Quadratic Mappings Research Articles

On an n-dimensional mixed type additive and quadratic functional equation

In this paper, we investigate the generalized Hyers–Ulam stability of the functional equation∑k2,…,kn=01fx1+∑i=2n(-1)kixi-2n-1f(x1)-2n-2∑i=2nf(xi)+f(-xi)=0for integer values of n such that n⩾2, where f is a function from a normed space X to a Banach space Y. The solutions of the equation are called additive–quadratic mappings.

On the Definition of Quadratic Mappings

In 1966, Gleason published some interesting results about quadratic mappings. He considered a mapping Q : V → W such that Q(u + v) + Q(u − v) = 2Q(u) + 2Q(v), and he proved that it was \({\mathbb{Z}}\) -quadratic, provided that the multiplication by 2 in W was injective. Then he answered this question negatively: if V and W are vector spaces over a field K, and if Q is \({\mathbb{Z}}\)-quadratic and K-homogeneous of order 2, is it always true that Q is K-quadratic? These considerations and other related topics are here revisited, without the systematic hypothesis about the multiplication by 2.

Change in the orbit orientation for a certain family of stochastic quadratic mappings

Nonlinear bijective mappings give many interesting families of discrete-time dynamical systems. This short note proposes a special orientation-changing mechanism for orbits of nonlinear plane mappings (in fact, these mappings have a probabilistic origin and can be interpreted as simplified models in genetics; however, such an interpretation is not discussed in the paper).

Fuzzy approximation of Euler-Lagrange quadratic mappings

Abstract In this article, we consider the Hyers-Ulam stability of the Euler-Lagrange quadratic functional equation f ( k x + l y ) + f ( k x − l y ) = k l [ f ( x + y ) + f ( x − y ) ] + 2 ( k − l ) [ k f ( x ) − l f ( y ) ] in fuzzy Banach spaces, where k, l are nonzero rational numbers with k ≠ l .

Approximate Multi-Jensen, Multi-Euler-Lagrange Additive and Quadratic Mappings in -Banach Spaces

We prove the generalized Hyers-Ulam stability of multi-Jensen, multi-Euler-Lagrange additive, and quadratic mappings in -Banach spaces, using the socalled direct method. The corollaries from our main results correct some outcomes from Park (2011).

Fixed point approach to the estimation of approximate general quadratic mappings

The purpose of this paper is to investigate the stability of a mixed functional equation with quadratic and Jensen additive mappings by using the fixed point method without dividing an approximate mapping by two approximate even and odd parts of the approximate mapping.

Singularities of homogeneous quadratic mappings

We consider the affine real variety \(V=F^{-1}(0)\) in \(\mathbf{R}^n\) where \(F\) is given by two quadratic forms. We announce several recent results, obtained in collaboration with various coauthors, about the topology of the generic \(V\) and of its deformations \(V_t=F^{-1}(t)\), especially those that are smooth. This work started many years ago [16] with the case where the two quadratic forms are simultaneously diagonalizable and with some restrictions. Now we have the result without restrictions and including the non-diagonalizable case so we have a complete topological description of \(V\) in all generic cases. We have also results about the topology of the intersection of \(V\) with a half-space \(\{x_i\ge 0\}\) and the topology of the various deformations \(V_t\). However, in these cases the results are for the moment not as complete as those obtained for the topology of \(V\). Proofs are referred to published articles or only outlined when complete proofs are still to appear.

Stability of general multi-Euler-Lagrange quadratic functional equations in non-Archimedean fuzzy normed spaces

In this paper we prove the generalized Hyers-Ulam stability of the system defining general Euler-Lagrange quadratic mappings in non-Archimedean fuzzy normed spaces over a field with valuation using the direct and the fixed point methods.MSC:39B82, 39B52, 46H25.

Approximate additive and quadratic mappings in 2-Banach spaces and related topics

Won{Gil Park [Won{Gil Park, J. Math. Anal. Appl., 376 (1) (2011) 193{202] proved the Hyers{ Ulam stability of the Cauchy functional equation, the Jensen functional equation and the quadratic functional equation in 2{Banach spaces. One can easily see that all results of this paper are incorrect. Hence the control functions in all theorems of this paper are not correct. In this paper, we correct these results.

Approximation of radical functional equations related to quadratic and quartic mappings

In this paper, we introduce and solve of the radical quadratic and radical quartic functional equations: f(ax2+by2)=af(x)+bf(y),f(ax2+by2)+f(|ax2−by2|)=2a2f(x)+2b2f(y). We also establish some stability results in 2-normed spaces and then the stability by using subadditive and subquadratic functions in p-2-normed spaces for these functional equations.

Nearly Quadratic Mappings over p‐Adic Fields

We establish some stability results over p‐adic fields for the generalized quadratic functional equation where n ∈ ℕ and n ≥ 2.

Stabilities of Cubic Mappings in Various Normed Spaces: Direct and Fixed Point Methods

In 1940 and 1964, Ulam proposed the general problem: “When is it true that by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?”. In 1941, Hyers solved this stability problem for linear mappings. According to Gruber (1978) this kind of stability problems are of the particular interest in probability theory and in the case of functional equations of different types. In 1981, Skof was the first author to solve the Ulam problem for quadratic mappings. In 1982–2011, J. M. Rassias solved the above Ulam problem for linear and nonlinear mappings and established analogous stability problems even on restricted domains. The purpose of this paper is the generalized Hyers‐Ulam stability for the following cubic functional equation: f(mx + y) + f(mx − y) = mf(x + y) + mf(x − y) + 2(m 3 − m)f(x), m ≥ 2 in various normed spaces.

A generalized Bernstein approximation theorem

ABSTRACT The present paper is concerned with some generalizations of Bernstein’s approximation theorem. One of the most elegant and elementary proofs of the classic result, for a function f(x) defined on the closed interval [0, 1], uses the Bernstein’s polynomials of f, We shall concern the m-dimensional generalization of the Bernstein’s polynomials and the Bernstein’s approximation theorem by taking an (m−1)-dimensional simplex in cube [0, 1]m. This is motivated by the fact that in the field of mathematical biology naturally arouse dynamic systems determined by quadratic mappings of “standard” (m − 1)-dimensional simplex to self. The last condition guarantees saving of the fundamental simplex. Then there are surveyed some other the m-dimensional generalizations of the Bern- stein’s polynomials and the Bernstein’s approximation theorem.

Solution and stability of mixed type functional equation in non-Archimedean random normed spaces

The generalized stability of the Euler-Lagrange quadratic mappings in the framework of non-Archimedean random normed spaces is proved. The interdisciplinary relation among the theory of random spaces, the theory of non-Archimedean spaces, and the theory of functional equations is presented.

STABILITY OF A QUADRATIC FUNCTIONAL EQUATION IN INTUITIONISTIC FUZZY NORMED SPACES

In this paper, we determine some stability results concern- ing the 2-dimensional vector variable quadratic functional equation f(x+ y;z + w) + f(x y;z w) = 2f(x;z) + 2f(y;w) in intuitionistic fuzzy normed spaces (IFNS). We dene the intuitionistic fuzzy continuity of the 2-dimensional vector variable quadratic mappings and prove that the ex- istence of a solution for any approximately 2-dimensional vector variable quadratic mapping implies the completeness of IFNS.

On two questions of optimization theory concerning quadratic mappings

In this paper we present solutions of two open problems from the theory of quadratic mappings arising from the optimization theory and formulated by Hiriart-Urruty (SIAM Rev 49(2):255–273, 2007). The first problem is concerned with the non-triviality of the set of common zeroes of finitely many of quadratic forms. The second problem deals with the positivity of the maximum of finitely many of quadratic forms.

QUADRATIC MAPPINGS ASSOCIATED WITH INNER PRODUCT SPACES

In [7], Th.M. Rassias proved that the norm defined over a real vector space V is induced by an inner product if and only if for a fixed integer <TEX>$n{\geq}2$</TEX> <TEX>$${\sum_{i=1}^{n}}\left\|x_i-{\frac{1}{n}}{\sum_{j=1}^{n}}x_j \right\|^2={\sum_{i=1}^{n}}{\parallel}x_i{\parallel}^2-n\left\|{\frac{1}{n}}{\sum_{i=1}^{n}}x_i \right\|^2$$</TEX> holds for all <TEX>$x_1$</TEX>, <TEX>${\cdots}$</TEX>, <TEX>$x_n{\in}V$</TEX>. Let V, W be real vector spaces. It is shown that if an even mapping <TEX>$f:V{\rightarrow}W$</TEX> satisfies <TEX>$$(0.1)\;{\sum_{i=1}^{2n}f}\(x_i-{\frac{1}{2n}}{\sum_{j=1}^{2n}}x_j\)={\sum_{i=1}^{2n}}f(x_i)-2nf\({\frac{1}{2n}}{\sum_{i=1}^{2n}}x_i\)$$</TEX> for all <TEX>$x_1$</TEX>, <TEX>${\cdots}$</TEX>, <TEX>$x_{2n}{\in}V$</TEX>, then the even mapping <TEX>$f:V{\rightarrow}W$</TEX> is quadratic. Furthermore, we prove the generalized Hyers-Ulam stability of the quadratic functional equation (0.1) in Banach spaces.

Fixed Points and Stability for Quadratic Mappings in β–Normed Left Banach Modules on Banach Algebras

The goal of the present paper is to investigate some new stability results by applying the alternative fixed point of generalized quadratic functional equation $$\begin{array}{ll}f\left(\sum\limits_{i=1}^{n}a_ix_i\right)+{\sum\limits_{i=1}^{n-1}}{\sum\limits_{j=i+1}^{n}}\left[f(a_ix_i+a_jx_j)+2f(a_ix_i-a_jx_j)\right]\\ \qquad \quad = (3n-2){\sum\limits_{i=1}^{n}}a^2_{i}f(x_{i})\end{array}$$ in β–Banach modules on Banach algebras, where $${a_{1},\dots,a_{n}\in \mathbb{Z}{\setminus}\{0\}}$$ and some $${\ell\in\{1 , 2 ,\dots, n-1\},}$$ a l ≠ ±1 and a n = 1, where n is a positive integer greater or at least equal to two.

Approximate Quartic and Quadratic Mappings in Quasi‐Banach Spaces

we establish the general solution for a mixed type functional equation of aquartic and a quadratic mapping in linear spaces. In addition, we investigate the generalized Hyers‐Ulam stability in p‐Banach spaces.

Fuzzy Stability of a Functional Equation Deriving from Quadratic and Additive Mappings

We investigate a fuzzy version of stability for the functional equation f(2x + y) + f(2x − y)+2f(x) − f(x + y) − f(x − y) − 2f(2x) = 0 in the sense of Mirmostafaee and Moslehian.