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A paradigm on the energy efficiency (EE) and spectral efficiency (SE) tradeoff named resource efficiency (RE) exploits the EE‐SE equilibrium, which is formulated as Pareto optimal set based on two objectives balancing power consumption against occupied bandwidth. In optical code division multiple access (OCDMA) networks, the EE‐SE tradeoff is discussed by formulating a multiobjective optimization problem, while solutions are compared by deploying the weighted sum (WS) and ‐Constraint (‐C) scalarization procedures combined with nonlinear programming methods. The generalized RE optimization problem with QoS guarantees for the OCDMA networks is formulated considering that such methods result in the same Pareto boundaries. Theoretical results demonstrate that the obtained numerical optimized solutions are efficient due to the multiobjective characteristics of the optimization problem. Also, considering the numerical tests performed, it is noteworthy that the ‐C procedure found a greater diversity of solutions inside the Pareto front when compared with the WS method.

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.

We consider in this work the fourth order equation with nonlinear boundary conditions. We present the result for the existence of multiple solutions based on the Avery-Peterson fixed-point theorem. This work is also a study for numerical solutions based on the Levenberg-Maquardt method with a heuristic strategy for initial points that proposes to numerically determine multiple solutions to the problem addressed. (2010): 34-XX, 34-BXX, 34-B15. Mathematics subject classification

This work presents conditions for the existence of multiple solutions for a sixth order equation with homogeneous boundary conditions using Avery Peterson's theorem. In addition, non-trivial examples are presented and a new numerical method based on the Banach's Contraction Principle is introduced.

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