Influenced by Higham, Mao and Stuart [10], several numerical methods have been developed to study the strong convergence of the numerical solutions to stochastic differential equations (SDEs) under the local Lipschitz condition. These numerical methods include the tamed Euler–Maruyama (EM) method, the tamed Milstein method, the stopped EM, the backward EM, the backward forward EM, etc. In this ...

Lipschitz self mappings of metric spaces appear in many branches of mathematics. In this paper we introduce a modification of the Lipschitz condition which takes into account not only the mapping itself but also the behaviour of a finite number of its iterates. We refer to such mappings as mean Lipschitzian. The study of this new class of mappings seems potentially interesting and leads to some...

We consider Calderón’s inverse problem on planar domains Ω with conductivities in fractional Sobolev spaces. When Ω is Lipschitz, the problem was shown to be stable in the L–sense in [18]. We remove the Lipschitz condition on the boundary. To this end, we analyse the Sobolev regularity of the characteristic function of Ω. For Ω a quasiball, we compute ‖χΩ‖W s,p(Rd) in terms of the δ–neighbourho...

We present a semilocal convergence analysis for a simplified NewtonLavrentiev regularization method for solving ill-posed problems in a Hilbert space setting. We use a center-Lipschitz instead of a Lipschitz condition in our convergence analysis. This way we obtain: weaker convergence criteria, tighter error bounds and more precise information on the location of the solution than in earlier stu...

We show that local minimizers of functionals of the form Z Ω [f(Du(x)) + g(x , u(x))] dx, u ∈ u0 + W 1,p 0 (Ω), are locally Lipschitz continuous provided f is a convex function with p − q growth satisfying a condition of qualified convexity at infinity and g is Lipschitz continuous in u. As a consequence of this, we obtain an existence result for a related nonconvex functional.