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Basic calculus on time scales and some of its applications. (English) Zbl 0927.39003
The main result of this paper is a unification of the continuous and discrete Taylor’s formulae which is at the same time an extension to the case of so-called time scale. A time scale $T$ is a closed subset of the reals, and for a function $g:T\to ℝ$ it is possible to introduce a derivative ${g}^{{\Delta }}$ and an integral ${\int }_{a}^{b}g\left(\tau \right){\Delta }\tau$ in a certain manner. Basic tools of calculus on time scales such as versions of Taylor’s formula, l’Hôspital’s rule, and Kneser’s theorem are developed. The established Taylor’s formula is applied to obtain some results in interpolation theory. Applications of these results in the study of asymptotic and oscillatory behavior of solutions of higher-order equations on time scales are given $\left({y}^{{\Delta }}=f\left(t,y,{y}^{{\Delta }},\cdots ,{y}^{{{\Delta }}^{n-1}}\right)$, ${y}^{{{\Delta }}^{i}}\left(\alpha \right)={\alpha }_{i}$ for $0\le i\le n-1\right)$.

##### MSC:
 39A10 Additive difference equations 39A11 Stability of difference equations (MSC2000) 26A24 Differentiation of functions of one real variable 28A25 Integration with respect to measures and other set functions