Sensitivity analysis for abstract equilibrium problems

We develop the general framework of sensitivity analysis for equilibrium problems in the setting of vector topological normed space. Our approach does not make any recourse to geometrical properties and the obtained result can be viewed as an extension and generalization of the well-known results (on variational inequalities) in the literature. Even though we have worked under arbitrary constraints Kλ with Hölder-property—that have been decisive in our treatment—we have obtained, in a similar spirit of Domokos [J. Math. Anal. Appl. 230 (1999) 382-389], the best lower bound for the continuity modulus despite of the properties of the boundary of Kλ.     

Stability of Runge-Kutta methods for abstract time-dependent parabolic problems: The Hölder case

We consider an abstract time-dependent, linear parabolic problem

where , , is a family of sectorial operators in a Banach space with time-independent domain . This problem is discretized in time by means of an A() strongly stable Runge-Kutta method, . We prove that the resulting discretization is stable, under the assumption

where and . Our results are applicable to the analysis of parabolic problems in the , , norms.

     

Well-posedness and stability for abstract spline problems

In this work well-posedness and stability properties of the abstract spline problem are studied in the framework of reflexive spaces. Tykhonov well-posedness is proved without restrictive assumptions. In the context of Hilbert spaces, also the stronger notion of Levitin-Polyak well-posedness is established. A sequence of parametric problems converging to the given abstract spline problem is considered in order to study stability. Under natural assumptions, convergence results for sequences of solutions of the perturbed problems are obtained.     

Perturbation theory for abstract optimization problems

A general perturbation theory is given for optimization problems in locally convex, linear spaces. Neither differentiability of the constraints nor regularity of the solutions of the unperturbed problem are assumed. Without reference to a particular multiplier rule, multipliers of the unperturbed problem are defined and used for characterizing solutions of a perturbed problem. In case of differentiable constraints or finite-dimensional spaces, the results exceed those known so far.     

A stability result for balanced dictatorships in Sn

We prove that a balanced Boolean function on S_n whose Fourier transform is highly concentrated on the first two irreducible representations of S_n, is close in structure to a dictatorship, a function which is determined by the image or pre‐image of a single element. As a corollary, we obtain a stability result concerning extremal isoperimetric sets in the Cayley graph on S_n generated by the transpositions. Our proof works in the case where the expectation of the function is bounded away from 0 and 1. In contrast, [6] deals with Boolean functions of expectation O(1/ n) whose Fourier transform is highly concentrated on the first two irreducible representations of S_n. These need not be close to dictatorships; rather, they must be close to a union of a constant number of cosets of point‐stabilizers. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 494–530, 2015     

A stability result for linear Markovian stochastic optimization problems

Mathematical Programming - In this paper, we propose a semi-metric for Markov processes that allows to bound optimal values of linear Markovian stochastic optimization problems. Similar to existing...     

On the stability of meshless symmetric collocation for boundary value problems

In this paper, we study the stability of symmetric collocation methods for boundary value problems using certain positive definite kernels. We derive lower bounds on the smallest eigenvalue of the associated collocation matrix in terms of the separation distance. Comparing these bounds to the well-known error estimates shows that another trade-off appears, which is significantly worse than the one known from classical interpolation. Finally, we show how this new trade-off can be overcome as well as how the collocation matrix can be stabilized by smoothing. AMS subject classification (2000)  65N12, 65N15, 65N35     

A fairly strong stability result for parabolic quasiminimizers

In this paper we consider parabolic Q‐quasiminimizers related to the p‐Laplace equation in . In particular, we focus on the stability problem with respect to the parameters p and Q. It is known that, if , then parabolic quasiminimizers with fixed initial‐boundary data on converge to the parabolic minimizer strongly in under suitable further structural assumptions. Our concern is whether or not we can obtain even stronger convergence. We will show a fairly strong stability result.     

Existence and asymptotic stability for viscoelastic problems with nonlocal boundary dissipation

We consider the damped semilinear viscoelastic wave equation

A sharp attainment result for nonconvex variational problems

We consider the problem of minimizing autonomous, multiple integrals like

A sharp global stability result for a discrete population model

We get a sharp global stability result for a first order difference equation modelling the growth of bobwhite quail populations. The corresponding higher-dimensional model is also discussed, and our stability conditions improve other recent results for the same equation.     

An optimization result for set-valued mappings and a stability property in vector problems with constraints

First, we prove an existence result relative to minimal points of set-valued mappings. Then, conditions about the upper and lower semicontinuity of constraint sets defined through set-valued mappings are given. Finally, a stability result relative to vector problems with abstract constraints is proved.The author thanks the referee for helpful comments on the first version of this paper.     

An existence result for anisotropic quasilinear problems

We study existence of solutions for a boundary degenerate (or singular) quasilinear equation in a smooth bounded domain under Dirichlet boundary conditions. We consider a weighted $p$-Laplacian operator with a coefficient that is locally bounded inside the domain and satisfying certain additional integrability assumptions. Our main result applies for boundary value problems involving continuous non-linearities having no growth restriction, but provided the existence of a sub and a supersolution is guaranteed. As an application, we present an existence result for a boundary value problem with a non-linearity $f\left(u\right)$ satisfying $f\left(0\right)\le 0$ and having $(p-1)$-sublinear growth at infinity.     

A second-order Magnus-type integrator for quasi-linear parabolic problems

In this paper, we consider an explicit exponential method of classical order two for the time discretisation of quasi-linear parabolic problems. The numerical scheme is based on a Magnus integrator and requires the evaluation of two exponentials per step. Our convergence analysis includes parabolic partial differential equations under a Dirichlet boundary condition and provides error estimates in Sobolev spaces. In an abstract formulation the initial boundary value problem is written as an initial value problem on a Banach space 

    given

involving the sectorial operator with domain independent of . Under reasonable regularity requirements on the problem, we prove the stability of the numerical method and derive error estimates in the norm of certain intermediate spaces between  and . Various applications and a numerical experiment illustrate the theoretical results.

     

Energy stability for a class of two-dimensional interface linear parabolic problems

We consider parabolic equations in two-dimensions with interfaces corresponding to concentrated heat capacity and singular own source. We give an analysis for energy stability of the solutions based on special Sobolev spaces (the energies also are given by the norms of these spaces) that are intrinsic to such problems. In order to define these spaces we study nonstandard spectral problems in which the eigenvalue appears in the interfaces (conjugation conditions) or at the boundary of the spatial domain. The introducing of appropriate spectral problems enable us to precise the values of the parameters which control the energy decay. In fact, in order for numerical calculation to be carried out effectively for large time, we need to know quantitatively this decay property.     

A tabu search scheme for abstract problems, with applications to the computation of fixed points

We present a general tabu search iterative algorithm to solve abstract problems on metric spaces. At each iteration, if the current solution turns out to be unacceptable then a neighborhood of unacceptable solutions is determined and excluded for further exploration, in such a way that, under mild assumptions, an acceptable solution is asymptotically reached. Thus our algorithm makes a crucial use of memory to avoid visiting unacceptable solutions more than once. We also present a specialization of our general method to the computation of fixed points.     

A characterization of well-posedness for the second order abstract Cauchy problems with finite delay

In this work, we introduce a strongly continuous one-parameter family of bounded linear operators that completely describes the well-posedness of a second order abstract differential delay equation in the initial history space $L^p([-r,0];X)$, $r>0$. This family, which satisfies a specific functional equation is applied to characterize the mild solution of the considered second order delay equation.