A system of Abel type integral equations (Q6614518)

The following system of Abel type integral equations is considered:\N\[\N\left\{ \begin{array}{ll} u(x)=\int\limits_{0}^{x}(x-s)^{\alpha-1}h(v(s))ds, & \\\Nv(x)=\int\limits_{0}^{x}(x-s)^{\beta-1}g(u(s))ds, & \end{array} \right.\N\]\Nwhere \(\alpha,\beta>0\), \(g(u)\) is nondecreasing and continuous for \(u\geq 0\), \(g(0)=0\), \(h(v)\) is nondecreasing, continuous and concave for \(v\geq 0\), \(h(0)=0\).\N\NThe existence of nontrivial solutions and their properties are discussed, focusing on its explosive behavior. The author shows that the existence of a nontrivial solution of this system that is governed by a generalized Osgood condition.