Existence of positive solutions to a non-positive elastic beam equation with both ends fixed

where λ is a positive parameter, f : [0,1] × R¹ → R¹ is continuous.

Fourth-order two-point BVPs are useful for material mechanics because the problems usually characterize the deflection of an elastic beam. The problem (P) describes the deflection of an elastic beam with both ends rigidly fixed. The existence and multiplicity of positive solutions for the elastic beam equations has been studied extensively when the non-linear term f : [0,1] × [0, +∞) → [0, +∞) is continuous, see for example [1–10] and references therein. Agarwal and Chow [1] investigated problem (P) by using of contraction mapping and iterative methods. Bai [3] applied upper and lower solution method and Yao [9] used Guo-Krasnosel'skii fixed point theorem of cone expansion-compression type. However, there are only a few articles concerned with the nonpositive or semipositive elastic beam equations. Yao [11] considered the existence of positive solutions of semipositive elastic beam equations by constructing control functions and a special cone and using fixed point theorem of cone expansion-compression type. In this article, we assume that f : [0,1] × R¹ → R¹, which implies the problem (P) is nonpositive (or semipositive particularly). By the topological degree and fixed point theory of superlinear operator on lattice (the definition of lattice will be given in Section 2), we obtain the existence of nontrivial solutions for the non-positive BVP (P) and the existence of positive solutions for the semipositive BVP (P).

Let E be an ordered Banach space in which the partial ordering ≤ is induced by a cone P ⊂ E, θ denote the zero element of E. P is called solid if int P ≠ ∅, i.e., P has nonempty interior. P is called a generating cone if E = P - P . For the concepts and properties about the cones we refer to [12–14].

which are called the positive and the negative part of x, respectively. Take |x| = x⁺ + x ^- , then |x| ∈ P , and |x| is called the module of x. One can see [15] for the definition and the properties about the lattice.

Remark 2.1 If E is a lattice, then P is a generating cone.

where x₊ and x _- are defined by (2.1).

Remark 2.2 We point out that the condition (2.2) appears naturally in the applications of nonlinear differential equations and integral equations.

and hence |x|(t) = |x(t)|. By Remark 3.1 in [16], we know that there exists y₀ ∈ C[a, b] such that Fx = Fx₊ + Fx _- + y₀, ∀x ∈ C[a, b].

which means that A is quasi-additive on lattice.

Lemma 2.1 [[18], Theorem 4.2.2, p. 122]. Let P be a generating cone and B a u₀- bounded completely continuous linear operator. Then the spectral radius r(B) ≠ 0 and r ^{- 1}(B) is the only positive eigenvalue corresponding to positive eigenvectors and B has no other eigenvectors except those corresponding to r ^{- 1}(B).

then P (g*, δ) is also a cone in E.

Definition 2.3 [[19], Definition, p. 528]. Let B be a positive linear operator. B is said to satisfy H condition, if there exist $\bar{\phi }\in P\backslash \left\{\theta \right\}$, g* ∈ P*\{θ} and δ > 0 such that (2.3) holds, and B maps P into P(g*, δ).

and v(x)g*(x) ≢ 0, then B satisfies H condition (see [19]).

Then there exists R₀> 0 such that deg(I - A, T _R , θ) = 0 for any R > R₀ , where T _R = {x ∈ C[0, 1] : ||x|| < R}.

Then deg(I - A, Ω, θ) = 1.

In the sequel we always take E = C[0,1] with the norm ||u|| = max_0≤t≤1|u(t)| and P = {u ∈ C[0, 1] | u(t) ≥ 0, 0 ≤ t ≤ 1}. Then P is a solid cone in E and E is a lattice under the partial ordering ≤ induced by P.

and

(G₁) G(t, s) ≥ 0, 0 ≤ t, s ≤ 1;

(G₂) G(t, s) = G(s, t);

(G₃) G(t, s) ≥ p(t)G(τ; s), 0 ≤ t, s, τ ≤ 1, where $p\left(t\right)=\frac{2}{3}\text{min}\left\{t^2,{\left(1-t\right)}^2\right\}$.

Similarly, (3.1) holds for 0 ≤ s ≤ t < 1 and t > 0. The proof is complete.    □

Lemma 3.2 Let B be defined by (3.3). Then B is a u₀- bounded linear operator.

Proof. Let $u_0\left(t\right)=H\left(t\right)=\frac{1}{2}{t}^2{\left(1-t\right)}^2$, t ∈ [0,1]. For any u ∈ P\{θ}, by Lemma 3.1

This indicates that B : E → E is a u₀- bounded linear operator. □

From Lemma 2.1 we have r(B) ≠ 0 and r ^{- 1}(B) is the only eigenvalue of B. Denote λ₁ = r ^{- 1}(B).

Now let us list the following conditions which will be used in this article:

(H₃) $\underset{u\to +\infty }{\text{lim}}\frac{f\left(t,u\right)}{u}=+\infty$.

Theorem 3.1 Suppose that (H₁) and (H₂) hold. Then for any $\mathrm{\lambda }\in \left(\frac{{\mathrm{\lambda }}_1}{\alpha },\frac{{\mathrm{\lambda }}_1}{\iota }\right)$, BVP (P) has at least one nontrivial solution, where λ₁ = r ^{- 1}(B) is the only eigenvalue of B, B is denoted by (3.3), ι = max{β, γ}.

Proof. Let (Fu)(t) = f(t, u(t)). Then A = BF, where A is denoted by (3.2). By Remark 2.2, F is quasi-additive on lattice. Applying the Arzela-Ascoli theorem and a standard argument, we can prove that A : E → E is a completely continuous operator.

Now we show that λA = λBF has at least one nontrivial fixed point, which is the nontrivial solution of BVP (P).

Notice that $Bu\left(t\right)={\int }_0^{1}G\left(t,s\right)u\left(s\right)ds$, where G(t, s) ≥ 0, G(t, s) ∈ C([0,1] × [0,1]). From Lemma 3.2 B is a u₀- bounded linear operator. By Lemma 2.1 we have r(B) ≠ 0 and λ₁ = r ^{- 1}(B) is the only eigenvalue of B. Then there exist $\bar{\phi }\in P\backslash \left\{\theta \right\}$ and g* ∈ P*\{θ} such that (2.3) holds. Notice that λ > 0, from Remark 2.3, λB satisfies H condition.

By (3.6) and (3.7), we have (2.4) and (2.5) hold, where a₁ = α - ε, a₂ = β + ε.

It is easy to see from (3.9) and (3.10) that λA has at least one nontrivial fixed point. Thus problem (P) has at least one nontrivial solution. □

Remark 3.1 If α = +∞, β = γ = 0, then for any λ > 0 problem (P) has at least one nontrivial solution.

Then for any $\mathrm{\lambda }\in \left(\frac{{\mathrm{\lambda }}_1}{\alpha },\frac{{\mathrm{\lambda }}_1}{\beta }\right)$ and $\mathrm{\lambda }\ne \frac{{\mathrm{\lambda }}_1}{\rho }$, BVP (P) has at least one nontrivial solution.

where κ is the sum of algebraic multiplicities for all eigenvalues of $\mathrm{\lambda }{A}_{\theta }^{\prime }$ lying in the interval (0, 1). From the proof of Theorem 3.1 we have that (3.9) holds for any $\mathrm{\lambda }\in \left(\frac{{\mathrm{\lambda }}_1}{\alpha },\frac{{\mathrm{\lambda }}_1}{\beta }\right)$. By (3.9) and (3.12), λA has at least one nontrivial fixed point. Thus problem (P) has at least one nontrivial solution. □

In many real problems, the positive solution is more significant. In this section we will study this kind of question.

Theorem 4.1 Suppose that (H₃) holds. Then there exists λ* > 0 such that for any 0 < λ < λ* BVP (P) has at least one positive solution.

Then A₁ : E → E is a completely continuous operator.

Since A₁ is completely continuous, then $\left\{u_{{\mathrm{\lambda }}_n}\right\}$ has a subsequence converging to u* ∈ C[0,1]. Assume, without loss of generality, that it is $\left\{u_{{\mathrm{\lambda }}_n}\right\}$. Taking n → +∞ in (4.5), we have u* = θ, which is a contradiction to $\left|\right|u_{{\mathrm{\lambda }}_n}\left|\right|>r>0$. Hence (4.4) holds.

And so u _λ (t) is a positive solution of problem (P). □

Remark 4.1 In Theorem 4.1, without assuming that f(t, u) ≥ 0 when u ≥ 0, we obtain the existence of positive solutions for the semipositive BVP (P).

Remark 4.2 Noticing that, in this article, we only study the existence of positive solutions for BVP (P), which is irrelevant to the value of f(t, u) when u ≤ 0, we only need to suppose that f(t, u) is bounded from below when u ≥ 0. In fact, f(t, u) may be unbounded from below when u ≤ 0.

In order to illustrate possible applications of Theorems 3.2 and 4.1, we give two examples.

Take α = 3π/2, β = π/2, ρ = π +1. Then (5.1), (5.2) indicate (H₁), (3.11) hold, repectively. Notice that α > ρ > β > 0 and f(t, 0) ≡ 0, ∀t ∈ [0,1], by Theorem 3.2 for any $\mathrm{\lambda }\in \left(\frac{{\mathrm{\lambda }}_1}{\alpha },\frac{{\mathrm{\lambda }}_1}{\beta }\right)$ and $\mathrm{\lambda }\ne \frac{{\mathrm{\lambda }}_1}{\rho }$, BVP (P₁) has at least one nontrivial solution.

(5.3) means (H₃) holds. By Theorem 4.1 there exists λ* > 0 such that for any 0 < λ < λ* BVP (P₂) has at least one positive solution.