near equality near convexity sums of maximally monotone operators and averages of firmly nonexpansive mappings we study nearly equal and nearly convex sets ranges of maximally monotone operators and ranges and fixed points of convex combinations of firmly nonexpansive mappings the main result states that the range of an average of firmly nonexpansive mappings is nearly equal to the average of the ranges of the mappings a striking application of this result yields that the average of asymptotically regular firmly nonexpansive mappings is also asymptotically regular throughout examples are provided to illustrate the theory we also obtain detailed information on the domain and range of the resolvent average

stable coexistence states in the volterra lotka competition model with diffusion we study the existence uniqueness and stability of coexistence states in the lotka volterra model with diffusion for two competing species we assume that the parameters describing the interactio

the generalized quasi linearization method for reaction diffusion equations on an unbounded domain the method of generalized quasi linearization has been well developed for ordinary differential equations in this paper we extend the method of generalized quasi linearization to reaction diffusion equations on an unbounded domain the iterates which are solutions of linear equations starting from lower and upper solutions converge uniformly and monotonically to the unique solution of the nonlinear reaction diffusion equation in an unbounded domain initially an existence theorem for the linear nonhomogeneous reaction diffusion equation in an unbounded domain has been proved under improved conditions the quadratic convergence has been proved by using a comparison theorem of reaction diffusion equations with ordinary differential equations this avoids the computational complexity of the quasi linearization method since the computation of green s function at each stage of the iterates is avoided

global existence and asymptotic stability of equilibria to reaction diffusion systems in this paper we study weakly coupled reaction diffusion systems in unbounded domains of or where the reaction terms are sums of quasimonotone nondecreasing and nonincreasing functions such systems are more complicated than those in many previous publications and little is known about them a comparison principle and global existence and boundedness theorems for solutions to these systems are established sufficient conditions on the nonlinearities ensuring the positively ljapunov stability of the zero solution with respect to h2 perturbations are also obtained as samples of applications these results are applied to an autocatalytic chemical model and a concrete problem whose nonlinearities are nonquasimonotone our results are novel in particular we present a solution to an open problem posed by escher and yin 2005 j nonlinear anal theory methods appl 60 1065 84

inconsistent signal feasibility problems least squares solutions in a product space presents parallel projection methods to find least squares solutions to inconsistent convex set theoretic signal synthesis problems the problem of finding a signal that minimizes a weighted average of the squares of the distances to constraint sets is reformulated in a product space where it is equivalent to that of finding a point that lies in a particular subspace and at minimum distance from the cartesian product of the original sets a solution is obtained in the product space via methods of alternating projections which naturally lead to methods of parallel projections in the original space the convergence properties of the proposed methods are analyzed and signal synthesis applications are demonstrated

the optimal value and optimal solutions of the proximal average of convex functions abstract the proximal average of a finite collection of convex functions is a parameterized convex function that provides a continuous transformation between the convex functions in the collection this paper analyzes the dependence of the optimal value and the minimizers of the proximal average on the weighting parameter concavity of the optimal value is established and implies further regularity properties of the optimal value boundedness outer semicontinuity single valuedness continuity and lipschitz continuity of the minimizer mapping are concluded under various assumptions sharp minimizers are given further attention several examples are given to illustrate our results

stability and attractivity of periodic solutions of parabolic systems with time delays abstract this paper is concerned with the existence stability and global attractivity of time periodic solutions for a class of coupled parabolic equations in a bounded domain the problem under consideration includes coupled system of parabolic and ordinary differential equations and time delays may appear in the nonlinear reaction functions our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations the existence of time periodic solutions is for a class of locally lipschitz continuous reaction functions without any quasimonotone requirement using schauder fixed point theorem while the stability and attractivity analysis is for quasimonotone nondecreasing and mixed quasimonotone reaction functions using the monotone iterative scheme the results for the general system are applied to the standard parabolic equations without time delay and to the corresponding ordinary differential system applications are also given to three lotka volterra reaction diffusion model problems and in each problem a sufficient condition on the reaction rates is obtained to ensure the stability and global attractivity of positive periodic solutions

the asymptotic behavior of solutions of a system of reaction diffusion equations which models the belousov zhabotinskii chemical reaction abstract we investigate the boundary value problem u t 2 u x 2 u 1 u rv v t 2 v x 2 buv u t v t 0 u t 1 and v t t 0 where r 0 b 0 0 and x r this system has been proposed by murray as a model for the propagation of wave fronts of chemical activity in the belousov zhabotinskii chemical reaction here u and v are proportional to the concentrations of bromous acid and bromide ion respectively we determine the global stability of the constant solution u v 1 0 furthermore we introduce a moving coordinate and for each fixed x r we investigate the asymptotic behavior of u x ct t and v x ct t as t for both large and small values of the wave speed c 0

positive steady states of reaction diffusion advection competition models in periodic environment abstract in this paper we consider the positive steady states for reaction diffusion advection competition models in the whole space with a spatially periodic structure under the spatially periodic setting we establish sufficient conditions for the existence of positive steady states of this model respectively by investigating the sign of the principal eigenvalue for some linearized eigenvalue problems as an application a lotka volterra reaction diffusion advection model for two competing species in a spatially periodic environment is considered finally some numerical simulations are presented to seek dynamical behaviors

krull dimension of generalized weyl algebras and iterated skew polynomial rings commutative coefficients abstract many rings that have enjoyed growing interest in recent years e g quantum enveloping algebras quantum matrices certain witten algebras can be presented as generalized weyl algebras in the paper we develop techniques for calculating dimensions here mainly the krull dimension in the sense of rentschler gabriel of such generalized weyl algebras and specify the results for some popular algebras including those mentioned above

existence uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks abstract the author is concerned with the existence and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks previous methods do not apply in solving these problems because there is no maximum principle or conservation laws available to the integral differential equations he applies fixed point theorems to prove the existence of the traveling waves then he makes use of linearization technique as well as eigenvalue functions to study the exponential stability of the waves

maximal monotonicity for the precomposition with a linear operator we give the weakest constraint qualification known to us that ensures the maximal monotonicity of the operator a circ t circ a when a is a linear continuous mapping between two reflexive banach spaces and t is a maximal monotone operator as a special case we get the weakest constraint qualification that guarantees the maximal monotonicity of the sum of two maximal monotone operators on a reflexive banach space then we give a weak constraint qualification assuring the bre zis haraux type approximation of the range of the subdifferential of the precomposition to a of a proper convex lower semicontinuous function in nonreflexive banach spaces extending and correcting in a special case an older result due to riahi

the resolvent average for positive semidefinite matrices we define a new average termed the resolvent average for positive semidefinite matrices for positive definite matrices the resolvent average enjoys self duality and it interpolates between the harmonic and the arithmetic averages which it approaches when taking appropriate limits we compare the resolvent average to the geometric mean some applications to matrix functions are also given

parabolic systems in unbounded domains i existence and dynamics abstract in this paper we investigate the existence uniqueness and asymptotic behavior of a solution for a class of coupled nonlinear parabolic equations in a general unbounded domain that includes the whole space r n the exterior of a bounded domain and a half space in r n the asymptotic behavior of the solution is with respect to a pair of quasi solutions of the corresponding elliptic system and when these two quasi solutions coincide the solution of the parabolic system converges to a unique solution of the elliptic system

existence of coupled quasi solutions of systems of nonlinear reaction diffusion equations systems of nonlinear parabolic initial boundary value problems arise in many applications such as epidemies ecology biochemistry biology chemical and nuclear engineering constructive methods of proving existence results for such problems which can also provide numerical procedures for the computation of solutions are of greater value than theoretical existence results the method of upper and lower solutions coupled with monotone iterative technique has been employed successfully to prove existence of multiple solutions of nonlinear reaction diffusion equations in special cases by various authors 335 10 11 15 181 recently in 6 171 weakly coupled systems of reaction diffusion equations when the nonlinear terms are independent of gradient terms are discussed and some special type of results are obtained we in this paper investigate general systems of nonlinear reaction diffusion problems when the nonlinear terms possess a mixed quasi monotone property we discuss a very general situation and obtain coupled extremal quasi solutions which in special cases reduce to minimal and maximal solutions we shall also indicate how one step cyclic monotone iterative schemes can be generated which yield accelerated rate of convergence of iterates this work is in the spirit of our recent paper 12 for elliptic systems

fixed points of averages of resolvents geometry and algorithms to provide generalized solutions if a given problem admits no actual solution is an important task in mathematics and the natural sciences it has a rich history dating back to the early 19th century when carl friedrich gauss developed the method of least squares of a system of linear equations its solutions can be viewed as fixed points of averaged projections onto hyperplanes a powerful generalization of this problem is to find fixed points of averaged resolvents i e firmly nonexpansive mappings this paper concerns the relationship between the set of fixed points of averaged resolvents and certain fixed point sets of compositions of resolvents it partially extends recent work for two mappings on a question of c byrne the analysis suggests a reformulation in a product space algorithmic consequences are also presented

large time behaviour of the solutions of a semilinear parabolic equation in rn abstract the asymptotic behaviour as t tends to of the solution of u t u u u p 1 0 in r n r p 1 was studied it was proved that the behaviour depends strongly on the sign of n 2 n p and also on the rate of decay of the admissible initial data u 0 x as x tends to

qualitative theory for a model of laminar flames with arbitrary nonnegative initial data abstract we consider a system of nonlinear parabolic partial differential equations arising as a model of laminar flames in a premixed reactive gas the existence of traveling wave solutions has been established for positive ignition temperature by berestycki nicolaenko and scheurer and by terman for zero ignition temperature stability and instability of these solutions have been established in various situations by sivashinsky clavin and recently terman our goal is to study the equations with initial data that are bounded uniformly continuous and nonnegative but otherwise arbitrary we establish the existence of unique global strong solutions satisfying appropriate a priori estimates with a positivity condition imposed on the initial data for the temperature we show that the concentration decays exponentially this result while easy to obtain plays an important role in results that follow of greatest physical interest are the cases where ignition occurs precisely at one end our main result is that if the average of the initial temperature values at the ends of the chamber is above ignition temperature then on any ray coming from the ignition end the temperature is uniformly above ignition temperature and the concentration decays uniformly to zero we can continually advance the endpoint of this zone of ignition thus roughly mimicking the motion of traveling wave solutions we also point out the appropriateness of the averaging condition on the initial temperature and discuss two examples where this averaging condition is not satisfied in one we eventually have flame propagation and in the other example we have eventual flame extinguishment on any ray coming from the cold end of the chamber

on nonlinear reaction diffusion systems abstract this paper presents a qualitative analysis for a coupled system of two reaction diffusion equations under various boundary conditions which arises from a number of physical problems the nonlinear reaction functions are classified into three basic types according to their relative quasi monotone property for each type of reaction functions an existence comparison theorem in terms of upper and lower solutions is established for the time dependent system as well as some boundary value problems three concrete physical systems arising from epidemics biochemistry and engineering are taken as representatives of the basic types of reacting problems through suitable construction of upper and lower solutions various qualitative properties of the solution for each system are obtained these include the existence and bounds of time dependent solutions asymptotic behavior of the solution stability and instability of nontrivial steady state solutions estimates of stability regions and finally the blowing up property of the solution special attention is given to the homogeneous neumann boundary condition

on positive solutions of the lotka volterra cooperating models with diffusion abstract in this paper using the existence and comparison conclusion for the time periodic quasimonotone nondecreasing systems given by pao c v and iterative method we investigate the time periodic lotka volterra cooperating model and present some existence uniqueness and nonexistence results we also use similar methods to treat the corresponding stationary system to give parallel results

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

nearly convex sets fine properties and domains or ranges of subdifferentials of convex functions nearly convex sets play important roles in convex analysis optimization and theory of monotone operators we give a systematic study of nearly convex sets and construct examples of subdifferentials of lower semicontinuous convex functions whose domains or ranges are nonconvex

spatially inhomogeneous steady state solutions for systems of equations describing interacting populations for x e i2 and t 0 where 0 is a bounded region in r n 1 2 3 and a denotes the laplacian models the interaction of two species co existing in q the densities of the species at time t and at the point x e q are given by u x t and v x t the constants d dz 0 give the rates at which the species diffuse the constants a and b give if positive the birth rates and if negative the mortality rates of the species so that system 1 1 has a unique solution we must specify initial conditions i e u x 0 and li x 0 and boundary conditions we shall be interested in the case of dirichlet boundary conditions i e in the case where u x f and t x t are specified functions of x on x co existing populations may interact in various ways e g they may compete or co operate with each other or one may be a predator preying on the other the nature of the interaction is reflected by the properties of the terms f u tl and g u c for example if the 1 population preys on the ii population an increase in l would lead to a decrease in u and so it would be natural to assume that zf k 0 throughout we shall assume that

on the range of the douglas rachford operator the problem of finding a minimizer of the sum of two convex functions or more generally that of finding a zero of the sum of two maximally monotone operators is of central importance in variational analysis perhaps the most popular method of solving this problem is the douglas rachford splitting method surprisingly little is known about the range of the douglas rachford operator in this paper we set out to study this range systematically we prove that for 3 monotone operators a very pleasing formula can be found that reveals the range to be nearly equal to a simple set involving the domains and ranges of the underlying operators a similar formula holds for the range of the corresponding displacement mapping we discuss applications to subdifferential operators to the infimal displacement vector and to firmly nonexpansive mappings various examples and counterexamples are presented including some concerning the celebrated brezis haraux theorem

dynamics of reaction diffusion systems and applications abstract the dynamics of a system of arbitrary number of nonlinear parabolic equations in terms of the quasisolutions of the corresponding system of elliptic equations is discussed the conclusions for the general system are used to study the coexistence and permanence of a three species predator prey model and a food chain model arising from ecology

instability of stationary solutions for a lotka volterra competition model with diffusion abstract we consider a lotka volterra competition model with diffusion onrwhich describes the dynamics of the population of two competing species and study the stability of positive stationary solutions of the model relative to the spacexof bounded uniformly continuous functions with the supremum norm in consideration of earlier results we shall arrive at the study of the distribution of the real eigenvalues of the linearized operator around the stationary solution and prove the existence of only one positive eigenvalue whose eigenfunction exponentially decays to 0 as this suggests that the instability of the stationary solution is not due to the spacex

parabolic systems in unbounded domains ii equations with time delays abstract this paper extends the results of an earlier article concerning the existence uniqueness comparison and dynamics problems for a coupled system of parabolic equations in unbounded domains including the whole space r n the present extension is for a system of functional parabolic ordinary equations where the reaction function is allowed to depend on the unknown function with time delay which may be discrete or continuous finite or infinite applications are made to some model problems with emphasis on the convergence fo the time dependent solution to a steady state solution and the decay property of the solution as x

the proximal average basic theory the recently introduced proximal average of two convex functions is a convex function with many useful properties in this paper we introduce and systematically study the proximal average for finitely many convex functions the basic properties of the proximal average with respect to the standard convex analytical notions domain fenchel conjugate subdifferential proximal mapping epi continuity and others are provided and illustrated by several examples

interior estimates for a class of reaction diffusion systems from l1 a priori estimates abstract we obtain interior estimates for a class of semilinear reaction diffusion systems from l 1 a priori estimates our results are applied to a predator prey model in which the species switch the role of predator and prey on given subsets of their domain of interaction and a one dimensional flame propagation model extensions of earlier results in morgan 14 15 follow from the analysis

coexistence and stability of a competition diffusion system in population dynamics abstract the coexistence and stability of the population densities of two competing species in a bounded habitat are investigated in the present paper where the effect of dispersion transportation is taken into consideration the mathematical problem involves a coupled system of lotka volterra type reaction diffusion equations together with some initial and boundary conditions including the dirichlet neumann and third type necessary and sufficient conditions for the coexistence and competitive exclusion are established and the effect of diffusion is explicitly given for the stability problem general criteria for the stability and instability of a steady state solution are established and then applied to various situations depending on the relative magnitude among the physical parameters also given are necessary and sufficient conditions for the existence of multiple steady state solutions and the stability or instability of each of these solutions special attention is given to the neumann boundary condition with respect to which some threshold results for the coexistence and stability or instability of the four uniform steady states are characterized it is shown in this situation that only one of the four constant steady states is asymptotically stable while the remaining three are unstable the stability or instability of these states depends solely on the relative magnitude among the various rate constants and is independent of the diffusion coefficients

the jespers van oystaeyen conjecture e g 3 8 121 the mother examples of all these generalizations were studied already in 1968 by fossum 6 i e maximal orders over krull domains one of the main problems in the theory is to find a suitable generalization of unique factorization domains and related to this question to find a proper definition of a class group several possible definitions were suggested e g the k theoretic classgroup w a by fossum 6 the normalizing classgroup c l by chamarie 3 and the central classgroup which has been studied extensively by jespers and wouters 7 81 van oystaeyen 22 and later jespers 7 asked whether for a maximal order n over a dedekind domain