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*new*Lyapunov exponent) associated to a continuous map. Our first result states that these new exponents coincide with the usual Lyapunov exponents if the map is differentiable. Then, we apply this concept to prove that there exists a $C^0$-dense subset of the set of the area-preserving homeomorphisms defined in a compact, connected and boundaryless surface such that any element inside this residual subset has zero

*new*Lyapunov exponents for Lebesgue almost every point. Finally, we prove that the function that associates an area-preserving homeomorphism, equipped with the $C^0$-topology, to the integral (with respect to area) of its top

*new*Lyapunov exponent over the whole surface cannot be upper-semicontinuous.

We give necessary integral conditions and sufficient ones for the existence of a general concept of $μ$-dichotomy for evolution operators defined on the half-line which includes as particular cases the well-known concepts of nonuniform exponential dichotomy and nonuniform polynomial dichotomy, and also contains new situations. Additionally, we consider an adapted notion of Lyapunov function and use our results to obtain necessary and sufficient conditions for the existence of nonuniform $μ$-dichotomies using these Lyapunov functions.

We study an optimal control problem for a non-autonomous SEIRS model with incidence given by a general function of the infective, the susceptible and the total population, and with vaccination and treatment as control variables. We prove existence and uniqueness results for our problem and, for the case of mass-action incidence, we present some simulation results designed to compare an autonomous and corresponding periodic model, as well as the controlled versus uncontrolled models.

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