In this paper, we consider an integrated pest management model which is impulsively controlled by means of biological and chemical controls. These controls are assumed to act in a periodic fashion, a nonlinear incidence rate being used to account for the dynamics of the disease caused by the application of the biological control. The Floquet theory for impulsive ordinary differential equations is employed to obtain a condition in terms of an inequality involving the total action of the nonlinear force of infection in a period, under which the susceptible pest-eradication solution is globally asymptotically stable. If the opposite inequality is satisfied, then it is shown that the system under consideration becomes uniformly persistent. A biological interpretation of the persistence condition is also provided.
In this paper a proximal point algorithm (PPA) for maximal monotone operators with appropriate regularization parameters is considered. A strong convergence result for PPA is stated and proved under the general condition that the error sequence tends to zero in norm. Note that Rockafellar (SIAM J Control Optim 14:877-898, 1976) assumed summability for the error sequence to derive weak convergence of PPA in its initial form, and this restrictive condition on errors has been extensively used so far for different versions of PPA. Thus this Note provides a solution to a long standing open problem and in particular offers new possibilities towards the approximation of the minimum points of convex functionals.
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