Abstract

We investigate the stability problems for the ๐‘›-dimensional mixed-type additive and quadratic functional equation โˆ‘2๐‘“(๐‘›๐‘—=1๐‘ฅ๐‘—โˆ‘)+1โ‰ค๐‘–,๐‘—โ‰ค๐‘›,๐‘–โ‰ ๐‘—๐‘“(๐‘ฅ๐‘–โˆ’๐‘ฅ๐‘—โˆ‘)=(๐‘›+1)๐‘›๐‘—=1๐‘“(๐‘ฅ๐‘—โˆ‘)+(๐‘›โˆ’1)๐‘›๐‘—=1๐‘“(โˆ’๐‘ฅ๐‘—) 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 ๐บ1 be a group, and let ๐บ2 be a metric group with a metric ๐‘‘(โ‹…,โ‹…). Given ๐œ€>0, does there exist a ๐›ฟ>0 such that if a function โ„Žโˆถ๐บ1โ†’๐บ2 satisfies the inequality ๐‘‘(โ„Ž(๐‘ฅ๐‘ฆ),โ„Ž(๐‘ฅ)โ„Ž(๐‘ฆ))<๐›ฟ for all ๐‘ฅ,๐‘ฆโˆˆ๐บ1, then there is a homomorphism ๐ปโˆถ๐บ1โ†’๐บ2 with ๐‘‘(โ„Ž(๐‘ฅ),๐ป(๐‘ฅ))<๐œ€ for all ๐‘ฅโˆˆ๐บ1?

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 ๐บ1 and ๐บ2 are assumed to be Banach spaces. This result of Hyers is stated as follows.

Let ๐‘“โˆถ๐ธ1โ†’๐ธ2 be a function between Banach spaces such thatโ€–๐‘“(๐‘ฅ+๐‘ฆ)โˆ’๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ฆ)โ€–โ‰ค๐›ฟ(1.1) for some ๐›ฟ>0 and for all ๐‘ฅ,๐‘ฆโˆˆ๐ธ1. Then the limit ๐ด(๐‘ฅ)=lim๐‘›โ†’โˆž2โˆ’๐‘›๐‘“(2๐‘›๐‘ฅ) exists for each ๐‘ฅโˆˆ๐ธ1, and ๐ดโˆถ๐ธ1โ†’๐ธ2 is the unique additive function such that โ€–๐‘“(๐‘ฅ)โˆ’๐ด(๐‘ฅ)โ€–โ‰ค๐›ฟ for every ๐‘ฅโˆˆ๐ธ1. Moreover, if ๐‘“(๐‘ก๐‘ฅ) is continuous in ๐‘ก for each fixed ๐‘ฅโˆˆ๐ธ1, then function ๐ด is linear.

We remark that the additive function ๐ด is directly constructed from the given function ๐‘“, 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 ๐‘“(๐‘ฅ+๐‘ฆ)=๐‘“(๐‘ฅ)+๐‘“(๐‘ฆ) is said to have the Hyers-Ulam stability on (๐ธ1,๐ธ2) if for every function ๐‘“โˆถ๐ธ1โ†’๐ธ2 satisfying the inequality (1.1) for some ๐›ฟโ‰ฅ0 and for all ๐‘ฅ,๐‘ฆโˆˆ๐ธ1, there exists an additive function ๐ดโˆถ๐ธ1โ†’๐ธ2 such that ๐‘“โˆ’๐ด is bounded on ๐ธ1.

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 Hyersโ€™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 ๐‘“โˆถ๐ธ1โ†’๐ธ2 be a function between Banach spaces. If ๐‘“ satisfies the functional inequality๎€ทโ€–๐‘“(๐‘ฅ+๐‘ฆ)โˆ’๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ฆ)โ€–โ‰ค๐œƒโ€–๐‘ฅโ€–๐‘+โ€–๐‘ฆโ€–๐‘๎€ธ(1.2)for some ๐œƒโ‰ฅ0, ๐‘ with 0โ‰ค๐‘<1 and for all ๐‘ฅ,๐‘ฆโˆˆ๐ธ1, then there exists a unique additive function ๐ดโˆถ๐ธ1โ†’๐ธ2 such that โ€–๐‘“(๐‘ฅ)โˆ’๐ด(๐‘ฅ)โ€–โ‰ค(2๐œƒ/(2โˆ’2๐‘))โ€–๐‘ฅโ€–๐‘ for each ๐‘ฅโˆˆ๐ธ1. If, in addition, ๐‘“(๐‘ก๐‘ฅ) is continuous in ๐‘ก for each fixed ๐‘ฅโˆˆ๐ธ1, then the function ๐ด 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 [6โ€“17]).

In this paper, applying the fixed point method, we prove the Hyers-Ulam-Rassias stability of the ๐‘›-dimensional mixed-type additive and quadratic functional equation ๎ƒฉ2๐‘“๐‘›๎“๐‘—=1๐‘ฅ๐‘—๎ƒช+๎“1โ‰ค๐‘–,๐‘—โ‰ค๐‘›,๐‘–โ‰ ๐‘—๐‘“๎€ท๐‘ฅ๐‘–โˆ’๐‘ฅ๐‘—๎€ธ=(๐‘›+1)๐‘›๎“๐‘—=1๐‘“๎€ท๐‘ฅ๐‘—๎€ธ+(๐‘›โˆ’1)๐‘›๎“๐‘—=1๐‘“๎€ทโˆ’๐‘ฅ๐‘—๎€ธ(1.3) in random normed spaces. Every solution of (1.3) is called a quadratic-additive function.

Throughout this paper, let ๐‘› 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 byฮ”+[]โ†’[]โˆถ={๐นโˆถ0,โˆž0,1โˆฃ๐นisleft-continuousandnondecreasing[on0,โˆž),๐น(0)=0,and๐น(โˆž)=1}.(2.1) Let us define ๐ท+โˆถ={๐นโˆˆฮ”+โˆฃlim๐‘กโ†’โˆž๐น(๐‘ก)=1}. The set ฮ”+ is partially ordered by the usual pointwise ordering of functions, that is, ๐นโ‰ค๐บ if and only if ๐น(๐‘ก)โ‰ค๐บ(๐‘ก) for all ๐‘กโ‰ฅ0. The maximal element for ฮ”+ in this order is the distribution function ๐œ€0โˆถ[0,โˆž]โ†’[0,1] given by๐œ€0๎‚ป0(๐‘ก)=if1๐‘ก=0,if๐‘ก>0.(2.2)

Definition 2.1 (See [18]). A function ๐œโˆถ[0,1]ร—[0,1]โ†’[0,1] is called a continuous triangular norm (briefly, continuous ๐‘ก-norm) if ๐œ satisfies the following conditions:(๐‘Ž)๐œ is commutative and associative;(๐‘)ฯ„ is continuous;(๐‘)๐œ(๐‘Ž,1)=๐‘Ž for all ๐‘Žโˆˆ[0,1];(๐‘‘)๐œ(๐‘Ž,๐‘)โ‰ค๐œ(๐‘,๐‘‘) for all ๐‘Ž,๐‘,๐‘,๐‘‘โˆˆ[0,1] with ๐‘Žโ‰ค๐‘ and ๐‘โ‰ค๐‘‘.
Typical examples of continuous ๐‘ก-norms are ๐œ๐‘ƒ(๐‘Ž,๐‘)=๐‘Ž๐‘, ๐œ๐‘€(๐‘Ž,๐‘)=min{๐‘Ž,๐‘}, and ๐œ๐ฟ(๐‘Ž,๐‘)=max{๐‘Ž+๐‘โˆ’1,0}.

Definition 2.2 (See [19]). Let ๐‘‹ be a vector space, ๐œ a continuous ๐‘ก-norm, and let ฮ›โˆถ๐‘‹โ†’๐ท+ be a function satisfying the following conditions:(๐‘…1)ฮ›๐‘ฅ(๐‘ก)=๐œ€0(๐‘ก) for all ๐‘ก>0 if and only if ๐‘ฅ=0;(๐‘…2)ฮ›๐›ผ๐‘ฅ(๐‘ก)=ฮ›๐‘ฅ(๐‘ก/|๐›ผ|) for all ๐‘ฅโˆˆ๐‘‹, ๐›ผโ‰ 0, and for all ๐‘กโ‰ฅ0;(๐‘…3)ฮ›๐‘ฅ+๐‘ฆ(๐‘ก+๐‘ )โ‰ฅ๐œ(ฮ›๐‘ฅ(๐‘ก),ฮ›๐‘ฆ(๐‘ )) for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹ and all ๐‘ก,๐‘ โ‰ฅ0. A triple (๐‘‹,ฮ›,๐œ) is called a random normed space (briefly, RN-space).
If (๐‘‹,โ€–โ‹…โ€–) is a normed space, we can define a function ฮ›โˆถ๐‘‹โ†’๐ท+ by ฮ›๐‘ฅ(๐‘ก๐‘ก)=๐‘ก+โ€–๐‘ฅโ€–(2.3) for all ๐‘ฅโˆˆ๐‘‹ and ๐‘ก>0. Then (๐‘‹,ฮ›,๐œ๐‘€) is a random normed space, which is called the induced random normed space.

Definition 2.3. Let (๐‘‹,ฮ›,๐œ) be an ๐‘…๐‘-space.(๐‘–)A sequence {๐‘ฅ๐‘›} in ๐‘‹ is said to be convergent to a point ๐‘ฅโˆˆ๐‘‹ if, for every ๐‘ก>0 and ๐œ€>0, there exists a positive integer ๐‘ such that ฮ›๐‘ฅ๐‘›โˆ’๐‘ฅ(๐‘ก)>1โˆ’๐œ€ whenever ๐‘›โ‰ฅ๐‘.(๐‘–๐‘–)A sequence {๐‘ฅ๐‘›} in ๐‘‹ is called a Cauchy sequence if, for every ๐‘ก>0 and ๐œ€>0, there exists a positive integer ๐‘ such that ฮ›๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘š(๐‘ก)>1โˆ’๐œ€ whenever ๐‘›โ‰ฅ๐‘šโ‰ฅ๐‘.(๐‘–๐‘–๐‘–)An RN-space (๐‘‹,ฮ›,๐œ) is called complete if and only if every Cauchy sequence in ๐‘‹ converges to a point in ๐‘‹.

Definition 2.4. Let ๐‘‹ be a nonempty set. A function ๐‘‘โˆถ๐‘‹2โ†’[0,โˆž] is called a generalized metric on ๐‘‹ if and only if ๐‘‘ satisfies(๐‘€1)๐‘‘(๐‘ฅ,๐‘ฆ)=0 if and only if ๐‘ฅ=๐‘ฆ;(๐‘€2)๐‘‘(๐‘ฅ,๐‘ฆ)=๐‘‘(๐‘ฆ,๐‘ฅ) for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹;(๐‘€3)๐‘‘(๐‘ฅ,๐‘ง)โ‰ค๐‘‘(๐‘ฅ,๐‘ฆ)+๐‘‘(๐‘ฆ,๐‘ง) for all ๐‘ฅ,๐‘ฆ,๐‘งโˆˆ๐‘‹.
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 (๐‘‹,๐‘‘) be a complete generalized metric space. Assume that ฮ›โˆถ๐‘‹โ†’๐‘‹ is a strict contraction with the Lipschitz constant ๐ฟ<1. If there exists a nonnegative integer ๐‘›0 such that ๐‘‘(ฮ›๐‘›0+1๐‘ฅ,ฮ›๐‘›0๐‘ฅ)<โˆž for some ๐‘ฅโˆˆ๐‘‹, then the following statements are true:(๐‘–)the sequence {ฮ›๐‘›๐‘ฅ} converges to a fixed point ๐‘ฅโˆ— of ฮ›;(๐‘–๐‘–)๐‘ฅโˆ— is the unique fixed point of ฮ› in ๐‘‹โˆ—={๐‘ฆโˆˆ๐‘‹โˆฃ๐‘‘(ฮ›๐‘›0๐‘ฅ,๐‘ฆ)<โˆž};(๐‘–๐‘–๐‘–)if ๐‘ฆโˆˆ๐‘‹โˆ—, then๐‘‘๎€ท๐‘ฆ,๐‘ฅโˆ—๎€ธโ‰ค11โˆ’๐ฟ๐‘‘(ฮ›๐‘ฆ,๐‘ฆ).(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 [23โ€“26]). Furthermore, Miheลฃ 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 ๐‘›-dimensional mixed-type additive and quadratic functional equation (1.3) by using the direct method. According to [28], a function ๐‘“โˆถ๐ธ1โ†’๐ธ2 is a quadratic-additive function, where ๐ธ1 and ๐ธ2 are vector spaces, if and only if there exist an additive function ๐‘Žโˆถ๐ธ1โ†’๐ธ2 and a quadratic function ๐‘žโˆถ๐ธ1โ†’๐ธ2 such that ๐‘“(๐‘ฅ)=๐‘Ž(๐‘ฅ)+๐‘ž(๐‘ฅ) for all ๐‘ฅโˆˆ๐ธ1.

3. Hyers-Ulam-Rassias Stability

Throughout this paper, let ๐‘‹ be a real vector space and let (๐‘Œ,ฮ›,๐œ๐‘€) be a complete RN-space. For a given function ๐‘“โˆถ๐‘‹โ†’๐‘Œ, we use the following abbreviation:๎€ท๐‘ฅ๐ท๐‘“1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›๎€ธ๎ƒฉโˆถ=2๐‘“๐‘›๎“๐‘—=1๐‘ฅ๐‘—๎ƒช+๎“1โ‰ค๐‘–,๐‘—โ‰ค๐‘›,๐‘–โ‰ ๐‘—๐‘“๎€ท๐‘ฅ๐‘–โˆ’๐‘ฅ๐‘—๎€ธโˆ’(๐‘›+1)๐‘›๎“๐‘—=1๐‘“๎€ท๐‘ฅ๐‘—๎€ธโˆ’(๐‘›โˆ’1)๐‘›๎“๐‘—=1๐‘“๎€ทโˆ’๐‘ฅ๐‘—๎€ธ(3.1) for all ๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹.

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 ๐‘‹ be a real vector space, (๐‘,ฮ›โ€ฒ,๐œ๐‘€) an RN-space, (๐‘Œ,ฮ›,๐œ๐‘€) a complete RN-space, and let ๐œ‘โˆถ(๐‘‹โงต{0})๐‘›โ†’๐‘ be a function. Assume that ๐œ‘ satisfies one of the following conditions:(๐‘–)ฮ›๎…ž๐›ผ๐œ‘(๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›)(๐‘ก)โ‰คฮ›๎…ž๐œ‘(๐‘›๐‘ฅ1,๐‘›๐‘ฅ2,โ€ฆ,๐‘›๐‘ฅ๐‘›)(t) for some 0<๐›ผ<๐‘›;(๐‘–๐‘–)ฮ›๎…ž๐œ‘(๐‘›๐‘ฅ1,๐‘›๐‘ฅ2,โ€ฆ,๐‘›๐‘ฅ๐‘›)(๐‘ก)โ‰คฮ›๎…ž๐›ผ๐œ‘(๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›)(๐‘ก) for some ๐›ผ>๐‘›2 for all ๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹โงต{0} and ๐‘ก>0. If a function ๐‘“โˆถ๐‘‹โ†’๐‘Œ satisfies ๐‘“(0)=0 and ฮ›๐ท๐‘“(๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›)(๐‘ก)โ‰ฅฮ›๎…ž๐œ‘(๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›)(๐‘ก)(3.2) for all ๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹โงต{0} and ๐‘ก>0, then there exists a unique function ๐นโˆถ๐‘‹โ†’๐‘Œ such that ๎€ท๐‘ฅ๐ท๐น1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›๎€ธ=0(3.3) for all ๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹โงต{0} and ฮ›๐‘“(๐‘ฅ)โˆ’๐น(๐‘ฅ)๎‚ป(๐‘ก)โ‰ฅ๐‘€(๐‘ฅ,2(๐‘›โˆ’๐›ผ)๐‘ก)if๐œ‘satis๏ฌes๐‘€๎€ท๎€ท(๐‘–),๐‘ฅ,2๐›ผโˆ’๐‘›2๎€ธ๐‘ก๎€ธif๐œ‘satis๏ฌes(๐‘–๐‘–)(3.4) for all ๐‘ฅโˆˆ๐‘‹โงต{0} and ๐‘ก>0, where ๐‘€(๐‘ฅ,๐‘ก)โˆถ=๐œ๐‘€(ฮ›โ€ฒ๐œ‘(ฬ‚๐‘ฅ)(๐‘ก),ฮ›โ€ฒ๐œ‘(๎‚Šโˆ’๐‘ฅ)(๐‘ก)), and ฬ‚๐‘ฅ=(๐‘ฅ,๐‘ฅ,โ€ฆ,๐‘ฅ).

Proof. We will first treat the case where ๐œ‘ satisfies the condition (๐‘–). Let ๐‘† be the set of all functions ๐‘”โˆถ๐‘‹โ†’๐‘Œ with ๐‘”(0)=0, and let us define a generalized metric on ๐‘† by ๐‘‘๎€ฝ[](๐‘”,โ„Ž)โˆถ=inf๐‘ขโˆˆ0,โˆžโˆฃฮ›๐‘”(๐‘ฅ)โˆ’โ„Ž(๐‘ฅ)๎€พ.(๐‘ข๐‘ก)โ‰ฅ๐‘€(๐‘ฅ,๐‘ก)โˆ€๐‘ฅโˆˆ๐‘‹โงต{0},๐‘ก>0(3.5) It is not difficult to show that (๐‘†,๐‘‘) is a complete generalized metric space (see [29] or [30, 31]).
Consider the operator ๐ฝโˆถ๐‘†โ†’๐‘† defined by ๐ฝ๐‘“(๐‘ฅ)โˆถ=๐‘“(๐‘›๐‘ฅ)โˆ’๐‘“(โˆ’๐‘›๐‘ฅ)+2๐‘›๐‘“(๐‘›๐‘ฅ)+๐‘“(โˆ’๐‘›๐‘ฅ)2๐‘›2.(3.6) Then we can apply induction on ๐‘š to prove ๐ฝ๐‘š๐‘“(๐‘ฅ)=๐‘“(๐‘›๐‘š๐‘ฅ)โˆ’๐‘“(โˆ’๐‘›๐‘š๐‘ฅ)2๐‘›๐‘š+๐‘“(๐‘›๐‘š๐‘ฅ)+๐‘“(โˆ’๐‘›๐‘š๐‘ฅ)2๐‘›2๐‘š(3.7) for all ๐‘ฅโˆˆ๐‘‹ and ๐‘šโˆˆโ„•.
Let ๐‘“,๐‘”โˆˆ๐‘† and let ๐‘ขโˆˆ[0,โˆž] be an arbitrary constant with ๐‘‘(๐‘”,๐‘“)โ‰ค๐‘ข. For some 0<๐›ผ<๐‘› satisfying the condition (๐‘–), it follows from the definition of ๐‘‘, (๐‘…2), (๐‘…3), and (๐‘–) that ฮ›๐ฝ๐‘”(๐‘ฅ)โˆ’๐ฝ๐‘“(๐‘ฅ)๎‚€๐›ผ๐‘ข๐‘ก๐‘›๎‚=ฮ›((๐‘›+1)(๐‘”(๐‘›๐‘ฅ)โˆ’๐‘“(๐‘›๐‘ฅ))/2๐‘›2)โˆ’((๐‘›โˆ’1)(๐‘”(โˆ’๐‘›๐‘ฅ)โˆ’๐‘“(โˆ’๐‘›๐‘ฅ))/2๐‘›2)๎‚€๐›ผ๐‘ข๐‘ก๐‘›๎‚โ‰ฅ๐œ๐‘€๎ƒฉฮ›(๐‘›+1)(๐‘”(๐‘›๐‘ฅ)โˆ’๐‘“(๐‘›๐‘ฅ))/2๐‘›2๎ƒฉ(๐‘›+1)๐›ผ๐‘ข๐‘ก๎€ท2๐‘›2๎€ธ๎ƒช,ฮ›(๐‘›โˆ’1)(๐‘”(โˆ’๐‘›๐‘ฅ)โˆ’๐‘“(โˆ’๐‘›๐‘ฅ))/2๐‘›2๎ƒฉ(๐‘›โˆ’1)๐›ผ๐‘ข๐‘ก๎€ท2๐‘›2๎€ธ๎ƒช๎ƒชโ‰ฅ๐œ๐‘€๎€ทฮ›๐‘”(๐‘›๐‘ฅ)โˆ’๐‘“(๐‘›๐‘ฅ)(๐›ผ๐‘ข๐‘ก),ฮ›๐‘”(โˆ’๐‘›๐‘ฅ)โˆ’๐‘“(โˆ’๐‘›๐‘ฅ)(๎€ธ๐›ผ๐‘ข๐‘ก)โ‰ฅ๐œ๐‘€๎‚€ฮ›๎…ž๐œ‘(๎‚Š๐‘›๐‘ฅ)(๐›ผ๐‘ก),ฮ›๎…ž๐œ‘(๎‚ฟโˆ’๐‘›๐‘ฅ)๎‚(๐›ผ๐‘ก)โ‰ฅ๐‘€(๐‘ฅ,๐‘ก)(3.8) for all ๐‘ฅโˆˆ๐‘‹โงต{0} and ๐‘ก>0, which implies that ๐‘‘๐›ผ(๐ฝ๐‘“,๐ฝ๐‘”)โ‰ค๐‘›๐‘‘(๐‘“,๐‘”).(3.9) That is, ๐ฝ is a strict contraction with the Lipschitz constant 0<๐›ผ/๐‘›<1.
Moreover, by (๐‘…2), (๐‘…3), and (3.2), we see that ฮ›๐‘“(๐‘ฅ)โˆ’๐ฝ๐‘“(๐‘ฅ)๎‚€๐‘ก๎‚2๐‘›=ฮ›(โˆ’(๐‘›+1)๐ท๐‘“(ฬ‚๐‘ฅ)+(๐‘›โˆ’1)๐ท๐‘“(๎‚Šโˆ’๐‘ฅ))/4๐‘›2๎‚€๐‘ก๎‚2๐‘›โ‰ฅ๐œ๐‘€๎‚ตฮ›(๐‘›+1)๐ท๐‘“(ฬ‚๐‘ฅ)/4๐‘›2๎‚ต(๐‘›+1)๐‘ก4๐‘›2๎‚ถ,ฮ›(๐‘›โˆ’1)๐ท๐‘“(๎‚Šโˆ’๐‘ฅ)/4๐‘›2๎‚ต(๐‘›โˆ’1)๐‘ก4๐‘›2๎‚ถ๎‚ถโ‰ฅ๐œ๐‘€๎‚€ฮ›๐ท๐‘“(ฬ‚๐‘ฅ)(๐‘ก),ฮ›๐ท๐‘“(๎‚Šโˆ’๐‘ฅ)๎‚(๐‘ก)โ‰ฅ๐‘€(๐‘ฅ,๐‘ก)(3.10) for all ๐‘ฅโˆˆ๐‘‹โงต{0} and ๐‘ก>0. Hence, it follows from the definition of ๐‘‘ that 1๐‘‘(๐‘“,๐ฝ๐‘“)โ‰ค2๐‘›<โˆž.(3.11) Now, in view of Theorem 2.5, the sequence {๐ฝ๐‘š๐‘“} converges to the unique โ€œfixed pointโ€ ๐นโˆถ๐‘‹โ†’๐‘Œ of ๐ฝ in the set ๐‘‡={๐‘”โˆˆ๐‘†โˆฃ๐‘‘(๐‘“,๐‘”)<โˆž} and ๐น is represented by ๐น(๐‘ฅ)=lim๐‘šโ†’โˆž๎‚ต๐‘“(๐‘›๐‘š๐‘ฅ)โˆ’๐‘“(โˆ’๐‘›๐‘š๐‘ฅ)2๐‘›๐‘š+๐‘“(๐‘›๐‘š๐‘ฅ)+๐‘“(โˆ’๐‘›๐‘š๐‘ฅ)2๐‘›2๐‘š๎‚ถ(3.12) for all ๐‘ฅโˆˆ๐‘‹.
By Theorem 2.5, (3.11), and the definition of ๐‘‘, we have 1๐‘‘(๐‘“,๐น)โ‰ค11โˆ’๐›ผ/๐‘›๐‘‘(๐‘“,๐ฝ๐‘“)โ‰ค2,(๐‘›โˆ’๐›ผ)(3.13) that is, the first inequality in (3.4) holds true.
We will now show that ๐น is a quadratic-additive function. It follows from (๐‘…3) and the definition of ๐œ๐‘€ that ฮ›๐ท๐น(๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›)๎‚†ฮ›(๐‘ก)โ‰ฅmin2(๐นโˆ’๐ฝ๐‘šโˆ‘๐‘“)(๐‘›๐‘—=1๐‘ฅ๐‘—)๎‚€๐‘ก5๎‚,๎‚ปฮ›min(๐นโˆ’๐ฝ๐‘š๐‘“)(๐‘ฅ๐‘–โˆ’๐‘ฅ๐‘—)๎‚ต๐‘ก(๎‚ถ๎‚ผ,๎‚ปฮ›5๐‘›(๐‘›โˆ’1))โˆฃ1โ‰ค๐‘–,๐‘—โ‰ค๐‘›,๐‘–โ‰ ๐‘—min(๐‘›+1)(๐ฝ๐‘š๐‘“โˆ’๐น)(๐‘ฅ๐‘—)๎‚ต๐‘ก๎‚ถ๎‚ผ,๎‚ปฮ›(5๐‘›)โˆฃ๐‘—=1,โ€ฆ,๐‘›min(๐‘›โˆ’1)(๐ฝ๐‘š๐‘“โˆ’๐น)(โˆ’๐‘ฅ๐‘—)๎‚ต๐‘ก๎‚ถ๎‚ผ,ฮ›(5๐‘›)โˆฃ๐‘—=1,โ€ฆ,๐‘›๐ท๐ฝ๐‘š๐‘“(๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›)๎‚€๐‘ก5๎‚๎‚‡(3.14) for all ๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹โงต{0}, ๐‘ก>0, and ๐‘šโˆˆโ„•. Due to the definition of ๐น, the first four terms on the right-hand side of the above inequality tend to 1 as ๐‘šโ†’โˆž.
By a somewhat tedious manipulation, we have ๐ท๐ฝ๐‘š๐‘“๎€ท๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›๎€ธ=12๐‘›2๐‘š๎€ท๐‘›๐ท๐‘“๐‘š๐‘ฅ1,โ€ฆ,๐‘›๐‘š๐‘ฅ๐‘›๎€ธ+12๐‘›2๐‘š๎€ท๐ท๐‘“โˆ’๐‘›๐‘š๐‘ฅ1,โ€ฆ,โˆ’๐‘›๐‘š๐‘ฅ๐‘›๎€ธ+12๐‘›๐‘š๎€ท๐‘›๐ท๐‘“๐‘š๐‘ฅ1,โ€ฆ,๐‘›๐‘š๐‘ฅ๐‘›๎€ธโˆ’12๐‘›๐‘š๎€ท๐ท๐‘“โˆ’๐‘›๐‘š๐‘ฅ1,โ€ฆ,โˆ’๐‘›๐‘š๐‘ฅ๐‘›๎€ธ.(3.15) Hence, it follows from (๐‘…2), (๐‘…3), definition of ๐œ๐‘€, (3.2), and (๐‘–) that ฮ›๐ท๐ฝ๐‘š๐‘“(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›)๎‚€๐‘ก5๎‚๎‚†ฮ›โ‰ฅmin๐ท๐‘“(๐‘›๐‘š๐‘ฅ1,โ€ฆ,๐‘›๐‘š๐‘ฅ๐‘›)/2๐‘›2๐‘š๎‚€๐‘ก๎‚20,ฮ›๐ท๐‘“(โˆ’๐‘›๐‘š๐‘ฅ1,โ€ฆ,โˆ’๐‘›๐‘š๐‘ฅ๐‘›)/2๐‘›2๐‘š๎‚€๐‘ก๎‚,ฮ›20๐ท๐‘“(๐‘›๐‘š๐‘ฅ1,โ€ฆ,๐‘›๐‘š๐‘ฅ๐‘›)/2๐‘›๐‘š๎‚€๐‘ก๎‚20,ฮ›๐ท๐‘“(โˆ’๐‘›๐‘š๐‘ฅ1,โ€ฆ,โˆ’๐‘›๐‘š๐‘ฅ๐‘›)/2๐‘›๐‘š๎‚€๐‘ก๎‚ปฮ›20๎‚๎‚‡โ‰ฅmin๐ท๐‘“(๐‘›๐‘š๐‘ฅ1,โ€ฆ,๐‘›๐‘š๐‘ฅ๐‘›)๎‚ต๐‘›2๐‘š๐‘ก๎‚ถ10,ฮ›๐ท๐‘“(โˆ’๐‘›๐‘š๐‘ฅ1,โ€ฆ,โˆ’๐‘›๐‘š๐‘ฅ๐‘›)๎‚ต๐‘›2๐‘š๐‘ก๎‚ถ,ฮ›10๐ท๐‘“(๐‘›๐‘š๐‘ฅ1,โ€ฆ,๐‘›๐‘š๐‘ฅ๐‘›)๎‚ต๐‘›๐‘š๐‘ก๎‚ถ10,ฮ›๐ท๐‘“(โˆ’๐‘›๐‘š๐‘ฅ1,โ€ฆ,โˆ’๐‘›๐‘š๐‘ฅ๐‘›)๎‚ต๐‘›๐‘š๐‘ก๎‚ปฮ›10๎‚ถ๎‚ผโ‰ฅmin๎…ž๐œ‘(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›)๎‚ต๐‘›2๐‘š๐‘ก(10๐›ผ๐‘š)๎‚ถ,ฮ›๎…ž๐œ‘(โˆ’๐‘ฅ1,โ€ฆ,โˆ’๐‘ฅ๐‘›)๎‚ต๐‘›2๐‘š๐‘ก(10๐›ผ๐‘š)๎‚ถ,ฮ›๎…ž๐œ‘(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›)๎‚ต๐‘›๐‘š๐‘ก(10๐›ผ๐‘š)๎‚ถ,ฮ›๎…ž๐œ‘(โˆ’๐‘ฅ1,โ€ฆ,โˆ’๐‘ฅ๐‘›)๎‚ต๐‘›๐‘š๐‘ก(10๐›ผ๐‘š),๎‚ถ๎‚ผ(3.16) which tends to 1 as ๐‘šโ†’โˆž for all ๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹โงต{0} and ๐‘ก>0. Therefore, (3.14) implies that ฮ›๐ท๐น(๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›)(๐‘ก)=1(3.17) for any ๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹โงต{0} and ๐‘ก>0. By (๐‘…1), this implies that ๐ท๐น(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›)=0 for all ๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹โงต{0}, which ends the proof of the first part.
Now, assume that ๐œ‘ satisfies the condition (๐‘–๐‘–). Let (๐‘†,๐‘‘) be the same as given in the first part. We now consider the operator ๐ฝโˆถ๐‘†โ†’๐‘† defined by ๐‘›๐ฝ๐‘”(๐‘ฅ)โˆถ=2๎‚€๐‘”๎‚€๐‘ฅ๐‘›๎‚๎‚€โˆ’๐‘ฅโˆ’๐‘”๐‘›+๐‘›๎‚๎‚22๎‚€๐‘”๎‚€๐‘ฅ๐‘›๎‚๎‚€โˆ’๐‘ฅ+๐‘”๐‘›๎‚๎‚(3.18) for all ๐‘”โˆˆ๐‘† and ๐‘ฅโˆˆ๐‘‹. Notice that ๐ฝ๐‘š๐‘›๐‘”(๐‘ฅ)=๐‘š2๎‚€๐‘”๎‚€๐‘ฅ๐‘›๐‘š๎‚๎‚€โˆ’๐‘ฅโˆ’๐‘”๐‘›๐‘š+๐‘›๎‚๎‚2๐‘š2๎‚€๐‘”๎‚€๐‘ฅ๐‘›๐‘š๎‚๎‚€โˆ’๐‘ฅ+๐‘”๐‘›๐‘š๎‚๎‚(3.19) for all ๐‘ฅโˆˆ๐‘‹ and ๐‘šโˆˆโ„•.
Let ๐‘“,๐‘”โˆˆ๐‘† and let ๐‘ขโˆˆ[0,โˆž] be an arbitrary constant with ๐‘‘(๐‘”,๐‘“)โ‰ค๐‘ข. From (๐‘…2), (๐‘…3), the definition of ๐‘‘, and (๐‘–๐‘–), we have ฮ›๐ฝ๐‘”(๐‘ฅ)โˆ’๐ฝ๐‘“(๐‘ฅ)๎‚ต๐‘›2๐‘ข๐‘ก๐›ผ๎‚ถ=ฮ›((๐‘›2+๐‘›)/2)(๐‘”(๐‘ฅ/๐‘›)โˆ’๐‘“(๐‘ฅ/๐‘›))+((๐‘›2โˆ’๐‘›)/2)(๐‘”(โˆ’๐‘ฅ/๐‘›)โˆ’๐‘“(โˆ’๐‘ฅ/๐‘›))๎‚ต๐‘›2๐‘ข๐‘ก๐›ผ๎‚ถโ‰ฅ๐œ๐‘€๎ƒฉฮ›((๐‘›2+๐‘›)/2)(๐‘”(๐‘ฅ/๐‘›)โˆ’๐‘“(๐‘ฅ/๐‘›))๎ƒฉ๎€ท๐‘›2๎€ธ+๐‘›๐‘ข๐‘ก๎ƒช,ฮ›(2๐›ผ)((๐‘›2โˆ’๐‘›)/2)(๐‘”(โˆ’๐‘ฅ/๐‘›)โˆ’๐‘“(โˆ’๐‘ฅ/๐‘›))๎ƒฉ๎€ท๐‘›2๎€ธโˆ’๐‘›๐‘ข๐‘ก(2๐›ผ)๎ƒช๎ƒช=๐œ๐‘€๎‚€ฮ›๐‘”(๐‘ฅ/๐‘›)โˆ’๐‘“(๐‘ฅ/๐‘›)๎‚€๐‘ข๐‘ก๐›ผ๎‚,ฮ›๐‘”(โˆ’๐‘ฅ/๐‘›)โˆ’๐‘“(โˆ’๐‘ฅ/๐‘›)๎‚€๐‘ข๐‘ก๐›ผ๎‚๎‚โ‰ฅ๐œ๐‘€๎‚€๐‘€๎‚€๐‘ฅ๐‘›,๐‘ก๐›ผ๎‚๎‚€โˆ’๐‘ฅ,๐‘€๐‘›,๐‘ก๐›ผ๎‚๎‚=๐œ๐‘€๎‚ตฮ›๎…ž๐œ‘(๎‚Š๐‘ฅ/๐‘›)๎‚€๐‘ก๐›ผ๎‚,ฮ›๎…ž๐œ‘(๎ƒณโˆ’๐‘ฅ/๐‘›)๎‚€๐‘ก๐›ผ๎‚๎‚ถ=๐œ๐‘€๎‚ตฮ›๎…ž๐›ผ๐œ‘(๎‚Š๐‘ฅ/๐‘›)(๐‘ก),ฮ›๎…ž๐›ผ๐œ‘(๎ƒณโˆ’๐‘ฅ/๐‘›)๎‚ถ(๐‘ก)โ‰ฅ๐œ๐‘€๎‚€ฮ›๎…ž๐œ‘(ฬ‚๐‘ฅ)(๐‘ก),ฮ›๎…ž๐œ‘(๎‚Šโˆ’๐‘ฅ)๎‚(๐‘ก)=๐‘€(๐‘ฅ,๐‘ก)(3.20) for all ๐‘ฅโˆˆ๐‘‹โงต{0}, ๐‘ก>0, and for some ๐›ผ>๐‘›2 satisfying (๐‘–๐‘–), which implies that ๐‘›๐‘‘(๐ฝ๐‘“,๐ฝ๐‘”)โ‰ค2๐›ผ๐‘‘(๐‘“,๐‘”).(3.21) That is, ๐ฝ is a strict contraction with the Lipschitz constant 0<๐‘›2/๐›ผ<1.
Moreover, by (๐‘…2), (3.2), and (๐‘–๐‘–), we see that ฮ›๐‘“(๐‘ฅ)โˆ’๐ฝ๐‘“(๐‘ฅ)๎‚ต๐‘ก๎‚ถ(2๐›ผ)=ฮ›(1/2)๐ท๐‘“(๎‚Š๐‘ฅ/๐‘›)๎‚ต๐‘ก๎‚ถ(2๐›ผ)โ‰ฅฮ›๎…ž๐œ‘(๎‚Š๐‘ฅ/๐‘›)๎‚€๐‘ก๐›ผ๎‚=ฮ›๎…ž๐›ผ๐œ‘(๎‚Š๐‘ฅ/๐‘›)(๐‘ก)โ‰ฅฮ›๎…ž๐œ‘(ฬ‚๐‘ฅ)(๐‘ก)โ‰ฅ๐‘€(๐‘ฅ,๐‘ก)(3.22) for all ๐‘ฅโˆˆ๐‘‹โงต{0} and ๐‘ก>0. This implies that ๐‘‘(๐‘“,๐ฝ๐‘“)โ‰ค1/(2๐›ผ)<โˆž by the definition of ๐‘‘. Therefore, according to Theorem 2.5, the sequence {๐ฝ๐‘š๐‘“} converges to the unique โ€œfixed pointโ€ ๐นโˆถ๐‘‹โ†’๐‘Œ of ๐ฝ in the set ๐‘‡={๐‘”โˆˆ๐‘†โˆฃ๐‘‘(๐‘“,๐‘”)<โˆž} and ๐น is represented by ๐น(๐‘ฅ)=lim๐‘šโ†’โˆž๎‚ต๐‘›๐‘š2๎‚€๐‘“๎‚€๐‘ฅ๐‘›m๎‚๎‚€โˆ’๐‘ฅโˆ’๐‘“๐‘›๐‘š+๐‘›๎‚๎‚2๐‘š2๎‚€๐‘“๎‚€๐‘ฅ๐‘›๐‘š๎‚๎‚€โˆ’๐‘ฅ+๐‘“๐‘›๐‘š๎‚ถ๎‚๎‚(3.23) for all ๐‘ฅโˆˆ๐‘‹. Since 1๐‘‘(๐‘“,๐น)โ‰ค1โˆ’๐‘›21/๐›ผ๐‘‘(๐‘“,๐ฝ๐‘“)โ‰ค2๎€ท๐›ผโˆ’๐‘›2๎€ธ,(3.24) the second inequality in (3.4) holds true.
Next, we will show that ๐น is a quadratic-additive function. As we did in the first part, we obtain the inequality (3.14). In view of the definition of ๐น, the first four terms on the right-hand side of the inequality (3.14) tend to 1 as ๐‘šโ†’โˆž. Furthermore, a long manipulation yields ๐ท๐ฝ๐‘š๐‘“๎€ท๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›๎€ธ=๐‘›2๐‘š2๎‚€๐‘ฅ๐ท๐‘“1๐‘›๐‘š๐‘ฅ,โ€ฆ,๐‘›๐‘›๐‘š๎‚+๐‘›2๐‘š2๎‚€โˆ’๐‘ฅ๐ท๐‘“1๐‘›๐‘š๐‘ฅ,โ€ฆ,โˆ’๐‘›๐‘›๐‘š๎‚+๐‘›๐‘š2๎‚€๐‘ฅ๐ท๐‘“1๐‘›๐‘š๐‘ฅ,โ€ฆ,๐‘›๐‘›๐‘š๎‚โˆ’๐‘›๐‘š2๎‚€โˆ’๐‘ฅ๐ท๐‘“1๐‘›๐‘š๐‘ฅ,โ€ฆ,โˆ’๐‘›๐‘›๐‘š๎‚.(3.25) Thus, it follows from (๐‘…2), (๐‘…3), definition of ๐œ๐‘€, (3.2), and (๐‘–๐‘–) that ฮ›๐ท๐ฝ๐‘š๐‘“(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›)๎‚€๐‘ก5๎‚๎‚†ฮ›โ‰ฅmin(๐‘›2๐‘š/2)๐ท๐‘“(๐‘ฅ1/๐‘›๐‘š,โ€ฆ,๐‘ฅ๐‘›/๐‘›๐‘š)๎‚€๐‘ก๎‚20,ฮ›(๐‘›2๐‘š/2)๐ท๐‘“(โˆ’๐‘ฅ1/๐‘›๐‘š,โ€ฆ,โˆ’๐‘ฅ๐‘›/๐‘›๐‘š)๎‚€๐‘ก๎‚,ฮ›20(๐‘›๐‘š/2)๐ท๐‘“(๐‘ฅ1/๐‘›๐‘š,โ€ฆ,๐‘ฅ๐‘›/๐‘›๐‘š)๎‚€๐‘ก๎‚20,ฮ›โˆ’(๐‘›๐‘š/2)๐ท๐‘“(โˆ’๐‘ฅ1/๐‘›๐‘š,โ€ฆ,โˆ’๐‘ฅ๐‘›/๐‘›๐‘š)๎‚€๐‘ก๎ƒฏฮ›20๎‚๎‚‡โ‰ฅmin๎…ž๐œ‘(๐‘ฅ1/๐‘›๐‘š,โ€ฆ,๐‘ฅ๐‘›/๐‘›๐‘š)๎ƒฉ๐‘ก๎€ท10๐‘›2๐‘š๎€ธ๎ƒช,ฮ›๎…ž๐œ‘(โˆ’๐‘ฅ1/๐‘›๐‘š,โ€ฆ,โˆ’๐‘ฅ๐‘›/๐‘›๐‘š)๎ƒฉ๐‘ก๎€ท10๐‘›2๐‘š๎€ธ๎ƒช,ฮ›๎…ž๐œ‘(๐‘ฅ1/๐‘›๐‘š,โ€ฆ,๐‘ฅ๐‘›/๐‘›๐‘š)๎‚ตt(10๐‘›๐‘š)๎‚ถ,ฮ›๎…ž๐œ‘(โˆ’๐‘ฅ1/๐‘›๐‘š,โ€ฆ,โˆ’๐‘ฅ๐‘›/๐‘›๐‘š)๎‚ต๐‘ก(10๐‘›๐‘š)๎ƒฏฮ›๎‚ถ๎‚ผโ‰ฅmin๎…ž๐›ผโˆ’๐‘š๐œ‘(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›)๎ƒฉ๐‘ก๎€ท10๐‘›2๐‘š๎€ธ๎ƒช,ฮ›๎…ž๐›ผโˆ’๐‘š๐œ‘(โˆ’๐‘ฅ1,โ€ฆ,โˆ’๐‘ฅ๐‘›)๎ƒฉ๐‘ก๎€ท10๐‘›2๐‘š๎€ธ๎ƒช,ฮ›๎…ž๐›ผโˆ’๐‘š๐œ‘(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›)๎‚ต๐‘ก(10๐‘›๐‘š)๎‚ถ,ฮ›๎…ž๐›ผโˆ’๐‘š๐œ‘(โˆ’๐‘ฅ1,โ€ฆ,โˆ’๐‘ฅ๐‘›)๎‚ต๐‘ก(10๐‘›๐‘š)๎ƒฏฮ›๎‚ถ๎‚ผ=min๎…ž๐œ‘(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›)๎ƒฉ๐›ผ๐‘š๐‘ก๎€ท10๐‘›2๐‘š๎€ธ๎ƒช,ฮ›๎…ž๐œ‘(โˆ’๐‘ฅ1,โ€ฆ,โˆ’๐‘ฅ๐‘›)๎ƒฉ๐›ผ๐‘š๐‘ก๎€ท10๐‘›2๐‘š๎€ธ๎ƒช,ฮ›๎…ž๐œ‘(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›)๎‚ต๐›ผ๐‘š๐‘ก(10๐‘›๐‘š)๎‚ถ,ฮ›๎…ž๐œ‘(โˆ’๐‘ฅ1,โ€ฆ,โˆ’๐‘ฅ๐‘›)๎‚ต๐›ผ๐‘š๐‘ก(10๐‘›๐‘š),๎‚ถ๎‚ผ(3.26) which tends to 1 as ๐‘šโ†’โˆž for all ๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹โงต{0} and ๐‘ก>0. Therefore, it follows from (3.14) that ฮ›๐ท๐น(๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›)(๐‘ก)=1(3.27) for any ๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹โงต{0} and ๐‘ก>0. By (๐‘…1), this implies that ๎€ท๐‘ฅ๐ท๐น1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›๎€ธ=0(3.28) for all ๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹โงต{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 ๐‘‹ be a real vector space, (๐‘,ฮ›โ€ฒ,๐œ๐‘€) an RN-space, (๐‘Œ,ฮ›,๐œ๐‘€) a complete RN-space, and let ๐œ‘โˆถ๐‘‹๐‘›โ†’๐‘ be a function. Assume that ๐œ‘ satisfies one of the conditions (๐‘–) and (๐‘–๐‘–) in Theorem 3.1 for all ๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹ and ๐‘ก>0. If a function ๐‘“โˆถ๐‘‹โ†’๐‘Œ satisfies ๐‘“(0)=0 and (3.2) for all ๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹ and ๐‘ก>0, then there exists a unique quadratic-additive function ๐นโˆถ๐‘‹โ†’๐‘Œ satisfying (3.4) for all ๐‘ฅโˆˆ๐‘‹ and ๐‘ก>0.

Now, we obtain general Hyers-Ulam stability results of (1.3) in normed spaces. If ๐‘‹ is a normed space, then (๐‘‹,ฮ›,๐œ๐‘€) is an induced random normed space. We get the following result.

Corollary 3.3. Let ๐‘‹ be a real vector space, ๐‘Œ a complete normed space, and let ๐œ‘โˆถ(๐‘‹โงต{0})๐‘›โ†’[0,โˆž) be a function. Assume that ๐œ‘ satisfies one of the following conditions:(๐‘–๐‘–๐‘–)๐œ‘(๐‘›๐‘ฅ1,โ€ฆ,๐‘›๐‘ฅ๐‘›)โ‰ค๐›ผ๐œ‘(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›) for some 1<๐›ผ<๐‘›;(๐‘–๐‘ฃ)๐œ‘(๐‘›๐‘ฅ1,โ€ฆ,๐‘›๐‘ฅ๐‘›)โ‰ฅ๐›ผ๐œ‘(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›) for some ๐›ผ>๐‘›2 for all ๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹โงต{0}. If a function ๐‘“โˆถ๐‘‹โ†’๐‘Œ satisfies ๐‘“(0)=0 and โ€–โ€–๎€ท๐‘ฅ๐ท๐‘“1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›๎€ธโ€–โ€–๎€ท๐‘ฅโ‰ค๐œ‘1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›๎€ธ(3.29) for all ๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹โงต{0}, then there exists a unique function ๐นโˆถ๐‘‹โ†’๐‘Œ such that ๎€ท๐‘ฅ๐ท๐น1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›๎€ธ=0(3.30) for all ๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹โงต{0} and โ€–โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ๎€ฝ๐œ‘๎€ท๐‘“(๐‘ฅ)โˆ’๐น(๐‘ฅ)โ€–โ‰คmax(ฬ‚๐‘ฅ),๐œ‘๎‚Šโˆ’๐‘ฅ๎€ธ๎€พ2(๐‘›โˆ’๐›ผ)if๐œ‘satis๏ฌes๎€ฝ๎€ท(๐‘–๐‘–๐‘–),max๐œ‘(ฬ‚๐‘ฅ),๐œ‘๎‚Šโˆ’๐‘ฅ๎€ธ๎€พ2๎€ท๐›ผโˆ’๐‘›2๎€ธif๐œ‘satis๏ฌes(๐‘–๐‘ฃ)(3.31) for all ๐‘ฅโˆˆ๐‘‹โงต{0}.

Proof. Let us put ๐‘โˆถ=โ„,ฮ›๐‘ฅ(๐‘ก๐‘ก)โˆถ=๐‘ก+โ€–๐‘ฅโ€–,ฮ›๎…ž๐‘ง(๐‘ก๐‘ก)โˆถ=๐‘ก+|๐‘ง|(3.32) for all ๐‘ฅ,๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹โงต{0}, ๐‘งโˆˆโ„โงต{0}, and ๐‘กโ‰ฅ0. If ๐œ‘ satisfies the condition (๐‘–๐‘–๐‘–) for all ๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹โงต{0} and for some 1<๐›ผ<๐‘›, then ฮ›๎…ž๐›ผ๐œ‘(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›)(๐‘ก๐‘ก)=๎€ท๐‘ฅ๐‘ก+๐›ผ๐œ‘1,โ€ฆ,๐‘ฅ๐‘›๎€ธโ‰ค๐‘ก๎€ท๐‘ก+๐œ‘๐‘›๐‘ฅ1,โ€ฆ,๐‘›๐‘ฅ๐‘›๎€ธ=ฮ›๎…ž๐œ‘(๐‘›๐‘ฅ1,โ€ฆ,๐‘›๐‘ฅ๐‘›)(๐‘ก)(3.33) for all ๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹โงต{0} and ๐‘ก>0, that is, ๐œ‘ satisfies the condition (๐‘–). In a similar way, we can show that if ๐œ‘ satisfies (๐‘–๐‘ฃ), then it satisfies the condition (๐‘–๐‘–).
Moreover, we get ฮ›๐ท๐‘“(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›)(๐‘ก๐‘ก)=โ€–โ€–๎€ท๐‘ฅ๐‘ก+๐ท๐‘“1,โ€ฆ,๐‘ฅ๐‘›๎€ธโ€–โ€–โ‰ฅ๐‘ก๎€ท๐‘ฅ๐‘ก+๐œ‘1,โ€ฆ,๐‘ฅ๐‘›๎€ธ=ฮ›๎…ž๐œ‘(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›)(๐‘ก)(3.34) for all ๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹โงต{0} and ๐‘ก>0, that is, ๐‘“ satisfies the inequality (3.2) for all ๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹โงต{0}.
According to Theorem 3.1, there exists a unique function ๐นโˆถ๐‘‹โ†’๐‘Œ such that๎€ท๐‘ฅ๐ท๐น1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›๎€ธ=0(3.35) for all ๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹โงต{0} and ฮ›๐‘“(๐‘ฅ)โˆ’๐น(๐‘ฅ)โŽงโŽชโŽจโŽชโŽฉ๐œ(๐‘ก)โ‰ฅ๐‘€๎‚€ฮ›๎…ž๐œ‘(ฬ‚๐‘ฅ)(2(๐‘›โˆ’๐›ผ)๐‘ก),ฮ›๎…ž๐œ‘(๎‚Šโˆ’๐‘ฅ)๎‚(2(๐‘›โˆ’๐›ผ)๐‘ก)if๐œ‘satis๏ฌes๐œ(๐‘–๐‘–๐‘–),๐‘€๎‚€ฮ›๎…ž๐œ‘(ฬ‚๐‘ฅ)๎€ท2๎€ท๐›ผโˆ’๐‘›2๎€ธ๐‘ก๎€ธ,ฮ›๎…ž๐œ‘(๎‚Šโˆ’๐‘ฅ)๎€ท2๎€ท๐›ผโˆ’๐‘›2๎€ธ๐‘ก๎€ธ๎‚if๐œ‘satis๏ฌes(๐‘–๐‘ฃ)(3.36) for all ๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹โงต{0} and ๐‘ก>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 ๐‘‹ be a real normed space, ๐‘โˆˆ[0,1)โˆช(2,โˆž), and let ๐‘Œ be a complete normed space. If a function ๐‘“โˆถ๐‘‹โ†’๐‘Œ satisfies ๐‘“(0)=0 and โ€–โ€–๎€ท๐‘ฅ๐ท๐‘“1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›๎€ธโ€–โ€–๎€ทโ€–โ€–๐‘ฅโ‰ค๐œƒ1โ€–โ€–๐‘+โ€–โ€–๐‘ฅ2โ€–โ€–๐‘โ€–โ€–๐‘ฅ+โ‹ฏ+๐‘›โ€–โ€–๐‘๎€ธ(3.37) for all ๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹ and for some ๐œƒโ‰ฅ0, then there exists a unique quadratic-additive function ๐นโˆถ๐‘‹โ†’๐‘Œ such that โ€–โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ๐‘“(๐‘ฅ)โˆ’๐น(๐‘ฅ)โ€–โ‰ค๐‘›๐œƒโ€–๐‘ฅโ€–๐‘2(๐‘›โˆ’๐‘›๐‘)if0โ‰คp<1,๐‘›๐œƒโ€–๐‘ฅโ€–๐‘2๎€ท๐‘›๐‘โˆ’๐‘›2๎€ธifp>2(3.38) for all ๐‘ฅโˆˆ๐‘‹.

Proof. If we put ๐œ‘๎€ท๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›๎€ธ๎€ทโ€–โ€–๐‘ฅโˆถ=๐œƒ1โ€–โ€–๐‘+โ€–โ€–๐‘ฅ2โ€–โ€–๐‘โ€–โ€–๐‘ฅ+โ‹ฏ+๐‘›โ€–โ€–๐‘๎€ธ,(3.39) then the induced random normed space (๐‘‹,ฮ›๐‘ฅ,๐œ๐‘€) satisfies the conditions stated in Theorem 3.2 with ๐›ผ=๐‘›๐‘.

Corollary 3.5. Let ๐‘‹ be a real normed space, ๐‘โˆˆ(โˆ’โˆž,0), and let ๐‘Œ be a complete normed space. If a function ๐‘“โˆถ๐‘‹โ†’๐‘Œ satisfies ๐‘“(0)=0 and โ€–โ€–๎€ท๐‘ฅ๐ท๐‘“1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›๎€ธโ€–โ€–๎“โ‰ค๐œƒ1โ‰ค๐‘–โ‰ค๐‘›,๐‘ฅ๐‘–โ‰ 0โ€–โ€–๐‘ฅ๐‘–โ€–โ€–๐‘(3.40) for all ๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹ and for some ๐œƒโ‰ฅ0, then there exists a unique quadratic-additive function ๐นโˆถ๐‘‹โ†’๐‘Œ satisfying โŽงโŽชโŽจโŽชโŽฉโ€–๐‘“(๐‘ฅ)โˆ’๐น(๐‘ฅ)โ€–โ‰ค๐‘›๐œƒโ€–๐‘ฅโ€–๐‘2(๐‘›โˆ’๐‘›๐‘)ifxโˆˆX0โงต{0},ifx=0.(3.41)

Proof. If we put ๐‘โˆถ=โ„, ๐›ผโˆถ=๐‘›๐‘, and define ฮ›๐‘ฅ(๐‘ก๐‘ก)โˆถ=๐‘ก+โ€–๐‘ฅโ€–,ฮ›๎…ž๐‘ง(๐‘ก๐‘ก)โˆถ=,๐œ‘๎€ท๐‘ฅ๐‘ก+|๐‘ง|1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›๎€ธโˆ‘โˆถ=๐œƒ1โ‰ค๐‘–โ‰ค๐‘›,๐‘ฅ๐‘–โ‰ 0โ€–โ€–๐‘ฅ๐‘–โ€–โ€–๐‘(3.42) for all ๐‘ฅ,๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹ and ๐‘งโˆˆ๐‘, then we have ฮ›๎…ž๐›ผ๐œ‘(๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›)(๐‘ก๐‘ก)=๎€ท๐‘ฅ๐‘ก+๐›ผ๐œ‘1,โ€ฆ,๐‘ฅ๐‘›๎€ธ=๐‘ก๎€ท๐‘ก+๐œ‘๐‘›๐‘ฅ1,โ€ฆ,๐‘›๐‘ฅ๐‘›๎€ธ=ฮ›๎…ž๐œ‘(๐‘›๐‘ฅ1,๐‘›๐‘ฅ2,โ€ฆ,๐‘›๐‘ฅ๐‘›)(๐‘ก),(3.43) that is, ๐œ‘ satisfies condition (๐‘–) given in Theorem 3.1 for all ๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹ and ๐‘ก>0. We moreover get ฮ›๐ท๐‘“(๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›)(๐‘ก๐‘ก)=โ€–โ€–๎€ท๐‘ฅ๐‘ก+๐ท๐‘“1,โ€ฆ,๐‘ฅ๐‘›๎€ธโ€–โ€–โ‰ฅ๐‘กโˆ‘๐‘ก+๐œƒ1โ‰ค๐‘–โ‰ค๐‘›,๐‘ฅ๐‘–โ‰ 0โ€–โ€–๐‘ฅ๐‘–โ€–โ€–๐‘=๐‘ก๎€ท๐‘ฅ๐‘ก+๐œ‘1,โ€ฆ,๐‘ฅ๐‘›๎€ธ=ฮ›๎…ž๐œ‘(๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›)(๐‘ก),(3.44) that is, ๐‘“ satisfies the inequality (3.2) for all ๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆˆ๐‘‹ and ๐‘ก>0.
According to Theorem 3.2, there exists a unique quadratic-additive function ๐นโˆถ๐‘‹โ†’๐‘Œ satisfying ๐‘ก๐‘ก+โ€–๐‘“(๐‘ฅ)โˆ’๐น(๐‘ฅ)โ€–=ฮ›๐‘“(๐‘ฅ)โˆ’๐น(๐‘ฅ)(๐‘ก)โ‰ฅ๐‘€(๐‘ฅ,2(๐‘›โˆ’๐‘›๐‘=โŽงโŽชโŽจโŽชโŽฉ)๐‘ก)2(๐‘›โˆ’๐‘›๐‘)๐‘ก2(๐‘›โˆ’๐‘›๐‘โ€–)๐‘ก+๐‘›๐œƒ๐‘ฅโ€–๐‘if1๐‘ฅโˆˆ๐‘‹โงต{0},if๐‘ฅ=0(3.45) for all ๐‘ก>0, or equivalently โ€–โ€–๐‘“(๐‘ฅ)โˆ’๐น(๐‘ฅ)๐‘กโ‰คโŽงโŽชโŽจโŽชโŽฉ๐‘›๐œƒโ€–๐‘ฅโ€–๐‘2(๐‘›โˆ’๐‘›๐‘)๐‘กif0๐‘ฅโˆˆ๐‘‹โงต{0},if๐‘ฅ=0(3.46) for all ๐‘ก>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).