A Variational Approach to Lyapunov Type Inequalities by Antonio Cañada & Salvador Villegas

Author:Antonio Cañada & Salvador Villegas
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

It is possible to define a minimizing sequence for β 1, n which shows that the lower bound found in (3.22) is, in fact, the infimum defined in (3.7). The relations (2.​35) motivate the definition of this minimizing sequence. Anyway, this is a very technical subject and the lector is referred to [5] for further details.

A minimizing sequence can be defined in the following way:

Let be sufficiently small. Let us define the function by

(3.24)

Then , the function belongs to and

(3.25)

Lastly, it can be proved that β 1, n is not attained. To see this, let be such that and let u be any nontrivial solution of (3.1) associated with the function a. Once more, we denote the zeros of u′ by 0 = x 0 < x 2 < … < x 2m  = L and the zeros of u by x 1 < x 3 < … < x 2m−1. Then,

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