scholarly journals On the Best Proximity Points for p–Cyclic Summing Contractions

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1060
Author(s):  
Miroslav Hristov ◽  
Atanas Ilchev ◽  
Boyan Zlatanov
Keyword(s):  
Fixed Point Theory ◽  
Complete Metric Space ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Point Theory ◽  
Picard Iteration ◽  
Best Proximity Points ◽  
Underlying Space ◽  
Cyclic Maps

We present a condition that guarantees the existence and uniqueness of fixed (or best proximity) points in complete metric space (or uniformly convex Banach spaces) for a wide class of cyclic maps, called p–cyclic summing maps. These results generalize some known results from fixed point theory. We find a priori and a posteriori error estimates of the fixed (or best proximity) point for the Picard iteration associated with the investigated class of maps, provided that the modulus of convexity of the underlying space is of power type. We illustrate the results with some applications and examples.

scholarly journals General Local Convergence Theorems about the Picard Iteration in Arbitrary Normed Fields with Applications to Super–Halley Method for Multiple Polynomial Zeros

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1599
Author(s):  
Stoil I. Ivanov
Keyword(s):  
Error Estimates ◽  
Local Convergence ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Sufficient Conditions ◽  
Posteriori Error ◽  
Picard Iteration ◽  
Polynomial Zeros ◽  
Convergence Theorems ◽  
Simple And Multiple Roots

In this paper, we prove two general convergence theorems with error estimates that give sufficient conditions to guarantee the local convergence of the Picard iteration in arbitrary normed fields. Thus, we provide a unified approach for investigating the local convergence of Picard-type iterative methods for simple and multiple roots of nonlinear equations. As an application, we prove two new convergence theorems with a priori and a posteriori error estimates about the Super-Halley method for multiple polynomial zeros.

scholarly journals Error estimates for approximating best proximity points for cyclic contractive maps

2016 ◽  
Vol 32 (2) ◽  
pp. 265-270 ◽  
Author(s):  
BOYAN ZLATANOV ◽  
Keyword(s):  
Error Estimates ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Convex Banach Space ◽  
Picard Iteration ◽  
Power Type ◽  
Cyclic Contraction ◽  
Contraction Map ◽  
Contractive Maps

We find a priori and a posteriori error estimates of the best proximity point for the Picard iteration associated to a cyclic contraction map, which is defined on a uniformly convex Banach space with modulus of convexity of power type.

scholarly journals A Priori and A Posteriori Error Estimates for a Crank Nicolson Type Scheme of an Elliptic Problem with Dynamical Boundary Conditions

2016 ◽  
Vol 8 (2) ◽  
pp. 1
Author(s):  
Rola Ali Ahmad ◽  
Toufic El Arwadi ◽  
Houssam Chrayteh ◽  
Jean-Marc Sac-Epee
Keyword(s):  
Error Estimates ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Time Discretization ◽  
Posteriori Error ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Dynamical Boundary Conditions ◽  
Error Indicators ◽  
Crank Nicolson

In this article we claim that we are going to give a priori and a posteriori error estimates for a Crank Nicolson type scheme. The problem is discretized by the finite elements in space. The main result of this paper consists in establishing two types of error indicators, the first one linked to the time discretization and the second one to the space discretization.

ERROR ESTIMATES FOR LINEAR PDEs SOLVED BY WAVELET BASED TAYLOR–GALERKIN SCHEMES

2009 ◽  
Vol 07 (02) ◽  
pp. 143-162 ◽  
Author(s):  
MANI MEHRA ◽  
B. V. RATHISH KUMAR
Keyword(s):  
Error Estimates ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Equivalent Norm ◽  
Posteriori Error ◽  
Discrete Problem ◽  
Fully Discrete ◽  
Time Stepping ◽  
Priori Estimates ◽  
Space Error

In this paper, we develop a priori and a posteriori error estimates for wavelet-Taylor–Galerkin schemes introduced in Refs. 6 and 7 (particularly wavelet Taylor–Galerkin scheme based on Crank–Nicolson time stepping). We proceed in two steps. In the first step, we construct the priori estimates for the fully discrete problem. In the second step, we construct error indicators for posteriori estimates with respect to both time and space approximations in order to use adaptive time steps and wavelet adaptivity. The space error indicator is computed using the equivalent norm expressed in terms of wavelet coefficients.

On a theorem of Brian Fisher in the framework of w-distance

2017 ◽  
Vol 33 (2) ◽  
pp. 199-205
Author(s):  
DARKO KOCEV ◽  
◽  
VLADIMIR RAKOCEVIC ◽  
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Complete Metric Space ◽  
Point Theory ◽  
Common Fixed Points ◽  
Complete Metric Spaces ◽  
Condition C ◽  
Relaxing Condition

In 1980. Fisher in [Fisher, B., Results on common fixed points on complete metric spaces, Glasgow Math. J., 21 (1980), 165–167] proved very interesting fixed point result for the pair of maps. In 1996. Kada, Suzuki and Takahashi introduced and studied the concept of w–distance in fixed point theory. In this paper, we generalize Fisher’s result for pair of mappings on metric space to complete metric space with w–distance. The obtained results do not require the continuity of maps, but more relaxing condition (C; k). As a corollary we obtain a result of Chatterjea.

scholarly journals A Class of Spectral Element Methods and Its A Priori/A Posteriori Error Estimates for 2nd-Order Elliptic Eigenvalue Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Jiayu Han ◽  
Yidu Yang
Keyword(s):  
Spectral Methods ◽  
Eigenvalue Problems ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Spectral Element ◽  
A Posteriori ◽  
Spectral Element Methods ◽  
A Posteriori Error ◽  
Elliptic Eigenvalue Problems

This paper discusses spectral and spectral element methods with Legendre-Gauss-Lobatto nodal basis for general 2nd-order elliptic eigenvalue problems. The special work of this paper is as follows. (1) We prove a priori and a posteriori error estimates for spectral and spectral element methods. (2) We compare between spectral methods, spectral element methods, finite element methods and their derivedp-version,h-version, andhp-version methods from accuracy, degree of freedom, and stability and verify that spectral methods and spectral element methods are highly efficient computational methods.

The a Posteriori Error Estimates for Chebyshev-Galerkin Spectral Methods in One Dimension

2015 ◽  
Vol 7 (2) ◽  
pp. 145-157 ◽  
Author(s):  
Jianwei Zhou
Keyword(s):  
A Priori ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
One Dimension ◽  
A Posteriori ◽  
Error Estimators ◽  
Poisson Equations ◽  
Spectral Approximations ◽  
Fourth Order Equations ◽  
Error Indicators

AbstractIn this paper, the Chebyshev-Galerkin spectral approximations are employed to investigate Poisson equations and the fourth order equations in one dimension. Meanwhile, p-version finite element methods with Chebyshev polynomials are utilized to solve Poisson equations. The efficient and reliable a posteriori error estimators are given for different models. Furthermore, the a priori error estimators are derived independently. Some numerical experiments are performed to verify the theoretical analysis for the a posteriori error indicators and a priori error estimations.

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