- Research
- Open Access

# A fixed point theorem and the Hyers-Ulam stability in Riesz spaces

- Bogdan Batko
^{1, 2}and - Janusz Brzdęk
^{1}Email author

**2013**:138

https://doi.org/10.1186/1687-1847-2013-138

© Batko and Brzdęk; licensee Springer 2013

**Received:**8 January 2013**Accepted:**23 April 2013**Published:**14 May 2013

## Abstract

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.

## Keywords

- Hyers-Ulam type stability
- fixed point
- Riesz space
- square symmetric groupoid

## 1 Introduction

The Hyers-Ulam stability for functional, but also for difference, differential and integral equations, is a very quickly growing area of investigations (for more details and further references, see, *e.g.*, [1–7]; examples of some recent results can be found in [8–20]). It is related to the notions of shadowing (see, *e.g.*, [21–23]) as well as to the theories of perturbation (see, *e.g.*, [24, 25]) and optimization. The first known result on such stability is due to Pólya and Szegö [26] and reads as follows.

*For every real sequence*
${({a}_{n})}_{n\in \mathbb{N}}$
*with*
${sup}_{n,m\in \mathbb{N}}|{a}_{n+m}-{a}_{n}-{a}_{m}|\le 1$
*, there is a real number*
*ω*
*such that*
${sup}_{n\in \mathbb{N}}|{a}_{n}-\omega n|\le 1$
*. Moreover,*
$\omega ={lim}_{n\to \mathrm{\infty}}{a}_{n}/n$
*.*

But it was Ulam (*cf.* [27, 28]) who in 1940 gave the main motivation for the study of that subject, at the University of Wisconsin, where he presented some unsolved problems and in particular the following one.

*Let*
${G}_{1}$
*be a group and*
$({G}_{2},d)$
*a metric group. Given*
$\epsilon >0$
*, does there exist*
$\delta >0$
*such that if*
$f:{G}_{1}\to {G}_{2}$
*satisfies*
$d(f(xy),f(x)f(y))<\delta $
*for all*
$x,y\in {G}_{1}$
*, then a homomorphism*
$T:{G}_{1}\to {G}_{2}$
*exists with*
$d(f(x),T(x))<\epsilon $
*for all*
$x,y\in {G}_{1}$
*?*

In 1941 Hyers [27] published an answer to it, which is presented below in a bit generalized form (see, *e.g.*, [4]).

*Let*$(X,+)$

*be a commutative semigroup,*

*Y*

*be a Banach space and*$\epsilon >0$

*. Then for every*$g:X\to Y$

*with*${sup}_{x,y\in X}\parallel g(x+y)-g(x)-g(y)\parallel \le \epsilon $

*there is a unique*$f:X\to Y$

*such that*${sup}_{x\in X}\parallel g(x)-f(x)\parallel \le \epsilon $

*and*

By now we express that result simply saying that *the Cauchy functional equation* (1) *is Hyers-Ulam stable (or has the Hyers-Ulam stability) in the class of functions* ${Y}^{X}$.

The Hyers-Ulam stability in Riesz spaces have already been studied in [29] (with a direct method) and in [11] (with an application of the spectral representation theorem). The main motivation for this kind of investigations follows from the pretty natural concept to pose the stability problem for a given functional equation in the settings of an ordered structure as an alternative for the topological or metric ones. To preserve the spirit of the Hyers-Ulam stability, we do that in a way which allows to sustain the full coincidence with the classical notion in the simplest case of real functions.

The fixed point approach has been already applied in the investigation of the Hyers-Ulam stability, *e.g.*, in [8, 12, 13, 30–37] and it seems that Baker (see [30]) has used this tool for the first time in this field; for a survey on this subject, see [38]. In this paper we continue this direction following the approach presented in [12, 13, 31]. We start our paper with a natural fixed point theorem and next derive some stability results from it.

## 2 Preliminaries

In this paper, as usual, ℕ, ${\mathbb{N}}_{0}$, ℝ and ${\mathbb{R}}_{+}$ denote the sets of all positive integers, nonnegative integers, real numbers and nonnegative real numbers, respectively.

For the readers’ convenience, we quote basic definitions and properties concerning Riesz spaces (see [39]).

**Definition 1** (*cf.* [39], Definition 11.1 and Definition 22.1)

*L*, endowed with a partial order $\le \subset {L}^{2}$, is a

*Riesz space*if $sup\{x,y\}$ exists for all $x,y\in L$ and

we define the absolute value of $x\in L$ by the formula $|x|:=sup\{x,-x\}\ge 0$.

A Riesz space *L* is called *Archimedean* if, for each $x\in L$, the inequality $x\le 0$ holds whenever the set $\{nx:n\in \mathbb{N}\}$ is bounded from above.

It is easily seen that in a Riesz space *L* we have $|v|\ge 0$ and $\alpha u\le \beta u$ for every $v,u\in L$, $u\ge 0$, and $\alpha ,\beta \in \mathbb{R}$, $\alpha \le \beta $.

There are several types of convergence that may be defined according to the order structure. One of them is the *relatively uniform convergence* defined as follows.

**Definition 2** (*cf.* [39], Definition 39.1)

Let *L* be a Riesz space and $e\in L$. A sequence ${\{{f}_{n}\}}_{n\in \mathbb{N}}$ in *L* is said to *converge* *e-uniformly* to an element $f\in L$ whenever, for every $\epsilon >0$, there exists a positive integer ${n}_{0}$ such that $|f-{f}_{n}|\le \epsilon e$ holds for all $n\ge {n}_{0}$. A sequence ${\{{f}_{n}\}}_{n\in \mathbb{N}}$ in *L* is called *e-uniform Cauchy sequence* whenever, for every $\epsilon >0$, there exists a positive integer ${n}_{1}$ such that $|{f}_{m}-{f}_{n}|\le \epsilon e$ holds for all $m,n\ge {n}_{1}$.

Let us point out (see [[39], p.252]) that in a Riesz space *L* that is Archimedean, the *e*-uniform limit of a sequence in it, if exists, is unique and the fact that ${\{{f}_{n}\}}_{n\in \mathbb{N}}$ converges *e*-uniformly to *f* will be denoted by ${lim}_{n\to \mathrm{\infty}}^{e}{f}_{n}=f$ (in particular, if $v\in L$ and ${f}_{n}\ge v$ for $n\in \mathbb{N}$, then $f\ge v$).

**Definition 3** The series ${\sum}_{i=0}^{\mathrm{\infty}}{a}_{i}$ in a Riesz space *L* is said to *converge* *e-uniformly* (for a given $e\in L$) to an element $S\in L$ whenever the sequence ${S}_{n}:={\sum}_{i=0}^{n}{a}_{i}$ converges *e*-uniformly to *S*. If *L* is Archimedean, then we write ${\sum}_{i=0}^{\mathrm{\infty}}{a}_{i}\stackrel{e}{=}S$.

**Definition 4** (*cf.* [39], Definition 39.3)

A Riesz space *L* is called *e-uniformly complete* (with a given $e\in L$) whenever every *e*-uniform Cauchy sequence has an *e*-uniform limit.

There is a large class of spaces satisfying the above conditions. In particular, every Dedekind *σ*-complete space (that is such that any non-empty at most countable subset which is bounded from above has a supremum) is an Archimedean *e*-uniformly complete space for every $e\ge 0$ (see [[39], pp.125, 252, 253]).

## 3 Main result

In what follows *X* is a nonempty set and *Y* is an Archimedean *u*-uniformly complete Riesz space for some given $u\in Y$.

We start that part with a lemma that is quite elementary. However, since we use it several times, we present it with a proof for the convenience of readers.

**Lemma 1** *Let* *L* *be an Archimedean Riesz space and* $e\in L$. *The sequence of rests* ${r}_{n}\stackrel{e}{:=}{\sum}_{i=n+1}^{\mathrm{\infty}}{a}_{i}$ *of an* *e*-*uniformly convergent series* ${\sum}_{i=0}^{\mathrm{\infty}}{a}_{i}$ *in* *L* *is* *e*-*uniformly convergent to* 0.

*Proof* Let us observe that $|{\sum}_{i=n+1}^{m}{a}_{i}-S+{S}_{n}|=|{S}_{m}-{S}_{n}-S+{S}_{n}|=|{S}_{m}-S|$. Consequently, ${\sum}_{i=n+1}^{\mathrm{\infty}}{a}_{i}\stackrel{e}{=}S-{S}_{n}\in L$ as ${S}_{m}$ converges *e*-uniformly to *S* and, therefore, ${r}_{n}$ is well defined. Moreover, we have ${lim}_{n\to \mathrm{\infty}}^{e}{r}_{n}={lim}_{n\to \mathrm{\infty}}^{e}(S-{S}_{n})=0\in L$. □

The following two hypotheses will be useful.

Observe that Λ is monotone with respect to the pointwise ordering (because ${L}_{i}$ is nonnegative). Our main theorem reads as follows.

**Theorem 1**

*Let us assume that hypotheses*(H1)

*and*(H2)

*are fulfilled and that functions*$\phi ,\epsilon ,S:X\to Y$

*satisfy the conditions*

*Then*$\mathcal{J}$

*has a unique fixed point*$\psi \in {Y}^{X}$

*with*

*Moreover*,

*Proof* The proof is somewhat similar to the proof of [[12], Theorem 1], but since there are also significant differences between them, we present it here.

*i.e.*, (3)), we have

*u*-uniform Cauchy sequence for any $t\in X$, and therefore

*u*-uniformly convergent, as

*Y*is

*u*-uniformly complete. Let us denote by $\psi (t)$ its relative uniform limit. Using (9) with $n=0$, we have

*u*-uniform convergence of ${({(\mathcal{J}\phi )}^{k}(t))}_{k\in \mathbb{N}}$ to $\psi (t)$ and ${({S}_{k-1}(t))}_{k\in \mathbb{N}}$ to $S(t)$, for $t\in X$, we derive

which results in (6) as *Y* is Archimedean.

*ψ*is a fixed point of $\mathcal{J}$. Having applied (2) with $a=\psi $ and $b={\mathcal{J}}^{n-1}\phi $, for any $t\in X$, one has

which means that $\mathcal{J}\psi =\psi $ as *Y* is Archimedean.

*ψ*, assume that ${\psi}_{1},{\psi}_{2}\in {Y}^{X}$ are two fixed points of $\mathcal{J}$ satisfying

Thus we have proved (15). Letting $n\to \mathrm{\infty}$ and taking into account Lemma 1, we infer that ${\psi}_{1}={\psi}_{2}$. □

**Remark 1** If in (2) we take $k=1$, ${f}_{1}(x)=x$ and ${L}_{1}(x)=\lambda \in [0,1)$ for $x\in X$, then Theorem 1 becomes a result corresponding to the classical Banach contraction principle.

**Remark 2**We can associate to a given $u\in Y$ an extended (

*i.e.*, admitting the infinite value) norm on

*Y*denoted by ${\parallel \cdot \parallel}_{u}$ and defined by

From Theorem 1 we can easily deduce the following corollary.

**Corollary 1**

*Let hypothesis*(H1)

*be fulfilled*, ${\mathrm{\Lambda}}_{0}:{\mathbb{R}}_{+}^{X}\to {\mathbb{R}}_{+}^{X}$

*be a linear operator given by*

*and functions*$\phi :X\to Y$

*and*${\epsilon}_{0}:X\to {\mathbb{R}}_{+}$

*satisfy the conditions*

*Then*$\mathcal{J}$

*has a unique fixed point*$\psi \in {Y}^{X}$

*with*

*Moreover*, *ψ* *is given by* (7).

*Proof* Define $S,\epsilon :X\to Y$ by $S(t):={S}_{0}(t)u$ and $\epsilon (t):={\epsilon}_{0}(t)u$ for $t\in X$. Next let $\mathrm{\Lambda}:{Y}^{X}\to {Y}^{X}$ be given by (3). Then it is easily seen that (4) and (5) are valid. Consequently, in view of Theorem 1, $\mathcal{J}$ has a unique fixed point $\psi \in {Y}^{X}$ such that (6) holds; moreover, *ψ* is given by (7). Clearly, (6) and (19) are equivalent, whence we derive the statement concerning the uniqueness of *ψ*. □

## 4 The Hyers-Ulam stability

in the class of functions ${Y}^{X}$, with given $F:X\times Y\to Y$ and $\xi :X\to X$.

**Corollary 2**

*Let*$F:X\times Y\to Y$, $\xi :X\to X$, $\lambda :X\to {\mathbb{R}}_{+}$, $\phi ,\epsilon ,S:X\to Y$,

*and*

*Then there is a unique solution*$\psi \in {Y}^{X}$

*of functional equation*(20)

*such that*

*Moreover*, *ψ* *is given by* (7) *with* $(\mathcal{T}g)(t)=F(t,g(\xi (t)))$ *for* $g\in {Y}^{X}$, $t\in X$.

*Proof* Use Theorem 1 with $k=1$, ${f}_{1}=\xi $, ${L}_{1}=\lambda $, and $(\mathcal{J}g)(t)=F(t,g(\xi (t)))$ for $g\in {Y}^{X}$, $t\in X$. □

**Example 1** If there is $\alpha \in {\mathbb{R}}_{+}$ such that (in Corollary 2) $\lambda (t)\le \alpha $ for $t\in X$, then clearly we get $S(t)\le {\sum}_{n=0}^{\mathrm{\infty}}{\alpha}^{n}\epsilon ({\xi}^{n}(t))$ for $t\in X$, provided ${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}^{n}\epsilon ({\xi}^{n}(t))$ is *u*-uniformly convergent in *Y*. Moreover, in the case where $\epsilon (f(t))\le \beta \epsilon (t)$ for all $t\in X$ and some $\beta \in {\mathbb{R}}_{+}$ with $\alpha \beta <1$ (*e.g.*, when $\alpha <1$ and *ε* is a constant function), $S(t)\le \frac{1}{1-\alpha \beta}\epsilon (t)$ for $t\in S$. This means that Corollary 2 implies some analogues of the results in [30] for the Riesz spaces.

Analogously, from Corollary 1 we obtain the following.

**Corollary 3**

*Let*$F:X\times Y\to Y$, $\xi :X\to X$, ${\epsilon}_{0},\lambda :X\to {\mathbb{R}}_{+}$, $\phi :X\to Y$, (21)

*holds*,

*Then there is a unique solution*$\psi \in {Y}^{X}$

*of equation*(20)

*such that*

*Moreover*, *ψ* *is given by* (7).

**Example 2**If (in Corollaries 2 and 3) $F(t,v)=b(t)v+H(t)$ for $t\in X$, $v\in Y$ with some $b:X\to \mathbb{R}$, $\xi :X\to X$ and $H:X\to Y$, then we get stability results for the equation

We also can use Theorem 1 and Corollary 1 to deal with the stability of a bit more involved equations with any number of unknown functions, *e.g.*, as follows.

**Corollary 4**

*Let*$k,m\in \mathbb{N}$, ${L}_{1},\dots ,{L}_{k}:X\to {\mathbb{R}}_{+}$, $G:X\times {Y}^{m}\to Y$, ${f}_{1},\dots ,{f}_{k},{g}_{1},\dots ,{g}_{m}:X\to X$, $\mathrm{\Phi}:X\times {Y}^{k}\to Y$

*and*

*Suppose that functions*$\phi ,\epsilon ,{\mu}_{1},\dots ,{\mu}_{m}:X\to Y$

*satisfy the inequality*

*and*(5)

*holds with*Λ

*given by*(3).

*Then there exists a unique*$\psi :X\to Y$

*satisfying inequality*(24),

*which solves the equation*

*Moreover*,

*where* $({T}_{0}h)(t):=\mathrm{\Phi}(t,h({f}_{1}(t)),\dots ,h({f}_{k}(t)))+G(t,{\mu}_{1}({g}_{1}(t)),\dots ,{\mu}_{m}({g}_{m}(t)))$ *for* $h\in {Y}^{X}$, $t\in X$.

*Proof* It is enough to use Theorem 1 with $\mathcal{J}={T}_{0}$. □

**Corollary 5**

*Let*$k,m\in \mathbb{N}$, ${L}_{1},\dots ,{L}_{k}:X\to {\mathbb{R}}_{+}$, $G:X\times {Y}^{m}\to Y$, ${f}_{1},\dots ,{f}_{k},{g}_{1},\dots ,{g}_{m}:X\to X$,

*and*$\mathrm{\Phi}:X\times {Y}^{k}\to Y$

*satisfy*(28).

*Suppose that functions*$\phi ,{\mu}_{1},\dots ,{\mu}_{m}:X\to Y$

*and*$\epsilon :X\to {\mathbb{R}}_{+}$

*are such that*

*and* (18) *holds with* ${\mathrm{\Lambda}}_{0}$ *given by* (16). *Then there exists a unique* $\psi :X\to Y$ *satisfying inequality* (27) *and equation* (29). *Moreover*, *ψ* *is given by* (30).

*Proof* It is enough to use Corollary 1 with $\mathcal{J}={T}_{0}$. □

In the next part we supply an example of a result concerning the Hyers-Ulam stability of functional equations in several variables.

## 5 Stability of the equation of homomorphism for the square symmetric groupoids

*T*be a nonempty set. A binary operation $\star :{T}^{2}\to T$ is square symmetric provided

*e.g.*, [18, 42–45]). If $(T,+)$ is a commutative semigroup, ${\gamma}_{0}\in T$ and $\alpha ,\beta :T\to T$ are endomorphisms with $\alpha \circ \beta =\beta \circ \alpha $ (for instance, $\alpha ={\beta}^{n}$ for an $n\in {\mathbb{N}}_{0}$), then it is easy to check that $\ast :{T}^{2}\to T$, given by

*T*is a linear space over a field $\mathbb{K}$ and ∗ is defined by

(for instance, with some fixed $A,B\in \mathbb{K}$, $c\in T$ and for functions $f\in {T}^{T}$) is also an example of a homomorphism equation for some square symmetric groupoids.

The next corollary provides an example of stability result for the functional equation of homomorphism for square symmetric groupoids, which corresponds to some outcomes in [18, 42–44].

**Corollary 6**

*Let*$C,D\in \mathbb{R}$, $C\ne -D$, $S,\phi :X\to Y$, $\chi :{X}^{2}\to Y$, $w\in Y$, $\star :{X}^{2}\to X$

*be a square symmetric binary operation*,

*and let*$\rho :X\to X$

*be given by*$\rho (t):=t\star t$

*for*$t\in X$,

*and*

*Then there exists a unique solution*$\mathrm{\Phi}\in {Y}^{X}$

*of the functional equation*

*such that*

*Moreover*,

*Proof*Take $s=t$ in (32). Then we get

such that condition (36) is fulfilled. Moreover, Φ is given by (37).

This means that Φ satisfies (35) (in view of (34) and (37)).

Since $\mathrm{\Phi}\in {X}^{S}$ is the unique solution of functional equation (41) fulfilling (36), and every solution of (35) satisfies (41), we also obtain the statement concerning the uniqueness of Φ. □

**Corollary 7**

*Let*$C,D\in \mathbb{R}$, $C\ne -D$, $\phi :X\to Y$, ${\chi}_{0}:{X}^{2}\to {\mathbb{R}}_{+}$, $w\in Y$, $\star :{X}^{2}\to X$

*be a square symmetric binary operation*,

*let*$\rho :X\to X$

*be given by*$\rho (t):=t\star t$

*for*$t\in X$,

*and*

*Then there is a unique solution*$\mathrm{\Phi}\in {Y}^{X}$

*of equation*(35)

*such that*

*Moreover*, Φ *is given by* (37).

*Proof* We argue as in the proof of Corollary 1. Namely, define $S:X\to Y$ and $\chi :{X}^{2}\to Y$ by $S(t):={S}_{0}(t)u$ and $\chi (t):={\chi}_{0}(t)u$ for $t\in X$. Then (42)-(44) imply (32)-(34). Hence, by Corollary 6, there exists a unique solution $\mathrm{\Phi}\in {Y}^{X}$ of functional equation (35) such that (36) holds; in particular, Φ is given by (37). Clearly (36) and (45) are equivalent, so we also obtain the statement on the uniqueness of Φ. □

In the very particular case where $C=D=1$ and functions *χ* and ${\chi}_{0}$ are constant, Corollaries 6 and 7 yield analogues of the classical result of Hyers [27].

## Declarations

## Authors’ Affiliations

## References

- Agarwal RP, Xu B, Zhang W: Stability of functional equations in single variable.
*J. Math. Anal. Appl.*2003, 288: 852-869. 10.1016/j.jmaa.2003.09.032MathSciNetView ArticleGoogle Scholar - Brillouët-Belluot N, Brzdęk J, Ciepliński K: On some recent developments in Ulam’s type stability.
*Abstr. Appl. Anal.*2012., 2012: Article ID 716936Google Scholar - Czerwik S:
*Functional Equations and Inequalities in Several Variables*. World Scientific, River Edge; 2002.View ArticleGoogle Scholar - Hyers DH, Isac G, Rassias TM Progress in Nonlinear Differential Equations and Their Applications 34. In
*Stability of Functional Equations in Several Variables*. Birkhäuser, Boston; 1998.View ArticleGoogle Scholar - Jung S-M:
*Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis*. Hadronic Press, Palm Harbor; 2001.Google Scholar - Jung S-M:
*Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis*. Springer, New York; 2011.View ArticleGoogle Scholar - Moszner Z: On the stability of functional equations.
*Aequ. Math.*2009, 77: 33-88. 10.1007/s00010-008-2945-7MathSciNetView ArticleGoogle Scholar - Badora R, Brzdęk J: A note on a fixed point theorem and the Hyers-Ulam stability.
*J. Differ. Equ. Appl.*2012, 18: 1115-1119. 10.1080/10236198.2011.559165View ArticleGoogle Scholar - Batko B: On the stability of an alternative functional equation.
*Math. Inequal. Appl.*2005, 8: 685-691.MathSciNetGoogle Scholar - Batko B: Stability of an alternative functional equation.
*J. Math. Anal. Appl.*2008, 339: 303-311. 10.1016/j.jmaa.2007.07.001MathSciNetView ArticleGoogle Scholar - Batko B, Tabor J: Stability of the generalized alternative Cauchy equation.
*Abh. Math. Semin. Univ. Hamb.*1999, 69: 67-73. 10.1007/BF02940863MathSciNetView ArticleGoogle Scholar - Brzdęk J, Chudziak J, Palés Z: A fixed point approach to stability of functional equations.
*Nonlinear Anal.*2011, 74: 6728-6732. 10.1016/j.na.2011.06.052MathSciNetView ArticleGoogle Scholar - Brzdęk J, Ciepliński K: A fixed point approach to the stability of functional equations in non-Archimedean metric spaces.
*Nonlinear Anal.*2011, 74: 6861-6867. 10.1016/j.na.2011.06.050MathSciNetView ArticleGoogle Scholar - Forti G-L: Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations.
*J. Math. Anal. Appl.*2004, 295: 127-133. 10.1016/j.jmaa.2004.03.011MathSciNetView ArticleGoogle Scholar - Forti G-L: Elementary remarks on Ulam-Hyers stability of linear functional equations.
*J. Math. Anal. Appl.*2007, 328: 109-118. 10.1016/j.jmaa.2006.04.079MathSciNetView ArticleGoogle Scholar - Forti G-L, Sikorska J: Variations on the Drygas equation and its stability.
*Nonlinear Anal.*2011, 74: 343-350. 10.1016/j.na.2010.08.004MathSciNetView ArticleGoogle Scholar - Jabłoński W, Reich L: Stability of the translation equation in rings of formal power series and partial extensibility of one-parameter groups of truncated formal power series.
*Osterreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II*2006, 215: 127-137.MathSciNetGoogle Scholar - Páles Z: Hyers-Ulam stability of the Cauchy functional equation on square-symmetric groupoids.
*Publ. Math. (Debr.)*2001, 58: 651-666.Google Scholar - Popa D: Hyers-Ulam-Rassias stability of a linear recurrence.
*J. Math. Anal. Appl.*2005, 309: 591-597. 10.1016/j.jmaa.2004.10.013MathSciNetView ArticleGoogle Scholar - Sikorska J: On a direct method for proving the Hyers-Ulam stability of functional equations.
*J. Math. Anal. Appl.*2010, 372: 99-109. 10.1016/j.jmaa.2010.06.056MathSciNetView ArticleGoogle Scholar - Hayes W, Jackson KR: A survey of shadowing methods for numerical solutions of ordinary differential equations.
*Appl. Numer. Math.*2005, 53: 299-321. 10.1016/j.apnum.2004.08.011MathSciNetView ArticleGoogle Scholar - Palmer K Mathematics and Its Applications 501. In
*Shadowing in Dynamical Systems. Theory and Applications*. Kluwer Academic, Dordrecht; 2000.View ArticleGoogle Scholar - Pilyugin SY Lectures Notes in Mathematics 1706. In
*Shadowing in Dynamical Systems*. Springer, Berlin; 1999.Google Scholar - Chang KW, Howes FA:
*Nonlinear Singular Perturbation Phenomena: Theory and Application*. Springer, Berlin; 1984.View ArticleGoogle Scholar - Lin Z-C, Zhou M-R:
*Perturbation Methods in Applied Mathematics*. Jiangsu Education Press, Nanjing; 1995.Google Scholar - Pólya G, Szegö G:
*Aufgaben und Lehrsätze aus der Analysis I*. Springer, Berlin; 1925.View ArticleGoogle Scholar - Hyers DH: On the stability of the linear functional equation.
*Proc. Natl. Acad. Sci. USA*1941, 27: 222-224. 10.1073/pnas.27.4.222MathSciNetView ArticleGoogle Scholar - Ulam SM:
*Problems in Modern Mathematics*. Wiley, New York; 1964. science ed.Google Scholar - Polat F: Approximate Riesz algebra-valued derivations.
*Abstr. Appl. Anal.*2012., 2012: Article ID 240258Google Scholar - Baker JA: The stability of certain functional equations.
*Proc. Am. Math. Soc.*1991, 112: 729-732. 10.1090/S0002-9939-1991-1052568-7View ArticleGoogle Scholar - Cǎdariu L, Găvruţa L, Găvruţa P: Fixed points and generalized Hyers-Ulam stability.
*Abstr. Appl. Anal.*2012., 2012: Article ID 712743. doi:10.1155/2012/712743Google Scholar - Cǎdariu L, Radu V: Fixed point methods for the generalized stability of functional equations in a single variable.
*Fixed Point Theory Appl.*2008., 2008: Article ID 749392. doi:10.1155/2008/749392Google Scholar - Găvruţa P, Găvruţa L: A new method for the generalized Hyers-Ulam-Rassias stability.
*Int. J. Nonlinear Anal. Appl.*2010, 1(2):11-18.Google Scholar - Jung S-M: A fixed point approach to the stability of isometries.
*J. Math. Anal. Appl.*2007, 329: 879-890. 10.1016/j.jmaa.2006.06.098MathSciNetView ArticleGoogle Scholar - Jung S-M, Kim T-S: A fixed point approach to the stability of the cubic functional equation.
*Bol. Soc. Mat. Mexicana*2006, 12: 51-57.MathSciNetGoogle Scholar - Jung Y-S, Chang I-S: The stability of a cubic type functional equation with the fixed point alternative.
*J. Math. Anal. Appl.*2005, 306: 752-760. 10.1016/j.jmaa.2004.10.017MathSciNetView ArticleGoogle Scholar - Mirzavaziri M, Moslehian MS: A fixed point approach to stability of a quadratic equation.
*Bull. Braz. Math. Soc.*2006, 37: 361-376. 10.1007/s00574-006-0016-zMathSciNetView ArticleGoogle Scholar - Ciepliński K: Applications of fixed point theorems to the Hyers-Ulam stability of functional equations - a survey.
*Ann. Funct. Anal.*2012, 3: 151-164.MathSciNetView ArticleGoogle Scholar - Luxemburg WAJ, Zaanen AC:
*Riesz Spaces*. North-Holland, Amsterdam; 1971.Google Scholar - Brzdęk J, Popa D, Xu B: Selections of set-valued maps satisfying a linear inclusion in a single variable.
*Nonlinear Anal.*2011, 74: 324-330. 10.1016/j.na.2010.08.047MathSciNetView ArticleGoogle Scholar - Trif T: On the stability of a general gamma-type functional equation.
*Publ. Math. (Debr.)*2002, 60: 47-61.MathSciNetGoogle Scholar - Kim GH: On the stability of functional equations with square-symmetric operation.
*Math. Inequal. Appl.*2001, 4: 257-266.MathSciNetGoogle Scholar - Kim GH: Addendum to ‘On the stability of functional equations on square-symmetric groupoid’.
*Nonlinear Anal.*2005, 62: 365-381. 10.1016/j.na.2004.12.003MathSciNetView ArticleGoogle Scholar - Páles Z, Volkmann P, Luce RD: Hyers-Ulam stability of functional equations with a square-symmetric operation.
*Proc. Natl. Acad. Sci. USA*1998, 95: 12772-12775. 10.1073/pnas.95.22.12772MathSciNetView ArticleGoogle Scholar - Tabor J, Tabor J: Stability of the Cauchy functional equation in metric groupoids.
*Aequ. Math.*2008, 76: 92-104. 10.1007/s00010-007-2912-8MathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.