Local and global theorems
34) Local and global theorems.
Theorem. For any point T there exists a fundamental matrix solution defined in some small neighborhood U of .
Proof. The linear vector-function (t, x) is holomorphic everywhere on T × . By the local existence theorem, for any initial condition (,) T × there exists a holomorphic vector solution defined on some neighborhood of , meeting the condition . Choose n solutions satisfying n linear independent initial conditions at , arranged as columns of a square matrix and considered on their common domain.
By construction, , hence the holomorphic matrix is holomorphically invertible in some neighborhood of the point .
Theorem. (global existence theorem). A linear system on a Riemann surface T admits a fundamental solution in any simply connected subdomain
Proof. Choose a base point and let be a local fundamental matrix solution at this point. We extend it to an arbitrary point .
Since is connected, there exists a compact piecewise smooth curve (path) γ connecting with . γ can be covered by carrying the respective local fundamental matrix solutions, such that are connected, and if and only if |i − j| 1.
Assume that satisfy , , . Then ,
Agree on the intersections:
the solution can be explicitly constructed:
This completes the proof of existence of analytic continuation of solutions along paths.
35) Analyticity and differentiation of solutions.
If f ‘(x0) exists, then for x close to x0, we have
This is the "linear approximation" done via the tangent line. Obviously this implies
which means that is continuous at. Thus there is a link between continuity and differentiability: If a function is differentiable at a point, it is also continuous there. Consequently, there is no need to investigate for differentiability at a point, if the function fails to be continuous at that point.
Note that a function may be continuous but not differentiable, the absolute value function at is the archetypical example.
This relationship between differentiability and continuity is local. But a global property also holds. Indeed, let be a differentiable function on an interval . Assume that is bounded on , that is there exists such that
The Mean Value Theorem will then imply that
for any . This is the definition of Lipschitz continuity. In other words, if is bounded then is a Lipschitzian function. Conversely, it is also true that Lipschitzian functions have bounded first derivatives, when they exist. Since Lipschitzian functions are uniformly continuous, then is uniformly continuous provided is bounded.
Nevertheless, a function may be uniformly continuous without having a bounded derivative. For example, is uniformly continuous on [0,1], but its derivative is not bounded on [0,1], since the function has a vertical tangent at 0.
36) Solution continuity of parameters and initial data.
Theorem. Suppose that ƒ and are continuous and bounded in a given region U. Let be a solution of (1) passing through , and be a solution of (1) passing through (. Suppose that ϕ and ψ exist on some integral I.
Then, for each ε > 0, there exists δ > 0 such that if |t — | < δ and |||| < δ, then
||ϕ(t) – ψ()|| < ε, for t, ∈ I.
Proof. Since ϕ is the solution of (1) through the point (), we have, for all t ∈ I,
ϕ(t)= + (2)
As ψ is the solution of (1) through the point (,), we have, for all t ∈ I,
ψ(t)= + (3)
subtracting (3) from (2) gives
|| ϕ(t) – ψ(t) || |||| + ||||+|| ||
Using the boundedness assumptions on ƒ and to evaluate the right hand side of the latter inequation, we obtain
|| ϕ(t) – ψ(t) || |||| +M || + K
If || < δ, |||| <δ, then we have
|| ϕ(t) – ψ(t) || +Mδ + K || || (4)
Applying Gronwall’s inequality to (4) gives
||ϕ(t) – ψ()|| ||ϕ(t) – ψ(t)|| ||ψ(t) – ψ( )||
Now, given ε > 0, we need only choose ε/[M + (1+] to obtain the desired inequality, completing the proof.
37. Fundamental system of solutions
Definition. System of n linearly independent solutions (t), (t), …, (t)
of system is called a fundamental system of solutions or basis.
Theorem. The system has a fundamental system of solutions. If
(t), (t), …, (t) is basis, then the general solution has the form
where are arbitrary constants.
Concept of the fundamental matrix. Ostrogradsky-Liouville formula.
We consider the system
of n arbitrary vector solutions of the vector equation and form a matrix of the order nxn
and the Wronskian. If the system of vectors is linearly independent, then detY(t)=W(t)does not vanish for any value t of the interval of continuity of the matrix A(t). In this case the matrix Y(t) is called an integral or a fundamental matrix for the system . If Y( )= E, where E is the unit matrix, the matrix is called integral normalized at the point t=.