Convergence proof techniques

Monotonicity (Lyapunov functions)

Every bounded monotonic sequence has a convergent subsequence. If it can be shown that all of the subsequences of $f$ have the same limit, such as by showing that there is a unique fixed point of the transformation $T$, then the initial sequence must also converge to that limit.

This approach can also be applied to sequences that are not monotonic. Instead, it is possible to define a function $V:\mathbb {R} ^{n}\rightarrow \mathbb {R}$ such that $V(f(k))$ is monotonic in $k$. If the $V$ satisfies the conditions to be a Lyapunov function then $f$ is convergent. Lyapunov's theorem is normally stated for ordinary differential equations, but can also be applied to sequences of iterates by replacing derivatives with discrete differences.

The basic requirements on $L$ are that

1. $V(f(k+1))-V(f(k))<0$ for $f(k)\neq 0$ and $V(0)=0$ (or ${\dot {V}}(x)<0$ for $x\neq 0$)
2. $V(x)>0$ for all $x\neq 0$ and $V(0)=0$
3. $V$ be "radially unbounded", so that $V(x)$ goes to infinity for any sequence with $||x||$ that tends to infinity.

In many cases, a Lyapunov function of the form $L(x)=x^{T}Ax$ can be found, although more complex forms are also used.

For delay differential equations, a similar approach applies with Lyapunov functions replaced by Lyapunov functionals also called Lyapunov-Krasovskii functionals.

If the inequality in the condition 1 is weak, LaSalle's invariance principle may be used.