Knowlpedia

Theorem

Small solution of an invertible linear equation with quadratic error

A contraction estimate solves an invertible linear equation perturbed by a bounded quadratic term.

Statement

Let XX be a , B:X→XB:X\to X a bounded invertible linear map with ∥B−1∥≤β\|B^{-1}\|\le\beta, and Q:X×X→XQ:X\times X\to X a bounded bilinear map with norm at most κ\kappa. If

8β2κ∥d∥≤1,8\beta^2\kappa\|d\|\le1,

then Bc+Q(c,c)=dBc+Q(c,c)=d has a unique solution in the closed ball ∥c∥≤2β∥d∥\|c\|\le2\beta\|d\|.

Contraction proof

Set r=2β∥d∥r=2\beta\|d\| and T(c)=B−1(d−Q(c,c))T(c)=B^{-1}(d-Q(c,c)). For ∥c∥≤r\|c\|\le r,

∥T(c)∥≤β∥d∥+βκr2≤34r.\|T(c)\|\le\beta\|d\|+\beta\kappa r^2\le\tfrac34r.

On this ball ∥T(c)−T(c′)∥≤2βκr∥c−c′∥≤12∥c−c′∥\|T(c)-T(c')\|\le2\beta\kappa r\|c-c'\|\le\tfrac12\|c-c'\|. The contraction theorem applies; if d=0d=0, the radius-zero case is immediate. When Q=0Q=0, no smallness restriction is needed.

Smooth parameter families

In finite dimensions, if B,Q,dB,Q,d vary smoothly over compact parameters with these uniform zeroth-order bounds, the selected solution is smooth. Its Jacobian is B+Q(c,⋅)+Q(⋅,c)B+Q(c,\cdot)+Q(\cdot,c), and after multiplication by B−1B^{-1} the perturbation of the identity has norm at most 1/21/2. The implicit function theorem applies. Higher derivatives follow from implicit differentiation; smoothness does not require simultaneous smallness at every derivative order.