A nonstandard combinatorial technique for integral equations (Q1902271)

The authors discuss the nonstandard combinatorial technique for integral equations: For \({}^*f, {}^*g\in \text{Map} (c, {}^*k)\), the functions from \(c\) into \({}^*k\), one has \[ {}^*g (x)= \sum_{0\leq y\leq x} {}^*f (y) {}^*\mu_1 (y, x) \iff {}^* f(x)= \sum_{0\leq y\leq x} {}^*g (y) {}^* \mu_2 (y, x) \qquad (x\in c), \] where \({}^*\mu_1\), \({}^*\mu_2\) are \({}^*\)-Möbius operators. This result may be used to solve the Volterra equation of the second kind (1) \(f(x)= u(x)- \rho \int_0^x k(y, x) u(y) dy\) by using a quasi-Duhamel principle. The authors give the following solution of (1): \(u(x)= f(x)+ \rho \int_0^x \Gamma (y, x, \rho) f(y) dy\).