We prove that the system $\dot{\theta}(t) =\Lambda(t)\theta(t)$
 , $t\in\mathbb{R}_{+}$
 , is Hyers-Ulam stable if and only if it is uniformly exponentially stable under certain conditions; we take the exact solutions of the Cauchy problem $\dot{\phi}(t)=\Lambda(t)\phi(t)+e^{i\gamma t}\xi(t)$
 , $t\in\mathbb{R}_{+}$
 , $\phi(0)=\theta_{0}$
 as the approximate solutions of $\dot{\theta}(t)=\Lambda(t)\theta(t)$
 , where γ is any real number, ξ is a 2-periodic, continuous, and bounded vectorial function with $\xi(0)=0$
 , and $\Lambda(t)$
 is a 2-periodic square matrix of order l.

/pdf/connections-between-hyers-ulam-stability-and-uniform-1bz8o8i76b.pdf

Connections between Hyers-Ulam stability and uniform exponential stability of 2-periodic linear nonautonomous systems

A new class of rational cubic spline fractal interpolation function and its constrained aspects

This paper investigates some univariate and bivariate constrained interpolation problems using fractal interpolation functions. First, we obtain rational cubic fractal interpolation functions lying above a prescribed straight line. Using a transfinite interpolation via blending functions, we extend the properties of the univariate rational cubic fractal interpolation function to generate surfaces that lie above a plane. In particular, the constrained bivariate interpolation discussed herein includes a method to construct fractal interpolation surfaces that preserve positivity inherent in a prescribed data set. Uniform convergence of the bivariate fractal interpolant to the original function which generates the data is proven.

Bicubic partially blended rational fractal surface for a constrained interpolation problem

The hidden variable fractal approximation is more diverse than classical fractal approximation. The main goal of the article is to establish new kind of hidden variable fractal approximants which possess convergence and non-differentiability simultaneously for any choice of the scaling factors. Without imposing any condition on the scaling vector, we establish the constrained approximation by the proposed hidden variable fractal approximants. By imposing suitable conditions on the scaling factors, we study the calculus of proposed hidden variable fractal approximants. We identify the suitable conditions for the parameters of hidden variable iterated function system so that the proposed hidden variable fractal functions preserve fundamental shape properties such as monotonicity and convexity in addition to the smoothness of the original function in the given compact interval.

Approximation by Hidden Variable Fractal Functions: A Sequential Approach

Fractal interpolation has become popular in 2D and 3D data visualization to model a wide variety of physical phenomena. The purpose of this paper is to present a promising new class of positivity preserving $$\mathcal {C}^1$$
 -rational cubic spline fractal interpolation surfaces (FISs) to stitch surface data arranged over a rectangular grid. We develop the rational cubic FIS as a combination of blending functions and the rational fractal boundary curves that are constructed along the grid lines of the interpolation domain. We investigate the stability results of the rational cubic FIS with respect to its independent and dependent variables, and the corresponding first order partial derivatives at knots. We identify the scaling factors and shape parameters so that the positivity feature of given data along the grid lines is translated to the corresponding rational fractal boundary curves. In particular, our rational cubic FIS is positive whenever the corresponding fractal boundary curves are positive. Numerical examples are provided to illustrate: (i) the construction of positive rational cubic FISs for a given positive surface data, (ii) the effects on the shape of the positive rational cubic FIS due to the changes in the scaling factors or/and shape parameters, and (iii) the stability results of the proposed rational cubic FIS.

Positivity and Stability of Rational Cubic Fractal Interpolation Surfaces

In this paper we consider the (inverse) problem of determining the iterated function system (IFS) which produces a shaped fractal interpolant. We develop a new type of rational IFS by using functions of the form , where are cubics and are preassigned quadratics having 3-shape parameters. The fixed point of the developed rational cubic IFS is in , but its derivative varies from a piecewise differentiable function to a continuous nowhere differentiable function. An upper bound of the uniform error between the fixed point of a rational IFS and an original function is deduced for the convergence results. The automatic generations of the scaling factors and shape parameters in the rational IFS are formulated so that its fixed point preserves the positive/monotonic features of prescribed data. The presence of scaling factors provides additional freedom to the shape of the fractal interpolant over its classical counterpart in the modeling of discrete data.

/pdf/rational-iterated-function-system-for-positive-monotonic-48jlch6h9q.pdf

Akb Chand

Papers

Rational iterated function system for positive/monotonic shape preservation