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α-Admissibility and Fixed Points
Published in Dhananjay Gopal, Poom Kumam, Mujahid Abbas, Background and Recent Developments of Metric Fixed Point Theory, 2017
Deepesh Kumar Patel, Wutiphol Sintunavarat
It is clear that f is not continuous at 1. The Banach contraction principle and also Theorem 3.1 are not applicable. Define the mapping α:X×X→[0,+∞) $ \alpha : X \times X \rightarrow [0,+\infty ) $ by α(x,y)=1ifx,y∈[0,1],0otherwise. $$ \begin{aligned} \alpha (x,y)= {\left\{ \begin{array}{ll} 1&\text{ if} x,y \in [0, 1], \\ 0&\text{ otherwise}. \end{array}\right.} \end{aligned} $$
Quantum Signal Processing
Published in Harish Parthasarathy, Advanced Probability and Statistics: Applications to Physics and Engineering, 2023
This transition probability is random since f(.) is a random process. Now in the limit as ϵ → 0, Pt (m|n, ϵ) → 0 and we wish to determine the rate at which this transition probability converges to zero. It is clear that Pt (m|n, ϵ) has a χ2- distribution since it is a quadratic functional of a zero mean Gaussian process. So we can apply the contraction principle to determine the rate function of this transition probability.
Fixed Point Theory
Published in Eberhard Malkowsky, Vladimir Rakočević, Advanced Functional Analysis, 2019
Eberhard Malkowsky, Vladimir Rakočević
We note that Kannan’s fixed point theorem, Theorem 10.7.1, is not an extension of Banach’s contraction principle, Theorem 10.2.2. It is also known that a metric space X is complete if and only if every Kannan mapping has a fixed point, while there exists a metric space X such that X is not complete and every contractive mapping on X has a fixed point ([46,195]).
Moderate deviations for stochastic variational inequalities
Published in Optimization, 2023
Mingjie Gao, Ka-Fai Cedric Yiu
The contraction principle ([19, Theorem 4.2.1]). Let and be two separable metric spaces and is continuous. If satisfies a large deviation principle with speed and with good rate function I, then satisfies the large deviation principle with the speed and with the good rate function , where the rate function given by The following delta method in large deviation theory is given by Gao and Zhao in [22].
An extended local principle of fixed points for weakly contractive set-valued mappings
Published in Optimization, 2022
M. Ait Mansour, A. El Bekkali, J. Lahrache
In this work we have presented an extension of the local contraction principle by Dontchev and Hager [6, Theorem 5E.2]. Our principle of fixed points makes appeal to weakly contracting maps, which allows us to extend and unify many classic fixed points assertions such as Nadler's Theorem [1] and many other results in [11, Theorem 2.1], Gordji et al.[11], Kannan [12], Chatterjea [13] and Reich [14]. In addition, as an application we have established two new results with respect to quantitative stability results for fixed points of set-valued maps. This improves some previous results on the stability topic such as the famous Lim's Lemma [17] and its extended version in [16, Theorem 3.2]. Consequently, according to Remark 3.4, our results in Corollary 3.3 have good perspectives concerning stability of local solutions to quasi-optimization problems, variational problems and differential inclusions and many other related mathematical problems.
Fixed point theorems for the Mann's iteration scheme in convex graphical rectangular b-metric spaces
Published in Optimization, 2021
Lili Chen, Ni Yang, Yanfeng Zhao
Banach's Contraction Principle [1] is one of the most widely applied fixed point theorems in all branches of Mathematics [2–10]. In recent decades, scholars have devoted themselves to extending above theorem to all kinds of generalized metric spaces [11–18]. In 1993, Czerwik [12] introduced the concepts of b-metric spaces by weakening the coefficient of the triangle inequality and generalized Banach's Contraction Principle to these spaces. In 2000, Branciari [19] presented the concepts of rectangular metric spaces by inserting two points in the right side of the triangle inequality. In 2015, the notations of rectangular b-metric spaces were introduced as the generalizations of metric spaces, rectangular metric spaces and b-metric spaces by George et al. [13].