block monotone iterative methods for numerical solutions of nonlinear elliptic equations two block monotone iterative schemes for a nonlinear algebraic system which is a finite difference approximation of a nonlinear elliptic boundary value problem are presented and are shown to converge monotonically either from above or from below to a solution of the system this monotone convergence result yields a computational algorithm for numerical solutions as well as an existence comparison theorem of the system including a sufficient condition for the uniqueness of the solution an advantage of the block iterative schemes is that the thomas algorithm can be used to compute numerical solutions of the sequence of iterations in the same fashion as for one dimensional problems the block iterative schemes are compared with the point monotone iterative schemes of picard jacobi and gauss seidel and various theoretical comparison results among these monotone iterative schemes are given these comparison results demonstrate that the sequence of iterations from the block iterative schemes converges faster than the corresponding sequence given by the point iterative schemes application of the iterative schemes is given to a logistic model problem in ecology and numerical ressults for a test problem with known analytical solution are given

accelerated monotone scheme for finite difference equations concerning steady state prey predator interactions abstract in previous work by adapting a suitable finite difference method to a particular monotone scheme the authors and a lazer have studied the numerical solution of a system of semilinear elliptic partial differential equations which determines the equilibria of the volterra lotka equations describing prey predator interactions with diffusion in this paper in order to improve the efficiency of the method we show how newton s method can be successfully combined with the previous scheme to greatly accelerate the convergence in some particularly difficult problems the new method reduces the average number of iterations necessary to generate each element of the monotone sequences from 15 to about 3

nonlinear fourth order elliptic equations with nonlocal boundary conditions this paper is concerned with a class of fourth order nonlinear elliptic equations with nonlocal boundary conditions including a multi point boundary condition in a bounded domain of rn also considered is a second order elliptic equation with nonlocal boundary condition and the usual multi point boundary problem in ordinary differential equations the aim of the paper is to show the existence of maximal and minimal solutions the uniqueness of a positive solution and the method of construction for these solutions our approach to the above problems is by the method of upper and lower solutions and its associated monotone iterations the monotone iterative schemes can be developed into computational algorithms for numerical solutions of the problem by either the finite difference method or the finite element method

a note on the galois correspondence for commutative rings introductionthe basic correspondence of galois theory for commutative rings de scribed in chapter v of the book by magid 8 is incorrect it relies on awxproperty of the category of profinite spaces lemma v 9 which unfortu nately does not hold in this note we show with an example that thisproperty is crucial for the correctness of the correspondence and weestablish the correct one corollary 5 freely using the work on general or abstract galois theory performed in the meantime by the second named author 2 5 whose first purpose was to generalize the results of chapterwxiv of 8 wxthe standard reference we use for general categorical concepts is thebook by mac lane 7 and for internal category theory the book bywxjohnstone 6 wx

stability for time varying linear dynamic systems on time scales we study conditions under which the solutions of a time varying linear dynamic system of the form x t a t x t are stable on certain time scales we give sufficient conditions for various types of stability including lyapunov type stability criteria and eigenvalue conditions on slowly varying systems that ensure exponential stability finally perturbations of the unforced system are investigated and an instability criterion is also developed

the method of lower and upper solutions for fourth order two point boundary value problems abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem u iv f x u x u x u 0 u 1 u 0 u 1 0 where f 0 1 r r r is continuous

a tannakian context for galois theory abstract strong similarities have been long observed between the galois and tannaka theories of the representation of groups in this paper we construct an explicit neutral tannakian context for the galois theory of atomic topoi and prove equivalence for the fundamental theorem since the theorem is known for the galois context this yields a proof of the fundamental recognition theorem for a new tannakian context this example is different from the additive cases or their generalization for which the theorem is known to hold and for which the unit of the tensor product is always an object of finite presentation which is not the case in our context

hyers ulam stability of non autonomous systems in terms of boundedness of cauchy problems in this paper the concepts of hyers ulam stability are generalized for non autonomous linear differential systems we prove that the k periodic linear differential matrix system z t a t z t t r is hyers ulam stable if and only if the matrix family l e k 0 has no eigenvalues on the unit circle i e we study the hyers ulam stability in terms of dichotomy of the differential matrix system z t a t z t t r furthermore we relate hyers ulam stability of the system z t a t z t t r to the boundedness of solution of the following cauchy problem y t a t y t t t 0 y 0 x x 0 where a t is a square matrix for any t r t is a bounded function and x x 0 c m

galois theory in symmetric monoidal categories abstract the galois theory presented here is at a level of generality essentially between that of g janelidze d schumacher and r street 1993 appl categ structures 1 103 110 and g janelidze 1991 lecture notes in mathematics vol 1488 pp 157 173 however our purpose is to concentrate on symmetric monoidal categories and so provide a new approach to tannaka duality

ulam hyers stability of dynamic equations on time scales via picard operators in this paper we study the ulam hyers stability of some linear and nonlinear dynamic equations and integral equations on time scales we use both direct and operatorial methods and we propose a unified approach to ulam hyers stability based on the theory of picard operators see 28 33 our results extend some recent results from 24 25 8 14 13 5 6 to dynamic equations and are more general than the results from 1 the operatorial point of view based on the theory of picard operators allows to discuss the ulam hyers stability of many types of differential and integral equations on time scales and also to obtain simple and structured proofs to the existing results but as we point out at our final remarks there are also a few disadvantages

monotone explicit iterations of the finite element approximations for the nonlinear boundary value problem in this paper we consider monotone explicit iterations of the finite element schemes for the nonlinear equations associated with the boundary value problem u bu 2 based on piecewise linear polynomials and the lumping operator these iterations construct the monotonically decreasing and increasing sequences and convergence proofs are given finally we present some numerical examples verifying the effectiveness of the theory

full order convergence of a mixed finite element method for fourth order elliptic equations abstract by using a special interpolation operator and an elaborate element analysis in this paper we improve the classical error estimates to full order for a mixed finite element method for the fourth order elliptic equations on the rectangular mesh therefore we obtain the truly optimal error estimates in view of the interpolation space for the first time

numerical solutions for some coupled systems of nonlinear boundary value problems in the well known volterra lotka model concerning two competing species with diffusion the densities of the species are governed by a coupled system of reaction diffusion equations the aim of this paper is to present an iterative scheme for the steady state solutions of a finite difference system which corresponds to the coupled nonlinear boundary value problems this iterative scheme is based on the method of upper lower solutions which leads to two monotone sequences from some uncoupled linear systems it is shown that each of the two sequences converges to a nontrivial solution of the discrete equations the model under consideration may have one two or three nonzero solutions and each of these solutions can be computed by a suitable choice of initial iteration numerical results are given for these solutions under both the dirichlet boundary condition and the mixed type boundary condition

ulam s type stability of impulsive ordinary differential equations abstract in this paper we introduce four ulam s type stability concepts for impulsive ordinary differential equations by applying the integral inequality of gronwall type for piecewise continuous functions ulam s type stability results for impulsive ordinary differential equations are obtained an example is also provided to illustrate our results

population dynamics control in regard to minimizing the time necessary for the regeneration of a biomass taken away from the population the dynamics of a population whose mathematical model is the equation of verhulst is considered from the population by discrete external effects in the form of impulses a certain quantity of biomass is taken away or added this process is described by means of differential equations with impulses at fixed moments the moments and the magnitudes of the impulses are determined so that the time of regeneration of the quantity of biomass taken away from the population be minimal

ulam s type stability of higher order nonlinear delay differential equations via integral inequality of gronwall bellman bihari s type abstract in this paper we prove the hyers ulam stability and the hyers ulam rassias stability of a class of higher order nonlinear delay differential equations with multiple bounded variable delays on a compact interval result of existence and uniqueness of solution is obtained by fixed point approach meanwhile for the first time integral inequality of gronwall bellman bihari s type with delay is applied to prove the stability theorem which made our results more generalized and interesting

existence uniqueness and stability of solution to mixed integral dynamic systems with instantaneous and noninstantaneous impulses on time scales abstract in this paper we study the existence and uniqueness of solution and stability results of mixed integral dynamic system with instantaneous and noninstantaneous impulses on time scales by using the fixed point method the main tools to establish our results are the gronwall s inequality on time scales picard operator and abstract gronwall lemma some assumptions are made to overcome the hurdles in achieving our results at the end an example is given that shows the validity of our main results

block monotone iterations for numerical solutions of fourth order nonlinear elliptic boundary value problems this paper is concerned with monotone iterative methods for numerical solutions of a class of nonlinear fourth order elliptic boundary value problems in a two dimensional domain the boundary value problem is discretized by the finite difference method and two iterative processes called block jacobi and block gauss seidel monotone iterations are presented for the computation of solutions of the finite difference system using either an upper solution or a lower solution as the initial iteration it is shown that the sequence of iterations converges monotonically to a maximal solution or a minimal solution if the nonlinear function is quasi monotone nondecreasing a sufficient condition is given to ensure that the maximal and minimal solutions coincide and their common value is the unique solution of the finite difference system similar results are obtained for quasi monotone nonincreasing functions an analytical comparison relation between the block jacobi and block gauss seidel monotone iterations is obtained it is also shown that the finite difference solution converges to the continuous solution as the mesh size tends to zero numerical results by the block monotone iterative schemes are given for two model problems and are compared with the known analytical solutions for accuracy also compared are the rates of convergence of the two types of block monotone iterations as well as similar types of pointwise monotone iterations

on fourth order elliptic boundary value problems with nonmonotone nonlinear function this paper is concerned with the fourth order elliptic boundary value problems with nonmonotone nonlinear function the existence and uniqueness of a solution is proven by the method of upper and lower solutions a monotone iteration is developed so that the iteration sequence converges monotonically to a maximal solution or a minimal solution depending on whether the initial iteration is an upper solution or a lower solution

the method of lower and upper solutions for second third fourth and higher order boundary value problems abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem u n t t u t u i a u i b i r i 0 n 1 where is a caratheodory function we obtain necessary and sufficient conditions in to guarantee the existence of solutions between a lower solution and an upper solution for n 2 if n 3 either or and n 4 if furthermore we obtain sufficient conditions in for n 2 k 6 when