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., [18]). Its main motivation was given by Ulam (cf. [911]) 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 [1214]) and is composed of the outcomes in [1517]. 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 [2325]). 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 [3035].

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.