Banach Limit and Fixed Point Approach in the Ulam Stability of the Quadratic Functional Equation

Abstract

We show how to obtain new results on the Ulam stability of the quadratic equation $q(a+b)+q(a-b)=2q\left(a\right)+2q\left(b\right)$ using the Banach limit and the fixed point theorem obtained quite recently for some function spaces. The equation is modeled on the parallelogram identity used by Jordan and von Neumann to characterize the inner product spaces. Our main results state that the maps, from the Abelian groups into the set of reals, that satisfy the equation approximately (in a certain sense) are close to its solutions. In this way, we generalize several previous similar outcomes, by giving much finer estimations of the distances between such solutions to the equation. We also present a simplified survey of the earlier related outcomes.

1. Introduction

This is an expository paper in which we present the possibilities of using the Banach limit in the fixed point approach to Ulam stability. We do this with the example of a quadratic function equation. We also provide a brief survey of related stability results, which could be useful to other authors.

The main subject of Ulam stability for equations (e.g., functional, differential, difference, integral) can be basically stated as follows: How much a mapping that roughly (in a particular sense) satisfies an equation differs from a solution of the equation. This type of stability is quite often also called the Ulam–Hyers stability or Hyers–Ulam stability (sometimes some other names are added to this term).

This problem is becoming increasingly popular as a research topic. For the historical background and methods used, see [1,2,3].

Let ${\Omega }_1$ and ${\Omega }_2$ be normed spaces over the set of real numbers $\mathbb{R}$. Examples of the most well-known findings on the Ulam stability of the additive Cauchy functional equation

$$\Psi (u+z)=\Psi \left(u\right)+\Psi \left(z\right),$$

for $\Psi :{\Omega }_1\to {\Omega }_2$, are included in the subsequent theorem (see, e.g., [3]):

Theorem 1.

Let ${\Omega }_0:={\Omega }_1\setminus \left\{0\right\}$, $\sigma \ne 1$ and $\lambda \ge 0$ be fixed real numbers, and $\Psi :{\Omega }_1\to {\Omega }_2$ satisfy

$$\begin{array}{c}\hfill \parallel \Psi (s+t)-\Psi \left(s\right)-\Psi \left(t\right)\parallel \le {\lambda (\parallel s\parallel }^{\sigma }+{\parallel t\parallel }^{\sigma }),s,t\in {\Omega }_0.\end{array}$$

(1)

Then, the following two statements are valid.

(i) 

If ${\Omega }_2$ is complete and $\sigma \ge 0$, then there exists exactly one mapping $G:{\Omega }_1\to {\Omega }_2$ such that

$$\begin{array}{c}\hfill G(a+b)=G\left(a\right)+G\left(b\right),a,b\in {\Omega }_1,\end{array}$$

(2)

and

$$\begin{array}{c}\hfill \parallel \Psi \left(a\right)-G\left(a\right)\parallel \le {\displaystyle \frac{\lambda }{|1-2^{\sigma -1}|}}{\parallel a\parallel }^{\sigma },a\in {\Omega }_0.\end{array}$$

(3)

(ii) 

If $\sigma <0$, then Ψ is a solution to functional equation (2).

In the case of $\sigma =0$, this outcome was first obtained in [4] as an answer to a problem posed by S.M. Ulam in 1940. Subsequently, its extension, for $\sigma \in [0,1)$, was proved by Aoki [5]. An analogous result (as that of Aoki), but with some additional conclusions, was published much later by Rassias [6], who was also the author of the observation that a similar reasoning works for $\sigma <0$. A similar result for $\sigma >1$ was proved by Gajda [7], with an example that for $\sigma =1$ an analogous result is not possible.

Statement (ii) (now called the hyperstability of Equation (2)) was first proven in [8] (we refer to [3,9] for further information and relevant references on hyperstability). If ${\Omega }_1$ is a real inner product space, then estimation (3) is optimal (see [9], Remark 2 and Proposition 1).

These results have been generalized and/or extended in various ways and directions (see, e.g., [1,2,3]). For example, Găvruţa [10] has replaced (1) with the following more general condition:

$$\begin{array}{c}\hfill \parallel \Psi (s+t)-\Psi \left(s\right)-\Psi \left(t\right)\parallel \le \psi (s,t),s,t\in {\Omega }_1,\end{array}$$

(4)

and proved the subsequent theorem.

Theorem 2.

Let $({\Omega }_1,+)$ be an Abelian group, ${\Omega }_2$ be a Banach space, and $\psi :{\Omega }_1^2\to [0,\infty )$ be such that

$${\displaystyle \tilde{\psi }(s,t):=\frac{1}{2}\sum _{n=0}^{\infty }{2}^{-n}\psi (2^{n}s,2^{n}t)<\infty ,s,t\in {\Omega }_1.}$$

If $\Psi :{\Omega }_1\to {\Omega }_2$ satisfies (4), then there exists exactly one mapping $G:{\Omega }_1\to {\Omega }_2$ that is additive (i.e., satisfies Equation (2)) and fulfills the inequality $\parallel \Psi \left(t\right)-G\left(t\right)\parallel \le \tilde{\psi }(t,t)$ for all $t\in {\Omega }_1$.

In [11] (Theorem 8 and Remark 7), it is shown that for ${\Omega }_2=\mathbb{R}$, the following finer results can be obtained for Equation (2) with the technique of a Banach limit:

Theorem 3.

Assume that ${\Omega }_1$ is a normed space, ${\Omega }_0:={\Omega }_1\setminus \left\{0\right\}$, and $\kappa ,\sigma ,\tau \in \mathbb{R}$, $\sigma \ne 1$, $\tau \le \kappa$. Let $h:E_1\to \mathbb{R}$ be such that

$$\begin{array}{c}\hfill {\tau (\parallel s\parallel }^{\sigma }+{\parallel t\parallel }^{\sigma }{)\le \Psi (s+t)-\Psi \left(s\right)-\Psi \left(t\right)\le \kappa (\parallel s\parallel }^{\sigma }+{\parallel t\parallel }^{\sigma }),s,t\in {\Omega }_0.\end{array}$$

(5)

Then, we have the subsequent two statements:

(i) 

If $\sigma \ge 0$, then there exists exactly one mapping $T:E_1\to \mathbb{R}$, which is additive and such that, in the case of $\sigma <1$,

$$\begin{array}{c}\hfill {\displaystyle \frac{\tau }{1-2^{\sigma -1}}}{\parallel s\parallel }^{\sigma }\le T\left(s\right)-\Psi \left(s\right)\le {\displaystyle \frac{\kappa }{1-2^{\sigma -1}}}{\parallel s\parallel }^{\sigma },s\in E_0,\end{array}$$

(6)

and, in the case of $\sigma >1$,

$$\begin{array}{c}\hfill {\displaystyle \frac{\tau }{2^{\sigma -1}-1}}{\parallel s\parallel }^{\sigma }\le \Psi \left(s\right)-T\left(s\right)\le {\displaystyle \frac{\kappa }{2^{\sigma -1}-1}}{\parallel s\parallel }^{\sigma },s\in E_0.\end{array}$$

(7)

(ii) 

If $\sigma <0$, then Ψ is additive (on account of (5); this situation is possible only when $\tau \le 0\le \kappa$).

More information on various other related results and other relevant references concerning Ulam stability can be found in monographs [1,2,3].

In this paper, we focus on the stability of the quadratic equation

$$\begin{array}{c}\hfill q(s+t)+q(s-t)=2q\left(s\right)+2q\left(t\right)\end{array}$$

(8)

for maps from a normed space into the set of reals $\mathbb{R}$. In Section 2, we provide more information on the solutions to the equation and various stability results related to it.

Here, let us only mention that Equation (8) is modeled on the parallelogram identity

$${\parallel s+t\parallel }^2{+\parallel s-t\parallel }^2={2\parallel s\parallel }^2+2{\parallel t\parallel }^2,$$

that was applied in [12] to characterize the inner product spaces. Namely (see [12] and Ch. 1.4 in [13]):

• A normed space (real or complex) is an inner product space if and only if the norm fulfills the parallelogram identity.

We show that an application of the fixed point approach and Banach limit technique provide new outcomes on the Ulam stability of (8) that are similar to those in Theorem 3. In this way, we complement and generalize several previous similar results by giving much finer estimations of the distances between the approximate and exact solutions to the equation.

We also obtain some new characterizations of inner product spaces generalizing the one given in [12] and mentioned above.

2. Solutions and Stability of the Quadratic Equation

Let us start with some information on the solutions to (8).

Let E be a linear space over a field F of characteristic 0. It is known (see, e.g., Ch. 11 in [14]) that a function $q:E\to F$ satisfies Equation (8) if and only if there is a biadditive symmetric $L:E^2\to F$ with $q\left(s\right)=L(s,s)$ for $s\in E$. This mapping L is unique.

Let us recall (see, e.g., [14]) that $L:E^2\to F$ is symmetric if $L(x,y)=L(y,x)$ for $x,y\in E$; $L:E^2\to F$ is biadditive if the mappings $L_z,L^z:E\to F$, $L_z\left(x\right)=L(z,x)$ and $L^z\left(x\right)=L(x,z)$ for $x\in E$, are additive for each $z\in E$.

For more general results and information, we refer to [15], where the author gives a description of all mappings $f:S\to H$ satisfying the following generalization of (8):

$$f(s+\sigma (t\left)\right)+f(s+\tau (t\left)\right)=2f\left(s\right)+2f\left(t\right),s,t\in S,$$

(9)

where H is an Abelian group that is uniquely divisible by 2, S is an Abelian semigroup, and $\sigma ,\tau$ are two involutions of S.

Now, we present a brief survey of various stability results that have already been published for the quadratic equation (8) and generalizations and/or modifications of it.

As a kind of analogue of Theorem 1, we could mention here the stability results obtained for (8) in [16]. They can be stated in the following way (cf. Theorems 8.3 and 8.4 in [2]):

Theorem 4.

Assume that $E_2$ is a Banach space, $E_1$ is a normed space, and $\sigma \ne 2$, $\lambda >0$, $\xi \ge 0$ are fixed real numbers. Let $\xi =0$ or $\sigma <2$. If $\varphi :E_1\to E_2$ satisfies the inequality

$$\parallel \varphi (s+t)+\varphi (s-t)-2\varphi \left(s\right)-2\varphi \left(t\right)\parallel \le {\xi +\lambda (\parallel s\parallel }^{\sigma }+{\parallel t\parallel }^{\sigma }),s,t\in E_0,$$

then there is exactly one solution $q:E_1\to E_2$ of Equation (8) such that

$$\parallel \varphi \left(s\right)-q\left(s\right)\parallel \le \gamma +{\displaystyle \frac{\lambda }{|2^{\sigma -1}-2|}}{\parallel s\parallel }^{\sigma },s\in E_0,$$

where

$$E_0=\left\{\begin{array}{cc}{E}_1\setminus \left\{0\right\}\hfill & if\sigma <2;\hfill \\ E_1\hfill & if\sigma >2,\hfill \end{array}\right.\gamma =\left\{\begin{array}{cc}{\displaystyle \frac{1}{3}}(\parallel f\left(0\right)\parallel +\xi )\hfill & if\sigma <2;\hfill \\ 0\hfill & if\sigma >2.\hfill \end{array}\right.$$

However, we should mention here that the first stability result for the quadratic equation was provided in [17] (see also [18,19] for the case where the domain of f is an Abelian group); it was actually Theorem 4 with $E_0=E_1$, $\lambda =0$ and $\gamma =\xi /2$.

Some generalizations and/or extensions of Theorem 4 can be found in [20,21,22,23] (see also [24,25] for related results in modular spaces). In [26], some hyperstability results can be found. In simplified form, one of them can be written as follows:

Theorem 5.

Assume that $E_1$ and $E_2$ are normed spaces, $E_0:=E_1\setminus \left\{0\right\}$, and ${\sigma }_1,{\sigma }_2,\lambda \in \mathbb{R}$ are such that $\lambda >0$ and ${\sigma }_1+{\sigma }_2<0$. Let $\varphi :E_1\to E_2$ satisfy the inequality

$$\parallel \varphi (s+t)+\varphi (s-t)-2\varphi \left(s\right)-2\varphi \left(t\right)\parallel \le {\lambda \parallel s\parallel }^{{\sigma }_1}{\parallel t\parallel }^{{\sigma }_2},s,t\in E_0.$$

Then, ϕ is a solution to (8).

These outcomes have been extended and/or generalized in various directions. Below, we briefly indicate some of such studies.

For instance, the stability of the pexiderized version of (8), i.e., of the equation

$$g_1(s+t)+g_2(s-t)=g_3\left(s\right)+g_4\left(t\right),$$

for mappings $g_1,g_2,g_3,g_4$ from a normed space into a Banach space, is investigated in [23,27].

The stability of the equations

$$g(s+t)+g(s+\rho (t\left)\right)=g\left(s\right)+g\left(t\right),$$

$$g_1(s+t)+g_2(s+\rho \left(t\right))=g_3\left(s\right)+g_4\left(t\right),$$

for $g,g_1,g_2,g_3,g_4$ mapping a normed space E into a Banach space, with a fixed additive involution $\rho :E\to E$, is studied in [28,29,30,31].

In [32], the authors considered the stability of the equation

$$g\left(st\right)+g\left(s\xi \right(t\left)\right)=2g\left(s\right)+h\left(t\right)$$

for mappings $g,h$ from a semigroup $(S,·)$ into an Abelian semigroup $(H,+)$, where $\xi :S\to S$ is an endomorphism of S and an involution.

Finally, let us add that numerous authors have studied in [33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57] the stability of many other functional equations, depicted with the term quadratic. These equations have various forms, but majority of them are particular cases of the equation

$$\sum _{i=1}^m{A}_{i}f\left(\sum _{j=1}^n{a}_{ij}\left(x_j\right)\right)=D(x_1,\dots ,x_n),$$

(10)

which can be considered for a mapping f from a semigroup $(X,+)$ into a Banach space Y over the field $K\in \{\mathbb{R},\mathbb{C}\}$, where $m>1$ and n are fixed positive integers, $A_1,\dots ,A_m\in K\setminus \left\{0\right\}$, $a_{ij}:X\to X$ for $i=1,\cdots ,m$, $j=1,\dots ,n$, and $D:X^n\to Y$ is given. These equations have mainly been considered in normed and Banach spaces, but also in Banach modules, non-Archimedean normed spaces, fuzzy Banach spaces, the spaces of Fourier hyperfunctions and tempered distributions, 2-Banach spaces, non-Archimedean $(n,\beta )$-normed spaces, intuitionistic random normed spaces, fuzzy normed spaces, non-Archimedean intuitionistic fuzzy normed spaces, and quasi-$\beta$-normed spaces. Some further stability results can be derived for them from the very general outcomes that were obtained for (10) in [58,59,60,61,62,63,64,65,66,67].

Moreover, note that the above-listed equations are special cases of (10), as well. This means that also for them we can infer some stability results from the outcomes in [58,59,60,61,62,63,64,65,66,67].

We should add here that, in fact, refs. [61,62,65,66,67] provide the most general stability results for (8). They are not actually stated there, but follow from very general theorems concerning (10), which are too complicated to present here. As an example, let us only present the following hyperstability theorem for (8), which can be deduced from Theorem 2.6 in [62].

Theorem 6.

Let $E_1$ be a normed space, $E_0:=E_1\setminus \left\{0\right\}$, $E_2$ be a Banach space, and $\sigma <0$, $\lambda >0$ be fixed real numbers. Let $\varphi :E_1\to E_2$ satisfy the inequality

$$\parallel \varphi (s+t)+\varphi (s-t)-2\varphi \left(s\right)-2\varphi \left(t\right)\parallel \le {\lambda (\parallel s\parallel }^{\sigma }+{\parallel t\parallel }^{\sigma }),\forall s,t\in E_0.$$

Then, ϕ fulfills Equation (8).

The authors of the mentioned papers used mainly two techniques. One of them is called the direct one and the other is the fixed point approach. For further information on this topic, we refer to [1,2,3]. In this paper, we show that the use of Banach limit in the fixed point approach to Ulam stability allows one to obtain much finer estimations in the case of functions taking real values.

3. Information on Banach Limit

The main tool that we use in the proof of our main result is the Banach limit. Let us recall that early information on it is given in [68]. Later results concerning the Banach limit can be found in [69,70] (see also [71,72,73] for further information and examples of possible applications).

By the Banach limit, we mean here a real linear functional on the space ${\ell }^{\infty }$ (of all real sequences that are bounded), usually denoted by $\mathrm{LIM}$, which fulfills the subsequent two conditions:

$$inf\{d_k:k\in \mathbb{N}\}\le \mathrm{LIM}\left({\left(d_k\right)}_{k\in \mathbb{N}}\right)\le sup\{d_k:k\in \mathbb{N}\},$$

(11)

$$\mathrm{LIM}\left({\left(d_{k+m}\right)}_{k\in \mathbb{N}}\right)=\mathrm{LIM}\left({\left(d_k\right)}_{k\in \mathbb{N}}\right)$$

(12)

for each sequence ${\left(d_k\right)}_{k\in \mathbb{N}}\in {\ell }^{\infty }$ and all $m\in \mathbb{N}$ ($\mathbb{N}$ stands for the set of positive integers). The existence of such functional results from the Hahn–Banach theorem, and therefore, it depends on the axiom of choice.

Note that (11) and (12) yield the inequalities

$$\underset{k\to \infty }{lim\; inf}{d}_k\le \mathrm{LIM}\left({\left(d_k\right)}_{k\in \mathbb{N}}\right)\le \underset{k\to \infty }{lim\; sup}{d}_k,{\left(d_k\right)}_{k\in \mathbb{N}}\in {\ell }^{\infty },$$

(13)

whence

$$\begin{array}{c}\hfill \mathrm{LIM}\left({\left(d_k\right)}_{k\in \mathbb{N}}\right)=\underset{k\to \infty }{lim}{d}_k\end{array}$$

(14)

for every convergent sequence ${\left(d_k\right)}_{k\in \mathbb{N}}\in {\ell }^{\infty }$.

The functional $\mathrm{LIM}$ is not unique, but condition (14) is always fulfilled. Moreover, there are also non-convergent sequences from ${\ell }^{\infty }$ for which the Banach limit is uniquely determined (they are called almost convergent) and a simple example of them is the sequence: $d_k={(-1)}^k$ for $k\in \mathbb{N}$, when $\mathrm{LIM}\left({\left(d_k\right)}_{k\in \mathbb{N}}\right)=0$ (because $\mathrm{LIM}\left({\left(d_k\right)}_{k\in \mathbb{N}}\right)=\mathrm{LIM}\left({(-d_{k+1})}_{k\in \mathbb{N}}\right)$ $=-\mathrm{LIM}\left({\left(d_{k+1}\right)}_{k\in \mathbb{N}}\right)=-\mathrm{LIM}\left({\left(d_k\right)}_{k\in \mathbb{N}}\right)$).

4. Auxiliary Fixed Point Result

Now, we present some auxiliary results that can easily be derived from [74]. If A and B are nonempty sets, then $B^A$ stands for the family of all maps from A to B, and, given $\varphi \in A^A$, we define ${\varphi }^n\in A^A$ for $n\in {\mathbb{N}}_0$ by

$${\varphi }^0\left(s\right)=s,{\varphi }^{n+1}\left(s\right)=\varphi \left({\varphi }^n\left(s\right)\right),s\in A,n\in {\mathbb{N}}_0,$$

where ${\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$.

Let Y be a nonempty set. Let ${\alpha }_1,{\alpha }_2\in {\mathbb{R}}^Y$ and $d\in \mathbb{R}$. Then, as usual, we define mappings $d{\alpha }_1,{\alpha }_1+d,-{\alpha }_1,{\alpha }_1\pm {\alpha }_2\in {\mathbb{R}}^Y$ by the formulas:

$$\left(d{\alpha }_1\right)\left(t\right)=d{\alpha }_1\left(t\right),({\alpha }_1+d)\left(t\right)={\alpha }_1\left(t\right)+d,$$

$$(-{\alpha }_1)\left(t\right)=-{\alpha }_1\left(t\right),({\alpha }_1\pm {\alpha }_2)\left(t\right)={\alpha }_1\left(t\right)\pm {\alpha }_2\left(t\right)$$

for every $t\in Y$. In addition, we write ${\alpha }_1\le {\alpha }_2$ if ${\alpha }_1\left(t\right)\le {\alpha }_2\left(t\right)$ for all $t\in Y$.

Given ${\Gamma }_1,{\Gamma }_2:{\mathbb{R}}^Y\to {\mathbb{R}}^Y$, we say that $T:{\mathbb{R}}^Y\to {\mathbb{R}}^Y$ is $({\Gamma }_1,{\Gamma }_2)$-contractive if

$$\begin{array}{c}\hfill {\Gamma }_1{\xi }_1\le T\alpha -T\beta \le {\Gamma }_2{\xi }_2\end{array}$$

(15)

for every ${\xi }_1,{\xi }_2,\alpha ,\beta \in {\mathbb{R}}^Y$ with ${\xi }_1\le \alpha -\beta \le {\xi }_2$.

Next, we say that a sequence ${\left(d_n\right)}_{n\in \mathbb{N}}$ in ${\mathbb{R}}^Y$ is pointwise bounded (for short: p.b.) if the sequence ${\left(d_n\left(t\right)\right)}_{n\in \mathbb{N}}$ is bounded in $\mathbb{R}$ for every $t\in Y$.

We need the following hypothesis on the operators $T:{\mathbb{R}}^Y\to {\mathbb{R}}^Y$.

(H)

For every p.b. sequence ${\left(d_n\right)}_{n\in \mathbb{N}}$ in ${\mathbb{R}}^Y$, the sequence ${\left(T\left(d_n\right)\right)}_{n\in \mathbb{N}}$ is p.b., and

$$\begin{array}{c}\hfill T\left(\widehat{\mathrm{LIM}}\left({\left(d_n\right)}_{n\in \mathbb{N}}\right)\right)=\widehat{\mathrm{LIM}}\left({\left(T\left(d_n\right)\right)}_{n\in \mathbb{N}}\right),\end{array}$$

(16)

where, for every p.b. sequence ${\left(c_n\right)}_{n\in \mathbb{N}}$ in ${\mathbb{R}}^Y$, $\widehat{\mathrm{LIM}}\left({\left(c_n\right)}_{n\in \mathbb{N}}\right)\in {\mathbb{R}}^Y$ is defined by

$$\widehat{\mathrm{LIM}}\left({\left(c_n\right)}_{n\in \mathbb{N}}\right)\left(t\right):=\mathrm{LIM}\left({\left(c_n\left(t\right)\right)}_{n\in \mathbb{N}}\right),t\in Y.$$

The next theorem is a simplified version of Theorem 5 in [74].

Theorem 7.

Let ${\Gamma }_1,{\Gamma }_2:{\mathbb{R}}^Y\to {\mathbb{R}}^Y$, $T:{\mathbb{R}}^Y\to {\mathbb{R}}^Y$ be $({\Gamma }_1,{\Gamma }_2)$-contractive, $\mu ,\tau \in {\mathbb{R}}^Y$, and ${\mu }_{n,k},{\tau }_{n,k}\in {\mathbb{R}}^Y$ for $k,n\in {\mathbb{N}}_0$ be given by

$${\mu }_{n,k}\left(t\right):=\sum _{j=k}^{k+n}\left({\Gamma }_1^j\mu \right)\left(t\right),{\tau }_{n,k}\left(t\right):=\sum _{j=k}^{k+n}\left({\Gamma }_2^j\tau \right)\left(t\right),k,n\in {\mathbb{N}}_0,t\in Y.$$

(17)

Assume that the sequences ${\left({\mu }_{n,0}\right)}_{n\in \mathbb{N}}$ and ${\left({\tau }_{n,0}\right)}_{n\in \mathbb{N}}$ are p.b., T fulfills hypothesis (H), and $\varphi :Y\to \mathbb{R}$ satisfies the inequalities

$$\mu \le T\varphi -\varphi \le \tau .$$

(18)

Then, the sequence ${\left(T^n\varphi \right)}_{n\in \mathbb{N}}$ is p.b. and the mapping $\Phi :Y\to \mathbb{R}$, given by

$$\begin{array}{c}\hfill \Phi \left(t\right)=\mathrm{LIM}\left({\left(\left(T^n\varphi \right)\left(t\right)\right)}_{n\in \mathbb{N}}\right),t\in Y,\end{array}$$

(19)

is a fixed point of T with

$$\begin{array}{cc}\hfill {\widehat{\mu }}_k\left(t\right)& :=\underset{n\to \infty }{lim\; inf}{\mu }_{n,k}\left(t\right)\le \Phi \left(t\right)-\left(T^k\varphi \right)\left(t\right)\hfill \\ & \le \underset{n\to \infty }{lim\; sup}{\tau }_{n,k}\left(t\right)=:{\widehat{\tau }}_k\left(t\right),t\in Y,k\in {\mathbb{N}}_0.\hfill \end{array}$$

(20)

Moreover, if ${\rho }_0,{\chi }_0\in {\mathbb{R}}^Y$ are such that

$$\underset{n\in \mathbb{N}}{sup}{\Gamma }_1^n({\chi }_0-{\rho }_0)\left(t\right)=0,\underset{n\in \mathbb{N}}{inf}{\Gamma }_2^n({\rho }_0-{\chi }_0)\left(t\right)=0,t\in Y,$$

(21)

then, for each ${\Phi }_0\in {\mathbb{R}}^Y$, T has at most one fixed point Φ fulfilling the condition

$$\begin{array}{c}\hfill {\chi }_0\le \Phi -{\Phi }_0\le {\rho }_0.\end{array}$$

(22)

Remark 1.

Let $m\in \mathbb{N}$, $f_1,\dots ,f_m\in Y^Y$, and $h,L_1,\dots ,L_m\in {\mathbb{R}}^Y$. Define $T:{\mathbb{R}}^Y\to {\mathbb{R}}^Y$ by

$$\begin{array}{c}\hfill T\psi \left(t\right)=\sum _{i=1}^m{L}_i\left(t\right)\psi \left(f_i\left(t\right)\right)+h\left(t\right),t\in Y,\psi \in {\mathbb{R}}^Y.\end{array}$$

(23)

Then, according to Remark 3 in [74], the operator T fulfills hypothesis (H).

Remark 2.

It follows from Remark 4 in [74] that every $T:{\mathbb{R}}^Y\to {\mathbb{R}}^Y$, which is additive (i.e., $T(\varphi +\psi )=T\varphi +T\psi$ for $\varphi ,\psi \in {\mathbb{R}}^Y$) and non-decreasing (i.e., $T\varphi \le T\psi$ for $\varphi ,\psi \in {\mathbb{R}}^Y$ with $\varphi \le \psi$), is $({\Gamma }_1,{\Gamma }_2)$-contractive with ${\Gamma }_1={\Gamma }_2=T$.

The next example has been added on the demand of the reviewers.

Example 1.

Let Y be a normed space. Consider $T:{\mathbb{R}}^Y\to {\mathbb{R}}^Y$ given by the following very simple formula:

$$\begin{array}{c}\hfill T\psi \left(s\right):={\displaystyle \frac{1}{4}}\psi \left(2s\right),\psi \in {\mathbb{R}}^Y,s\in Y,\end{array}$$

as in the next theorem (which is our main result). Then, in view of Remarks 1 and 2, T fulfills hypothesis (H) and is $({\Gamma }_1,{\Gamma }_2)$-contractive with ${\Gamma }_1={\Gamma }_2=T$.

Fix ${\xi }_1,{\xi }_2,{\theta }_1,{\theta }_2\in \mathbb{R}$ and ${\sigma }_1,{\sigma }_2\in [0,2)$ such that

$${\xi }_1+{\theta }_1{\parallel t\parallel }^{{\sigma }_1}\le {\xi }_2+{\theta }_2{\parallel t\parallel }^{{\sigma }_2},t\in Y.$$

Let $\mu \left(t\right)={\xi }_1+{\theta }_1{\parallel t\parallel }^{{\sigma }_1}$ and $\tau \left(t\right)={\xi }_2+{\theta }_2{\parallel t\parallel }^{{\sigma }_2}$ for $t\in Y$. Then, with ${\Gamma }_1={\Gamma }_2=T$, (18) is

$$\begin{array}{cc}\hfill {\xi }_1+{\theta }_1{\parallel t\parallel }^{{\sigma }_1}& \le T\varphi \left(t\right)-\varphi \left(t\right)\le {\xi }_2+{\theta }_2{\parallel t\parallel }^{{\sigma }_2},t\in Y,\hfill \end{array}$$

and (17) can be written as

$$\begin{array}{cc}\hfill {\mu }_{n,k}\left(t\right)& :=\sum _{j=k}^{k+n}\left(T^j\mu \right)\left(t\right)=\sum _{j=k}^{k+n}{\displaystyle \frac{1}{4^j}}\left({\xi }_1+{\parallel 2^{j}t\parallel }^{{\sigma }_1}\right),\hfill \\ \hfill {\tau }_{n,k}\left(t\right)& :=\sum _{j=k}^{k+n}\left(T^j\tau \right)\left(t\right)=\sum _{j=k}^{k+n}{\displaystyle \frac{1}{4^j}}\left({\xi }_2+{\parallel 2^{j}t\parallel }^{{\sigma }_2}\right),k,n\in {\mathbb{N}}_0,t\in Y,\hfill \end{array}$$

whence

$$\begin{array}{cc}\hfill {\widehat{\mu }}_k\left(t\right)& :=\underset{n\to \infty }{lim\; inf}{\mu }_{n,k}\left(t\right)=4^{1-k}{\displaystyle \frac{{\xi }_1}{3}}+2^{({\sigma }_1-2)k}{\displaystyle \frac{{\theta }_1}{1-2^{{\sigma }_1-2}}}{\parallel t\parallel }^{{\sigma }_1},\hfill \\ \hfill {\widehat{\tau }}_k\left(t\right)& :=\underset{n\to \infty }{lim\; sup}{\tau }_{n,k}\left(t\right)=4^{1-k}{\displaystyle \frac{{\xi }_2}{3}}+2^{({\sigma }_2-2)k}{\displaystyle \frac{{\theta }_2}{1-2^{{\sigma }_2-2}}}{\parallel t\parallel }^{{\sigma }_2},t\in Y,k\in {\mathbb{N}}_0.\hfill \end{array}$$

Consequently, for $k=0$, condition (20) takes the form

$$\begin{array}{c}\hfill {\displaystyle \frac{4{\xi }_1}{3}}+{\displaystyle \frac{{\theta }_1{\parallel t\parallel }^{{\sigma }_1}}{1-2^{{\sigma }_1-2}}}\le \Phi \left(t\right)-\varphi \left(t\right)\le {\displaystyle \frac{4{\xi }_2}{3}}+{\displaystyle \frac{{\theta }_2{\parallel t\parallel }^{{\sigma }_2}}{1-2^{{\sigma }_2-2}}},t\in Y.\end{array}$$

(24)

For instance, if ${\xi }_1={\theta }_1=0$, then (24) yields

$$\begin{array}{c}\hfill 0\le \Phi \left(t\right)-\varphi \left(t\right)\le {\displaystyle \frac{4{\xi }_2}{3}}+{\displaystyle \frac{{\theta }_2{\parallel t\parallel }^{{\sigma }_2}}{1-2^{{\sigma }_2-2}}},t\in Y;\end{array}$$

if ${\xi }_1=-3$, ${\xi }_2=-3/4$ and ${\theta }_1={\theta }_2=0$, then from (24), we obtain

$$\begin{array}{c}\hfill -4\le \Phi \left(t\right)-\varphi \left(t\right)\le -1,t\in Y.\end{array}$$

5. The Main Results

Let $(E,+)$ be a commutative group.

The next corollary shows that a partial analogue to Theorem 3 can also be obtained for the quadratic equation. We start with a somewhat more general situation to be able to also obtain some other outcomes.

Theorem 8.

Let $T:{\mathbb{R}}^E\to {\mathbb{R}}^E$ be given by

$$\begin{array}{c}\hfill T\psi \left(s\right):={\displaystyle \frac{1}{4}}\psi \left(2s\right),\psi \in {\mathbb{R}}^E,s\in E.\end{array}$$

(25)

Let $\gamma ,\delta :E^2\to \mathbb{R}$ and $\varphi :E\to \mathbb{R}$ fulfill the inequalities

$$\begin{array}{c}\hfill \delta (s,t)\le \varphi (s+t)+\varphi (s-t)-2\varphi \left(t\right)-2\varphi \left(s\right)\le \gamma (s,t),s,t\in E,\end{array}$$

(26)

and let ${\gamma }_n,{\delta }_n\in {\mathbb{R}}^E$ for $n\in \mathbb{N}$ be defined by

$$\begin{array}{c}\hfill {\gamma }_n\left(s\right)={\displaystyle \frac{1}{4}}\sum _{j=0}^n{T}^j\delta (s,s),{\delta }_n\left(s\right)={\displaystyle \frac{1}{4}}\sum _{j=0}^n{T}^j\gamma (s,s),n\in \mathbb{N},s\in E.\end{array}$$

(27)

Let the sequences ${\left({\gamma }_n\right)}_{n\in \mathbb{N}}$ and ${\left({\delta }_n\right)}_{n\in \mathbb{N}}$ be p.b. and

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{T}^n\delta (s,t)=\underset{n\to \infty }{lim}{T}^n\gamma (s,t)=0,s,t\in E.\end{array}$$

(28)

Then, the sequence ${\left(T^n\varphi \right)}_{n\in \mathbb{N}}$ is p.b., and the mapping $\Phi :E\to \mathbb{R}$, given by

$$\begin{array}{c}\hfill \Phi \left(t\right):=\mathrm{LIM}\left({\left(T^n\varphi \left(t\right)\right)}_{n\in \mathbb{N}}\right),t\in E,\end{array}$$

(29)

fulfills the quadratic Equation (8), i.e., the equation

$$\Phi (s+t)+\Phi (s-t)=2\Phi \left(s\right)+2\Phi \left(t\right),s,t\in E,$$

(30)

and

$$\widehat{\delta }\left(t\right):=\underset{k\to \infty }{lim\; inf}{\delta }_k\left(t\right)\le \Phi \left(t\right)-\varphi \left(t\right)+{\displaystyle \frac{1}{3}}\varphi \left(0\right)\le \underset{k\to \infty }{lim\; sup}{\gamma }_k\left(t\right)=:\widehat{\gamma }\left(t\right),t\in E.$$

(31)

Moreover, if

$$\underset{n\in \mathbb{N}}{inf}{T}^n\left(\widehat{\gamma }-\widehat{\delta }\right)\left(t\right)=0,t\in E,$$

(32)

then $\Phi :E\to \mathbb{R}$ is the only solution of (30), which fulfills (31).

Proof. 

Taking $t=s$ in (26), we obtain the inequalities

$$\begin{array}{c}\hfill \delta (s,s)\le \varphi \left(2s\right)+\varphi \left(0\right)-4\varphi \left(s\right)\le \gamma (s,s),s\in E,\end{array}$$

(33)

whence

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{4}}\left(\delta (s,s)-\varphi \left(0\right)\right)\le {\displaystyle \frac{1}{4}}\varphi \left(2s\right)-\varphi \left(s\right)\le {\displaystyle \frac{1}{4}}\left(\gamma (s,s)-\varphi \left(0\right)\right),s\in E,\end{array}$$

(34)

which means that

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{4}}\left(\delta (s,s)-\varphi \left(0\right)\right)\le T\varphi \left(s\right)-\varphi \left(s\right)\le {\displaystyle \frac{1}{4}}\left(\gamma (s,s)-\varphi \left(0\right)\right),s\in E.\end{array}$$

(35)

Let $\mu :=\delta -\varphi \left(0\right)$ and $\tau :=\gamma -\varphi \left(0\right)$. Then, it is easily seen that (18) holds and the sequences ${\left({\mu }_{n,0}\right)}_{n\in \mathbb{N}}$ and ${\left({\tau }_{n,0}\right)}_{n\in \mathbb{N}}$ are p.b., where ${\mu }_{n,k},{\tau }_{n,k}\in {\mathbb{R}}^Y$ are given by (17).

Moreover, according to Remarks 1 and 2, T is $({\Gamma }_1,{\Gamma }_2)$-contractive with ${\Gamma }_1={\Gamma }_2=T$, and hypothesis (H) holds. Therefore, by Theorem 7 (with ${\Gamma }_1={\Gamma }_2=T$), the sequence ${\left(T^n\varphi \right)}_{n\in \mathbb{N}}$ is p.b., and the mapping $\Phi :E\to \mathbb{R}$, given by (29), is a fixed point of T satisfying inequalities (20). Note that (20) (with $k=0$) yields

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim\; inf}{\delta }_k\left(s\right)-{\displaystyle \frac{1}{3}}\varphi \left(0\right)\le \Phi \left(s\right)-\varphi \left(s\right)\le \underset{k\to \infty }{lim\; sup}{\gamma }_k\left(s\right)-{\displaystyle \frac{1}{3}}\varphi \left(0\right),s\in E,\end{array}$$

which implies (31).

Now, we prove that $\Phi$ is a solution of (30). To this end, we first show by induction that

$$\begin{array}{c}\hfill T^n\delta (s,t)\le T^n\varphi (s+t)+T^n\varphi (s-t)-2T^n\varphi \left(s\right)-2T^n\varphi \left(t\right)\le T^n\gamma (s,t)\end{array}$$

(36)

for every $s,t\in E$, $n\in {\mathbb{N}}_0$.

Note that the case $n=0$ is (26). So, let $n\in \mathbb{N}$ be fixed and assume that (36) is valid for $s,t\in E$. Then, for all $s,t\in E$,

$$\begin{array}{c}{T}^{n+1}\varphi (s+t)+T^{n+1}\varphi (s-t)-2T^{n+1}\varphi \left(s\right)-2T^{n+1}\varphi \left(t\right)\hfill \\ \hspace{1em}={\displaystyle \frac{1}{4}}\left(T^n\varphi \left(2(s+t)\right)+T^n\varphi \left(2(s-t)\right)-2T^n\varphi \left(2s\right)-2T^n\varphi \left(2t\right)\right),\hfill \end{array}$$

which means that

$$\begin{array}{cc}\hfill T^{n+1}\delta (s,t)& ={\displaystyle \frac{1}{4}}{T}^n\mu (2s,2t)\hfill \\ \hfill & \le {\displaystyle \frac{1}{4}}\left(T^n\varphi (2s+2t)+T^n\varphi (2s-2t)-2T^n\varphi \left(2s\right)-2T^n\varphi \left(2t\right)\right)\hfill \\ \hfill & =T^{n+1}\varphi (s+t)+T^{n+1}\varphi (s-t)-2T^{n+1}\varphi \left(s\right)-2T^{n+1}\varphi \left(t\right)\hfill \\ \hfill & \le {\displaystyle \frac{1}{4}}{T}^n\gamma (2s,2t)=T^{n+1}\gamma (s,t).\hfill \end{array}$$

This completes the proof that (36) is valid for all $s,t\in E$ and $n\in {\mathbb{N}}_0$.

Note that, by the linearity of the Banach limit and (29), for every $s,t\in E$, we have

$$\begin{array}{cc}\hfill \Phi (s+t)& +\Phi (s-t)-2\Phi \left(s\right)-2\Phi \left(t\right)\hfill \\ \hfill & =\mathrm{LIM}\left({\left(T^n\varphi (s+t)+T^n\varphi (s-t)-2T^n\varphi \left(s\right)-2T^n\varphi \left(t\right)\right)}_{n\in \mathbb{N}}\right),\hfill \end{array}$$

whence (13) and (36) imply that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; inf}{T}^n\delta (s,t)\le \Phi (s+t)+\Phi (s-t)-2\Phi \left(s\right)-2\Phi \left(t\right)\le \underset{n\to \infty }{lim\; sup}{T}^n\gamma (s,t).\end{array}$$

(37)

Consequently, in view of (28), we obtain that

$$\begin{array}{c}\hfill \Phi (s+t)+\Phi (s-t)-2\Phi \left(s\right)-2\Phi \left(t\right)=0,s,t\in E.\end{array}$$

(38)

Finally, observe that $T^n\left(\widehat{\delta }-\widehat{\gamma }\right)=-T^n\left(\widehat{\gamma }-\widehat{\delta }\right)$ for every $n\in \mathbb{N}$. Therefore, by (32),

$$\begin{array}{c}\hfill \underset{n\in \mathbb{N}}{sup}{T}^n\left(\widehat{\delta }-\widehat{\gamma }\right)\left(s\right)=-\underset{n\in \mathbb{N}}{inf}{T}^n\left(\widehat{\gamma }-\widehat{\delta }\right)\left(s\right)=0,s\in E,\end{array}$$

which means that (21) holds with ${\Gamma }_1={\Gamma }_2=T$, ${\chi }_0=\widehat{\delta }$ and ${\rho }_0=\widehat{\gamma }$. Since every solution to (30) is a fixed point of T, from Theorem 7 (with ${\Phi }_0=\varphi -{\displaystyle \frac{1}{3}}\varphi \left(0\right)$) we also obtain the uniqueness of $\Phi$. □

Theorem 8 yields the following partial generalization of Theorem 4.

Corollary 1.

Let E be a real normed space. Let $\varphi :E\to \mathbb{R}$ and ${\xi }_i,{\theta }_i,{\eta }_i,{\sigma }_i,{\rho }_i\in \mathbb{R}$ for $i=1,2$ be such that ${\sigma }_1,{\sigma }_2,{\rho }_1,{\rho }_2\in [0,2)$ and

$$\begin{array}{cc}\hfill {\xi }_1+{\theta }_1{\parallel s\parallel }^{{\rho }_1}+{\eta }_1{\parallel t\parallel }^{{\sigma }_1}& \le \varphi (s+t)+\varphi (s-t)-2\varphi \left(s\right)-2\varphi \left(t\right)\hfill \\ \hfill & \le {\xi }_2+{\theta }_2{\parallel s\parallel }^{{\rho }_2}+{\eta }_2{\parallel t\parallel }^{{\sigma }_2},s,t\in E.\hfill \end{array}$$

Then, there exists exactly one solution $\Phi :E\to \mathbb{R}$ of (30) with

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\xi }_1}{3}}+{\displaystyle \frac{{\theta }_1{\parallel s\parallel }^{{\rho }_1}}{4-2^{{\rho }_1}}}& +{\displaystyle \frac{{\eta }_1{\parallel s\parallel }^{{\sigma }_1}}{4-2^{{\sigma }_1}}}\le \Phi \left(s\right)-\varphi \left(s\right)+{\displaystyle \frac{1}{3}}\varphi \left(0\right)\hfill \\ \hfill & \le {\displaystyle \frac{{\xi }_2}{3}}+{\displaystyle \frac{{\theta }_2{\parallel s\parallel }^{{\rho }_2}}{4-2^{{\rho }_2}}}+{\displaystyle \frac{{\eta }_2{\parallel s\parallel }^{{\sigma }_2}}{4-2^{{\sigma }_2}}},s\in E.\hfill \end{array}$$

(39)

Proof. 

According to Theorem 8 (with $\delta (s,t)={\xi }_1+{\theta }_1{\parallel s\parallel }^{{\rho }_1}+{\eta }_1{\parallel t\parallel }^{{\sigma }_1}$ and $\gamma (s,t)={\xi }_2+{\theta }_2{\parallel s\parallel }^{{\rho }_2}+{\eta }_2{\parallel t\parallel }^{{\sigma }_2}$ for $s,t\in E$), there exists a unique solution $\Phi :E\to \mathbb{R}$ of Equation (30), which fulfills inequalities (31). In this case, (31) is exactly (39). □

We also have the subsequent.

Corollary 2.

Let E be a real normed space. Let $\varphi :E\to \mathbb{R}$ and ${\xi }_i,{\theta }_i,{\sigma }_i,{\rho }_i\in \mathbb{R}$ for $i=1,2$ be such that ${\sigma }_1+{\rho }_1,{\sigma }_2+{\rho }_2\in [0,2)$ and

$$\begin{array}{cc}\hfill {\xi }_1+{\theta }_1{\parallel s\parallel }^{{\sigma }_1}{\parallel t\parallel }^{{\rho }_1}& \le \varphi (s+t)+\varphi (s-t)-2\varphi \left(s\right)-2\varphi \left(t\right)\hfill \\ \hfill & \le {\xi }_2+{\theta }_2{\parallel s\parallel }^{{\sigma }_2}{\parallel t\parallel }^{{\rho }_2},s,t\in E.\hfill \end{array}$$

Then, there exists exactly one solution $\Phi :E\to \mathbb{R}$ of (30) with

$$\begin{array}{c}\hfill {\displaystyle \frac{{\xi }_1}{3}}+{\displaystyle \frac{{\theta }_1{\parallel s\parallel }^{{\sigma }_1+{\rho }_1}}{4-2^{{\sigma }_1+{\rho }_1}}}\le \Phi \left(s\right)-\varphi \left(s\right)+{\displaystyle \frac{1}{3}}\varphi \left(0\right)\le {\displaystyle \frac{{\xi }_2}{3}}+{\displaystyle \frac{{\theta }_2{\parallel s\parallel }^{{\sigma }_2+{\rho }_2}}{4-2^{{\sigma }_2+{\rho }_2}}},s\in E.\end{array}$$

(40)

Proof. 

We again use Theorem 8 with $\delta (s,t)={\xi }_1+{\theta }_1{\parallel s\parallel }^{{\sigma }_1}{\parallel t\parallel }^{{\rho }_1}$ and $\gamma (s,t)={\xi }_2+{\theta }_2{\parallel s\parallel }^{{\sigma }_2}{\parallel t\parallel }^{{\rho }_2}$ for $s,t\in E$. Now, (31) is exactly (40). □

6. Applications

In this part, we show how to use Corollaries 1 and 2 to obtain some new characterizations of inner product spaces that generalize the result of Jordan and von Neumann [12].

Namely, we have the following two corollaries.

Corollary 3.

Let E be a real or complex normed space. Assume that there exist ${\xi }_i,{\theta }_i,{\eta }_i,{\sigma }_i,{\rho }_i\in \mathbb{R}$ for $i=1,2$ such that ${\sigma }_1,{\sigma }_2,{\rho }_1,{\rho }_2\in [0,2)$ and

$$\begin{array}{cc}\hfill {\xi }_1+{\theta }_1{\parallel s\parallel }^{{\rho }_1}+{\eta }_1{\parallel t\parallel }^{{\sigma }_1}& \le {\parallel s+t\parallel }^2{+\parallel s-t\parallel }^2-{2\parallel s\parallel }^2-2{\parallel t\parallel }^2\hfill \\ \hfill & \le {\xi }_2+{\theta }_2{\parallel s\parallel }^{{\rho }_2}+{\eta }_2{\parallel t\parallel }^{{\sigma }_2},s,t\in E.\hfill \end{array}$$

Then, the norm is induced by an inner product.

Proof. 

According to Corollary 1 with $\varphi \left(s\right)={\parallel s\parallel }^2$ for $s\in E$, there is a solution $\Phi :E\to \mathbb{R}$ of Equation (30) such that

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\xi }_1}{3}}+{\displaystyle \frac{{\theta }_1{\parallel s\parallel }^{{\rho }_1}}{4-2^{{\rho }_1}}}& +{\displaystyle \frac{{\eta }_1{\parallel s\parallel }^{{\sigma }_1}}{4-2^{{\sigma }_1}}}\le \Phi \left(s\right)-{\parallel s\parallel }^2\hfill \\ \hfill & \le {\displaystyle \frac{{\xi }_2}{3}}+{\displaystyle \frac{{\theta }_2{\parallel s\parallel }^{{\rho }_2}}{4-2^{{\rho }_2}}}+{\displaystyle \frac{{\eta }_2{\parallel s\parallel }^{{\sigma }_2}}{4-2^{{\sigma }_2}}},s\in E.\hfill \end{array}$$

(41)

Next, it follows from (30) that $\Phi \left(2^{n}s\right)=4^n\Phi \left(s\right)$ for $s\in E$, $n\in \mathbb{N}$. Hence, replacing s by $2^{n}s$ in (41), for each $n\in \mathbb{N}$, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\xi }_1}{3}}+{\displaystyle \frac{2^{n{\rho }_1}{\theta }_1{\parallel s\parallel }^{{\rho }_1}}{4-2^{{\rho }_1}}}& +{\displaystyle \frac{2^{n{\sigma }_1}{\eta }_1{\parallel s\parallel }^{{\sigma }_1}}{4-2^{{\sigma }_1}}}\le 4^n\Phi \left(s\right)-4^n{\parallel s\parallel }^2\hfill \\ \hfill & \le {\displaystyle \frac{{\xi }_2}{3}}+{\displaystyle \frac{2^{n{\rho }_2}{\theta }_2{\parallel s\parallel }^{{\rho }_2}}{4-2^{{\rho }_2}}}+{\displaystyle \frac{2^{n{\sigma }_2}{\eta }_2{\parallel s\parallel }^{{\sigma }_2}}{4-2^{{\sigma }_2}}},s\in E,\hfill \end{array}$$

whence (dividing by $4^n$) we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\xi }_1}{4^{n}3}}+{\displaystyle \frac{2^{n({\rho }_1-2)}{\theta }_1{\parallel s\parallel }^{{\rho }_1}}{4-2^{{\rho }_1}}}& +{\displaystyle \frac{2^{n({\sigma }_1-2)}{\eta }_1{\parallel s\parallel }^{{\sigma }_1}}{4-2^{{\sigma }_1}}}\le \Phi \left(s\right)-{\parallel s\parallel }^2\hfill \\ \hfill & \le {\displaystyle \frac{{\xi }_2}{4^{n}3}}+{\displaystyle \frac{2^{n({\rho }_2-2)}{\theta }_2{\parallel s\parallel }^{{\rho }_2}}{4-2^{{\rho }_2}}}+{\displaystyle \frac{2^{n({\sigma }_2-2)}{\eta }_2{\parallel s\parallel }^{{\sigma }_2}}{4-2^{{\sigma }_2}}},s\in E.\hfill \end{array}$$

Consequently, letting $n\to \infty$, we obtain ${\parallel s\parallel }^2=\Phi \left(s\right)$ for $s\in E$, which means that the norm fulfills the parallelogram identity and, therefore, must be induced by an inner product (see, e.g., [12] and Ch. 1.4 in [13]). □

Corollary 4.

Let E be a real or complex normed space. Assume that there exist ${\xi }_i,{\theta }_i,{\sigma }_i,{\rho }_i\in \mathbb{R}$ for $i=1,2$ such that ${\sigma }_1+{\rho }_1,{\sigma }_2+{\rho }_2\in [0,2)$ and

$$\begin{array}{cc}\hfill {\xi }_1+{\theta }_1{\parallel s\parallel }^{{\sigma }_1}{\parallel t\parallel }^{{\rho }_1}& \le {\parallel s+t\parallel }^2{+\parallel s-t\parallel }^2-{2\parallel s\parallel }^2-2{\parallel t\parallel }^2\hfill \\ \hfill & \le {\xi }_2+{\theta }_2{\parallel s\parallel }^{{\sigma }_2}{\parallel t\parallel }^{{\rho }_2},s,t\in E.\hfill \end{array}$$

Then, the norm is induced by an inner product.

Proof. 

It is enough to argue as in the proof of Corollary 3, replacing Corollary 1 by Corollary 2. □

7. Final Observations

From Corollary 1 and Theorem 6, we can derive the following corollary that is an analogue of Theorems 1 for the quadratic Equation (30). It also partly improves Theorem 4.

Corollary 5.

Let $\sigma \in \mathbb{R}$, $\sigma <2$, E be a real normed space and

$$E_0=\left\{\begin{array}{cc}E\setminus \left\{0\right\}\hfill & if\sigma <0;\hfill \\ E\hfill & if\sigma \ge 0.\hfill \end{array}\right.$$

Let $\varphi :E\to \mathbb{R}$ and ${\xi }_1,{\xi }_2,\theta ,\eta \in \mathbb{R}$ be such that

$$\begin{array}{cc}\hfill {\xi }_1+\eta \left({\parallel s\parallel }^{\sigma }+{\parallel t\parallel }^{\sigma }\right)& \le \varphi (s+t)+\varphi (s-t)-2\varphi \left(s\right)-2\varphi \left(t\right)\hfill \\ \hfill & \le {\xi }_2+\theta \left({\parallel s\parallel }^{\sigma }+{\parallel t\parallel }^{\sigma }\right),s,t\in E_0.\hfill \end{array}$$

Then, the following two statements are valid.

(i) 

If $\sigma \ge 0$, then there is a unique solution $\Phi :E\to \mathbb{R}$ of Equation (30) such that

$$\begin{array}{c}\hfill {\displaystyle \frac{{\xi }_1}{3}}+{\displaystyle \frac{{\eta \parallel s\parallel }^{\sigma }}{2-2^{\sigma -1}}}\le \Phi \left(s\right)-\varphi \left(s\right)+{\displaystyle \frac{1}{3}}\varphi \left(0\right)\le {\displaystyle \frac{{\xi }_2}{3}}+{\displaystyle \frac{{\theta \parallel s\parallel }^{\sigma }}{2-2^{\sigma -1}}},s\in E.\end{array}$$

(42)

(ii) 

If $\sigma <0$ and ${\xi }_1={\xi }_2=0$, then ϕ is a solution of Equation (30).

Proof. 

If $\sigma \ge 0$, then according to Corollary 1 (with ${\theta }_1={\eta }_1=\eta$, ${\theta }_2={\eta }_2=\theta$ and ${\rho }_1={\sigma }_1={\rho }_2={\sigma }_2=\sigma$), there exists a unique solution $\Phi :E\to \mathbb{R}$ of Equation (30), which fulfills inequalities (39). In this case, (39) is exactly (42).

If $\sigma <0$ and ${\xi }_1={\xi }_2=0$, then

$$\begin{array}{cc}\hfill -\left|\eta \right|\left({\parallel s\parallel }^{\sigma }+{\parallel t\parallel }^{\sigma }\right)& \le \eta \left({\parallel s\parallel }^{\sigma }+{\parallel t\parallel }^{\sigma }\right)\le \varphi (s+t)+\varphi (s-t)-2\varphi \left(s\right)-2\varphi \left(t\right)\hfill \\ \hfill & \le \theta \left({\parallel s\parallel }^{\sigma }+{\parallel t\parallel }^{\sigma }\right)\le \left|\theta \right|\left({\parallel s\parallel }^{\sigma }+{\parallel t\parallel }^{\sigma }\right),s,t\in E_0,\hfill \end{array}$$

and it is enough to use Theorem 6 with $\lambda =max\left\{\right|\eta |,|\theta \left|\right\}$. □

Example 2.

Note that (42) with ${\xi }_1=\eta =0$ gives the estimation

$$\begin{array}{c}\hfill 0\le \Phi \left(s\right)-\varphi \left(s\right)+{\displaystyle \frac{1}{3}}\varphi \left(0\right)\le {\displaystyle \frac{{\xi }_2}{3}}+{\displaystyle \frac{{\theta \parallel s\parallel }^{\sigma }}{2-2^{\sigma -1}}},s\in E.\end{array}$$

(43)

Next, (42) with ${\xi }_1={\xi }_2=0$ and $0<\theta =c\eta$, with some real $c\ge 1$, becomes the condition

$$\begin{array}{c}\hfill {\displaystyle \frac{{\eta \parallel s\parallel }^{\sigma }}{2-2^{\sigma -1}}}\le \Phi \left(s\right)-\varphi \left(s\right)+{\displaystyle \frac{1}{3}}\varphi \left(0\right)\le {\displaystyle \frac{{c\eta \parallel s\parallel }^{\sigma }}{2-2^{\sigma -1}}},s\in E.\end{array}$$

(44)

On the other hand, from Theorem 4 (for $E_2=\mathbb{R}$), we can only obtain analogous inequalities of the following symmetric form:

$$\begin{array}{c}\hfill -{\displaystyle \frac{{\xi }_2}{3}}-{\displaystyle \frac{{\theta \parallel s\parallel }^{\sigma }}{2-2^{\sigma -1}}}\le \Phi \left(s\right)-\varphi \left(s\right)+{\displaystyle \frac{1}{3}}\varphi \left(0\right)\le {\displaystyle \frac{{\xi }_2}{3}}+{\displaystyle \frac{{\theta \parallel s\parallel }^{\sigma }}{2-2^{\sigma -1}}},s\in E,\end{array}$$

i.e., (42) with ${\xi }_1=-{\xi }_2$ and $\eta =-\theta$.

This shows that our results provide much finer estimations for functions that take real values.

8. Conclusions

Using the quadratic functional equation

$$q(a+b)+q(a-b)=2q\left(a\right)+2q\left(b\right)$$

as an example, we have shown how to apply the Banach limit and the fixed point approach to obtain new Ulam stability results for mappings that take real values.

In particular, our findings provide new characterizations of inner product spaces that extend the classical result of Jordan and von Neumann [12]. The technique for proving these characterizations is based on the results in [12] and also on the Ulam stability results for the quadratic equation, obtained by us.

The natural limitations we encountered were related to the form of the fixed point theorem we used. Perhaps further analogous (or even more general) results are possible using other fixed point theorems (see, e.g., [75]).

Potential further research could also aim to obtain similar results for some other functional equations, in particular those mentioned in Section 2, which are direct generalizations of the quadratic equation. Moreover, it would be interesting to consider an analogous approach for mappings taking values in Banach spaces (not only in $\mathbb{R}$) and/or defined on non-Abelian groups or some non-Archimedean spaces (e.g., non-Archimedean normed spaces).