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

#### DOI NO:

https://doi.org/10.26782/jmcms.2018.12.00017

#### Keywords:

Arbitrary two-pointboundary conditions,Nonlinear fourthorder ordinary differential equation,Unique symmetric positive solutions,Fixed point theorem,

#### Abstract

In this paper, we explore the existence and uniqueness of positive solutions for the following nonlinear fourth order ordinary differential equation        (4) u t  f t,u t , t 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 Ca,b0,,0,.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.

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Author(s): Md. Asaduzzaman, Md. Zulfikar Ali