A nonlocal boundary value problem with singularities in phase variables

The singular differential equation (g(χ′))′ = ƒ(t, χ, χ′) together with the nonlocal boundary conditions χ(0) = χ(T) = −, γ min{χ(t) : t ∈ [0,T]} is considered. Here g ∈ C0(ℝ) is an increasing and odd function, positive ƒ satisfying the local Carathéodory conditions on [0, T] × (ℝ s0})su2 may be singular at the value 0 in all its phase variables and γ ∈ (o, ∞). The existence result for the above boundary value problem is proved by the regularization and sequential techniques. Proofs use the topological transversality principle and the Vitaliʹs convergent theorem.