on nonstability of the linear recurrence of order one abstract we show that under some assumptions the linear recurrence or difference equation of order one in a banach space is nonstable in the hyers ulam sense our results are also connected with the notion of shadowing in dynamical systems and computer sciences

stability of multi additive mappings in β banach spaces abstract in this paper we prove the generalized hyers ulam stability of the system defining multi additive mappings and an equation which characterizes these mappings in banach spaces we thus generalize some previous results obtained in classical banach spaces in order to get our outcomes we use both direct and fixed point methods

on stability of the monomial functional equation in normed spaces over fields with valuation a stability theorem is proved for the monomial functional equation where the functions map a normed space over a field of characteristic zero with an arbitrary valuation into a banach space over a field of characteristic zero with a valuation some regularity properties of the monomial functions are also discussed

a fixed point theorem and the hyers ulam stability in non archimedean spaces abstract we prove a fixed point theorem in complete non archimedean normed spaces and show its applications in the hyers ulam stability of some functional equations

addendum to on the stability of functional equations on square symmetric groupoid abstract let x be a square symmetric groupoid and y d a complete metric divisible square symmetric groupoid in this paper we obtain the hyers ulam stability problem of functional inequality d g x y g x g y e x y for approximate mapping g x y of functional equation f x y f x f y differing by e x 2 0 in particular we have investigated the case of f x f y h f x 1 t f y 1 t on some set y

hyers ulam stability of functional equations with a square symmetric operation abstract the stability of the functional equation f x y h f x f y x y s is investigated where h is a homogeneous function and is a square symmetric operation on the set s the results presented include and generalize the classical theorem of hyers obtained in 1941 on the stability of the cauchy functional equation

necessary and sufficient conditions for optimal offers in electricity markets in this paper we consider the optimal policy for a generator offering power into a wholesale electricity market operating under a pool arrangement anderson and philpott math oper res 27 2002 pp 82 100 recently discussed necessary conditions for an optimal offer curve when there is uncertainty in the demand and in the behavior of other participants in the market they show that the objective function in these circumstances can be expressed as a line integral along the offer curve of a profit function integrated with respect to a market distribution function in this paper we prove the existence of an optimal offer stack and we extend the analysis of math oper res 27 2002 pp 82 100 to include necessary conditions of a higher order in the presence of horizontal and or vertical sections in an offer curve finally we establish sufficient conditions for an offer curve to be locally optimal

cone metric spaces and fixed point theorems of contractive mappings in this paper we introduce cone metric spaces prove some fixed point theorems of contractive mappings on cone metric spaces

positive line bundles on arithmetic varieties p 5 u integers is defined to count the volume of the lattice i l n of integral sections in the space j7 lon of real sections with supremum norm we want to prove that the leading term of this hilbert function as n f oo is given in terms of the height of x in as 1 this is known as a theorem of gillet and soule gs2 if x has a regular generic fiber beside this known result our proof uses hironaka s theorem on resolutions of singularities and minkowski s theorem on successive minima by hironaka s theorem we may construct

on the generalized hyers ulam stability of multi quadratic mappings a mapping f v email protected w where v is a commutative group w is a linear space and n 2 is an integer is called multi quadratic if it is quadratic in each variable in this paper we prove the generalized hyers ulam stability of multi quadratic mappings in banach spaces and complete non archimedean spaces

on approximate roots of polynomials and systems of polynomials in ultrametric banach algebras abstract we present methods that allow to estimate the distance between the approximate and exact zeros of some polynomial equations and systems of them also infinite in ultrametric banach algebras to make our results more useful we consider that issue in a more general situation i e for some functional equations of polynomial form moreover we do it almost everywhere with respect to a given ideal as an auxiliary tool we prove an ultrametric version of a fixed point theorem in some function spaces also almost everywhere our investigations have been motivated by several previous outcomes and the notion of ulam stability

a fixed point technique for investigating the stability of α β γ derivations on lie c algebras abstract we apply a fixed point theorem to investigate some new stability results for derivations on lie c algebras associated with the generalized cauchy jensen type additive functional equation i 1 n f x i 1 n 1 j 1 j i n x j 2 i 1 n f x i

optimal offer construction in electricity markets in this paper we study strategies for generators making offers into electricity markets in circumstances where both the demand for electricity and the behaviour of competing generators is unknown but can be represented by a probability distribution given this probability distribution we derive necessary optimality conditions for a broad class of supply offer curves we show how these conditions can be used to construct an optimal solution for a simple example we also consider the case in which a generator is restricted in the number of prices at which power can be offered

on the rational approximations to the powers of an algebraic number solution of two problems of mahler and mendes france about fifty years ago mahler ma proved that i f 1 is rational but not an integer and i f 0 l all of whose conjugates different from cr have absolute value less than 1 note that rational integers larger than 1 are pisot numbers according to our definition mahler also added that it would be of some interest to know which algebraic numbers have the same property as the rationals in the theorem now it seems that even replacing ridout s theorem with the modern versions of roth s theorem valid for several valuations and approximations in any given number field the method of mahler does not lead to a complete solution to his question one of the objects of the present paper is to answer mahler s question completely our methods will involve a suitable version of the schmidt subspace theorem which may be considered as a multi dimensional extension of the results mentioned by roth mahler and ridout we state at once our first theorem where as usual we denote by iix the distance of the complex number x from the nearest integer in z i e

a new type of approximation for the radical quintic functional equation in non archimedean 2 β banach spaces abstract let r be the set of real numbers in this paper we first introduce the notions of non archimedean 2 normed spaces x and we will reformulate the fixed point theorem 10 theorem 1 in this space after it we introduce and solve the radical quintic functional equation f x 5 y 5 5 f x f y x y r also under some weak natural assumptions on the function r r x 0 we show that this theorem is a very efficient and convenient tool for proving the hyperstability results when f r x satisfy the following radical quintic inequality f x 5 y 5 5 f x f y z x y z x y r 0 z x with x y

a quantitative version of the absolute subspace theorem the celebrated subspace theorem of w m schmidt 12 says the following subspace theorem let l1 ln be linearly independent linear forms in n variables with real or complex algebraic coefficients suppose 0 consider the inequality 1 1 l1 x ln x 0 put 1 5 q c x r li x qi 1 i n then the parametric subspace theorem can be stated as follows parametric subspace theorem let c be a fixed tuple satisfying 1 3 let 0 then there are finitely many proper linear subspaces t1 tb of q n such that for any q which is sufficiently large there exists a subspace ti t1 tb with 1 6 q q c z ti

fixed point theorems and the ulam hyers stability in non archimedean cone metric spaces abstract let x p be a metric space with a k valued non archimedean metric p in this paper we prove the existence and approximation of a fixed point for operators f x x satisfying the contractive condition in the form p f x f y q p x y where q k k is an increasing operator then we study the generalized ulam hyers stability of fixed point equations we next obtain an extension of the krasnoselskii fixed point theorem for the sum of two operators finally an application to functional equations is given

a note on cone metric fixed point theory and its equivalence abstract the main aim of this paper is to investigate the equivalence of vectorial versions of fixed point theorems in generalized cone metric spaces and scalar versions of fixed point theorems in general metric spaces in usual sense we show that the banach contraction principles in general metric spaces and in t v s cone metric spaces are equivalent our theorems also extend some results in huang and zhang 2007 l g huang x zhang cone metric spaces and fixed point theorems of contractive mappings j math anal appl 332 2007 1468 1476 rezapour and hamlbarani 2008 sh rezapour r hamlbarani some notes on the paper cone metric spaces and fixed point theorems of contractive mappings j math anal appl 345 2008 719 724 and others

a fixed point theorem in n banach spaces and ulam stability abstract motivated by some issues in ulam stability we prove a fixed point theorem for operators acting on some classes of functions with values in n banach spaces we also present applications of it to ulam stability of eigenvectors and some functional and difference equations

finding supply function equilibria with asymmetric firms firms compete in supply functions when they offer a schedule of prices and quantities into a market for example this occurs in many wholesale electricity markets we study the equilibrium behaviour when firms differ both with regard to their costs and their capacities we characterize strong equilibrium solutions in which given the other players supply functions optimal profits are achieved for every demand realisation if the demand can be low enough for it to be met economically with supply from just one firm then the supply function equilibria are ordered in a natural way we consider equilibria in which for the highest levels of demand all but one of the firms have reached their capacity limit we show that there can be at most one supply function equilibrium with this property we also propose a new numerical method to find asymmetric supply function equilibria using piecewise linear approximations and a discretization of the demand distribution we show that this approach has good theoretical convergence behaviour finally we present numerical results from an implementation using gams to demonstrate that the approach is effective in practice

pricing with markups in industries with increasing marginal costs we study a game that models a market in which heterogeneous producers of perfect substitutes make pricing decisions in a first stage followed by consumers that select a producer that sells at lowest price as opposed to cournot or bertrand competition producers select prices using a supply function that maps prices to production levels solutions of this type of models are normally referred to as supply function equilibria we consider a market where producers convex costs functions are proportional to each other depending on the efficiency of each particular producer we provide necessary and sufficient conditions for the existence of an equilibrium that uses simple supply functions that replicate the cost structure we then specialize the model to monomial cost functions with exponent q 0 which allows us to reinterpret the simple supply functions as a markup applied to the production cost we prove that an equilibrium for the markups exists if and only if the number of producers in the market is strictly larger than 1 q and if an equilibrium exists it is unique the main result for monomials is that the equilibrium nearly minimizes the total production cost when the market is competitive the result holds because when there is enough competition markups are bounded thus preventing prices to be significantly distorted from costs focusing on the case of linear unit cost functions on the production quantities we characterize the equilibrium accurately and refine the previous result to establish an almost tight bound on the worst case inefficiency of equilibria finally we derive explicitly the producers best response for series parallel networks with linear unit cost functions extending our previous result to more general topologies we prove that a unique equilibrium exists if and only if the network that captures the market structure is 3 edge connected for non series parallel markets we provide an example that does not admit an equilibrium on markups

on a direct method for proving the hyers ulam stability of functional equations abstract we study the stability of an equation in a single variable of the form f x a f h x b f h x with some conditions imposed on constants a b and function h the results are later applied for proving the stability by use of a direct method of a couple of functional equations in several variables they let us not only shorten various approaches obtained during the last years but also improve the approximating constants

a fixed point approach to the stability of isometries in this paper we apply a fixed point theorem to the proof of hyers ulam rassias stability property for isometries from a normed space into a banach space in which the parallelogram law holds

selections of set valued maps satisfying a linear inclusion in a single variable abstract we prove that a set valued map satisfying a linear functional inclusion in a single variable admits in appropriate conditions a unique selection satisfying a linear functional equation in a single variable as a consequence there follow results on the hyers ulam stability of the linear functional equation in a single variable and the equation of homomorphism of square symmetric groupoids

a fixed point theorem and the ulam stability in generalized dq metric spaces abstract we prove a fixed point theorem for function spaces that is a very efficient and convenient tool for the investigations of various operator inequalities connected to ulam stability issues in classes of functions taking values in various spaces e g in ultrametric spaces dq metric spaces quasi banach spaces and p banach spaces the theorem is a natural generalization and extension of the classical banach contraction principle and some other more recent results

stability for a family of equations generalizing the equation of p wright affine functions we prove some general stability results for a family of equations which generalizes the equation of p wright affine functions in this way we obtain some hyperstability properties for those equations as well we also provide some applications of those outcomes in proving inequalities characterizing the inner product spaces and stability of homomorphisms of c algebras the main tool in the proofs is a fixed point result in brzd k chudziak pales 2011

local existence of lipschitz continuous solutions of systems of nonlinear functional equations with iterated deviations abstract some conditions which guarantee the local existence of lipschitz continuous solutions of some systems of nonlinear functional equations with complicated deviations dependent of unknown functions are presented

on the ulam hyers stability of first order differential systems with nonlocal initial conditions abstract in this paper we investigate the ulam hyers stability of first order differential systems with nonlocal initial conditions on compact intervals in w 1 1 0 k and on noncompact intervals in w w 1 2 0 with suitable weight function w the novelty consists in the simultaneous use of vectorial norms convergent to zero matrices sobolev spaces for studying the ulam hyers stability of equations with nonlocal conditions on noncompact intervals

varepsilon optimal bidding in an electricity market with discontinuous market distribution function this paper investigates the optimal bidding strategy supply function for a generator offering power into a wholesale electricity market the model has three characteristics the uncertainties facing the generator are described by a single probability function namely the market distribution function the supply function to be chosen is nondecreasing but need not be smooth the objective function is the expected profit which can be formulated as a stieltjes integral along the generator s supply curve in previous work the market distribution function has been assumed smooth but in practice this assumption may not be satisfied this paper focuses on the case that the market distribution function is not continuous and hence an optimal supply function may not exist we consider a modified optimization problem and show the existence of an optimal solution for this problem then we show constructively how such an optimum can be approximated with an varepsilon optimal supply function by undercutting when

on cone metric spaces a survey abstract using an old m krein s result and a result concerning symmetric spaces from s radenovic z kadelburg quasi contractions on symmetric and cone symmetric spaces banach j math anal 5 1 2011 38 50 we show in a very short way that all fixed point results in cone metric spaces obtained recently in which the assumption that the underlying cone is normal and solid is present can be reduced to the corresponding results in metric spaces on the other hand when we deal with non normal solid cones this is not possible in the recent paper m a khamsi remarks on cone metric spaces and fixed point theorems of contractive mappings fixed point theory appl 2010 7 pages article id 315398 doi 10 1115 2010 315398 the author claims that most of the cone fixed point results are merely copies of the classical ones and that any extension of known fixed point results to cone metric spaces is redundant also that underlying banach space and the associated cone subset are not necessary in fact khamsi s approach includes a small class of results and is very limited since it requires only normal cones so that all results with non normal cones which are proper extensions of the corresponding results for metric spaces cannot be dealt with by his approach

on the existence of supply function equilibria supply function equilibria are used in the analysis of divisible good auctions with a large number of identical objects to be sold or bought an important example occurs in wholesale electricity markets despite the substantial literature on supply function equilibria the existence of a pure strategy nash equilibria for a uniform price auction in asymmetric cases has not been established in a general setting in this paper we prove the existence of a supply function equilibrium for a duopoly with asymmetric firms having convex non decreasing marginal costs with decreasing concave demand subject to an additive demand shock provided that the second derivative of the demand function is small enough and not increasing the proof is constructive and also gives insight into the structure of the equilibrium solutions

comments on the core of the direct method for proving hyers ulam stability of functional equations in this short paper the core of the direct method for proving stability of functional equations is described in a clear way and in a quite general form

on approximate solutions of the linear functional equation of higher order abstract we show that under some assumptions every approximate solution of the linear functional equation of higher order in single variable generates a solution of the equation that is close to it we also give a description of a procedure that yields such a solution estimate the distance between those approximate and exact solutions to the equation and discuss the problem of uniqueness moreover as a consequence we obtain some results concerning the hyers ulam stability of the equation

stability of multi additive mappings in non archimedean normed spaces abstract a function f v n w where v is a commutative semigroup w is a linear space and n 1 is an integer is called multi additive if it is additive in each variable in this paper we prove the generalized hyers ulam stability of multi additive mappings in non archimedean normed spaces using the so called direct method

a conditional exponential functional equation and its stability abstract we study the conditional functional equation x y x y f x y f x f y with a given function if is a norm in a real linear space then the equation becomes the well known isosceles orthogonally exponential functional equation we give the descriptions of its solutions and investigate the stability of it we also consider the pexiderized version of the equation

stability of additive mappings in non archimedean fuzzy normed spaces in this paper we introduce a notion of a non archimedean fuzzy norm and study the stability of the cauchy equation in the context of non archimedean fuzzy spaces in the spirit of hyers ulam rassias gavruta as a corollary the stability of the jensen equation is established we indeed present an interdisciplinary relation between the theory of fuzzy spaces the theory of non archimedean spaces and the theory of functional equations

stability of functional equations in single variable abstract this paper discusses hyers ulam stability for functional equations in single variable including the forms of linear functional equation nonlinear functional equation and iterative equation surveying many known and related results we clarify the relations between hyers ulam stability and other senses of stability such as iterative stability continuous dependence and robust stability which are used for functional equations applying results of nonlinear functional equations we give the hyers ulam stability of bottcher s equation we also prove a general result of hyers ulam stability for iterative equations

a fixed point approach to the stability of functional equations in non archimedean metric spaces abstract in this note we prove a simple fixed point theorem for a special class of complete metric spaces namely complete non archimedean metric spaces which are connected with some problems coming from quantum physics p adic strings and superstrings we also show that this theorem is a very efficient and convenient tool for proving the hyers ulam stability of a quite wide class of functional equations in a single variable

on approximate isometries and application to stability of a functional equation abstract in this paper we show a generalized stability of isometries between banach spaces making use of this result we prove the corresponding stability of the functional equation f x y f x f y

stability of the multi jensen equation abstract in this paper we study the stability in the sense of th m rassias of multi jensen equation

finiteness of integral values for the ratio of two linear recurrences let f n n n g n n n be linear recurrent sequences in this paper we are concerned with the well known diophantine problem of the finiteness of the set of natural numbers n such that f n g n is an integer in this direction we have for instance a deep theorem of van der poorten solving a conjecture of pisot he established that if coincides with n then f n g n n n is itself a linear recurrence sequence here we shall prove that if is an infinite set then there exists a nonzero polynomial p such that p n f n g n coincides with a linear recurrence for all n in a suitable arithmetic progression examples like f n 2n 2 g n n 2n 2 n show that our conclusion is in a sense best possible in the proofs we introduce a new method to cope with a notorious crucial difficulty related to the existence of a so called dominant root in an appendix we shall also prove a zero density result for in the cases when the polynomial p cannot be taken a constant

the stability of a cubic type functional equation with the fixed point alternative abstract in this note we investigate the generalized hyers ulam rassias stability for the new cubic type functional equation f x y 2 z f x y 2 z f 2 x f 2 y 2 f x y 2 f x z 2 f x z 2 f y z 2 f y z by using the fixed point alternative the first systematic study of fixed point theorems in nonlinear analysis is due to g isac and th m rassias internat j math math sci 19 1996 219 228

variations on the drygas equation and its stability abstract we study the stability of the drygas functional equation g x y g x y 1 2 g x g y g y 1 mixing the direct method of the proof with the method of the invariant means dropping the assumption about the domain to be an abelian group we assume that the function we are dealing with is central i e g x y g y x is approximatively central i e g x y g y x satisfies the kannappan condition i e g x y z g x z y or that the group is amenable