منابع مشابه
The Contraction Mapping Theorem
A mapping f : X → X from a set X to itself has a fixed point if there is an x ∈ X such that f(x) = x. The simplest fixed point theorem is that a continuous function f : [a, b] → [a, b] has at least one fixed point. This is a consequence of the intermediate value theorem from calculus, as follows. Since f(a) ≥ a and f(b) ≤ b, we have f(b) − b ≤ 0 ≤ f(a) − a. The difference f(x)−x is continuous, ...
متن کاملA Stochastic Contraction Mapping Theorem
The submartingale convergence theorem is often described as a stochastic analogue to the bounded convergence theorem for monotone sequences of real numbers. We define an alternative process which is a stochastic analogue to an iterative contraction process. A general convergence theorem is presented, with an extension to multivariate processes. The convergence theorem functions in a manner simi...
متن کاملThe Contraction Mapping Theorem and the Implicit Function Theorem
denote the open ball of radius a centred on the origin in IR. If the function ~g : Ba → IR d obeys there is a constant G < 1 such that ‖~g(~x)− ~g(~y)‖ ≤ G ‖~x− ~y‖ for all ~x, ~y ∈ Ba (H1) ‖~g(~0)‖ < (1−G)a (H2) then the equation ~x = ~g(~x) has exactly one solution. Discussion of hypothesis (H1): Hypothesis (H1) is responsible for the word “Contraction” in the name of the theorem. Because G <...
متن کاملA \converse" of the Banach Contraction Mapping Theorem
We prove a type of converse of the Banach contraction mapping theorem for metric spaces: if X is a T 1 topological space and f : X ! X is a function with unique xed point a such that f n (x) converges to a for each x 2 X , then there exists a distance function d on X such that f is a contraction on the complete ultrametric space (X; d) with contractivity factor 1 2. We explore properties of the...
متن کاملThe Lebesgue Monotone Convergence Theorem
For simplicity, we adopt the following rules: X is a non empty set, S is a σ-field of subsets of X, M is a σ-measure on S, E is an element of S, F , G are sequences of partial functions from X into R, I is a sequence of extended reals, f , g are partial functions from X to R, s1, s2, s3 are sequences of extended reals, p is an extended real number, n, m are natural numbers, x is an element of X...
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ژورنال
عنوان ژورنال: Journal of Mathematics
سال: 2020
ISSN: 2314-4629,2314-4785
DOI: 10.1155/2020/2879283