ulam stability problem for generalized a quadratic mappings abstract it is the purpose of this paper to investigate the stability problem of ulam for an approximate mapping for the following generalized quadratic functional equation i 1 n q j i x j n 1 x i n q i 1 n x i n 2 i 1 n q x i for all x i i 1 n where n 1

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

moduli spaces and formal operads let mg l be the moduli space of stable algebraic curves of genus g with l marked points with the operations that relate the different moduli spaces identifying marked points the family mg l g l is a modular operad of projective smooth deligne mumford stacksm in this paper we prove that the modular operad of singular chains s m q is formal so it is weakly equivalent to the modular operad of its homology h m q as a consequence the up to homotopy algebras of these two operads are the same to obtain this result we prove a formality theorem for operads analogous to the deligne griffiths morgan sullivan formality theorem the existence of minimal models of modular operads and a characterization of formality for operads which shows that formality is independent of the ground field

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

remarks on chern simons invariants the perturbative chern simons theory is studied in a finite dimensional version or assuming that the propagator satisfies certain properties as is the case e g with the propagator defined by axelrod and singer it turns out that the effective bv action is a function on cohomology with shifted degrees that solves the quantum master equation and is defined modulo certain canonical transformations that can be characterized completely out of it one obtains invariants

stability of an alternative functional equation abstract let f s x map an abelian semigroup s into a banach space x we deal with stability of the following alternative functional equation f x y f x f y 0 f x y f x f y we assume that f x y f x f y 1 x y f x y f x f y 2 x y for all x y s where 1 2 s r are given functions and prove that under some additional assumptions on 1 2 there exists a unique additive mapping a s x such that f x a x x for x s where s r is a function which can be explicitly computed starting from 1 and 2

on the general solution and hyperstability of the general radical quintic functional equation in quasi β banach spaces abstract the goal of this paper is to study the general solution of the following general radical quintic functional equation f a x 5 b y 5 5 r f x s f y for f a mapping from the field of real numbers into a vector space where a b r s are fixed nonzero reals also we prove the generalized hyperstability results for the general radical quintic functional equation by using the fixed point theorem cf dung and hang 2018 15 theorem 2 1 in quasi banach spaces namely we show under some weak natural assumptions functions satisfying the above equation approximately in some sense must be actually solutions to it

the lambrechts stanley model of configuration spaces we prove the validity over mathbb r of a commutative differential graded algebra model of configuration spaces for simply connected closed smooth manifolds answering a conjecture of lambrechts stanley we get as a result that the real homotopy type of such configuration spaces only depends on the real homotopy type of the manifold we moreover prove if the dimension of the manifold is at least 4 that our model is compatible with the action of the fulton macpherson operad weakly equivalent to the little disks operad when the manifold is framed we use this more precise result to get a complex computing factorization homology of framed manifolds our proofs use the same ideas as kontsevich s proof of the formality of the little disks operads

stability of multi jensen mappings in non archimedean normed spaces in this paper we study the stability of the multi jensen mappings in non archimedean normed spaces the results improve and extend some recent results

note on nonstability of the linear functional equation of higher order we provide a complete solution of the problem of hyers ulam stability for a large class of higher order linear functional equations in single variable with constant coefficients we obtain this by showing that such an equation is nonstable in the case where at least one of the roots of the characteristic equation is of module 1 our results are related to the notions of shadowing in dynamical systems and computer science and controlled chaos they also correspond to some earlier results on approximate solutions of functional equations in single variable

on approximation of approximate solutions of dhombres equation abstract let f s x map an abelian semigroup s into a banach space x we deal with stability of dhombres equation f x f y 0 f x y f x f y we assume that f x f y 1 x y f x y f x f y 2 x y for all x y s where 1 2 s r are given functions and prove that under some additional assumptions on 1 2 there exists a solution of dhombres equation a s x such that f x a x x for x s where s r is a function which can be explicitly computed starting from 1 and 2

stability of a functional equation deriving from cubic and quadratic functions abstract in this paper we establish the general solution of the functional equation 6 f x y 6 f x y 4 f 3 y 3 f x 2 y 3 f x 2 y 9 f 2 y and investigate the hyers ulam rassias stability of this equation

on the hyers ulam stability of functional equations connected with additive and quadratic mappings abstract we investigate some inequalities connected with the hyers ulam stability of three functional equations which have a solution of the form a q where a is an additive mapping and q is a quadratic one

on an equation characterizing multi additive quadratic mappings and its hyers ulam stability in this paper we unify the system of functional equations defining a multi additive quadratic mapping to obtain a single equation we also prove using the fixed point method the generalized hyers ulam stability of this equation thus generalizing some known results

hyers ulam stability of a generalized apollonius type quadratic mapping abstract let x y be linear spaces it is shown that if a mapping q x y satisfies the following functional equation 0 1 q i 1 n z i i 1 n x i q i 1 n z i i 1 n y i 1 2 q i 1 n x i i 1 n y i 2 q i 1 n z i i 1 n x i i 1 n y i 2 then the mapping q x y is quadratic we moreover prove the hyers ulam stability of the functional equation 0 1 in banach spaces

on the quotient stability of a family of functional equations abstract we show some stability results for a family of functional equations that contains the exponential and the golab schinzel functional equations

on the hyers ulam stability of the functional equations that have the quadratic property abstract the hyers ulam stability of the quadratic functional equation 1 on a restricted domain shall be investigated and the result shall be applied to the study of an asymptotic behavior of that equation furthermore the hyers ulam stability problems of another quadratic equation 4 on a restricted domain shall also be treated under the approximately even or odd condition and some asymptotic behaviors of the quadratic mappings and the additive mappings shall be investigated

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

stability of a mixed functional equation in several variables on banach modules abstract in this paper we establish the general solution of the functional equation i 1 n f x i 1 n j 1 n x j i 1 n f x i n f 1 n i 1 n x i n 2 and use a fixed point method to prove its generalized hyers ulam stability in banach modules over a unital banach algebra

continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems we consider a dynamic system containing uncertain elements only the set of possible values of these uncertainties is known based on this information a class of state feedback controls is proposed in order to guarantee uniform ultimate boundedness of every system response within an arbitrarily small neighborhood of the zero state these feedback controls are continuous functions of the state

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

on the hyers ulam stability of the linear differential equation abstract we obtain some results on generalized hyers ulam stability of the linear differential equation in a banach space as a consequence we improve some known estimates of the difference between the perturbed and the exact solutions

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

hyers ulam rassias stability of a linear recurrence abstract in this paper we give a hyers ulam rassias stability result for the first order linear recurrence in banach spaces

remarks on the hyers ulam stability of some systems of functional equations abstract in this paper we present a method that allows to study the hyers ulam stability of some systems of functional equations connected with the cauchy jensen and quadratic equations in particular we generalize and extend some already known results

a generalization of the hyers ulam rassias stability of jensen s equation in this paper we prove a generalization of the stability of the jensen s equation in the spirit of hyers ulam rassias and g vruta

remarks on stability and non stability of the linear functional equation of the first order abstract we prove some results on stability and non stability of the linear functional equation of the first order in single variable moreover we suggest some possible definitions of stability and non stability

on the stability of the first order linear recurrence in topological vector spaces abstract suppose that x is a sequentially complete hausdorff locally convex space over a scalar field k v is a bounded subset of x a n n 0 is a sequence in k 0 with the property lim inf n a n 1 and b n n 0 is a sequence in x we show that for every sequence x n n 0 in x satisfying x n 1 a n x n b n v n 0 there exists a unique sequence y n n 0 satisfying the recurrence y n 1 a n y n b n n 0 and for every q with 1 q lim inf n a n there exists n 0 n such that x n y n 1 q 1 c o n v v b n n 0

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

on the stability of a quadratic jensen type functional equation abstract in this paper we obtain the general solution of the quadratic jensen type functional equation 9f x y z 3 f x f y f z 4 f x y 2 f y z 2 f z x 2 and prove the stability of this equation in the spirit of hyers ulam rassias and g vruta

the generalized hyperstability of general linear equations in quasi banach spaces abstract in this paper we study the hyperstability for the general linear equation in the setting of quasi banach spaces we first extend the fixed point result of brzdek et al 5 theorem 1 in metric spaces to b metric spaces in particular to quasi banach spaces then we use this result to generalize the main results on the hyperstability for the general linear equation in banach spaces to quasi banach spaces we also show that we can not omit the assumption of completeness in 5 theorem 1 as a consequence we conclude that we need more explanations to replace a normed space by its completion in the proofs of some results in the literature

a fixed point approach to stability of functional equations abstract we prove a simple fixed point theorem for some not necessarily linear operators and derive from it several quite general results on the stability of a very wide class of functional equations in single variable

decentralized variable structure control for tracking in non linear systems the problem of tracking time varying trajectories in non linear systems is considered these trajectories could be generated on line by a reference model the problem is tackled by considering the overall system as a set of decentralized linear sub systems which can be modelled as a set of double integrators and treating the non linearities as uncertainties with known bounds on their magnitudes a variable structure controller is designed based on the theory of uncertain dynamical systems the system is shown to track a class of trajectories generated by the reference mode a set of double integrators to within a specified tolerance some analysis is presented to explain the dynamic behaviour of the closed loop system and bounds are derived linking the rate of convergence and the tracking ability to system parameters such as available control and magnitude of the uncertainty simulation results are presented to validate the conclusions

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

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

a generalization of the hyers ulam rassias stability of approximately additive mappings abstract in this paper we prove a generalization of the stability of approximately additive mappings in the spirit of hyers ulam and rassias

hyers ulam stability of the linear differential operator with nonconstant coefficients abstract we obtain some stability results for the linear differential operator of order one in banach spaces as a consequence we derive a result on hyers ulam stability for the linear differential operator of higher order with nonconstant coefficients

construction of continuous solutions and stability for the polynomial like iterative equation abstract most of known results such as existence uniqueness and stability for polynomial like iterative equations were given under the assumption that the coefficient of the first order iteration term does not vanish the existence with a non zero leading coefficient was therefore raised as an open problem it was positively answered for local c 1 solutions later in this paper this problem is answered further by constructing c 0 solutions moreover we discuss the stability of those c 0 solutions which consequently implies a result of the stability for iterative roots

m kontsevich s graph complex and the grothendieck teichmuller lie algebra we show that the zeroth cohomology of m kontsevich s graph complex is isomorphic to the grothendieck teichmuller lie algebra mathfrak grt 1 the map is explicitly described this result has applications to deformation quantization and duflo theory we also compute the homotopy derivations of the gerstenhaber operad they are parameterized by mathfrak grt 1 up to one class or two depending on the definitions more generally the homotopy derivations of the non unital e n operads may be expressed through the cohomology of a suitable graph complex our methods also give a second proof of a result of h furusho stating that the pentagon equation for mathfrak grt 1 elements implies the hexagon equation

hyersulam stability of the iterative equation with a general boundary restriction hyersulam stability has played an important role not only in the theory of functional equations but also in a variety of branches of mathematics such as differential equations integral equations and linear operators in the present paper we will discuss the hyersulam stability of the iterative equation with a general boundary restriction by the construction of a uniformly convergent sequence of functions we prove that for every approximate solution of such an equation there exists an exact solution near it

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

on the rationality homotopy type of configuration spaces in this article we prove that the rational homotopy type of the configuration space of n distinct points on a smooth projective variety x over c is determined by the rational cohomology ring of x together with its canonical orientation class this strengthens a previous result of fulton and macpherson fm see previous issue in this volume a few words are due on the history of differential graded algebras by a dga we shall mean a graded commutative differential graded algebra fulton and macpherson fm gave a model f n for the space f x n by resolving the singularities of the inclusion f x n c xn and using a result of morgan m on the rational homotopy type of smooth quasiprojective varieties over c surprisingly the dga f n depended not only on h x and its orientation class but also on the chern classes of x fulton suggested that one should be able to find another explicit model of f x n independent of chern classes the dga used here e n was found independently by the author and by totaro t totaro s idea was to study the leray spectral sequence of the inclusion f x n c xn by a weight filtration argument he showed that the only nontrivial differential in this spectral sequence is d2m the e2 term with the differential d2m is isomorphic to e n thus by a result of deligne d2 h e n h f x n as rings finally the relations 1 3 of theorem 1 2 are much older than any of the results mentioned above they were discovered by f cohen c see also clma in the study of configuration spaces of ir what happens could be described by saying that h f x n for a smooth projective variety x is related to h f lr2m n in the simplest possible way

ulam s stability of a generalization of the frechet functional equation abstract we prove some general stability and hyperstability results for a generalization of the well known frechet equation stemming from the characterization of the inner product spaces due to jordan and von neumann the main result yields several stability outcomes for various other well known functional equations thus we obtain in particular some new inequalities characterizing the inner product spaces as the main tool we use a fixed point theorem for some function spaces

on the generalized a quadratic mappings associated with the variance of a discrete type distribution abstract in this paper we prove the hyers ulam rassias stability of the following generalized quadratic functional equation of euler lagrange type i 1 n r i q j 1 n r j x i x j i 1 n r i q i 1 n r i x i i 1 n r i 2 i 1 n r i q x i in banach modules over a unital c algebra where n 1 is a positive integer and r i 0 i 1 n is arbitrary given

continuity of iteration and approximation of iterative roots motivated by computing iterative roots for general continuous functions in this paper we prove the continuity of the iteration operators t n defined by t nf f n we apply the continuity and introduce the concept of continuity degree to answer positively the approximation question if lim m f m f can we find an iterative root f m of f m of order n for each email protected n such that the sequence f m tends to the iterative root of f of order n associated with a given initial function we not only give the construction of such an approximating sequence f m but also illustrate the approximation of iterative roots with an example some remarks are presented in order to compare our approximation with the hyers ulam stability

general stability of functional equations of linear type making use of a dynamical systems notion called shadowing we prove a stability result for linear functional equations in metric groups as a corollary we obtain stability of the quadratic functional equation in the case when the target space is a metric group satisfying some local 2 divisibility condition

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

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

generalization of functional equation for the square root spiral abstract in the paper we give the general solution of the linear functional equation p 1 p x c x h x furthermore we discuss its hyers ulam rassias stability and the stability in the sense of ger for the equation p 1 p x c x our results are applied to the square root spiral so that the related results by heuvers et al k j heuvers d s moak b boursaw the functional equation of the square root spiral in th m rassias ed functional equations and inequalities kluwer 2000 pp 111 117 and results by jung and sahoo stability of functional equation for square root spirals appl math lett 15 2002 435 438 are generalized

fixed points of a mapping and hyers ulam stability abstract we show that many general results on hyers ulam stability of some functional equations in a single variable follow immediately from a simple fixed point theorem the theorem is formulated for self maps of some subsets of the space of functions from a nonempty set into the set of reals we also give some applications of that theorem e g in investigations of solutions of some difference equations and functional inequalities

properties of min max controllers in uncertain dynamical systems we discuss properties of min max controllers previously obtained for uncertain dynamical systems in particular we present a new model for switching action and re examine asymptotic stability we demonstrate the attractiveness of switching surfaces and comment on the insensitivity of solutions in those surfaces to the uncertainty

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 the stability of the linear differential equation of higher order with constant coefficients abstract we obtain a result on stability of the linear differential equation of higher order with constant coefficients in aoki rassias sense as a consequence we obtain the hyers ulam stability of the above mentioned equation a connection with dynamical sytems perturbation is established

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

a dynamic programming approach to trajectory planning of robotic manipulators this paper presents a solution to the problem of minimizing the cost of moving a robotic manipulator along a specified geometric path subject to input torque force constraints taking the coupled nonlinear dynamics of the manipulator into account the proposed method uses dynamic programming dp to find the positions velocities accelerations and torques that minimize cost since the use of parametric functions reduces the dimension of the state space from 2n for an n jointed manipulator to two the dp method does not suffer from the curse of dimensionality while maintaining the elegance of our previous trajectory planning method we have developed the dp method for the general case where 1 the actuator torque limits are dependent on one another 2 the cost functions can have an arbitrary form and 3 there are constraints on the jerk or derivative of the acceleration also we have shown that the dp solution converges as the grid size decreases as numerical examples the trajectory planning method is simulated for the first three joints of the pacs arm which is a cylindrical arm manufactured by the bendix corporation

approximate solution of radical quartic functional equation related to additive mapping in 2 banach spaces abstract the aim of this paper is first to reformulate the fixed point theorem 4 theorem 1 in 2 banach spaces after it we introduce and solve the radical quartic functional equation f x 4 y 4 4 f x f y we also show that the fixed point methods allow to investigate ulam s type stability of radical quartic functional equation in 2 banach spaces

new class of control laws for robotic manipulators part 1 non adaptive case a new class of exponentially stabilizing control laws for joint level control of robot arms is introduced it has recently been recognized that the non linear dynamics associated with robotic manip

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