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In this paper, based on the centrosymmetric and skew-centrosymmetric split-ting (CSS) and circulant and skew-circulant splitting (CSCS) of the Toeplitz coefficient matrix, combined fixed-point iteration, we investigate the behavior of the splitting combined iterative methods for Toeplitz weakly nonlinear equations.The nonlinearity exists widely in the natural science, engineering and eco-nomics, and so on. The nonlinear science is an important research direction in the development of science. In many cases, The nonlinear differential equations used to describe the phenomenon is weakly nonlinear equations, and the non-linear algebraic equations generated by numerical discrete is weakly nonlinear equations, that is, the linear terms is strong dominant than the nonlinear terms in some norms.Generally, we can not solve the solution of the nonlinear equations exactly and often search its approximate solution by iterative method. Many iterative methods are given for several centuries, but after the test of practice, a number of shortcomings of classical iterative methods began to be discovered. In addition, the method for solving general nonlinear equations can be used to solve the weakly nonlinear equations with the special structure, but it is usually not the most effective.Therefore, for Toeplitz weakly nonlinear systems, by using the separability and strongly dominance between the linear and the nonlinear terms and using the centrosymmetric and skew-centrosymmetric splitting (CSS) iteration tech-nique, we establish three nonlinear composite iteration schemes, called Newton-CSS, Picard-CSS and nonlinear CSS-like iteration methods, and by using the circulant and skew-circulant splitting (CSCS) iteration technique, we establish three nonlinear composite iteration schemes, called Newton-CSCS, Picard-CSCS and nonlinear CSCS-like iteration methods, respectively. Theoretical analysis show that Newton-CSS and Newton-CSCS iteration methods are not always fea-sible and the other four iteration methods are local convergence under suitable conditions, and the convergence rate depends only on the spectral radius of the iteration matrix. Numerical results are provided, further show that Picard-CSS, nonlinear CSS-like, Picard-CSCS and nonlinear CSCS-like iteration methods are all feasible and very effective.
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