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.

1. Introduction

In the literature there are many characterizations of inner product spaces. The first norm characterization of inner product space was given by Fréchet [1] in 1935. He proved that a normed space is an inner product space if and only if, for all , In the same year Jordan and von Neumann [2] gave the celebrated parallelogram law characterization of an inner product space. Since then numerous further conditions, characterizing the inner product spaces among the normed spaces, have been shown. More than 300 such conditions have been collected in the book of Amir [3]. Many geometrical characterizations are presented in the book by Alsina et al. [4]; for some other see, for example, [58].

The results that we obtain are motivated by the notion of hyperstability of functional equations (see, e.g., [914]), which has been introduced in connection with the issue of stability of functional equations (for more details see, e.g., [15, 16]).

The main tool in the proof of the main theorem is a fixed point result for function spaces from [17] (for related outcomes see [18, 19]). Similar method of the proof has been already applied in [11, 20].

To present the fixed point theorem we introduce the following necessary hypotheses ( stands for the set of nonnegative reals and denotes the family of all functions mapping a set into a set ).(H1) is a nonempty set, is a Banach space, and functions and are given.(H2) is an operator satisfying the inequality (H3) is defined by

Now we are in a position to present the above mentioned fixed point theorem for function spaces (see [17]).

Theorem 1. Let hypotheses (H1)–(H3) be valid and functions and fulfil the following two conditions: Then there exists a unique fixed point of with Moreover,

We start our considerations from the functional equation that is patterned on (1) and therefore quite often named after Fréchet (see, e.g., [21]).

Note that (7) can be written in the form where denotes the Fréchet difference operator defined (for functions mapping a commutative semigroup into a group) by It is easy to check that

Such operators were first considered by Fréchet in [22, 23] (we refer to [24] for more information and further references concerning this subject); so, it is still another motivation for (7) to be called the Fréchet equation.

Let us yet observe (see [25]) that, alternatively, (7) can be written in the form where and ; that is, is the Cauchy difference of of the second order.

We prove the subsequent theorem, which corresponds to [26, Theorem 3.1], where the equation has been investigated (the author has named it the superstability result, which is not a precise description, because according to the terminology applied in [1014] it should be rather called the hyperstability result). For some analogous investigations see [2729]. Let us mention yet that stability of (7) has been already studied in [3033] and our results complement the outcomes included there.

It is easy to show that every solution of (7), mapping a commutative group into a real linear space , must be of the form with some additive and quadratic (see, e.g., [21]). Namely, with from (7) we deduce that , and, next, taking in (7), we obtain that the even part of is quadratic while the odd part is a solution of the Jensen equation, whence it is additive.

2. Main Results

The next theorem and corollary are the main results of the paper ( and stand, as usual, for the sets of all positive integers and integers, respectively; moreover, ).

Theorem 2. Let be a commutative group, , a Banach space, and , and satisfying the following three conditions: Then there is a unique function satisfying (7) for all and such that where

Proof. Replacing by and taking in (15) we get Further, put Then, and inequality (18) takes the form Define an operator for by for and . Then it is easily seen that, for each , the operator has the form described in (H3) with , , , and Moreover, for every , , , where for . It is easy to check that, in view of (14), Therefore, since the operator is linear, we have Thus, by Theorem 1 (with and ), for each there exists a function with Moreover, Define by and for and . Then it is easily seen that, by (20),
Next, we show that for every , , .
Fix . For , the condition (30) is simply (15). So, take and suppose that (30) holds for and . Then, for every , which completes the proof of (30).
Letting in (30), we obtain that So, we have proved that for each there exists a function satisfying (7) for and such that
Now, we show that for all . So, fix . Note that satisfies (32) with replaced by . Hence, replacing by and taking in (32), we obtain that for and whence, by the linearity of and (25), for every and . Now, letting we get .
Thus, in view of (33), we have proved that whence we derive (16).
Since (in view of (32)) it is easy to notice that is a solution to (7) (i.e., (7) holds for all ), it remains to prove the statement concerning the uniqueness of . So, let be also a solution of (7) and for . Then, Further, for each . Consequently, with a fixed , for and . Next, analogously as (25), by induction we get for , . This implies that .

Theorem 2 yields at once the following hyperstability result.

Corollary 3. Let be a commutative group, , let be a normed space, , , , and let the conditions (13), (14), and (15) be valid. Assume that Then satisfies (7) for all .

Proof. Note that without loss of generality we may assume that is complete, because otherwise we can replace it by its completion. Next, in view of (40), for each , where is defined by (17). Hence, from Theorem 2, we easily derive that is a solution to (7).

3. Final Remarks

Remark 4. Note that if, in Theorem 2, (this is the case when, e.g., ), then (13) holds and
Further, let be a normed space and with some reals and . Then, the condition (14) is valid, for instance, with for . Obviously, (40) holds, and there exists such that so we obtain (13), as well. Consequently, by Corollary 3, every function , fulfilling the inequality (15), satisfies (7) for all . In this way we have obtained a hyperstability result that corresponds to the recent hyperstability outcomes in [11, 20] and some classical stability results concerning the Cauchy equation (see, e.g., [9, page 3], [15, page 15, 16], and [16, page 2]).

Below, we provide two further simple and natural examples of functions and satisfying the conditions (13) and (14). The first one, clearly, includes the case just described. (a) for with some such that and for , and , where . (b) for with some reals and , , such that .

Clearly, if two functions satisfy the condition (14), then so do their sum and product, with suitable functions . Therefore, we can easily produce numerous examples of such functions. Of course there are some other such examples that are a bit more artificial; for instance, for , where , and are functions with and for , and .

We end the paper with a simple example of applications of our main result.

Corollary 5. Let be a normed space and . Write for . Assume that one of the following two hypotheses is valid. (i)There exist such that for and (ii)There exist reals such that and Then is an inner product space.

Proof. Write for . Then, with and of the forms described in Remark 4 (with ), from Corollary 3 we easily derive that is a solution to (7), which yields the statement.