• East Asian mathematical journal
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• v.28 no.1
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• pp.63-71
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• 2012

The aim of this article is to study an implicit iteration process with errors for a finite family of non-Lipschitzian asymptotically non expansive mappings in the intermediate sense in Banach spaces. Also we establish some strong convergence theorems and a weak convergence theorem for said scheme to converge to a common fixed point for non Lipschitzian asymptotically nonexpansive mappings in the intermediate sense. The results presented in this paper extend and improve the corresponding results of [1], [3]-[8], [10]-[11], [13]-[14], [16] and many others.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.22 no.1
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• pp.15-28
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• 2018

Recently Machine Learning algorithms are widely used to process Big Data in various applications and a lot of these applications are executed in run time. Therefore the speed of Machine Learning algorithms is a critical issue in these applications. However the most of modern iteration Machine Learning algorithms use a successive iteration technique well-known in Numerical Linear Algebra. But this technique has a very low convergence, needs a lot of iterations to get solution of considering problems and therefore a lot of time for processing even on modern multi-core computers and clusters. Tchebychev iteration technique is well-known in Numerical Linear Algebra as an attractive candidate to decrease the number of iterations in Machine Learning iteration algorithms and also to decrease the running time of these algorithms those is very important especially in run time applications. In this paper we consider the usage of Tchebychev iterations for acceleration of well-known K-Means and SVM (Support Vector Machine) clustering algorithms in Machine Leaning. Some examples of usage of our approach on modern multi-core computers under Apache Spark framework will be considered and discussed.

• Journal of applied mathematics & informatics
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• v.7 no.2
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• pp.605-615
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• 2000

Let E be a real Banach space with property (U,${\lambda}$,m+1,m);${\lambda}{\ge}$0; m${\in}N$, and let C be a nonempty closed convex and bounded subset of E. Suppose T: $C{\leftrightarro}C$ is a strongly accretive map, It is proved that each of the two well known fixed point iteration methods( the Mann and Ishikawa iteration methods.), under suitable conditions , converges strongly to a solution of the equation Tx=f.

• Kyungpook Mathematical Journal
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• v.59 no.4
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• pp.665-678
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• 2019

We construct an iteration scheme involving a hybrid pair of mappings, one a single-valued asymptotically nonexpansive mapping t and the other a multivalued nonexpansive mapping T, in a complete CAT(0) space. In the process, we remove a restricted condition (called the end-point condition) from results of Akkasriworn and Sokhuma [1] and and use this to prove some convergence theorems. The results concur with analogues for Banach spaces from Uddin et al. [16].

• Korean Journal of Mathematics
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• v.22 no.2
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• pp.235-252
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• 2014

In this paper, weak and strong convergence theorems of three step iteration process with errors are established for two weakly inward and non-self asymptotically nonexpansive mappings in Banach spaces. The results obtained in this paper extend and improve the several recent results in this area.

• Structural Engineering and Mechanics
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• v.8 no.6
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• pp.547-560
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• 1999

In this paper an asymptotic iteration method is adopted to analyze non-linear free vibration of reticulated circular plates composed of beam members placed in two orthogonal directions. For the resulting linear ordinary differential equations in the process of iteration, the power series with rapid convergence has been applied to obtain an analytical solution for non-linear characteristic relation between the amplitude and frequency of the structure. Numerical examples are given, and the phenomena indicating hardening of such structures have been presented for the (immovable or movable) simply-supported and clamped circular plates.

• Korean Journal of Mathematics
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• v.31 no.3
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• pp.269-294
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• 2023

The purpose of this paper is to introduce a new three-step iterative process and show that our iteration scheme is faster than other existing iteration schemes in the literature. We provide a numerical example supported by graphs and tables to validate our proofs. We also prove convergence and stability results for the approximation of fixed points of the contractive-like mapping in the framework of uniformly convex Banach space. In addition, we have established some weak and strong convergence theorems for nonexpansive mappings.

• Journal of applied mathematics & informatics
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• v.30 no.1_2
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• pp.71-82
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• 2012

In this paper, we first show the weak convergence of the modified Ishikawa iteration process with errors of two total asymptotically nonexpansive mappings, which generalizes the result due to Khan and Fukhar-ud-din [1]. Next, we show the strong convergence of the modified Ishikawa iteration process with errors of two total asymptotically nonexpansive mappings satisfying Condition ($\mathbf{A}^{\prime}$), which generalizes the result due to Fukhar-ud-din and Khan [2].

• East Asian mathematical journal
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• v.28 no.1
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• pp.81-92
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• 2012

The purpose of this paper is to establish strong convergence of an implicit iteration process to a common fixed point for a finite family of asymptotically quasi-nonexpansive type mappings in CAT(0) spaces. Our results improve and extend the corresponding results of Fukhar-ud-din et al. [15] and some others from the current literature.

• East Asian mathematical journal
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• v.29 no.1
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• pp.23-37
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• 2013

The purpose of this paper is to study an implicit iteration process with errors and establish weak and strong convergence theorems to converge to common fixed points for a finite family of generalized asymptotically quasi-nonexpansive mappings in the framework of uniformly convex Banach spaces. Our results extend, improve and generalize some known results from the existing literature.