Solvability for a nonlinear three-point boundary value problem with \(p\)-Laplacian-like operator at resonance (Q1347470)

Two theorems are presented for the existence of \(C^1\)-solutions to the three-point boundary value problem \[ (\Phi(u'))'= f(t,u,u'),\quad u'(a)= 0,\quad u(\mu)= u(b), \] at resonance, where the operator \((\Phi(u'))'\) is nonlinear, in general. For the particular case, the \(p\)-Laplace operator, the corresponding results are mentioned in the introduction to show the power of the obtained criteria. For the special structure of the right-hand side, namely \(f(t,u,u')= q(t)- g(t,u)\), an existence result is obtained also alternatively via time mappings.