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)     

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.     

The asymptotic behavior of nonoscillatory solutions of nonlinear neutral type difference equations

In this paper, the authors study the asymptotic behavior of solutions of second-order neutral type difference equations of the form

Δ²(y_n+py_n−k)+f(n,y_n−ℓ)=0,n

, n ε, and

Δ²(y_n+py_n−k)+f(n,y_n−ℓ,Δy_n−ℓ)=0,n

,n ε using some difference inequalities. We establish conditions under which all nonoscillatory solutions are asymptotic to an + b as n → ∞ with a and b ε .     

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.     

Anti-periodic solutions for a class of nonlinear th-order differential equations with delays

In this paper, we use the Leray–Schauder degree theory to establish new results on the existence and uniqueness of anti-periodic solutions for a class of nonlinear nth-order differential equations with delays of the form

x⁽ⁿ⁾(t)+f(t,x⁽ⁿ⁻¹⁾(t))+g(t,x(t−τ(t)))=e(t).

     

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 .     

A comparative study of functional equations characterising sine and cosine

A comparative study of the functional equationsf(x+y)f(x–y)=f ²(x)–f ²(y),f(y){f(x+y)+f(x–y)}=f(x)f(2y) andf(x+y)+f(x–y)=2f(x){1–2f ²(y/2)} which characterise the sine function has been carried out. The zeros of the functionf satisfying any one of the above equations play a vital role in the investigations. The relation of the equationf(x+y)+f(x–y)=2f(x){1–2f ²(y/2)} with D'Alembert's equation,f(x+y)+f(x–y)=2f(x)f(y) and the sine-cosine equationg(x–y)=g(x)g(y) +f(x)f(y) has also been investigated.     

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)]     

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 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)     

On existence of singular solutions

In the paper sufficient conditions are given under which the differential equation y⁽ⁿ⁾=f(t,y,…,y⁽ⁿ⁻²⁾)g(y⁽ⁿ⁻¹⁾) has a singular solution y :[T,τ)→R, τ<∞ fulfilling

     

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)     

A connection between a generalized Pascal matrix and the hypergeometric function

The n × n generalized Pascal matrix P(t) whose elements are related to the hypergeometric function ₂F₁(a, b; c; x) is presented and the Cholesky decomposition of P(t) is obtained. As a result, it is shown that

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is the solution of the Gauss's hypergeometric differential equation,

x(1 − x)y″ + [1 + (a + b − 1)x]y′ − ABY = 0

. where a and b are any nonnegative integers. Moreover, a recurrence relation for generating the elements of P(t) is given.     

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)].     

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),     

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.     

Long time behaviour for generalized complex Ginzburg–Landau equation

In this paper, the two-dimensional generalized complex Ginzburg–Landau equation (CGL)

u_t=ρu−Δφ(u)−(1+iγ)Δu−νΔ²u−(1+iμ)|u|^2σu+αλ₁(|u|²u)+β(λ₂)|u|²

is studied. The existence of global attractor for this equation with periodic boundary condition is established and upper bounds of Hausdorff and fractal dimensions of attractor are obtained.     

Construction Techniques for Highly Accurate Quasi-Interpolation Operators

Under mild additional assumptions this paper constructs quasi-interpolants in the form

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with approximation order ℓ−1, where_h(x) is a linear combination of translatesψ(x−jh) of a functionψinC^ℓ( ). Thus the order of convergence of such operators can be pushed up to a limit that only depends on the smoothness of the functionψ. This approach can be generalized to the multivariate setting by using discrete convolutions with tensor products of odd-degreeB-splines.     

On stability of a cubic functional equation in intuitionistic fuzzy normed spaces

In this paper, we determine some stability results concerning the cubic functional equation

f(2x+y)+f(2x-y)=2f(x+y)+2f(x-y)+12f(x)