Approximation theory is an intensive research area, developed in different directions by many mathematicians.
For example, approximation and iteration processes arise in a very natural way in many problems dealing with the constructive approximation of functions as well as solutions to (partial) differential equations and integral equations. Moreover, approximation theory can be successfully applied in fixed point theory, in computer aided geometric design, in artificial neural networks, in the study of evolution problems, and in function algebras.
The goal of this special issue is to attract original research and review articles that highlight recent advances in operator methods within approximation theory and related applications.
The interest aroused by the mathematicians who work in this area is remarkable, as evidenced by the thirty-six submissions received.
The papers that have been accepted for the publication in the issue recover the following topics:
In what follows we give a brief description of the contents of this special issue.
In the review article titled “On Sequences of J. P. King-Type Operators,” T. Acar et al. provide an essential exposition of a series of investigations developed in the last fifteen years after the release of a paper written by J. P. King, where a modification of the classical Bernstein operators was considered in order to get better approximation properties than the original ones. After a brief history devoted to the sequences of positive linear operators fixing certain (polynomial, exponential, or more general) functions obtained by applying King’s approach, the authors illustrate certain King-type modifications of the well-known Bernstein and Szász-Mirakjan operators.
A. A. Bakery and M. M. Mohammed in the paper “Small Pre-Quasi Banach Operator Ideals of Type Orlicz-Cesáro Mean Sequence Spaces” deal with determining sufficient conditions on an Orlicz-Cesàro mean sequence space
The purpose of the paper
S. Z. Ullah et al. generalize and improve some known results concerning integral majorization type and generalized Favard’s inequalities for the class of strongly convex functions in the paper titled “Integral Majorization Type Inequalities for the Functions in the Sense of Strong Convexity.”
In the paper “Bivariate Chlodowsky-Stancu Variant of (
N. Özmen in the paper titled “New Generating Function Relations for the q-Generalized Cesàro Polynomials” examines a
H. J. Lee proves that the
In the paper “Convergence Analysis of an Accelerated Iteration for Monotone Generalized
C. Zhang and S. Wang in the paper titled “Structure Properties for Binomial Operators” discussed some structures properties of the binomial operators, such as moments representation, derivatives representation, and binary representation. As applications, the authors prove that the binomial operators considered preserve increasing functions, convex functions, and Hölder (continuous) functions.
The purpose of the paper “On New Picard-Mann Iterative Approximations with Mixed Errors for Implicit Midpoint Rule and Applications” written by T. Li and H. Lan is to introduce and study a new class of Picard-Mann iteration processes with mixed errors for the implicit midpoint rules and to analyze the convergence and stability of the proposed method. Some numerical examples and applications to optimal control problems with elliptic boundary value constraints are presented, and they show that the Picard-Mann iteration process discussed in the article is more effective than other related iterative processes.
J.-L. Wang et al. in the paper titled “On Approximating the Toader Mean by Other Bivariate Means” provide several sharp bounds for the Toader mean by using certain combinations of the arithmetic, quadratic, contraharmonic, and Gaussian arithmetic geometric means.
The paper “
In the paper “On a New Stability Problem of Radical
In the context of singular Hadamard fractional boundary value problems, J. Mao et al. establish, by using an iterative algorithm, the existence and uniqueness of the exact iterative solution in the paper titled “The Unique Positive Solution for Singular Hadamard Fractional Boundary Value Problems.” Moreover, they show the iterative sequences converge uniformly to the exact solution, and they provide estimation of the approximation error and the convergence rate.
The Guest Editors declare that they have no conflicts of interest regarding the publication of this special issue.
We are very grateful to the mathematical community for the great interest shown in this special issue. In particular, we want to thank all the authors of the published papers for their contribution in this field. Also, our gratitude goes to the reviewers, for their precious help in handling all the submitted manuscripts, and to Hindawi who supported us throughout the development of this special issue. Finally, the Lead Guest Editor sincerely thanks her Guest Editors for agreeing to join her in this project.