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Extending the applicability of Newton's method for a class of boundary value problems using the shooting method

Abstract:

We use Newton's method to approximate locally unique solutions for a class of boundary value problems by applying the shooting method. The utilized operator is Fréchet-differentiable between Banach spaces. These conditions are more general than those that appear in previous works. In particular, we show that the old semilocal and local convergence criteria for Newton's method involving Banach space value operators can be replaced by weaker ones. Hence, extending the applicability of the method. Several numerical examples are developed to test the new convergence criteria and also compare them to the old ones.