EXISTENCE, STABILITY AND BOUNDEDNESS OF PERIODIC SOLUTIONS FOR CERTAIN NONLINEAR BOUNDARY VALUE PROBLEM OF FOURTH ORDER ORDINARY DIFFERENTIAL EQUATIONS

In this paper, some new conditions on the existence, stability and boundedness of periodic solutions for certain nonlinear boundary value problems were investigated. Construction of a complete Lyapunov function for higher order differential equations were not all that easy and therefore a new method for investigating and proving the existence, stability and boundedness was considered. The method of Leray-Schauder fixed point theorem provided existence of periodic solutions which depended on the availability of suitable boundedness results. In some cases, boundedness results were very difficult to establish due to the nature of the Lyapunov function involved and the method of the integrated equation was used as a mode for estimating apriori bounds for the fourth order differential equation. The aim of using integrated equation was to amelioriate the technical problems arising from the construction of Lyapunov function for higher order differential equations which was considered to be cumbersome and complex. However, our results generalize and complement some existing results in literature.