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This paper is concerned with the following partially dissipative viscoelastic Timoshenko system { ρ 1 ϕ t t − κ (ϕ x + ψ) x + κ ∫ 0 t g (t − s) (ϕ x + ψ) x d s = 0 in (0 , L) × R + , ρ 2 ψ t t − b ψ x x + κ (φ x + ψ) − κ ∫ 0 t g (t − s) (ϕ x + ψ) d s = 0 in (0 , L) × R + , with damping mechanism acting only on the shear force, and with Dirichlet boundary conditions. We consider a very general relaxation function g ′ (t) ≤ − ξ (t) H (g (t)) , ∀ t ≥ 0. Under appropriate conditions on ξ and H, we establish a general stability result. The result is obtained under the assumption of equal speed of wave propagation. This work extends and generalizes many results in literature such as Alves et al. [On modeling and uniform stability of a partially dissipative viscoelastic Timoshenko system. SIAM J Math Anal. 2019;51(6):4520–4543]. [ABSTRACT FROM AUTHOR]

For solving strongly convex optimization problems, we propose and study the global convergence of variants of the accelerated hybrid proximal extragradient (A-HPE) and large-step A-HPE algorithms of R.D.C. Monteiro and B.F. Svaiter [An accelerated hybrid proximal extragradient method for convex optimization and its implications to second-order methods, SIAM J. Optim. 23 (2013), pp. 1092–1125.]. We prove linear and the superlinear O (k − k (p − 1 p + 1) ) global rates for the proposed variants of the A-HPE and large-step A-HPE methods, respectively. The parameter p ≥ 2 appears in the (high-order) large-step condition of the new large-step A-HPE algorithm. We apply our results to high-order tensor methods, obtaining a new inexact (relative-error) tensor method for (smooth) strongly convex optimization with iteration-complexity O (k − k (p − 1 p + 1) ) . In particular, for p = 2, we obtain an inexact proximal-Newton algorithm with fast global O (k − k / 3) convergence rate. [ABSTRACT FROM AUTHOR]