On the best constant in Hyers–Ulam stability of some positive linear operators

We obtain the infimum of the Hyers–Ulam stability constants for Stancu, Bernstein and Kantorovich operators and prove that in a class of certain positive linear operators this infimum for Bernstein operator has a minimality property.     

Fixed points of a mapping and Hyers–Ulam stability

We show that many general results on Hyers–Ulam stability of some functional equations in a single variable follow immediately from a simple fixed point theorem. The theorem is formulated for self-maps of some subsets of the space of functions from a nonempty set into the set of reals. We also give some applications of that theorem, e.g., in investigations of solutions of some difference equations and functional inequalities.     

A fixed point approach to the stability of functional equations in non-Archimedean metric spaces

In this note, we prove a simple fixed point theorem for a special class of complete metric spaces (namely, complete non-Archimedean metric spaces which are connected with some problems coming from quantum physics, p-adic strings and superstrings). We also show that this theorem is a very efficient and convenient tool for proving the Hyers–Ulam stability of a quite wide class of functional equations in a single variable.     

Hyers–Ulam stability of a system of first order linear differential equations with constant coefficients

In this paper, using matrix method, we prove the Hyers–Ulam stability of a system of first order linear differential equations with constant coefficients.     

Integral-type operators on continuous function spaces on the real line

In this paper we introduce some new sequences of positive linear operators, acting on a sufficiently large space of continuous functions on the real line, which generalize Gauss–Weierstrass operators.We study their approximation properties and prove an asymptotic formula that relates such operators to a second order elliptic differential operator of the form Lu?αu^′′+βu^′+γu.Shape-preserving and regularity properties are also investigated.     

A Mazur–Ulam theorem in non-Archimedean normed spaces

The classical Mazur–Ulam theorem which states that every surjective isometry between real normed spaces is affine is not valid for non-Archimedean normed spaces. In this paper, we establish a Mazur–Ulam theorem in the non-Archimedean strictly convex normed spaces.     

Waist of the sphere for maps to manifolds

We generalize the sphere waist theorem of Gromov and the Borsuk–Ulam type measure partition lemma of Gromov–Memarian for maps to manifolds.     

New results for best approximation on Banach lattices

The purpose of this paper is to introduce and to discuss the concept of approximation preserving operators on Banach lattices with a strong unit. We show that every lattice isomorphism is an approximation preserving operator. Also we give a necessary and sufficient condition for uniqueness of the best approximation by closed normal subsets of X⁺$X^{+}$, and show that this condition is characterized by some special operators.     

ON THE RATES OF CONVERGENCE FOR PROBABILITY TYPE OPERATORS SEQUENCE

Abstract. In this paper, the rates of convergence for some probability type operators sequence are obtained. The quantitative Poisson type limit theorem is established as an application.     

Simultaneous Approximation for Generalized Baskakov-Durrmeyer-Type Operators

The present paper deals with the study of a Durrmeyer-type integral modification of certain modified Baskakov operators. Here we study simultaneous approximation properties for these operators by using the iterative combinations. We obtain an asymptotic formula and an error estimation in terms of higher order modulus of continuity for these operators.      

概率型算子在Lp空间的渐近性质

In 1985, Khan, R. A. established the asymptotic formulas of operators of probabilistic type inL1, space by introducing a newLp-norm. The purpose of this paper is to study the asymptotic rate of these operators, inLp (p>1) spaces. Project supported by the National Natural Science Foundation of China     

Convergence rate of a new Bezier variant of Chlodowsky operators to bounded variation functions

In the present paper, we estimate the rate of pointwise convergence of the Bézier Variant of Chlodowsky operators _Cn,α$C_{n,\alpha }$ for functions, defined on the interval extending infinity, of bounded variation. To prove our main result, we have used some methods and techniques of probability theory.     

Local approximation results for Szász-Mirakjan type operators

In this study, motivating our earlier work [O. Duman and M.A. ?zarslan, Szász-Mirakjan type operators providing a better error estimation. Appl. Math. Lett. 20, 1184–1188 (2007)], we investigate the local approximation properties of Szász-Mirakjan type operators. The second modulus of smoothness and Petree’s K-functional are considered in proving our results. Received: 17 September 2007     

On a Sequence of Positive Linear Operators Associated with a Continuous Selection of Borel Measures

In this paper we introduce and study a new sequence of positive linear operators acting on the space of Lebesgue-integrable functions on the unit interval. These operators are defined by means of continuous selections of Borel measures and generalize the Kantorovich operators. We investigate their approximation properties by presenting several estimates of the rate of convergence by means of suitable moduli of smoothness. Some shape preserving properties are also shown. Dedicated to the memory of Professor Aldo Cossu     

On the iterates of Jackson type operator GS,N in Hilbert space

In this paper we study the limit of the iterates of Jackson type operator. Our results continue the works of Badea [2] and Nagler et al. [9, 10]. The proofs are based on spectral theory of linear operators and are performed at first for Hilbert space and then are extended for some Banach spaces.     

Best constant in stability of some positive linear operators

We prove that the kernels of Bernstein, Stancu and Kantorovich operators are proximinal sets, therefore the infimum of Hyers–Ulam constants is also a Hyers–Ulam constant for the above mentioned operators. Moreover, we investigate what happens when the supremum norm is replaced by the L₁-norm.     

Recurrence relations for a Newton-like method in Banach spaces

The convergence of iterative methods for solving nonlinear operator equations in Banach spaces is established from the convergence of majorizing sequences. An alternative approach is developed to establish this convergence by using recurrence relations. For example, the recurrence relations are used in establishing the convergence of Newton's method [L.B. Rall, Computational Solution of Nonlinear Operator Equations, Robert E. Krieger, New York, 1979] and the third order methods such as Halley's, Chebyshev's and super Halley's [V. Candela, A. Marquina, Recurrence relations for rational cubic methods I: the Halley method, Computing 44 (1990) 169–184; V. Candela, A. Marquina, Recurrence relations for rational cubic methods II: the Halley method, Computing 45 (1990) 355–367; J.A. Ezquerro, M.A. Hernández, Recurrence relations for Chebyshev-type methods, Appl. Math. Optim. 41 (2000) 227–236; J.M. Gutiérrez, M.A. Hernández, Third-order iterative methods for operators with bounded second derivative, J. Comput. Appl. Math. 82 (1997) 171–183; J.M. Gutiérrez, M.A. Hernández, Recurrence relations for the Super–Halley method, Comput. Math. Appl. 7(36) (1998) 1–8; M.A. Hernández, Chebyshev's approximation algorithms and applications, Comput. Math. Appl. 41 (2001) 433–445 [10]].     

Approximation by boolean sums of positive linear operators III: Estimates for some numerical approximation schemes

In the present note we intröduce and investigate certain sequences of discrete positive linear operators and Boolean sum modifications of them. The mappings considered are obtained by discretizing a class of transformed convolution-type operators using Gaussian quadrature of appropriate order. For our operators and their modifications we prove pointwise Jackson-type theorems involving the first and second order moduli of smoothness, thus providing new and elegant proofs of earlier results by Timan, Telyakowskii, Gopengauz and DeVore. Due to their discrete structure, optimal order of approximation and ease of computation, the operators appear to be useful for numerical approximation. In an intermediate step we solve an old problem in Approximation Theory; its importance was only recently emphasized in a paper of Butzer.