Existence Conditions For Second Order Neumann Boundary Value Problems Of Functional Differential Equations With Upper And Lower Solutions In The Reverse Order

In this paper, we show that the method of monotone iterative technique is valid to obtain two monotone sequences converges uniformly to extremal solutions of second order functional differential equation and Laplace equation with Neumann boundary value conditions.This paper is mainly concerned with the second order functional differential equation with Neumann boundary value conditions of the formwhere f(t, u, v) : I × R2 → R is a continuous function,τ ∈ C(I, I).To develop a monotone method, we will use the continuation theory of Gaines and Mawhin to prove the existence of solutions for the following two nonlinear Neumann boundary value problems at first, i.e.,The existence of solutions for the problem that we will study is given via anti-maximum principles. Such a comparison principles are fundamental since if the lower and upper solutions are given in the reverse order, the monotone method is not valid in general. So the comparison principles ensure both the existence and the approximation of extremal solutions of problems via the monotone method. Thus,they are the key point in this paper.where f(t, u, v) : I × R3 → R is a continuous function, r G C(I, I).This paper is also concerned with the Laplacian with Neumann boundary value conditions of the form...