Existence of n symmetric positive solutions for singular second order two-point and four-point boundary value problems
Abstract
In this dissertation, we find positive symmetric solution of singular second order two-point boundary value problem using fixed point theory, monotone iterative technique and then extend this method to find solution for singular second order four point boundary value problem. By symmetric solution we mean a solution w is positive w(t) > 0 and w(t) = w(1 − t), 0 ≤ t ≤ 1. Here we consider the coefficient h(t) is singular at end points t = 0 and t = 1 and w(t) is symmetric and concave whereas w′(t) is skew-symmetric on interval [0, 1]. We consider ordered Banach space having an appropriate norm and cone k of continuous differentiable functions. Then we construct an integral operator t (w) using Green’s function and take its mapping on cone (on k into itself) and use Arzela-Ascoli theorem to show that this operator is completely continuous. This operator is symmetric and concave whereas its derivative is skew-symmetric in [0, 1]. Then by selecting some suitable constants and imposing conditions on norm we construct the monotone iterative technique to obtain the fixed point of the operator and this fixed point is positive symmetric solution of our considered boundary value problem. First we do it for one symmetric positive solution and then extend this technique to find n symmetric positive solutions of second order two-point and four-point boundary value problems.