Using the $q$-Bernstein basis, we construct a new sequence $\{ L_{n} \}$ of positive linear operators in $C[0,1].$ We study its approximation properties and the rate of convergence in terms of modulus of continuity.
In this paper we give some new results concerning solvability of the 1-dimensional differential equation $y^{\prime } = f(x,y)$ with initial conditions. We study the basic theorem due to Picard. First we prove that the existence and uniqueness result remains true if $f$ is a Lipschitz function with respect to the first argument. In the second part we give a contractive method for the proof of Picard theorem. These considerations allow us to develop two new methods for finding an approximation sequence for the solution. Finally, some applications are given.
It is not the purpose of this paper to construct approximations but to establish a class of almost periodic functions which can be approximated, with an arbitrarily prescribed accuracy, by continuous periodic functions uniformly on ${\mathbb R} = (-\infty ; +\infty )$.
We introduce modified (p, q)-Bernstein-Durrmeyer operators. We discuss approximation properties for these operators based on Korovkin type approximation theorem and compute the order of convergence using usual modulus of continuity. We also study the local approximation property of the sequence of positive linear operators D∗ n,p,q and compute the rate of convergence for the function f belonging to the class LipM(γ).
In the paper, we discuss convergence properties and Voronovskaja type theorem for bivariate $q$-Bernstein polynomials for a function analytic in the polydisc $D_{R_{1}}\times D_{R_{2}}=\{z\in C\colon \vert z\vert <R_{1}\} \times \{ z\in C\colon \vert z\vert <R_{1}\}$ for arbitrary fixed $q>1$. We give quantitative Voronovskaja type estimates for the bivariate $q$-Bernstein polynomials for $q>1$. In the univariate case the similar results were obtained by S. Ostrovska: $q$-Bernstein polynomials and their iterates. J. Approximation Theory 123 (2003), 232–255. and S. G. Gal: Approximation by Complex Bernstein and Convolution Type Operators. Series on Concrete and Applicable Mathematics 8. World Scientific, New York, 2009.
The paper deals with a class of discrete-time stochastic control processes under a discounted optimality criterion with random discount rate, and possibly unbounded costs. The state process { x_{t} } and the discount process { \alpha_{t} } evolve according to the coupled difference equations x_{t+1}=F(x_{t},\alpha _{t},a_{t},\xi _{t}), \alpha_{t+1}=G(\alpha _{t},\eta _{t}) where the state and discount disturbance processes { \xi _{t} } and { \eta _{t} } are sequences of i.i.d. random variables with densities \rho^{\xi } and \rho^{\eta } respectively. The main objective is to introduce approximation algorithms of the optimal cost function that lead up to construction of optimal or nearly optimal policies in the cases when the densities \rho ^{\xi } and \rho ^{\eta } are either known or unknown. In the latter case, we combine suitable estimation methods with control procedures to construct an asymptotically discounted optimal policy.
In this paper, we show two applications of the Banach's Fixed-Point Theorem: first, to approximate the ultimate ruin probability in the classical risk model or Cramér-Lundberg model when claim sizes have some arbitrary continuous distribution and second, we propose an algorithm based in this theorem and some conditions to guarantee the continuity of the ruin probability with respect to the weak metric (Kantorovich). In risk theory literature, there is no methodology based in the Banach's Fixed-Point Theorem to calculate the ruin probability. Numerical results in this paper, guarantee a good approximation to the analytic solution of the ruin probability problem. Finally, we present numerical examples when claim sizes have distribution light and heavy-tailed.
In the present paper, using a Picard type method of approximation, we investigate the global existence of mild solutions for a class of Ito type stochastic differential equations whose coefficients satisfy conditions more general than the Lipschitz and linear growth ones.
The aim of this paper is to present some ideas how to relax the notion of the optimal solution of the stochastic optimization problem. In the deterministic case, ε-minimal solutions and level-minimal solutions are considered as desired relaxations. We call them approximative solutions and we introduce some possibilities how to combine them with randomness. Relations among random versions of approximative solutions and their consistency are presented in this paper. No measurability is assumed, therefore, treatment convenient for nonmeasurable objects is employed.
An invasive sawfly Aproceros leucopoda Takeuchi, 1939, which originates from East Asia, has colonized elms (Ulmus spp.) in Austria, Hungary, Poland, Romania, Slovakia and the Ukraine, at least since 2003. In Europe, the larvae can completely defoliate native and non-native elm trees and may cause at least partial dieback. Field observations indicate that elms are infested independent of their age and site characteristics. The life cycle of A. leucopoda is described based on material reared in Hokkaido, Japan. Parthenogenetic reproduction, short life cycle of summer generations and the ability to produce four generations per year result in the production of numerous progeny. The evolution of a seasonal dimorphism in head morphology, a simple cocoon that is attached directly to the host plant and a short period spent in the cocoon stage during summer, are putative apomorphies shared by Aproceros Takeuchi, 1939 and Aprosthema Konow, 1899. These traits reduce developmental costs and contribute to the proliferation of A. leucopoda. No specialized parasitoid, that can effectively reduce outbreaks of this species, is known. It is likely that this pest will spread into central and south-western Europe. Further monitoring of A. leucopoda is required to assess future range extensions in Europe, its exacerbating effect on Dutch elm disease and to find a suitable biocontrol agent. Concise keys to imaginal and larval stages are presented that will facilitate the identification of A. leucopoda.