Abstract

The aim of this paper is to investigate fuzzy Hyers-Ulam-Rassias stability of the general case of quadratic functional equation 𝑓(𝑎𝑥+𝑏𝑦)+𝑓(𝑎𝑥𝑏𝑦)=(𝑎/2)𝑓(𝑥+𝑦)+(𝑎/2)𝑓(𝑥𝑦)+(2𝑎2𝑎)𝑓(𝑥)+(2𝑏2𝑎)𝑓(𝑦), where 𝑎,𝑏1 and fixed integers with 𝑎2𝑏2. These functional equations are equivalent. This has been proven by Ulam, 1964.

1. Introduction and Preliminaries

The stability problem of functional equations was raised by Ulam [1] in 1964. In fact he posed the question “Assume that a function satisfies a functional equation approximately according to some convention. Is it then possible to find near this function a function satisfying the equation accurately?” In 1941 Hyers gave a significant partial solution to this problem in his paper [2].

Hyers’ result was generalized by Aoki [3] for additive mappings. In 1978, Rassias and Song [4] generalized Hyers’ result, a fact which rekindled interest in the field. Such type of stability is now called the Ulam-Hyers-Rassias stability of functional equations.We refer the curious readers for further information on such problems to, for example, [57].

The functional equation 𝑓(𝑥+𝑦)+𝑓(𝑥𝑦)=2𝑓(𝑥)+2𝑓(𝑦)(1.1) is said to be a simple quadratic functional equation. The first person that investigated the stability of the simple quadratic equation was Skof [8]. He proved that, if 𝑓 is a mapping from a normed space 𝑋 into a Banach space 𝑌 satisfying 𝑓(𝑥+𝑦)+𝑓(𝑥𝑦)2𝑓(𝑥)2𝑓(𝑦)𝜖forsome𝜖>0,(1.2) then there is a unique simple quadratic function 𝑔𝑋𝑌 such that 𝜖𝑓(𝑥)𝑔(𝑥)2.(1.3) In 1984, Katsaras [9] defined a fuzzy norm on a linear space to construct a fuzzy vector topological structure on the space. Later, some mathematicians have defined fuzzy norms on a linear space from various points of view [10, 11]. In particular, in 2003, Bag and Samanta [12], following Cheng and Mordeson [13], gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [14]. They also established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces.

Recently, considerable attention has been increasing to the problem of fuzzy stability of functional equations. Several fuzzy stability results concerning Cauchy, Jensen, simple quadratic, and cubic functional equations have been investigated [1518].

Definition 1.1. Let 𝑋 be a real vector space. A function 𝑁𝑋×𝑅[0,1] is called fuzzy normed on 𝑋 if for all 𝑥,𝑦𝑋 and all 𝑠,𝑡𝑅(N1)𝑁(𝑥,𝑡)=0 for 𝑡0,(N2)𝑥=0 if and only if 𝑁(𝑥,𝑡)=1 for all 𝑡>0,(N3)𝑁(𝑐𝑥,𝑡)=𝑁(𝑥,𝑡/|𝑐|) if 𝑐0,(N4)𝑁(𝑥+𝑦,𝑡+𝑠)min{𝑁(𝑥,𝑡),𝑁(𝑦,𝑠)}, (N5)𝑁(𝑥,.) is a nondecreasing function of 𝑅 and lim𝑡𝑁(𝑥,𝑡)=1,(N6)for 𝑥0,𝑁(𝑥,.) is continuous on 𝑅,the pair (𝑋,𝑁) is called a fuzzy normed vector space.

Example 1.2. Let (𝑋,) be a normed linear space. One can easily verify that, for each 𝑘>0, 𝑁𝑘𝑡(𝑥,𝑡)=𝑡+𝑘𝑥,if𝑡>0,0,if𝑡0,(1.4) defines a fuzzy norm on 𝑋.

Definition 1.3. Let (𝑋,𝑁) be a fuzzy normed vector space. A sequence {𝑥𝑛} in 𝑋 is said to be convergent or converges if there exists an 𝑥𝑋 such that lim𝑛𝑁(𝑥𝑛𝑥,𝑡)=1 for all 𝑡>0. In this case, 𝑥 is called the limit of the sequence {𝑥𝑛}, and one denotes it by 𝑁lim𝑛𝑁𝑥𝑛𝑥,𝑡=𝑥.(1.5)

Definition 1.4. Let (𝑋,𝑁) be a fuzzy normed vector space. A sequence {𝑥𝑛} in 𝑋 is said to be Cauchy if for each 𝜖>0 and each 𝛿>0 there exists an 𝑛0𝑁 such that 𝑁𝑥𝑚𝑥𝑛,𝛿>1𝜖𝑚,𝑛𝑛0.(1.6)
It is well known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.

2. Main Results

Let 𝑋 be a linear space and (𝑍,𝑁) a fuzzy normed space. Let (𝑌,𝑁) be a fuzzy Banach space and 𝑓𝑋𝑌 a function satisfying 𝑁𝑎𝑓(𝑎𝑥+𝑏𝑦)+𝑓(𝑎𝑥𝑏𝑦)2𝑎𝑓(𝑥+𝑦)2𝑓(𝑥𝑦)2𝑎2𝑎𝑓(𝑥)2𝑏2𝑎𝑓(𝑦),𝑡+𝑠min{𝑁(𝜑(𝑥),𝑡𝑞),𝑁(𝜑(𝑦),𝑠𝑞)},(2.1) where 𝑎,𝑏1,𝑎2𝑏2 and for all 𝑡,𝑠>0,𝑞>1/2, such that 𝜑𝑋𝑍 is a function, and 𝜑(𝑎𝑥)=𝛼𝜑(𝑥),𝑥𝑋,(2.2) for some number 𝛼 with 0<|𝛼|<𝑎. Then, there exists a unique quadratic functional equation 𝑄𝑋𝑌 such that 𝑁𝑎(𝑄(𝑥)𝑓(𝑥),𝑡)𝑁𝜑(𝑥),2𝛼𝑝𝑡2𝑞.(2.3)

Proof. Putting 𝑦=0 and 𝑡=𝑠 in (2.1), we have that 𝑁2𝑓(𝑎𝑥)2𝑎2𝑓(𝑥),2𝑡𝑁(𝜑(𝑥),𝑡𝑞)(𝑥𝑋,𝑡>0).(2.4) Therefore 𝑁1𝑎2𝑡𝑓(𝑎𝑥)𝑓(𝑥),𝑎2𝑁(𝜑(𝑥),𝑡𝑞)(𝑥𝑋,𝑡>0),(2.5) and 𝑁1𝑎2𝑡𝑓(𝑎𝑥)𝑓(𝑥),𝑝𝑎2𝑁(𝜑(𝑥),𝑡).(2.6) Now, replacing 𝑥=𝑎𝑥 in (2.6), 𝑁1𝑎4𝑓𝑎2𝑥1𝑎2𝑡𝑓(𝑎𝑥),𝑝𝑎4𝑁(𝜑(𝑎𝑥),𝑡),(2.7) and then by the assumption that 𝜑(𝑎𝑥)=𝛼𝜑(𝑥) and property (𝑁3) of Definition 1.1 we obtain that 𝑁1𝑎4𝑓𝑎2𝑥1𝑎2𝑓(𝑎𝑥),(𝛼𝑡)𝑝𝑎4𝑁(𝜑(𝑥),𝑡).(2.8) By comparing (2.6) and (2.8) and using property (𝑁4) we obtain that 𝑁1𝑎4𝑓𝑎2𝑥𝑡𝑓(𝑥),𝑝𝑎2+(𝛼𝑡)𝑝𝑎4𝑁(𝜑(𝑥),𝑡)(𝑥𝑋,𝑡>0).(2.9) Again, by replacing 𝑥=𝑎𝑥, in (2.9), 𝑁1𝑎6𝑓𝑎3𝑥1𝑎2𝑡𝑓(𝑎𝑥),𝑝𝑎4+(𝛼𝑡)𝑝𝑎6𝑁(𝜑(𝑎𝑥),𝑡).(2.10) Thus 𝑁1𝑎6𝑓𝑎3𝑥1𝑎2𝑓(𝑥),(𝛼𝑡)𝑝𝑎4+𝛼2𝑝𝑡𝑝𝑎6𝑁(𝜑(𝑥),𝑡).(2.11) By comparing (2.6), and (2.11) we obtain that 𝑁1𝑎6𝑓𝑎3𝑥𝑡𝑓(𝑥),𝑝𝑎2+(𝛼𝑡)𝑝𝑎4+𝛼2𝑝𝑡𝑝𝑎6𝑁(𝜑(𝑥),𝑡)(𝑥𝑋,𝑡>0).(2.12) With following this process we obtain that 𝑁1𝑎2𝑛𝑓(𝑎𝑛𝑥)𝑓(𝑥),𝑛𝑘=1𝛼(𝑘1)𝑝𝑎2𝑘𝑡𝑝𝑁(𝜑(𝑥),𝑡).(2.13) If 𝑚𝑁, 𝑛>𝑚>0, then 𝑛𝑚𝑁. Replacing 𝑛 by 𝑛𝑚 in (2.13) gives 𝑁𝑎2𝑚2𝑛𝑓(𝑎𝑛𝑚𝑥)𝑓(𝑥),𝑛𝑚𝑘=1𝛼(𝑘1)𝑝𝑎2𝑘𝑡𝑝𝑁(𝜑(𝑥),𝑡)(𝑥𝑋,𝑡>0,𝑛𝑁).(2.14) By replacing 𝑥=𝑎𝑚𝑥 in (2.14) we obtain that 𝑁𝑎2𝑛𝑓(𝑎𝑛𝑥)𝑎2𝑚𝑓(𝑎𝑚𝑥),𝑎2𝑚𝑛𝑚𝑘=1𝛼(𝑘1)𝑝𝑎2𝑘𝑡𝑝𝑁(𝜑(𝑎𝑚𝑥),𝑡).(2.15) Thus 𝑁𝑎2𝑛𝑓(𝑎𝑛𝑥)𝑎2𝑚𝑓(𝑎𝑚𝑥),𝑎2𝑚𝑛𝑚𝑘=1𝛼(𝑘1)𝑝𝑎2𝑘𝛼𝑚𝑝𝑡𝑝𝑁(𝜑(𝑥),𝑡).(2.16) It follows that 𝑁𝑎2𝑛𝑓(𝑎𝑛𝑥)𝑎2𝑚𝑓(𝑎𝑚𝑥),𝑛𝑘=𝑚+1𝛼(𝑘1)𝑝𝑎2𝑘𝑡𝑝𝑁(𝜑(𝑥),𝑡)(𝑥𝑋,𝑡>0).(2.17) Let 𝑐>0, and let 𝜖 be given. Since lim𝑡𝑁(𝜑(𝑥),𝑡)=1, there is some 𝑡0>0 such that 𝑁𝜑(𝑥),𝑡01𝜖.(2.18) Fix some 𝑡>𝑡0. The convergence of series 𝑘=1(𝛼𝑝/𝑎2)𝑘𝛼𝑝𝑡𝑝 guarantees that there exists some 𝑛00 such that, for each 𝑛>𝑚>𝑛0, the inequality 𝑛𝑘=𝑚+1(𝛼𝑝/𝑎2)𝑘𝛼𝑝𝑡𝑝<𝑐 holds. It follows that 𝑁𝑎2𝑛𝑓(𝑎𝑛𝑥)𝑎2𝑚𝑓(𝑎𝑚𝑥),𝑐𝑁𝑎2𝑛𝑓(𝑎𝑛𝑥)𝑎2𝑚𝑓(𝑎𝑚𝑥),𝑛𝑘=𝑚+1𝛼𝑝𝑎2𝑘𝛼𝑝𝑡𝑝0𝑁𝜑(𝑥),𝑡01𝜖.(2.19) Hence {𝑓(𝑎𝑛𝑥)/𝑎2𝑛} is a Cauchy sequence in fuzzy Banach space (𝑌,𝑁), and thus this sequence converges to some 𝑄(𝑥)𝑌. It means that 𝑄(𝑥)=𝑁lim𝑛𝑓(𝑎𝑛𝑥)𝑎2𝑛.(2.20) Furthermore by putting 𝑚=0 in (2.17), 𝑁𝑎2𝑛𝑓(𝑎𝑛𝑥)𝑓(𝑥),𝑛𝑘=1𝛼𝑝𝑎2𝑘𝛼𝑝𝑡𝑝𝑁𝑁(𝜑(𝑥),𝑡),(2.21)𝑎2𝑛𝑓(𝑎𝑛𝑡𝑥)𝑓(𝑥),𝑡𝑁𝜑(𝑥),𝑞𝑛𝑘=1𝛼𝑝/𝑎2𝑘𝛼𝑝𝑞(𝑥𝑋,𝑡>0).(2.22) Next we will show that 𝑄 is quadratic. Let 𝑥,𝑦𝑋, and then we have that 𝑁𝑎𝑄(𝑎𝑥+𝑏𝑦)+𝑄(𝑎𝑥𝑏𝑦)2𝑎𝑄(𝑥+𝑦)2(𝑥𝑦)2𝑎2𝑎𝑄(𝑥)2𝑏2𝑏𝑄(𝑦),𝑡𝑁𝑎𝑄(𝑎𝑥+𝑏𝑦)+𝑄(𝑎𝑥𝑏𝑦)2𝑎𝑄(𝑥+𝑦)2(𝑥𝑦)2𝑎2𝑎𝑄(𝑥)2𝑏2𝑁𝑏𝑄(𝑦),𝑡min𝑓𝑄(𝑎𝑥+𝑏𝑦)(𝑎𝑛(𝑎𝑥+𝑏𝑦))𝑎2𝑛,𝑡7,𝑁𝑄(𝑎𝑥𝑏𝑦)𝑓(𝑎𝑛(𝑎𝑥𝑏𝑦))𝑎2𝑛,𝑡7,𝑁𝑎2𝑓(𝑎𝑛(𝑥+𝑦))𝑎2𝑛𝑎2𝑄𝑡(𝑥+𝑦),7,𝑁𝑎2𝑓(𝑎𝑛(𝑥𝑦))𝑎2𝑛𝑎2𝑡𝑄(𝑥𝑦),7,𝑁2𝑎2𝑎𝑓(𝑎𝑛(𝑥))𝑎2𝑛2𝑎2𝑡𝑎𝑄(𝑥),7,𝑁2𝑏2𝑎𝑓(𝑎𝑛(𝑦))𝑎2𝑛2𝑏2𝑡𝑎𝑄(𝑦),7,𝑁𝑓(𝑎𝑛(𝑎𝑥+𝑏𝑦))𝑎2𝑛+𝑓(𝑎𝑛(𝑎𝑥𝑏𝑦))𝑎2𝑛𝑎2𝑓(𝑎𝑛(𝑥+𝑦))𝑎2𝑛𝑎2𝑓(𝑎𝑛(𝑥𝑦))𝑎2𝑛2𝑎2𝑎𝑓(𝑎𝑛(𝑥))𝑎2𝑛2𝑏2𝑎𝑓(𝑎𝑛(𝑦))𝑎2𝑛,𝑡7.(2.23) The first six terms on the right-hand side of the above inequality tend to 1 as 𝑛, and the seventh term, by (2.1), is greater than or equal to 𝑁𝑎min𝜑(𝑥),2𝑞𝛼𝑛𝑡14𝑞𝑎,𝑁𝜑(𝑦),2𝑞𝛼𝑛𝑡14𝑞,(2.24) which tends to 1 as 𝑛. Therefore 𝑁𝑎𝑄(𝑎𝑥+𝑏𝑦)+𝑄(𝑎𝑥𝑏𝑦)2𝑎𝑄(𝑥+𝑦)2𝑄(𝑥𝑦)2𝑎2𝑎𝑄(𝑥)2𝑏2𝑎𝑄(𝑦),𝑡=1(2.25) for each 𝑥,𝑦𝑋 and 𝑡>0. So by property (𝑁2), we have that 𝑄𝑎(𝑎𝑥+𝑏𝑦)+𝑄(𝑎𝑥𝑏𝑦)2𝑄𝑎(𝑥+𝑦)2𝑄(𝑥𝑦)2𝑎2𝑄𝑎(𝑥)2𝑏2𝑎𝑄(𝑦)=0,𝑥,𝑦𝑋.(2.26) Therefore 𝑄 is quadratic function. For every 𝑥𝑋 and 𝑡,𝑠>0, by (2.22), for large enough 𝑛, we have that 𝑁𝑁(𝑄(𝑥)𝑓(𝑥),𝑡)min𝑄(𝑥)𝑓(𝑎𝑛𝑥)𝑎2𝑛,𝑡2,𝑁𝑓(𝑎𝑛𝑥)𝑎2𝑛𝑡𝑓(𝑥),2𝑁𝜑(𝑥),(𝑡/2)𝑞𝑛𝑘=1𝛼𝑝/𝑎2𝑘𝛼𝑝𝑞𝑎=𝑁𝜑(𝑥),2𝛼𝑝𝑡2𝑞.(2.27) Let 𝑄 be another quadratic function from 𝑋 to 𝑌 which satisfies (2.3). Since, for each 𝑛𝑁, 𝑄(𝑎𝑛𝑥)=𝑎2𝑛𝑄(𝑥),𝑄(𝑎𝑛𝑥)=𝑎2𝑛𝑄(𝑥),(2.28) We have that 𝑁𝑄(𝑥)𝑄(𝑥),𝑡=𝑁𝑄(𝑎𝑛𝑥)𝑄(𝑎𝑛𝑥),𝑎2𝑛𝑡𝑁min𝑄(𝑎𝑛𝑥)𝑓(𝑎𝑛𝑎𝑥),2𝑛𝑡2,𝑁𝑓(𝑎𝑛𝑥)𝑄(𝑎𝑛𝑎𝑥),2𝑛𝑡2𝑎𝑁𝜑(𝑥),2𝛼𝑝2𝑞𝑎2𝑛𝑞𝛼𝑛𝑡𝑞2𝑞(2.29) for each 𝑛𝑁. Due to 𝑞>1/2, lim𝑛𝑁𝑎𝜑(𝑥),2𝛼𝑝2𝑞𝑎2𝑛𝑞𝛼𝑛𝑡𝑞2𝑞=1(2.30) for each 𝑥𝑋 and 𝑡>0. Therefore 𝑄=𝑄.