Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces

In this paper we establish the general solution of the functional equation

f(2x+y)+f(2x−y)=f(x+y)+f(x−y)+2f(2x)−2f(x)     

On the stability of a quartic functional equation

In this paper, we prove the generalized Hyers-Ulam stability for the following quartic functional equation

f(2x+y)+f(2x−y)=4f(x+y)+4f(x−y)+24f(x)−6f(y).     

Quartic functional equations

In this paper, we solve a new functional equation

f(2x+y)+f(2x−y)=4f(x+y)+4f(x−y)+24f(x)−6f(y)     

Stability of a mixed additive and cubic functional equation in quasi-Banach spaces

In this paper we establish the general solution and investigate the Hyers-Ulam-Rassias stability of the following functional equation

f(2x+y)+f(2x−y)=2f(x+y)+2f(x−y)+2[f(2x)−2f(x)]     

Stability for quadratic functional equation in the spaces of generalized functions

In this paper, we consider the general solution of quadratic functional equation

f(ax+y)+f(ax−y)=f(x+y)+f(x−y)+2(a²−1)f(x)     

A functional equation originating from quadratic forms

In this paper, we obtain the general solution and the stability of the 2-variable quadratic functional equation

f(x+y,z+w)+f(x−y,z−w)=2f(x,z)+2f(y,w).     

The stability of d'Alembert and Jensen type functional equations

The aim of this paper is to study the stability problem of the d'Alembert type and Jensen type functional equations:

f(x+y)+f(x+σy)=2g(x)f(y),     

Stability of Cubic and Quartic Functional Equations in Non-Archimedean Spaces

We prove generalized Hyers-Ulam–Rassias stability of the cubic functional equation f(kx+y)+f(kx?y)=k[f(x+y)+f(x?y)]+2(k ³?k)f(x) for all \(k\in \Bbb{N}\) and the quartic functional equation f(kx+y)+f(kx?y)=k ²[f(x+y)+f(x?y)]+2k ²(k ²?1)f(x)?2(k ²?1)f(y) for all \(k\in \Bbb{N}\) in non-Archimedean normed spaces.     

On the Stability of a k-Cubic Functional Equation in Intuitionistic Fuzzy n-Normed Spaces

In this paper, we investigate some stability results concerning the k-cubic functional equation f(kx + y) + f(kx?y) = kf(x + y) + kf(x?y) + 2k(k²?1)f(x) in the intuitionistic fuzzy n-normed spaces.     

Local convergence of some iterative methods for generalized equations

We study generalized equations of the following form:

(render)

0f(x)+g(x)+F(x),

where f is Fréchet differentiable in a neighborhood of a solution x^* of (*) and g is Fréchet differentiable at x^* and where F is a set-valued map acting in Banach spaces. We prove the existence of a sequence (x_k) satisfying

which is super-linearly convergent to a solution of (*). We also present other versions of this iterative procedure that have superlinear and quadratic convergence, respectively.     

Remark on a double-inequality for the Riemann zeta function

Let ζ be the Riemann zeta function and δ(x)=1/(2^x-1). For all x>0 we have

(1-δ(x))ζ(x)+αδ(x)<ζ(x+1)<(1-δ(x))ζ(x)+βδ(x),

with the best possible constant factors

This improves a recently published result of Cerone et al., J. Inequalities Pure Appl. Math. 5(2) (43) (2004), who showed that the double-inequality holds with and .     

Quadrature rules using first derivatives for oscillatory integrands

We consider the integral of a function and its approximation by a quadrature rule of the form

i.e., by a rule which uses the values of both y and its derivative at nodes of the quadrature rule. We examine the cases when the integrand is either a smooth function or an ω dependent function of the form y(x)=f₁(x) sin(ωx)+f₂(x) cos(ωx) with smoothly varying f₁ and f₂. In the latter case, the weights w_k and α_k are ω dependent. We establish some general properties of the weights and present some numerical illustrations.     

Decomposer and associative functional equations

In the previous researches [2,3] b-integer and b-decimal parts of real numbers were introduced and studied by M.H. Hooshmand. The b-parts real functions have many interesting number theoretic explanations, analytic and algebraic properties, and satisfy the functional equation f (f(x) + y - f(y)) = f(x). These functions have led him to a more general topic in semigroups and groups (even in an arbitrary set with a binary operation [4] and the following functional equations have been introduced: Associative equations:

f(xf(yz))=f(f(xy)z),f(xf(yz))=f(f(xy)z)=f(xyz)

. Decomposer equations:

f(f^*(x)f(y))=f(y),f(f(x)f_*(y))=f(x)

.Strong decomposer equations:

f(f^*(x)y)=f(y),f(xf_*(y))=f(x)

.Canceler equations:

f(f(x)y)=f(xy),f(xf(y))=f(xy),f(xf(y)z)=f(xyz)

, where f^*(x) f(x) = f (x) f_* (x) = x. In this paper we solve them and introduce the general solution of the decomposer and strong decomposer equations in the sets with a binary operation and semigroups respectively and also associative equations in arbitrary groups. Moreover we state some equivalent equations to them and study the relations between the above equations. Finally we prove that the associative equations and the system of strong decomposer and canceler equations do not have any nontrivial solutions in the simple groups.     

Functional equations with exotic addition

We deal with the following “exotic” addition:

x⊕y:=xf(y)+yf(x)     

On some composite functional inequalities

The aim of the paper is to deal with the following composite functional inequalities

Stability of Jensen functional equation in intuitionistic fuzzy normed space

In this paper, we determine some stability results concerning the Jensen functional equation 2f((x+y)/2)=f(x)+f(y) in intuitionistic fuzzy normed spaces (IFNS). We define the intuitionistic fuzzy continuity of the Jensen mappings and prove that the existence of a solution for any approximately Jensen mapping implies the completeness of IFNS.     

A functional equation having monomials as solutions

For each n=1,2,3, we obtain the general solution and the stability of the functional equation

f(2x+y)+f(2x-y)=2^n-2[f(x+y)+f(x-y)+6f(x)].     

On the stability of Euler-Lagrange type cubic mappings in quasi-Banach spaces

In this paper, we solve the generalized Hyers-Ulam-Rassias stability problem for Euler-Lagrange type cubic functional equations

f(ax+y)+f(x+ay)=(a+1)²(a−1)[f(x)+f(y)]+a(a+1)f(x+y)     

Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces

In this paper, we achieve the general solution and the generalized Hyers–Ulam–Rassias stability of the following functional equation

f(x+ky)+f(x−ky)=k²f(x+y)+k²f(x−y)+2(1−k²)f(x)

Generalized Hyers-Ulam Stability for a General Mixed Functional Equation in Quasi-��-normed Spaces

In this paper, we establish the general solution and investigate the generalized Hyers-Ulam stability of the following mixed additive and quadratic functional equation