Monotone Iterative Technique for a Class of Four Point BVPs with Reversed Ordered Upper and Lower Solutions

Consider the class of four point nonlinear BVPs -w ''(x) = f(x, w, w'), x is an element of I, w'(0) = 0, w(1) = delta(1)w(eta(1)) + delta(2)w(eta(2)), where f is an element of (I x R x R, R) is continuous, I = [0, 1], eta(1), eta(2) is an element of (0, 1) such that eta(1 )<= eta(2) and delta(1),delta(2) >= 0. In this paper, we demonstrate an iterative technique. The iterative scheme is deduced by using quasilinearization. Then we consider upper-lower solutions in well ordered and reverse ordered cases and prove existence of solutions under some sufficient conditions. We show that under certain conditions, generated sequences are monotone, uniformly convergent and converges to the solution of the above problem. We also provide examples which validate that all the conditions derived in this paper, are realistic and can be satisfied. We have also plotted upper and lower solutions for the test examples and have shown that under the conditions, the derived upper and lower solutions are monotonic in nature.