## Artigo

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## versão On-line ISSN 0719-0646

### Cubo vol.17 no.3 Temuco  2015

#### http://dx.doi.org/10.4067/S0719-06462015000300001

Right General Fractional Monotone Approximation

George A. Anastassiou

Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, U.S.A., ganastss@memphis.edu

ABSTRACT

Here is introduced a right general fractional derivative Caputo style with respect to a base absolutely continuous strictly increasing function g. We give various examples of such right fractional derivatives for different g. Let f be p-times continuously differentiable function on [a, b], and let L be a linear right general fractional differential operator such that L (f) is non-negative over a critical closed subinterval J of [a, b]. We can find a sequence of polynomials Qn of degree less-equal n such that L (Qn) is non-negative over J, furthermore f is approximated uniformly by Qn over [a, b] . The degree of this constrained approximation is given by an inequality using the first modulus of continuity of f(p). We finish we applications of the main right fractional monotone approximation theorem for different g.

Keywords and Phrases: Right Fractional Monotone Approximation, general right fractional derivative, linear general right fractional differential operator, modulus of continuity.
2010 AMS Mathematics Subject Classification: 26A33, 41A10, 41A17, 41A25, 41A29.

RESUMEN

Aquí introducimos una derivada fraccional derecha general al estilo de Caputo con respecto a una base de funciones absolutamente continuas estrictamente crecientes g. Damos varios ejemplos de dichas derivadas fraccionales derechas para diferentes g. Sea f una función p-veces continuamente diferenciable en [a, b], y sea L un operador diferencial fraccional derecho general tal que L(f) es no-negativo en un subintervalo cerrado crítico J de [a, b]. Podemos encontrar una sucesión de polinomios L (Qn) de grado menor o igual a n tal que L (Qn) es no-negativo en J, más aún f es aproximada uniformemente por Qn en [a, b] . El grado de esta aproximación restringida es dada por una desigualdad usando el primer módulo de continuidad de f(p). Concluimos con aplicaciones del teorema principal de aproximación monótona fraccional derecha para diferentes g.

References

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Received: April 2015. Accepted: July 2015.

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