Janusz Brzdek

Janusz Brzdek
AGH University of Science and Technology in Kraków | AGH · Faculty of Applied Mathematics

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Publications (162)
Article
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Ulam stability is motivated by the following issue: how much an approximate solution of an equation differs from the exact solutions to the equation. It is connected to some other areas of investigation, e.g., optimization, approximation theory and shadowing. In this paper, we present and discuss the published results on such stability for function...
Article
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We prove new results on Ulam stability of the nonhomogeneous Cauchy functional equation f(x+y)=f(x)+f(y)+d(x,y) in the class of mappings f from a square symmetric groupoid (H,+) into the set of reals R. The mapping d:H2→R is assumed to be given and satisfy some weak natural assumption. The equation arises naturally, e.g., in the theory of informati...
Article
We show how to use the Banach limit to obtain a fixed point theorem for function spaces. We also present some applications of this result in Ulam stability.
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We investigate Ulam stability of a general delayed differential equation of a fractional order. We provide formulas showing how to generate the exact solutions of the equation using functions that satisfy it only approximately. Namely, the approximate solution ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsf...
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We prove and discuss several fixed point results for nonlinear operators, acting on some classes of functions with values in a b-metric space. Thus we generalize and extend a recent theorem of Dung and Hang (J Math Anal Appl 462:131–147, 2018), motivated by several outcomes in Ulam type stability. As a simple consequence we obtain, in particular, t...
Article
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Citation: Sarwar, M.; Ullah, M.; Aydi, H.; De La Sen, M. Near-Fixed Point Results via Z-Contractions in Metric Interval and Normed Interval Spaces. Symmetry 2021, 13, 2320. https://doi.
Article
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The theory of Ulam stability was initiated by a problem raised in 1940 by S. Ulam and concerning approximate solutions to the equation of homomorphism in groups. It is somehow connected to various other areas of investigation such as, e.g., optimization and approximation theory. Its main issue is the error that we make when replacing functions sati...
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We show how to get new results on Ulam stability of some functional equations using the Banach limit. We do this with the examples of the linear functional equation in single variable and the Cauchy equation.
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We present some applications of the Banach limit in the study of the stability of the linear functional equation in a single variable.
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We study the Ulam-type stability of a generalization of the Fréchet functional equation. Our aim is to present a method that gives an estimate of the difference between approximate and exact solutions of this equation. The obtained estimate depends on the values of the coefficients of the equation and the form of the control function. In the proofs...
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Let S denote the unit circle on the complex plane and ★:S2→S be a continuous binary, associative and cancellative operation. From some already known results, it can be deduced that the semigroup (S,★) is isomorphic to the group (S,·); thus, it is a group, where · is the usual multiplication of complex numbers. However, an elementary construction of...
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We present some hyperstability results for the well-known additive Cauchy functional equation f(x+y)=f(x)+f(y) in n-normed spaces, which correspond to several analogous outcomes proved for some other spaces. The main tool is a recent fixed-point theorem.
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In (Brzdęk and Schwaiger in Aeq Math 92: 975–991, 2018) solutions of far reaching generalizations of the so-called radical functional equationf(p(π(x)+π(y)))=f(x)+f(y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setl...
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We investigate functions that satisfy the condition defining generalized Lie derivations only asymptotically approximately, in a neighbourhood of the origin in a Banach algebra. We show that, under suitable assumptions, they are close to generalized Lie derivations.
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Let \((G,+)\) be a commutative semigroup, \(\tau \) be an endomorphism of G and involution, D be a nonempty subset of G, and \((H,+)\) be an abelian group, uniquely divisible by 2. Motivated by the extension problem of J. Aczél and the stability problem of S.M. Ulam, we show that if the set D is “sufficiently large”, then each function \(g{:} D\rig...
Article
We show how some Ulam stability issues can be approached for functions taking values in 2-Banach spaces. We use the example of the well-known Cauchy equation $f(x+y)=f(x)+f(y)$ , but we believe that this method can be applied for many other equations. In particular we provide an extension of an earlier stability result that has been motivated by a...
Book
This book is an outcome of two Conferences on Ulam Type Stability (CUTS) organized in 2016 (July 4-9, Cluj-Napoca, Romania) and in 2018 (October 8-13, 2018, Timisoara, Romania). It presents up-to-date insightful perspective and very resent research results on Ulam type stability of various classes of linear and nonlinear operators; in particular on...
Article
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During the 16th International Conference on Functional Equations and Inequalities a talk was given concerning the stability of the so-called radical functional equation f(x2+y2)=f(x)+f(y). The second author’s question about the general solution of the equation itself was answered later by the first one. Contrary to some assertions in the literature...
Article
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.
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We prove a fixed point result for nonlinear operators, acting on some classes of functions with values in a dq-metric space, and show some applications of it. The result has been motivated by some issues arising in Ulam stability. We use a restricted form of a contraction condition.
Article
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...
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Let X be a complex linear space, endowed with an extended (that is, admitting infinite values) norm. We prove a fixed point theorem for operators of the form p3L3+p2L2+p1L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \...
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We study a generalization of the Fréchet functional equation, stemming from a characterization of inner product spaces. We show, in particular, that under some weak additional assumptions each solution of such an equation is additive. We also obtain a theorem on the Ulam type stability of the equation. In its proof we use a fixed point result to sh...
Article
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 i...
Article
Motivated by the notion of Ulam stability, we investigate some inequalities connected with the functional equation $$\begin{eqnarray}f(xy)+f(x\unicode[STIX]{x1D70E}(y))=2f(x)+h(y),\quad x,y\in G,\end{eqnarray}$$ for functions $f$ and $h$ mapping a semigroup $(G,\cdot )$ into a commutative semigroup $(E,+)$ , where the map $\unicode[STIX]{x1D70E}:G\...
Article
The aim of this article is to prove a fixed point theorem in 2-Banach spaces and show its applications to the Ulam stability of functional equations. The obtained stability results concern both some single variable equations and the most important functional equation in several variables, namely, the Cauchy equation. Moreover, a few corollaries cor...
Article
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This is an expository paper containing remarks on solutions to some functional equations of a form, that could be called of the radical type. Simple natural examples of them are the following two functional equations fn √xn + yn= f(x) + f(y),f n √xn + yn+ fn p|xn −yn|= 2f(x) + 2f(y) considered recently in several papers, for real functions and with...
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We prove a fixed point theorem for nonlinear operators, acting on some function spaces (of set-valued maps), which satisfy suitable inclusions. We also show some applications of it in the Ulam type stability.
Article
We present results on approximate solutions to the biadditive equation f(x+y,z-w)+f(x-y,z+w)=2f(x,z)-2f(y,w) on a restricted domain. The proof is based on a quite recent fixed point theorem in some function spaces. Our main results state that, under some weak natural assumptions, functions satisfying the equation approximately (in some sense) must...
Article
Composite type functional equations in several variables play a significant role in various branches of mathematics and they have several interesting applications. Therefore their stability properties are of interest. The aim of this paper is to present a survey of some results and methods concerning stability of several of such equations; especial...
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It is a survey on functional equations of a certain type, for functions in two complex variables, which often arise in queueing models. They share a common pattern despite their apparently different forms. In particular, they invariably characterize the probability generating function of the bivariate distribution characterizing a two-queue system...
Article
We suggest a somewhat new approach to the issue of Hyers-Ulam stability. Namely, let A, B be (real or complex) linear spaces, be a linear operator, , and and be semigauges on A and B, respectively. We say that L is HU-stable with constant if for each such that there exists with .
Article
We prove a result on hyperstability (in normed spaces) of the equation that defines the p-Wright affine functions and show that it yields a simple characterization of complex inner product spaces. We also obtain in this way some inequalities describing derivations, Lie derivations and Lie homomorphisms.
Article
Assume that $(G,+)$ is a commutative semigroup, $\unicode[STIX]{x1D70F}$ is an endomorphism of $G$ and an involution, $D$ is a nonempty subset of $G$ and $(H,+)$ is an abelian group uniquely divisible by two. We prove that if $D$ is ‘sufficiently large’, then each function $g:D\rightarrow H$ satisfying $g(x+y)+g(x+\unicode[STIX]{x1D70F}(y))=2g(x)$...
Book
This book presents current research on Ulam stability for functional equations and inequalities. Contributions from renowned scientists emphasize fundamental and new results, methods and techniques. Detailed examples are given to theories to further understanding at the graduate level for students in mathematics, physics, and engineering. Key topic...
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We show that some multifunctions F: K → n(Y), satisfying functional inclusions of the form (formula presented) admit near-selections f: K → Y, fulfilling the functional equation (formula presented) where functions G: K → n(Y), Ψ: K × Yⁿ → Y and ξ1,…, ξn ∈ KK are given, n is a fixed positive integer, K is a nonempty set, (Y, ·) is a group and n(Y) d...
Article
In this paper we present a simple (fixed point) method that yields various results concerning approximate solutions of some difference equations. The results are motivated by the notion of Ulam stability.
Chapter
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During the last five decades, various functional equations possessing a certain structure popped up in many modern disciplines like queuing theory, communication and networks. There is no universal solution methodology available for them, and the closed-form solutions are known only in some particular cases. We address several issues concerning sol...
Article
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This is a survey presenting the most significant results concerning approximate (generalized) derivations, motivated by the notions of Ulam and Hyers-Ulam stability. Moreover, the hyperstability and superstability issues connected with derivations are discussed. In the section before the last one we highlight some recent outcomes on stability of co...
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We discuss some issues concerning solutions of the functional equation $$\begin{aligned} (M(x,y)-xy)P(x,y)=&\;(1-y)(M(x,0)+\widehat{r}_{1}\xi_{2}xy)P(x,0) \\ &+(1-x)(M(0,y)+\widehat{r}_{2}\xi_{1}xy)P(0,y)\\ &-(1-x)(1-y)M(0,0)P(0,0) \end{aligned}$$in the class of analytic functions P mapping \({\overline{D}^2}\)(\({\overline{D}}\) stands for the clo...
Article
We prove some general stability and hyperstability results for a generalization of the well known Fréchet 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 i...
Article
We prove some stability results for the equation $$Af(px * ry) + Bf(qx * sy) = Cf(x) + Df(y),$$ in the class of functions mapping a groupoid (X, ∗) into a Banach space Y , where \({p, q, r, s: X \rightarrow X}\) are endomorphisms of the groupoid, and A, B, C, D are fixed scalars. Particular cases of the equation are the equation of the p-Wright aff...
Article
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∗-homom...
Article
In this paper we propose an approach to Ulam’s stability of some fractional differential equations with a delay. In particular, we show that, under suitable assumptions, every approximate solution of such an equation is close to a unique exact solution to it. We also obtain an auxiliary result on Ulam’s stability of a Volterra integral equation.
Article
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In this paper, we investigate functions that are approximate fixed points of some (possibly nonlinear) operators almost everywhere, with respect to some ideals of sets. We prove that (under suitable assumptions) there exist fixed points of the operators that are “near” those functions. The results are applied to obtain some general stability result...
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We prove that the superstability of some functional equations (e.g., of Cauchy, d’Alembert, Wilson, Reynolds, and homogeneity) is a consequences of two simple theorems. In this way we generalize several classical superstability results.
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We study the Hyers-Ulam stability in a Banach space of the system of first order linear difference equations of the form for (nonnegative integers), where is a given matrix with real or complex coefficients, respectively, and is a fixed sequence in . That is, we investigate the sequences in such that (with the maximum norm in ) and show that, in th...
Article
We present a general method for investigation of the Hyers-Ulam stability of linear equations (differential, difference, functional, integral) of higher orders. It is shown that in many cases, that kind of stability for such equations is a consequence of a similar property of the corresponding first-order equations. Some particular examples of appl...
Article
The issue of Ulam's type stability of an equation is understood in the following way: when a mapping which satisfies the equation approximately (in some sense), it is "close" to a solution of it. In this expository paper, we present a survey and a discussion of selected recent results concerning such stability of the equations of homomorphisms, foc...
Article
We prove some results on approximate solutions of the generalized Volterra integral equation ψ(x)=∫axN(x,t,ψ(α(x,t)))dt+G(x) for continuous functions mapping a real interval II, of the form [a,b)[a,b) or [a,b][a,b] or [a,∞)[a,∞), into a Banach space. We show that, under suitable assumptions, they generate exact solutions of the equation. In particu...
Chapter
We present a survey of selected recent results of several authors concerning stability of the following polynomial functional equation (in single variable)$$\varphi(x)=\sum_{i=1}^m a_i(x)\varphi(\xi_i(x))^{p(i)}+F(x),$$in the class of functions ϕ mapping a nonempty set S into a Banach algebra X over a field \(\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}...
Chapter
We present a survey of several results on selections of some set-valued functions satisfying some inclusions and also on stability of those inclusions. Moreover, we show their consequences concerning stability of the corresponding functional equations.
Chapter
Let \((G,\star)\) and \((H,\circ)\) be square symmetric groupoids and \(S\subset G\) be nonempty. We present some remarks on stability of the following conditional equation of homomorphism$$f(x\star y)=f(x)\circ f(y) \qquad x,y\in S, x\star y\in S\;,$$in the class of functions mapping S into H. In particular, we consider the situation where \(H=\ma...
Chapter
Let W be a Banach space, \((V,+)\) be a commutative group, p be an endomorphism of V, and \(\overline{p}:V\to V\) be defined by \(\overline{p}(x):=x-p(x)\) for \(x\in V\). We present some results on the Hyers–Ulam type stability for the following functional equation$$f(p(x)+\overline{p}(x))+f(\overline{p}(x)+p(y))=f(x)+f(y),$$in the class of functi...
Article
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Let A be a subgroup of a commutative group (G, +) and P be a quadratically closed field. We give the full description of all pairs of functions f : G → P and g : A → P such that f(x + y) + f(x − y) = 2f(x)g(y) for (x, y) ∈ G × A.
Article
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This is a survey paper concerning stability results for the linear functional equation in single variable. We discuss issues that have not been considered or have been treated only briefly in other surveys concerning stability of the equation. In this way, we complement those surveys.
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The fixed point method has been applied for the first time, in proving the stability results for functional equations, by Baker (1991); he used a variant of Banach’s fixed point theorem to obtain the stability of a functional equation in a single variable. However, most authors follow the approaches involving a theorem of Diaz and Margolis. The mai...
Chapter
We study continuous at a point functions that take values in a Riesz space and satisfy some systems of two simultaneous functional inequalities. In this way we obtain in particular generalizations and extensions of some earlier results of Krassowska, Matkowski, Montel, and Popoviciu.
Article
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.
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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 invest...
Article
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We prove a general result on Ulam's type stability of the functional equation f(x + y) = f(x) + f(y), in the class of functions mapping a commutative group into a commutative group. As a consequence, we deduce from it some hyperstability outcomes. Moreover, we also show how to use that result to improve some earlier stability estimations given by I...
Article
We prove a hyperstability result for the Cauchy functional equation \$f(x+ y)= f(x)+ f(y)\$, which complements some earlier stability outcomes of J. M. Rassias. As a consequence, we obtain the slightly surprising corollary that for every function \$f\$, mapping a normed space \${E}_{1} \$ into a normed space \${E}_{2} \$, and for all real numbers \...
Article
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This is an expository paper in which we present some simple observations on the stability of some inhomogeneous functional equations. In particular, we state several stability results for the inhomogeneous Cauchy equation $$f(x+y)=f(x)+f(y)+d(x,y)
Article
We give an answer to a problem formulated by Th. M. Rassias in 1991 concerning stability of the Cauchy equation; we also disprove a conjecture of Th. M. Rassias and J. Tabor. In particular, we present a new method for proving stability results for functional equations.
Article
We present some simple observations on hyperstability for the Cauchy equation on a restricted domain. Namely, we show that (under some weak natural assumptions) functions that satisfy the equation approximately (in some sense), must be actually solutions to it. In this way we demonstrate in particular that hyperstability is not a very exceptional p...
Article
We show that the fixed point methods allow to investigate Ulam’s type stability of additivity quite efficiently and precisely. Using them we generalize, extend and complement some earlier classical results concerning the stability of the additive Cauchy equation. MSC: 39B82, 47H10.
Article
We show that a very classical result, proved by T. Aoki, Z. Gajda and Th. M. Rassias and concerning the Hyers–Ulam stability of the Cauchy equation f(x+y)=f(x)+f(y), can be significantly improved. We also provide some immediate applications of it (among others for the cocycle equation, which is useful in characterizations of information measures)....
Article
Let \({\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}, I = (d, \infty), \phi : I \to I}\) be unbounded continuous and increasing, X be a normed space over \({\mathbb{K}, \mathcal{F} : = \{f \in X^I : {\rm lim}_{t \to \infty} f(t) {\rm exists} \, {\rm in} X\},\hat{a} \in \mathbb{K}, \mathcal{A}(\hat{a}) : = \{\alpha \in \mathbb{K}^I : {\rm lim}_{t \to \i...
Article
Motivated by the notion of Ulam’s type stability and some recent results of S.-M. Jung, concerning the stability of zeros of polynomials, we prove a stability result for functional equations that have polynomial forms, considerably improving the results in the literature.
Article
We prove some stability results for the equation of the p-Wright affine functions.
Article
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We prove a fixed point theorem and show its applications in investigations of the Hyers-Ulam type stability of some functional equations (in single and many variables) in Riesz spaces. MSC: 39B82, 47H10.
Article
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.
Article
We study the generalized Hyers-Ulam stability of Lie homomorphisms on Banach algebras. We show that such homomorphisms can be generated by functions which satisfy quite natural and simple assumptions.
Article
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Let A be a subgroup of a commutative group (G, + ) and P be a commutative ring. We give the full description of functions ${g: G \rightarrow P}$ satisfying $$g(x + y) + g(x - y) = 2g(x)g(y) \quad (x, y) \in A \times G. \quad\quad\quad\quad (A)$$ Thus we obtain a family of functions depicting evolutions of quite arbitrary functions ${g_0 : G \t...
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We prove some results for mappings taking values in ultrametric spaces and satisfying approximately a generalization of the equation of p-Wright affine functions. They are motivated by the notion of stability for functional equations.
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We prove some stability and hyperstability results for the well-known Fréchet equation stemming from one of the characterizations of the inner product spaces. As the main tool, we use a fixed point theorem for the function spaces. We finish the paper with some new inequalities characterizing the inner product spaces.
Article
We study the real (measurable and continuous at a point) functions that satisfy, almost everywhere, some systems of two simultaneous functional inequalities. In particular, we obtain generalizations and extensions of some earlier results of D. Krassowska, J. Matkowski, P. Montel, and T. Popoviciu.
Article
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This is a survey paper concerning the notions of hyperstability and superstability, which are connected to the issue of Ulam’s type stability. We present the recent results on those subjects.
Article
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.
Article
We present a survey of some selected recent developments (results and methods) in the theory of Ulam's type stability. In particular we provide some information on hyperstability and the fixed point methods.
Article
We present some observations concerning stability of the following linear functional equation (in single variable) $$\varphi\bigl(f^m(x) \bigr)=\sum_{i=1}^m a_i(x)\varphi\bigl(f^{m-i}(x) \bigr)+F(x), $$ in the class of functions φ mapping a nonempty set S into a Banach space X over a field \(\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}\), where m is a f...
Article
We prove two general theorems, which appear to be very useful in the investigation of the Hyers-Ulam stability of a higher-order linear functional equation in single variable, with constant coefficients. We give several examples of their applications. In particular, we show that we obtain in this way several fixed point results for a particular ope...
Article
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.
Article
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 stab...
Article
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 re...
Article
Let C be a convex symmetric subset of a real Banach space F and K be a subgroup of the group (F,+). Let E be a real linear space, h:E→F, and h(x+y)−h(x)−h(y)∈K+C for x,y∈E. We prove that under some additional assumptions h can be represented in the form: h=A+γ+κ with an additive (or linear) A:E→F and some γ:E→C, κ:E→K.
Article
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,...
Article
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...
Article
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We prove some Hyers-Ulam stability results for an operator linear equation of the second order that is patterned on the difference equation, which defines the Lucas sequences (and in particular the Fibonacci numbers). In this way, we obtain several results on stability of some linear functional and differential and integral equations of the second...
Article
We prove some stability results for linear recurrences with constant coefficients in normed spaces. As a consequence we obtain a complete solution of the problem of the Hyers–Ulam stability for such recurrences.

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