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A STUDY OF GENERALIZED OF QUADRATIC FUNCTIONAL OF

FUZZY EQUATION IN SPACES STABILITY

ANUDEEP NAIN

RESEARCH SCHOLAR OPJS UNIVERSITY DR. RAJEEV KUMAR

ASSOCIATE PROFESSOR OPJS UNIVERSITY...

ABSTRACT

The stability study results in the quadratic and cubic functional equations in arbitrary t-norms in

random normed spaces (in the case of Sherstnev). Functional equations are a contemporary field

of mathematics that offers a strong method for working on major concepts and relationships of

symmetry, linearity and equivalence in analysis and algebra. Although the systematic study of

these equations is a relatively recent area of mathematical research, mathematicians such as

Euler in the 18th and Cauchy in the 19th centuries have considered them in various ways before.

Functional equations theory is a growing branch of mathematics that has greatly contributed to

the development of the powerful instruments of modern mathematics. Many new issues and

hypotheses have inspired the development of new techniques and methods in functional

equations. The great mathematicians who work on functional equations and solutions to these

problems included D'Alembert, Euler, Gauss, Cauchy, Abel, Weierstrass, Darboux and Hilbert.

Functional equations are an alternative way to model physics problems. The interest in modeling

physical problems with functional equations is that the difference between function f is not

appropriate. Functional equations therefore tend to lead to solutions different from those given

by partial differential equations, and physicists can consider these other solutions of interest. The

functional equation's ability to construct mathematical models is the most attractive feature.

Mathematics now known as functional equations theory, but in many situations it has remained a

challenging task to find specific solutions for a given functional equation since then . The now

famous "babbage equation " α(μ(x))= x, for one of its example , whose ̈ solutions are known as

identity root and the more general equation ± (̈) ̈ ̈ f(x), which describes a sort of a squeeze root of

certain functions f. f. This form of equations were solved roughly with a special topologic and

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rule by the authors Lars Kindermann, Achim Lewandowski and Peter Protzel, using neural

networks.

KEYWORDS:Quadratic Functional, Fuzzy Equation, Spaces Stability, cubic functional

equations, mathematical models

INTRODUCTION

In mathematics, a function is defined in a secret style as a functional equation. The value of

afunction(s) is (or is) expressed at some point by the equation. For example, FES is used to

evaluate functional properties. FEs is φ(x + y) = φ(x) + φ(y) are the most common FE. Cauchy

(1821) achieved his solution for real variables. The growths of other FEs are protected by the

properties of this FE. This FE is called Cauchy FE. In more or less any field of the natural and

social sciences, the properties of the Additive Cauchy FE are dominant devices. This new FE

theory is growing quickly. There are an increasing number of FE mathematical documents and

mathematicians day after day. Work on FEs in other topics, such as difference geometry,

iteration and analytic functionality, differential equations, theory of numbers, abstract algebra,

shows that FEs are becoming increasingly important. This theory thus acquired a personality of

its own. Because analytical methods in many branches of mathematics are already to a certain

excess exhaustion, FEs are interested in mathematicians all over the world. The use of the simple

methods allows one to obtain far more profound and general results than the application of

traditional methods of mathematical analysis.

The stability problem of functional equations originated from a question of Ulam in 1940,

concerning the stability of group homomorphisms. Let be a group, and let be a

metric group with the metric Given does there exist such that if a mapping

satisfies the inequality then there

exists a homomorphism In other

words, under what condition does there exist a homomorphism near an approximate

homomorphism?

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When we replace the functional equation with inequality, the definition of equilibrium arises as a

distortion of the equation. The equation becomes skewed. 1940, S. S. The problem of group

homomorphism stability was asked by M. Ulam[11]. The first positive response to Ulam's spaces

for Banach was given next year by Hyers[12]. Th. M. 1978. Rassias[13] extended the Hyers

theorem that enables the gap between the cauchy to be infinite. This result attracted and inspired

many mathematicians to explore the stability issues of functional equations. In particular, various

functional equations in different spaces were investigated in the stability probelems.

In 1941, Hyers gave a first affirmative answer to the question of Ulam for Banach spaces. Let

be a mapping between Banach spaces such that

for all x, y ∈ E and for some δ > 0. Then there exists a unique additive mapping T : E → Esuch

that

for all x ∈ E. Additionally on the off chance that ftx is ceaseless in t for each fixed x ∈ E, at that

point T is linear see moreover. In 1950, Aoki summed up Hyers' theorem for around added

substance mappings. In 1978, Th. M. Rassias gave a speculation of Hyers' theorem which

permits the Cauchy distinction to be unbounded. This new idea is known as Hyers-Ulam-Rassias

soundness of functional equations.

The functional equation

is related to symmetric biadditive function. In the real case it has among its

solutions. Thus, it has been called quadratic functional equation, and each of its solutions is said

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to be a quadratic function. Hyers-Ulam-Rassias stability for the quadratic functional equation

1.3was proved by Skof for functions f : A → B, where A is normed space and B Banach space.

The following cubic functional equation was introduced by the third author of this paper, J. M.

Rassias

Jun and Kim introduced the following cubic functional equation:

and they established the general solution and the generalized Hyers-Ulam-Rassias stability for

the functional equation (1.5).

The function satisfies the functional equation (1.5), which explains why it is called

cubic functional equation.

Jun and Kim demonstrated that a function f between genuine vector spaces X and Y is an answer

of (1.5) if and just if there exists a one of a kind function C : X × X × X → Y with the end goal

that fx Cx, x, x for all x ∈ X, and C is symmetric for each fixed one variable and is added

substance for fixed two variables

We manage the accompanying functional condition getting from added substance, cubic and

quadratic functions:

It is easy to see that the function is a solution of the functional

equation 1.6. In the present paper we investigate the general solution and the generalized Hyers-

Ulam-Rassias stability of the functional equation (1.6).

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Recently C.Park and D.Y.Shin[1] presented Hyers-Ulam Stability of a class of Quadratic, Cubic

and Quartic functional equations in paranormed spaces.

The functional equation

f (3x y) f (3xy) f (xy) f (xy) 16 f (x)

is a quadratic functional equation and every solution of the quadratic functional equation is said

to be a quadratic function. The Functional equation

is a cubic functional equation and every solution of the cubic functional equation is said to be a

cubic function. In this paper, we investigate the Hyers-Ulam Stability of the Quadratic Equation

and Cubic equation in Paranormed spaces.

STABILITY OF QUADRATIC FUNCTIONAL EQUATIONS

In this section, we deal with the stability problem for the following quadratic functional equation

in paranormed spaces.

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for all nonnegative integers m and l with m > l and for all x Y. It follows from (3.3) that the

sequence

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for all x Y.

Moreover, letting l = 0 and passing the limit m in (3.3), we get (3.2)

It follows from (3.1) that

for all x, y Y.

Hence

for all x, y Y. and so the mapping Q2 :Y X is quadratic. Now let T :Y X be another

quadratic mapping satisfying (3.2). Then we have

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which tends to zero as n for all x Y. So we can conclude that ( ) ( ) 2 Q x T x for all x

Y. This proves the uniqueness of Q2 . Thus the mapping Q2 :Y X is a unique quadratic

mapping satisfying (3.2).

STABILITY OF CUBIC FUNCTIONAL EQUATION

In this section we prove the stability of the following cubic functional equation in paranormed

spaces.

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Moreover, letting l = 0 and passing the limit m in (4.3), we get (4.2).

It follows from (4.1) that

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which tends to zero as n for all x Y. So we can conclude that C(x) T(x) for all x Y.

This proves the uniqueness of C . Thus the mapping C :Y X is a unique cubic mapping

satisfying (4.2).

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for all nonnegative integers m and l with m > l and for all x X . It follows from (4.6) that the

sequence

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which tends to zero as n for all x X. So we can conclude that C(x) T(x) for all xX .

This proves the uniqueness of C . Thus the mapping C: X Y is a unique cubic mapping

satisfying (4.5).

PROBLEM FOR FUNCTIONAL EQUATION

The stability of FEs is a fascinating subject that has been managed throughout the previous six

decades. In mathematics, a stipulation where a slight unsettling influence in a framework doesn't

make an extensive upsetting result on that framework. An equation is said to be steady if a

somewhat unique arrangement is near the specific arrangement of that equation. In numerical

demonstrating of physical issues, the devations in estimations will result with blunders and these

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deviations can be managed the stability of equations. Subsequently the stability of equations is

basic in scientific models. A steady arrangement will be reasonable disregarding such deviations.

An intriguing and famous talk given by Ulam (97) in 1940, roused to consider the examination

of stability of FEs. The ensuing question was asked by him pertaining to the permanence of

homomorphisms in group theory:

Let X , Y be a group and a metric group with metric d(·, ·), respectively. Suppose e > 0 is a fixed

cosntant. Then, whether there occurs a constant δ > 0 so that if a mapping g : X → Y fulfills

d(g(uv), g(u)g(v)) < δ

∀u, v ∈X , and there occurs a homomorphism a : X → Y with the condition

If the answer is confirmative, then the FE for homomorphism is said to be stable.

CONCLUSION

The study of functional equations is a contemporary region of mathematics that furnishes an

incredible way to deal with working with significant ideas and connections in examination and

variable based math, for example, evenness, linearity and identicalness. In spite of the fact that

the methodical investigation of such equations is a moderately late territory of numerical

examination. The theory of functional equations is a developing part of mathematics which has

contributed a great deal to the advancement of the solid devices in the present mathematics.

Numerous new applied issues and speculations have propelled functional equations to grow new

methodologies and strategies. The FEs considered by the creator right now broad renditions of

numerous FEs. The creator has expressed and demonstrated different stability theorems of

blended sort FEs. In future, these stability results will be useful to analyze stability consequences

of other kind of FEs in various spaces. The creator is likewise intrigued to discover uses of new

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FEs presented and explored right now different fields of Mathematics, Science and Engineering.

The exhaustive HUR lastingness of numerous FEs are examined in various spaces in the theory

of Ulam stability of FEs by numerous mathematicians. In our pack, we have tackled different

types of blended kind FEs for their general arrangement. Likewise, in our examination, by

demonstrating the comprehsive HUR permanance of arranged kind addtive-quadratic FEs in

semi β-normed spaces, far reaching HUR permanance of grouped sort added substance quadratic

and quadratic-cubic FEs in paranormed spaces, complete HUR permanance of added substance

quartic FEs in semi Banach spaces and non-Archimedean spaces, and exhaustive HUR

permanance of arranged kind added substance quadratic-cubic-quartic FE in Banach spaces, we

have indicated that the FEs considered in our pack are steady in the feeling of Ulam, Hyers,

Rassias and Gavruta. The thought of probabilistic measurement spaces was presented by

Menger. Menger proposed moving the probabilistic ideas of quantum mechanics from material

science to geometry. Probabilistic normed spaces are genuine linear spaces in which the standard

of every vector is a proper likelihood conveyance function as opposed to a number. The theory

of probabilistic normed spaces was presented by Serstnev in 1963. In Alsina, Schweizer and

Sklar gave another meaning of probabilistic normed spaces which incorporates Serstnevs as a

unique case and leads normally to the distinguishing proof of the rule class of probabilistic

normed spaces, the Menger spaces.

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