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Existence of periodic solutions for higher order ordinary differential equations by the use of Leray- Schauder fixed-point technique depended mainly on the availability of suitable boundedness results. In some cases however, boundedness results were very difficult to establish because of the nature of the Lyapunov functions involved. In this thesis, existence results have been obtained using the Leray- Schauder fixed point technique and the integrated equations for the more general equations of the form: ... x + f (x]) + a1x? + a2x = p (t) x(4) + g ( ... x) + b1x] + b2x? + b3x = p (t) x(5) +...

The text foundation of differential equations has the basic rudiments needed in the studying differential equations. Partial fractions, review of fundamental calculus, Laplace transform and its inverse, etc. have been treated in details in the preliminary chapter. In the main body many examples have been done with illustrations. Also exercises have been included to enable the reader access himself/herself and he/she progresses