This analysis demonstrates rapid convergence of monotone sequences to unique solutions in initial value problems, suggesting a reliable approach for solving differential equations.
We apply the generalized quasilinearization method of V. Lakshmikantham (Nonlinear Anal. 27 (1996), 223-227) and the extended method of R. N. Mohapatra, K. Vajravelu, and Y. Yin (J. Optim. Theory Appl., in press) to the first order initial value problem (IVP for short) x'(t) = f(f, x), x(0) = x(0), t is an element of J with wider class functions f and obtain rapid convergence of monotone sequences which converge uniformly to the unique solution of the IVP.
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A 1997 study studied this question.
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