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.
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