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Definitions and Concepts
Published in Daniel Zwillinger, Vladimir Dobrushkin, Handbook of Differential Equations, 2021
Daniel Zwillinger, Vladimir Dobrushkin
A contraction mapping is a functional iteration, say yn+1=N[yn], that converges to the solution of the fixed point equation y=F[y]. The Picard iteration (see page 453) is such a mapping.
Stability and Convergence
Published in Randal Douc, Eric Moulines, David S. Stoffer, Nonlinear Time Series, 2014
Randal Douc, Eric Moulines, David S. Stoffer
A kernel P on X×X defines a mapping on the Polish space M1(X),dTV. If ΔTV(P)<1, then P is a contraction mapping. The Banach fixed point theorem, also known as the contraction mapping theorem or contraction mapping principle (see Theorem 6.39), guarantees the existence and uniqueness of fixed points of contraction mapping and provides a constructive method to find those fixed points: given any ξ∈M1(X), the sequence ξ,ξP,ξP2,… converges to the fixed point. Clearly, a fixed point π of P is an invariant probability measure, πP=π. Therefore, if ΔTV(P)<1, then P admits a unique invariant probability measure. The Banach fixed point theorem also provides an estimate of the rate of convergence to those fixed points. A slight improvement of this argument leads to the following elementary but very powerful theorem, proved by Dobrushin (1956).
Nonlinear Equations
Published in Jeffery J. Leader, Numerical Analysis and Scientific Computation, 2022
The Fixed Point Theorem is a special case of a more general result known as the (Banach) Contraction Mapping Theorem. If D is a closed subset of ℝ⋉ and T:D→D then T is said to be a contraction (or to be contractive) on D if there is an s, 0<s<1, such that |T(x)−T(y)|≤s|x−y| for all x,y∈D. (We interpret |·| as a vector norm if necessary.) The constant s is called the contractivity factor. Show that if a function meets the hypotheses of the Fixed Point Theorem then it is a contraction on some D⊆ℝ. What is the contractivity factor s?Prove that a contraction T must be a continuous function.The Contraction Mapping Theorem states that every contraction mapping has a unique fixed point on its domain D, often called an invariant set or attractor in this context. Prove the Contraction Mapping Theorem.Show that the function on ℝ2 defined by T(x,y)→(12x+1,12y) is a contraction. What is its contractivity factor? Find its invariant set analytically, then verify that fixed point iteration converges to that fixed point.
Modification of ARL for detecting changes on the double EWMA chart in time series data with the autoregressive model
Published in Connection Science, 2023
Kotchaporn Karoon, Yupaporn Areepong, Saowanit Sukparungsee
Letbe a complete metric space and letbe a contraction mapping on Y. Then T has a unique fixed point(such that) with contraction constantsuch that, for all . Then there exists a unique such that , i.e. a unique fixed-point in (Sofonea et al., 2005).
Optimal contraception control for a size-structured population model with extra mortality
Published in Applicable Analysis, 2020
Obviously, if is a fixed point of the map , then it is a solution of Equation (11) and vice versa. Define a new norm in by for some , which is equivalent to the usual norm. It is clear that is a Banach space; that is, is a complete metric space. Here for , . It is easy to see that maps into itself. Now, we show that is a contraction mapping on the complete metric space . For any , , we have Choose λ such that . Then is a contraction mapping on the complete metric space . By the Banach fixed-point theorem, owns a unique fixed point , which is the solution of Equation (11).
Voltage Stability Online Monitoring and Analysis Method Based on the Fixed-Point Principle
Published in IETE Technical Review, 2021
For the analysis of voltage stability, this paper mainly uses the Banach fixed-point theorem, which is also called the contraction-mapping principle. For the general nonlinear self-mapping Equation (9), the Banach fixed-point theorem can be explained briefly as follows: In the interval , if the matrix norm then Equation (9) has a unique fixed-point in this interval, and this fixed-point can be obtained by a standard fixed-point iterative method, i.e. For a specific time section , define In practical engineering, we cannot specify the load type. However, the load is of constant impedance type at the specific time section. Obviously, is the equivalent admittance of the jth load at the specific time section. From Equation (10), we can obtain Thus, applying the Banach fixed-point principle to Equation (10) for voltage stability analysis, we can draw the following conclusions: if then the system is voltage stable at the specific time . Equation (16) is the most primitive voltage stability criterion proposed in this paper. If condition (16) is satisfied, then according to the norm theory, we can derive the following result: This means that if criterion (16) is satisfied, the load node voltage will be bounded and deterministic, that is, stable.