Solvability of nonlocal multipoint boundary value problems. (Q2783612)

Here, the multipoint value problem \( -u''=f(t, u, u')\), \(0<t<1\), \( u(0)=\sum_{i=1}^{m} a_i u(\xi_i), \) \( u(1)=\sum_{j=1}^{n} b_j u(\eta_j)\) is studied, where \(a_1,\dots, a_m\) and \(b_1,\dots, b_n\) are nonnegative real numbers and \(\xi_1 ,\dots, \xi_m\) and \( \eta_1 ,\dots, \eta_n \) belong to \((0,1)\). This work is an extention of the above reviewed work, where an analogous problem has been considered with \(f(t,u)\) not depending on \(u'\). The authors apply the method of upper and lower solutions to prove the existence of at least one solution of the considered problem under the Nagumo-Wintner condition for \(f(t,x,y)\).