On the unique solvability of the Runge-Kutta equations.
The two-point boundary value problem is considered, where , and for . The existence of positive solutions is well-known. Several sufficient conditions have been obtained for the uniqueness of positive solutions. On the other hand, a non-uniqueness example was given by Moore and Nehari in 1959. In this paper, new uniqueness results are presented.
It is proved that nonincreasing and satisfying the Volterra condition right-hand side of a functional differential equation does not guarantee the uniqueness of solutions.
For the differential equation where the vector function has nonintegrable singularities with respect to the first argument, sufficient conditions for existence and uniqueness of the Vallée–Poussin problem are established.
We propose a theoretical framework for solving a class of worst scenario problems. The existence of the worst scenario is proved through the convergence of a sequence of approximate worst scenarios. The main convergence theorem modifies and corrects the relevant results already published in literature. The theoretical framework is applied to a particular problem with an uncertain boundary value problem for a nonlinear ordinary differential equation with an uncertain coefficient.