Geodesic
A Note on Operator Equations Describing the Integral
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 9 (2013) no. 1, pp. 51-58 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study operator equations generalizing the chain rule and the substitution rule for the integral and the derivative of the type \begin{equation} f\circ g + c = I\ (Tf\circ g\cdot Tg), \quad f,g\in C^1(\mathbb{R}),\tag{1} \end{equation} where $T\!: C^1(\mathbb{R})\to C(\mathbb{R})$ and where $I$ is defined on $C(\mathbb{R})$. We consider suitable conditions on $I$ and $T$ such that (1) is well-defined and, after reformulating (1) as \begin{equation} V(f\circ g)=Tf\circ g\cdot Tg, \quad f,g\in C^1(\mathbb{R})\tag{2} \end{equation} with $V\!: C^1(\mathbb{R})\to C(\mathbb{R})$, give the general form of $T$, $V$ and $I$. Simple initial conditions then guarantee that the derivative and the integral are the only solutions for $T$ and $I$. We also consider an analogue of the Leibniz rule and study surjectivity properties there. 
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     title = {A {Note} on {Operator} {Equations} {Describing} the {Integral}},
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H. König; V. Milman. A Note on Operator Equations Describing the Integral. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 9 (2013) no. 1, pp. 51-58. http://geodesic.mathdoc.fr/item/JMAG_2013_9_1_a3/

[AKM] S. Artstein-Avidan, H. König, V. Milman, “The Chain Rule as a Functional Equation”, J. Funct. Anal., 259 (2010), 2999–3024 | DOI | MR | Zbl

[H] Ph. Hartman, Ordinary Differential Equations, $2^{nd}$ ed., Birkhäuser, 1982 | MR | Zbl

[KM1] H. König, V. Milman, “Characterizing the Derivative and the Entropy Function by the Leibniz Rule”, with an Appendix by D. Faifman, J. Funct. Anal., 261 (2011), 1325–1344 | DOI | MR | Zbl

[KM2] H. König, V. Milman, “Rigidity and stability of the Leibniz and the chain rule”, Proc. Steklov Inst. Math., 280, 2013 (to appear) | DOI | Zbl