# zbMATH — the first resource for mathematics

Weighted derivative and differential equations. (Russian) Zbl 1054.26006
Let $$\alpha=\alpha(t)$$ be a positive bounded single-valued function on an interval and let $$p\in[0,1]$$. The weighted derivative of a function $$f$$ with respect to the function $$\alpha^p$$ at a point $$t$$ is the function $D_{\alpha,p}f(t)=\lim\limits_{\Delta t\to 0} \frac{\alpha^p(t+\alpha^{1-p}(t)\Delta t)f(t+\alpha^{1-p}(t)\Delta t)- \alpha^p(t)f(t)}{\Delta t}.$ The author establishes some properties of the weighted derivatives and presents a special differential equation associated with these derivatives.
##### MSC:
 26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
##### Keywords:
weighted derivative
Full Text: