THE GENERALIZED HYPERSTABILITY OF GENERAL LINEAR EQUATION IN QUASI-2-BANACH SPACE*

Ravinder Kumar SHARMA Sumit CHANDOK

School of Mathematics， Thapar Institute of Engineering and Technology， Patiala-147004， India E-mail: ravindersharma880@gmail.com; sumit.chandok@thapar.edu

Abstract In this paper， we study the hyperstability for the general linear equation

Key words Hyperstability; quasi-2-Banach spaces; fixed point; general linear equation

1 Introduction and Preliminaries

The main motivation for the investigation of the stability of functional equations was given by Ulam in 1940 in his talk at the university of Wisconsin [27]. Hyers [17] gave affirmative partial answer to the question of Ulam in Banach space. After that Aoki[3] generalized Hyers’Theorem for additive mapping and Rassias [26] generalized Hyers theorem for linear mapping by considering an unbounded Cauchy difference. In 1994， G˘avruta [14] generalized Rassias’theorem and discuss the stability of linear functional equation.

During the last four decades，many results concering the Hyers-Ulam stability of important functional equations have been obtained by several mathematicians (see [1， 19， 22， 28–30] and references cited therein).

A functional equation is hyperstable if a function satisfying the equation approximately is a true solution of it. In 1949， Bourgin [5] gave the first hyperstability result and concerned the ring homomorphisms. The hyperstability results of the several functional equation in the literature have been studied by many authors in recent years(see[4，6，7， 10–12， 15，20， 23，24]and references cited therein).

In this paper， we extend Theorem 1.1 in the setting of quasi 2-Banach space. Also， we obtian some results for the hyperstability of the general linear equation (1.1) in the setting of quasi-2-Banach spaces.

2 Properties of Quasi-2-Normed Space

In this section， we present some definitions and properties of quasi-2-normed space which will be used in the sequel.

In 1965， G¨ahler [16] introduced 2-normed spaces. One of the axioms of the 2-norm is the parallelepiped inequality， which is a fundamental in the theory of 2-normed spaces. Park [21]replaced precisely this inequality(analogously as in the normed spaces)with a new more general condition， and gave the concept of quasi-2-normed spaces and quasi-(2;p)-normed spaces.

Definition 2.1 Let X be a real linear space of dimension greater than 1 and ‖·，·‖ be a real valued function on X×X satisfying the following four conditions:

(N1) ‖x，y‖ = 0 if and only if x and y are linearly dependent in X;

(N2) ‖x，y‖ = ‖y，x‖ for all x，y ∈X;

(N3) ‖x，αy‖ = |α|.‖x，y‖ for every real number α;(N4) ‖x，y+z‖≤‖x，y‖+‖x，z‖ for all x，y，z ∈X.

Then function ‖·，·‖ is called 2-norm on X and the pair (X，‖·，·‖) is a called 2-normed space.

Definition 2.2 Let X be a linear space. A quasi-2-norm is a real-valued function on X ×X satisfying (N1)，(N2)，(N3) of Definition 2.1 and

(N5) ‖x+y，z‖≤K‖x，z‖+K‖y，z‖，for all x，y，z ∈X and K ≥1.

If ‖·，·‖ is a quasi-2-norm on X， then (X，‖·，·‖，K) is called quasi-2-normed space. The smallest possible K is called the modulus of concavity of ‖·，·‖.

A quasi-2-norm ‖·，·‖ is called quasi-p-norm (0 ＜p ＜1) if ‖x+y，z‖p≤‖x，z‖p+‖y，z‖p，for all x，y，z ∈X.

Every 2-normed space is a special case of quasi-2-normed spaces (for K =1).

Example 2.3 Assume that X = R3and x = x1i+x2j+x3k， y = y1i+y2j+y3k ∈R3.Define

Proposition 2.6 Every quasi-2-Cauchy sequence in a quasi-2-normed space is quasi-2-bounded.

Proof Suppose that {xn} is a quasi-2-Cauchy sequence in a quasi-2-normed space X.Thus by the definition， for all ε=1 ＞0， there exists N ∈N such that

Now， we have to show that {xn} is quasi-2-bounded; that is ‖xn，z‖ ＜M. Using triangle inequality， we have

and this is true for all n ＞N. By choosing the maximum of either 1 + ‖xN+1，z‖ or the maximum of the aforementioned set， we can find our M which bounds all the terms in the sequence. Hence， the result is obtained. □

Definition 2.7 Let X be a real linear space of dimension greater than 1. Quasi-2-norms|||·，·||| and ||·，·|| define on X are equivalent if it exists m，M ＞0 such that

Lemma 2.8 Let (X，‖·，· ‖) be quasi-2-normed space. There is a p(0 ＜p ≤1) and equivalent quasi-2-norm |||·，·||| on X satisfying |||x + y，z|||p≤|||x，z|||p+ |||y，z|||p， for all x，y，z ∈X.

Here， it is to note that quasi-2-norm is not continuous but its equivalent norm defined by(2.1) is continuous.

3 Fixed Point

The basic tool in this paper is the following theorem which assert the existence of a unique fixed point of the function T : YX→YX. This result is the extension of Theorem 1.1 from 2-Banach space to quasi-2-Banach space using Lemmas 2.8 and 2.9.

By combining (3.9) and (3.3)， the sequence {(Tnφ)(x)} is a quasi-2-Cauchy sequence in the equivalent quasi-2-normed (Y， |||·，·|||). Using (2.2)， {(Tnφ)(x)} is also quasi-2-Cauchy in(Y，‖·，·‖，K). Since Y is complete， there exists the following limit in (Y，‖·，·‖，K):

4 Hyperstability of Functional Equation

In this section， we obtain the hyperstability of functional equation (1.1) in the setting of quasi-2-Banach spaces using Theorem 3.1.

Theorem 4.1 (1)Let(X，‖·，·‖，K)be a quasi-2-normed space over the field F，(Y，‖·，·‖，K)be a quasi-2-Banach space over the field K and Z0be a subset of Y containing two linearly independent vectors.

for all x，y ∈X{0} with ax+by /=0.