Weighted derivative and differential equations. (Russian) Zbl 1054.26006

Bichegkuev, M. S.

Vladikavkaz. Mat. Zh. 5, No. 4, 32-42 (2003).

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.

Reviewer: N. A. Kudryavtseva (Novosibirsk)

MSC:

26A24

Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems

Keywords:

weighted derivative