Ordinary differential equations on closed subsets of locally convex spaces with applications to fixed point theorems

AbstractThe construction and convergence of an approximate solution to the initial value problem x′ = f(t, x), x(0) = x0, defined on closed subsets of locally convex spaces are given. Sufficient conditions that guarantee the existence of an approximate solution are analyzed in relation to the Nagumo boundary condition used in the Banach space case. It is also shown that the Nagumo boundary condition does not guarantee the existence of an approximate solution. Applications to fixed point theorems for weakly inward mappings are given