Existence theorems of solutions for two-point boundary value problem of second order ordinary differential equations in Banach spaces (Q1914581)

The authors establish certain existence and uniqueness results for the Dirichlet problem \((*)\) \(u''= f(t, u, u)\), \(u(0)= u(1)= 0\), \(t\in [0, T]\), \(u\in E\), where \(E\) is a weakly sequentially complete Banach space with normal cone. Assuming the existence of two comparison functions \(v_0\leq w_0\) and some monotonicity conditions on the functions like \(f(t, x, y)+ Mx- Ly\), the authors find an iterative sequence \(u_n(t)\) uniformly converging to a unique solution \(u^*(t)\) of \((*)\) from the order interval \([v_0, w_0]\).