Kontraktiivne

Kontraktiivne is a mathematical term used in Estonian to describe a contraction. A function f from a metric space (X, d) to itself is kontraktiivne if there exists a constant c with 0 ≤ c < 1 such that for all x, y in X, the inequality d(f(x), f(y)) ≤ c · d(x, y) holds. Equivalently, f is a Lipschitz map with Lipschitz constant c < 1. This property means that the function brings points closer together, shrinking distances under each application.

Contraction mappings have important consequences. If the space X is complete, a kontraktiivne function f has a unique fixed point x*, satisfying f(x*) = x*. For any starting point x0 in X, the iterative sequence x_{n+1} = f(x_n) converges to x*. The convergence is geometric, with the distance to the fixed point decreasing at a rate governed by c (for example, d(x_n, x*) ≤ (c^n)/(1 − c) · d(x_1, x_0) under standard estimates).

Examples often cited include linear contractions on real numbers, such as f(x) = a x + b with |a| < 1, which is a contraction on R with fixed point at x* = b/(1 − a). Contractions are central to the Banach fixed-point theorem, widely used in analysis and numerical methods to guarantee existence and computability of solutions to equations and integral or differential problems.

In Estonian mathematical usage, kontraktiivne forms the basis for discussing stability and convergence properties of iterative processes, and the concept is foundational in topics such as fixed-point theory and iterative schemes.