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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
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