Remarks on stability of some inhomogeneous functional equations

This is an expository paper in which we present some simple observations on the stability of some inhomogeneous functional equations. In particular, we state several stability results for the inhomogeneous Cauchy equation f(x+y)=f(x)+f(y)+d(x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x+y)=f(x)+f(y)+d(x,y)$$\end{document}and for the inhomogeneous forms of the Jensen and linear functional equations.

Let us recall that the problem of stability of functional equations was motivated by a question of Ulam asked in 1940 and an answer to it published by Hyers in [21]. Since then numerous papers on this subject have been published and we refer to [4,11,22,[25][26][27] for more details, some discussions, further references and examples of very recent results.
One of the most classical results, concerning the stability of the Cauchy equation can be stated as follows.
J. Brzdęk AEM Theorem 1.1. Let E 1 and E 2 be normed spaces and c ≥ 0 and p = 1 be fixed real numbers. Assume also that f : E 1 → E 2 is a mapping satisfying If p ≥ 0 and E 2 is complete, then there exists a unique solution T : E 1 → E 2 of (1.1) such that If p < 0, then f is additive [i.e., it is a solution to (1.1)].
It is composed of the results in [1,8,20,21,29]. Also, it is known (see [20,22]) that for p = 1 an analogous result is not valid. Moreover, it has been proved in [5] that estimation (1.3) is optimal (for E 1 There arises a natural question whether analogous results can be proved for the inhomogenous Cauchy equation The equation has drawn the attention of several authors and been studied already for various spaces and forms of d in, e.g., [3,[12][13][14][15][16][17]19,23,24].
In this expository paper we present some simple remarks motivated by that issue. We believe that they are new and can be of some interest for researchers investigating that field and related areas.
In particular, we show that the following result is valid.
there exists a unique solution g : Moreover, that estimation is optimal when E 1 = R; namely there exists a function f : Vol. 89 (2015) Stability of some inhomogeneous equations 85 for each solution h : R → R of (1.4).

General observations
We start with some general observations. In what follows S is a nonempty set, (X, +) is a commutative group, and we define a binary operation + in X S (the family of all functions mapping S into X) in the usual way by: Let us introduce the following technical definition (2 X stands for the family of all subsets of X).
and U ⊂ D be nonempty. We say that the conditional equation is (Φ, Ψ) -stable in U provided for any ϕ 0 ∈ U with there exists a solution ϕ ∈ D of Eq. (2.1) such that Moreover, if for every ϕ 0 ∈ U, satisfying (2.2), there is exactly one solution ϕ ∈ D of (2.1), fulfilling (2.3), then we say that Eq.
If U = D, then we omit the part 'in U' and simply say '(Φ, Ψ) -stable'.
Let n ∈ N, P ⊂ S n be nonempty, U ⊂ D be two subgroups of the group (X S , +) and H : D → X P be additive, i.e., We have the following (very simple, but very useful) observation.
Suppose that the equation admits a solution f 0 ∈ U. Then the equation Write g 0 := g + f 0 . Then g 0 ∈ U and Hence, there exists a solution h 0 ∈ D of Eq. (2.4) such that The proof of the necessary condition is analogous. But for the convenience of readers, we present it below. So, assume that Eq. (2.7) Write g := g 0 − f 0 . Then Hence, there exists a solution h ∈ D of Eq. (2.5) such that Clearly, h 0 := h + f 0 is a solution to (2.4) and Assume now that Eq. (2.4) is (Φ, Ψ)-stable in U with uniqueness. Let g ∈ U satisfy condition (2.6) and h, h ∈ D be solutions to (2.5) such that Vol. 89 (2015) Stability of some inhomogeneous equations 87 Write g 0 := g + f 0 , h 0 := h + f 0 and h 0 := h + f 0 . Then (2.7) holds, h 0 and h 0 are solutions to (2.4) and Consequently, h 0 = h 0 , whence h = h . This completes the proof of the sufficient condition concerning uniqueness. The proof of the converse implication is analogous. ; in such a case we could speak of trivial (Φ, Ψ)-nonstability. Clearly, without this assumption, we could deduce from Theorem 2.2 that the existence of a function g 0 ∈ U satisfying (2.7) implies the existence of a solution f ∈ D of (2.4). The subsequent example shows that sometimes this is not the case, which makes the necessity of the assumption more convincing.
Example. Let E 1 and E 2 be normed spaces, d : E 2 1 → E 2 , c, p, r, s ∈ R, p < 0, c > 0, s + r < 0, d(x, x) = 0 for some x = 0, and let one of the following two inequalities be fulfilled:

Proof of Theorem 1.2
The proof is actually a routine in view of Theorems 1.1 and 2.2. But for the convenience of readers we present the main steps. In particular, we need the following simple observation.
Thus we have proved that h is additive. Now, we are ready to prove the theorem. First we show statement (a). So, fix p ≥ 0, p = 1. In view of Theorem 1.1 the conditional functional equation Hence, according to Theorem 2.2 (β) (with n = 2, whence, by Lemma 3.1, it is additive, which means that g is a solution to Eq. (1.4).
To complete the proof of (a), assume that E 1 = R. Take u 0 ∈ E 2 with u 0 = 1 and define functions f 1 : R → R and f : R → E 2 by Then (1.7) holds and This ends the proof of (a), because by [5, Theorem 2] Vol. 89 (2015) Stability of some inhomogeneous equations 89 Statement (b) follows at once from Lemma 3.1 and Theorems 1.1 and 2.2 (β) (analogously to statement (a)), with Finally, we prove (c). Let E 1 = E 2 = R. Fix c 0 > 0. According to the results in [20], there is f 1 : R → R such that x,y∈ R, Take a solution h : R → R of (1.4). Then h 1 := h − f 0 is additive and This ends the proof.

Further consequences
In  Suppose that g : E → F is continuous at a point x 0 ∈ E and satisfies the condition Then there exists an additive h : E → F such that Moreover, if C is bounded, then h is unique.
there exists a solution h : Moreover, if C is bounded, then h is unique.
for every x, y ∈ E and U being the family of all functions f ∈ F E that are continuous at x 0 ; moreover that stability is with uniqueness when C is bounded. So, it is enough to use Theorem 2.2 with n = 2, S = E, X = F , and B = ∅.
Now we present several examples of hyperstability results (see [11] for further information on this issue). Let us start with the following remark.
The next theorem can be easily deduced from [9, Proposition 2.2].
Theorem 4.4. Let E and Y be normed spaces, dim E > 2, g : E → Y , and p = 1 and L 0 be positive real numbers with Then g is additive.
It yields the following.
Then g is a solution to (1.4).
for every x, y ∈ E.
The next result was proved in [28,Theorem 2] and concerns the linear functional equation (in two variables). Then We will derive from it a hyperstability result for the inhomogeneous version of the linear equation. To this end we need yet the following.

Lemma 4.7. Assume that E is a linear space over
Then f satisfies the equation for every x, y ∈ E.
Proof. Let e := f (0). Then, in view of (4.7), we get This implies that Consequently, by (4.7) and (4.9), for every x, y ∈ E \ {0} We can find in [4,11,22,[25][26][27] numerous further examples of stability results for various equations that can be extended to their inhomogeneous versions, in a similar way as in this paper (by applying Theorem 2.2). In general, it is very easy to find suitable reasonings.
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