oscillations of nonlinear partial difference systems abstract this paper studies the following two dimensional nonlinear partial difference systems t 1 2 x mn b mn g y mn 0 t 1 2 y mn a mn f x mn 0 where m n n0 0 1 2 t 1 2 1 2 i t 1 2 1 2 i 1ymn ym 1 n ymn 2ymn ym n 1 ymn iymn ymn 1ymn ym 1 n ymn 2ymn ym n 1 ymn amn and bmn are real sequences m n n0 and f g r r are continuous with uf u 0 and ug u 0 for all u 0 a solution xmn ymn of the system is oscillatory if both components are oscillatory some sufficient conditions for all solutions of this system to be oscillatory are derived

measures for oscillatory sequences abstract the classical concept of oscillation is not sufficient to fully describe the oscillatory behavior of a real sequence by introducing the concepts of frequency measures of subsets of integers and investigating their properties we are able to strengthen the original concept of an oscillatory sequence and thus new oscillation criteria for difference equations can be derived these criteria differ from the classical ones in that they are described by means of the measures of the level sets of the involved parameter sequences a brief description of the theory of frequency measures for multiple sequences is also given

eventually positive and monotonely decreasing solutions of partial difference equations we establish criteria for the nonexistence of eventually positive negative and monotonely decreasing increasing solutions of the partial difference inequality where gi m and hi n 1 i are some deviating arguments several examples are presented to dwell upon the importance of the results obtained

an iterative multi level algorithm for solving nonlinear ill posed problems the convergence analysis of landweber s iteration for the solution of nonlinear ill posed problem has been developed recently by hanke neubauer and scherzer in concrete applications sufficient conditions for convergence of the landweber iterates developed there although quite natural turned out to be complicated to verify analytically however in numerical realizations when discretizations are considered sufficient conditions for local convergence can usually easily be stated this paper is motivated by these observations initially a discretization is fixed and a discrete landweber iteration is implemented until an appropriate stopping criterion becomes active the output is used as an initial guess for a finer discretization an advantage of this method is that the convergence analysis can be considered in a family of finite dimensional spaces the numerical performance of this multi level algorithm is compared with landweber s iteration

estimates on translations and taylor expansions in fractional sobolev spaces abstract in this paper we study how the normalised gagliardo semi norms u w s p r n control translations in particular we prove that u y u l p r n c u w s p r n y s for n 1 s 0 1 and p 1 where c depends only on n we then obtain a corresponding higher order version of this result we get fractional rates of the error term in the taylor expansion we also present relevant implications of our two results first we obtain a direct proof of several compact embedding of w s p r n where the frechet kolmogorov theorem is applied with known rates we also derive fractional rates of convergence of the convolution of a function with suitable mollifiers thirdly we obtain fractional rates of convergence of finite difference discretisations for w s p r n

oscillation of partial difference equations with variable coefficients for the partial difference equations am 1 n am n 1 am n pm nam k n 1 0 and am 1 n am n 1 am n i 1upi m n fi am ki n li 0 we shall obtain sufficient conditions for the oscillation of all solutions

a second degree method for nonlinear inverse problems the paper is concerned with the solution of nonlinear ill posed problems by methods that utilize the second derivative a general predictor corrector approach is developed one which avoids solving quadratic equations during the iteration process combining regularization of each iteration step with an adequate stopping condition leads to a general regularization scheme for nonlinear equations possible implementations and discussion of the performance of this method are illustrated by applications to some well known inverse problems

oscillation theorems for second order nonlinear partial difference equations 479 abstract the oscillation of the second order nonlinear partial difference equation t 1 2 c mn 1 y mn i 1 s a i m n f i y m 1 n 1 y mn 0 is investigated some sufficient conditions for the oscillation of all solutions of the above equation are obtained

notes on limits of sobolev spaces and the continuity of interpolation scales we extend lemmas by bourgain brezis mironescu 2001 and maz ya shaposhnikova 2002 on limits of sobolev spaces to the setting of interpolation scales this is achieved by means of establishing the continuity of real and complex interpolation scales at the end points a connection to extrapolation theory is developed and a new application to limits of sobolev scales is obtained we also give a new approach to the problem of how to recognize constant functions via sobolev conditions

oscillation for nonlinear delay partial difference equations with positive and negative coefficients abstract we construct an important transform to obtain sufficient conditions for the oscillation of all solutions of the delay partial difference equations with positive and negative coefficients of the form m n a m n a m u n v a m 1 n a m n 1 a mn pa m k n l qa m k n l 0 and mn m n a m n a m u n v a m 1 n a m n 1 a mn p mn a m k n l q mn a m k n l 0 where m n 0 1 and k k l l are nonnegative integers p q 0 and u v are positive integers the coefficients mn qmn and pmn are assumed to be positive

a new characterization of sobolev spaces on mathbb r n in this paper we present a new characterization of sobolev spaces on mathbb r n our characterizing condition is obtained via a quadratic multiscale expression which exploits the particular symmetry properties of euclidean space an interesting feature of our condition is that depends only on the metric of mathbb r n and the lebesgue measure so that one can define sobolev spaces of any order of smoothness on any metric measure space

generalized gagliardo nirenberg inequalities using lorentz spaces bmo holder spaces and fractional sobolev spaces abstract the main purpose of this paper is to prove some generalized gagliardo nirenberg interpolation inequalities involving the lorentz spaces l p bmo and the fractional sobolev spaces w s p including also c holder spaces although some of the results can be alternatively obtained by using interpolation spaces specifically the reiteration theorem the precise form of the inequalities stated here appears to be novel and moreover the proofs given in the present paper are self contained save for the use of the john nirenberg inequality for the bmo result in contrast to the other mentioned approach the use of c holder spaces in such gagliardo nirenberg inequalities seems to be new in the literature

nonlocal discrete diffusion equations and the fractional discrete laplacian regularity and applications abstract the analysis of nonlocal discrete equations driven by fractional powers of the discrete laplacian on a mesh of size h 0 h s u f for u f z h r 0 s 1 is performed the pointwise nonlocal formula for h s u and the nonlocal discrete mean value property for discrete s harmonic functions are obtained we observe that a characterization of h s as the dirichlet to neumann operator for a semidiscrete degenerate elliptic local extension problem is valid regularity properties and schauder estimates in discrete holder spaces as well as existence and uniqueness of solutions to the nonlocal dirichlet problem are shown for the latter the fractional discrete sobolev embedding and the fractional discrete poincare inequality are proved which are of independent interest we introduce the negative power fundamental solution u h s f which can be seen as the neumann to dirichlet map for the semidiscrete extension problem we then prove the discrete hardy littlewood sobolev inequality for h s as applications the convergence of our fractional discrete laplacian to the continuous fractional laplacian as h 0 in holder spaces is analyzed indeed uniform estimates for the error of the approximation in terms of h under minimal regularity assumptions are obtained we finally prove that solutions to the poisson problem for the fractional laplacian s u f in r can be approximated by solutions to the dirichlet problem for our fractional discrete laplacian with explicit uniform error estimates in terms of h

nonexistence of unbounded nonoscillatory solutions of partial difference equations abstract we develop criteria for the nonexistence of eventually positive negative and nondecreasing nonincreasing solutions of the partial difference equation m n y m n p m n y m k n l q m n y m k n l and m n y m n i 1 p i m n y m k i n l i i 1 q i m n y m k i n l i several examples which dwell upon the importance of our results are also included

a two stage method for nonlinear inverse problems in this work we are interested in the solution of nonlinear inverse problems of the form f x y we consider a two stage method which is third order convergent for well posed problems combining the method with levenberg marquardt regularization of the linearized problems at each stage and using the discrepancy principle as a stopping criterion we obtain a regularization method for ill posed problems numerical experiments on some parameter identification and inverse acoustic scattering problems are presented to illustrate the performance of the method

on convergence rates of inexact newton regularizations reginn is an algorithm of inexact newton type for the regularization of nonlinear ill posed problems inverse problems 15 1999 pp 309 327 in the present article convergence is shown under weak smoothness assumptions source conditions moreover convergence rates are established some computational illustrations support the theoretical results

on the bourgain brezis and mironescu theorem concerning limiting embeddings of fractional sobolev spaces the article is concerned with the bourgain brezis and mironescu theorem on the asymptotic behaviour of the norm of the sobolev type embedding operator ws p lpn n sp as s 1 and s n p thei

oscillation in nonlinear difference equations abstract in this paper we shall discuss oscillatory behavior of the solutions of difference equations including the self adjoint second order linear equation and the discrete version of the nonlinear wave equation our work is to give sufficient conditions such that every nontrivial solution of the equations oscillates

the oscillation of partial difference equations with continuous arguments abstract for the partial difference equations and we shall obtain sufficient conditions for the oscillation of all solutions of these equations

robust numerical methods for nonlocal and local equations of porous medium type part ii schemes and experiments we develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations partial t u mathfrak l varphi u f x t in math

oscillatory behavior of delay partial difference equations with positive and negative coefficients abstract we construct an important transform to obtain sufficient conditions for the oscillation of all solutions of delay partial difference equations with positive and negative coefficients of the form a m 1 n a m n 1 a mn p mn a m k 1 q mn a m k n l 0 where m n 0 1 and k k l l are nonnegative integers p q 0 the coefficients q mn and p mn are sequences of nonnegative real numbers