Alienation and stability of Jensen’s and other functional equations

Abstract

Let S be a semigroup and $\mathbb {K}$ be the field of real or complex numbers. We deal with the stability and alienation of Cauchy’s multiplicative (resp. additive) and Jensen’s functional equations, starting from the inequalities

$$\begin{aligned} \left| f(xy)+f(x\sigma y)+g(xy)-2f(x)-g(x)g(y)\right|\le & {} \varepsilon ,\ \;x,y\in S, \\ \left| f(xy)+f(x\sigma y)+g(xy)-2f(x)-g(x)-g(y)\right|\le & {} \varepsilon ,\ \;x,y\in S, \end{aligned}$$

where $f,g:S\rightarrow \mathbb {K}$ and $\sigma$ is an involutive automorphism on S. We also consider analogous problems for Jensen’s and the quadratic (resp. Drygas) functional equations with an involutive automorphism.

1 Introduction

The notions of alienation and strong alienation, introduced by Dhombres [7], describe the phenomenon that a functional equation, resulting from adding up two functional equations side by side, splits back into the system of these two equations. More precisely given two functional equations $E_{1}(f)=0$ and $E_{2}(g)=0$ for two (possibly the same) functions f and g, we add the equations side by side obtaining

$$\begin{aligned} E_{1}(f)+E_{2}(g)=0, \end{aligned}$$

(1.1)

and we ask, if f and g solve the system

$$\begin{aligned} \left\{ \begin{array}{l} E_{1}(f)=0 \\ E_{2}(g)=0 \end{array} \right. . \end{aligned}$$

(1.2)

In such a situation, we say that the equations in (1.2) are strongly alien to each other.

Then the problem of alienation was studied by many authors, we cite here just a few of them [2, 11,12,13,14, 17, 18, 20, 22, 23].

Let S be a semigroup and $\sigma$ be an involutive automorphism of $S\$(that is $\sigma (xy)=\sigma (x) \sigma (y)$ and $\sigma \circ \sigma (x)=x$ for any $x,y\in S)$. For any complex-valued function f on S we use the notation

$$\begin{aligned} f_{e}=\frac{f+f\circ \sigma }{2}\ \ ,\ \ \ \ f_{o}=\frac{f-f\circ \sigma }{2} . \end{aligned}$$

We say that f is $\sigma$-odd if $f\circ \sigma =-f.$

The functional equations

$$\begin{aligned} f(x+y)=f(x)+f(y),\;x,y\in S, \end{aligned}$$

(1.3)

and

$$\begin{aligned} f(x+y)=f(x)f(y),\;x,y\in S, \end{aligned}$$

are called, respectively, the additive Cauchy and the exponential Cauchy functional equations, while the functional equations

$$\begin{aligned} f(x+y)+f(x+ \sigma y)=2f(x),\;x,y\in S, \end{aligned}$$

and

$$\begin{aligned} f(x+y)+f(x+ \sigma y)=2f(x)+2f(y),\;x,y\in S, \end{aligned}$$

(1.4)

mean here Jensen’s and the quadratic functional equations. We also deal with Drygas functional equation

$$\begin{aligned} f(x+y)+f(x+ \sigma y)=2f(x)+f(y)+f( \sigma y),\;x,y\in S. \end{aligned}$$

An additive function is a solution of (1.3). The monographs by Aczél and Dhombres [1] and by Stetkær [21] contain many references about applications and numerous references concerning these functional equations.

Sobek [20] gave the solutions of the functional equation

$$\begin{aligned} f(x+y)+f(x+ \sigma y)+g(x+y)=2f(x)+g(x)g(y),\,x,y\in S, \end{aligned}$$

(1.5)

where $f,g:S\rightarrow \mathbb {K}$, S is a commutative semigroup, and $\mathbb {K}$ is a field of characteristic different from 2. She proved that Jensen’s and the exponential Cauchy equations are strongly alien.

In [3], the solution of the functional equation

$$\begin{aligned} f(x+y)+f(x-y)+g(x+y)=2f(x)-f(y)-f(-y)-g(x)g(y),\;x,y\in G, \nonumber \\ \end{aligned}$$

(1.6)

was described by Aissi et al. They proved that Drygas and the exponential functional equations are strongly alien on a 2-divisible group G (not necessarily commutative). The same authors [4] proved that Jensen’s equation and the quadratic equation are strongly alien.

Starting from two inequalities (for functions f, g with values in normed spaces)

$$\begin{aligned} \Vert E_1(f)\Vert \le \varepsilon _1\; \;\; \text {and }\;\;\; \Vert E_2(g)\Vert \le \varepsilon _2, \end{aligned}$$

(1.7)

we have that

$$\begin{aligned} \Vert E_{1}(f)+E_{2}(g)\Vert \le \varepsilon _{1}+\varepsilon _{2}, \end{aligned}$$

(1.8)

where $E_{1}$ and $E_{2}$ are two functional equations. Inspired by the alienation phenomenon and stability results (see, e.g., [15, 16]) we discuss the converse case. Now, assume that functions f, g satisfy (1.8 ). The basic question is whether or not there exist nonnegative $\varepsilon _{1}$, $\varepsilon _{2}$ such that the two inequalities in (1.7) hold. If the answer is “yes”, we say that the conditions in (1.7) are alien to each other. Results in this direction were achieved by Ger in [10], Sikorska in [19], and others.

Bourgin [6] studied the following system of inequalities.

$$\begin{aligned} \left\{ \begin{array}{l} \Vert f(x+y)-f(x)-f(y)\Vert \le \varepsilon _{1} \\ \Vert f(xy)-f(x)f(y)\Vert \le \varepsilon _{2} \end{array} \right. , \end{aligned}$$

(1.9)

for f acting between two unital Banach algebras. Assuming the surjectivity of f, he obtained that f had to be a ring homomorphism, that is, f satisfies both $f(x+y)=f(x)+f(y)$ and $f(xy)=f(x)f(y)$. Badora [5] generalized this result getting rid of the surjectivity assumption.

In [10], Ger studied the inequality

$$\begin{aligned} \Vert f(x+y)+f(xy)-f(x)-f(y)-f(x)f(y)\Vert \le \varepsilon , \end{aligned}$$

(1.10)

assuming that f acts from a unital ring $\mathcal {R}$ into a unital commutative Banach algebra $\mathcal {A}$ under some conditions on f. In the case $\mathcal {R}$ is a field and $\mathcal {A}=\mathbb {C}$, he derived that either f is bounded or it is a ring homomorphism, i.e. it satisfies the equations

$$\begin{aligned} f(x+y)=f(x)+f(y)\ \text { and }\ \ f(xy)=f(x)f(y), \end{aligned}$$

which combines the classical superstability effect and the alienation phenomenon.

Another contribution in this direction was made by Adam [2]. He proved a stability result starting from the following inequality

$$\begin{aligned} \left\| f(x+y)+f(x-y)+g(x+y)-2f(x)+2f(y)+g(x)+g(y)\right\| \le \varepsilon , \end{aligned}$$

(1.11)

for functions acting on a 2-divisible abelian group and with values in a Banach space.

Sikorska [19] studied the stability problem and the problem of alienation of the approximate additivity condition from the condition of approximate exponentiality. Starting from the inequality

$$\begin{aligned} \left| f(x+y)-f(x)-f(y)+g(x+y)-g(x)g(y)\right| \le \varepsilon , \end{aligned}$$

(1.12)

where f is a complex valued function defined on an abelian monoid S, she proved that either

• (i) g is bounded and there exists an additive function $a:S\rightarrow \mathbb {C}$ such that $f-a$ is bounded, or

• (ii) g is unbounded and there exists a function ${\tilde{f}}:S\rightarrow \mathbb {C}$ such that

  $$\begin{aligned} {\tilde{f}}(x+y)+g(x+y)={\tilde{f}}(x)+{\tilde{f}}(y)+g(x)g(y),\;\;\;x,y\in S \end{aligned}$$

  and

  $$\begin{aligned} \Vert f(x)-{\tilde{f}}(x)\Vert \le \varepsilon . \end{aligned}$$

Let S be a semigroup that need not be abelian and $\sigma$ be an involutive automorphism on S.The aim of the present paper is to study the problem of stability and alienation phenomenon for some functional equations. Firstly, we will prove that the two inequalities

$$\begin{aligned} \left| f(xy)+f(x \sigma y)-2f(x)\right| \le \varepsilon ,\,x,y\in S, \end{aligned}$$

(1.13)

and

$$\begin{aligned} \left| g(xy)-g(x)g(y)\right| \le \varepsilon ,\,x,y\in S, \end{aligned}$$

(1.14)

on the one hand, and that (1.13) and the inequality

$$\begin{aligned} \left| g(xy)+g(x \sigma y)-2g(x)-2g(y)\right| \le \varepsilon ,\,x,y\in S, \end{aligned}$$

(1.15)

on the other, are alien to each other, under the condition that g is central i.e. $g(xy)=g(yx)$ for all $x,y\in S$. We provide a similar result about the stability and alienation of the additive Cauchy equation and Jensen’s functional equation. Secondly, we will study the stability problem and alienation of Jensen’s and Drygas functional equations, i.e. the alienation of the two inequalities (1.13) and

$$\begin{aligned} \left| g(xy)+g(x \sigma y)-2g(x)-g(y)-g( \sigma y)\right| \le \varepsilon ,\,x,y\in S. \end{aligned}$$

(1.16)

2 Alienation and stability problem of Jensen’s and the exponential Cauchy functional equations

Theorem 1

Let S be a semigroup and $\varepsilon$ be a nonnegative real number. Let $f,g:S\rightarrow \mathbb {K}$, such that g is central, satisfy

$$\begin{aligned} \left| f(xy)+f(x\sigma y)+g(xy)-2f(x)-g(x)g(y)\right| \le \varepsilon , \end{aligned}$$

(2.1)

for all $x,y\in S.$Then

$$\begin{aligned} \left| f(xy)+f(x\sigma y)-2f(x))\right| \le 2\varepsilon , \end{aligned}$$

and

$$\begin{aligned} \left| g(xy)-g(x)g(y)\right| \le 3\varepsilon , \end{aligned}$$

for all $x,y\in S.$ In other words, inequalities (1.13) and (1.14) are alien to each other.

Proof

Let $f,g:S\rightarrow \mathbb {K}$ satisfy (2.1). Making the substitutions (xy, z), $(x\sigma y,z),(x,yz),\$and $(x,\sigma yz)$ in (2.1) we get, respectively, the four inequalities

$$\begin{aligned}{} & {} \mid f(xyz)+f(xy\sigma z)+g(xyz)-2f(xy)-g(xy)g(z)\mid \le \varepsilon , \\{} & {} \mid f(x\sigma yz)+f(x\sigma y\sigma z)+g(x\sigma yz)-2f(x\sigma y)-g(x\sigma y)g(z)\mid \le \varepsilon , \\{} & {} \mid -f(xyz)-f(x\sigma yz)-g(xyz)+2f(x)+g(x)g(yz)\mid \le \varepsilon , \\{} & {} \mid -f(x\sigma yz)-f(xy\sigma z)-g(x\sigma yz)+2f(x)+g(x)g(\sigma yz)\mid \le \varepsilon , \end{aligned}$$

at

$$\begin{aligned}{} & {} \left. \mid 2(f(xy)+f(x\sigma y))-2f(x))-g(x)(g(yz)+g(\sigma yz))\right. \\{} & {} \quad \quad \left. +g(z)(g(x\sigma y)+g(xy))\mid \le 4\varepsilon , \right. \end{aligned}$$

for all $x,y,z\in S$. Thus, setting in the last inequality $z=x$ and taking the centrality of g into account, we find that

$$\begin{aligned} \left| f(xy)+f(x\sigma y)-2f(x))\right| \le 2\varepsilon . \end{aligned}$$

Moreover from the last inequality and (2.1), we derive that

$$\begin{aligned} \mid g(xy)-g(x)g(y)\mid \le 3\varepsilon \;\text{ for }\;x,y\in S. \end{aligned}$$

(2.2)

Therefore we deduce that (1.13) and (1.14) are alien to each other with $\varepsilon _{1}=2\varepsilon$ and $\varepsilon _{2}=3\varepsilon$. $\square$

As a main application, we get from Theorem 1 and the superstability of the functional equation $g(xy)=g(x)g(y)$ on semigroups, the following corollary

Corollary 1

Let S be a semigroup and $\varepsilon$ be a nonnegative real number. Let $f,g:S\rightarrow \mathbb {K}$, such that g is central, satisfy

$$\begin{aligned} \left| f(xy)+f(x\sigma y)+g(xy)-2f(x)-g(x)g(y)\right| \le \varepsilon , \end{aligned}$$

(2.3)

for all $x,y\in S$. Then, either g is bounded or

$$\begin{aligned} g(xy)=g(x)g(y), \end{aligned}$$

and

$$\begin{aligned} \left| f(xy)+f(x\sigma y)-2f(x))\right| \le \varepsilon , \end{aligned}$$

(2.4)

for all $x,y\in S.$

Proof

From Theorem 1 we have $\left| g(xy)-g(x)g(y)\right| \le 3\varepsilon \$for all $x,y\in S$. So putting $f=g$ and $\sigma$ the identity map in [24, Proposition 3.1] we find that either g is bounded or $g(xy)=g(x)g(y)$ for all $x,y\in S.$ Therefore inequality (2.4) follows immediately from (2.3). $\square$

On putting $f=g$ in Theorem 1 and Corollary 1 we get the following corollary about the “stability and alienation” of Jensen’s and the exponential Cauchy equations

Corollary 2

Let S be a semigroup and $\varepsilon$ be a nonnegative real number. Let $f:S\rightarrow \mathbb {K}$ be a central function satisfying

$$\begin{aligned} \left| 2f(xy)+f(x\sigma y)-2f(x)-f(x)f(y)\right| \le \varepsilon ,\;\; \end{aligned}$$

(2.5)

for all $x,y\in S.$ Then

$$\begin{aligned} \left| f(xy)+f(x\sigma y)-2f(x))\right| \le 2\varepsilon , \end{aligned}$$

and

$$\begin{aligned} \left| f(xy)-f(x)f(y)\right| \le 3\varepsilon , \end{aligned}$$

for all $x,y\in S$. Moreover, if f is unbounded then $f(xy)=f(x)f(y)$ and

$$\begin{aligned} \left| f(xy)+f(x\sigma y)-2f(x))\right| \le \varepsilon , \end{aligned}$$

for all $x,y\in S.$

Corollary 3

Let S be a semigroup and $\varepsilon$ be a nonnegative real number. Let $f:S\rightarrow \mathbb {K}$ be a central function satisfying

$$\begin{aligned} \left| 3f(xy)-2f(x)-f(x)f(y)\right| \le \varepsilon , \end{aligned}$$

(2.6)

for all $x,y\in S.$Then

$$\begin{aligned} \left| f(xy)-f(x))\right| \le \varepsilon \ \text { and } \ \left| f(xy)-f(x)f(y)\right| \le 3\varepsilon , \end{aligned}$$

(2.7)

for all $x,y\in S$. Moreover, f is a bounded function.

Proof

On replacing $\sigma$ by the identity map in Corollary 2 we get the inequalities in (2.7). Suppose that f is unbounded. Again using Corollary 2 we get that $f(xy)=f(x)f(y),$ which implies, by using ( 2.6) that

$$\begin{aligned} \left| f(xy)-f(x)\right| \le \frac{\varepsilon }{2}, \end{aligned}$$

for all $x,y\in S$. This gives that

$$\begin{aligned} \left| f(x)(f(y)-1)\right| \le \frac{\varepsilon }{2}\ \text {for all }x,y\in S. \end{aligned}$$

Since f is unbounded we can choose $a\in S$ such that $f(a)\ne 1$, and then we get from the last inequality that

$$\begin{aligned} \left| f(x)\right| \le \frac{\varepsilon }{2\left| f(a)-1\right| }, \end{aligned}$$

for all $x\in S.$ But this contradicts that f is unbounded. So f is bounded. $\square$

3 Alienation and stability problem of Jensen’s and the additive Cauchy functional equations

Theorem 2

Let S be a semigroup and $\varepsilon$ be a nonnegative real number. Let $f,g:S\rightarrow \mathbb {K}$, such that g is central, satisfy

$$\begin{aligned} \left| f(xy)+f(x\sigma y)+g(xy)-2f(x)-g(x)-g(y)\right| \le \varepsilon , \end{aligned}$$

(3.1)

for all $x,y\in S$. Then

$$\begin{aligned} \left| f(xy)+f(x\sigma y)-2f(x))\right| \le 2\varepsilon ,\ \;x,y\in S, \end{aligned}$$

and

$$\begin{aligned} \left| g(xy)-g(x)-g(y)\right| \le 3\varepsilon ,\ \;x,y\in S. \end{aligned}$$

In other words, inequalities (1.13) and (1.3) are alien to each other.

Proof

Let $f,g:S\rightarrow \mathbb {K}$ satisfy (3.1). Making the substitutions (xy, x), $(x\sigma y,x),(x,yx)$,and $(x,\sigma yx)$ in (3.1), we get respectively

$$\begin{aligned}{} & {} \left| f(xyx)+f(xy\sigma x)+g(xyx)-2f(xy)-g(xy)-g(x)\right| \le \varepsilon , \\{} & {} \left| f(x\sigma yx)+f(x\sigma y\sigma x)+g(x\sigma yx)-2f(x\sigma y)-g(x\sigma y)-g(x)\right| \le \varepsilon ,\\{} & {} \left| -f(xyx)-f(x\sigma y\sigma x)-g(xyx)+2f(x)+g(x)+g(yx)\right| \le \varepsilon ,\\{} & {} \left| -f(x\sigma yx)-f(xy\sigma x)-g(x\sigma yx)+2f(x)+g(x)+g(\sigma yx)\right| \le \varepsilon , \end{aligned}$$

for all $x,y\in S$. From the above inequalities, we derive that

$$\begin{aligned}{} & {} \left. \left| 2(-f(xy)-f(x\sigma y)+2f(x))-(g(xy)-g(yx))\right. \right. \\{} & {} \left. \left. -(g(x\sigma y)-g(\sigma yx))\right| \le 4\varepsilon . \right. \end{aligned}$$

Considering that g is central we get that

$$\begin{aligned} \mid f(xy)+f(x\sigma y)-2f(x))\mid \le 2\varepsilon . \end{aligned}$$

(3.2)

Moreover from (3.1) and (3.2), we find that

$$\begin{aligned} \left| g(xy)-g(x)-g(y)\right| \le 3\varepsilon \;\text{ for }\;x,y\in S. \end{aligned}$$

We deduce that (1.13) and (1.3) are alien to each other with $\varepsilon _{1}=2\varepsilon$ and $\varepsilon _{2}=3\varepsilon$. $\square$

4 Alienation and stability problem of Jensen’s and the quadratic functional equations

Theorem 3

Let S be a semigroup and $\varepsilon$ be a nonnegative real number. Let $f,g:S\rightarrow \mathbb {K}$, such that g is central, satisfy

$$\begin{aligned} \left| f(xy)+f(x\sigma y)+g(xy)+g(x\sigma y)-2f(x)-2g(x)-2g(y)\right| \le \varepsilon , \end{aligned}$$

(4.1)

for all $x,y\in S$. Then

$$\begin{aligned} \left| f(xy)+f(x\sigma y)-2f(x)\right| \le 2\varepsilon ,\ \;x,y\in S, \end{aligned}$$

and

$$\begin{aligned} \left| g(xy)+g(x\sigma y)-2g(x)-2g(y)\right| \le 3\varepsilon ,\ \;x,y\in S. \end{aligned}$$

In other words, inequalities (1.13) and (1.15) are alien to each other.

Proof

Let $f,g:S\rightarrow \mathbb {K}$ satisfy (4.1). Making the substitutions (xy, z), $(x\sigma y,z),(x,yz),\$and $(x,\sigma yz)$ in (4.1) we get, respectively the following inequalities

$$\begin{aligned}{} & {} \mid f(xyz)+f(xy\sigma z)+g(xyz)+g(xy\sigma z)-2f(xy)\\{} & {} \quad \quad -2g(xy)-2g(z) \mid \le \varepsilon ,\\{} & {} \mid f(x\sigma yz)+f(x\sigma y\sigma z)+g(x\sigma yz)+g(x\sigma y\sigma z)\\{} & {} \quad \quad -2f(x\sigma y)-2g(x\sigma y)-2g(z)\mid \le \varepsilon ,\\{} & {} \mid -f(xyz)-f(x\sigma y\sigma z)-g(xyz)-g(x\sigma y\sigma z)\\{} & {} \quad \quad +2f(x)+2g(x)+2g(yz) \mid \le \varepsilon ,\\{} & {} \mid -f(x\sigma yz)-f(xy\sigma z)-g(x\sigma yz)-g(xy\sigma z)\\{} & {} \quad \quad +2f(x)+2g(x)+2g(\sigma yz)\mid \le \varepsilon , \end{aligned}$$

for all $x,y,z\in S.\$From the above inequalities, we obtain

$$\begin{aligned}{} & {} \mid 2(-f(xy)-f(x\sigma y)+2f(x))-2(g(xy)-g(yz))\\{} & {} \quad \quad -2(g(x\sigma y)-g(\sigma yz))-4(g(z)-g(x))\mid \le 4\varepsilon . \end{aligned}$$

Setting in the last inequality $z=x$, we find, considering that g is central, that

$$\begin{aligned} \mid f(xy)+f(x\sigma y)-2f(x))\mid \le 2\varepsilon . \end{aligned}$$

(4.2)

Moreover from (4.2) and (4.1), we obtain that

$$\begin{aligned} \mid g(xy)+g(x\sigma y)-2g(x)-2g(y)\mid \le 3\varepsilon \;\text{ for } \;x,y\in S. \end{aligned}$$

(4.3)

We deduce that (1.13) and (1.15) are alien to each other with $\varepsilon _{1}=2\varepsilon$ and $\varepsilon _{2}=3\varepsilon$. $\square$

As an application of Theorem 3, we derive the following result

Corollary 4

Let S be a semigroup and let $\varepsilon$ be a nonnegative real number. Let $f:S\rightarrow \mathbb {K}$ be a central function satisfying

$$\begin{aligned} \mid f(xy)+f(x\sigma y)-2f(x)-f(y)\mid \le \varepsilon , \end{aligned}$$

(4.4)

for all $x,y\in S$. Then

$$\begin{aligned} \mid f(xy)+f(x\sigma y)-2f(x))\mid \le 2\varepsilon ,\;x,y\in S, \end{aligned}$$

(4.5)

and

$$\begin{aligned} \mid f(xy)+f(x\sigma y)-2f(x)-2f(y)\mid \le 3\varepsilon ,\;\text{ for } \;x,y\in S. \end{aligned}$$

(4.6)

Furthermore, f is bounded and

$$\begin{aligned} \left| f(x)\right| \le 3\varepsilon \ \text { for all }x\in S. \end{aligned}$$

(4.7)

Proof

Inequalities (4.5) and (4.6) Follow on puting $f=g$ in Theorem 3, while (4.7) follows from (4.4) combined with (4.5). $\square$

5 Alienation and stability problem of Jensen’s and Drygas functional equations

Theorem 4

Let S be a semigroup and $\varepsilon$ be a nonnegative real number. Let $f,g:S\rightarrow \mathbb {K}$, such that g is central, satisfy

$$\begin{aligned}{} & {} \left| f(xy)+f(x\sigma y)+g(xy)+g(x \sigma y)-2f(x)\right. \end{aligned}$$

(5.1)

$$\begin{aligned}{} & {} \left. \left. -2g(x)-g(y)-g( \sigma y)\right| \le \varepsilon ,\right. \end{aligned}$$

(5.2)

for all $x,y\in S$. Then

$$\begin{aligned} \left| \left( f+g_{e}\right) (xy)+\left( f+g_{e}\right) (x\sigma y)-2\left( f+g_{e}\right) (x))\right| \le 2\varepsilon ,\ \;x,y\in S, \end{aligned}$$

(5.3)

and

$$\begin{aligned} \left| g_{o}(xy)+g_{o}(x\sigma y)-2g_{o}(x)-2g_{o}(y)\right| \le 3\varepsilon ,\ \;x,y\in S. \end{aligned}$$

(5.4)

Proof

Let $f,g:S\rightarrow \mathbb {K}$ satisfy (5.1). Making the substitutions $(xy,z), (x\sigma y,z),(x,yz)$ and $(x,\sigma yz)$ in (5.1) we get

$$\begin{aligned}{} & {} \mid f(xyz)+f(xy\sigma z)+g(xyz)+g(xy\sigma z)-2f(xy)\\{} & {} \quad \quad -2g(xy)-2g_{o}(z)\mid \le \varepsilon ,\\{} & {} \mid f(x\sigma yz)+f(x\sigma y\sigma z)+g(x\sigma yz)+g(x\sigma y\sigma z)\\{} & {} \quad \quad -2f(x\sigma y)-2g(x\sigma y)-2g_{o}(z)\mid \le \varepsilon ,\\{} & {} \mid -f(xyz)-f(x\sigma y\sigma z)-g(xyz)-g(x\sigma y\sigma z)\\{} & {} \quad \quad +2f(x)+2g(x)+2g_{o}(yz)\mid \le \varepsilon ,\\{} & {} \mid -f(x\sigma yz)-f(xy\sigma z)-g(x\sigma yz)-g(xy\sigma z)\\{} & {} \quad \quad +2f(x)+2g(x)+2g_{o}(\sigma yz)\mid \le \varepsilon . \end{aligned}$$

For all $x,y,z\in S.\$From the above inequalities, we obtain

$$\begin{aligned}{} & {} \mid 2(f(xy)+f(x\sigma y)-2f(x))+2(g(xy)-g_{o}(yz))\\{} & {} \quad \quad +2(g(x\sigma y)-g_{o}(\sigma yz))+4(g_{o}(z)-g(x))\mid \le 4\varepsilon . \end{aligned}$$

Setting in the last inequality $z=x$, we find, on taking account that g is central, that

$$\begin{aligned} \left| 2(f(xy)+f(x\sigma y)-2f(x))+2(g_{e}(xy)+g_{e}(x\sigma y)-2g_{e}(x))\right| \le 4\varepsilon , \end{aligned}$$

meaning

$$\begin{aligned} \left| \left( f+g_{e}\right) (xy)+\left( f+g_{e}\right) (x\sigma y)-2\left( f+g_{e}\right) (x))\right| \le 2\varepsilon . \end{aligned}$$

(5.5)

Moreover from (5.5) and (5.1), we get that

$$\begin{aligned} \left| g_{o}(xy)+g_{o}(x\sigma y)-2g_{o}(x)-2g_{o}(y)\right| \le 3\varepsilon \;\text{ for }\;x,y\in S. \end{aligned}$$

(5.6)

$\square$

Remark 1

The stability of Jensen’s functional equation on semigroups (i.e. solutions of inequalities like (5.3)) were discussed in e.g. [8, 9].

As an immediate consequence of Theorem 4, we get the following result

Corollary 5

Let S be a semigroup and $\varepsilon$ be a nonnegative real number. Let $f,g:S\rightarrow \mathbb {K}$, such that g is central and $\sigma$-odd, satisfy

$$\begin{aligned} \left| f(xy)+f(x\sigma y)+g(xy)+g(x \sigma y)-2f(x)-2g(x)\right| \le \varepsilon , \end{aligned}$$

for all $x,y\in S$. Then,

$$\begin{aligned} \left| f(xy)+f(x\sigma y)-2f(x))\right| \le 2\varepsilon ,\ \;x,y\in S, \end{aligned}$$

and

$$\begin{aligned} \left| g(xy)+g(x\sigma y)-2g(x)-2g(y)\right| \le 3\varepsilon ,\ \;x,y\in S. \end{aligned}$$