Abstract
We study the boundary value problems of second-order singular differential equations. At first, we reduce the BVPs to initial value problems of first-order singular integrodifferential equations and then we employ the quasilinearization method in studying the IVPs and obtain two monotone iterative sequences, which converge uniformly and quadratically to the unique solution of the IVPs. Finally, we get the similar result for the given BVPs.
Highlights
It is well known that quasilinearization method is a powerful tool for proving the existence of approximate solutions of nonlinear systems and the converge quadratically to the unique solution of the given problems
We develop the following result which is important for the final result
From (H2.3) we find that fxx(t, X, U) ≥ 0, and f (t, X1, U) ≥ f (t, X2, U) + fx (t, X2, U) (X1 − X2), (8)
Summary
It is well known that quasilinearization method is a powerful tool for proving the existence of approximate solutions of nonlinear systems and the converge quadratically to the unique solution of the given problems (see [1]). By using the existence result [4] for linear singular systems and the comparison result [5],we investigate two monotone iterative sequences which converge uniformly and quadratically to the solution of the problem. In order to obtain two monotone sequences, we introduce an existence result for the corresponding linear singular systems and a comparison result. The existence of the solution of the linear initial value problem of the form
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