Monotone update bound below one

Read the equation

The inequality means the next iterate is larger than the current one but no larger than the fixed point one. Repeatedly applying it gives a monotone bounded sequence within the specified interval. Identifying its limit then uses continuity and the fixed-point equation.

Terms & definitions

How it works

Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.

Assumptions & limits

Source recovery establishes that the expression appears in this record. Its mathematical validity, implementation behavior, and interpretation must be assessed separately. See the linked source and research notes for the checks available for this version.

Original context

Two factorizations establish the result:

Fz−z=1−z2z+17z,  Fz−1=5z−1z−17z.

For z>1, these show 1<Fz<z. The sequence decreases while bounded below by one. For 1/5≤z<1, they show z<Fz≤1, so the sequence increases while bounded above by one. For 0<z<1/5, the next value is greater than one and thereafter belongs to the first case. A bounded monotone sequence has a limit; continuity and the positive fixed-point equation force that limit to be one.

For z0=2, local computation gives:

Iteration

State

z<F\left(z\right)\le 1