A FIXED POINT APPROACH TO THE STABILITY OF GENERAL QUADRATIC EULER-LAGRANGE FUNCTIONAL EQUATIONS IN INTUITIONISTIC FUZZY SPACES

In this paper, we prove the generalized Hyers-Ulam stability of a general k-quadratic Euler-Lagrange functional equation: for any fixed positive integer in intuitionistic fuzzy normed spaces using a fixed point method.


INTRODUCTION
The notion of fuzzy sets was first introduced by Zadeh [30] in 1965 which is a powerful hand set for modeling uncertainty and vagueness in various problems arising in the field of science and engineering.After that, fuzzy theory has become very active area of research in various fields, e.g.population dynamics [5], chaos control [10,11], computer programming [12], nonlinear dynamical systems [13], nonlinear operators [18], statistical convergence [19] and a lot of developments have been made in the theory of fuzzy sets to find the fuzzy analogues of the classical set theory.The notion of intuitionistic fuzzy norm is also useful one to deal with the inexactness and vagueness arising in modeling.The concept and properties intuitionistic fuzzy metric spaces and normed spaces have been investigated of a number of the authors [8,14,17,20,25].
The stability problem of functional equations originated from a stability question of Ulam [28] concerning the stability of group homomorphisms.''When is it true that by slightly changing the hypotheses of a theorem one can still assert that thesis of the theorem remains true or approximately true?"In [15], Hyers gave the first affirmative partial answer to the question of Ulam for Banach spaces.Hyers' theorem was generalized by Aoki [2] for additive mappings.In 1978, Rassias [22] generalized Hyers theorem by obtaining a unique linear mapping near an approximately additive mapping.The paper of Rassias has provided a lot of influence in the development of what we call the generalized Hyers-Ulam-Rassias stability of functional equations.In 1996, Issac and Rassias [16] were the first to provide applications of the stability theory of functional equations for the proof new fixed point theorems with applications.Cădariu and Radu [6] used the fixed point method to the investigation of the Jensen functional equation.

Let be a set. A function is a called a generalized metric on if satisfies (i) if and only if (ii) for all ;
(iii) for all Then is a generalized metric space.
We recall the following fixed point theorem which was proved by Diaz and Margolis [9]: Let ( be a complete generalized metric space and be a strictly contractive mapping with Lipshitz constant .Then, for any , either for all nonnegative integers or other exists a natural number such that (i) for all ; (ii) the sequence is convergent to a fixed point of ; By using fixed point methods, the stability problems of various functional equations in intuitionistic fuzzy normed spaces have been extensively investigated by a number of authors (see, [1], [7], [21], [23], [24], [26], [27], [29]).Now, we consider the functional equation

PRELIMINARIES
In this section, we recall some definitions, notations and conventions of theory of intuitionistic fuzzy normed spaces which are needed to prove our main results.We denote its units by and .Classically, a triangular norm on is defined as an increasing, commutative, associative mapping satisfying for all .A triangular conorm is defined as an increasing, commutative, associative mapping satisfying for all .

Definition 2.3. A triangular norm (shortly, -norm) on
is a mapping satisfying the following conditions: (i) for all (: boundary condition); (ii) for all (: commutativity); (iii) for all (: associativity); (iv) and ⇒ ) for all (: monotonicity).The triple is said to be an intuitionistic fuzzy normed space if is a linear space, is a continuous and is a mapping satisfying the following conditions: (i) ; (ii) if and only if ; (iii) for all ; ; (iv) is continuous; (v) for all and .In this case, is called an intuitionistic fuzzy norm and we write and for all and Example 2.7.Let be a normed space and for all and let and be the membership and the non-membership degrees of an intuitionistic fuzzy set from to [0,1].Let be the intuitionistic fuzzy set on defined as follows: for all and Then is an intuitionistic fuzzy normed space.
(i) A sequence is said to be intuitionistic fuzzy convergent to if as and for all (ii) A sequence is said to be a intuitionistic fuzzy Cauchy sequence in if, for all and there exists an such that for all , where is the standard negator.
(iii) is complete if every intuitionistic fuzzy Cauchy sequence in is intuitionistic fuzzy convergent in .A complete intuitionistic fuzzy normed space is called an intuitionistic fuzzy Banach space.

MAIN RESULTS
Throughout this section, for any mapping let us define a general -quadratic Euler-Lagrange difference operator for any fixed positive integers .
We prove the intuitionistic fuzzy stability of general -quadratic Euler-Lagrange functional equation (1.1) in the setting intuitionistic fuzzy normed space.
Theorem 3.1.Let be a linear space, be an intuitionistic fuzzy normed space and be a function such that for some (iii) is the unique fixed point of in the set D e c e m b e r 2016 w w w .c i r w o r l d .c o m (iv) for all

(1. 1 )
for any fixed positive integer .The functional equation (1.1) is said to be a general -quadratic Euler-Lagrange functional equation.It is easy to see that the mapping is a solution of the functional equation (1.1).Every solution of the general -quadratic Euler-Lagrange functional equation is said to be a quadratic mapping.Note that, if we replace in (1.1), then we get .Letting in (1.1), is even.Letting in (1.1), we obtain .In this paper, we study some stability results concerning the functional equation (1.1) with the help of the notion of continuous -representable in the setting of intuitionistic fuzzy normed spaces by the fixed point alternative.

Lemma 2 . 1 [ 8 ]Definition 2 . 2 . [ 3 ]
Consider the set and the order relation defined by and for all Then the pair is a complete lattice.An intuitionistic fuzzy set in a universal set is an object where , and are called the membership degree and the non-membership degree, respectively, of and they satisfy

Definition 2 . 4 .ForDefinition 2 . 5 .
A continuous -norm on is said to be continuous -representable if there exist a continuous -norm and a continuous -conorm on [0,1] such that, for all A negation on is any strictly decreasing mapping satisfying and .If for all , then is called an involutive negation.A negator on [0,1] is a decreasing mapping satisfying and .denotes the standard negator on defined by for all Definition 2.6.[3] Let and be the membership and the non-membership degrees of an intuitionistic fuzzy set from I S S N 2 3 4 7 -3487 V o l u m e 1 2 N u m b e r 4 J o u r n a l o f A d v a n c e s i n P h y s i c s 4432 | P a g e D e c e m b e r 2016 w w w .c i r w o r l d .c o m to [0,1] such that for all and .
there is a unique -quadratic Euler-Lagrange mapping such that(3.4)for all and .