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Functional Analysis, Fixed Points Theory and Iterative Schemes are key areas of research in Mathematics today. This work introduces the readers to introductory part of functional analysis, fixed points theory and some iterative schemes and applications in solving differential equations. It is interesting to see how the iterative schemes work in obtaining solutions to initial value problems. Several maps of interest are explained and their relationship given concrete examples to illustrate the idea. Much attention is given to a special class of problems in non-linear functional analysis namely: iterative approximation of k-strictly pseudo-contractive maps in Hilbert spaces using Modified Picard Iteration.