A note on operator equations describing the integral (Q2848852)

Motivated by the classical chain rule and the substitution rule for the integral and the derivative, the authors investigate the operator equations NEWLINE\[NEWLINE f\circ g+c = I\bigl((Tf\circ g)\cdot Tg\bigr);\quad V(f\circ g) = (Tf\circ g)\cdot Tg. NEWLINE\]NEWLINE Here \(T,V:\mathcal{C}^1(\mathbb R)\to\mathcal{C}(\mathbb R)\) and \(I:\mathcal{C}(\mathbb R)\to\mathbb R\). The main results of the paper guarantee that, under some simple and reasonable assumptions, the only solutions for \(T\) and \(I\) are the derivative and the integral. The analogue of the Leibniz rule is also investigated.