Page 17 - Textos de Matemática Vol. 36

1.2. Boundary value problems 7 then the boundary value problem (1.10) has a unique solution.
Proof. If the initial value problem (1.11) has a solution in [t0,tf], then we let Y (t) ∈ Rn×n denote the solution of
Y˙ = (M22 + M21W)Y, Y (t0) = In (1.12) X(t) := W(t)Y (t) ∈ Rm×n. (1.13)
in [t0, tf ] and define
Notice that (1.12) implies det Y (t) ̸= 0 for all t ∈ [t0 , tf ]. We then infer that
X (t) (m+n)×n Y (t) ∈ R
solves the linear differential equation in (1.10) and consists
of n linearly independent solutions of that system with X 0
Y (t0)= I . n
Let
problem in (1.10) are determined by
x c1 ψ (t)=φ(t) c ,
2 for some constant vectors c1 ∈ Rm, c2 ∈ Rn.
Im 0
denote a fundamental matrix of (1.10); then all solutions of the boundary value
φ11 X
φ(t):= φ21 Y , φ(t0)= ∗ In
We shall now prove that c1,c2 are uniquely determined. At t0 we obtain x x0Im 0c1
ψ(t0)=ψ(t)=∗I c, 0n2
hence c1 = x0. At tf , from the equality
Kfx(tf)=Kf(φ11(tf)x0 +X(tf)c2)=ψ(tf)=φ21(tf)x0 +Y(tf)c2,
we obtain the equation
(In −KfW(tf))Y(tf)c2 = (Kfφ11(tf)−φ21(tf))x0.
Since W(t) = X(t)Y−1(t) exists in [t0,tf] we infer together with (1.12) that Y (tf ) is regular and since (In − Kf W (tf )) is assumed to be regular too, c2 is also uniquely determined. Hence the boundary value problem has a unique solution.