Stability under Galerkin truncation of A-stable Runge--Kutta discretizations in time

We consider semilinear evolution equations for which the linear part is normal and generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of Hilbert spaces. We approximate their semiflow by an implicit, A-stable Runge--Kutta discretization in time and a spectral Galerkin truncation in space. We show regularity of the Galerkin-truncated semiflow and its time-discretization on open sets of initial values with bounds that are uniform in the spatial resolution and the initial value. We also prove convergence of the space-time discretization without any condition that couples the time step to the spatial resolution. Then we estimate the Galerkin truncation error for the semiflow of the evolution equation, its Runge--Kutta discretization, and their respective derivatives, showing how the order of the Galerkin truncation error depends on the smoothness of the initial data. Our results apply, in particular, to the semilinear wave equation and to the nonlinear Schr\"odinger equation

Solvability of geometrically nonlinear boundary-value problems for shallow shells of Timoshenko type with pivotally supported edges

© 2015, Allerton Press, Inc. We study solvability of a geometrically nonlinear, physically linear boundary-value problems for elastic shallow homogeneous isotropic shells with pivotally supported edges in the framework of S. P. Timoshenko’s shear model. The purpose of work is the proof of the theorem on existence of solutions. Research method consists in reducing the original system of equilibrium equations to one nonlinear differential equation for the deflection. The method is based on integral representations for displacements, which are built with the help of the general solutions of the nonhomogeneous Cauchy-Riemann equation. The solvability of equation relative to deflection is established with the use of principle of contraction mappings

Fixed point theorems for α\alpha--contractive mappings of Meir--Keeler type and applications

In this paper, we introduce the notion of $\alpha$--contractive mapping of Meir--Keeler type in complete metric spaces and prove new theorems which assure the existence, uniqueness and iterative approximation of the fixed point for this type of contraction. The presented theorems extend, generalize and improve several existing results in literature. To validate our results, we establish the existence and uniqueness of solution to a class of third order two point boundary value problems

n-Harmonic mappings between annuli

The central theme of this paper is the variational analysis of homeomorphisms h\colon \mathbb X \onto \mathbb Y between two given domains $\mathbb X, \mathbb Y \subset \mathbb R^n$. We look for the extremal mappings in the Sobolev space $\mathscr W^{1,n}(\mathbb X,\mathbb Y)$ which minimize the energy integral $$\mathscr E_h=\int_{\mathbb X} ||Dh(x)||^n dx.$$ Because of the natural connections with quasiconformal mappings this $n$-harmonic alternative to the classical Dirichlet integral (for planar domains) has drawn the attention of researchers in Geometric Function Theory. Explicit analysis is made here for a pair of concentric spherical annuli where many unexpected phenomena about minimal $n$-harmonic mappings are observed. The underlying integration of nonlinear differential forms, called free Lagrangians, becomes truly a work of art.Comment: 120 pages, 22 figure

A-stable Runge-Kutta methods for semilinear evolution equations

We consider semilinear evolution equations for which the linear part generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of Hilbert spaces. In this setting, we prove the existence of solutions which are temporally smooth in the norm of the lowest rung of the scale for an open set of initial data on the highest rung of the scale. Under the same assumptions, we prove that a class of implicit, $A$-stable Runge--Kutta semidiscretizations in time of such equations are smooth as maps from open subsets of the highest rung into the lowest rung of the scale. Under the additional assumption that the linear part of the evolution equation is normal or sectorial, we prove full order convergence of the semidiscretization in time for initial data on open sets. Our results apply, in particular, to the semilinear wave equation and to the nonlinear Schr\"odinger equation