Existence and Uniqueness Analysis for a Class of Singular Non-Linear Two-Point Boundary Value Problems by an Optimal Iterative Sequence

In this paper, a constructive proof is presented to study the existence and uniqueness of the solutions to the following singular problem y ′′(x) + ny ′(x) + m x y ′(x) = f(x; y(x)); 0 < x  1; with the boundary conditions y ′(0) = 0; ay(1) + by ′ (1) = c: it is assumed, in general, f (x; y(x)) be non-singular with respect to the independent variable x but it is allowed to be singular with respect to y. we apply the picard iterative sequence by constructing integral equation whose green's function is not negative. the convergence of this iterative sequence is then controlled by an embedded parameter. the fastest convergence occurs for an optimal embedded parameter which maximizes a special function. this optimization problem brings a sequence withhigh rate of the convergence to the unique solution in the finite region where @f@y has to be positive. some illustrative examples are given to conrm the validity and reliability of this constructive theory.