡∈𝕋 and (𝜂,𝜉)∈ℝ². Consider the following problem:Δ²𝑢(𝑡−1)+𝜆₁𝑢(𝑡)+𝑔₀((𝑡)(𝑡(𝑡)𝑢(0)()𝑡,𝑢,Δ𝑢))=𝑓,𝑡∈𝕋,=𝑢𝑇+1=0. (2.3)(2.4)

Definition 2.1. If 𝑥∈𝑋 satisfies Δ²𝑥(𝑡−1)+𝜆₁𝑥(𝑡)≥𝑓(𝑡)−𝑔₀((𝑡)(𝑡𝑥(0)()𝑡,𝑥,Δ𝑥)),𝑡∈𝕋,≤0,𝑥𝑇+1≤0, (2.5) then one says 𝑥(𝑡) is a lower solution of (2.3), (2.4). If 𝑦∈𝑋 satisfies Δ²𝑦(𝑡−1)+𝜆₁𝑦(𝑡)≤𝑓(𝑡)−𝑔₀(𝑡)(𝑡𝑦(0)()𝑡,𝑦,Δ𝑦),𝑡∈𝕋,≥0,𝑦𝑇+1≥0, (2.6) then one says 𝑦(𝑡) is an upper solution of (2.3), (2.4).

Theorem 2.2. Suppose that 𝑥(𝑡), 𝑦(𝑡) are the lower and upper solutions of (2.3), (2.4), respectively, and 𝑥(𝑡)≤𝑦(𝑡), 𝑡∈𝕋. Then BVP (2.3) and (2.4) have at least one solution 𝑢(𝑡) satisfies 𝑥(𝑡)≤𝑢(𝑡)≤𝑦(𝑡). (2.7)

Proof. Define the function 𝑝∶𝕋×ℝ→ℝ by 𝑝((𝑡=𝑥(𝑡)(𝑡)(𝑡),𝑢(𝑡)(𝑡)(𝑡)(𝑡),𝑦(𝑡)(𝑡)(𝑡).𝑡,𝑢)),𝑢<𝑥,𝑥≤𝑢≤𝑦,𝑢>𝑦 (2.8) Set 𝑓^∗(𝑡,𝑢,𝑣)=𝑓(𝑡)−𝑔₀(𝑡,𝑢,𝑣)−𝜆₁𝑢. Consider the auxiliary problems: Δ²𝑢(𝑡−1)=𝑓^∗()(𝑢(0)()𝑡,𝑝𝑡,𝑢,Δ𝑝𝑡,𝑢),𝑡∈𝕋,=𝑢𝑇+1=0. (2.9) From (2.8) and the boundness of 𝑔₀, we know 𝑓^∗(𝑡,𝑝(𝑡,𝑢),Δ𝑝(𝑡,𝑢)) is bounded. So, by the Schauder fixed point theorem, (2.9) has a solution 𝑢∈𝑋.
Now, we only prove 𝑢(𝑡)≤𝑦(𝑡), the other case 𝑢(𝑡)≥𝑥(𝑡) is similar.
Set 𝑧(𝑡)=𝑢(𝑡)−𝑦(𝑡). Suppose that 𝑧(𝑡)>0, for 𝑡∈{𝑡₀+1,𝑡₀+2,…,𝑡₀+𝑝}, and 𝑧(𝑡₀)≤0, 𝑧(𝑡₀+𝑝+1)≤0, where 𝑡₀∈{0,1,…,𝑇}, 𝑝∈{1,2,…,𝑇}.
On the other hand, by the definition of upper solution, for 𝑡∈{𝑡₀+1,𝑡₀+2,…,𝑡₀+𝑝},Δ²𝑦(𝑡−1)≤𝑓^∗(𝑡)(𝑡(𝑡)𝑡,𝑦,Δ𝑦)=𝑓−𝜆₁𝑦(𝑡)−𝑔₀(𝑡)(𝑡)(𝑡)𝑡,𝑦,Δ𝑦=𝑓−𝜆₁𝑝(𝑡,𝑢)−𝑔₀𝑡,𝑝(𝑡,𝑢(𝑡)),Δ𝑝(𝑡,𝑢(𝑡))=Δ²𝑢(𝑡−1). (2.10) Then Δ²𝑧(𝑡−1)≥0,𝑡∈𝑡₀+1,𝑡₀+2,…,𝑡₀,𝑧𝑡+𝑝₀≤0,𝑧𝑡₀+𝑝+1≤0. (2.11) Now, by the convexity of 𝑧 on {𝑡₀+1,𝑡₀+2,…,𝑡₀+𝑝}, we get 𝑧(𝑡)≤0, 𝑡∈{𝑡₀+1,𝑡₀+2,…,𝑡₀+𝑝}, that is, 𝑢(𝑡)≤𝑦(𝑡), 𝑡∈{𝑡₀+1,𝑡₀+2,…,𝑡₀+𝑝}. This contradicts 𝑢(𝑡)>𝑦(𝑡), 𝑡∈{𝑡₀+1,𝑡₀+2,…,𝑡₀+𝑝}. Thus, 𝑢(𝑡)≤𝑦(𝑡), 𝑡∈𝕋.

Lemma 2.3. See ∑^𝑇_𝑡=1sin²(𝜋𝑡/(𝑇+1))=(𝑇+1)/2.

Proof. Let 𝜔=cos(2𝜋/(𝑇+1))+𝑖sin(2𝜋/(𝑇+1)). Then 𝜔^𝑇+1=1 and (1−𝜔)(1+𝜔+𝜔²+⋯+𝜔^𝑇)=0. Since 1−𝜔≠0, we have ∑^𝑇_𝑡=1cos(2𝜋𝑡/(𝑇+1))=−1. This together with the fact that ∑^𝑇_𝑡=1sin²(𝜋𝑡/(𝑇+1))=∑^𝑇_𝑡=1(1−cos(2𝜋𝑡/(𝑇+1)))/2 implies the assertion holds.

Now, let 𝜓₁(𝑡)∶=2/(𝑇+1)sin(𝜋𝑡/(𝑇+1)), 𝑡∈𝕋, denote the positive eigenfunction corresponding to the first eigenvalue 𝜆₁=4sin²(𝜋/2(𝑇+1)) of (1.7). Then by Lemma 2.3, ∑^𝑇_𝑡=1𝜓²₁(𝑡)=1.

Since 𝜓₁(𝑡) is located on 2/(𝑇+1)sin𝑡,𝑡∈[0,𝜋], by the direct computation, we can obtain the following result.

Lemma 2.4. If 𝑇 is an odd number, then Δ𝜓₁(𝑡)𝑇>0,for𝑡∈0,…2,Δ𝜓₁(𝑡)𝑇<0,for𝑡∈2+1,…,𝑇, (2.12) if 𝑇 is an even number, then Δ𝜓₁(𝑇/2)=0, Δ𝜓₁(𝑡)𝑇>0,for𝑡∈0,…2−1,Δ𝜓₁(𝑡)𝑇<0,for𝑡∈2+1,…,𝑇. (2.13)

Define the operator 𝐿∶𝐷(𝐿)⊂𝑋→𝑌 by𝐿𝑢(𝑡)=Δ²𝑢(𝑡−1)+𝜆₁𝑢(𝑡), (2.14)

where 𝐷(𝐿)={𝑢∈𝑋∣𝑢(0)=𝑢(𝑇+1)=0}.

Define 𝑁∶𝑋→𝑌 by(𝑁𝑢)(𝑡)=𝑓(𝑡)−𝑔₀(𝑡,𝑢(𝑡),Δ𝑢(𝑡)). (2.15)

Then (2.3), (2.4) is equivalent to the operator equation 𝐿𝑢=𝑁𝑢.

In Theorem 2.2, we established the methods of lower and upper solutions under well order. Now, we can also develop the methods of lower and upper solutions for (2.3), (2.4) when 𝑥(𝑡)≤𝑦(𝑡) is not necessary, its proofs are based on the following lemma, that is, the connectivity properties of the solution sets of parameterized families of compact vector fields, they are a direct consequence of Mawhin [22, Lemma  2.3].

Lemma 2.5 (see [22, Lemma  2.3]). Let 𝐸 be a Banach space and 𝐶⊂𝐸 a nonempty, bounded, closed convex subset. Suppose that 𝑇∶[𝑎,𝑏]×𝐶→𝐶 is completely continuous. Then the set 𝑆={(𝜆,𝑥)∣𝑇(𝜆,𝑥)=𝑥,𝜆∈[𝑎,𝑏]} (2.16) contains to be a closed connected subset Σ which connects {𝑎}×𝐶 to {𝑏}×𝐶.

Theorem 2.6. Assume that 𝑥(𝑡), 𝑦(𝑡) are the lower solution and the upper solution of (2.3), (2.4), respectively. Then (2.3) and (2.4) have at least one solution.

Proof. Define the projections 𝑃∶𝑋→𝑋, 𝑄∶𝑌→ℝ by (𝑃𝑢)(𝑡)=^𝑇_𝑡=1𝑢(𝑡)𝜓₁(𝑡)𝜓₁(𝑡),𝑄𝑦(𝑡)=^𝑇_𝑡=1𝑦(𝑡)𝜓₁(𝑡). (2.17) Then Im𝑃=Ker𝐿, Ker𝑄=Im𝐿, and 𝑋=(Ker𝑃⊕Ker𝐿),𝑌=(Im𝐿⊕Im𝑄). Now, the operator equation 𝐿𝑢=𝑁𝑢 is equivalent to the alternative system 𝑢−𝑃𝑢=𝐾(𝐼−𝑄)𝑁𝑢,𝑄𝑁𝑢=0, (2.18) where 𝐾 is the inverse of mapping 𝐿∶(𝐷(𝐿)∩Ker𝑃)→Im𝐿.
Writing 𝑢∈𝐷(𝐿) in the form 𝑢(𝑡)=𝑐𝜓₁(𝑡)+𝑤(𝑡), 𝑐∈ℝ, ∑^𝑇_𝑡=1𝜓₁(𝑡)𝑤(𝑡)=0, (2.3) and (2.4) are equivalent to the system 𝑤=𝐾(𝐼−𝑄)𝑁𝑐𝜓₁(⋅)+𝑤, (2.19)𝑄𝑁𝑐𝜓₁(⋅)+𝑤=0. (2.20) Since 𝑋 is finite dimensional, it is easy to see that 𝐾(𝐼−𝑄)𝑁 is completely continuous, by the Schauder fixed point theorem and the fact 𝑁(𝑐𝜓₁(𝑡)+𝑤(𝑡)) is bounded, we get that for any fixed 𝑐∈ℝ, 𝑊(𝑐)∶={𝑤∈𝑋∩Ker𝑃∣(𝑐,𝑤)satisfies(2.19)}≠∅ and 𝑊(𝑐) is bounded. Then there exist positive constants 𝛼>𝜏 such that −𝜏𝜓₁≤𝑤≤𝜏𝜓₁ for all 𝑤∈𝑊(𝑐), (𝛼−𝜏)𝜓₁≥𝑥(𝑡) and −(𝛼−𝜏)𝜓₁≤𝑦(𝑡). Let 𝛾(𝑐,𝑤)=^𝑇_𝑡=1𝑁𝑐𝜓₁(𝑡)+𝑤(𝑡)𝜓₁(𝑡), (2.21) for all (𝑐,𝑤)∈ℝ×𝑊(𝑐). Observe that Lemma 2.5 is applicable. Hence there exists a connected subset of {(𝑐,𝑤)∈𝑅×(𝑋∩Ker𝑃)∣(𝑐,𝑤)satisfies(2.19)}, Σ_𝛼(𝜓₁), which connected {−𝛼}×𝑊(−𝛼) and {𝛼}×𝑊(𝛼). Since 𝛾∶Σ_𝛼(𝜓₁)→ℝ is continuous, 𝐼∶=𝛾(Σ_𝛼(𝜓₁)) is an interval. If 0∈𝐼, then (2.3) and (2.4) have a solution. If 𝐼⊂(0,∞), then every 𝑐𝜓₁+𝑤 with (𝑐,𝑤)∈Σ_𝛼(𝜓₁) is an upper solution. Indeed, it is obvious that 𝐿𝑐𝜓₁(𝑡)+𝑤(𝑡)−𝑁𝑐𝜓₁(𝑡)+𝑤(𝑡)=−𝛾(𝑐,𝑤)≤0,𝑡∈𝕋,𝑐𝜓₁+𝑤(0)=𝑐𝜓₁+𝑤(𝑇+1)=0. (2.22) By construction, 𝛼𝜓₁+𝑤 with (𝛼,𝑤)∈Σ_𝛼(𝜓₁) satisfies 𝑥(𝑡)≤𝛼𝜓₁+𝑤. Hence, from Theorem 2.2, (2.3) and (2.4) have a solution. A similar argument applies if 𝐼⊂(−∞,0).

Theorem 2.7. Suppose that 𝑓 satisfies 𝑓(𝑡)=𝑠𝜓₁(𝑡)+𝑓(𝑡),𝑠∈ℝ, (2.23) where 𝑓 satisfies ^𝑇_𝑡=1𝑓(𝑡)𝜓₁(𝑡)=0. (2.24) Then, there exists a nonempty, connected, and bounded set 𝐽_𝑓⊂ℝ such that (2.3) and (2.4) have at least one solution 𝑢∈𝑋 if and only if𝑠∈𝐽_𝑓.

Proof. As the proof of Theorem 2.6, (2.3) and (2.4) are equivalent to the system (2.19), (2.20). Since 𝑁 is bounded, applying the Schauder fixed point theorem we obtain that for any fixed 𝑐∈ℝ, there exits at least one 𝑤_𝑐∈𝑋 such that (2.19) holds.
Now, (2.20) becomes ^𝑇_𝑡=1𝑔₀𝑡,𝑐𝜓₁(𝑡)+𝑤_𝑐(𝑡),𝑐Δ𝜓₁(𝑡)+Δ𝑤_𝑐(𝑡)𝜓₁(𝑡)=𝑠. (2.25) Hence, for a given 𝑓, ∑^𝑇_𝑡=1𝑓(𝑡)𝜓₁(𝑡)=0, (2.3), (2.4) with 𝑓(𝑡)=𝑐𝜓₁(𝑡)+𝑓(𝑡) has at least one solution if and only if 𝑠 belongs to the range of the (multivalued, in general) function Γ_𝑓∶ℝ→Γ_𝑓(ℝ), Γ_𝑓(𝑐)=^𝑇_𝑡=1𝑔₀𝑡,𝑐𝜓₁(𝑡)+𝑤_𝑐(𝑡),𝑐Δ𝜓₁(𝑡)+Δ𝑤_𝑐(𝑡)𝜓₁(𝑡), (2.26) where 𝑤_𝑐∈{𝑤∈𝐷(𝐿)∶𝑤isasolutionof(2.19)forfixed𝑐}. But 𝐽_𝑓≡Γ_𝑓(ℝ) is a connected set. In fact, let 𝑠₁ and 𝑠₂ belong to 𝐽_𝑓 and 𝑠₁≤𝑠₂. Then (2.3), (2.4) with 𝑓₁=𝑠₁𝜓₁+𝑓 and 𝑓₂=𝑠₂𝜓₁+𝑓 has solutions 𝑢₁ and 𝑢₂, respectively. If we consider (2.3), (2.4) with 𝑓=𝑠𝜓₁+𝑓, where 𝑠∈[𝑠₁,𝑠₂], then 𝑢₁ is an upper solution and 𝑢₂ is a lower solution to this problem. By Theorem 2.6, there exists at least one solution, that is, 𝑠 belongs to 𝐽_𝑓. Moreover, since 𝑔 is bounded, the range of Γ_𝑓 is bounded.

3. Main Results

In this section, we deal with (1.6). First, let us make the following assumptions:

(H1) 𝑔∶ℝ→ℝ is a bounded and continuous function and satisfies 𝑔(+∞)=𝑔(−∞) and 𝑔(𝜉)<𝑔(+∞) for any 𝜉∈ℝ,

(H2) 𝑓∶𝕋→ℝ satisfies𝑓(𝑡)=𝑠𝜓₁(𝑡)+𝑓(𝑡),𝑠∈ℝ,^𝑇_𝑡=1𝑓(𝑡)𝜓₁(𝑡)=0. (3.1)

Theorem 3.1. Suppose that (H1),  (H2) hold. Then there exists a real number 𝑔_𝑓, 𝑔_𝑓<2/(𝑇+1)𝑔(+∞)∑^𝑇_𝑡=1sin(𝜋𝑡/(𝑇+1)), such that (1.6) has at least one solution 𝑢∈𝑋 if and only if 𝑠∈𝑔_𝑓,2𝑔()𝑇+1+∞^𝑇_𝑡=1sin𝜋𝑡.𝑇+1 (3.2)

Proof. Note that 𝜓₁(𝑡)=2/(𝑇+1)sin(𝜋𝑡/(𝑇+1)). Due to the consideration in the the proof of Theorem 2.7. It is sufficient to show that for a given 𝑓 with ∑^𝑇_𝑡=1𝑓(𝑡)𝜓₁(𝑡)=0, we have Γ_𝑓(ℝ)=𝑔_𝑓,2𝑔()𝑇+1+∞^𝑇_𝑡=1sin𝜋𝑡.𝑇+1 (3.3) The (possibly multivalued) function Γ_𝑓 has the following form: Γ_𝑓(𝑐)=2𝑇+1^𝑇_𝑡=1𝑔𝑐Δ𝜓₁(𝑡)+Δ𝑤_𝑐(𝑡)sin𝜋𝑡,𝑇+1 (3.4) where 𝑐∈ℝ and 𝑤_𝑐 verify (2.19). From the boundedness of 𝑔 and (2.19), there exists a constant 𝐷>0 (independent of 𝑐) such that ‖𝑤_𝑐‖_𝑋≤𝐷 for any 𝑐∈ℝ, furthermore, max_{{}𝑡∈0,1,…,𝑇}Δ𝑤_𝑐(𝑡)≤2𝐷. (3.5) Now, we divide the proof into two cases.Case 1. 𝑇 is an odd number. By Lemma 2.4, we obtain that ^𝑇_𝑡=1𝑔2𝑐2𝜋𝑇+1sin2()𝜋()𝑇+1cos2𝑡+12(𝑇+1)+Δ𝑤_𝑐(𝑡)sin𝜋𝑡=𝑇+1_{[𝑇/2]𝑡=1}𝑔2𝑐2𝜋𝑇+1sin2()𝜋()𝑇+1cos2𝑡+12(𝑇+1)+Δ𝑤_𝑐(𝑡)sin𝜋𝑡+𝑇+1_{𝑇[]𝑡=𝑇/2+1}𝑔2𝑐2𝜋𝑇+1sin2()𝜋()𝑇+1cos2t+12(𝑇+1)+Δ𝑤_𝑐(𝑡)sin𝜋𝑡⟶𝑇+1_{[𝑇/2]𝑡=1}𝑔(±∞)sin𝜋𝑡+𝑇+1^𝑇_{[]𝑠=𝑇/2+1}𝑔(∓∞)sin𝜋𝑡,𝑇+1 (3.6) as 𝑐→±∞. Due to 𝑔(+∞)=𝑔(−∞), we get Γ_𝑓(𝑐)→2𝑔()𝑇+1+∞^𝑇_𝑡=1sin𝜋𝑡.𝑇+1 (3.7) The assumption 𝑔(𝜉)<𝑔(+∞), 𝜉∈ℝ, and (3.5) yields Γ_𝑓(𝑐)<2𝑔()𝑇+1+∞^𝑇_𝑡=1sin𝜋𝑡𝑇+1 (3.8) for any 𝑐∈ℝ.Case 2. 𝑇 is an even number. By Lemma 2.4, we know that Δ𝜓₁𝑇2=2𝑇+1Δsin𝜋𝑇2(𝑇+1)=22𝜋𝑇+1sin2()𝜋()𝑇+1cos2𝑇/2+12(𝑇+1)=0. (3.9) Hence, ^𝑇_𝑡=1𝑔2𝑐2𝜋𝑇+1sin2()𝜋()𝑇+1cos2𝑡+12(𝑇+1)+Δ𝑤_𝑐(𝑡)sin𝜋𝑡=𝑇+1_{(𝑇/2)−1𝑡=1}𝑔2𝑐2𝜋𝑇+1sin2()𝜋()𝑇+1cos2𝑡+12(𝑇+1)+Δ𝑤_𝑐(𝑡)sin𝜋𝑡+𝑇+1_{𝑇()𝑡=𝑇/2+1}𝑔2𝑐2𝜋𝑇+1sin2()𝜋()𝑇+1cos2𝑡+12(𝑇+1)+Δ𝑤_𝑐(𝑡)sin𝜋𝑡𝑇+1+𝑔Δ𝑤_𝑐𝑇2sin𝜋𝑇2().𝑇+1 (3.10)By (3.5) and the assumption 𝑔(𝜉)<𝑔(+∞), 𝜉∈ℝ, we know that for any 𝑐∈ℝ, 𝑔(Δ𝑤_𝑐(𝑇/2))<𝑔(+∞). Thus, for any 𝑐∈ℝ, Γ_𝑓(𝑐)<2𝑔()𝑇+1+∞^𝑇_𝑡=1sin𝜋𝑡.𝑇+1 (3.11)
It is sufficient to prove that this infimum is achieved. Let us denote 𝑔_𝑓=inf_𝑐∈ℝΓ_𝑓(𝑐). (3.12) Suppose that {𝑠_𝑛}⊂Γ_𝑓(ℝ) satisfies 𝑠_𝑛→𝑔_𝑓 and {𝑐_𝑛} is the corresponding minimizing sequence, that is, 𝑢_𝑛(𝑡)=𝑐_𝑛2/(𝑇+1)sin(𝜋𝑡/(𝑇+1))+𝑤_𝑐𝑛(𝑡), are the solution of (1.1), with the right-hand sides 𝑓_𝑛(𝑡)=𝑠_𝑛2/(𝑇+1)sin(𝜋𝑡/(𝑇+1))+𝑓(𝑡).
We claim that {𝑐_𝑛} is bounded. In fact, if 𝑐_𝑛→∞ as 𝑛→∞, then we can get two contradictions in the following two cases.
Case 1. If 𝑇 is an odd number, then by (2.20), 2𝑇+1^𝑇_𝑡=1𝑔2𝑐_𝑛2𝜋𝑇+1sin2()𝜋()𝑇+1cos2𝑡+12(𝑇+1)+Δ𝑤_𝑐𝑛(𝑡)sin𝜋𝑡𝑇+1=𝑠_𝑛, (3.13) letting 𝑛→∞ in (3.13), we get 2𝑇+1^𝑇_𝑡=1𝑔(+∞)sin𝜋𝑡𝑇+1=𝑔_𝑓. (3.14)From (H1), we arrive for any 𝜉∈ℝ, 𝑔(𝜉)<𝑔(+∞), which together with (3.14) implies that 2𝑇+1^𝑇_𝑡=1𝑔2𝑐2𝜋𝑇+1sin2()𝜋()𝑇+1cos2𝑡+12(𝑇+1)+Δ𝑤_𝑐(𝑡)sin𝜋𝑡𝑇+1<𝑔_𝑓. (3.15) This contradicts (3.12).Case 2. If 𝑇 is an even number, then by (2.20) and Δ𝜓₁(𝑇/2)=0, we get 𝑠_𝑛=2𝑇+1^(𝑇/2)−1_𝑡=1𝑔2𝑐_𝑛2𝜋𝑇+1sin2()𝜋()𝑇+1cos2𝑡+12(𝑇+1)+Δ𝑤_𝑐𝑛(𝑡)sin𝜋𝑡+𝑇+12𝑇+1^𝑇_{()𝑡=𝑇/2+1}𝑔2𝑐_𝑛2𝜋𝑇+1sin2()𝜋()𝑇+1cos2𝑡+12(𝑇+1)+Δ𝑤_𝑐𝑛(𝑡)sin𝜋𝑡+𝑇+12𝑔𝑇+1Δ𝑤_𝑐𝑛𝑇2sin𝜋𝑇2().𝑇+1 (3.16)This implies that 𝑔_𝑓≥2𝑇+1^(𝑇/2)−1_𝑡=1𝑔(+∞)sin𝜋𝑡+𝑇+12𝑇+1^𝑇_{()𝑡=𝑇/2+1}𝑔(+∞)sin𝜋𝑡+𝑇+12𝑇+1inf_{𝑤𝑐𝑛∈𝑋}𝑔Δ𝑤_𝑐𝑛𝑇2sin𝜋𝑇2().𝑇+1 (3.17) On the other hand, by (3.12) and (H1), we get that for any fixed 𝑛∈ℕ, 𝑔_𝑓≤2𝑇+1^(𝑇/2)−1_𝑡=1𝑔2𝑐_𝑛2𝜋𝑇+1sin2()𝜋()𝑇+1cos2𝑡+12(𝑇+1)+Δ𝑤_𝑐𝑛(𝑡)sin𝜋𝑡+𝑇+12𝑇+1^𝑇_{()𝑡=𝑇/2+1}𝑔2𝑐_𝑛2𝜋𝑇+1sin2()𝜋()𝑇+1cos2𝑡+12(𝑇+1)+Δ𝑤_𝑐𝑛(𝑡)sin𝜋𝑡+𝑇+12𝑇+1inf_{𝑤𝑐𝑛∈𝑋}𝑔Δ𝑤_𝑐𝑛𝑇2sin𝜋𝑇2()<𝑇+12𝑇+1^(𝑇/2)−1_𝑡=1𝑔(+∞)sin𝜋𝑡+𝑇+12𝑇+1^𝑇_{()𝑡=𝑇/2+1}𝑔(+∞)sin𝜋𝑡+𝑇+12𝑇+1inf_{𝑤𝑐𝑛∈𝑋}𝑔Δ𝑤_𝑐𝑛𝑇2sin𝜋𝑇2().𝑇+1 (3.18) Now, we obtain a contradiction. Thus, 𝑐_𝑛 is bounded.
Since 𝑋 is finite dimensional and 𝑤_𝑐𝑛 is bounded, we obtain that 𝑐_𝑛→𝑐, 𝑤_𝑐𝑛→𝑤_𝑐 (at least for a subsequence), and 𝑢(𝑡)=𝑐2/(𝑇+1)sin(𝜋𝑡/(𝑇+1))+𝑤_𝑐(𝑡) is a solution of (1.1) with 𝑓(𝑡)=𝑔_𝑓2/(𝑇+1)sin(𝜋𝑡/(𝑇+1))+𝑓(𝑡). Hence, the infimum is achieved in 𝑐.

Acknowledgments

This paper is supported by NSFC (11061030, 11101335, 11126296) and the Fundamental Research Funds for the Gansu Universities.