Stability of functional equations connected with quadrature rules

We study the stability properties of the equation 0.1F(y)-F(x)=(y-x)∑i=1naif(αix+βiy)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(y) - F(x) = (y - x) \sum_{i=1}^{n}a_{i}f(\alpha_i x + \beta_{i}y)$$\end{document}which is motivated by numerical integration. In Szostok and Wa̧sowicz (Appl Math Lett 24(4):541–544, 2011) the stability of the simplest equation of the type (0.1) was investigated thus the inequality F(y)-F(x)-(y-x)fx+y≤ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left|F(y) - F(x) - (y - x)f\left(x + y\right)\right| \leq \varepsilon$$\end{document}was studied. In the current paper we present a somewhat different approach to the problem of stability of (0.1). Namely, we deal with the inequality F(y)-F(x)y-x-∑i=1naif(αix+βiy)≤ε.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left|\frac{F(y) - F(x)}{y - x} - \sum_{i=1}^{n}a_{i}f(\alpha_{i}x + \beta_{i}y)\right| \leq \varepsilon.$$\end{document}


Introduction
In this paper we study the stability properties of the equation (1.1) Equation (1.1) is a profound generalization of the well known Aczél equation which was motivated by the Lagrange mean value theorem (see [1]).
It can be proved (under some assumptions) that solutions of (1.1) are polynomial functions (see [4]). By a polynomial function of order n we mean any solution of the functional equation Δ n+1 h f (x) = 0, where Δ n h stands for the nth iterate of the difference operator The stability of (1.2) was studied in [10] where the inequality In the current paper we present a somewhat different approach to the stability of (1.1). Equation (1.2) is known as the Aczél equation and was inspired by the Lagrange mean value theorem. Therefore it is natural to write (1.2) in the form Now, we may consider the following inequality Moreover, we shall study in this setting the stability properties of the more general equation (1.1).

Results
First we prove a technical lemma concerning a pexiderized version of (1.1) Vol. 90 (2016)

Stability of functional equations 165
Proof. Taking x + h instead of y in (2.2), we get further, taking x + 2h, x + h in place of y, x resp. in (2.2), we get and these two equations give us On the other hand, taking x + 2h in place of y in (2.2), we get This, together with (2.4), yields In the next part of the paper we are going to use a result proved by Baker in [2]. For the sake of completeness we shall cite this theorem.
and there exists a polynomial function p k : V → B of order at most m − 1 and a constant c k such that Although Lemma 1 was stated for Eq. (2.1), in the remaining part of the paper we shall work with (1.1) with an additional assumption that α k +β k = 1. There are several reasons to restrict ourselves to this case. Our equations stem from numerical integration and quadratures used to approximate the integral take exactly the form used in (1.1) (with α k + β k = 1). Further if we want to prove that functions satisfying our equations are continuous we have to make some assumptions of this kind. Now we shall state the main result of the paper.

Theorem 2.2. Let n ∈ N and let functions
Then there exist a constant M > 0 and a polynomial function p of order at most 3n − 2 such that Further there exist a polynomial P of degree at most 3n − 1 and a constant K > 0 such that the function x → (F (x) − P (x)) is Lipschitz continuous and functions P, p satisfy (1.1). Finally, if a 1 + · · · + a n = 0 then also f must be continuous.
Consequently, condition (2.5) is satisfied. We only have to check that not all numbers b i are zero.
To this end, we observe that points of the form First we consider the case n = 1. In this case inequality (2.8) contains values of f at the points: At least one of these points is different from the others. Thus its term cannot vanish after our simplification. Further if n = 2 and at least one of the numbers α 1 , α 2 is different from 0 and 1 then we have four different points of the forms x + (1 − α i )h, x + (2 − α i )h, and only two of the shape x + (2 − 2α 1 )h. Like before, it means that some of the values of f from (2.8) do not vanish. In the case: α 1 = 0, α 2 = 1 we have a concrete form of (2.8) and it is easy to check that the left-hand side of this inequality is nontrivial. To finish this part of the proof assume that n ≥ 3. In this case the system contains at least 2n − 2 different points. Thus there must be an i 0 ∈ {1, . . . , n} such that In view of Theorem 2.1, this means that some function a i0 f is close to a polynomial function. Thus there exists p(x) = p 0 + p 1 (x) + · · · + p 3n−2 (x) such that functions p k are monomial of orders k and the inequality is satisfied with some M > 0. Therefore we may write where |r(x)| ≤ Mε. Now we shall use this equality in (2.6). Without loss of generality we may assume that F (0) = 0 thus, taking y = 0 in (2.6), we may write

T. Szostok AEM
Using here the boundedness of r, we get for some K > 0 is a monomial function of order k and |R(x)| ≤ Kε. Using (2.11), (2.10) and the boundedness of r in (2.6), we get for some L > 0 (2.12) Now we observe that yP0−xP0 (2.13) In the next step of the proof we substitute 2x and 2y instead of x and y, respectively, and we arrive at (2.14) Dividing both sides of (2.14) by 2 3n−2 , we can see that the only terms which remain unchanged are those of order 3n − 2, all others are divided by powers of two. If we repeat this operation then all expressions with orders smaller than 3n − 2 tend to zero. This yields Now we may use a result from [5] which states that the mapping x → xP 3n−2 (x) is an ordinary polynomial. This means that P 3n−2 (x) = b 3n−1 x 3n−1 , for some real number b 3n−1 . Further, using (2.15) in (2.13), we get Repeating this procedure sufficiently many times, we show that F is of the form (2.7). Moreover, we have for all x = y. Note also that the last inequality which we obtain is of the form Now let A be a k-additive and symmetric function such that then (2.17) may be rewritten in the form (2.18) Now, let q j be a sequence of rational numbers tending to 1 and different from 1. Then, taking y = q j x in (2.18), we get

T. Szostok AEM
Tending here with j to infinity, we get thus also p k is continuous, provided that n i=1 a i = 0. Remark 1. A careful inspection of the proof of Theorem 2.2 shows that it is possible to obtain the exact values of M and K but the formulas expressing them would be very complicated. Moreover these values rely strongly on our method and, therefore, they are probably far from being optimal.
The following corollary will show that the degrees of f and F obtained in Theorem 2.2 may, in the case of concrete equations be lower. The inequality considered in this corollary is motivated by the Simpson quadrature rule.

Corollary 1. If functions f, F satisfy the inequality
and Proof. Using Theorem 2.2 we can see that there exist functions r and R such that where P is a polynomial of degree at most 5 and where p is a polynomial of degree at most 4. However from Theorem 2.2 we know that functions P, p satisfy the equation It is well known (see for example [7]) that in such a case the degree of p is not greater than 3. Further using the continuity of p and tending with y to x, we get that P = p, as claimed.
The next example will show that it is impossible to get the superstability result (as it was the case in the paper [10]). Example. Assume that n i=1 a i = 1, take function F as any function satisfying the Lipschitz condition with constant ε/2 and f as a function bounded by ε/2. Then functions F, f satisfy inequality (2.6).
Remark 2. As it is easy to observe, inequalities considered in [10] and in the present paper have a joint generalization which is given by (2.20) In view of results contained in [10] and of Theorem 2.2, we may say that the stability problem posed in this way has a satisfactory solution for p = 1 and a partial solution for p = 0.
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