The author considers a class of nonlocal convolution-type ordinary differential equations the form − A ∗ ( g ∘ u ) 1 ′ t = λ f , 0 < $$\begin{equation*} \quad -A{\left({\left(a*(g\,\circ\, u)\right)}(1)\right)}u^{\prime \prime }(t)=\lambda f{\left(t,u(t)\right)}, 1,\nonumber \end{equation*}$$ where $g$ is continuous function satisfying p q $(p,q)$ growth — that is, there exist constants c $c_1$...

This note gives a method of presenting axiom systems for various induced theories like the natural numbers theory of the real numbers theory and the pure set theory (with no constants except G and = and with set variables only) induced by the Zermelo-Fraenkel set theory with the axion of foundation, an operation symbol cr(x), an axiom x^0Z)<r(x)Gx and all the instances of the axiom schemata con...