DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

NECESSARY AND SUFFICIENT CONDITIONS FOR THE
EXISTENCE OF NONCONSTANT TWICE CONTINUOUSLY
DIFFERENTIABLE SOLUTIONS OF x ′′ = f (x)
RODRIGO LÓPEZ POUSO
We deduce necessary and sufficient conditions for having nonconstant classical solutions
of the scalar equation x ′′ = f (x).
Copyright © 2006 Rodrigo López Pouso. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
In this note, we complement the results obtained in [1] concerning necessary and sufficient
conditions for the existence of nontrivial Carathéodory solutions of the initial value
problem
x ′′ = f (x), x(0) = x 0 , x ′ (0) = x 1 , (1.1)
where f :Dom(f ) ⊂ R → R ∪{−∞,+∞} and x 0 ,x 1 ∈ R.
In order to present the main result in [1] we need some preliminaries. First, a Carathéodory
solution of (1.1)isanymappingx : I ⊂ R → R,whereI is a nontrivial interval
that contains 0, such that x ′ exists and is locally absolutely continuous on I, x(0) = x 0 ,
x ′ (0) = x 1 ,andx satisfies the differential equation almost everywhere on I (in Lebesgue’s
measure sense). Let us remark that classical (twice continuously differentiable) solutions
of (1.1)areCarathéodory solutions but the converse is not true in general.
Let J ⊂ Dom( f ) be a nontrivial interval that contains x 0 (if no such interval exists,
then (1.1) would have no nonconstant solution). Without further assumptions over f ,
we formally define the time map
τ : y ∈ J
∫ y
dr
τ(y):= √
x 0 x
2
1 +2 ∫ r
x 0
f (s)ds , (1.2)
and now we are in a position to state the main result in [1], which gives a characterization
of the nontrivial solvability of (1.1).
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 1195–1199