Abstract

In this paper, first we give a theorem which generalizes the Banach contraction principle and fixed point theorems given by many authors, and then a fixed point theorem for a multi-valued $(\theta, L)$-weak contraction. We extend the notion of $(\theta,L)$-weak contraction to fuzzy mappings and obtain some fixed point theorems. A coincidence point theorem for a hybrid pair of mappings $f:X\to X$ and $T:X\to W(X)$ is established. Later on we prove a fixed point theorem for a different type of fuzzy mapping.

Keywords: Contraction mapping; fixed point; Banach contraction principle; Hausdorff metric; fuzzy set; fuzzy mapping.

MSC: 03E72, 47H10, 54H25

Abstract

The present paper deals with certain Bézier-Durrmeyer type sequence of linear positive operators $M_{n,\alpha}(f,x)$, having different basis functions in summation and integration. We estimate the rate of convergence of these operators $M_{n,\alpha}(f,x)$, for functions having derivatives of bounded variation.

Keywords: Linear positive operators; Baskakov and Szász basis functions; summation-integral type operators; Bézier variant; rate of convergence.

MSC: 41A25, 41A30

Abstract

We study under which conditions weighted composition operators acting on weighted Bergman spaces of infinite order are bounded from below or, equivalently, have closed range.

Keywords: Weighted composition operator; weighted Bergman space of infinite order.

MSC: 47B33, 47B38

Abstract

In this note we give an example of a Hausdorff, star compact space with a $G_\delta$-diagonal which is not metrizable, which answers negatively a question of van Mill, Tkachuk and Wilson (Problem 4.8 in [J. van Mill, V.V. Tkachuk, R.G. Wilson, Classes defined by stars and neighbourhood assignments, Topology Appl. 154 (2007), 2127--2134]).

Keywords: Star compact space, $G_\delta$-diagonal.

MSC: 54D20

Abstract

In this paper some generating functions for some sets of polynomials in several variables are established. In these classes of generating functions an arbitrary sequence of multivariable functions is considered. The generating functions so derived are shown here to lead some known results of Raina, Raina and Bajpai and Zeitlin and are capable to provide as special cases, a large number of new summation formulas and generating functions for simpler sequences, extended polynomials and generalized Lauricella functions.

Keywords: Lagrange's expansions, generating functions, summation formulae, Lauricella function.

MSC: 33C70, 33C47, 33C65

Abstract

For analytic functions $f$ normalized by $f(0)=f'(0)-1=0$ in the open unit disk $U$, a class $P_{\alpha}(\lambda)$ of $f$ defined by $|D^{\alpha}_{z}(\frac{z}{f(z)})|\leq \lambda$, where $D^{\alpha}_{z}$ denotes the fractional derivative of order $\a$, $m \leq \alpha < m+1$, $m \in N_{0} $, is introduced. In this article, we study the problem when $\frac{1}{r} f(rz) \in P_{\alpha}(\lambda)$, $3 \leq \alpha < 4$.

Keywords: Analytic functions, univalent functions, Cauchy-Schwarz inequality, fractional differential operator.

MSC: 30C45

Abstract

This paper deals with a modification of the classical Szász-Mirakjan type operators of two variables. It introduces a new sequence of non-polynomial linear operators which hold fixed the polynomials $x^{2}+\alpha x$ and $y^{2}+\beta y$ with $\alpha ,\beta \in [0,\infty)$ and we study the convergence properties of the new approximation process. Also, we compare it with Szász-Mirakjan type operators and show an improvement of the error of convergence in $[0,1] \times [0,1]$. Finally, we study statistical convergence of this modification.

Keywords: Szász-Mirakjan type operators, $A$-statistical convergence for double sequences, Korovkin-type approximation theorem, modulus of contiunity.

MSC: 41A25, 41A36

Abstract

In this work, we construct a family of optimal fourth order iterative methods requiring three evaluations. During each iterative step, methods need evaluation of two derivatives and one function. According to the Kung and Traub conjecture an optimal iterative method without memory based on $3$ evaluations could achieve an optimal convergence order of $4$. The proposed iterative family of methods are especially appropriate for finding zeros of functions whose derivative is easy to evaluate. For example, polynomial functions and functions defined via integrals.

Keywords: Iterative method; fourth order; Newton method; convergence; nonlinear; optimal; derivative.

MSC: 41H25, 65D99

Abstract

Let $\Cal{Q}_{b}(\Phi ,\Psi ;\alpha )$ be the class of normalized analytic functions defined in the open unit disk and satisfying $$ \RE\left\{ 1+\frac{1}{b}\left( \frac{f(z)\ast \Phi (z)}{f(z)\ast \Psi (z)}-1\right) \right\} >\alpha $$ for nonzero complex number $b$ and for $0\leq \alpha <1$. Sufficient condition, involving coefficient inequalities, for $f(z)$ to be in the class $\Cal{Q}_{b}(\Phi ,\Psi ;\alpha )$ is obtained. Our main result contains some interesting corollaries as special cases.

Keywords: Analytic functions; Starlike and convex functions of complex order; Hadamard product; Coefficient inequalities.

MSC: 30C45