Disconjugacy conditions and spectrum structure of clamped beam equations with two parameters

In this work, we apply the 'disconjugacy theory' and Elias's spectrum theory to study the disconjugacy \begin{document}$ u^{(4)} + \beta u''-\alpha u = 0 $\end{document} with two parameters \begin{document}$ \alpha,\beta\in\mathbb{R} $\end{document} and the spectrum structure of the linear operator \begin{document}$ u^{(4)} + \beta u''-\alpha u $\end{document} coupled with the clamped beam conditions \begin{document}$ u(0) = u'(0) = u(1) = u'(1) = 0 $\end{document} . As the application of our results, we obtain the global structure of nodal solutions of the corresponding nonlinear analogue based on the bifurcation theory.

Recently, Cabada and Enguiça [3] made an exhaustive study of the fourth order linear operator u (4) + M u coupled with the clamped beam conditions (2). They obtained the exact values on the real parameter M for which this operator satisfies an anti-maximum principle. Such a property is equivalent to the fact that the related Green's function is nonnegative in [0, 1]×[0, 1]. When M < 0 they obtained the best estimate by means of the spectral theory and for M > 0 they attained the optimal value by studying the oscillation properties of the solutions of the homogeneous equation u (4) + M u = 0. By using the method of lower and upper solutions they deduced the existence of solutions for nonlinear problems coupled with (2). More precisely, they proved the following Ma et al. [13] also obtained the exact values on the real parameter M ∈ (−m 4 1 , m 4 0 ) for which confirm the positivity of linear equation u (4) + M u = 0 by using the ' disconjugacy theory' [6] and the spectrum theory Elias [7,8] and Rynne [18]. In addition, they obtained the spectrum structure of the linear operator u (4) + M u coupled with the clamped beam conditions (2).
A natural question is whether or not exist the optimal condition of the positivity and spectrum structure for the general equation u (4) + βu − αu = 0. Note that [4] give the description of the interval of parameters for the general linear nth-order ordinary differential equation is disconjugate on a given bounded interval I. For example, let . This is an important result of disconjugacy theory, however, the choice ofM limits the widespread application. It is the purpose of this paper to apply the ' disconjugacy theory' [4,6] and the spectrum theory Elias [7,8] and Rynne [18] to study the choice of α to provide the disconjugacy of the equation u (4) +βu −αu = 0 and the spectrum structure of the linear operator u (4) + βu − αu coupled with the clamped beam conditions (2). As the applications of our results, we show the global structure of nodal solutions of the corresponding nonlinear problem (1), (2). For other results on the existence of solutions for fourth-order boundary value problems, see [2,10,12,14,15,20] and the reference therein.
The rest of the paper is arranged as follows: in Section 2, we state some preliminary results about disconjugacy and the spectrum of some general linear operators. Section 3 is devoted to showing that the sufficient conditions and necessary and sufficient conditions for the equation u (4) + βu − αu = 0 is disconjugate on [0, 1]. And we also obtain the spectrum structure of the corresponding fourth-order linear operator. In Section 4, we apply our results on linear problems to show the global structure of nodal solutions of the nonlinear problem (1), (2).
). The functions y 1 , · · · , y n ∈ C n [a, b] are said to form a Markov system if the n Wronskians where D = d/dt, and , · · · , n. Suppose the equation (7) is disconjugate on [a, b]. Let f be a continuous function on [a, b]. Let k be a positive integer, the (k, n−k) two-point boundary value problem has a unique solution y, since the corresponding homogeneous problem has no nontrivial solution. The solution can be represented in the form where the Green's function G(t, s) is defined by the following properties (1) As a function of t, G(t, s) is a solution of (7) on [a, s) and on (s, b] and satisfies the n boundary conditions (10); (2) As a function of t, G(t, s) and its first n − 2 derivatives are continuous at t = s, while G (n−1) (s + 0, s) − G (n−1) (s − 0, s) = 1. Suppose the equation (7) is disconjugate on [a, b]. Then the Green's function of (k, n − k) two-point boundary value problem satisfies (−1) n−k G(t, s) > 0, a < s < b, a < t < b.

YANQIONG LU AND RUYUN MA
Note that Cabada and Saavedra[5, Theorem 3.1] also obtain the sign of the Green's function of (k, n − k) two-point boundary value problem.
For 2mth order (m ≥ 1), self-adjoint, disconjugate differential operator, Rynne [18] showed a spectrum theorem which will be the foundation of this paper.
Suppose that for each i = 1, · · · , 2m, we have a function The functions L i u, i = 0, · · · , 2m will be called the quasi-derivatives of u. We consider the Banach spaces with the norm || · || 2m−1 which, for convenience, we will write as || · ||. Define an operator L : For each integer k ≥ 1 and ν ∈ {+, −}, let S k,ν denote the set of functions u ∈ E such that: (1) u has only simple zeros in (a, b) and no quasi-derivative of u is zero at a or b, other than those specified in (10); (2) u has exactly k − 1 zeros in (a, b); (3) νu > 0 in a deleted neighborhood of t = 0. , and the boundary conditions (10) are such that L is formally self-adjoint, that is where ·, · denotes the standard L 2 (a, b) inner product. Assume that (H0) p ∈ C 0 [a, b], and p ≥ 0 on [a, b], while p ≡ 0 on any interval of [a, b]. Then for each k ≥ 1 and each ν ∈ {+, −}, problem Lu = µp(t)u, t ∈ (a, b), y (i) (a) = 0, i = 0, 1, · · · , k − 1; has a unique solution (µ k , ψ k ) ∈ R + × S k,ν with ||ψ k || = 1. In addition: Let y k (t, a) be the solution of (7) which satisfies at t = a for k = 1, · · · , n − 1 the initial conditions y (n−k) k (a) = 1, y (n−j) k (a) = 0 (j = 1, · · · n; j = k). Denote the Wronskian of y 1 (t, a), · · · , y k (t, a). Remark 1. It may be note that W n (s, a), s ∈ [a, b] never vanishes because the solutions y 1 , · · · , y n are linearly independent.  . Let L n and L m be two general linear differential operators of order n and m respectively. If the equations L n u(t) = 0 and L m u(t) = 0 are disconjugate on the interval I, then the composite n + mth order equation L n (L m u(t)) = 0 is also disconjugate on I.
(2) λ 2 < 0 is the maximum of the biggest negative eigenvalues on T n [M ] in E k , with n − k odd.
3. Disconjugacy for the equation u (4) + βu − αu = 0. Let us consider the linear boundary value problem where α, β ∈ R are two parameters. First, we consider the case α = 0. In this case, (11) reduces to The problem (12) has nontrivial solution if and only if β solves the equation Moreover, the first positive solution of the equation (13) is β 1 = 4π 2 . Proof. We divide the proof into two cases. Case 1 β ∈ (0, β 1 ). In the virtue of Lemma 2.3, we only need to find a Markov fundamental system of solutions of u (4) + βu = 0.
Next, we consider the case α > 0. In this case, by computing, the problem (11) has a nontrivial solution if and only if (α, β) solve the equation where Especially, if β = 0, then (11) degenerate the boundary value problem where C i , i = 1, 2, 3, 4 are arbitrary constants. Let u 1 be the unique solution of the initial value problem Then Let u 2 be the unique solution of the initial value problem Then u 2 (t) = cosh λt − cos µt, t ∈ [0, 1]. Let u 3 be the unique solution of the initial value problem Then u 3 (t) = µ 3 sinh λt + λ 3 sin µt, t ∈ [0, 1].

YANQIONG LU AND RUYUN MA
Let u 4 be the unique solution of the initial value problem Then Then where σ ∈ (0, 1) is small enough.
To show that the functions {z 1 , z 2 , ; z 3 , z 4 } form a Markov fundamental system of solutions of u (4) + βu − αu = 0 on [0, 1], we only need to provide the value range of α, β to ensure For convenience, in the sequel of the proof we replace For any fixed σ ∈ (0, 1) is small enough, by the sign of W 1 (t) and W 3 (t), it is not difficult to verify that W 1 (t), W 3 (t) is positive on [0, 1], so we only discuss the positivity of W 2 (t). Clearly, we can get that the first derivative and second derivative of W 2 as follows: Thus, we only need to show that W 2 (t) has at most one zero point on [0, 1], which is equivalent to solve the equation It is very difficult to solve the above equation since the parameter λ, µ is not given. Hence, we give some sufficient conditions to ensure the disconjugacy of u (4) + βu − αu = 0 on [0, 1].
Proof. To this end, in the virtue of Lemma 2.3, we only need to find a Markov fundamental system of solutions of u (4) + βu − α 0 u = 0.
Subcase (c) α < − β 2 4 . Let a, b > 0 be the real number and satisfy Then Let us consider the following equation here It is worthy to point out that if β = 0, then (25) is degenerated (6). So the equation u (4) + αu = 0 has nontrivial solution if and only if α < 0 solve the equation (6), i.e.
Remark 7. From Corollary 3, the disconjugacy of the equation u (4) +βu −αu = 0 is an important property to obtain the spectrum structure of the linear operator u (4) + βu − αu = 0 coupled with the clamped beam conditions u(0) = u(1) = u (0) = u (1) = 0, which provides an important theoretical basis for studying the steady-state solution of elastic beam equations in engineering and mechanics.