In this paper, we considered the application of Leray-Schauder fixed point theorem with further augmentations by Scheafer’s Lemma for an integrated mode for investigating apriori-bounds of solutions of Duffing’s Equation. The results obtained showed boundedness of the solution with different apriori-bounds pointing to optimal solution which is continuous, closed, bounded and integrable. We concluded that the bounded solution of Duffing’s Equation has a cyclic relationship other qualitative properties of solutions such as stability, periodicity and uniform convergence. This has opened the window for further research, using Abstract Implicit function theorem in Banach spaces.
Keywords: Apriori-Bounds, Cauchy –Schwartz’s Inequality, Duffings Equation, Leray-Schauder Fixed Point, Scheafer’s Lemma
