7) Theorem about existence and uniqueness of Cauchy problem.
Existence theorem. If in the equation у’=ƒ(х,у) the function ƒ is defined and continuous in a bounded domain D of the plane (x, y), then for any point (х0, у0)∈D exists a solution y (x) of the initial problem
dy/dx=f(x,y), y(x0)=y0 (1)
defined on some interval containing х0.
Existence and uniqueness theorem. If the function ƒ is defined and continuous in a bounded domain D of the plane (x, y), and it satisfies in D to Lipschitz condition in the variable y, i.e.
| ƒ(х,у1)- ƒ(х,у2)| ≤ L|у2-у1|, (2)
where, L is positive constant, then for any point (х0,у0)∈D exists a unique solution у(х) of the initial problem (1), defined on some interval containing х0.
Extension Theorem. At the conditions of the existence theorem or the theorem of the existence and uniqueness an any solution of the Cauchy problem (1) with initial data (х0,у0)∈D can be extended to a point arbitrarily close to the boundary of D. In the first case, the continuation, in general, is not necessarily unique; in the second case it is unique.
8) General properties of solutions. Continuity, differentiation of solutions of parameter axis and initial data.
Consider the IVP with a parameter s ∈ Rm
(1)
where
f : Ω → Rn and Ω is an open subset of
. Here the triple (t, x, s) is identified as a point in
follows:
(t, x, s) = (t, x1, .., xn, s1, ..., sm).
For
any s
∈
,
consider the open set
Ωs
={(t, x) ∈
:
(t, x, s) ∈ Ω}
Denote by S the set of those s, for which Ωs contains (t0, x0),
that is,
S = {s ∈ : (t0, x0) ∈ Ωs}
= {s ∈ : (t0, x0, s) ∈ Ω}
Then the IVP can be considered in the domain Ωs for any s ∈ S. We always assume that the set S is non-empty. Assume also in the sequel that f (t, x, s) is a continuous function in (t, x, s) ∈ Ω and is locally Lipschitz in x for any s ∈ S. For any s ∈ S,
denote by x (t, s) the maximal solution of (1) and let Is be its domain (that is, Is is an
open interval on the axis t). Hence, x (t, s) as a function of (t, s) is defined in the set
U
={(t, s) ∈
: s ∈ S, t ∈ Is}
Theorem 1. Under the above assumptions, the set U is an open subset of and
the
function x
(t, s) : U →
is
continuous.
Differentiability Of solutions.
A property of solutions of differential equations, that the solutions posses a specific number of continuous derivatives with respect to the independent variable and the parameter appearing in the equation.
Consider an equation of the type (x may also be a vector):
|
(1) |
=
f(t, x, μ)
(1)where μ is a parameter (usually also a vector), and let x(t, μ) be a solution of (1) defined by the initial condition
x| t=t0=x0 (2) First differentiability of the solution with respect to t is considered. If f is continuous with respect to t and x , the theorem on the existence of a continuous solution of the problem (1)–(2) is applicable in some domain, and then it follows from the identity which is obtained after substitution of x(t, μ) in (1) that the continuous derivative xt also exists. The presence of n continuous derivatives of f with respect to t and x means that there exist n+1 continuous derivatives of the solution with respect to t; xt(n) may be found (expressed in terms of x(t, μ) ) by successive differentiation of the identity obtained by substituting x(t, μ) in (1). |