Quasilinearization of Resonant Boundary-value Problems with Mixed Boundary Conditions

We consider resonant problems of the form (i) x″ + p(t)x′ + q(t)x = f(t, x, x′), (ii) x′(0) = 0, x(T) = 0, where p, q, and f are continuous functions, and a homogeneous problem (iii) x″ + p(t)x′ + q(t)x = 0 with the boundary conditions (ii), which has a nontrivial solution. The problem is studied by modifying the linear part and applying the procedure of quasilinearization to the modified problem. © 2015, Springer Science+Business Media New York.