Classical analysis of linear multistep formulae (LMFs) for initial-value problems in ordinary diierential equations (ODEs) has concentrated on problems satisfying uniform Lipschitz or one-sided Lipschitz conditions, and corresponding stability models. We here contribute insight into the robustness of such formulae for equations with a stronger nonlinearity and of related formulae for functional...

We generalise the current theory of optimal strong convergence rates for implicit Euler-based methods by allowing for Poisson-driven jumps in a stochastic differential equation (SDE). More precisely, we show that under one-sided Lipschitz and polynomial growth conditions on the drift coefficient and global Lipschitz conditions on the diffusion and jump coefficients, three variants of backward E...

Based on A-stable one-leg methods and linear interpolations, we introduce four algorithms for solving neutral differential equations with variable delay. A natural question is which algorithm is better. To answer this question, we analyse the error behavior of the four algorithms and obtain their error bounds under a one-sided Lipschitz condition and some classical Lipschitz conditions. After e...

Relaxed one-sided Lipschitz conditions play an important role when analyzing ordinary differential inclusions. They allow to derive a-priori estimates of solutions and convergence estimates for explicit and implicit time discretizations. In this paper we consider Galerkin finite element discretizations of semilinear elliptic inclusions that satisfy a relaxed one-sided Lipschitz condition. It is...

We introduce one-sided Lipschitz (OSL) conditions of set-valued maps with respect to given set differences. The existence of selections of such maps that pass through any point of their graphs and inherit uniformly their OSL constants is studied. We show that the OSL property of a convex-valued set-valued map with respect to the Demyanov difference with a given constant is characterized by the ...

State estimation of systems satisfying some special nonlinearities have been important topics in nonlinear system theory. In this paper, we discuss the problem ofobserver design for one-sided Lipschitz nonlinear generalized systems basedon generalized quadratic stability and Lyapunov function, by using the linear matrix inequality (LMI) andlinear matrix equality (LME) approach, we propose some ...