On the theory of the continual integro-differentiation operator (Q1776258)

The author considers the operator \[ D_{0x}^{[\alpha, \beta]}u(x)\equiv \int_{\alpha}^{\beta}D_{0x}^tu(x)\, dt, \tag{1} \] known as continual integro-differentiation operator, where \[ D_{0x}^t g(x)=\begin{cases} \frac{1}{\Gamma(-t)}\int_0^x g(\xi)(x-\xi)^{-t-1}\, d\xi\quad &\text{if } t<0\\ g(x)&\text{if } t=0\\ \frac{d^n}{dx^n}D_{0x}^{t-n}g(x) &\text{if } n-1<t\leq n, n\in\mathbb N, \end{cases} \] and extends earlier results by \textit{A. M. Nakhushev} [Sov. Math., Dokl. 37, No.~3, 729--732 (1988); translation from Dokl. Akad. Nauk SSSR 300, No.~4, 796--799 (1988; Zbl 0684.34015)] who obtained the solvability of \[ D_{0x}^{[\alpha, \beta]}u(x)=v(x) \tag{2} \] in which \(\beta\geq0\). This paper offers the constructions of the inverse of the operator (1), the solvability of (2) for \(\beta \leq 0\), as well as of its Cauchy problem for \(\beta >0\). Analogs of the Newton-Leibniz formula for (1) are also dervied.