Ads
related to: fixed point theorem math problems examples problem solvingkutasoftware.com has been visited by 10K+ users in the past month
Search results
Results from the WOW.Com Content Network
The Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point. [2]By contrast, the Brouwer fixed-point theorem (1911) is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, [3] but it doesn ...
In numerical analysis, fixed-point iteration is a method of computing fixed points of a function.. More specifically, given a function defined on the real numbers with real values and given a point in the domain of , the fixed-point iteration is + = (), =,,, … which gives rise to the sequence,,, … of iterated function applications , (), (()), … which is hoped to converge to a point .
A fixed-point theorem is a result saying that at least one fixed point exists, under some general condition. [1] For example, the Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, fixed-point iteration will always converge to a fixed point.
In mathematics, the common fixed point problem is the conjecture that, for any two continuous functions that map the unit interval into itself and commute under functional composition, there must be a point that is a fixed point of both functions.
The Banach fixed point theorem is then invoked to show that there exists a unique fixed point, which is the solution of the initial value problem. An older proof of the Picard–Lindelöf theorem constructs a sequence of functions which converge to the solution of the integral equation, and thus, the solution of the initial value problem.
In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces and provides a constructive method to find those fixed points.
For example, the Iimura-Murota-Tamura theorem states that (in particular) if is a function from a rectangle subset of to itself, and is hypercubic direction-preserving, then has a fixed point. Let f {\displaystyle f} be a direction-preserving function from the integer cube { 1 , … , n } d {\displaystyle \{1,\dots ,n\}^{d}} to itself.
Nicomachus's theorem (number theory) Nielsen fixed-point theorem (fixed points) Nielsen–Ninomiya theorem (quantum field theory) Nielsen realization problem (geometric topology) Nielsen–Schreier theorem (free groups) Niven's theorem (number theory) No-broadcasting theorem (quantum information theory) No-cloning theorem (quantum computation)
Ads
related to: fixed point theorem math problems examples problem solvingkutasoftware.com has been visited by 10K+ users in the past month