On approximation of approximately generalized quadratic functional equation via Lipschitz criteria

Let G be an Abelian group with a metric d and E ba a normed space. For any f : G → E we define the generalized quadratic difference of the function f by the formulaQkf(x, y) := f(x + ky) + f(x - ky) - f(x + y) - f(x - y) - 2(k2 - 1) f(y)for all x, y ∈ G and for any integer k with k ≠ 1, -1. In this paper, we achieve the general solution of equation Qkf(x, y) = 0; after it, we show that if Qkf is Lipschitz, then there exists a quadratic function K : G → E such that f - K is Lipschitz with the same constant. Moreover, some results concerning the stability of the generalized quadratic functional equation in the Lipschitz norms are presented. In the particular case, if k = 0 we obtain the main result that is in [7].Mathematics Subject Classification (2010): Primary 39B82, 39B52.Keywords: Generalized quadratic functional equation, stability, Lipschitz spac

On the superstability of generalized d’Alembert harmonic functions

The aim of this paper is to study the superstability problem of the d’Alembert type functional equation f(x+y+z)+f(x+y+σ(z))+f(x+σ(y)+z)+f(σ(x)+y+z)=4f(x)f(y)f(z) $$f(x + y + z) + f(x + y + \sigma (z)) + f(x + \sigma (y) + z) + f(\sigma (x) + y + z) = 4f(x)f(y)f(z)$$ for all x, y, z ∈ G, where G is an abelian group and σ : G → G is an endomorphism such that σ(σ(x)) = x for an unknown function f from G into ℂ or into a commutative semisimple Banach algebra

On approximate solution of Drygas functional equation according to the Lipschitz criteria

Let G be an Abelian group with a metric d and E be a normed space. For any f : G → E we define the Drygas difference of the function f by the formul

On the hyperstability of a pexiderised σ-quadratic functional equation on semigroups

DOI: 10.1017/S000497271700113

Stability of the Equation of <i>q</i>-Wright Affine Functions in Non-Archimedean (<i>n</i>,<i>β</i>)-Banach Spaces

In this article, we employ a version of some fixed point theory (FPT) to obtain stability results for the symmetric functional equation (FE) of q-Wright affine functions in non-Archimedean (n,β)-Banach spaces (nArch(n,β)-BS). Furthermore, we give some interesting consequences of our results. In this way, we generalize several earlier outcomes

Stability of the Equation of q-Wright Affine Functions in Non-Archimedean (n,&beta;)-Banach Spaces

In this article, we employ a version of some fixed point theory (FPT) to obtain stability results for the symmetric functional equation (FE) of q-Wright affine functions in non-Archimedean (n,&beta;)-Banach spaces (nArch(n,&beta;)-BS). Furthermore, we give some interesting consequences of our results. In this way, we generalize several earlier outcomes

The hyperstability of AQ-Jensen functional equation on 2-divisible abelian group and inner product spaces

In this paper, we prove the hyperstability of the following mixed additive-quadratic-Jensen functional equation 2f(x+y2)+f(x−y2)+f(y−x2)=f(x)+f(y)$$2f({{x + y} \over 2}) + f({{x - y} \over 2}) + f({{y - x} \over 2}) = f(x) + f(y)$$ in the class of functions from an 2-divisible abelian group G into a Banach space