and Applied Analysis 3 In the light of (H1)–(H4), it is easy to see that problems (1)-(2) admit an equilibrium point u = 0. Definition 1. The equilibrium point u = 0 of problems (1)(2) is said to be globally exponentially stable if there exist constants κ > 0 andM ≥ 1 such that 󵄩󵄩󵄩󵄩u (t, x; t0, φ) 󵄩󵄩󵄩󵄩Ω ≤ M 󵄩󵄩󵄩󵄩φ 󵄩󵄩󵄩󵄩Ω e −κ(t−t 0 ) , t ≥ t 0 , (10) where ‖φ‖ 2 Ω = sup t 0 −τ≤s≤t 0 ∑ n i=1 ∫ Ω φ...

Robustness analysis of fuzzy cellular neural networks with deviating arguments and stochastic disturbances is the main topic discussion in this paper. The issue at hand what upper bounds intervals for network can withstand before losing its stability. We solve these problems by using Gronwall-Bellman lemma some inequality techniques. theoretical results point that an exponentially stable networ...

In this paper we study the stability for all positive time of the fully implicit Euler scheme for the two-dimensional Navier–Stokes equations. More precisely, we consider the time discretization scheme and with the aid of the discrete Gronwall lemma and the discrete uniform Gronwall lemma we prove that the numerical scheme is stable.

In this paper, some new Gronwall type inequalities involving iterated integrals are given.

In this paper we introduce a new family of refined WatanabeSobolev spaces that capture in a fine way integrability in time of the Malliavin derivative. We consider duality in these spaces and derive a Burkholder type inequality in a dual norm. The theory we develop allows us to prove weak convergence with essentially optimal rate for numerical approximations in space and time of semilinear para...

Error estimates for the Crank-Nicolson in time, finite element in space (CNFE) discretization of the Navier-Stokes equations require a discrete version of the Gronwall inequality, which leads to a time-step restriction. We prove herein that no restriction on the time-step is necessary for a linear, fully implicit variation of CN-FE obtained by extrapolation of the convecting velocity. Previous ...