Solvability of the \(\varPhi \)-Laplacian with nonlocal boundary conditions (Q732382)

Sufficient conditions are obtained for the existence of at least one positive solution to a nonlinear boundary value problem with nonlocal linear boundary conditions \[ [\Phi(x'(t))]'+ c(t)(Fx)(t)= 0,\qquad t\in (0,1), \] \[ x(0)- L_0(x)= x(1)- L_1(x)= 0, \] where \(\Phi\) is an increasing homeomorphism of the real lline onto itself, \(F\) is a nonlinear operator, \(L_0\), \(L_1\) are bounded linear positive operators. The proof is based on the fixed-point theorem in cones by Krasnoselskii. Some examples are given.