Hyperstability of Some Functional Equations on Restricted Domain

Hindawi Journal of Function Spaces Volume 2017, Article ID 1946394, 6 pages https://doi.org/10.1155/2017/1946394

Research Article Hyperstability of Some Functional Equations on Restricted Domain Anna Bahyrycz and Jolanta Olko Faculty of Applied Mathematics, AGH University of Science and Technology, Mickiewicza 30, 30-059 Kraków, Poland Correspondence should be addressed to Anna Bahyrycz; [email protected] Received 20 April 2017; Accepted 30 May 2017; Published 16 July 2017 Academic Editor: Adrian Petrusel Copyright © 2017 Anna Bahyrycz and Jolanta Olko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The paper concerns functions which approximately satisfy, not necessarily on the whole linear space, a generalization of linear functional equation. A Hyers-Ulam stability result is proved and next applied to give conditions implying the hyperstability of the equation. The results may be used as tools in stability studies on restricted domains for various functional equations. We use the main theorem to obtain a few hyperstability results of Fréchet equation on restricted domain for different control functions.

than or equal to 𝑛 − 1 (cf. [2, 4]). As a generalization we can consider the functional equation

1. Introduction The famous Cauchy equation 𝑓 (𝑥 + 𝑦) = 𝑓 (𝑥) + 𝑓 (𝑦)

(1)

has a very long history, which started with A. M. Legendre in 1791 and C. F. Gauss in 1809, but the first important result about the solutions of (1) was proved by Cauchy (see [1]) and due to this outcome (1) is today named after him, as is mentioned in [2]. In the meantime, many authors have dealt with this equation, and next Fréchet in [3] studied an important generalization of Cauchy equation, characterizing the polynomials among the continuous mappings, namely, the equation Δ𝑛ℎ1 ⋅⋅⋅ℎ𝑛 𝑓 (𝑥) = 0,

(2)

where Δ denotes the difference operator defined as usual by Δ ℎ 𝑓 (𝑥) = Δ1ℎ 𝑓 (𝑥) fl 𝑓 (𝑥 + ℎ) − 𝑓 (𝑥) , 𝑛 Δ𝑛+1 ℎ1 ⋅⋅⋅ℎ𝑛+1 𝑓 (𝑥) = Δ ℎ𝑛+1 (Δ ℎ1 ⋅⋅⋅ℎ𝑛 𝑓 (𝑥)) ,

(3)

for 𝑥, ℎ, ℎ1 , . . . , ℎ𝑛+1 ∈ R, 𝑛 ∈ N. From the result of Fréchet it follows that a continuous function 𝑓 : R → R satisfies (2) for all 𝑥 and ℎ𝑗 if and only if 𝑓 is a polynomial of degree smaller

𝑚

𝑛

𝑖=1

𝑗=1

∑𝐴 𝑖 𝑔 ( ∑ 𝑎𝑖𝑗 𝑥𝑗 ) + 𝐴 = 0,

(4)

where 𝑔 : 𝑋 → 𝑌 and 𝑋, 𝑌 are the linear spaces over a field F ∈ {R, C}, 𝐴, 𝑎𝑖𝑗 ∈ F, and 𝐴 𝑖 ∈ F \ {0}, 𝑖 ∈ {1, . . . , 𝑚}, 𝑗 ∈ {1, . . . , 𝑛}. The stability and hyperstability of the above functional equation and its particular cases were studied by many mathematicians (cf., e.g., [5–16]). Let us recall (see [9]) that the theory of Hyers-Ulam stability, in general, was motivated by a problem raised in 1940 by Ulam [17] and a partial answer to it was provided by Hyers [18]. The term hyperstability, in the meaning used here, was introduced presumably in [19] (cf. also [20]). Our basic assumptions are the following. 𝑋 is a normed space over a field F ∈ {R, C}, 𝐸1 , 𝐸2 are the nonempty subsets of 𝑋 such that 𝐸2 ⊆ 𝐸1 ⊆ 𝑋, and 𝐷 is a nonempty subset of F such that 1 ∈ 𝐷, R+ fl [0, +∞). Denote N≤𝑘 fl {1, . . . , 𝑘} and N𝑘 fl {𝑙 ∈ N : 𝑙 ≥ 𝑘}, for 𝑘 ∈ N, and a sum of numbers over an empty set is defined to be zero. In the sequel, P(𝐴) = {𝐵 ⊂ 𝐴 : 𝐴 ≠ 0} for nonempty set 𝐴.

2

Journal of Function Spaces Let 𝜃 : 𝐸2𝑛 → R+ satisfy the inequality

We say that the equation 𝑚

𝑛

𝑖=1

𝑗=1

𝜃 (𝛽𝑖 𝑥1 , . . . , 𝛽𝑖 𝑥𝑛 ) ≤ 𝜔 (𝛽𝑖 ) 𝜃 (𝑥1 , . . . , 𝑥𝑛 ) ,

∑𝐴 𝑖 𝑔 ( ∑𝑎𝑖𝑗 𝑥𝑗 ) + 𝐴 = 0,

𝑖 ∈ N≤𝑚 , 𝑥1 , . . . , 𝑥𝑛 ∈ 𝐸2 . 𝑛

𝑛

𝑗=1

𝑗=1

(5)

𝑥1 , . . . , 𝑥𝑛 ∈ 𝐸2 , ∑𝑎1𝑗 𝑥𝑗 , . . . , ∑𝑎𝑚𝑗 𝑥𝑗 ∈ 𝐸1 is 𝜃-hyperstable in the class of functions 𝑔 : 𝐸1 → 𝑌 (with a control function 𝜃 : 𝐸2𝑛 → R+ ), if 𝑔 : 𝐸1 → 𝑌 satisfies the inequality 󵄩󵄩 𝑚 󵄩󵄩 𝑛 󵄩󵄩 󵄩 󵄩󵄩∑𝐴 𝑔 ( ∑ 𝑎 𝑥 ) + 𝐴󵄩󵄩󵄩 ≤ 𝜃 (𝑥 , . . . , 𝑥 ) , 󵄩󵄩 󵄩󵄩 𝑖 𝑖𝑗 𝑗 1 𝑛 󵄩󵄩𝑖=1 󵄩󵄩 𝑗=1 󵄩 󵄩

(6)

for all 𝑥1 , . . . , 𝑥𝑛 ∈ 𝐸2 , and ∑𝑛𝑗=1 𝑎1𝑗 𝑥𝑗 , . . . , ∑𝑛𝑗=1 𝑎𝑚𝑗 𝑥𝑗 ∈ 𝐸1 fulfills (5). The above notion of hyperstability for functions defined on subsets of a linear space 𝑋 (not necessarily on the whole space but on the restricted domain) is a counterpart of hyperstability considered in [8]. In this paper we obtain, applying the fixed point theorem, some results concerning stability and hyperstability for (4) on restricted domain. Our outcomes allow studying the behavior of functions approximately fulfilling functional equations of this linear type on different subsets of the linear space, and this also applies to domain restrictions for control functions. This way we give a tool for the further Hyers-Ulam stability research. Its application is presented in the last section where some new as well as generalizations of known results on hyperstability of Cauchy and Fréchet equation are obtained.

2. Auxiliary Result First we establish a counterpart of stability result from [7] on restricted domain, and to do this we use the fixed point theorem. This result will be needed to show our main theorem which provides criteria for hyperstability of many functional equations of the linear type on a restricted domain. Lemma 1. Assume that 𝑌 is a Banach space, 𝐴 = 0 or “𝐴 ≠ 0 and ∑𝑚 𝑖=1 𝐴 𝑖 ≠ 0,” and 𝜔 : 𝐷 → R+ . Suppose that there exist 0 ≠ 𝐼 ⊂ N≤𝑚 and 𝑐1 , . . . , 𝑐𝑛 ∈ 𝐷 such that 𝑛

𝑗=1

𝐴 𝐼 fl ∑𝐴 𝑙 ≠ 0, 𝑙∈𝐼

󵄨󵄨 𝐴 󵄨󵄨 󵄨 󵄨 𝛾 fl ∑ 󵄨󵄨󵄨 𝑖 󵄨󵄨󵄨 𝜔 (𝛽𝑖 ) < 1. 󵄨 󵄨 𝑖∉𝐼 󵄨 𝐴 𝐼 󵄨

󵄩󵄩 𝑚 󵄩󵄩 𝑛 󵄩󵄩 󵄩 󵄩󵄩∑𝐴 𝑔 ( ∑𝑎 𝑥 ) + 𝐴󵄩󵄩󵄩 ≤ 𝜃 (𝑥 , . . . , 𝑥 ) , 󵄩󵄩 𝑖 𝑖𝑗 𝑗 1 𝑛 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩𝑖=1 𝑗=1 󵄩 𝑛

𝑛

𝑗=1

𝑗=1

(9)

𝑥1 , . . . , 𝑥𝑛 ∈ 𝐸2 , ∑𝑎1𝑗 𝑥𝑗 , . . . , ∑ 𝑎𝑚𝑗 𝑥𝑗 ∈ 𝐸1 , then there exists a unique function 𝐺 : 𝐸2 → 𝑌 satisfying (5) such that 󵄩󵄩 󵄩 𝜃 (𝑐1 𝑥, . . . , 𝑐𝑛 𝑥) , 𝑥 ∈ 𝐸2 . 󵄩󵄩𝑔 (𝑥) − 𝐺 (𝑥)󵄩󵄩󵄩 ≤ 󵄨󵄨 󵄨󵄨 󵄨󵄨𝐴 𝐼 󵄨󵄨 (1 − 𝛾)

(10)

Proof. Take 0 ≠ 𝐼 ⊂ N≤𝑚 and 𝑐1 , . . . , 𝑐𝑛 ∈ 𝐷 such that assumptions (7) and (8) hold. First consider the case 𝐴 = 0. Taking 𝑥 ∈ 𝐸2 and substituting 𝑥𝑗 = 𝑐𝑗 𝑥, 𝑗 ∈ N≤𝑛 , in (9) we have 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩𝑔 (𝑥) − ∑ −𝐴 𝑖 𝑔 (𝛽𝑖 𝑥)󵄩󵄩󵄩 ≤ 𝜃 (𝑐1 𝑥,󵄨 . . .󵄨 , 𝑐𝑛 𝑥) , 󵄩󵄩 󵄩󵄩 󵄨󵄨𝐴 𝐼 󵄨󵄨 󵄩󵄩 󵄩󵄩 󵄨 󵄨 𝑖∉𝐼 𝐴 𝐼

𝑥 ∈ 𝐸2 . (11)

Define −𝐴 𝑖 𝜉 (𝛽𝑖 𝑥) , 𝜉 ∈ 𝑌𝐸2 , 𝑥 ∈ 𝐸2 , 𝐴 𝐼 𝑖∉𝐼

(T𝜉) (𝑥) fl ∑

󵄨󵄨 −𝐴 󵄨󵄨 󵄨 󵄨 (Λ𝛿) (𝑥) fl ∑ 󵄨󵄨󵄨 𝑖 󵄨󵄨󵄨 𝛿 (𝛽𝑖 𝑥) , 󵄨 󵄨 𝑖∉𝐼 󵄨 𝐴 𝐼 󵄨 𝜀 (𝑥) fl

𝛿 ∈ R+ 𝐸2 , 𝑥 ∈ 𝐸2 ,

𝜃 (𝑐1 𝑥, . . . , 𝑐𝑛 𝑥) , 𝑥 ∈ 𝐸2 . 󵄨󵄨 󵄨󵄨 󵄨󵄨𝐴 𝐼 󵄨󵄨

(12)

(13) (14)

Then (11) may be rewritten as follows: 󵄩󵄩 󵄩 󵄩󵄩𝑔 (𝑥) − T𝑔 (𝑥)󵄩󵄩󵄩 ≤ 𝜀 (𝑥) ,

𝑥 ∈ 𝐸2 .

(15)

By induction, for every 𝑥 ∈ 𝐸2 and 𝑘 ∈ N0 Λ𝑘 𝜀 (𝑥) ≤ 𝜀 (𝑥) 𝛾𝑘 .

∞

∞

𝑛=0

𝑛=0

𝜀∗ (𝑥) fl ∑ (Λ𝑛 𝜀) (𝑥) ≤ 𝜀 (𝑥) ∑ 𝛾𝑛 =

𝑐𝑗 𝑥 ∈ 𝐸2 , 𝛽𝑖 𝑥 ∈ 𝐸2 , 𝑖 ∈ N≤𝑚 , 𝑗 ∈ N≤𝑛 , 𝑥 ∈ 𝐸2 , 𝑙 ∈ 𝐼,

If 𝑔 : 𝐸1 → 𝑌 fulfills the estimation

(16)

Since 𝛾 < 1, we see that

𝛽𝑖 fl ∑ 𝑎𝑖𝑗 𝑐𝑗 ∈ 𝐷,

𝛽𝑙 = 1,

(8)

(7)

𝜀 (𝑥) , 1−𝛾

(17) 𝑥 ∈ 𝐸2 .

Notice that all assumptions of Theorem 1 in [21] are satisfied for operators (12) and (13) and 𝑋 = 𝐸2 . According to this fixed point theorem, the function 𝐺 : 𝐸2 → 𝑌 given by 𝐺(𝑥) = lim𝑛→∞ (T𝑛 𝑔)(𝑥) for 𝑥 ∈ 𝐸2 is a fixed point of T fulfilling (10).

Journal of Function Spaces

3

It is sufficient to show that 𝐺 satisfies (5) (with 𝐴 = 0). First we prove that for every 𝑘 ∈ N∪{0} and every 𝑥1 , . . . , 𝑥𝑛 ∈ 𝐸2 such that ∑𝑛𝑗=1 𝑎1𝑗 𝑥𝑗 , . . . , ∑𝑛𝑗=1 𝑎𝑚𝑗 𝑥𝑗 ∈ 𝐸1 󵄩󵄩 𝑚 󵄩󵄩 𝑛 󵄩󵄩 󵄩 󵄩󵄩∑𝐴 (T𝑘 𝑔) ( ∑𝑎 𝑥 )󵄩󵄩󵄩 ≤ 𝛾𝑘 𝜃 (𝑥 , . . . , 𝑥 ) . 󵄩󵄩 𝑖 𝑖𝑗 𝑗 󵄩 1 𝑛 󵄩󵄩 󵄩󵄩𝑖=1 󵄩󵄩 𝑗=1 󵄩

(18)

𝑛

𝑖=1

𝑗=1

𝑛

𝑖=1

𝑗=1

= ∑𝐴 𝑖 ∑ 𝑖=1

=∑ 𝑝∉𝐼

−𝐴 𝑝

𝑝∉𝐼

−𝐴 𝑝 𝐴𝐼

𝐴𝐼

(iv) 𝜃4 (𝑥1 , . . . , 𝑥𝑛 ) = 𝐶⊓𝑛𝑗=1 ‖𝑥𝑗 ‖𝑘𝑗 ;

󵄨 󵄨max{𝑘1 ,...,𝑘𝑛 ,𝑡} {󵄨󵄨󵄨𝛽󵄨󵄨󵄨 𝜔5 (𝛽) fl { min{𝑘 ,...,𝑘 ,𝑡} 󵄨󵄨 󵄨󵄨 1 𝑛 {󵄨󵄨𝛽󵄨󵄨

𝑗=1

𝑚

[𝑖=1

𝑗=1

]

󵄩󵄩 𝑚 󵄩󵄩 𝑛 󵄩󵄩 󵄩 󵄩󵄩∑𝐴 (T𝑘+1 𝑔) ( ∑𝑎 𝑥 )󵄩󵄩󵄩 󵄩󵄩 𝑖 𝑖𝑗 𝑗 󵄩 󵄩󵄩 󵄩󵄩𝑖=1 󵄩󵄩 𝑗=1 󵄩 󵄨󵄨 −𝐴 󵄨󵄨 󵄨 𝑝 󵄨󵄨 𝑘 󵄨󵄨 𝛾 𝜃 (𝛽𝑝 𝑥1 , . . . , 𝛽𝑝 𝑥𝑛 ) ≤ ∑ 󵄨󵄨󵄨󵄨 󵄨 𝑝∉𝐼 󵄨󵄨 𝐴 𝐼 󵄨󵄨

We can now state a counterpart (on restricted domain) of Theorem 2.1 in [8], concerning 𝜃-hyperstability of (5).

(20)

Theorem 3. Let 𝑌 be a normed space over F, 𝐴 = 0 or “𝐴 ≠ 0 and ∑𝑚 𝑖=1 𝐴 𝑖 ≠ 0,” 𝑔 : 𝐸1 → 𝑌, 𝜔 : 𝐷 → R+ , and 𝜃 : 𝐸2𝑛 → R+ satisfy (9). If there exist 0 ≠ 𝐼 ⊂ N≤𝑚 and a sequence {(𝑐1𝑘 , . . . , 𝑐𝑛𝑘 )}𝑘∈N of elements of 𝐷𝑛 such that 𝑛

𝛽𝑖𝑘 fl ∑ 𝑎𝑖𝑗 𝑐𝑗𝑘 ∈ 𝐷, 𝑗=1

𝜃 (𝑥1 , . . . , 𝑥𝑛 ) ,

𝑐𝑗𝑘 𝑥 ∈ 𝐸2 ,

which finishes the proof of (18). Letting 𝑘 → ∞ in (18), we prove that 𝐺 satisfies (5), which completes the proof in the first case. In the case where 𝐴 ≠ 0 and ∑𝑚 𝑖=1 𝐴 𝑖 ≠ 0, the function 𝐴 , 𝑥 ∈ 𝐸 , satisfies ℎ(𝑥) fl 𝑔(𝑥) + 𝐴/ ∑𝑚 1 𝑖=1 𝑖 󵄩󵄩 󵄩󵄩 𝑚 𝑛 󵄩 󵄩󵄩 󵄩󵄩∑𝐴 ℎ ( ∑𝑎 𝑥 )󵄩󵄩󵄩 ≤ 𝜃 (𝑥 , . . . , 𝑥 ) , 󵄩󵄩 𝑖 𝑖𝑗 𝑗 󵄩 1 𝑛 󵄩󵄩 󵄩󵄩𝑖=1 󵄩󵄩 𝑗=1 󵄩

Obviously 𝐺(𝑥) fl 𝐻(𝑥) − desired function.

𝑖 ∈ N≤𝑚 , 𝑗 ∈ N≤𝑛 , 𝑘 ∈ N, 𝑥 ∈ 𝐸2 , 𝛽𝑙𝑘 = 1, 𝑙 ∈ 𝐼, 𝐴 𝐼 fl ∑𝐴 𝑙 ≠ 0, 𝑙∈𝐼

(26)

󵄨󵄨 𝐴 󵄨󵄨 󵄨 󵄨 lim ∑ 󵄨󵄨󵄨 𝑖 󵄨󵄨󵄨 𝜔 (𝛽𝑖𝑘 ) < 1, 󵄨 󵄨 𝑘→∞ 𝑖∉𝐼 󵄨 𝐴 𝐼 󵄨 𝜃 (𝛽𝑖𝑘 𝑥1 , . . . , 𝛽𝑖𝑘 𝑥𝑛 ) ≤ 𝜔 (𝛽𝑖𝑘 ) 𝜃 (𝑥1 , . . . , 𝑥𝑛 ) 𝑖 ∈ N≤𝑚 , 𝑘 ∈ N, 𝑥1 , . . . , 𝑥𝑛 ∈ 𝐸2 ,

(22)

𝐴 𝑖 and 𝑥 ∈ 𝐸2 is the

(25)

𝛽𝑖𝑘 𝑥 ∈ 𝐸2 ,

(21)

for 𝑥1 , . . . , 𝑥𝑛 ∈ 𝐸2 , and ∑𝑛𝑗=1 𝑎1𝑗 𝑥𝑗 , . . . , ∑𝑛𝑗=1 𝑎𝑚𝑗 𝑥𝑗 ∈ 𝐸1 . Consequently, we conclude that there exists a unique function 𝐻 : 𝐸2 → 𝑌 such that 𝜃 (𝑐1 𝑥, . . . , 𝑐𝑛 𝑥) , 𝑥 ∈ 𝐸2 . ‖ℎ (𝑥) − 𝐻 (𝑥)‖ ≤ 󵄨󵄨 󵄨󵄨 󵄨󵄨𝐴 𝐼 󵄨󵄨 (1 − 𝛾)

(24)

3. The Main Result

󵄨󵄨 −𝐴 󵄨󵄨 󵄨 𝑝 󵄨󵄨 𝑘 󵄨󵄨 𝛾 𝜔 (𝛽𝑝 ) 𝜃 (𝑥1 , . . . , 𝑥𝑛 ) ≤ ∑ 󵄨󵄨󵄨󵄨 󵄨 𝑝∉𝐼 󵄨󵄨 𝐴 𝐼 󵄨󵄨

𝐴/ ∑𝑚 𝑖=1

󵄨 󵄨 for 󵄨󵄨󵄨𝛽󵄨󵄨󵄨 ≥ 1 󵄨 󵄨 for 󵄨󵄨󵄨𝛽󵄨󵄨󵄨 < 1,

where 𝑡 = ∑𝑛𝑗=1 𝑡𝑗 . If there exists at least one 𝑗0 ∈ N≤𝑛 such that 𝑘j0 < 0 or 𝑡𝑗0 < 0 then 0 ∉ 𝐷.

𝑛

Thus by induction hypothesis and (8) we have

=𝛾

(iii) 𝜃3 (𝑥1 , . . . , 𝑥𝑛 ) = 𝐶 max {‖𝑐𝑗 𝑥𝑗 ‖𝑘𝑗 : 𝑗 ∈ {1, . . . , 𝑛}};

(T 𝑔) (𝛽𝑝 ∑𝑎𝑖𝑗 𝑥𝑗 )

[∑𝐴 𝑖 (T𝑘 𝑔) ( ∑ 𝑎𝑖𝑗 (𝛽𝑝 𝑥𝑗 ))] .

𝑘+1

(ii) 𝜃2 (𝑥1 , . . . , 𝑥𝑛 ) = 𝐶 ∑𝑛𝑗=1 ‖𝑐𝑗 𝑥𝑗 ‖𝑘𝑗 ;

(19)

𝑛

𝑘

Remark 2. It is easily seen that functions

with some 𝐶, 𝑆 ∈ [0, +∞), 𝐶 + 𝑆 > 0, 𝑐𝑗 ∈ F0 , and 𝑘𝑗 , 𝑡𝑗 ≠ 0 fulfill condition (8) for some function 𝜔. For instance, in case 𝑛 (iv) we can use 𝜔4 (𝛽) = |𝛽|∑𝑗=1 𝑘𝑗 for 𝛽 ∈ 𝐷, and in case (v) we may take

= ∑𝐴 𝑖 T (T𝑘 𝑔) ( ∑𝑎𝑖𝑗 𝑥𝑗 ) 𝑚

(23)

F0 fl F \ {0} .

(v) 𝜃5 (𝑥1 , . . . , 𝑥𝑛 ) = 𝐶⊓𝑛𝑗=1 ‖𝑥𝑗 ‖𝑡𝑗 + 𝑆 ∑𝑛𝑗=1 ‖𝑐𝑗 𝑥𝑗 ‖𝑘𝑗 ;

∑𝐴 𝑖 (T𝑘+1 𝑔) ( ∑ 𝑎𝑖𝑗 𝑥𝑗 ) 𝑚

𝑋0 fl 𝑋 \ {0} ,

(i) 𝜃1 (𝑥1 , . . . , 𝑥𝑛 ) = 𝐶;

Obviously, the case 𝑘 = 0 is just (9). Next, fix 𝑘 ∈ N ∪ {0}, 𝑥1 , . . . , 𝑥𝑛 ∈ 𝐸2 , and ∑𝑛𝑗=1 𝑎1𝑗 𝑥𝑗 , . . . , ∑𝑛𝑗=1 𝑎𝑚𝑗 𝑥𝑗 ∈ 𝐸1 and assume that (18) holds. Then 𝑚

Denote

lim 𝜃 (𝑐1𝑘 𝑥, . . . , 𝑐𝑛𝑘 𝑥) = 0,

𝑘→∞

then 𝑔 satisfies (5).

(27)

(28)

4


Proof. We can certainly assume that 𝑌 is a Banach space, for if not we replace it by its completion. Take 0 ≠ 𝐼 ⊂ N≤𝑚 and the sequence {(𝑐1𝑘 , . . . , 𝑐𝑛𝑘 )}𝑘∈N of the elements of 𝐷𝑛 such that conditions (25), (26), and (27) hold. Hence we can find 𝜀 ∈ (0, 1) and 𝑘0 ∈ N such that 󵄨󵄨 𝐴 󵄨󵄨 󵄨 󵄨 𝛾𝑘 fl ∑ 󵄨󵄨󵄨 𝑖 󵄨󵄨󵄨 𝜔 (𝛽𝑖𝑘 ) < 1 − 𝜀, 𝑘 ≥ 𝑘0 . 󵄨 󵄨 𝑖∉𝐼 󵄨 𝐴 𝐼 󵄨

󵄩 󵄩󵄩 󵄩󵄩𝑔 (𝑥) − 𝐺𝑘 (𝑥)󵄩󵄩󵄩 ≤

, 󵄨󵄨 󵄨󵄨 󵄨󵄨𝐴 𝐼 󵄨󵄨 (1 − 𝛾𝑘 )

{𝑘 + 𝑙0 𝑐𝑗𝑘 fl { 1 − 𝑘 − 𝑙0 {

(29)

Therefore, according to Lemma 1, for each 𝑘 ∈ N𝑘0 there exists a unique function 𝐺𝑘 : 𝐸2 → 𝑌 satisfying equation (5) such that 𝜃 (𝑐1𝑘 𝑥, . . . , 𝑐𝑛𝑘 𝑥)

It follows easily that the assumptions of Theorem 3 are satisfied with 𝑚 = 2𝑛 and 𝐼 = {2𝑛 }. In case (b) we can find 𝑗0 ∈ {2, . . . , 𝑛} such that 𝑘𝑗0 ≥ 0. Define for 𝑗 ≠ 𝑗0

(35)

for 𝑗 = 𝑗0 .

There exists 𝑖0 ∈ {1, . . . , 2𝑛 } such that 𝑎𝑖0 1 = 𝑎𝑖0 𝑗0 = 1 and 𝑎𝑖0 𝑗 = 0 for 𝑗 ∈ {2, . . . , 𝑛} \ {1, 𝑗0 }. We will verify that we can apply Theorem 3 for 𝐼 = {𝑖0 }. Namely, 𝑛

∑𝑎𝑖0 𝑗 𝑐𝑗𝑘 = 𝑘 + 𝑙0 + 1 − 𝑘 − 𝑙0 = 1,

𝑥 ∈ 𝐸2 .

(30)

In this way we obtain the sequence {𝐺𝑘 }𝑘∈N𝑘 of functions 0 satisfying (5). Letting in (30) 𝑘 → ∞, we get, using (28), that 𝑔 satisfies (5).

4. Applications Using the above outcome we can get the results for the particular form of (5), as, for instance, for Fréchet functional equation, presented in this section. Corollary 4. Assume that 𝑌 is a normed space over F, 𝑛 ∈ N2 , 𝐸 is a nonempty subset of 𝑋0 such that there exists a positive integer 𝑙0 with −𝑥, 𝑙𝑥 ∈ 𝐸, 𝑥 ∈ 𝐸, 𝑙 ∈ N𝑙0 .

(31)

𝑗=1

𝑘 + 𝑙0 ≤

𝑛

∑𝑎𝑖𝑗 𝑐𝑗𝑘 𝑗=1

(36) ≤ (𝑘 + 𝑙0 ) (𝑛 − 1)

for 𝑖 ≠ 𝑖0 .

Therefore 󵄨󵄨 𝑛 󵄨󵄨∑𝑛𝑗=1 𝑘𝑗 󵄨󵄨 󵄨󵄨 𝑘 lim ∑ 󵄨󵄨󵄨󵄨 ∑𝑎𝑖𝑗 𝑐𝑗 󵄨󵄨󵄨󵄨 = 0, 𝑘→∞ 󵄨󵄨 𝑖=𝑖̸ 0 󵄨󵄨󵄨𝑗=1 󵄨 󵄩 󵄩𝑘𝑗 lim 𝐶⊓𝑛𝑗=1 󵄩󵄩󵄩󵄩𝑐𝑗𝑘 𝑥󵄩󵄩󵄩󵄩

(37)

𝑘→∞

󵄨 󵄨∑ 𝑘 󵄨 󵄨𝑘 = lim 󵄨󵄨󵄨𝑘 + 𝑙0 󵄨󵄨󵄨 𝑗=𝑗̸ 0 𝑗 󵄨󵄨󵄨𝑘 + 𝑙0 − 1󵄨󵄨󵄨 𝑗0 ‖𝑥‖ 𝑘→∞

∑𝑛𝑗=1

𝑘𝑗

𝑛 󵄨 󵄨∑𝑛 𝑘 ≤ lim 󵄨󵄨󵄨𝑘 + 𝑙0 󵄨󵄨󵄨 𝑗=1 𝑗 ‖𝑥‖∑𝑗=1 𝑘𝑗 = 0. 𝑘→∞

Thus conditions (26) and (28) hold, which finishes the proof.

If 𝑔 : 𝐸 → 𝑌 fulfills the inequality 󵄩 󵄩 󵄩𝑘 󵄩󵄩 𝑛−1 󵄩󵄩Δ 𝑥 ⋅⋅⋅𝑥 𝑔 (𝑥1 )󵄩󵄩󵄩 ≤ 𝐶⊓𝑛𝑗=1 󵄩󵄩󵄩𝑥𝑗 󵄩󵄩󵄩 𝑗 , 󵄩 󵄩 󵄩 󵄩 2 𝑛

(32)

∑ 𝑥𝑗 ∈ 𝐸, 𝑇 ∈ P (N≤𝑛 ) ,

𝑗∈𝑇

with 𝐶 >

0 and ∑𝑛𝑗=1

𝑤

𝑘𝑗 < 0, then 𝑔 is a solution of the equation

Δ𝑛−1 𝑥2 ⋅⋅⋅𝑥𝑛 𝑔 (𝑥1 )

Corollary 5. Assume that 𝑌 is a normed space over F, 𝑛 ∈ N2 , and 𝐸 is a nonempty subset of 𝑋0 such that there exists a positive integer 𝑙0 with (31). If 𝑔 : 𝐸 → 𝑌 fulfills the inequality 𝑛 󵄩󵄩 𝑛−1 󵄩 󵄩𝑘 󵄩 󵄩󵄩Δ 𝑥 ⋅⋅⋅𝑥 𝑔 (𝑥1 )󵄩󵄩󵄩 ≤ ( ∑𝐶𝑗 󵄩󵄩󵄩𝑥𝑗 󵄩󵄩󵄩 𝑗 ) , 󵄩 2 𝑛 󵄩 󵄩 󵄩 𝑗=1

= 0,

∑ 𝑥𝑗 ∈ 𝐸, 𝑇 ∈ P (N≤𝑛 ) ,

(33)

∑ 𝑥𝑗 ∈ 𝐸, 𝑇 ∈ P (N≤𝑛 ) .

𝑗∈𝑇

𝑛

Proof. Put 𝐸 fl 𝐸2 = 𝐸1 , 𝐷 fl Z0 , and 𝜔(𝛽) fl |𝛽|∑𝑗=1 𝑘𝑗 for 𝛽 ∈ Z0 and consider two cases: (a) ∑𝑛𝑗=2 𝑘𝑗 < 0,

𝑗∈𝑇

with 𝐶𝑗 > 0 and 𝑤𝑘𝑗 < 0, for 𝑗 ∈ {1, . . . , 𝑛}, then 𝑔 is a solution of (33). Proof. Putting 𝐸 fl 𝐸2 = 𝐸1 , 𝐷 fl Z0 , and 𝜔(𝛽) fl |𝛽|−V|𝑤| for 𝛽 ∈ Z0 , where V fl min{|𝑘1 |, . . . , |𝑘𝑛 |}, 𝐼 fl {1}, and {(𝑛 − 1) (𝑘 + 𝑙0 ) + 1 𝑐𝑗𝑘 fl { −𝑘 − 𝑙0 {

(b) ∑𝑛𝑗=2 𝑘𝑗 ≥ 0. In the case (a) we define the sequence {(𝑐1𝑘 , . . . , 𝑐𝑛𝑘 )}𝑘∈N as follows: 𝑐𝑗𝑘

{𝑘 + 𝑙0 fl { 1 {

for 𝑗 ≠ 1 for 𝑗 = 1.

(34)

(38)

for 𝑗 = 1 for 𝑗 > 1,

(39)

we get that 𝛽1𝑘 = 1, 𝑘 + 𝑙0 + 1 ≤ 𝛽𝑖𝑘 ≤ (𝑛 − 1) (𝑘 + 𝑙0 ) + 1

for 𝑖 > 1,

(40)


5

and therefore lim𝑘→∞ |𝛽𝑖𝑘 | = +∞ for 𝑖 ≠ 1, and consequently condition (26) is satisfied. Also, condition (28) holds, because 𝑘 󵄨 󵄨 lim (𝐶1 (󵄨󵄨󵄨(𝑛 − 1) (𝑘 + 𝑙0 ) + 1󵄨󵄨󵄨 ‖𝑥‖) 1

𝑘→∞

(41)

𝑤

𝑛

𝑘 󵄨 󵄨 + ∑ 𝐶𝑗 (󵄨󵄨󵄨𝑘 + 𝑙0 󵄨󵄨󵄨 ‖𝑥‖) 𝑗 ) = 0,

We end the paper with an example, which shows that the additional assumption on 𝐸, appearing in the above corollaries, is essential.

𝑗=2

which with Theorem 3 completes the proof. Denote 𝐹 (𝑓, 𝑥1 , . . . , 𝑥𝑛 ) = Δ𝑛𝑥1 ⋅⋅⋅𝑥𝑛 𝑓 (0) + (−1)𝑛+1 𝑓 (0) , 𝑓 ∈ 𝑌𝑋 , 𝑥1 , . . . , 𝑥𝑛 ∈ 𝑋.

(42)

Corollary 6. Assume that 𝑌 is a normed space over F, 𝑛 ∈ N2 , and 𝐸 is a nonempty subset of 𝑋0 such that there exists a positive integer 𝑙0 with 𝑙𝑥 ∈ 𝐸, 𝑥 ∈ 𝐸, 𝑙 ∈ N𝑙0 .

∑ 𝑥𝑗 ∈ 𝐸, 𝑇 ∈ P (N≤𝑛 ) ,

(44)

𝑗∈𝑇

with 𝐶 > 0 and ∑𝑛𝑗=1 𝑘𝑗 < 0, then 𝑔 is a solution of the equation ∑ 𝑥𝑗 ∈ 𝐸, 𝑇 ∈ P (N≤𝑛 ) .

(48)

𝑗=1

𝑗∈𝑇

Proof. Put 𝐸 fl 𝐸2 = 𝐸1 , 𝐷 fl Z0 , and 𝜔(𝛽) fl |𝛽| for 𝛽 ∈ Z0 . There exist 𝑗0 ∈ {1, . . . , 𝑛} and 𝑖0 ∈ {1, . . . , 2𝑛 − 1}, such that ∑𝑛𝑗=𝑗̸ 0 𝑘𝑗 < 0, 𝑎𝑖0 𝑗0 = 1, and 𝑎𝑖0 𝑗 = 0 for 𝑗 ≠ 𝑗0 . as follows:

{𝑘 + 𝑙0 for 𝑗 ≠ 𝑗0 (46) 𝑐𝑗𝑘 fl { 1 for 𝑗 = 𝑗0 . { The assumptions of Theorem 3 are fulfilled with 𝑚 = 2𝑛 − 1 and 𝐼 = {𝑖0 }, as is easy to check, which finishes the proof. In the same way as Corollary 5 we can prove the following corollary. Corollary 7. Assume that 𝑌 is a normed space over F, 𝑛 ∈ N2 , and 𝐸 is a nonempty subset of 𝑋0 such that there exists a positive integer 𝑙0 with (31). If 𝑔 : 𝐸 → 𝑌 fulfills the inequality 𝑤

𝑛 󵄩 󵄩𝑘𝑗 󵄩󵄩 󵄩 󵄩󵄩𝐹 (𝑔, 𝑥1 , . . . , 𝑥𝑛 )󵄩󵄩󵄩 ≤ ( ∑ 𝐶𝑗 󵄩󵄩󵄩󵄩𝑥𝑗 󵄩󵄩󵄩󵄩 ) , 𝑗=1

𝑗=1

for ∑𝑗∈𝑇 𝑥𝑗 ∈ 𝐸, and 𝑇 ∈ P(N≤𝑛 ) but 𝑓 satisfies neither (33) nor (45).

Conflicts of Interest (45)

∑𝑛𝑗=1 𝑘𝑗

Define the sequence

𝑛 󵄩󵄩 𝑛−1 󵄩 󵄩−1 󵄩 󵄩󵄩Δ 𝑥 ⋅⋅⋅𝑥 𝑓 (𝑥1 )󵄩󵄩󵄩 < 1 ≤ ∑ 󵄩󵄩󵄩𝑥𝑗 󵄩󵄩󵄩 , 󵄩 2 𝑛 󵄩 󵄩 󵄩 𝑛 󵄩 󵄩−1 󵄩 󵄩󵄩 󵄩󵄩𝐹 (𝑓, 𝑥1 , . . . , 𝑥𝑛 )󵄩󵄩󵄩 < 1 ≤ ∑ 󵄩󵄩󵄩󵄩𝑥𝑗 󵄩󵄩󵄩󵄩 ,

󵄩󵄩 󵄩 𝑛 󵄩 󵄩𝑘𝑗 󵄩󵄩𝐹 (𝑔, 𝑥1 , . . . , 𝑥𝑛 )󵄩󵄩󵄩 ≤ 𝐶⊓𝑗=1 󵄩󵄩󵄩󵄩𝑥𝑗 󵄩󵄩󵄩󵄩 ,

{(𝑐1𝑘 , . . . , 𝑐𝑛𝑘 )}𝑘∈N

Example 9. Let 𝑋 = 𝑌 = R, 𝐸 = [−1, 0) ∪ (0, 1], and 𝑓 : 𝐸 → R be defined by 𝑓(𝑥) = sin 𝑥/2𝑛 , 𝑥 ∈ 𝐸. Then 𝑓 satisfies the estimations 󵄩󵄩󵄩Δ𝑛−1 𝑓 (𝑥 )󵄩󵄩󵄩 < 1 ≤ ⊓𝑛 󵄩󵄩󵄩𝑥 󵄩󵄩󵄩−1 , 󵄩󵄩 𝑥2 ⋅⋅⋅𝑥𝑛 1 󵄩 𝑗=1 󵄩 󵄩 󵄩 𝑗 󵄩󵄩 󵄩󵄩 󵄩 𝑛 󵄩 󵄩−1 󵄩󵄩𝐹 (𝑓, 𝑥1 , . . . , 𝑥𝑛 )󵄩󵄩󵄩 < 1 ≤ ⊓𝑗=1 󵄩󵄩󵄩󵄩𝑥𝑗 󵄩󵄩󵄩󵄩 ,

(43)

If 𝑔 : 𝐸 → 𝑌 fulfills the inequality

𝐹 (𝑔, 𝑥1 , . . . , 𝑥𝑛 ) = 0,

Remark 8. Observe that putting in the above two corollaries 𝑛 = 2 we get hyperstability results for Cauchy equation. Corollary 7 generalizes the result from [9] (see Theorem 1.2), where 𝑛 = 2 and 𝑘1 = 𝑘2 < 0 (in our considerations 𝑘1 , 𝑘2 < 0 may be different). Corollaries 6 and 7 generalize the outcomes from [6].

(47)

∑ 𝑥𝑗 ∈ 𝐸, 𝑇 ∈ P (N≤𝑛 )

𝑗∈𝑇

with 𝐶𝑗 > 0 and 𝑤𝑘𝑗 < 0 for 𝑗 ∈ {1, . . . , 𝑛}, then 𝑔 is a solution of (45).

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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