• Korean Journal of Mathematics
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• v.18 no.3
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• pp.213-227
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• 2010

Convergence of an implicit iterative process is investigated for nonexpansive mappings. Strong and weak convergence theorems of common fixed points of two finite families of nonexpansive mappings are established in the framework of Banach spaces.

• Bulletin of the Korean Mathematical Society
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• v.47 no.6
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• pp.1259-1268
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• 2010

In this paper, we consider an implicit iterative process with errors for an in nite family of strict pseudocontractions. Strong convergence theorems are established in the framework of Banach spaces. The results presented in this paper improve and extend the recent ones announced by many others.

• East Asian mathematical journal
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• v.31 no.3
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• pp.379-382
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• 2015

The purpose of this note is to study the estimation of errors of the implicit Mann iterative process with random errors.

• Communications of the Korean Mathematical Society
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• v.26 no.1
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• pp.23-35
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• 2011

In the present work we construct a composite implicit random iterative process with errors for a finite family of asymptotically nonexpansive random operators and discuss a necessary and sufficient condition for the convergence of this process in an arbitrary real Banach space. It is also proved that this process converges to the common random fixed point of the finite family of asymptotically nonexpansive random operators in the setting of uniformly convex Banach spaces. The present work also generalizes a recently established result in Banach spaces.

• Communications of the Korean Mathematical Society
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• v.25 no.4
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• pp.571-581
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• 2010

Some convergence theorems for approximating to a common fixed point of an infinite family of strictly pseudocontractive mappings of Browder-Petryshyn type are proved in the setting of Banach spaces by using a new composite implicit iterative process with errors. The results presented in the paper generalize and improve the main results of Bai and Kim [1], Gu [4], Osilike [5], Su and Li [7], and Xu and Ori [8].

• East Asian mathematical journal
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• v.24 no.1
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• pp.35-43
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• 2008

Let E be a uniformly convex Banach space and K be a nonempty closed convex subset of E. Let ${\{T_i\}}^N_{i=1}$ be N nonexpansive self-mappings of K with $F\;=\;{\cap}^N_{i=1}F(T_i)\;{\neq}\;{\theta}$ (here $F(T_i)$ denotes the set of fixed points of $T_i$). Suppose that one of the mappings in ${\{T_i\}}^N_{i=1}$ is semi-compact. Let $\{{\alpha}_n\}\;{\subset}\;[{\delta},\;1-{\delta}]$ for some ${\delta}\;{\in}\;(0,\;1)$ and $\{{\beta}_n\}\;{\subset}\;[\tau,\;1]$ for some ${\tau}\;{\in}\;(0,\;1]$. For arbitrary $x_0\;{\in}\;K$, let the sequence {$x_n$} be defined iteratively by $\{{x_n\;=\;{\alpha}_nx_{n-1}\;+\;(1-{\alpha}_n)T_ny_n,\;\;\;\;\;\;\;\;\; \atop {y_n\;=\;{\beta}nx_{n-1}\;+\;(1-{\beta}_n)T_nx_n},\;{\forall}_n{\geq}1,}$, where $T_n\;=\;T_{n(modN)}$. Then {$x_n$} convergence strongly to a common fixed point of the mappings family ${\{T_i\}}^N_{i=1}$. The result presented in this paper generalized and improve the corresponding results of Chidume and Shahzad [C. E. Chidume, N. Shahzad, Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings, Nonlinear Anal. 62(2005), 1149-1156] even in the case of ${\beta}_n\;{\equiv}\;1$ or N=1 are also new.

• Nuclear Engineering and Technology
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• v.56 no.8
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• pp.3139-3143
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• 2024

Stratified flow in horizontal tubes is frequently observed in gas-liquid two-phase flow system. In the two-fluid modeling, it is important to define the interface shape in solving the balance equations to determine the key parameters such as the interfacial transfer terms, void fraction, and pressure drop. A double-circle model is usually introduced to depict the concave-down interface in a horizontal circular tube under the stratified-wavy flow condition. However, calculation of the central angle in the double-circle model, which represents the interfacial curvature, requires an appropriate iterative numerical root-finding scheme to solve the implicit transcendental equation. In this study, an explicit approximate equation has been proposed without requirement of the iterative scheme and numerical instability, which is expected to improve the coding process and computation efficiency in the analysis code with the two-fluid model.

• Structural Engineering and Mechanics
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• v.56 no.4
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• pp.549-567
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• 2015

A first-order moment method (FORM) reliability analysis is commonly used for structural stability analysis. It requires the values and partial derivatives of the performance to function with respect to the random variables for the design. These calculations can be cumbersome when the performance functions are implicit. A Gaussian process (GP)-based response surface is adopted in this study to approximate the limit state function. By using a trained GP model, a large number of values and partial derivatives of the performance functions can be obtained for conventional reliability analysis with a FORM, thereby reducing the number of stability analysis calculations. This dynamic renewed knowledge source can provide great assistance in improving the predictive capacity of GP during the iterative process, particularly from the view of machine learning. An iterative algorithm is therefore proposed to improve the precision of GP approximation around the design point by constantly adding new design points to the initial training set. Examples are provided to illustrate the GP-based response surface for both structural and non-structural reliability analyses. The results show that the proposed approach is applicable to structural reliability analyses that involve implicit performance functions and structural response evaluations that entail time-consuming finite element analyses.

• Journal of the Korean Geotechnical Society
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• v.22 no.9
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• pp.45-59
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• 2006

The successful performance of any numerical geotechnical simulation depends on the accuracy and efficiency of the numerical implementation of constitutive model used to simulate the stress-strain (constitutive) response of the soil. The corner stone of the numerical implementation of constitutive models is the numerical integration of the incremental form of soil-plasticity constitutive equations over a discrete sequence of time steps. In this paper a well known two-surface soil plasticity model is implemented using a generalized implicit return mapping algorithm to arbitrary convex yield surfaces referred to as the Closest-Point-Projection method (CPPM). The two-surface model describes the nonlinear behavior of coarse-grained materials by incorporating a bounding surface concept together with isotropic and kinematic hardening as well as fabric formulation to account for the effect of fabric formation on the unloading response. In the course of investigating the performance of the CPPM integration method, it is proven that the algorithm is an accurate, robust, and efficient integration technique useful in finite element contexts. It is also shown that the algorithm produces a consistent tangent operator $\frac{d\sigma}{d\varepsilon}$ during the iterative process with quadratic convergence rate of the global iteration process.

• 한국전산유체공학회:학술대회논문집
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• 1998.05a
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• pp.110-115
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• 1998

An axisymmetric Navier-Stokes procedure has been developed to analyze the pulse jet flow in a ceramic filter unit for the dust dislodging process. Using Baldwin-Lomax turbulence model as a closure relationship, the SIAF(Scalar Implicit Approximate Factorization) algorithm together with the ${\delta}^k-Correction$ iterative time marching scheme is adopted to solve the unsteady compressible Navier-Stokes equations. After some validation tests, the code has been applied to solve the pulse jet flow and examine the effects of geometry and reservoir pressure condition on the pressure level inside the filter unit. To avoid dealing with the uncertainty of such factors as the cohesion of the collected dust and the adhesion of the dust to the medium and also to simplify the analysis, the filter wall is assumed to be impermeable. The results for various test cases are presented.