Existence of solutions for a class of nonlinear Neumann boundary value problems in the presence of upper and lower solutions

In this article, we explore the monotone iterative technique (MI-technique) to study the existence of solutions for a class of nonlinear Neumann four-point, boundary value problems (BVPs) defined as, -phi((2))(z) = G(z, phi, phi((1))), 0 < z < 1, phi((1))(0) = lambda phi((1))(beta(1)), phi((1))(1) = delta phi((1))(beta(2)), where 0 < beta(1) <= beta(2) < 1 and lambda, delta is an element of (0,1). The nonlinear term G(z, phi, phi((1)())) : Omega -> R, where = [0, 1] x R-2, is Lipschitz in phi((1))(z) and one-sided Lipschitz in phi(z). Using lower solution l(z) and upper solutions u(z), we develop MI-technique, which is based on quasilinearization. To construct the sequences of upper and lower solutions which are monotone, we prove maximum principle as well as anti-maximum principle. Then under certain assumptions, we prove that these sequences converge uniformly to the solution phi(z) in the specific region, where (partial derivative G/partial derivative phi) < 0 or (partial derivative G/partial derivative phi) > 0. To demonstrate that the proposed technique is effective, we compute the solution of the nonlinear multi-point BVPs. We don't require sign restriction on G which is very common and strict condition.