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Erschienen in:
2010 | OriginalPaper | Buchkapitel
Fixed Point Theory and Induction
verfasst von : María Tomás-Rodríguez, Stephen P. Banks
Erschienen in: Linear, Time-varying Approximations to Nonlinear Dynamical Systems
Verlag: Springer London
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In this chapter we shall show that we can obtain results on various aspects of systems of the form
$$ \dot{x} = A(x)x, \: x(0)=x_{0} ~(8.1) $$
by using a sequence of approximations
$$ \dot{x}^{[i]}(t) = A(x^{[i-1]}(t))x^{[i]}(t), \: x^{[0]}(0)=x_{0} ~(8.2) $$
as before and a combination of fixed point theorems and induction. The induction will proceed in the following way: suppose we want to prove some property
P
of Equation 8.1, and assume we can find a function
x
[0]
(
t
) which has this property. Suppose also that if
x
[
i
− 1]
(
t
) has the property, then the solution
x
[
i
]
(
t
) of Equation 8.2 also has the property. Then if the sequence {
x
[
i
]
(
t
) } converges on some interval [0,
T
], it follows by induction that the nonlinear system (8.1) (or the solutions thereof) also have the property
P
.
We shall see that this can be applied to stability of nonlinear systems and the existence of periodic solutions. The same idea can, however, be applied to many other situations.