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Journal of Applied Mathematics
VolumeΒ 2012, Article IDΒ 547865, 15 pages
http://dx.doi.org/10.1155/2012/547865
Research Article

Stability of an 𝑛-Dimensional Mixed-Type Additive and Quadratic Functional Equation in Random Normed Spaces

1Department of Mathematics Education, Gongju National University of Education, Gongju 314-711, Republic of Korea
2Mathematics Section, College of Science and Technology, Hongik University, Jochiwon 339-701, Republic of Korea

Received 5 August 2011; Accepted 28 November 2011

Academic Editor: XianhuaΒ Tang

Copyright Β© 2012 Yang-Hi Lee and Soon-Mo Jung. 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.

Abstract

We investigate the stability problems for the 𝑛-dimensional mixed-type additive and quadratic functional equation βˆ‘2𝑓(𝑛𝑗=1π‘₯π‘—βˆ‘)+1≀𝑖,𝑗≀𝑛,𝑖≠𝑗𝑓(π‘₯π‘–βˆ’π‘₯π‘—βˆ‘)=(𝑛+1)𝑛𝑗=1𝑓(π‘₯π‘—βˆ‘)+(π‘›βˆ’1)𝑛𝑗=1𝑓(βˆ’π‘₯𝑗) in random normed spaces by applying the fixed point method.

1. Introduction

In 1940, Ulam [1] gave a wide-ranging talk before a mathematical colloquium at the University of Wisconsin, in which he discussed a number of important unsolved problems. Among those was the following question concerning the stability of homomorphisms.

Let 𝐺1 be a group, and let 𝐺2 be a metric group with a metric 𝑑(β‹…,β‹…). Given πœ€>0, does there exist a 𝛿>0 such that if a function β„ŽβˆΆπΊ1→𝐺2 satisfies the inequality 𝑑(β„Ž(π‘₯𝑦),β„Ž(π‘₯)β„Ž(𝑦))<𝛿 for all π‘₯,π‘¦βˆˆπΊ1, then there is a homomorphism 𝐻∢𝐺1→𝐺2 with 𝑑(β„Ž(π‘₯),𝐻(π‘₯))<πœ€ for all π‘₯∈𝐺1?

If the answer is affirmative, we say that the functional equation for homomorphisms is stable. Hyers [2] was the first mathematician to present the result concerning the stability of functional equations. He answered the question of Ulam for the case where 𝐺1 and 𝐺2 are assumed to be Banach spaces. This result of Hyers is stated as follows.

Let π‘“βˆΆπΈ1→𝐸2 be a function between Banach spaces such that‖𝑓(π‘₯+𝑦)βˆ’π‘“(π‘₯)βˆ’π‘“(𝑦)‖≀𝛿(1.1) for some 𝛿>0 and for all π‘₯,π‘¦βˆˆπΈ1. Then the limit 𝐴(π‘₯)=limπ‘›β†’βˆž2βˆ’π‘›π‘“(2𝑛π‘₯) exists for each π‘₯∈𝐸1, and 𝐴∢𝐸1→𝐸2 is the unique additive function such that ‖𝑓(π‘₯)βˆ’π΄(π‘₯)‖≀𝛿 for every π‘₯∈𝐸1. Moreover, if 𝑓(𝑑π‘₯) is continuous in 𝑑 for each fixed π‘₯∈𝐸1, then function 𝐴 is linear.

We remark that the additive function 𝐴 is directly constructed from the given function 𝑓, and this method is called the direct method. The direct method is a very powerful method for studying the stability problems of various functional equations. Taking this famous result into consideration, the additive Cauchy equation 𝑓(π‘₯+𝑦)=𝑓(π‘₯)+𝑓(𝑦) is said to have the Hyers-Ulam stability on (𝐸1,𝐸2) if for every function π‘“βˆΆπΈ1→𝐸2 satisfying the inequality (1.1) for some 𝛿β‰₯0 and for all π‘₯,π‘¦βˆˆπΈ1, there exists an additive function 𝐴∢𝐸1→𝐸2 such that π‘“βˆ’π΄ is bounded on 𝐸1.

In 1950, Aoki [3] generalized the theorem of Hyers for additive functions, and in the following year, Bourgin [4] extended the theorem without proof. Unfortunately, it seems that their results failed to receive attention from mathematicians at that time. No one has made use of these results for a long time.

In 1978, Rassias [5] addressed the Hyers’s stability theorem and attempted to weaken the condition for the bound of the norm of Cauchy difference and generalized the theorem of Hyers for linear functions.

Let π‘“βˆΆπΈ1→𝐸2 be a function between Banach spaces. If 𝑓 satisfies the functional inequality‖𝑓(π‘₯+𝑦)βˆ’π‘“(π‘₯)βˆ’π‘“(𝑦)β€–β‰€πœƒβ€–π‘₯‖𝑝+‖𝑦‖𝑝(1.2)for some πœƒβ‰₯0, 𝑝 with 0≀𝑝<1 and for all π‘₯,π‘¦βˆˆπΈ1, then there exists a unique additive function 𝐴∢𝐸1→𝐸2 such that ‖𝑓(π‘₯)βˆ’π΄(π‘₯)‖≀(2πœƒ/(2βˆ’2𝑝))β€–π‘₯‖𝑝 for each π‘₯∈𝐸1. If, in addition, 𝑓(𝑑π‘₯) is continuous in 𝑑 for each fixed π‘₯∈𝐸1, then the function 𝐴 is linear.

This result of Rassias attracted a number of mathematicians who began to be stimulated to investigate the stability problems of functional equations. By regarding a large influence of Ulam, Hyers, and Rassias on the study of stability problems of functional equations, the stability phenomenon proved by Rassias is called the Hyers-Ulam-Rassias stability. For the last thirty years, many results concerning the Hyers-Ulam-Rassias stability of various functional equations have been obtained (see [6–17]).

In this paper, applying the fixed point method, we prove the Hyers-Ulam-Rassias stability of the 𝑛-dimensional mixed-type additive and quadratic functional equation 2𝑓𝑛𝑗=1π‘₯𝑗ξƒͺ+1≀𝑖,𝑗≀𝑛,𝑖≠𝑗𝑓π‘₯π‘–βˆ’π‘₯𝑗=(𝑛+1)𝑛𝑗=1𝑓π‘₯𝑗+(π‘›βˆ’1)𝑛𝑗=1π‘“ξ€·βˆ’π‘₯𝑗(1.3) in random normed spaces. Every solution of (1.3) is called a quadratic-additive function.

Throughout this paper, let 𝑛 be an integer larger than 1.

2. Preliminaries

We introduce some terminologies, notations, and conventions usually used in the theory of random normed spaces (see [18, 19]). The set of all probability distribution functions is denoted byΞ”+[]β†’[]∢={𝐹∢0,∞0,1∣𝐹isleft-continuousandnondecreasing[on0,∞),𝐹(0)=0,and𝐹(∞)=1}.(2.1) Let us define 𝐷+∢={πΉβˆˆΞ”+∣limπ‘‘β†’βˆžπΉ(𝑑)=1}. The set Ξ”+ is partially ordered by the usual pointwise ordering of functions, that is, 𝐹≀𝐺 if and only if 𝐹(𝑑)≀𝐺(𝑑) for all 𝑑β‰₯0. The maximal element for Ξ”+ in this order is the distribution function πœ€0∢[0,∞]β†’[0,1] given byπœ€0ξ‚»0(𝑑)=if1𝑑=0,if𝑑>0.(2.2)

Definition 2.1 (See [18]). A function 𝜏∢[0,1]Γ—[0,1]β†’[0,1] is called a continuous triangular norm (briefly, continuous 𝑑-norm) if 𝜏 satisfies the following conditions:(π‘Ž)𝜏 is commutative and associative;(𝑏)Ο„ is continuous;(𝑐)𝜏(π‘Ž,1)=π‘Ž for all π‘Žβˆˆ[0,1];(𝑑)𝜏(π‘Ž,𝑏)β‰€πœ(𝑐,𝑑) for all π‘Ž,𝑏,𝑐,π‘‘βˆˆ[0,1] with π‘Žβ‰€π‘ and 𝑏≀𝑑.
Typical examples of continuous 𝑑-norms are πœπ‘ƒ(π‘Ž,𝑏)=π‘Žπ‘, πœπ‘€(π‘Ž,𝑏)=min{π‘Ž,𝑏}, and 𝜏𝐿(π‘Ž,𝑏)=max{π‘Ž+π‘βˆ’1,0}.

Definition 2.2 (See [19]). Let 𝑋 be a vector space, 𝜏 a continuous 𝑑-norm, and let Ξ›βˆΆπ‘‹β†’π·+ be a function satisfying the following conditions:(𝑅1)Ξ›π‘₯(𝑑)=πœ€0(𝑑) for all 𝑑>0 if and only if π‘₯=0;(𝑅2)Λ𝛼π‘₯(𝑑)=Ξ›π‘₯(𝑑/|𝛼|) for all π‘₯βˆˆπ‘‹, 𝛼≠0, and for all 𝑑β‰₯0;(𝑅3)Ξ›π‘₯+𝑦(𝑑+𝑠)β‰₯𝜏(Ξ›π‘₯(𝑑),Λ𝑦(𝑠)) for all π‘₯,π‘¦βˆˆπ‘‹ and all 𝑑,𝑠β‰₯0. A triple (𝑋,Ξ›,𝜏) is called a random normed space (briefly, RN-space).
If (𝑋,β€–β‹…β€–) is a normed space, we can define a function Ξ›βˆΆπ‘‹β†’π·+ by Ξ›π‘₯(𝑑𝑑)=𝑑+β€–π‘₯β€–(2.3) for all π‘₯βˆˆπ‘‹ and 𝑑>0. Then (𝑋,Ξ›,πœπ‘€) is a random normed space, which is called the induced random normed space.

Definition 2.3. Let (𝑋,Ξ›,𝜏) be an 𝑅𝑁-space.(𝑖)A sequence {π‘₯𝑛} in 𝑋 is said to be convergent to a point π‘₯βˆˆπ‘‹ if, for every 𝑑>0 and πœ€>0, there exists a positive integer 𝑁 such that Ξ›π‘₯π‘›βˆ’π‘₯(𝑑)>1βˆ’πœ€ whenever 𝑛β‰₯𝑁.(𝑖𝑖)A sequence {π‘₯𝑛} in 𝑋 is called a Cauchy sequence if, for every 𝑑>0 and πœ€>0, there exists a positive integer 𝑁 such that Ξ›π‘₯π‘›βˆ’π‘₯π‘š(𝑑)>1βˆ’πœ€ whenever 𝑛β‰₯π‘šβ‰₯𝑁.(𝑖𝑖𝑖)An RN-space (𝑋,Ξ›,𝜏) is called complete if and only if every Cauchy sequence in 𝑋 converges to a point in 𝑋.

Definition 2.4. Let 𝑋 be a nonempty set. A function π‘‘βˆΆπ‘‹2β†’[0,∞] is called a generalized metric on 𝑋 if and only if 𝑑 satisfies(𝑀1)𝑑(π‘₯,𝑦)=0 if and only if π‘₯=𝑦;(𝑀2)𝑑(π‘₯,𝑦)=𝑑(𝑦,π‘₯) for all π‘₯,π‘¦βˆˆπ‘‹;(𝑀3)𝑑(π‘₯,𝑧)≀𝑑(π‘₯,𝑦)+𝑑(𝑦,𝑧) for all π‘₯,𝑦,π‘§βˆˆπ‘‹.
We now introduce one of the fundamental results of the fixed point theory. For the proof, we refer to [20] or [21].

Theorem 2.5 (See [20, 21]). Let (𝑋,𝑑) be a complete generalized metric space. Assume that Ξ›βˆΆπ‘‹β†’π‘‹ is a strict contraction with the Lipschitz constant 𝐿<1. If there exists a nonnegative integer 𝑛0 such that 𝑑(Λ𝑛0+1π‘₯,Λ𝑛0π‘₯)<∞ for some π‘₯βˆˆπ‘‹, then the following statements are true:(𝑖)the sequence {Λ𝑛π‘₯} converges to a fixed point π‘₯βˆ— of Ξ›;(𝑖𝑖)π‘₯βˆ— is the unique fixed point of Ξ› in π‘‹βˆ—={π‘¦βˆˆπ‘‹βˆ£π‘‘(Λ𝑛0π‘₯,𝑦)<∞};(𝑖𝑖𝑖)if π‘¦βˆˆπ‘‹βˆ—, then𝑑𝑦,π‘₯βˆ—ξ€Έβ‰€11βˆ’πΏπ‘‘(Λ𝑦,𝑦).(2.4)

In 2003, Radu [22] noticed that many theorems concerning the Hyers-Ulam stability of various functional equations follow from the fixed point alternative (Theorem 2.5). Indeed, he applied the fixed point method to prove the existence of a solution of the inequality (1.1) and investigated the Hyers-Ulam stability of the additive Cauchy equation (see also [23–26]). Furthermore, MiheΕ£ and Radu [27] applied the fixed point method to prove the stability theorems of the additive Cauchy equation in random normed spaces.

In 2009, Towanlong and Nakmahachalasint [28] established the general solution and the stability of the 𝑛-dimensional mixed-type additive and quadratic functional equation (1.3) by using the direct method. According to [28], a function π‘“βˆΆπΈ1→𝐸2 is a quadratic-additive function, where 𝐸1 and 𝐸2 are vector spaces, if and only if there exist an additive function π‘ŽβˆΆπΈ1→𝐸2 and a quadratic function π‘žβˆΆπΈ1→𝐸2 such that 𝑓(π‘₯)=π‘Ž(π‘₯)+π‘ž(π‘₯) for all π‘₯∈𝐸1.

3. Hyers-Ulam-Rassias Stability

Throughout this paper, let 𝑋 be a real vector space and let (π‘Œ,Ξ›,πœπ‘€) be a complete RN-space. For a given function π‘“βˆΆπ‘‹β†’π‘Œ, we use the following abbreviation:ξ€·π‘₯𝐷𝑓1,π‘₯2,…,π‘₯π‘›ξ€Έξƒ©βˆΆ=2𝑓𝑛𝑗=1π‘₯𝑗ξƒͺ+1≀𝑖,𝑗≀𝑛,𝑖≠𝑗𝑓π‘₯π‘–βˆ’π‘₯π‘—ξ€Έβˆ’(𝑛+1)𝑛𝑗=1𝑓π‘₯π‘—ξ€Έβˆ’(π‘›βˆ’1)𝑛𝑗=1π‘“ξ€·βˆ’π‘₯𝑗(3.1) for all π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹.

We will now prove the stability of the functional equation (1.3) in random normed spaces by using fixed point method.

Theorem 3.1. Let 𝑋 be a real vector space, (𝑍,Ξ›β€²,πœπ‘€) an RN-space, (π‘Œ,Ξ›,πœπ‘€) a complete RN-space, and let πœ‘βˆΆ(𝑋⧡{0})𝑛→𝑍 be a function. Assume that πœ‘ satisfies one of the following conditions:(𝑖)Ξ›ξ…žπ›Όπœ‘(π‘₯1,π‘₯2,…,π‘₯𝑛)(𝑑)β‰€Ξ›ξ…žπœ‘(𝑛π‘₯1,𝑛π‘₯2,…,𝑛π‘₯𝑛)(t) for some 0<𝛼<𝑛;(𝑖𝑖)Ξ›ξ…žπœ‘(𝑛π‘₯1,𝑛π‘₯2,…,𝑛π‘₯𝑛)(𝑑)β‰€Ξ›ξ…žπ›Όπœ‘(π‘₯1,π‘₯2,…,π‘₯𝑛)(𝑑) for some 𝛼>𝑛2 for all π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹β§΅{0} and 𝑑>0. If a function π‘“βˆΆπ‘‹β†’π‘Œ satisfies 𝑓(0)=0 and Λ𝐷𝑓(π‘₯1,π‘₯2,…,π‘₯𝑛)(𝑑)β‰₯Ξ›ξ…žπœ‘(π‘₯1,π‘₯2,…,π‘₯𝑛)(𝑑)(3.2) for all π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹β§΅{0} and 𝑑>0, then there exists a unique function πΉβˆΆπ‘‹β†’π‘Œ such that ξ€·π‘₯𝐷𝐹1,π‘₯2,…,π‘₯𝑛=0(3.3) for all π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹β§΅{0} and Λ𝑓(π‘₯)βˆ’πΉ(π‘₯)ξ‚»(𝑑)β‰₯𝑀(π‘₯,2(π‘›βˆ’π›Ό)𝑑)ifπœ‘satisfies𝑀(𝑖),π‘₯,2π›Όβˆ’π‘›2𝑑ifπœ‘satisfies(𝑖𝑖)(3.4) for all π‘₯βˆˆπ‘‹β§΅{0} and 𝑑>0, where 𝑀(π‘₯,𝑑)∢=πœπ‘€(Ξ›β€²πœ‘(Μ‚π‘₯)(𝑑),Ξ›β€²πœ‘(ξ‚Šβˆ’π‘₯)(𝑑)), and Μ‚π‘₯=(π‘₯,π‘₯,…,π‘₯).

Proof. We will first treat the case where πœ‘ satisfies the condition (𝑖). Let 𝑆 be the set of all functions π‘”βˆΆπ‘‹β†’π‘Œ with 𝑔(0)=0, and let us define a generalized metric on 𝑆 by 𝑑[](𝑔,β„Ž)∢=infπ‘’βˆˆ0,βˆžβˆ£Ξ›π‘”(π‘₯)βˆ’β„Ž(π‘₯)ξ€Ύ.(𝑒𝑑)β‰₯𝑀(π‘₯,𝑑)βˆ€π‘₯βˆˆπ‘‹β§΅{0},𝑑>0(3.5) It is not difficult to show that (𝑆,𝑑) is a complete generalized metric space (see [29] or [30, 31]).
Consider the operator π½βˆΆπ‘†β†’π‘† defined by 𝐽𝑓(π‘₯)∢=𝑓(𝑛π‘₯)βˆ’π‘“(βˆ’π‘›π‘₯)+2𝑛𝑓(𝑛π‘₯)+𝑓(βˆ’π‘›π‘₯)2𝑛2.(3.6) Then we can apply induction on π‘š to prove π½π‘šπ‘“(π‘₯)=𝑓(π‘›π‘šπ‘₯)βˆ’π‘“(βˆ’π‘›π‘šπ‘₯)2π‘›π‘š+𝑓(π‘›π‘šπ‘₯)+𝑓(βˆ’π‘›π‘šπ‘₯)2𝑛2π‘š(3.7) for all π‘₯βˆˆπ‘‹ and π‘šβˆˆβ„•.
Let 𝑓,π‘”βˆˆπ‘† and let π‘’βˆˆ[0,∞] be an arbitrary constant with 𝑑(𝑔,𝑓)≀𝑒. For some 0<𝛼<𝑛 satisfying the condition (𝑖), it follows from the definition of 𝑑, (𝑅2), (𝑅3), and (𝑖) that Λ𝐽𝑔(π‘₯)βˆ’π½π‘“(π‘₯)𝛼𝑒𝑑𝑛=Ξ›((𝑛+1)(𝑔(𝑛π‘₯)βˆ’π‘“(𝑛π‘₯))/2𝑛2)βˆ’((π‘›βˆ’1)(𝑔(βˆ’π‘›π‘₯)βˆ’π‘“(βˆ’π‘›π‘₯))/2𝑛2)𝛼𝑒𝑑𝑛β‰₯πœπ‘€ξƒ©Ξ›(𝑛+1)(𝑔(𝑛π‘₯)βˆ’π‘“(𝑛π‘₯))/2𝑛2(𝑛+1)𝛼𝑒𝑑2𝑛2ξ€Έξƒͺ,Ξ›(π‘›βˆ’1)(𝑔(βˆ’π‘›π‘₯)βˆ’π‘“(βˆ’π‘›π‘₯))/2𝑛2(π‘›βˆ’1)𝛼𝑒𝑑2𝑛2ξ€Έξƒͺξƒͺβ‰₯πœπ‘€ξ€·Ξ›π‘”(𝑛π‘₯)βˆ’π‘“(𝑛π‘₯)(𝛼𝑒𝑑),Λ𝑔(βˆ’π‘›π‘₯)βˆ’π‘“(βˆ’π‘›π‘₯)(𝛼𝑒𝑑)β‰₯πœπ‘€ξ‚€Ξ›ξ…žπœ‘(ξ‚Šπ‘›π‘₯)(𝛼𝑑),Ξ›ξ…žπœ‘(ξ‚Ώβˆ’π‘›π‘₯)(𝛼𝑑)β‰₯𝑀(π‘₯,𝑑)(3.8) for all π‘₯βˆˆπ‘‹β§΅{0} and 𝑑>0, which implies that 𝑑𝛼(𝐽𝑓,𝐽𝑔)≀𝑛𝑑(𝑓,𝑔).(3.9) That is, 𝐽 is a strict contraction with the Lipschitz constant 0<𝛼/𝑛<1.
Moreover, by (𝑅2), (𝑅3), and (3.2), we see that Λ𝑓(π‘₯)βˆ’π½π‘“(π‘₯)𝑑2𝑛=Ξ›(βˆ’(𝑛+1)𝐷𝑓(Μ‚π‘₯)+(π‘›βˆ’1)𝐷𝑓(ξ‚Šβˆ’π‘₯))/4𝑛2𝑑2𝑛β‰₯πœπ‘€ξ‚΅Ξ›(𝑛+1)𝐷𝑓(Μ‚π‘₯)/4𝑛2ξ‚΅(𝑛+1)𝑑4𝑛2ξ‚Ά,Ξ›(π‘›βˆ’1)𝐷𝑓(ξ‚Šβˆ’π‘₯)/4𝑛2ξ‚΅(π‘›βˆ’1)𝑑4𝑛2ξ‚Άξ‚Άβ‰₯πœπ‘€ξ‚€Ξ›π·π‘“(Μ‚π‘₯)(𝑑),Λ𝐷𝑓(ξ‚Šβˆ’π‘₯)(𝑑)β‰₯𝑀(π‘₯,𝑑)(3.10) for all π‘₯βˆˆπ‘‹β§΅{0} and 𝑑>0. Hence, it follows from the definition of 𝑑 that 1𝑑(𝑓,𝐽𝑓)≀2𝑛<∞.(3.11) Now, in view of Theorem 2.5, the sequence {π½π‘šπ‘“} converges to the unique β€œfixed point” πΉβˆΆπ‘‹β†’π‘Œ of 𝐽 in the set 𝑇={π‘”βˆˆπ‘†βˆ£π‘‘(𝑓,𝑔)<∞} and 𝐹 is represented by 𝐹(π‘₯)=limπ‘šβ†’βˆžξ‚΅π‘“(π‘›π‘šπ‘₯)βˆ’π‘“(βˆ’π‘›π‘šπ‘₯)2π‘›π‘š+𝑓(π‘›π‘šπ‘₯)+𝑓(βˆ’π‘›π‘šπ‘₯)2𝑛2π‘šξ‚Ά(3.12) for all π‘₯βˆˆπ‘‹.
By Theorem 2.5, (3.11), and the definition of 𝑑, we have 1𝑑(𝑓,𝐹)≀11βˆ’π›Ό/𝑛𝑑(𝑓,𝐽𝑓)≀2,(π‘›βˆ’π›Ό)(3.13) that is, the first inequality in (3.4) holds true.
We will now show that 𝐹 is a quadratic-additive function. It follows from (𝑅3) and the definition of πœπ‘€ that Λ𝐷𝐹(π‘₯1,π‘₯2,…,π‘₯𝑛)Λ(𝑑)β‰₯min2(πΉβˆ’π½π‘šβˆ‘π‘“)(𝑛𝑗=1π‘₯𝑗)𝑑5,ξ‚»Ξ›min(πΉβˆ’π½π‘šπ‘“)(π‘₯π‘–βˆ’π‘₯𝑗)𝑑(ξ‚Άξ‚Ό,ξ‚»Ξ›5𝑛(π‘›βˆ’1))∣1≀𝑖,𝑗≀𝑛,𝑖≠𝑗min(𝑛+1)(π½π‘šπ‘“βˆ’πΉ)(π‘₯𝑗)𝑑,ξ‚»Ξ›(5𝑛)βˆ£π‘—=1,…,𝑛min(π‘›βˆ’1)(π½π‘šπ‘“βˆ’πΉ)(βˆ’π‘₯𝑗)𝑑,Ξ›(5𝑛)βˆ£π‘—=1,…,π‘›π·π½π‘šπ‘“(π‘₯1,π‘₯2,…,π‘₯𝑛)𝑑5(3.14) for all π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹β§΅{0}, 𝑑>0, and π‘šβˆˆβ„•. Due to the definition of 𝐹, the first four terms on the right-hand side of the above inequality tend to 1 as π‘šβ†’βˆž.
By a somewhat tedious manipulation, we have π·π½π‘šπ‘“ξ€·π‘₯1,π‘₯2,…,π‘₯𝑛=12𝑛2π‘šξ€·π‘›π·π‘“π‘šπ‘₯1,…,π‘›π‘šπ‘₯𝑛+12𝑛2π‘šξ€·π·π‘“βˆ’π‘›π‘šπ‘₯1,…,βˆ’π‘›π‘šπ‘₯𝑛+12π‘›π‘šξ€·π‘›π·π‘“π‘šπ‘₯1,…,π‘›π‘šπ‘₯π‘›ξ€Έβˆ’12π‘›π‘šξ€·π·π‘“βˆ’π‘›π‘šπ‘₯1,…,βˆ’π‘›π‘šπ‘₯𝑛.(3.15) Hence, it follows from (𝑅2), (𝑅3), definition of πœπ‘€, (3.2), and (𝑖) that Ξ›π·π½π‘šπ‘“(π‘₯1,…,π‘₯𝑛)𝑑5Λβ‰₯min𝐷𝑓(π‘›π‘šπ‘₯1,…,π‘›π‘šπ‘₯𝑛)/2𝑛2π‘šξ‚€π‘‘ξ‚20,Λ𝐷𝑓(βˆ’π‘›π‘šπ‘₯1,…,βˆ’π‘›π‘šπ‘₯𝑛)/2𝑛2π‘šξ‚€π‘‘ξ‚,Ξ›20𝐷𝑓(π‘›π‘šπ‘₯1,…,π‘›π‘šπ‘₯𝑛)/2π‘›π‘šξ‚€π‘‘ξ‚20,Λ𝐷𝑓(βˆ’π‘›π‘šπ‘₯1,…,βˆ’π‘›π‘šπ‘₯𝑛)/2π‘›π‘šξ‚€π‘‘ξ‚»Ξ›20β‰₯min𝐷𝑓(π‘›π‘šπ‘₯1,…,π‘›π‘šπ‘₯𝑛)𝑛2π‘šπ‘‘ξ‚Ά10,Λ𝐷𝑓(βˆ’π‘›π‘šπ‘₯1,…,βˆ’π‘›π‘šπ‘₯𝑛)𝑛2π‘šπ‘‘ξ‚Ά,Ξ›10𝐷𝑓(π‘›π‘šπ‘₯1,…,π‘›π‘šπ‘₯𝑛)ξ‚΅π‘›π‘šπ‘‘ξ‚Ά10,Λ𝐷𝑓(βˆ’π‘›π‘šπ‘₯1,…,βˆ’π‘›π‘šπ‘₯𝑛)ξ‚΅π‘›π‘šπ‘‘ξ‚»Ξ›10ξ‚Άξ‚Όβ‰₯minξ…žπœ‘(π‘₯1,…,π‘₯𝑛)𝑛2π‘šπ‘‘(10π›Όπ‘š)ξ‚Ά,Ξ›ξ…žπœ‘(βˆ’π‘₯1,…,βˆ’π‘₯𝑛)𝑛2π‘šπ‘‘(10π›Όπ‘š)ξ‚Ά,Ξ›ξ…žπœ‘(π‘₯1,…,π‘₯𝑛)ξ‚΅π‘›π‘šπ‘‘(10π›Όπ‘š)ξ‚Ά,Ξ›ξ…žπœ‘(βˆ’π‘₯1,…,βˆ’π‘₯𝑛)ξ‚΅π‘›π‘šπ‘‘(10π›Όπ‘š),ξ‚Άξ‚Ό(3.16) which tends to 1 as π‘šβ†’βˆž for all π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹β§΅{0} and 𝑑>0. Therefore, (3.14) implies that Λ𝐷𝐹(π‘₯1,π‘₯2,…,π‘₯𝑛)(𝑑)=1(3.17) for any π‘₯1,…,π‘₯π‘›βˆˆπ‘‹β§΅{0} and 𝑑>0. By (𝑅1), this implies that 𝐷𝐹(π‘₯1,…,π‘₯𝑛)=0 for all π‘₯1,…,π‘₯π‘›βˆˆπ‘‹β§΅{0}, which ends the proof of the first part.
Now, assume that πœ‘ satisfies the condition (𝑖𝑖). Let (𝑆,𝑑) be the same as given in the first part. We now consider the operator π½βˆΆπ‘†β†’π‘† defined by 𝑛𝐽𝑔(π‘₯)∢=2𝑔π‘₯π‘›ξ‚ξ‚€βˆ’π‘₯βˆ’π‘”π‘›+𝑛22𝑔π‘₯π‘›ξ‚ξ‚€βˆ’π‘₯+𝑔𝑛(3.18) for all π‘”βˆˆπ‘† and π‘₯βˆˆπ‘‹. Notice that π½π‘šπ‘›π‘”(π‘₯)=π‘š2𝑔π‘₯π‘›π‘šξ‚ξ‚€βˆ’π‘₯βˆ’π‘”π‘›π‘š+𝑛2π‘š2𝑔π‘₯π‘›π‘šξ‚ξ‚€βˆ’π‘₯+π‘”π‘›π‘šξ‚ξ‚(3.19) for all π‘₯βˆˆπ‘‹ and π‘šβˆˆβ„•.
Let 𝑓,π‘”βˆˆπ‘† and let π‘’βˆˆ[0,∞] be an arbitrary constant with 𝑑(𝑔,𝑓)≀𝑒. From (𝑅2), (𝑅3), the definition of 𝑑, and (𝑖𝑖), we have Λ𝐽𝑔(π‘₯)βˆ’π½π‘“(π‘₯)𝑛2𝑒𝑑𝛼=Ξ›((𝑛2+𝑛)/2)(𝑔(π‘₯/𝑛)βˆ’π‘“(π‘₯/𝑛))+((𝑛2βˆ’π‘›)/2)(𝑔(βˆ’π‘₯/𝑛)βˆ’π‘“(βˆ’π‘₯/𝑛))𝑛2𝑒𝑑𝛼β‰₯πœπ‘€ξƒ©Ξ›((𝑛2+𝑛)/2)(𝑔(π‘₯/𝑛)βˆ’π‘“(π‘₯/𝑛))𝑛2ξ€Έ+𝑛𝑒𝑑ξƒͺ,Ξ›(2𝛼)((𝑛2βˆ’π‘›)/2)(𝑔(βˆ’π‘₯/𝑛)βˆ’π‘“(βˆ’π‘₯/𝑛))𝑛2ξ€Έβˆ’π‘›π‘’π‘‘(2𝛼)ξƒͺξƒͺ=πœπ‘€ξ‚€Ξ›π‘”(π‘₯/𝑛)βˆ’π‘“(π‘₯/𝑛)𝑒𝑑𝛼,Λ𝑔(βˆ’π‘₯/𝑛)βˆ’π‘“(βˆ’π‘₯/𝑛)𝑒𝑑𝛼β‰₯πœπ‘€ξ‚€π‘€ξ‚€π‘₯𝑛,π‘‘π›Όξ‚ξ‚€βˆ’π‘₯,𝑀𝑛,𝑑𝛼=πœπ‘€ξ‚΅Ξ›ξ…žπœ‘(ξ‚Šπ‘₯/𝑛)𝑑𝛼,Ξ›ξ…žπœ‘(ξƒ³βˆ’π‘₯/𝑛)𝑑𝛼=πœπ‘€ξ‚΅Ξ›ξ…žπ›Όπœ‘(ξ‚Šπ‘₯/𝑛)(𝑑),Ξ›ξ…žπ›Όπœ‘(ξƒ³βˆ’π‘₯/𝑛)ξ‚Ά(𝑑)β‰₯πœπ‘€ξ‚€Ξ›ξ…žπœ‘(Μ‚π‘₯)(𝑑),Ξ›ξ…žπœ‘(ξ‚Šβˆ’π‘₯)(𝑑)=𝑀(π‘₯,𝑑)(3.20) for all π‘₯βˆˆπ‘‹β§΅{0}, 𝑑>0, and for some 𝛼>𝑛2 satisfying (𝑖𝑖), which implies that 𝑛𝑑(𝐽𝑓,𝐽𝑔)≀2𝛼𝑑(𝑓,𝑔).(3.21) That is, 𝐽 is a strict contraction with the Lipschitz constant 0<𝑛2/𝛼<1.
Moreover, by (𝑅2), (3.2), and (𝑖𝑖), we see that Λ𝑓(π‘₯)βˆ’π½π‘“(π‘₯)𝑑(2𝛼)=Ξ›(1/2)𝐷𝑓(ξ‚Šπ‘₯/𝑛)𝑑(2𝛼)β‰₯Ξ›ξ…žπœ‘(ξ‚Šπ‘₯/𝑛)𝑑𝛼=Ξ›ξ…žπ›Όπœ‘(ξ‚Šπ‘₯/𝑛)(𝑑)β‰₯Ξ›ξ…žπœ‘(Μ‚π‘₯)(𝑑)β‰₯𝑀(π‘₯,𝑑)(3.22) for all π‘₯βˆˆπ‘‹β§΅{0} and 𝑑>0. This implies that 𝑑(𝑓,𝐽𝑓)≀1/(2𝛼)<∞ by the definition of 𝑑. Therefore, according to Theorem 2.5, the sequence {π½π‘šπ‘“} converges to the unique β€œfixed point” πΉβˆΆπ‘‹β†’π‘Œ of 𝐽 in the set 𝑇={π‘”βˆˆπ‘†βˆ£π‘‘(𝑓,𝑔)<∞} and 𝐹 is represented by 𝐹(π‘₯)=limπ‘šβ†’βˆžξ‚΅π‘›π‘š2𝑓π‘₯𝑛mξ‚ξ‚€βˆ’π‘₯βˆ’π‘“π‘›π‘š+𝑛2π‘š2𝑓π‘₯π‘›π‘šξ‚ξ‚€βˆ’π‘₯+π‘“π‘›π‘šξ‚Άξ‚ξ‚(3.23) for all π‘₯βˆˆπ‘‹. Since 1𝑑(𝑓,𝐹)≀1βˆ’π‘›21/𝛼𝑑(𝑓,𝐽𝑓)≀2ξ€·π›Όβˆ’π‘›2ξ€Έ,(3.24) the second inequality in (3.4) holds true.
Next, we will show that 𝐹 is a quadratic-additive function. As we did in the first part, we obtain the inequality (3.14). In view of the definition of 𝐹, the first four terms on the right-hand side of the inequality (3.14) tend to 1 as π‘šβ†’βˆž. Furthermore, a long manipulation yields π·π½π‘šπ‘“ξ€·π‘₯1,π‘₯2,…,π‘₯𝑛=𝑛2π‘š2ξ‚€π‘₯𝐷𝑓1π‘›π‘šπ‘₯,…,π‘›π‘›π‘šξ‚+𝑛2π‘š2ξ‚€βˆ’π‘₯𝐷𝑓1π‘›π‘šπ‘₯,…,βˆ’π‘›π‘›π‘šξ‚+π‘›π‘š2ξ‚€π‘₯𝐷𝑓1π‘›π‘šπ‘₯,…,π‘›π‘›π‘šξ‚βˆ’π‘›π‘š2ξ‚€βˆ’π‘₯𝐷𝑓1π‘›π‘šπ‘₯,…,βˆ’π‘›π‘›π‘šξ‚.(3.25) Thus, it follows from (𝑅2), (𝑅3), definition of πœπ‘€, (3.2), and (𝑖𝑖) that Ξ›π·π½π‘šπ‘“(π‘₯1,…,π‘₯𝑛)𝑑5Λβ‰₯min(𝑛2π‘š/2)𝐷𝑓(π‘₯1/π‘›π‘š,…,π‘₯𝑛/π‘›π‘š)𝑑20,Ξ›(𝑛2π‘š/2)𝐷𝑓(βˆ’π‘₯1/π‘›π‘š,…,βˆ’π‘₯𝑛/π‘›π‘š)𝑑,Ξ›20(π‘›π‘š/2)𝐷𝑓(π‘₯1/π‘›π‘š,…,π‘₯𝑛/π‘›π‘š)𝑑20,Ξ›βˆ’(π‘›π‘š/2)𝐷𝑓(βˆ’π‘₯1/π‘›π‘š,…,βˆ’π‘₯𝑛/π‘›π‘š)𝑑Λ20β‰₯minξ…žπœ‘(π‘₯1/π‘›π‘š,…,π‘₯𝑛/π‘›π‘š)𝑑10𝑛2π‘šξ€Έξƒͺ,Ξ›ξ…žπœ‘(βˆ’π‘₯1/π‘›π‘š,…,βˆ’π‘₯𝑛/π‘›π‘š)𝑑10𝑛2π‘šξ€Έξƒͺ,Ξ›ξ…žπœ‘(π‘₯1/π‘›π‘š,…,π‘₯𝑛/π‘›π‘š)ξ‚΅t(10π‘›π‘š)ξ‚Ά,Ξ›ξ…žπœ‘(βˆ’π‘₯1/π‘›π‘š,…,βˆ’π‘₯𝑛/π‘›π‘š)𝑑(10π‘›π‘š)ξƒ―Ξ›ξ‚Άξ‚Όβ‰₯minξ…žπ›Όβˆ’π‘šπœ‘(π‘₯1,…,π‘₯𝑛)𝑑10𝑛2π‘šξ€Έξƒͺ,Ξ›ξ…žπ›Όβˆ’π‘šπœ‘(βˆ’π‘₯1,…,βˆ’π‘₯𝑛)𝑑10𝑛2π‘šξ€Έξƒͺ,Ξ›ξ…žπ›Όβˆ’π‘šπœ‘(π‘₯1,…,π‘₯𝑛)𝑑(10π‘›π‘š)ξ‚Ά,Ξ›ξ…žπ›Όβˆ’π‘šπœ‘(βˆ’π‘₯1,…,βˆ’π‘₯𝑛)𝑑(10π‘›π‘š)ξƒ―Ξ›ξ‚Άξ‚Ό=minξ…žπœ‘(π‘₯1,…,π‘₯𝑛)ξƒ©π›Όπ‘šπ‘‘ξ€·10𝑛2π‘šξ€Έξƒͺ,Ξ›ξ…žπœ‘(βˆ’π‘₯1,…,βˆ’π‘₯𝑛)ξƒ©π›Όπ‘šπ‘‘ξ€·10𝑛2π‘šξ€Έξƒͺ,Ξ›ξ…žπœ‘(π‘₯1,…,π‘₯𝑛)ξ‚΅π›Όπ‘šπ‘‘(10π‘›π‘š)ξ‚Ά,Ξ›ξ…žπœ‘(βˆ’π‘₯1,…,βˆ’π‘₯𝑛)ξ‚΅π›Όπ‘šπ‘‘(10π‘›π‘š),ξ‚Άξ‚Ό(3.26) which tends to 1 as π‘šβ†’βˆž for all π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹β§΅{0} and 𝑑>0. Therefore, it follows from (3.14) that Λ𝐷𝐹(π‘₯1,π‘₯2,…,π‘₯𝑛)(𝑑)=1(3.27) for any π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹β§΅{0} and 𝑑>0. By (𝑅1), this implies that ξ€·π‘₯𝐷𝐹1,π‘₯2,…,π‘₯𝑛=0(3.28) for all π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹β§΅{0}, which ends the proof.

By a similar way presented in the proof of Theorem 3.1, we can also prove the preceding theorem if the domains of relevant functions include 0.

Theorem 3.2. Let 𝑋 be a real vector space, (𝑍,Ξ›β€²,πœπ‘€) an RN-space, (π‘Œ,Ξ›,πœπ‘€) a complete RN-space, and let πœ‘βˆΆπ‘‹π‘›β†’π‘ be a function. Assume that πœ‘ satisfies one of the conditions (𝑖) and (𝑖𝑖) in Theorem 3.1 for all π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹ and 𝑑>0. If a function π‘“βˆΆπ‘‹β†’π‘Œ satisfies 𝑓(0)=0 and (3.2) for all π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹ and 𝑑>0, then there exists a unique quadratic-additive function πΉβˆΆπ‘‹β†’π‘Œ satisfying (3.4) for all π‘₯βˆˆπ‘‹ and 𝑑>0.

Now, we obtain general Hyers-Ulam stability results of (1.3) in normed spaces. If 𝑋 is a normed space, then (𝑋,Ξ›,πœπ‘€) is an induced random normed space. We get the following result.

Corollary 3.3. Let 𝑋 be a real vector space, π‘Œ a complete normed space, and let πœ‘βˆΆ(𝑋⧡{0})𝑛→[0,∞) be a function. Assume that πœ‘ satisfies one of the following conditions:(𝑖𝑖𝑖)πœ‘(𝑛π‘₯1,…,𝑛π‘₯𝑛)β‰€π›Όπœ‘(π‘₯1,…,π‘₯𝑛) for some 1<𝛼<𝑛;(𝑖𝑣)πœ‘(𝑛π‘₯1,…,𝑛π‘₯𝑛)β‰₯π›Όπœ‘(π‘₯1,…,π‘₯𝑛) for some 𝛼>𝑛2 for all π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹β§΅{0}. If a function π‘“βˆΆπ‘‹β†’π‘Œ satisfies 𝑓(0)=0 and β€–β€–ξ€·π‘₯𝐷𝑓1,π‘₯2,…,π‘₯𝑛‖‖π‘₯β‰€πœ‘1,π‘₯2,…,π‘₯𝑛(3.29) for all π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹β§΅{0}, then there exists a unique function πΉβˆΆπ‘‹β†’π‘Œ such that ξ€·π‘₯𝐷𝐹1,π‘₯2,…,π‘₯𝑛=0(3.30) for all π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹β§΅{0} and β€–βŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©ξ€½πœ‘ξ€·π‘“(π‘₯)βˆ’πΉ(π‘₯)‖≀max(Μ‚π‘₯),πœ‘ξ‚Šβˆ’π‘₯ξ€Έξ€Ύ2(π‘›βˆ’π›Ό)ifπœ‘satisfiesξ€½ξ€·(𝑖𝑖𝑖),maxπœ‘(Μ‚π‘₯),πœ‘ξ‚Šβˆ’π‘₯ξ€Έξ€Ύ2ξ€·π›Όβˆ’π‘›2ξ€Έifπœ‘satisfies(𝑖𝑣)(3.31) for all π‘₯βˆˆπ‘‹β§΅{0}.

Proof. Let us put π‘βˆΆ=ℝ,Ξ›π‘₯(𝑑𝑑)∢=𝑑+β€–π‘₯β€–,Ξ›ξ…žπ‘§(𝑑𝑑)∢=𝑑+|𝑧|(3.32) for all π‘₯,π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹β§΅{0}, π‘§βˆˆβ„β§΅{0}, and 𝑑β‰₯0. If πœ‘ satisfies the condition (𝑖𝑖𝑖) for all π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹β§΅{0} and for some 1<𝛼<𝑛, then Ξ›ξ…žπ›Όπœ‘(π‘₯1,…,π‘₯𝑛)(𝑑𝑑)=ξ€·π‘₯𝑑+π›Όπœ‘1,…,π‘₯𝑛≀𝑑𝑑+πœ‘π‘›π‘₯1,…,𝑛π‘₯𝑛=Ξ›ξ…žπœ‘(𝑛π‘₯1,…,𝑛π‘₯𝑛)(𝑑)(3.33) for all π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹β§΅{0} and 𝑑>0, that is, πœ‘ satisfies the condition (𝑖). In a similar way, we can show that if πœ‘ satisfies (𝑖𝑣), then it satisfies the condition (𝑖𝑖).
Moreover, we get Λ𝐷𝑓(π‘₯1,…,π‘₯𝑛)(𝑑𝑑)=β€–β€–ξ€·π‘₯𝑑+𝐷𝑓1,…,π‘₯𝑛‖‖β‰₯𝑑π‘₯𝑑+πœ‘1,…,π‘₯𝑛=Ξ›ξ…žπœ‘(π‘₯1,…,π‘₯𝑛)(𝑑)(3.34) for all π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹β§΅{0} and 𝑑>0, that is, 𝑓 satisfies the inequality (3.2) for all π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹β§΅{0}.
According to Theorem 3.1, there exists a unique function πΉβˆΆπ‘‹β†’π‘Œ such thatξ€·π‘₯𝐷𝐹1,π‘₯2,…,π‘₯𝑛=0(3.35) for all π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹β§΅{0} and Λ𝑓(π‘₯)βˆ’πΉ(π‘₯)⎧βŽͺ⎨βŽͺ⎩𝜏(𝑑)β‰₯π‘€ξ‚€Ξ›ξ…žπœ‘(Μ‚π‘₯)(2(π‘›βˆ’π›Ό)𝑑),Ξ›ξ…žπœ‘(ξ‚Šβˆ’π‘₯)(2(π‘›βˆ’π›Ό)𝑑)ifπœ‘satisfies𝜏(𝑖𝑖𝑖),π‘€ξ‚€Ξ›ξ…žπœ‘(Μ‚π‘₯)ξ€·2ξ€·π›Όβˆ’π‘›2𝑑,Ξ›ξ…žπœ‘(ξ‚Šβˆ’π‘₯)ξ€·2ξ€·π›Όβˆ’π‘›2𝑑ifπœ‘satisfies(𝑖𝑣)(3.36) for all π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹β§΅{0} and 𝑑>0, which ends the proof.

We now prove the Hyers-Ulam-Rassias stability of (1.3) in the framework of normed spaces.

Corollary 3.4. Let 𝑋 be a real normed space, π‘βˆˆ[0,1)βˆͺ(2,∞), and let π‘Œ be a complete normed space. If a function π‘“βˆΆπ‘‹β†’π‘Œ satisfies 𝑓(0)=0 and β€–β€–ξ€·π‘₯𝐷𝑓1,π‘₯2,…,π‘₯𝑛‖‖‖‖π‘₯β‰€πœƒ1‖‖𝑝+β€–β€–π‘₯2‖‖𝑝‖‖π‘₯+β‹―+𝑛‖‖𝑝(3.37) for all π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹ and for some πœƒβ‰₯0, then there exists a unique quadratic-additive function πΉβˆΆπ‘‹β†’π‘Œ such that β€–βŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©π‘“(π‘₯)βˆ’πΉ(π‘₯)β€–β‰€π‘›πœƒβ€–π‘₯‖𝑝2(π‘›βˆ’π‘›π‘)if0≀p<1,π‘›πœƒβ€–π‘₯‖𝑝2ξ€·π‘›π‘βˆ’π‘›2ξ€Έifp>2(3.38) for all π‘₯βˆˆπ‘‹.

Proof. If we put πœ‘ξ€·π‘₯1,π‘₯2,…,π‘₯𝑛‖‖π‘₯∢=πœƒ1‖‖𝑝+β€–β€–π‘₯2‖‖𝑝‖‖π‘₯+β‹―+𝑛‖‖𝑝,(3.39) then the induced random normed space (𝑋,Ξ›π‘₯,πœπ‘€) satisfies the conditions stated in Theorem 3.2 with 𝛼=𝑛𝑝.

Corollary 3.5. Let 𝑋 be a real normed space, π‘βˆˆ(βˆ’βˆž,0), and let π‘Œ be a complete normed space. If a function π‘“βˆΆπ‘‹β†’π‘Œ satisfies 𝑓(0)=0 and β€–β€–ξ€·π‘₯𝐷𝑓1,π‘₯2,…,π‘₯π‘›ξ€Έβ€–β€–ξ“β‰€πœƒ1≀𝑖≀𝑛,π‘₯𝑖≠0β€–β€–π‘₯𝑖‖‖𝑝(3.40) for all π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹ and for some πœƒβ‰₯0, then there exists a unique quadratic-additive function πΉβˆΆπ‘‹β†’π‘Œ satisfying ⎧βŽͺ⎨βŽͺβŽ©β€–π‘“(π‘₯)βˆ’πΉ(π‘₯)β€–β‰€π‘›πœƒβ€–π‘₯‖𝑝2(π‘›βˆ’π‘›π‘)ifx∈X0⧡{0},ifx=0.(3.41)

Proof. If we put π‘βˆΆ=ℝ, π›ΌβˆΆ=𝑛𝑝, and define Ξ›π‘₯(𝑑𝑑)∢=𝑑+β€–π‘₯β€–,Ξ›ξ…žπ‘§(𝑑𝑑)∢=,πœ‘ξ€·π‘₯𝑑+|𝑧|1,π‘₯2,…,π‘₯π‘›ξ€Έβˆ‘βˆΆ=πœƒ1≀𝑖≀𝑛,π‘₯𝑖≠0β€–β€–π‘₯𝑖‖‖𝑝(3.42) for all π‘₯,π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹ and π‘§βˆˆπ‘, then we have Ξ›ξ…žπ›Όπœ‘(π‘₯1,π‘₯2,…,π‘₯𝑛)(𝑑𝑑)=ξ€·π‘₯𝑑+π›Όπœ‘1,…,π‘₯𝑛=𝑑𝑑+πœ‘π‘›π‘₯1,…,𝑛π‘₯𝑛=Ξ›ξ…žπœ‘(𝑛π‘₯1,𝑛π‘₯2,…,𝑛π‘₯𝑛)(𝑑),(3.43) that is, πœ‘ satisfies condition (𝑖) given in Theorem 3.1 for all π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹ and 𝑑>0. We moreover get Λ𝐷𝑓(π‘₯1,π‘₯2,…,π‘₯𝑛)(𝑑𝑑)=β€–β€–ξ€·π‘₯𝑑+𝐷𝑓1,…,π‘₯𝑛‖‖β‰₯π‘‘βˆ‘π‘‘+πœƒ1≀𝑖≀𝑛,π‘₯𝑖≠0β€–β€–π‘₯𝑖‖‖𝑝=𝑑π‘₯𝑑+πœ‘1,…,π‘₯𝑛=Ξ›ξ…žπœ‘(π‘₯1,π‘₯2,…,π‘₯𝑛)(𝑑),(3.44) that is, 𝑓 satisfies the inequality (3.2) for all π‘₯1,π‘₯2,…,π‘₯π‘›βˆˆπ‘‹ and 𝑑>0.
According to Theorem 3.2, there exists a unique quadratic-additive function πΉβˆΆπ‘‹β†’π‘Œ satisfying 𝑑𝑑+‖𝑓(π‘₯)βˆ’πΉ(π‘₯)β€–=Λ𝑓(π‘₯)βˆ’πΉ(π‘₯)(𝑑)β‰₯𝑀(π‘₯,2(π‘›βˆ’π‘›π‘=⎧βŽͺ⎨βŽͺ⎩)𝑑)2(π‘›βˆ’π‘›π‘)𝑑2(π‘›βˆ’π‘›π‘β€–)𝑑+π‘›πœƒπ‘₯‖𝑝if1π‘₯βˆˆπ‘‹β§΅{0},ifπ‘₯=0(3.45) for all 𝑑>0, or equivalently ‖‖𝑓(π‘₯)βˆ’πΉ(π‘₯)π‘‘β‰€βŽ§βŽͺ⎨βŽͺβŽ©π‘›πœƒβ€–π‘₯‖𝑝2(π‘›βˆ’π‘›π‘)𝑑if0π‘₯βˆˆπ‘‹β§΅{0},ifπ‘₯=0(3.46) for all 𝑑>0, which ends the proof.

Acknowledgment

The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2011-0004919).

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