The Solvability and Numerical Simulation for the Elastic Beam Problems with Nonlinear Boundary Conditions

Abstract

We study the existence of multiple solutions for a fourth-order nonlinear boundary value problem. We give some new criteria for guaranteeing that the fourth-order elastic beam equation with a perturbed term has at least three solutions. The proof is based on some three critical points theorems of B. Ricceri. Furthermore, numerical simulations are also presented.

1. Introduction

In this paper, we consider the following elastic beam equation with nonlinear boundary conditions: 𝑢⁽⁴⁾()()𝑢(0)=𝜆𝑓𝑡,𝑢+𝜇𝑔𝑡,𝑢,0<𝑡<1,=𝑢^′(0)𝑢=0,_′′(1)=0,𝑢^′′′(1)=ℎ(𝑢(1)), (1) where 𝜆, 𝜇 are two positive parameters, 𝑓, 𝑔 are two 𝐿¹-Carathéodory’s functions, and ℎ∈𝐶(ℝ) is real function. This kind of problem arises in the study of deflections of elastic beams on nonlinear elastic foundations. The problem has the following physical description: a thin flexible elastic beam of length 1 is clamped at its left end 𝑡=0 and resting on an elastic device at its right end 𝑡=1, which is given by ℎ. Then, the problem models the static equilibrium of the beam under a load, along its length, characterized by 𝑓 and ℎ. The derivation of the model can be found in [1, 2].

In recent years, fourth-order boundary value problems modeling bending equilibria of elastic beams have been extensively studied by many researchers. We refer the reader to [1–8]. Some of them are concerned with nonlinear equations with null boundary conditions; see [3–6]. Others are concerned with nonlinear equations with nonzero or nonlinear boundary conditions, which can model beams resting on elastic bearings located in their extremities; see, for instance, [1, 2, 7, 8] and the references therein.

Let us briefly comment the known results related to problem (1). In [2], Ma studied the existence of positive solutions for problem (1) with 𝜆=1 and 𝜇=0 using mountain pass theorem. Later, Yang et al. [8] considered the existence of two solutions for problem (1) with 𝜇=0, which are generated by the function ℎ. Very recently, Li and Sun [9] obtained the existence of infinitely many solutions for problem (1) with 𝜆=1 and 𝜇=0 by using the fountain theorem.

Motivated by the above works, in the present paper, we establish some existence results of three solutions for problem (1) under rather different assumptions on functions 𝑓, ℎ, and 𝑔. We require that 𝑓 satisfies the asymptotically linear or sublinear conditions at infinity on 𝑢. The proof is based on some three critical points theorems due to Ricceri. It is worth noting that these critical points theorems have been extensively applied in the study of the existence of solutions for nonlinear differential equations; see [10–13].

The remainder of this paper is organized as follows. In Section 2, some preliminary results are presented. In Section 3, we give the proofs of our main results. In addition, some numerical simulations are presented at the end of the paper.

2. Variational Setting and Preliminaries

If 𝑋 is a real Banach space, denote by 𝒲_𝑋 the class of all functionals Φ∶𝑋→ℝ possessing the following property: if {𝑢_𝑛} is a sequence in 𝑋 converging weakly to 𝑢∈𝑋 and liminf_𝑛→∞Φ(𝑢_𝑛)≤Φ(𝑢), then {𝑢_𝑛} has a subsequence converging strongly to 𝑢.

For example, if 𝑋 is uniformly convex and 𝑔∶[0,+∞)→ℝ is a continuous, strictly increasing function, then, by a classical result, the functional 𝑢→𝑔(‖𝑢‖) belongs to the class 𝒲_𝑋.

Theorem 1 (see [14]). Let 𝑋 be a separable and reflexive real Banach space; let Φ∶𝑋→ℝ be a coercive, sequentially weakly lower semicontinuous  𝐶¹ functional, belonging to 𝒲_𝑋, bounded on each bounded subset of 𝑋 and whose derivative admits a continuous inverse on 𝑋^∗; 𝐽∶𝑋→ℝ a 𝐶¹ functional with compact derivative. Assume that Φ has a strict local minimum 𝑢₀ with Φ(𝑢₀)=𝐽(𝑢₀)=0. Finally, setting 𝛼=max0,limsup_{‖𝑢‖→+∞}𝐽(𝑢)Φ(𝑢),limsup_𝑢→𝑢0𝐽(𝑢)Φ(𝑢),𝛽=sup_{𝑢∈Φ−1((0,+∞))}𝐽(𝑢)Φ(𝑢), (2) assume that 𝛼<𝛽.
Then, for each compact interval [𝑎,𝑏]⊂(1/𝛽,1/𝛼) (with the conventions 1/0=+∞,1/+∞=0), there exists 𝑅>0 with the following property: for every 𝜆∈[𝑎,𝑏] and every 𝐶¹ functional Ψ∶𝑋→ℝ with compact derivative, there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], Φ^′(𝑥)=𝜆𝐽^′(𝑥)+𝜇Ψ^′(𝑥) (3) has at least three solutions in 𝑋 whose norms are less than 𝑅.

The following two results of Ricceri guarantee the existence of three solutions for a given equation.

Theorem 2 (see [15]). Let 𝑋 be a reflexive real Banach space; 𝐼⊆ℝ an interval; let Φ∶𝑋→ℝ be a sequentially weakly lower semicontinuous 𝐶¹ functional, bounded on each bounded subset of 𝑋, whose derivative admits a continuous inverse on 𝑋^∗; −𝐽∶𝑋→ℝ a 𝐶¹ functional with compact derivative. Assume that lim_{‖𝑥‖→+∞}(Φ(𝑥)−𝜆𝐽(𝑥))=+∞, (4) for all 𝜆∈𝐼, and that there exists 𝜌∈ℝ such that sup_𝜆∈𝐼inf_𝑥∈𝑋Φ(𝑥)+𝜆𝜌−𝐽(𝑥)<inf_𝑥∈𝑋sup_𝜆∈𝐼Φ(𝑥)+𝜆𝜌−𝐽(𝑥). (5) Then, there exist a nonempty open set 𝐴⊆𝐼 and a positive number 𝑅 with the following property: for every 𝜆∈𝐴 and every 𝐶¹ functional −Ψ∶𝑋→ℝ with compact derivative, there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], Φ^′(𝑥)−𝜆𝐽^′(𝑥)−𝜇Ψ^′(𝑥)=0 (6) has at least three solutions in 𝑋 whose norms are less than 𝑅.

Proposition 3 (see [16]). Let 𝑋 be a nonempty set and Φ, 𝐽 two real functions on 𝑋. Assume that there are 𝑟>0 and 𝑥₀, 𝑥₁∈𝑋 such that Φ𝑥₀=𝐽𝑥₀=0,Φ𝑥₁>𝑟,sup_{𝑥∈Φ−1((−∞,𝑟])}𝐽(𝑥)𝐽𝑥<𝑟¹Φ𝑥₁. (7) Then, for each 𝜌 satisfying sup_{𝑥∈Φ−1((−∞,𝑟])}𝐽(𝑥)𝐽𝑥<𝜌<𝑟¹Φ𝑥₁, (8) one has sup_𝜆≥0inf_𝑥∈𝑋Φ(𝑥)+𝜆𝜌−𝐽(𝑥)<inf_𝑥∈𝑋sup_𝜆≥0Φ(𝑥)+𝜆𝜌−𝐽(𝑥). (9)

Now, we begin describing the variational formulation of problem (1), which is based on the function space 𝐸=𝑢∈𝐻²(0,1);𝑢(0)=𝑢^′(0)=0, (10) where 𝐻²(0,1) is the Sobolev space of all functions 𝑢∶[0,1]→ℝ such that 𝑢 and its distributional derivative 𝑢^′ are absolutely continuous and 𝑢^′′ belongs to 𝐿²([0,1]). Then, 𝐸 is a Hilbert space equipped with the inner product and norm ⟨𝑢,𝑣⟩=¹₀𝑢^′′(𝑡)𝑣^′′(𝑡)𝑑𝑡,‖𝑢‖=𝑢^′′₂, (11) where ‖⋅‖_𝑝 denotes the standard 𝐿^𝑝-norm. In addition, 𝐸 is compactly embedded in the spaces 𝐿²([0,1]) and 𝐶([0,1]), and, therefore, there exist immersion constants 𝑆₂,  𝑆>0, such that ‖𝑢‖₂≤𝑆₂‖𝑢‖,‖𝑢‖_∞≤𝑆‖𝑢‖. (12)

We recall that 𝑓∶[0,1]×ℝ→ℝ is an 𝐿¹-Carathéodory function if(a)the mapping 𝑡↦𝑓(𝑡,𝑢) is measurable for every 𝑢∈ℝ;(b)the mapping 𝑢↦𝑓(𝑡,𝑢) is continuous for almost every 𝑡∈[0,1];(c)for every 𝜌>0 there exists a function 𝑙_𝜌∈𝐿¹([0,1]) such that sup_{|𝑢|≤𝜌}𝑓(𝑡,𝑢)≤𝑙_𝜌(𝑡), (13) for almost every 𝑡∈[0,1].

Definition 4. One says that a function 𝑢∈𝐸 is a weak solution of problem (1) if ¹₀𝑢^′′(𝑡),𝑣^′′(𝑡)𝑑𝑡+ℎ(𝑢(1))𝑣(1)=𝜆¹₀𝑓(𝑡,𝑢)𝑣𝑑𝑡+𝜇¹₀𝑔(𝑡,𝑢)𝑣𝑑𝑡 (14) holds for any 𝑣∈𝐸.

In order to study problem (1), we will use the functionals Φ, 𝐽, and Ψ∶𝐸→ℝ defined by putting Φ(𝑢)=12‖𝑢‖²+₀^𝑢(1)ℎ(𝑠)(𝑢)(𝑢),𝐽(𝑢)𝑑𝑠=𝜙+𝜓∶=¹₀𝐹(𝑡,𝑢)𝑑𝑡,Ψ(𝑢)∶=¹₀𝐺(𝑡,𝑢)𝑑𝑡, (15) respectively, for every 𝑢∈𝐸, where, 𝜙(𝑢)=12‖𝑢‖²,𝜓(𝑢)=₀^𝑢(1)ℎ(𝑡)𝐹()𝑑𝑡,𝑡,𝑢=^𝑢₀𝑓(𝑡,𝑠)𝑑𝑠,𝐺(𝑡,𝑢)=^𝑢₀𝑔(𝑡,𝑠)𝑑𝑠. (16) By the continuity of  ℎ, we get that functional Φ is a continuous Gâteaux differential functional whose Gâteaux derivative is the functional Φ^′(𝑢), given by ⟨Φ^′(𝑢),𝑣⟩=¹₀𝑢^′′(𝑡),𝑣^′′(𝑡)𝑑𝑡+ℎ(𝑢(1))𝑣(1), (17) for any 𝑢, 𝑣∈𝐸. Since 𝑓, 𝑔 are two 𝐿¹-Carathéodory functions, it implies that 𝐽 and Ψ are well defined and continuously Gâteaux differentiable in 𝐸. More precisely, their Gâteaux derivatives are ⟨𝐽^′(𝑢),𝑣⟩=¹₀𝑓(𝑡,𝑢)𝑣𝑑𝑡,⟨Ψ^′(𝑢),𝑣⟩=¹₀𝑔(𝑡,𝑢)𝑣𝑑𝑡, (18) respectively, for every 𝑢, 𝑣∈𝐸.

Lemma 5. Assume that the following condition holds: (H₁)ℎ(𝑠) are nondecreasing on 𝑠∈ℝ and ℎ(𝑠)𝑠≥0, for any 𝑠∈ℝ.
Then, Φ is sequentially weakly lower semicontinuous, bounded on each bounded subset of 𝐸, and its derivative admits a continuous inverse.

Proof. Obviously, 𝜙 is weakly lower semicontinuous in 𝐸. Therefore, it suffices to show that 𝜓 is weakly continuous in 𝐸. In fact, if {𝑢_𝑛}⊂𝐸 and 𝑢_𝑛⇀𝑢 in 𝐸, {𝑢_𝑛} converges uniformly to 𝑢 on [0,1]. Then, there exists 𝐶₁>0 such that 𝑢_𝑛∞≤𝐶₁,∀𝑛∈ℕ. (19) Therefore, we have 𝜓𝑢_𝑛−𝜓(𝑢)=^𝑢𝑛₀⁽¹⁾ℎ(𝑡)𝑑𝑡−₀^𝑢(1)ℎ(𝑡)=𝑑𝑡_{𝑢𝑛𝑢(1)(1)}ℎ(𝑡)𝑑𝑡≤𝐶₂𝑢_𝑛−𝑢_∞⟶0, (20) where 𝐶₂=max_{|𝑡|≤𝐶1}|ℎ(𝑡)|. Thus, 𝜓 is weakly continuous. Therefore, Φ is weakly lower semicontinuous in 𝐸.
Moreover, let Ω be a bounded subset of 𝐸; that is, there exists a constant 𝑎>0 such that ‖𝑢‖≤𝑎 for 𝑢∈Ω. Then, |Φ(𝑢=1)|2‖𝑢‖²+₀^𝑢(1)ℎ(𝑡)≤1𝑑𝑡2𝑎²+max_{|𝑡|≤𝑆𝑎}|ℎ(𝑡)|⋅‖𝑢‖_∞≤12𝑎²+𝑆max_{|𝑡|≤𝑆𝑎}|ℎ(𝑡⋅‖𝑢‖≤1)|2𝑎²+𝑆𝑎max_{|𝑡|≤𝑆𝑎}|ℎ(𝑡)|. (21) Hence, Φ is bounded on each bounded subset of 𝐸.
Next, we show that Φ^′ admits a continuous inverse. For any 𝑢∈𝐸⧵{0}, by (H₁), we have Φ^′(𝑢),𝑢=¹₀𝑢^′′(𝑡),𝑢^′′(𝑡)(𝑢(1𝑢(1)≥𝑑𝑡+ℎ))₁₀𝑢^′′(𝑡),𝑢^′′(𝑡)𝑑𝑡=‖𝑢‖². (22) So, lim_{‖𝑢‖→+∞}⟨Φ^′(𝑢),𝑢⟩/‖𝑢‖=+∞; that is, Φ^′ is coercive.
For any 𝑢,𝑣∈𝐸, by (H₁), one has Φ^′(𝑢)−Φ^′(𝑣)=,𝑢−𝑣₁₀𝑢^′′−𝑣^′′,𝑢^′′−𝑣^′′+(ℎ(𝑢(1(𝑣(1𝑢(1)(1≥‖‖𝑑𝑡))−ℎ)))(−𝑣))𝑢−𝑣². (23) So, Φ^′ is uniformly monotone. By [17, Theorem 26.A(d)], we have that (Φ^′)⁻¹ exists and is continuous.

Remark 6. If (H₁) is replaced by the following condition: (H₁)^′𝑔(𝑠) are odd and nondecreasing on 𝑠∈ℝ.
Then, we can obtain the same conclusion.

Lemma 7. 𝐽 and Ψ are continuously Gateaux differentiable in 𝐸 and their derivatives are compact.

Proof. It is easy to verify that 𝐽 and Ψ are continuously Gâteaux differentiable in 𝐸. Now, we proof that their derivatives are compact.
For any 𝑢, 𝑣∈𝐸, ⟨𝐽^′(𝑢),𝑣⟩=¹₀𝑓(𝑡,𝑢)𝑣𝑑𝑡. (24) Let 𝑢_𝑛⇀𝑢 as 𝑛→∞ in 𝐸. Then, 𝑢_𝑛→𝑢 in 𝐶([0,1]). Since 𝑓(𝑡,𝑢) is continuous in 𝑢, one has 𝑓(𝑡,𝑢_𝑛)→𝑓(𝑡,𝑢) as 𝑛→∞. So 𝐽^′(𝑢_𝑛)→𝐽^′(𝑢) as 𝑛→∞. That is, 𝐽^′ is strongly continuous on 𝐸, which implies that 𝐽^′ is a compact operator by [17, Proposition 26.2]. Moreover, 𝐽^′ is continuous since it is strongly continuous.
Analogously, we have that Ψ^′(𝑢) is a compact operator for any 𝑢.

3. Main Results

In this section, we establish the main abstract results of this paper. Put 𝜆₁‖𝑢‖=inf²+2∫₀^𝑢(1)ℎ(𝑡)𝑑𝑡2∫^𝑇₀𝐹(𝑡,𝑢(𝑡))𝑑𝑡∶𝑢∈𝐸,_𝑇0𝐹((𝑡,𝜆𝑡,𝑢))𝑑𝑡>0₂₋₁=max0,limsup_{‖𝑢‖→+∞}2∫^𝑇₀𝐹(𝑡,𝑢(𝑡))𝑑𝑡‖𝑢‖²+2∫₀^𝑢(1)ℎ(𝑡),𝑑𝑡limsup_|𝑢|→02∫^𝑇₀𝐹(𝑡,𝑢(𝑡))𝑑𝑡‖𝑢‖²+2∫₀^𝑢(1)ℎ(𝑡).𝑑𝑡 (25)

Theorem 8. Suppose that 𝑓∶[0,1]×ℝ→ℝ is an 𝐿¹-Carathéodory function and (𝐻₁) (or (𝐻₁)^′) holds. Moreover, assume that the following conditions hold:(R₁)there exists a constant 𝑝>0 such that maxlimsup_𝑢→0max_𝑡∈[0,1]𝐹(𝑡,𝑢)|𝑢|²,limsup_{|𝑢|→+∞}max_𝑡∈[0,1]𝐹(𝑡,𝑢)|𝑢|²<𝑝; (26)(R₂)there exists a function 𝑐∈𝐶²([0,1]) satisfying 𝑐(0)=𝑐^′(0)=0 and ¹₀𝑐^′′2𝑑𝑡+2₀^𝑐(1)ℎ(𝑡)𝑑𝑡≠0, (27)
such that 𝑝𝑆²<∫¹₀𝐹(𝑡,𝑐(𝑡))𝑑𝑡∫¹₀𝑐^′′2𝑑𝑡+2∫₀^𝑐(1)ℎ(𝑡).𝑑𝑡 (28) Then, for each compact interval [𝑎,𝑏]⊂(𝜆₁,𝜆₂), there exists 𝑅>0 with the following property: for every 𝜆∈[𝑎,𝑏] and for any 𝐿¹-Carathéodory function ℎ∶[0,1]×ℝ→ℝ, there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], problem (1) has at least three weak solutions whose norms are less than 𝑅.

Proof. Obviously, 𝐸 is a separable and uniformly convex Banach space. By Lemmas 5 and 7, we obtain that Φ is a continuous Gâteaux derivative, sequentially weakly lower semicontinuous, bounded on each bounded subset of 𝐸, and its derivative admits a continuous inverse; 𝐽, Ψ are the Gâteaux derivative functionals whose derivatives are compact. It is clear that 𝜙∈𝒲_𝐸 (see the beginning of Section 2). Now, we show that Φ∈𝐸. For a sequence {𝑢_𝑛}⊂𝐸, if {𝑢_𝑛}⇀𝑢∈𝐸 and liminf_𝑛→∞Φ(𝑢_𝑛)≤Φ(𝑢), in view of the weak continuity of 𝜓, one has liminf_𝑛→∞𝜙(𝑢_𝑛)≤𝜙(𝑢). Thus, {𝑢_𝑛} has a subsequence converging strongly to 𝑢. Therefore, Φ∈𝒲_𝐸. Moreover, Φ is coercive and has a strict local minimum 0 with Φ(0)=𝐽(0)=0.
In view of (R₁), there exist 𝜏₁, 𝜏₂ with 0<𝜏₁<𝜏₂ such that 𝐹(𝑡,𝑢)≤𝑝|𝑢|², (29) for any 𝑡∈[0,1] and 𝑢 with |𝑢|∈[0,𝜏₁)∪(𝜏₂,+∞). Since 𝑓 is a 𝐿¹-Carathéodory function, 𝐹(𝑡,𝑢) is bounded on 𝑡∈[0,1] and 𝑢 with |𝑢|∈[𝜏₁,𝜏₂]; we can choose 𝜌>0 and 𝜎>2 in such a way that 𝐹(𝑡,𝑢)≤𝑝|𝑢|²+𝜌|𝑢|^𝜎, (30) for all (𝑡,𝑢)∈[0,1]×ℝ. So, by (12), one has 𝐽(𝑢)≤𝑝𝑆²‖𝑢‖²+𝜌𝑆^𝜎‖𝑢‖^𝜎, (31) for all 𝑢∈𝐸. Hence, using (H₁) and (31), we have limsup_|𝑢|→0𝐽(𝑢)Φ(𝑢)≤2𝑝𝑆². (32) Furthermore, using (29) again, for each 𝑢∈𝐸⧵{0}, we obtain 2∫¹₀𝐹(𝑡,𝑢(𝑡))𝑑𝑡‖𝑢‖²+2∫₀^𝑢(1)ℎ(𝑡)=𝑑𝑡2∫_{|𝑢|≤𝜏2}𝐹(𝑡,𝑢(𝑡))𝑑𝑡‖𝑢‖²+2∫₀^𝑢(1)ℎ(𝑡)+𝑑𝑡2∫_|𝑢|>𝜏2𝐹(𝑡,𝑢(𝑡))𝑑𝑡‖𝑢‖²+2∫₀^𝑢(1)ℎ(𝑡)≤𝑑𝑡2sup_{𝑡∈[0,1],|𝑢|∈[0,𝜏2]}𝐹(𝑡,𝑢)‖𝑢‖²+2∫₀^𝑢(1)ℎ(𝑡)+𝑑𝑡2𝑝𝑆²‖𝑢‖²‖𝑢‖²+2∫₀^𝑢(1)ℎ(𝑡)≤𝑑𝑡2sup_{𝑡∈[0,1],|𝑢|∈[0,𝜏2]}𝐹(𝑡,𝑢)‖𝑢‖²+2∫₀^𝑢(1)ℎ(𝑡)𝑑𝑡+2𝑝𝑆². (33) So, we get limsup_{‖𝑢‖→+∞}𝐽(𝑢)Φ(𝑢)≤2𝑝𝑆². (34) Combining (32) with (34), one has 𝛼∶=max0,limsup_{‖𝑢‖→+∞}𝐽(𝑢)Φ(𝑢),limsup_|𝑢|→0𝐽(𝑢)Φ(𝑢)≤2𝑝𝑆². (35) By (H₁), (R₂), and (35), we have 𝛽=sup_{𝑢∈Φ−1((0,+∞))}𝐽(𝑢)Φ(𝑢)=sup_{𝑢∈𝐸⧵{0}}𝐽(𝑢)Φ(𝑢)≥2∫¹₀𝐹(𝑡,𝑐(𝑡))𝑑𝑡∫¹₀𝑐^′′2𝑑𝑡+2∫₀^𝑐(1)ℎ(𝑡)𝑑𝑡>2𝑝𝑆²≥𝛼. (36) Then, for each compact interval [𝑎,𝑏]⊂(𝜆₁,𝜆₂), there exists 𝑅>0 with the following property: for every 𝜆∈[𝑎,𝑏] and for any 𝐿¹-Carathéodory function ℎ∶[0,1]×ℝ→ℝ, there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], problem (1) has at least three weak solutions whose norms are less than 𝑅.

Theorem 9. Suppose that 𝑓∶[0,1]×ℝ→ℝ is an 𝐿¹-Carathéodory function, 0<𝑡<𝑡̂<1, and (𝐻₁) (or (𝐻₁)^′ holds. Furthermore, assume that the following conditions hold: (M₁)there exist two functions 𝑑∈𝐶²([0,𝑡]) and 𝑒∈𝐶²([𝑡,̂1]) satisfying 𝑑(0)=𝑑^′(0)=0,𝑑𝑡𝑡̂𝑑=𝑒=1,_′𝑡=𝑒^′𝑡̂=0, (37) and two positive constants 𝛾, 𝛿 with 𝛿>𝐷∶=𝛾𝑆∫^𝑡₀|𝑑^′′|²𝑑𝑡+∫¹_̂𝑡|𝑒^′′|²𝑑𝑡 such that max_{𝑡∈[0,1],|𝑢|≤𝛿}𝐹()<∫𝑡,𝑢_̂𝑡𝑡𝐹𝑡,𝛾𝑑𝑡∫^𝑡₀𝑑^′′2𝑑𝑡+∫¹_̂𝑡𝑒^′′2𝑑𝑡∫^𝑡₀𝑑^′′2𝑑𝑡+∫¹_̂𝑡𝑒^′′2𝑑𝑡+2/𝛾²∫₀^𝑒(1)𝛾ℎ(𝑡);𝑑𝑡 (38)(M₂)𝐹(𝑡,𝑢)≥0 for all 𝑡∈[0,𝑡]∪[𝑡,̂1] and 𝑢∈ℝ;(M₃)there exist 𝑤, 𝑞∈𝐿¹(0,𝑇;ℝ⁺) and a positive constant 𝛼∈[0,1) such that |𝑓(𝑡,𝑢)|≤𝑤(𝑡)|𝑢|^𝛼+𝑞(𝑡), for all (𝑡,𝑢)∈[0,1]×ℝ.
Then, there exist a nonempty open set 𝐴⊂[0,+∞) and a positive number 𝑅 with the following property: for every 𝜆∈𝐴 and for any 𝐿¹-Carathéodory function ℎ∶[0,1]×ℝ→ℝ, there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], problem (1) has at least three weak solutions whose norms are less than 𝑅.

Proof. Obviously, Φ(0)=𝐽(0)=0. For any 𝜆≥0 and 𝑢∈𝐸, by (H₁) and (M₃), we have Φ(𝑢)(𝑢)=1−𝜆𝐽2‖𝑢‖²+₀^𝑢(1)ℎ(𝑡)𝑑𝑡−𝜆¹₀𝐹()≥1𝑡,𝑢𝑑𝑡2‖𝑢‖²−𝜆¹₀𝑤(𝑡)𝛼|𝑢|^𝛼+1(𝑡𝑢|≥1+𝑞)|𝑑𝑡2‖𝑢‖²−𝜆𝑆^𝛼+1𝛼‖𝑤‖₁‖𝑢‖_𝛼+1−𝜆𝑞₁‖𝑢‖. (39) Since 𝛼<1, one has lim_{‖𝑢‖→+∞}(Φ(𝑢)−𝜆𝐽(𝑢))=+∞ for all 𝜆≥0.
For each 𝑟>0 and Φ(𝑢)≤𝑟, we have ‖𝑢‖_∞≤𝑆√2𝑟; that is, Φ⁻¹((−∞,𝑟])⊆𝑢∈𝐸∶max_𝑡∈[0,1]|𝑢(𝑡)|≤𝑆√2𝑟. (40) Therefore, sup_{𝑢∈Φ−1((−∞,𝑟])}𝐽(𝑢)≤max_{|𝑢|≤𝑆√2𝑟}𝐽(𝑢)=max_{|𝑢|≤𝑆√2𝑟}¹₀𝐹(𝑡,𝑢)𝑑𝑡≤max_{𝑡∈[0,1],|𝑢|≤𝑆√2𝑟}𝐹(𝑡,𝑢). (41) Define the function 𝜂(𝑡)𝑑(𝑡)∶=𝛾,𝑡∈0,𝑡,𝛾,𝑡∈𝑡̂,𝑒(𝑡)̂.𝑡,𝛾,𝑡∈𝑡,1 (42) From condition (M₁), it is easy to verify that 𝜂∈𝐸 and Φ𝜂=12𝜂²+₀^𝜂(1)ℎ(𝑡)=𝛾𝑑𝑡₂2^𝑡₀𝑑^′′2𝑑𝑡+¹_̂𝑡𝑒^′′2𝑑𝑡+₀^𝑒(1)𝛾ℎ(𝑡)𝑑𝑡. (43) Choose 𝑟=(𝛾²/2)[∫^𝑡₀|𝑑^′′|²𝑑𝑡+∫¹_̂𝑡|𝑒^′′|²𝑑𝑡]; then, using (43) and (H₁), we have Φ(𝜂)>𝑟. Furthermore, using (M₁), (M₂), (35), and (43), we have 𝑟𝐽𝜂Φ𝜂∫=𝑟¹⁰𝐹𝑡,𝜂𝑑𝑡(1/2)𝜂²+∫₀^𝜂(1)ℎ(𝑡)∫𝑑𝑡≥𝑟^̂𝑡𝑡𝐹𝑡,𝛾𝑑𝑡𝛾²/2∫^𝑡₀𝑑^′′2𝑑𝑡+∫¹_̂𝑡𝑒^′′2𝑑𝑡+∫₀^𝑒(1)𝛾ℎ(𝑡)=∫𝑑𝑡_̂𝑡𝑡𝐹𝑡,𝛾𝑑𝑡∫^𝑡₀𝑑^′′2𝑑𝑡+∫¹_̂𝑡𝑒^′′2𝑑𝑡∫^𝑡₀𝑑^′′2𝑑𝑡+∫¹_̂𝑡𝑒^′′2𝑑𝑡+2/𝛾²∫₀^𝑒(1)𝛾ℎ(𝑡)𝑑𝑡>max_{𝑡∈[0,1],|𝑢|≤𝛿}𝐹(𝑡,𝑢)>max_{𝑡∈[0,1],|𝑢|≤𝐷}𝐹(𝑡,𝑢)=max_{𝑡∈[0,1],|𝑢|≤𝑆√2𝑟}𝐹(𝑡,𝑢)≥sup_{𝑢∈Φ−1((−∞,𝑟])}𝐽(𝑢). (44) Therefore, we can fix 𝜌 such that sup_{𝑢∈Φ−1((−∞,𝑟])}𝐽(𝑢)𝐽𝜂<𝜌<𝑟Φ𝜂. (45) From Proposition 3, we obtain sup_𝜆≥0inf_𝑢∈𝐸Φ(𝑢)+𝜆𝜌−𝐽(𝑢)<inf_𝑢∈𝐸sup_𝜆≥0Φ(𝑢)+𝜆𝜌−𝐽(𝑢). (46) Therefore, using Theorem 2, for each compact interval [𝑎,𝑏]⊂(𝜆₁,𝜆₂), there exists 𝑅>0 with the following property: for every 𝜆∈[𝑎,𝑏] and for any 𝐿¹-Carathéodory function ℎ∶[0,1]×ℝ→ℝ, there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], Φ^′(𝑢)−𝜆𝐽^′(𝑢)−𝜇Ψ^′(𝑢)=0 has at least three solutions in 𝐸. Hence, problem (1) has at least three weak solutions whose norms are less than 𝑅.

Remark 10. Clearly, Theorem 9 gives the result of at least three solutions for problem (1) with 𝑓(𝑡,𝑢) being of sublinear growth.

4. Numerical Solutions

In this section, we consider a numerical algorithm for the problem (1) using the finite differences method. Let 0=𝑡₀<𝑡₁<⋯<𝑡_𝑛=1 be a discretization of the interval [0,1] with mesh size 𝑚=𝑡_𝑖−𝑡_𝑖−1. Then, putting 𝑢_𝑖=𝑢(𝑡_𝑖) and 𝑓_𝑖=𝑓(𝑡_𝑖,𝑢_𝑖) and using central differences formula, the first equation of problem (1) becomes 𝑢_𝑖−2−4𝑢_𝑖−1+6𝑢_𝑖−4𝑢_𝑖+1+𝑢_𝑖+2=𝑚⁴𝑓_𝑖, (47)2≤𝑖≤𝑛−2. Taking into account the boundary conditions of problem (1), we infer that 𝑢₀=0,𝑢₋₁=𝑢₁,𝑢_𝑛+1=−𝑢_𝑛−1+2𝑢_𝑛,𝑢_𝑛+2=2𝑚³ℎ𝑢_𝑛+𝑢_𝑛−2−4𝑢_𝑛−1+4𝑢_𝑛. (48) Then, we can compute 𝑢₁,…,𝑢_𝑛 by solving the nonlinear system 𝐴𝑈=𝐵, with 𝐴=_⋱^{7−41−46−411−46−46}_{1−46−411−45−22−42},𝑢𝑈=¹𝑢₂⋮𝑢_𝑛−1𝑢_𝑛𝑚,𝐵=⁴𝑓₁⋮𝑚⁴𝑓_𝑛−1𝑚⁴𝑓_𝑛−2𝑚³𝑔𝑢_𝑛. (49) Since the system is nonlinear, we use a linearization through an initial approximation 𝑈₀ and iterative methods for solving it. The quantities 𝑓_𝑖 are updated after each iteration.

Let 𝜇=0 and fix 𝜆>0. Choose a 𝑓 satisfying all conditions. Using 𝑢(𝑡)=𝑡 as initial approximation and mesh size 𝑚=0.1, it is easy to obtain the approximate solution with Error∶=max|𝑢_𝑖−𝑢(𝑡_𝑖)| by the Gauss-Seidel method.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

Q. Gao was supported by the Humanities and Social Sciences Foundation of the Ministry of Education in China (Grant no. 13YJAZH112).