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On the Ulam stability of mixed type QA mappings in IFN-spaces

Abdulrahman S Al-Fhaid and Syed Abdul Mohiuddine*

Author Affiliations

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

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Advances in Difference Equations 2013, 2013:203  doi:10.1186/1687-1847-2013-203

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/203


Received:13 March 2013
Accepted:15 June 2013
Published:8 July 2013

© 2013 Al-Fhaid and Mohiuddine; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We give Ulam-type stability results concerning the quadratic-additive functional equation in intuitionistic fuzzy normed spaces.

Keywords:
t-norm; t-conorm; quadratic-additive functional equation; intuitionistic fuzzy normed space; Hyers-Ulam stability

1 Introduction

In 1940, Ulam [1] proposed the following stability problem: ‘When is it true that a function which satisfies some functional equation approximately must be close to one satisfying the equation exactly?’. Hyers [2] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Aoki [3] presented a generalization of Hyers results by considering additive mappings, and later on Rassias [4] did for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias has significantly influenced the development of what we now call the Hyers-Ulam-Rassias stability of functional equations. Various extensions, generalizations and applications of the stability problems have been given by several authors so far; see, for example, [5-24] and references therein.

The notion of intuitionistic fuzzy set introduced by Atanassov [25] has been used extensively in many areas of mathematics and sciences. Using the idea of intuitionistic fuzzy set, Saadati and Park [26] presented the notion of intuitionistic fuzzy normed space which is a generalization of the concept of a fuzzy metric space due to Bag and Samanta [27]. The authors of [28-34] defined and studied some summability problems in the setting of an intuitionistic fuzzy normed space.

In the recent past, several Hyers-Ulam stability results concerning the various functional equations were determined in [35-46], respectively, in the fuzzy and intuitionistic fuzzy normed spaces. Quite recently, Alotaibi and Mohiuddine [47] established the stability of a cubic functional equation in random 2-normed spaces, while the notion of random 2-normed spaces was introduced by Goleţ [48] and further studied in [49-51].

The Hyers-Ulam stability problems of quadratic-additive functional equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M1">View MathML</a>

under the approximately even (or odd) condition were established by Jung [52] and the solution of the above functional equation where the range is a field of characteristic 0 was determined by Kannappan [53]. In this paper we determine the stability results concerning the above functional equation in the setting of intuitionistic fuzzy normed spaces. This work indeed presents a relationship between two various disciplines: the theory of fuzzy spaces and the theory of functional equations.

2 Definitions and preliminaries

We shall assume throughout this paper that the symbol ℕ denotes the set of all natural numbers.

A binary operation <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M2">View MathML</a> is said to be a continuoust-norm if it satisfies the following conditions:

(a) ∗ is associative and commutative, (b) ∗ is continuous, (c) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M3">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M4">View MathML</a>, (d) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M5">View MathML</a> whenever <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M6">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M7">View MathML</a> for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M8">View MathML</a>.

A binary operation <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M9">View MathML</a> is said to be a continuoust-conorm if it satisfies the following conditions:

(a′) ♢ is associative and commutative, (b′) ♢ is continuous, (c′) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M10">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M11">View MathML</a>, (d′) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M12">View MathML</a> whenever <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M6">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M7">View MathML</a> for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M8">View MathML</a>.

The five-tuple <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M16">View MathML</a> is said to be intuitionistic fuzzy normed spaces (for short, IFN-spaces) [26] if X is a vector space, ∗ is a continuous t-norm, ♢ is a continuous t-conorm, and μ, ν are fuzzy sets on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M17">View MathML</a> satisfying the following conditions. For every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M18">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M19">View MathML</a>,

(i) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M20">View MathML</a>,

(ii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M21">View MathML</a>,

(iii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M22">View MathML</a> if and only if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M23">View MathML</a>,

(iv) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M24">View MathML</a> for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M25">View MathML</a>,

(v) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M26">View MathML</a>,

(vi) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M27">View MathML</a> is continuous,

(vii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M28">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M29">View MathML</a>,

(viii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M30">View MathML</a>,

(ix) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M31">View MathML</a> if and only if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M23">View MathML</a>,

(x) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M33">View MathML</a> for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M25">View MathML</a>,

(xi) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M35">View MathML</a>,

(xii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M36">View MathML</a> is continuous,

(xiii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M37">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M38">View MathML</a>.

In this case <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M39">View MathML</a> is called an intuitionistic fuzzy norm. For simplicity in notation, we denote the intuitionistic fuzzy normed spaces by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M40">View MathML</a> instead of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M16">View MathML</a>. For example, let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M42">View MathML</a> be a normed space, and let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M43">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M44">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M45">View MathML</a>. For all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a> and every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a>, consider

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M48">View MathML</a>

Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M40">View MathML</a> is an intuitionistic fuzzy normed space.

The notions of convergence and Cauchy sequence in the setting of IFN-spaces were introduced by Saadati and Park [26] and further studied by Mursaleen and Mohiuddine [30].

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M40">View MathML</a> be an intuitionistic fuzzy normed space. Then the sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M51">View MathML</a> is said to be:

(i) Convergent to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M52">View MathML</a> with respect to the intuitionistic fuzzy norm <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M39">View MathML</a> if, for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M54">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a>, there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M56">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M57">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M58">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M59">View MathML</a>. In this case, we write <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M60">View MathML</a> or <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M61">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M62">View MathML</a>.

(ii) Cauchysequence with respect to the intuitionistic fuzzy norm <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M39">View MathML</a> if, for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M54">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a>, there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M56">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M67">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M68">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M69">View MathML</a>. An IFN-space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M40">View MathML</a> is said to be complete if every Cauchy sequence in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M40">View MathML</a> is convergent in the IFN-space. In this case, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M40">View MathML</a> is called an intuitionistic fuzzy Banach space.

3 Stability of a quadratic-additive functional equation in the IFN-space

We shall assume the following abbreviation throughout this paper:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M73">View MathML</a>

Theorem 3.1LetXbe a linear space and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M40">View MathML</a>be an IFN-space. Suppose thatfis an intuitionistic fuzzyq-almost quadratic-additive mapping from<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M40">View MathML</a>to an intuitionistic fuzzy Banach space<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M76">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M77">View MathML</a>

(3.1)

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M79">View MathML</a>, whereqis a positive real number with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M80">View MathML</a>. Then there exists a unique quadratic-additive mapping<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M81">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M82">View MathML</a>

(3.2)

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a>and all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a>with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M85">View MathML</a>, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M86">View MathML</a>.

Proof Putting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M87">View MathML</a> in (3.1), it follows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M88">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M89">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a>. Using the definition of IFN-space, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M91">View MathML</a>. Now we are ready to prove our theorem for three cases. We consider the cases as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M92">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M93">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M94">View MathML</a>.

Case 1. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M92">View MathML</a>. Consider a mapping <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M96">View MathML</a> to be such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M97">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a>. Notice that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M99">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M100">View MathML</a>

(3.3)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M102">View MathML</a>. Using the definition of IFN-space and (3.1), this equation implies that if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M103">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M104">View MathML</a>

(3.4)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M105">View MathML</a>

(3.5)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M108">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M109">View MathML</a>. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M54">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M111">View MathML</a> be given. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M112">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M113">View MathML</a>, there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M114">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M115">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M116">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a>. We observe that for some <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M118">View MathML</a>, the series <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M119">View MathML</a> converges for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M120">View MathML</a>, there exists some <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M121">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M122">View MathML</a> for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M123">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M124">View MathML</a>. Using (3.4) and (3.5), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M125">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M126">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M111">View MathML</a>. Hence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M129">View MathML</a> is a Cauchy sequence in the fuzzy Banach space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M76">View MathML</a>. Thus, we define a mapping <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M81">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M132">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a>. Moreover, if we put <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M134">View MathML</a> in (3.4) and (3.5), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M135">View MathML</a>

(3.6)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a>. Now we have to show that T is quadratic additive. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M138">View MathML</a>. Then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M139">View MathML</a>

(3.7)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M140">View MathML</a>

(3.8)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M142">View MathML</a>. Taking the limit as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M143">View MathML</a> in the inequalities (3.7) and (3.8), we can see that first seven terms on the right-hand side of (3.7) and (3.8) tend to 1 and 0, respectively, by using the definition of T. It is left to find the value of the last term on the right-hand side of (3.7) and (3.8). By using the definition of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M144">View MathML</a>, write

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M145">View MathML</a>

(3.9)

and, similarly,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M146">View MathML</a>

(3.10)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M142">View MathML</a>. Also, from (3.1), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M150">View MathML</a>

(3.11)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M151">View MathML</a>

(3.12)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M142">View MathML</a>. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M92">View MathML</a>, therefore (3.9) tends to 1 as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M143">View MathML</a> with the help of (3.11) and (3.12). Similarly, by proceeding along the same lines as in (3.11) and (3.12), we can show that (3.10) tends to 0 as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M143">View MathML</a>. Thus, inequalities (3.7) and (3.8) become

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M158">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a>. Accordingly, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M161">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78">View MathML</a>. Now we approximate the difference between f and T in a fuzzy sense. Choose <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M163">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M164">View MathML</a>. Since T is the intuitionistic fuzzy limit of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M129">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M166">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M142">View MathML</a>. From (3.6), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M170">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M171">View MathML</a>

Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M163">View MathML</a> is arbitrary, we get the inequality (3.2) in this case.

To prove the uniqueness of T, assume that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M173">View MathML</a> is another quadratic-additive mapping from X into Y, which satisfies the required inequality, i.e., (3.2). Then, by (3.3), for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M175">View MathML</a>

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M176">View MathML</a>

(3.13)

Therefore

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M177">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M178">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M142">View MathML</a>. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M182">View MathML</a> and taking limit as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M143">View MathML</a> in the last two inequalities, we get <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M184">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M185">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a>. Hence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M188">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a>.

Case 2. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M93">View MathML</a>. Consider a mapping <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M96">View MathML</a> to be such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M192">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a>. Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M99">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M195">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M197">View MathML</a>. Thus, for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M103">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M199">View MathML</a>

where ∏ and ∐ are the same as in Case 1. Proceeding along a similar argument as in Case 1, we see that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M129">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M76">View MathML</a>. Thus, we define <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M202">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a>. Putting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M134">View MathML</a> in the last two inequalities, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M205">View MathML</a>

(3.14)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a>. To prove that t is a quadratic-additive function, it is enough to show that the last term on the right-hand side of (3.7) and (3.8) tends to 1 and 0, respectively, as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M143">View MathML</a>. Using the definition of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M144">View MathML</a> and (3.1), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M210">View MathML</a>

(3.15)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M211">View MathML</a>

(3.16)

for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M142">View MathML</a>. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M215">View MathML</a> and taking the limit as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M143">View MathML</a>, we see that (3.15) and (3.16) tend to 1 and 0, respectively. As in Case 1, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M161">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78">View MathML</a>. Using the same argument as in Case 1, we see that (3.2) follows from (3.14). To prove the uniqueness of T, assume that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M173">View MathML</a> is another quadratic-additive mapping from X into Y satisfying (3.2). Using (3.2) and (3.13), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M220">View MathML</a>

(3.17)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M221">View MathML</a>

(3.18)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M142">View MathML</a>. Letting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M143">View MathML</a> in (3.17) and (3.18), and using the fact that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M226">View MathML</a> together with the definition of IFN-space, we get <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M227">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M185">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a>. Hence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M188">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a>.

Case 3. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M94">View MathML</a>. Define a mapping <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M96">View MathML</a> by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M235">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a>. In this case, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M99">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M238">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M197">View MathML</a>. Thus, for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M103">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M242">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a>. Proceeding along a similar argument as in the previous cases, we see that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M129">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M76">View MathML</a>. Thus, we define <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M132">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a>. Putting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M134">View MathML</a> in the last two inequalities, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M250">View MathML</a>

(3.19)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a>. Write

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M253">View MathML</a>

(3.20)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M254">View MathML</a>

(3.21)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M142">View MathML</a>. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M215">View MathML</a> and taking the limit as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M143">View MathML</a>, we see that (3.20) and (3.21) tend to 1 and 0, respectively. As in the previous cases, we have that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M161">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78">View MathML</a>. By the same argument as in previous cases, we can see that (3.2) follows from (3.19). To prove the uniqueness of T, assume that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M173">View MathML</a> is another quadratic-additive mapping from X into Y satisfying (3.2). From (3.2) and (3.13), for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a>, write

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M265">View MathML</a>

and, similarly,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M266">View MathML</a>

for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M142">View MathML</a>. Letting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M143">View MathML</a> in (3.17) and (3.18), and using the fact that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M269">View MathML</a> together with the definition of IFN-space, we get <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M227">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M185">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a>. Hence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M188">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a>. □

Remark 3.2 Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M40">View MathML</a> be an IFN-space and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M40">View MathML</a> be an intuitionistic fuzzy Banach space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M76">View MathML</a>. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M279">View MathML</a> be a mapping satisfying (3.1) with a real number <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M280">View MathML</a> and for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a>. If we choose a real number α with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M282">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M283">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M280">View MathML</a>. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M280">View MathML</a>, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M288">View MathML</a>. This implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M289">View MathML</a>

Thus, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M290">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M291">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a>. Hence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M294">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78">View MathML</a>. In other words, if f is an intuitionistic fuzzy q-almost quadratic-additive mapping for the case <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M280">View MathML</a>, then f is itself a quadratic-additive mapping.

Corollary 3.3Suppose thatfis an even mapping satisfying the conditions of Theorem 3.1. Then there exists a unique quadratic mapping<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M81">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M298">View MathML</a>

(3.22)

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a>, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M86">View MathML</a>.

Proof Since f is an even mapping, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M302">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M304">View MathML</a> is defined as in Theorem 3.1. In this case, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M305">View MathML</a>. For all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M307">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M308">View MathML</a>

Proceeding along the same lines as in Theorem 3.1, we obtain that T is a quadratic-additive function satisfying (3.22). Notice that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M132">View MathML</a>, T is even and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M161">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78">View MathML</a>. Hence, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M312">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M18">View MathML</a>. It follows that T is a quadratic mapping. □

Corollary 3.4Suppose thatfis an even mapping satisfying the conditions of Theorem 3.1. Then there exists a unique additive mapping<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M81">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M315">View MathML</a>

(3.23)

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M47">View MathML</a>, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M86">View MathML</a>.

Proof Since f is an odd mapping, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M319">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M304">View MathML</a> is defined as in Theorem 3.1. Here <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M305">View MathML</a>. For all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M307">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M325">View MathML</a>

Proceeding along the same lines as in Theorem 3.1, we obtain that T is a quadratic-additive function satisfying (3.23). Here <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M132">View MathML</a>, T is odd and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M161">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M78">View MathML</a>. Hence, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M329">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/203/mathml/M18">View MathML</a>. It follows that T is an additive mapping. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors contributed equally and significantly in writing this paper. Both the authors read and approved the final manuscript.

Acknowledgements

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. (405/130/1433). The authors, therefore, acknowledge with thanks DSR technical and financial support.

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