The unique symmetric positive solutions for nonlinear fourth order arbitrary two-point boundary value problems: A fixed point theory approach

In this paper, we explore the existence and uniqueness of positive solutions for the following nonlinear fourth order ordinary differential equation u((4))(t) = f(t, u(t)), t is an element of[a, b], withthe following arbitrary two-point boundary conditions: u(a) = u(b) = u'(a) = u'(b) = 0, where, a, b are two arbitrary constants satisfying b > 0, a = 1 - b and f is an element of C([a, b]x [0, infinity, [0, infinity)). Here we also demonstrate that under certain assumptions the above boundary value problem exist a unique symmetric positive solution. The analysis of this paper is based on a fixed point theorem in partially ordered metric spaces due to Amini-Harandi and Emami. The results of this paper generalize the results of several authors in literature. Finally, we provide some illustrative examples to support our analytic proof.