Multipoint boundary value problems for a second-order ordinary differential equation (Q1304689)

The existence of a solution to the differential equation \[ (\varphi(u'))'= f(t,u,u'),\quad 0< t< 1,\tag{\(*\)} \] subject to one of the following nonlinear boundary conditions \[ g(u(0), u(\alpha_1),\dots, u(\alpha_n))= A,\;h(u(1), u(\beta_1),\dots, u(\beta_n))= B,\tag{i} \] \[ u'(0)= A,\;h(u(1),u(\beta_1),\dots, u(\beta_n))= B,\;\alpha_i, \beta_j\in (0,1),\quad 1\leq i\leq m,\quad 1\leq j\leq n,\tag{ii} \] \[ k(u(0), u(1))= A,\quad \ell(u'(0), u'(1))= B,\tag{iii} \] is established under the assumption that the boundary value problems admit upper and lower solutions, where \(\varphi: \mathbb{R}\to \mathbb{R}\) is continuous, strictly increasing, onto and the right-hand side being a Carathéodory function. The conditions imposed on the right-hand side of \((*)\) are in the form of growth restrictions or some sign conditions on \(f\) about the origin or the Nagumo-type conditions.