Abstract
This paper presents a recursive method which yields necessary and sufficient conditions for the existence of solutions of a class of n-th order linear focal boundary value problems in the interior of a given interval, in the form of integral inequalities. Some results on the sign of the derivatives of the Green functions of n-th order linear focal boundary value problems will also be provided.
Highlights
IntroductionThe purpose of this paper is to describe a method that allows asserting the existence (or not) of solutions of (1.1)–(1.3) in extremes a , b interior to a given interval [a, b] ⊂ I, under certain conditions to be determined
Let I be a compact interval in R, let k, n ∈ N be such that 1 ≤ k < n and let us consider the n-th order boundary value problem y(n) + pn−1(x)y(n−1) + ... + p0(x)y = 0, x ∈]a, b [, y(i)(a ) = 0, i = 0, 1, 2, ..., k − 1, y(βi)(b ) = 0, i = 1, 2, . . . , n − k, 0 ≤ β1 < β2 < . . . < βn−k = n − 1, (1.1) (1.2) (1.3)
The purpose of this paper is to describe a method that allows asserting the existence of solutions of (1.1)–(1.3) in extremes a, b interior to a given interval [a, b] ⊂ I, under certain conditions to be determined
Summary
The purpose of this paper is to describe a method that allows asserting the existence (or not) of solutions of (1.1)–(1.3) in extremes a , b interior to a given interval [a, b] ⊂ I, under certain conditions to be determined Examples of these types of problems appear in the study of the deflections of beams, both straight ones with non-homogeneous cross-sections in free vibration, which are subject to the fourth-order linear Euler-Bernoulli equation, and curved ones with different shapes. The procedure to prove it will make use of cone theory and in particular [2, Theorem 2], as well as certain properties of the sign of the derivatives of the Green function of the problem (1.9) In this way, this paper can be considered as an extension of [2] and [3] to boundary value problems with right focal conditions (these papers requested βn−k < n − 1 in (1.6) as a fundamental assumption).
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