Research Article | Open Access

# On Ulam's Type Stability of the Cauchy Additive Equation

**Academic Editor:**N. Kallur

#### Abstract

We prove a general result on Ulam's type stability of the functional equation , 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 Isaac and Rassias.

#### 1. Introduction

The issue of stability of functional equations has been a very popular subject of investigations for the last nearly fifty years (see, e.g., [1–8]). Its main motivation was given by Ulam (cf. [9–11]) in 1940 in his talk at the University of Wisconsin. For instance, we can introduce the following definition, which somehow describes the main ideas of such stability notion for equations in two variables ( stands for the set of nonnegative reals).

*Definition 1. *Let be a nonempty set, be a metric space, be nonempty, be an operator mapping into , and be operators mapping nonempty into . We say that the equation
is -stable provided for every and with
there exists a solution of (1) such that

(As usual, denotes the family of all functions mapping a set into a set .) Roughly speaking, -stability of (1) means that every approximate (in the sense of (2)) solution to (1) is always close (in the sense of (3)) to an exact solution to (1). The next theorem is an example of the most classical results.

Theorem 2. *Let and be two normed spaces and let and be fixed real numbers. Let be an operator such that
**
If and is complete, then there is a unique operator that is additive (i.e., for ) and such that
**
If , then is additive.*

It has been motivated by Rassias (see [12–14]) and is composed of the outcomes in [15–17]. Note that Theorem 2 with yields the result of Hyers [9] and it is known (see [17]; cf. also [18, 19]) that for an analogous result is not valid. Moreover, it has been shown in [20] that estimation (5) is optimum for in the general case.

The second statement of Theorem 2, for , can be described as the -hyperstability of the additive Cauchy equation for (for further information on hyperstability see, e.g., [1, 16, 21, 22]; some other recent results can be found in [23–25]). It seems to be of interest that such result does not remain valid if we restrict the domain of to a subsemigroup of the group . The subsequent remark shows this.

*Remark 3. *Let , , , and be given by and for . Then clearly
Moreover,
In fact, suppose, for instance, that . Then , whence .

In this paper we prove a quite general result that allows us to generalize and extend Theorem 2 in various directions.

#### 2. An Auxiliary Result

In the proof of the main theorem in this paper, we use the following fixed point result that can be easily derived from [26, Theorem 2] (cf. [27, Theorem 1] and [28]). For a survey on applications of the fixed point methods for similar issues, see [29].

Theorem 4. *Assume that is a nonempty set, is a complete metric space, , is an operator satisfying the inequality
**
and is an operator defined by
**
Suppose that there exist functions and such that
**
where denotes the th iterate of (i.e., for and for ). Then there exists a unique fixed point of with
**
Moreover,
*

#### 3. The Main Theorem

Given a group , we denote by the family of all automorphisms of . Moreover, for each we write for and we define by .

The next theorem is the main result of this paper.

Theorem 5. *Let and be commutative groups, be a complete metric in that is invariant (i.e., for ), , and
**
where
**
for . Assume that satisfies the inequality
**
Then, for each nonempty such that
**
there exists a unique additive fulfilling the inequality
**
where
*

*Proof. *Let be nonempty and let (16) be valid. Write . Note that (15), with replaced by and , gives

Given , we define operators and by
It is easily seen that, for each , has form (9) with , , and . Moreover, (19) can be written in the following way
(Here and in the sequel, the restriction of to the set is also denoted by ; we believe that this will not cause any confusion.) And
for every , , and . Consequently, for each , also (8) is valid with , , and .

Note that, in view of the definition of ,
So, it is easy to show by induction on that
for , (nonnegative integers), and . Hence,
Now, we can use Theorem 4 with , , , and . According to it, the limit
exists for each and ,
and the function defined by
is a solution of the equation
because is a fixed point of .

Now we show that
for every , , , and .

Since the case is just (15), take and assume that (31) holds for and every , , and . Then, by (24),
Thus, by induction, we have shown that (31) holds for every , , , and . Letting in (31), we obtain the equality
From this we can deduce that is additive for each . The reasoning is very simple, but for the convenience of readers we present it here.

In view of (33), it is only enough to consider the situation . So take and (the case is trivial). Then, by (33),
which yields and consequently .

Next, we prove that each additive satisfying the inequality
with some and , is equal to for each . To this end fix , , and an additive satisfying (35). Note that, by (28) and (35), there is such that
for . Observe yet that and are solutions to (30) for all , because they are additive.

We show that, for each ,
The case is exactly (36). So fix and assume that (37) holds for . Then, in view of (24),
Thus we have shown (37). Now, letting in (37), we get
Since and are additive, we have .

In this way, we also have proved that for each (on account of (28)), which yields
This implies (17) with ; clearly, equality (39) means the uniqueness of , as well.

Thus we have completed the proof of Theorem 5.

#### 4. Some Consequences

Theorem 5 yields the subsequent corollary.

Corollary 6. *Let , , , and be as in Theorem 5. Suppose that there exists a nonempty such that (16) holds and
**
Then every satisfying (15) is additive.*

*Proof. *Suppose that satisfies (15). Then, by Theorem 5, there exists an additive such that (17) holds. Since, in view of (41), for , this means that for , whence
which implies that is additive (see the proof of (34)).

The next corollary corresponds to the results on the inhomogeneous Cauchy equation (44) in [30–35].

Corollary 7. *Let , , , and be as in Theorem 5 and . Suppose that
** for some , and there exists a nonempty such that (16) and (41) hold. Then the inhomogeneous Cauchy equation
**
has no solutions in the class of functions .*

*Proof. *Suppose that is a solution to (44). Then
Consequently, by Corollary 6, is additive, whence , which is a contradiction.

*Remark 8. *We have excluded and from the domain of , in Theorem 5, because of the reason which can be easily deduced from the subsequent natural example.

In the rest of this paper, we assume that and are normed spaces, is a subgroup of the group , with , and . For each define by for . Let be given by
with some real , , , and . Then
for every , , and . Hence,
and there is such that
So, it is easily seen that conditions (41) are fulfilled with
and therefore (by Corollary 6) every satisfying (15), with given by (46) is additive.

Clearly, the above reasoning also works (after an easy modification) when the function has the following a bit more involved form
with some real , , , and and additive injections (or ). So, we have the following corollary corresponding to the hyperstability results in [16, 21, 24] (see also [1, 22, 23, 25]).

Corollary 9. *Let be given by (51) with some real , , , and and some additive injections (, resp.). Then every satisfying (15) is additive.*

We also get an analogous conclusion when is given by with some real and such that and some additive injections (or ), because for every , , and . So we have the following hyperstability result, as well (it generalizes to some extend the main outcome in [36]).

Corollary 10. *Let be given by (52) with some , , , and some additive injections (, resp.). Then every satisfying (15) is additive.*

It is easily seen that another example of the function satisfying (41) is given by with some real , , , , , and some additive injections (, resp.), because for every , , and . So, we have yet the following.

Corollary 11. *Let be given by (54) with some real , , , , , and some additive injections (, resp.). Then every satisfying (15) is additive.*

We finish the paper with an example of corollary that generalizes some results in [37] and improves the estimations obtained there.

Corollary 12. *Let be divisible by 2 and let be given by (51) with some real numbers , , and and some additive injections (, resp.). Then, for every satisfying (15), there exists an additive mapping such that
*

*Proof. *Let satisfy (15) and be given by
Then and . Consequently, by Theorem 5 with , there is a unique additive such that

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### References

- N. Brillouët-Belluot, J. Brzdęk, and K. Ciepliñski, “On some recent developments in Ulam's type stability,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 716936, 41 pages, 2012. View at: Publisher Site | Google Scholar - S. Czerwik,
*Functional Equations and Inequalities in Several Variables*, World Scientific, London, UK, 2002. - V. A. Faĭziev, T. M. Rassias, and P. K. Sahoo, “The space of $\left(\psi ,\gamma \right)$-additive mappings on semi-groups,”
*Transactions of the American Mathematical Society*, vol. 354, pp. 4455–4472, 2002. View at: Google Scholar - D. H. Hyers, G. Isac, and T. M. Rassias,
*Stability of Functional Equations in Several Variables*, Birkhäuser, Boston, Mass, USA, 1998. - S.-M. Jung,
*Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis*, Hadronic Press, Palm Harbor, Fla, USA, 2001. - S.-M. Jung,
*Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis*, vol. 48 of*Springer Optimization and Its Applications*, Springer, London, UK, 2011. - Z. Moszner, “On the stability of functional equations,”
*Aequationes Mathematicae*, vol. 77, no. 1-2, pp. 33–88, 2009. View at: Publisher Site | Google Scholar - Z. Moszner, “On stability of some functional equations and topology of their target spaces,”
*Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica*, vol. 11, pp. 69–94, 2012. View at: Google Scholar - D. H. Hyers, “On the stability of the linear functional equation,”
*Proceedings of the National Academy of Sciences of the USA*, vol. 27, pp. 222–224, 1941. View at: Google Scholar - S. M. Ulam,
*A Collection of Mathematical Problems*, Interscience Publishers, New York, NY, USA, 1960. - S. M. Ulam,
*Problems in Modern Mathematics*, John Wiley and Sons, New York, NY, USA, 1964. - T. M. Rassias, “On the stability of the linear mapping in Banach spaces,”
*Proceedings of the American Mathematical Society*, vol. 72, pp. 297–300, 1978. View at: Google Scholar - T. M. Rassias, “Problem,”
*Aequationes Mathematicae*, vol. 39, p. 309, 1990. View at: Google Scholar - T. M. Rassias, “On a modified Hyers-Ulam sequence,”
*Journal of Mathematical Analysis and Applications*, vol. 158, no. 1, pp. 106–113, 1991. View at: Google Scholar - T. Aoki, “On the stability of the linear transformation in Banach spaces,”
*Journal of the Mathematical Society of Japan*, vol. 2, pp. 64–66, 1950. View at: Google Scholar - J. Brzdęk, “Hyperstability of the Cauchy equation on restricted domains,”
*Acta Mathematica Hungarica*, vol. 141, pp. 58–67, 2013. View at: Google Scholar - Z. Gajda, “On stability of additive mappings,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 14, pp. 431–434, 1991. View at: Google Scholar - T. M. Rassias and P. Semrl, “On the behavior of mappings which do not satisfy Hyers-Ulam stability,”
*Proceedings of the American Mathematical Society*, vol. 114, pp. 989–993, 1992. View at: Google Scholar - T. M. Rassias and P. Semrl, “On the Hyers-Ulam Stability of Linear Mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 173, no. 2, pp. 325–338, 1993. View at: Publisher Site | Google Scholar - J. Brzdęk, “A note on stability of additive mappings,” in
*Stability of Mappings of Hyers-Ulam Type*, T. M. Rassias and J. Tabor, Eds., pp. 19–22, Hadronic Press, Palm Harbour, Fla USA, 1994. View at: Google Scholar - J. Brzdęk, “Remarks on hyperstability of the Cauchy functional equation,”
*Aequationes Mathematicae*, vol. 86, pp. 255–267, 2013. View at: Google Scholar - J. Brzdęk and K. Ciepliñski, “Hyperstability and superstability,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 401756, 13 pages, 2013. View at: Publisher Site | Google Scholar - A. Bahyrycz and M. Piszczek, “Hyperstability of the Jensen functional equation,”
*Acta Mathematica Hungarica*, vol. 142, pp. 353–365, 2014. View at: Publisher Site | Google Scholar - M. Piszczek, “Remark on hyperstability of the general linear equation,”
*Aequationes Mathematicae*. View at: Publisher Site | Google Scholar - M. Piszczek and J. Szczawińska, “Hyperstability of the Drygas functional equation,”
*Journal of Function Spaces and Applications*, vol. 2013, Article ID 912718, 4 pages, 2013. View at: Publisher Site | Google Scholar - J. Brzdęk and K. Ciepliñski, “A fixed point approach to the stability of functional equations in non-Archimedean metric spaces,”
*Nonlinear Analysis: Theory, Methods and Applications*, vol. 74, no. 18, pp. 6861–6867, 2011. View at: Publisher Site | Google Scholar - J. Brzdȩk, J. Chudziak, and Z. Páles, “A fixed point approach to stability of functional equations,”
*Nonlinear Analysis: Theory, Methods and Applications*, vol. 74, no. 17, pp. 6728–6732, 2011. View at: Publisher Site | Google Scholar - L. Cădariu, L. Găvruţa, and P. G. Găvruţa, “Fixed points and generalized Hyers-Ulam stability,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 712743, 10 pages, 2012. View at: Publisher Site | Google Scholar - K. Ciepliñski, “Applications of fixed point theorems to the Hyers-Ulam stability of functional equations: a survey,”
*Annals of Functional Analysis*, vol. 3, pp. 151–164, 2012. View at: Google Scholar - C. Borelli Forti, “Solutions of a nonhomogeneous Cauchy equation,”
*Radovi Matematicki*, vol. 5, pp. 213–222, 1989. View at: Google Scholar - B. Ebanks, “Generalized Cauchy difference functional equations,”
*Aequationes Mathematicae*, vol. 70, no. 1-2, pp. 154–176, 2005. View at: Publisher Site | Google Scholar - B. Ebanks, “Generalized Cauchy difference equations. II,”
*Proceedings of the American Mathematical Society*, vol. 136, no. 11, pp. 3911–3919, 2008. View at: Publisher Site | Google Scholar - B. Ebanks, P. Kannappan, and P. K. Sahoo, “Cauchy differences that depend on the product of arguments,”
*Glasnik Matematicki*, vol. 27 (47), pp. 251–261, 1992. View at: Google Scholar - I. Fenyö and G. L. Forti, “On the inhomogeneous Cauchy functional equation,”
*Stochastica*, vol. 5, pp. 71–77, 1981. View at: Google Scholar - A. Járai, G. Maksa, and Z. Páles, “On Cauchy-differences that are also quasisums,”
*Publicationes Mathematicae*, vol. 65, no. 3-4, pp. 381–398, 2004. View at: Google Scholar - J. Brzdęk, “A hyperstability result for the Cauchy equation,”
*Bulletin of the Australian Mathematical Society*, vol. 89, no. 1, 2014. View at: Publisher Site | Google Scholar - G. Isac and Th. M. Rassias, “Functional inequalities for approximately additive mappings,” in
*Stability of Mappings of Hyers-Ulam Type*, T. M. Rassias and J. Tabor, Eds., pp. 117–125, Hadronic Press, Palm Harbour, Fla, USA, 1994. View at: Google Scholar

#### Copyright

Copyright © 2014 Janusz Brzdęk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.