For the spectral radius of weighted composition operators with positive weight e ϕ T α , ϕ ∈ C(X ), acting in the spaces L p (X, μ) the following variational principle holdswhere X is a Hausdorff compact space, α : X → X is a continuous mapping and τ α some convex and lower semicontinuous functional defined on the set M 1 α of all Borel probability and α-invariant measures on X . In other words τ α p is the LegendreFenchel conjugate of ln r (e ϕ T α ). In this paper we consider the polynomials with positive coefficients of weighted composition operator of the form A ϕ,c = n k=0 e c k (e ϕ T α ) k , c = (c k ) ∈ R n+1 . We derive two formulas on the Legendre-Fenchel transform of the spectral exponent ln r (A ϕ,c ) considering it firstly depending on the function ϕ and the variable c and secondly depending only on the function ϕ, by fixing c.
This work is an attempt at studying leader-following model on the arbitrary time scale. The step size is treated as a function of time. Our purpose is establishing conditions ensuring a leader-following consensus for any time scale basing on the Grönwall inequality. We give some examples illustrating the obtained results.
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