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Register a new userWeak and strong convergence theorems for generalized nonexpansive mappings
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We consider a class of generalized nonexpansive mappings introduced by Karapinar [5] and seen as a generalization of Suzuki (C)-condition. We prove some weak and strong convergence theorems for approximating fixed points of such mappings under suitable conditions in uniformly convex Banach spaces. Our results generalize those of Khan and Suzuki [4] to the case of this kind of mappings and, in turn, are related to a famous convergence theorem of Reich [2] on nonexpansive mappings.
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We show that the typical nonexpansive mapping on a small enough subset of a CAT($kappa$)-space is a contraction in the sense of Rakotch. By typical we mean that the set of nonexpansive mapppings without this property is a $sigma$-porous set and therefore also of the first Baire category. Moreover, we exhibit metric spaces where strict contractions are not dense in the space of nonexpansive mappings. In some of these cases we show that all continuous self-mappings have a fixed point nevertheless.
In this paper, we first introduce an iterative process in modular function spaces and then extend the idea of a {lambda}-firmly nonexpansive mapping from Banach spaces to modular function spaces. We call such mappings as ({lambda},{rho})-firmly nonexpansive mappings. We incorporate the two ideas to approximate fixed points of ({lambda},{rho})-firmly nonexpansive mappings using the above mentioned iterative process in modular function spaces. We give an example to validate our results.
Generalized smooth functions are a possible formalization of the original historical approach followed by Cauchy, Poisson, Kirchhoff, Helmholtz, Kelvin, Heaviside, and Dirac to deal with generalized functions. They are set-theoretical functions defined on a natural non-Archimedean ring, and include Colombeau generalized functions (and hence also Schwartz distributions) as a particular case. One of their key property is the closure with respect to composition. We review the theory of generalized smooth functions and prove both the local and some global inverse function theorems.
The main objective of this work is to study generalized Browders and Weyls theorems for the multiplication operators $L_A$ and $R_B$ and for the elementary operator $tau_{A,B}=L_AR_B$.
We estimate convergence rates for fixed-point iterations of a class of nonlinear operators which are partially motivated from solving convex optimization problems. We introduce the notion of the generalized averaged nonexpansive (GAN) operator with a positive exponent, and provide a convergence rate analysis of the fixed-point iteration of the GAN operator. The proposed generalized averaged nonexpansiveness is weaker than the averaged nonexpansiveness while stronger than nonexpansiveness. We show that the fixed-point iteration of a GAN operator with a positive exponent converges to its fixed-point and estimate the local convergence rate (the convergence rate in terms of the distance between consecutive iterates) according to the range of the exponent. We prove that the fixed-point iteration of a GAN operator with a positive exponent strictly smaller than 1 can achieve an exponential global convergence rate (the convergence rate in terms of the distance between an iterate and the solution). Furthermore, we establish the global convergence rate of the fixed-point iteration of a GAN operator, depending on both the exponent of generalized averaged nonexpansiveness and the exponent of the H$ddot{text{o}}$lder regularity, if the GAN operator is also H$ddot{text{o}}$lder regular. We then apply the established theory to three types of convex optimization problems that appear often in data science to design fixed-point iterative algorithms for solving these optimization problems and to analyze their convergence properties.
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