Disconjugacy characterization by means of spectral (k,n−k) problems

Abstract

This paper is devoted to the description of the interval of parameters for which the general linear n th -order equationwith ai ∈ C n−i (I), is disconjugate on I. Such interval is characterized by the closed to zero eigenvalues of this problem coupled with (k, n − k) boundary conditions, given by u(a) = · · · = u (k−1) (a) = u(b) = · · · = u (n−k−1) (b) = 0 , 1 ≤ k ≤ n − 1 .(2)

Constant sign solution for a simply supported beam equation

“…In this section, we are going to stablish a relation between the strongly inverse positive (negative) character of the operator T [p, c] on the setX, defined in (3), and the strongly inverse positive (negative) character of T [p, c] in X, defined in (5). First we introduce a previous result, which proof follows directly from the uniqueness of the homogeneous problem (6), (1).…”

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

“…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).…”

“…Fourth-order equations appear as model equations of elastic beams, see Gupta [9]. The deformation of an elastic beam whose both ends clamped may be described by u (4) (t) + βu (t) − αu = f (t, u(t)), t ∈ (0, 1),…”

“…Existence of solutions and positive solutions of the boundary value problem (1)- (2), or its generalizations, has been studied by several authors, see for examples, Agarwal and Chow [1], Cabada and Enguiça [3], Ma et al [13], Rynne [17], Webb et al [19] and Xu and Han [21]. 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).…”