On Multivariate Fractional Taylor's and Cauchy' Mean Value Theorem

Abstract

In this paper, a generalized multivariate fractional Taylor's and Cauchy's  mean value theorem of the kind
$$f(x,y) = \sum\limits_{j = 0}^n {\frac{{{D^{j\alpha }}f({x_{0,}}{y_0})}}{{\Gamma (j\alpha  + 1)}}}  + R_n^\alpha (\xi,\eta),\qquad\frac{{f(x,y) - \sum\limits_{j = 0}^n {\frac{{{D^{j\alpha }}f({x_{0,}}{y_0})}}{{\Gamma (j\alpha  + 1)}}} }}{{g(x,y) - \sum\limits_{j = 0}^n {\frac{{{D^{j\alpha }}g({x_{0,}}{y_0})}}{{\Gamma (j\alpha  + 1)}}} }} = \frac{{R_n^\alpha (\xi ,\eta )}}{{T_n^\alpha (\xi ,\eta )}},$$

where $0<\alpha \le 1$, is established. Such expression is precisely the classical Taylor's and Cauchy's mean value theorem in the particular case $\alpha=1$. In addition, detailed expressions for $R_n^\alpha (\xi,\eta)$ and $T_n^\alpha (\xi,\eta)$ involving the sequential Caputo fractional derivative are also given.