Reich's strong convergence theorem of approximate fixed points states that for every nonexpansive mapping T defined on a nonempty closed bounded convex set C of a uniformly smooth Banach space X, the approximate f...
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Reich's strong convergence theorem of approximate fixed points states that for every nonexpansive mapping T defined on a nonempty closed bounded convex set C of a uniformly smooth Banach space X, the approximate fixed points {x(lambda): 0 < lambda < 1} strongly converge to a fixed point of T as lambda -> 0(+), where x(lambda) is the unique fixed point of the contraction T-lambda = lambda(u0 )+ (1-lambda)T for every given u(0) is an element of C. In this paper, we first give this theorem and Xu's version of it localized settings: for nonempty super weakly compact convex set C of a Banach space X, there is an equivalent norm & Vert;|& sdot;& Vert;| on X so that both Reich's strong convergence theorem of approximate fixed points and Xu's version hold for every nonexpansive mapping T defined on (C,& Vert;|& sdot;& Vert;|). In particular, there is an equivalent norm & Vert;|& sdot;& Vert;| on L-1(mu) so that theorems mentioned above hold on every weakly compact convex set of (L-1(mu),& Vert;|& sdot;& Vert;|). They are done by showing that the theorems hold on a closed bounded convex set C subset of X if the norm of X is C-uniformly G & acirc;teaux smooth, and for every super weakly compact convex set C there is an equivalent C-uniformly Gateaux smooth norm on X.
In the context of finding most nearly compatible probability distributions under the discrete set-up, the role(s) of divergence measures as pseudo-distance (equivalently as measures of dissimilarity) measures are of p...
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In the context of finding most nearly compatible probability distributions under the discrete set-up, the role(s) of divergence measures as pseudo-distance (equivalently as measures of dissimilarity) measures are of paramount importance. For a detailed discussion on various measures of statistical divergence and its properties, see Pardo (2018) and the references cited therein. Recently, Ghosh and Balakrishnan (2015) have utilized several of such divergence measures, such as the Power divergence, and several other measures. However, in search for the most nearly compatible (or incompatible) distributions, the convergence of the iterativealgorithms remains a challenging issue. In this article, we put forward a sketch of the proof regarding the convergence of the iterativealgorithm(s) for certain divergence measures under some mild conditions. The proof for the more general case still remains an open problem.& COPY;2023 Elsevier B.V. All rights reserved.
This paper studies transmission power control algorithms for cellular networks. One of the challenges in commonly used iterative mechanisms to achieve this is to identify if the iteration will converge since convergen...
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This paper studies transmission power control algorithms for cellular networks. One of the challenges in commonly used iterative mechanisms to achieve this is to identify if the iteration will converge since convergence indicates feasibility of transmit power allocation under prevailing network conditions. The convergence criterion should also be simple to calculate given the time constraints in a real-time wireless network. Towards this goal, this paper derives simple sufficient conditions for convergence of an iterative power control algorithm using existing bounds from matrix theory. With the help of suitable numerical examples, it is shown that the allocated transmit powers of the nodes converge when sufficient conditions are satisfied, and diverge when they are not satisfied. This forms the basis for an efficient link data-rate based admission control mechanism for wireless networks. The mechanism considers parameters such as signal strength requirement, link datarate requirement, and number of nodes in the system. Simulation based analysis shows that existing links are able to maintain their desired datarates despite the addition of new wireless links.
An iterativealgorithm is proposed for the constrained minimization of a convex nonsmooth function on a set given as a convex smooth surface. The convergence of the algorithm in the sense of necessary conditions for a...
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An iterativealgorithm is proposed for the constrained minimization of a convex nonsmooth function on a set given as a convex smooth surface. The convergence of the algorithm in the sense of necessary conditions for a local minimum is proved.
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