On the solvability of a second-order singular initial value problem

Abstract

Using the Topological transversality theorem and the barrier strips technique, we study the solvability of the initial value problem x″ = f(t,x,x′),x(0) = A,x′(0) = B, where the scalar function f(t,x,p) may be unbounded as p→B. Obtained results guarantee the existence of at least one solution in C1[0,T]∩C2(0,T]. Under additional assumptions it is monotone and does not change its sign.