Peano existence theorem

The set of Peano is a sentence from the theory of ordinary differential equations. He gives a simple condition under which the initial value problem (at least) has a local solution. This set was published in 1886 by the mathematician Giuseppe Peano with a faulty proof. In 1890 he delivered after a correct proof.

Compared to the existence and uniqueness theorem of Picard - Lindelöf the existence theorem of Peano has the advantage that it has weaker requirements. But he makes no statement regarding the uniqueness of the solution.

If one possesses only once a (local) solution, one can conclude from this in a second step, the existence of a non- continuable solution. In this respect, the set of Peano is a first step for the existence theory of a differential equation.

Formulation

Let be a continuous function. Your domain is a rich subset of the. The closed ball of radius to call, ie

Then, for every initial value problem of the differential equation at least one local solution. Specifically, this means that there are one and a continuously differentiable function that satisfies two conditions:

• For all the point is.
• For all the differential equation is satisfied.

Such a one can specify exactly: on the closed and bounded amount of the continuous function has a maximum value, set

This number is a bound on the slope of a possible solution. We now choose

Then there exists ( at least) one solution of the initial value problem

On the interval with values ​​in.

Note: Analog, complex differential equations are considered, by considering real and imaginary parts of a complex component as an independent real component, ie, by the complex multiplication forgetting identified with the will.

For real Banach spaces

Let X be a real Banach space and steady and compact. For each initial value then exist a and a solution of the ordinary differential equation

With.

Note: In the case follows from the continuity of the compactness.

Sketch of proof of the finite-dimensional case

This theorem is proved in two parts. The first step is concerned with the help of Euler's Polygonzugverfahrens for every special - approximate solutions of this differential equation, more precisely: We construct a piecewise continuously differentiable function with which

Satisfied in each Differenzierbarkeitspunkt and equal continuity condition

For everyone.

In the second step shows with the help of the set of Arzelà - Ascoli that there is a uniformly convergent subsequence. From their limit function is then shows that they integral equation which

Met. From the Fundamental Theorem of Calculus then follows that is continuously differentiable and satisfies the differential equation.

Sketch of proof for real Banach spaces

We consider the corresponding Volterra integral equation for

We define the operator

This operator is continuous with respect to the supremum norm, as compact and is therefore limited. Furthermore, it is. By means of the theorem of Arzela Ascoli, one can show that is relatively compact with respect to the supremum norm in. Thus T is a continuous function that maps from a closed, convex subset in a compact subset. Thus T has at least one fixed point on the fixed point theorem of Schauder. Each of these checkpoints is a solution of the Volterra integral equation and so that the differential equation.