Abstract
We present a local convergence of two-step solvers for solving nonlinear operator equations under the generalized Lipschitz conditions for the first- and second-order derivatives and for the first order divided differences. In contrast to earlier works, we use our new idea of center average Lipschitz conditions, through which, we define a subset of the original domain that also contains the iterates. Then, the remaining average Lipschitz conditions are at least as tight as the corresponding ones in earlier works. This way, we obtain weaker sufficient convergence criteria, larger radius of convergence, tighter error estimates, and better information on the solution. These extensions require the same effort, since the new Lipschitz functions are special cases of the ones in earlier works. Finally, we give a numerical example that confirms the theoretical results, and compares favorably to the results from previous works.
Highlights
IntroductionLet X, Y stand for Banach spaces, D ⊂ X denote an open and convex set. where F : D → Y, G : D → Y ; F is a differentiable operator in the sense of Fréchet, and G is a continuous operator
Yk+1 = xk+1 − [ H]−1 H, k = 0, 1, 2, . . . , where H is the first-order divided difference, which can be applied for solving nonlinear equations with nondifferentiable operator (Equation (1))
I.e., if the inequalities in (20) or (21) or (22) are strict, the following advantages are available: larger convergence ball, tighter estimates on k xk − x∗ k, and better information regarding the location of the solution since all new constants and functions are more precise
Summary
Let X, Y stand for Banach spaces, D ⊂ X denote an open and convex set. where F : D → Y, G : D → Y ; F is a differentiable operator in the sense of Fréchet, and G is a continuous operator. We like to bring to the attention of the motivated reader the works in References [23,24,25], which constitute subdivision-based solvers seeking all roots in the domain of interest using Newton’s method provided the domain is sufficiently small. These solvers can be used to include a non-differentiable part. In References [26,27], the continuation method has been used as another powerful tool for finding roots These methods involve systems of equations on Ri and cannot provide information about solutions of Equation (1) in a Banach space setting.
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