Existence and uniqueness results for some nonlinear boundary value problems (Q1916764)

The authors are interested in the existence and uniqueness of solutions to the equation (1) \((\varphi (u'))' + k(t) \varphi (u') + f(t,u,u') = 0\), \(a < t < b\), subject to one of the following sets of boundary conditions: (2) \(u(a) = u(b) = 0\) (Dirichlet), (3) \(u'(a) = u'(b) = 0\) (Neumann), (4) \(u(a) = u(b)\) and \(u'(a) = u'(b)\) (periodic). Here \(\varphi : \mathbb{R} \to \mathbb{R}\) is an odd increasing homeomorphism, \(k : [a,b] \to \mathbb{R}\), and \(f : [a,b] \times \mathbb{R}^2 \to \mathbb{R}\). The authors establish existence results for (1) with each of the above boundary conditions, (2), (3), and (4) under various growth conditions on \(f\).