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In this paper we are considering a third-order three-point equation with nonhomogeneous conditions in the boundary. Using Krasnoselskii's Theorem and Leray-Schauder Alternative we provide existence results of positive solutions for this problem. Nontrivials examples are given and a numerical method is introduced.

Given an algorithm A for solving some mathematical problem based on the iterative solution of simpler subproblems, an Outer Trust-Region (OTR) modification of A is the result of adding a trust-region constraint to each subproblem. The trust-region size is adaptively updated according to the behavior of crucial variables. The new subproblems should not be more complex than the original ones and the convergence properties of the OTR algorithm should be the same as those of Algorithm A. In the present work, the OTR approach is exploited in connection with the "greediness phenomenon" of Nonlinear Programming. Convergence results for an OTR version of an Augmented Lagrangian method for nonconvex constrained optimization are proved and numerical experiments are presented.

Nonlinear programming, Greediness, Augmented Lagrangian method, Regularization,

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ABSTRACT. The focus of this paper is to address some classical results for a class of hypercomplex numbers. More specifically we present an extension of the Square of the Error Theorem and a Bessel inequality for octonions.

A nonlinear boundary value problem related to an equation of Kirchhoff type is considered. The existence of multiple positive solutions is proved through Avery-Peterson Fixed Point Theorem. A numerical method based on Levenberg-Marquadt algorithm combined with a heuristic process is present in order to align numerical and theoretical aspects.