Advances in Analysis

5

Authors: Wensheng Yang

An optimal control problem is studied for a predator-prey model with a general monotonic and non-monotonic functional response for prey. The control functions represent the rate of mixture of the populations and the cost functional is of Mayer type. The goal of this paper is to maximize the total density of the two populations at a fixed time moment. The number of switching points of the optimal control is determined by the threshold value c, which is the rate of translating the prey population into the predator population. It is shown that general monotonic or non-monotonic functional response has no influence on the number of switching points of the optimal control.

9

Authors: Valery Volchkov, Vitaly Volchkov

A non-Euclidean analog of the generalized Darboux equation is considered. For the case where its solutions are radial functions of the second variable we obtain an uniqueness result (Theorem 1), which deals with zero sets of these solutions. The example of the function in Theorem 2 of the paper shows that Theorem 1 cannot be essentially reinforced.

10

Authors: Steeve Burnet, Célia Jean-Alexis, Alain Piétrus

We deal with a perturbed version of a Hummel-Seebeck type method to approximate a solution of variational inclusions of the form: 0∈Φ(z) + F(z) where is a single-valued function twice continuously Fréchet differentiable and F is a set-valued map from ℜn to the closed subsets of ℜn. This framework is convenient to treat in a unified way standard sequential quadratic programming, its stabilized version, sequential quadratically constrained quadratic programming, and linearly constrained Lagrangian methods (see [1]). We obtain, thanks to some semistability and another property (which is close to the hemistability) of the solution ¯z of the previous inclusion, the local existence of a sequence that is superquadratically or cubically convergent.