Asymptotic Analysis - Volume 106, issue 3-4

Authors: Vodev, Georgi

Article Type: Research Article

Abstract: We study the location of the transmission eigenvalues in the isotropic case when the restrictions of the refraction indices on the boundary coincide. Under some natural conditions we show that there exist parabolic transmission eigenvalue-free regions.

Keywords: Transmission eigenvalues, eigenvalue-free regions

DOI: 10.3233/ASY-171443

Citation: Asymptotic Analysis, vol. 106, no. 3-4, pp. 147-168, 2018

Authors: Wang, Yejuan | Sui, Meiyu

Article Type: Research Article

Abstract: In this paper, we consider the dynamics of multi-valued processes generated by nonautonomous lattice systems with delays. In particular, the effects of small delays on the asymptotic behavior of multi-valued nonautonomous lattice systems and finite lattice approximation of infinite delay lattice systems are presented. We do not assume any Lipschitiz condition on the nonlinear term, just a continuity assumption together with growth condition, so that uniqueness of solutions of the problem fails to be true.

Keywords: Pullback attractors, lattice systems with delays, set-valued dynamical system, upper semicontinuity

DOI: 10.3233/ASY-171444

Citation: Asymptotic Analysis, vol. 106, no. 3-4, pp. 169-203, 2018

Authors: Ahuja, Saran | Ren, Weiluo | Yang, Tzu-Wei

Article Type: Research Article

Abstract: In this paper, we consider a mean field game (MFG) model perturbed by small common noise. Our goal is to give an approximation of the Nash equilibrium strategy of this game using a solution from the original no common noise MFG whose solution can be obtained through a coupled system of partial differential equations. We characterize the first order approximation via linear mean-field forward-backward stochastic differential equations whose solution is a centered Gaussian process with respect to the common noise. The first order approximate strategy can be described as follows: at time t ∈ [ 0 , T ] …, applying the original MFG optimal strategy for a sub game over [ t , T ] with the initial being the current state and distribution. We then show that this strategy gives an approximate Nash equilibrium of order ϵ 2 . Show more

Keywords: Mean field games, common noise, asymptotic, forward-backward stochastic differential equations

DOI: 10.3233/ASY-171446

Citation: Asymptotic Analysis, vol. 106, no. 3-4, pp. 205-232, 2018

Authors: Alzer, Horst | Kwong, Man Kam

Article Type: Research Article

Abstract: In 1974, Askey and Steinig showed that for n ⩾ 0 and x ∈ ( 0 , 2 π ) , S n ( x ) = ∑ k = 0 n sin ( ( k + 1 / 4 ) x ) k + 1 > 0 and C n ( x ) = ∑ k = 0 n cos ( ( k + 1 / 4 ) x ) …k + 1 > 0 . We prove that (0.1) S n ( x ) + C n ( x ) ⩾ 1 2 and that the alternating sums S n ∗ ( x ) = ∑ k = 0 n ( − 1 ) k sin ( ( k + 1 / 4 ) x ) k + 1 and C n ∗ ( x ) = ∑ k = 0 n ( − 1 ) k cos ( ( k + 1 / 4 ) x ) k + 1 satisfy (0.2) S n ∗ ( x ) + C n ∗ ( x ) ⩾ 1 200 ( 13 − 85 ) 300 + 20 85 = 0.41601 … . Both inequalities hold for all n ⩾ 0 and x ∈ [ 0 , 2 π ] . The constant lower bounds given in (0.1 ) and (0.2 ) are best possible. The asymptotic behaviour of both sums is also investigated. Show more

Keywords: Trigonometric sums, inequalities, asymptotics

DOI: 10.3233/ASY-171447

Citation: Asymptotic Analysis, vol. 106, no. 3-4, pp. 233-249, 2018