We suggest and study iteration procedures converging from below and above to robust stable solutions and to robust stable continuous branches of solutions for quasilinear boundary-value problems with continuous non-monotone non-linearities. The iterations are constructed by modifications of the shuttle iteration method, which is used in problems with monotone operators leaving invariant a cone interval. 1) The general theory of Banach spaces with cone semiordering can be found in [4, 11] . We

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... ly use some notation and simple facts. 2) This set is called a cone interval. 3) We consider perturbations of the non-linearity only. Also, it is possible to consider perturbations both of the linear and nonlinear parts of the problem.