International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 503164 | 8 pages | https://doi.org/10.5402/2011/503164

Fuzzy Stability of Quadratic Functional Equations in General Cases

Academic Editor: W. Sun
Received21 Feb 2011
Accepted17 Apr 2011
Published17 Jul 2011

Abstract

The aim of this paper is to investigate fuzzy Hyers-Ulam-Rassias stability of the general case of quadratic functional equation 𝑓(ğ‘Žğ‘¥+𝑏𝑦)+𝑓(ğ‘Žğ‘¥âˆ’ğ‘ğ‘¦)=(ğ‘Ž/2)𝑓(𝑥+𝑦)+(ğ‘Ž/2)𝑓(𝑥−𝑦)+(2ğ‘Ž2âˆ’ğ‘Ž)𝑓(𝑥)+(2𝑏2âˆ’ğ‘Ž)𝑓(𝑦), where ğ‘Ž,𝑏≥1 and fixed integers with ğ‘Žâ‰ 2𝑏2. 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, [5–7].

The functional equation 𝑓(𝑥+𝑦)+𝑓(𝑥−𝑦)=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(𝑥−𝑦)−2𝑓(𝑥)−2𝑓(𝑦)‖≤𝜖forsome𝜖>0,(1.2) then there is a unique simple quadratic function 𝑔∶𝑋→𝑌 such that 𝜖‖𝑓(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 [15–18].

Definition 1.1. Let 𝑋 be a real vector space. A function 𝑁∶𝑋×𝑅→[0,1] is called fuzzy normed on 𝑋 if for all 𝑥,𝑦∈𝑋 and all 𝑠,𝑡∈𝑅(N1)𝑁(𝑥,𝑡)=0 for 𝑡≤0,(N2)𝑥=0 if and only if 𝑁(𝑥,𝑡)=1 for all 𝑡>0,(N3)𝑁(𝑐𝑥,𝑡)=𝑁(𝑥,𝑡/|𝑐|) if 𝑐≠0,(N4)𝑁(𝑥+𝑦,𝑡+𝑠)≥min{𝑁(𝑥,𝑡),𝑁(𝑦,𝑠)}, (N5)𝑁(𝑥,.) is a nondecreasing function of 𝑅 and limğ‘¡â†’âˆžğ‘(𝑥,𝑡)=1,(N6)for 𝑥≠0,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, ğ‘ğ‘˜âŽ§âŽªâŽ¨âŽªâŽ©ğ‘¡(𝑥,𝑡)=𝑡+𝑘‖𝑥‖,if𝑡>0,0,if𝑡≤0,(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 𝑥∈𝑋 such that limğ‘›â†’âˆžğ‘(𝑥𝑛−𝑥,𝑡)=1 for all 𝑡>0. In this case, 𝑥 is called the limit of the sequence {𝑥𝑛}, and one denotes it by 𝑁−limğ‘›â†’âˆžğ‘î€·ğ‘¥ğ‘›î€¸âˆ’ğ‘¥,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 𝑛0∈𝑁 such that 𝑁𝑥𝑚−𝑥𝑛,𝛿>1−𝜖𝑚,𝑛≥𝑛0.(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,ğ‘î…ž) be a fuzzy Banach space and 𝑓∶𝑋→𝑌 a function satisfying ğ‘î…žî‚€ğ‘Žğ‘“(ğ‘Žğ‘¥+𝑏𝑦)+𝑓(ğ‘Žğ‘¥âˆ’ğ‘ğ‘¦)−2ğ‘Žğ‘“(𝑥+𝑦)−2𝑓(𝑥−𝑦)−2ğ‘Ž2î€¸âˆ’î€·âˆ’ğ‘Žğ‘“(𝑥)2𝑏2î€¸î‚âˆ’ğ‘Žğ‘“(𝑦),𝑡+𝑠≥min{𝑁(𝜑(𝑥),ğ‘¡ğ‘ž),𝑁(𝜑(𝑦),ğ‘ ğ‘ž)},(2.1) where ğ‘Ž,𝑏≥1,ğ‘Žâ‰ 2𝑏2 and for all 𝑡,𝑠>0,ğ‘ž>1/2, such that 𝜑∶𝑋→𝑍 is a function, and 𝜑(ğ‘Žğ‘¥)=𝛼𝜑(𝑥),∀𝑥∈𝑋,(2.2) for some number 𝛼 with 0<|𝛼|<ğ‘Ž. Then, there exists a unique quadratic functional equation 𝑄∶𝑋→𝑌 such that ğ‘î…žîƒ©îƒ©î€·ğ‘Ž(𝑄(𝑥)−𝑓(𝑥),𝑡)≥𝑁𝜑(𝑥),2−𝛼𝑝𝑡2îƒªğ‘žîƒª.(2.3)

Proof. Putting 𝑦=0 and 𝑡=𝑠 in (2.1), we have that ğ‘î…žî€·2𝑓(ğ‘Žğ‘¥)−2ğ‘Ž2𝑓(𝑥),2𝑡≥𝑁(𝜑(𝑥),ğ‘¡ğ‘ž)(∀𝑥∈𝑋,𝑡>0).(2.4) Therefore ğ‘î…žî‚€1ğ‘Ž2𝑡𝑓(ğ‘Žğ‘¥)−𝑓(𝑥),ğ‘Ž2≥𝑁(𝜑(𝑥),ğ‘¡ğ‘ž)(∀𝑥∈𝑋,𝑡>0),(2.5) and ğ‘î…žî‚µ1ğ‘Ž2𝑡𝑓(ğ‘Žğ‘¥)−𝑓(𝑥),ğ‘ğ‘Ž2≥𝑁(𝜑(𝑥),𝑡).(2.6) Now, replacing 𝑥=ğ‘Žğ‘¥ in (2.6), ğ‘î…žî‚µ1ğ‘Ž4ğ‘“î€·ğ‘Ž2𝑥−1ğ‘Ž2𝑡𝑓(ğ‘Žğ‘¥),ğ‘ğ‘Ž4≥𝑁(𝜑(ğ‘Žğ‘¥),𝑡),(2.7) and then by the assumption that 𝜑(ğ‘Žğ‘¥)=𝛼𝜑(𝑥) and property (𝑁3) of Definition 1.1 we obtain that ğ‘î…žî‚µ1ğ‘Ž4ğ‘“î€·ğ‘Ž2𝑥−1ğ‘Ž2𝑓(ğ‘Žğ‘¥),(𝛼𝑡)ğ‘ğ‘Ž4≥𝑁(𝜑(𝑥),𝑡).(2.8) By comparing (2.6) and (2.8) and using property (𝑁4) we obtain that ğ‘î…žî‚µ1ğ‘Ž4ğ‘“î€·ğ‘Ž2𝑥𝑡−𝑓(𝑥),ğ‘ğ‘Ž2+(𝛼𝑡)ğ‘ğ‘Ž4≥𝑁(𝜑(𝑥),𝑡)(∀𝑥∈𝑋,𝑡>0).(2.9) Again, by replacing 𝑥=ğ‘Žğ‘¥, in (2.9), ğ‘î…žî‚µ1ğ‘Ž6ğ‘“î€·ğ‘Ž3𝑥−1ğ‘Ž2𝑡𝑓(ğ‘Žğ‘¥),ğ‘ğ‘Ž4+(𝛼𝑡)ğ‘ğ‘Ž6≥𝑁(𝜑(ğ‘Žğ‘¥),𝑡).(2.10) Thus ğ‘î…žî‚µ1ğ‘Ž6ğ‘“î€·ğ‘Ž3𝑥−1ğ‘Ž2𝑓(𝑥),(𝛼𝑡)ğ‘ğ‘Ž4+𝛼2ğ‘ğ‘¡ğ‘ğ‘Ž6≥𝑁(𝜑(𝑥),𝑡).(2.11) By comparing (2.6), and (2.11) we obtain that ğ‘î…žî‚µ1ğ‘Ž6ğ‘“î€·ğ‘Ž3𝑥𝑡−𝑓(𝑥),ğ‘ğ‘Ž2+(𝛼𝑡)ğ‘ğ‘Ž4+𝛼2ğ‘ğ‘¡ğ‘ğ‘Ž6≥𝑁(𝜑(𝑥),𝑡)(∀𝑥∈𝑋,𝑡>0).(2.12) With following this process we obtain that ğ‘î…žîƒ©1ğ‘Ž2𝑛𝑓(ğ‘Žğ‘›ğ‘¥)−𝑓(𝑥),𝑛𝑘=1𝛼(𝑘−1)ğ‘ğ‘Žâˆ’2𝑘𝑡𝑝≥𝑁(𝜑(𝑥),𝑡).(2.13) If 𝑚∈𝑁, 𝑛>𝑚>0, then 𝑛−𝑚∈𝑁. Replacing 𝑛 by 𝑛−𝑚 in (2.13) gives ğ‘î…žîƒ©ğ‘Ž2𝑚−2𝑛𝑓(ğ‘Žğ‘›âˆ’ğ‘šğ‘¥)−𝑓(𝑥),𝑛−𝑚𝑘=1𝛼(𝑘−1)ğ‘ğ‘Žâˆ’2𝑘𝑡𝑝≥𝑁(𝜑(𝑥),𝑡)(∀𝑥∈𝑋,𝑡>0,𝑛∈𝑁).(2.14) By replacing 𝑥=ğ‘Žğ‘šğ‘¥ in (2.14) we obtain that ğ‘î…žîƒ©ğ‘Žâˆ’2𝑛𝑓(ğ‘Žğ‘›ğ‘¥)âˆ’ğ‘Žâˆ’2𝑚𝑓(ğ‘Žğ‘šğ‘¥),ğ‘Žâˆ’2𝑚𝑛−𝑚𝑘=1𝛼(𝑘−1)ğ‘ğ‘Žâˆ’2𝑘𝑡𝑝≥𝑁(𝜑(ğ‘Žğ‘šğ‘¥),𝑡).(2.15) Thus ğ‘î…žîƒ©ğ‘Žâˆ’2𝑛𝑓(ğ‘Žğ‘›ğ‘¥)âˆ’ğ‘Žâˆ’2𝑚𝑓(ğ‘Žğ‘šğ‘¥),ğ‘Žâˆ’2𝑚𝑛−𝑚𝑘=1𝛼(𝑘−1)ğ‘ğ‘Žâˆ’2𝑘𝛼𝑚𝑝𝑡𝑝≥𝑁(𝜑(𝑥),𝑡).(2.16) It follows that ğ‘î…žîƒ©ğ‘Žâˆ’2𝑛𝑓(ğ‘Žğ‘›ğ‘¥)âˆ’ğ‘Žâˆ’2𝑚𝑓(ğ‘Žğ‘šğ‘¥),𝑛𝑘=𝑚+1𝛼(𝑘−1)ğ‘ğ‘Žâˆ’2𝑘𝑡𝑝≥𝑁(𝜑(𝑥),𝑡)(∀𝑥∈𝑋,𝑡>0).(2.17) Let 𝑐>0, and let 𝜖 be given. Since limğ‘¡â†’âˆžğ‘(𝜑(𝑥),𝑡)=1, there is some 𝑡0>0 such that 𝑁𝜑(𝑥),𝑡0≥1−𝜖.(2.18) Fix some 𝑡>𝑡0. The convergence of series âˆ‘âˆžğ‘˜=1(𝛼𝑝/ğ‘Ž2)𝑘𝛼−𝑝𝑡𝑝 guarantees that there exists some 𝑛0≥0 such that, for each 𝑛>𝑚>𝑛0, the inequality ∑𝑛𝑘=𝑚+1(𝛼𝑝/ğ‘Ž2)𝑘𝛼−𝑝𝑡𝑝<𝑐 holds. It follows that ğ‘î…žî€·ğ‘Žâˆ’2𝑛𝑓(ğ‘Žğ‘›ğ‘¥)âˆ’ğ‘Žâˆ’2𝑚𝑓(ğ‘Žğ‘šî€¸ğ‘¥),ğ‘â‰¥ğ‘î…žîƒ©ğ‘Žâˆ’2𝑛𝑓(ğ‘Žğ‘›ğ‘¥)âˆ’ğ‘Žâˆ’2𝑚𝑓(ğ‘Žğ‘šğ‘¥),𝑛𝑘=𝑚+1î‚µğ›¼ğ‘ğ‘Ž2𝑘𝛼−𝑝𝑡𝑝0≥𝑁𝜑(𝑥),𝑡0≥1−𝜖.(2.19) Hence {𝑓(ğ‘Žğ‘›ğ‘¥)/ğ‘Ž2𝑛} is a Cauchy sequence in fuzzy Banach space (𝑌,ğ‘î…ž), and thus this sequence converges to some 𝑄(𝑥)∈𝑌. It means that 𝑄(𝑥)=ğ‘î…žâˆ’limğ‘›â†’âˆžğ‘“(ğ‘Žğ‘›ğ‘¥)ğ‘Ž2𝑛.(2.20) Furthermore by putting 𝑚=0 in (2.17), ğ‘î…žîƒ©ğ‘Žâˆ’2𝑛𝑓(ğ‘Žğ‘›ğ‘¥)−𝑓(𝑥),𝑛𝑘=1î‚µğ›¼ğ‘ğ‘Ž2𝑘𝛼−𝑝𝑡𝑝𝑁≥𝑁(𝜑(𝑥),𝑡),(2.21)î…žî€·ğ‘Žâˆ’2𝑛𝑓(ğ‘Žğ‘›î€¸âŽ›âŽœâŽœâŽğ‘¡ğ‘¥)−𝑓(𝑥),𝑡≥𝑁𝜑(𝑥),ğ‘žî‚€âˆ‘ğ‘›ğ‘˜=1𝛼𝑝/ğ‘Ž2î€¸ğ‘˜ğ›¼âˆ’ğ‘î‚ğ‘žâŽžâŽŸâŽŸâŽ (∀𝑥∈𝑋,𝑡>0).(2.22) Next we will show that 𝑄 is quadratic. Let 𝑥,𝑦∈𝑋, and then we have that ğ‘î…žî‚€ğ‘Žğ‘„(ğ‘Žğ‘¥+𝑏𝑦)+𝑄(ğ‘Žğ‘¥âˆ’ğ‘ğ‘¦)−2ğ‘Žğ‘„(𝑥+𝑦)−2(𝑥−𝑦)−2ğ‘Ž2î€¸î€·âˆ’ğ‘Žğ‘„(𝑥)−2𝑏2−𝑏𝑄(𝑦),ğ‘¡â‰¥ğ‘î…žî‚€ğ‘Žğ‘„(ğ‘Žğ‘¥+𝑏𝑦)+𝑄(ğ‘Žğ‘¥âˆ’ğ‘ğ‘¦)−2ğ‘Žğ‘„(𝑥+𝑦)−2−(𝑥−𝑦)2ğ‘Ž2î€¸î€·âˆ’ğ‘Žğ‘„(𝑥)−2𝑏2𝑁−𝑏𝑄(𝑦),𝑡≥minî…žî‚µğ‘“ğ‘„(ğ‘Žğ‘¥+𝑏𝑦)−(ğ‘Žğ‘›(ğ‘Žğ‘¥+𝑏𝑦))ğ‘Ž2𝑛,𝑡7,ğ‘î…žî‚µğ‘„(ğ‘Žğ‘¥âˆ’ğ‘ğ‘¦)−𝑓(ğ‘Žğ‘›(ğ‘Žğ‘¥âˆ’ğ‘ğ‘¦))ğ‘Ž2𝑛,𝑡7,ğ‘î…žî‚µğ‘Ž2𝑓(ğ‘Žğ‘›(𝑥+𝑦))ğ‘Ž2ğ‘›âˆ’ğ‘Ž2𝑄𝑡(𝑥+𝑦),7,ğ‘î…žî‚µğ‘Ž2𝑓(ğ‘Žğ‘›(𝑥−𝑦))ğ‘Ž2ğ‘›âˆ’ğ‘Ž2𝑡𝑄(𝑥−𝑦),7,ğ‘î…žî‚µî€·2ğ‘Ž2î€¸âˆ’ğ‘Žğ‘“(ğ‘Žğ‘›(𝑥))ğ‘Ž2𝑛−2ğ‘Ž2î€¸ğ‘¡âˆ’ğ‘Žğ‘„(𝑥),7,ğ‘î…žî‚µî€·2𝑏2î€¸âˆ’ğ‘Žğ‘“(ğ‘Žğ‘›(𝑦))ğ‘Ž2𝑛−2𝑏2î€¸ğ‘¡âˆ’ğ‘Žğ‘„(𝑦),7,ğ‘î…žî‚µğ‘“(ğ‘Žğ‘›(ğ‘Žğ‘¥+𝑏𝑦))ğ‘Ž2𝑛+𝑓(ğ‘Žğ‘›(ğ‘Žğ‘¥âˆ’ğ‘ğ‘¦))ğ‘Ž2ğ‘›âˆ’ğ‘Ž2𝑓(ğ‘Žğ‘›(𝑥+𝑦))ğ‘Ž2ğ‘›âˆ’ğ‘Ž2𝑓(ğ‘Žğ‘›(𝑥−𝑦))ğ‘Ž2𝑛−2ğ‘Ž2î€¸âˆ’ğ‘Žğ‘“(ğ‘Žğ‘›(𝑥))ğ‘Ž2𝑛−2𝑏2î€¸âˆ’ğ‘Žğ‘“(ğ‘Žğ‘›(𝑦))ğ‘Ž2𝑛,𝑡7.(2.23) The first six terms on the right-hand side of the above inequality tend to 1 as ğ‘›â†’âˆž, and the seventh term, by (2.1), is greater than or equal to î‚»ğ‘î‚µî‚µğ‘Žmin𝜑(𝑥),2ğ‘žğ›¼î‚¶ğ‘›î‚€ğ‘¡î‚14ğ‘žî‚¶î‚µî‚µğ‘Ž,𝑁𝜑(𝑦),2ğ‘žğ›¼î‚¶ğ‘›î‚€ğ‘¡î‚14ğ‘žî‚¶î‚¼,(2.24) which tends to 1 as ğ‘›â†’âˆž. Therefore ğ‘î…žî‚€ğ‘Žğ‘„(ğ‘Žğ‘¥+𝑏𝑦)+𝑄(ğ‘Žğ‘¥âˆ’ğ‘ğ‘¦)−2ğ‘Žğ‘„(𝑥+𝑦)−2−𝑄(𝑥−𝑦)2ğ‘Ž2î€¸î€·âˆ’ğ‘Žğ‘„(𝑥)−2𝑏2î€¸î‚âˆ’ğ‘Žğ‘„(𝑦),𝑡=1(2.25) for each 𝑥,𝑦∈𝑋 and 𝑡>0. So by property (𝑁2), we have that ğ‘„ğ‘Ž(ğ‘Žğ‘¥+𝑏𝑦)+𝑄(ğ‘Žğ‘¥âˆ’ğ‘ğ‘¦)−2ğ‘„ğ‘Ž(𝑥+𝑦)−2𝑄(𝑥−𝑦)−2ğ‘Ž2î€¸ğ‘„âˆ’î€·âˆ’ğ‘Ž(𝑥)2𝑏2î€¸âˆ’ğ‘Žğ‘„(𝑦)=0,∀𝑥,𝑦∈𝑋.(2.26) Therefore 𝑄 is quadratic function. For every 𝑥∈𝑋 and 𝑡,𝑠>0, by (2.22), for large enough 𝑛, we have that ğ‘î…žî‚»ğ‘(𝑄(𝑥)−𝑓(𝑥),𝑡)≥minî…žî‚µğ‘„(𝑥)−𝑓(ğ‘Žğ‘›ğ‘¥)ğ‘Ž2𝑛,𝑡2,ğ‘î…žî‚µğ‘“(ğ‘Žğ‘›ğ‘¥)ğ‘Ž2𝑛𝑡−𝑓(𝑥),2âŽ›âŽœâŽœâŽœâŽœâŽî‚¶î‚¼â‰¥ğ‘ğœ‘(𝑥),(𝑡/2)ğ‘žî‚µğ‘›âˆ‘ğ‘˜=1𝛼𝑝/ğ‘Ž2î€¸ğ‘˜ğ›¼âˆ’ğ‘î‚¶ğ‘žâŽžâŽŸâŽŸâŽŸâŽŸâŽ îƒ©îƒ©î€·ğ‘Ž=𝑁𝜑(𝑥),2−𝛼𝑝𝑡2îƒªğ‘žîƒª.(2.27) Let ğ‘„î…ž be another quadratic function from 𝑋 to 𝑌 which satisfies (2.3). Since, for each 𝑛∈𝑁, 𝑄(ğ‘Žğ‘›ğ‘¥)=ğ‘Ž2𝑛𝑄(𝑥),ğ‘„î…ž(ğ‘Žğ‘›ğ‘¥)=ğ‘Ž2ğ‘›ğ‘„î…ž(𝑥),(2.28) We have that ğ‘î…žî€·ğ‘„(𝑥)âˆ’ğ‘„î…žî€¸(𝑥),𝑡=ğ‘î…žî€·ğ‘„(ğ‘Žğ‘›ğ‘¥)âˆ’ğ‘„î…ž(ğ‘Žğ‘›ğ‘¥),ğ‘Ž2𝑛𝑡𝑁≥minî…žî‚µğ‘„(ğ‘Žğ‘›ğ‘¥)−𝑓(ğ‘Žğ‘›ğ‘Žğ‘¥),2𝑛𝑡2,ğ‘î…žî‚µğ‘“(ğ‘Žğ‘›ğ‘¥)âˆ’ğ‘„î…ž(ğ‘Žğ‘›ğ‘Žğ‘¥),2𝑛𝑡2î‚µî‚µğ‘Žî‚¶î‚¼â‰¥ğ‘ğœ‘(𝑥),2−𝛼𝑝2î‚¶ğ‘žğ‘Ž2ğ‘›ğ‘žğ›¼ğ‘›ğ‘¡ğ‘ž2ğ‘žî‚¶(2.29) for each 𝑛∈𝑁. Due to ğ‘ž>1/2, limğ‘›â†’âˆžğ‘î‚µî‚µğ‘Žğœ‘(𝑥),2−𝛼𝑝2î‚¶ğ‘žğ‘Ž2ğ‘›ğ‘žğ›¼ğ‘›ğ‘¡ğ‘ž2ğ‘žî‚¶=1(2.30) for each 𝑥∈𝑋 and 𝑡>0. Therefore 𝑄=ğ‘„î…ž.

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Copyright © 2011 Ehsan Movahednia. 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.


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