FUZZY STABILITY OF A GENERALIZED QUADRATIC FUNCTIONAL EQUATION

In this work, we investigate the generalized Hyers-Ulam stability of the Apollonius type additive functional equation in modular spaces with or without Δ 2 -conditions. We study the same problem in fuzzy Banach spaces and β -homogeneous Banach spaces. We show the hyperstability of the functional equation associated with the Jordan triple product in fuzzy Banach algebras. The obtained results can be applied to differential and integral equations with kernels of non-power types.

In this paper we investigate Hyers-Ulam-Rassias stability of certain nonlinear functional equations. Considerations of such stabilities in different branches of mathematics have been very extensive. Again the fuzzy concepts along with their several extensions have appeared in almost all branches of mathematics. Here we work on intuitionistic fuzzy real Banach spaces, which is obtained by combining together the concepts of fuzzy Banach spaces with intuitionistic fuzzy sets. We establish that pexiderized quadratic functional equations defined on such spaces are stable in the sense of Hyers-Ulam-Rassias stability. We adopt a fixed point approach to the problem. Precisely, we use a generxalized contraction mapping principle. The result is illustrated with an example.

8 - THE SYSTEM OF RATIONAL NUMBERS

Irreducible polynomials and full elasticity in rings of integer-valued polynomials

In this paper, we prove the generalized Hyers-Ulam stability of generalized mixed type cubic, quadratic, and additive functional equation, in fuzzy Banach spaces. 2010 Mathematics Subject Classification: 39B82; 39B52.

In , Th.M. Rassias introduced the following equality \sum_{i,j=1}^m \|x_i - x_j \|^2 = 2m \sum_{i=1}^m\|x_i\|^2, \qquad \sum_{i=1}^m x_i =0 for a fixed integer $m \ge 3$. Let $V, W$ be real vector spaces. It is shown that if a mapping $f : V \rightarrow W$ satisfies \sum_{i,j=1}^m f(x_i - x_j ) = 2m \sum_{i=1}^m f(x_i) for all $x_1, \ldots, x_{m} \in V$ with $\sum_{i=1}^m x_i =0$, then the mapping $f : V \rightarrow W$ is realized as the sum of an additive mapping and a quadratic mapping. From the above equality we can define the functional equation f(x-y) +f(2x+y) + f(x+2y)= 3f(x)+ 3f(y) + 3f(x+y) , which is called a {\it quadratic functional equation}. Every solution of the quadratic functional equation is said to be a {\it quadratic mapping}. Using fixed point theorem we prove the Hyers-Ulam stability of the functional equation () in fuzzy Banach spaces.

Let p≥3 be a prime number. A Fermat curve over Q of exponent p is defined by an equation of the form axp+byp+czp=0, where a, b, c are non-zero rational numbers. We prove in this article that there exist infinitely many Fermat curves defined over Q, of exponent p, pairwise non Q-isomorphic, contradicting the Hasse principle.

Rational products of singular moduli

The degrees-of-freedom of a K-user Gaussian interference channel (GIFC) has been defined to be the multiple of (1/2) log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> P at which the maximum sum of achievable rates grows with increasing P. In this paper, we establish that the degrees-of-freedom of three or more user, real, scalar GIFCs, viewed as a function of the channel coefficients, is discontinuous at points where all of the coefficients are non-zero rational numbers. More specifically, for all K > 2, we find a class of K-user GIFCs that is dense in the GIFC parameter space for which K/2 degrees-of-freedom are exactly achievable, and we show that the degrees-of-freedom for any GIFC with non-zero rational coefficients is strictly smaller than K/2. These results are proved using new connections with number theory and additive combinatorics.

We treat two different equations involving powers of singular moduli. On the one hand, we show that, with two possible (explicitly specified) exceptions, two distinct singular moduli [Formula: see text] such that the numbers [Formula: see text], [Formula: see text] and [Formula: see text] are linearly dependent over [Formula: see text] for some positive integers [Formula: see text], must be of degree at most [Formula: see text]. This partially generalizes a result of Allombert, Bilu and Pizarro-Madariaga, who studied CM-points belonging to straight lines in [Formula: see text] defined over [Formula: see text]. On the other hand, we show that, with obvious exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number. This generalizes a result of Bilu, Luca and Pizarro-Madariaga, who studied CM-points belonging to a hyperbola [Formula: see text], where [Formula: see text].

In [32, 33], the fuzzy stability problems for the Cauchy additive functional equation and the Jensen additive functional equation in fuzzy Banach spaces have been investigated. Using the fixed point method, we prove the Hyers-Ulam stability of the following additive-quadraticcubic-quartic functional equation f(x+2y)+f(x-2y)=4f(x+y)+4f(x-y)-6f(x)+f(2y)+f(-2y)-4f(y)-4f(-y) (1) in fuzzy Banach spaces.

In this research article, we discuss the stabilities of mixed AQ2CQ4 functional equation in Fuzzy Banach spaces with the help of the Hyers method.

It is known that there exists a multiple-unicast network which has a rate 1 linear network coding solution if and only if the characteristic of the finite field belongs to a given finite or co-finite set of primes. In this paper, we present a generalization of this result for a linear network coding solution of any rate. Specifically, we show that for any non-zero positive rational number k/n, there exists a multiple-unicast network which has a rate k/n fractional linear network coding solution if and only if the characteristic of the finite field belongs to a given finite or co-finite set of primes.

A nonzero rational number is called a {\it cube sum} if it is of the form $a^3+b^3$ with $a,b\in{\Bbb Q}^\times$. In this paper, we prove that for any odd integer $k\geq 1$, there exist infinitely many cube-free odd integers $n$ with exactly $k$ distinct prime factors such that $2n$ is a cube sum (resp. not a cube sum). We present also a general construction of Heegner points and obtain an explicit Gross-Zagier formula which is used to prove the Birch and Swinnerton-Dyer conjecture for certain elliptic curves related to the cube sum problem.

In this paper, I work on expanding the Quadratic ( ), -function inequa- lities by relying on the general quadratic functional equation with 2k-variables on the fuzzy Banach space.That's the main result of this.