Some matrix functional equations

We investigate the pair of matrix functional equations G(x)F(y) = G(xy) and G(x)G(y) = F(y/x), featuring the two independent scalar variables x and y and the two N×N matrices F(z) andG(z) (with N an arbitrary positive integer and the elements of these two matrices functions of the scalar variable z). We focus on the simplest class of solutions, i.e., on matrices all of whose elements are analytic functions of the independent variable. While in the scalar (N = 1) case this pair of functional equations only possess altogether trivial constant solutions, in the matrix (N > 1) case there are nontrivial solutions. These solutions satisfy the additional pair of functional equations F(x)G(y) = G(y/x) andF(x)F(y) = F(xy), and an endless hierarchy of other functional equations featuring more than two independent variables.     

On the Surface Integral Approximation of the Second Derivatives of the Potential of a Bulk Charge Located in a Layer of Small Thickness

We consider a bulk charge potential of the form

$$u(x) = \int\limits_\Omega {g(y)F(x - y)dy,x = ({x_1},{x_2},{x_3}) \in {\mathbb{R}^3},} $$

where Ω is a layer of small thickness h > 0 located around the midsurface Σ, which can be either closed or open, and F(x ? y) is a function with a singularity of the form 1/|x ? y|. We prove that, under certain assumptions on the shape of the surface Σ, the kernel F, and the function g at each point x lying on the midsurface Σ (but not on its boundary), the second derivatives of the function u can be represented as

$$\frac{{{\partial ^2}u(x)}}{{\partial {x_i}\partial {x_j}}} = h\int\limits_\Sigma {g(y)\frac{{{\partial ^2}F(x - y)}}{{\partial {x_i}\partial {x_j}}}} dy - {n_i}(x){n_j}(x)g(x) + {\gamma _{ij}}(x),i,j = 1,2,3,$$

where the function γ_ij(x) does not exceed in absolute value a certain quantity of the order of h², the surface integral is understood in the sense of Hadamard finite value, and the n_i(x), i = 1, 2, 3, are the coordinates of the normal vector on the surface Σ at a point x.

     

On the two-dimensional stationary Schrödinger equation with a singular potential

We study the equation

−△u(x,y)+ν(x,y)u(x,y)=0     

Stability of the preservation of the equality of distance

We study the stability problem for mappings satisfying the equation

‖f(x−y)‖=‖f(x)−f(y)‖.     

Diophantine inequality involving binary forms

Let d ? 3 be an integer, and set r = 2^d?1 + 1 for 3 ? d ? 4, \(\tfrac{{17}}{{32}} \cdot 2^d + 1\) for 5 ? d ? 6, r = d²+d+1 for 7 ? d ? 8, and r = d²+d+2 for d ? 9, respectively. Suppose that Φ _i (x, y) ∈ ?[x, y] (1 ? i ? r) are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ₁, λ₂,..., λ _r are nonzero real numbers with λ₁/λ₂ irrational, and λ₁Φ₁(x₁, y₁) + λ₂Φ₂(x₂, y₂) + · · · + λ _r Φ _r (x _r , y _r ) is indefinite. Then for any given real η and σ with 0 < σ < 2^2?d, it is proved that the inequality

$$\left| {\sum\limits_{i = 1}^r {{\lambda _i}\Phi {}_i\left( {{x_i},{y_i}} \right) + \eta } } \right| < {\left( {\mathop {\max \left\{ {\left| {{x_i}} \right|,\left| {{y_i}} \right|} \right\}}\limits_{1 \leqslant i \leqslant r} } \right)^{ - \sigma }}$$

has infinitely many solutions in integers x₁, x₂,..., x _r , y₁, y₂,..., y _r . This result constitutes an improvement upon that of B. Q. Xue.

     

Christensen measurable solutions of some functional equation

Let X be a separable F-space over the field K of reals or complex numbers. We characterize solutions of the equation

f(x+M(f(x))y)=f(x)f(y)     

The zeros of a quadratic form at square-free points

Let F(x₁,…,xn) be a nonsingular indefinite quadratic form, n=3 or 4. For n=4, suppose the determinant of F is a square. Results are obtained on the number of solutions of

F(x₁,…,xn)=0     

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).     

Average value problems in ordinary differential equations

Using Schauder's fixed point theorem, with the help of an integral representation in ‘Sharp conditions for weighted 1-dimensional Poincaré inequalities’, Indiana Univ. Math. J., 49 (2000) 143-175, by Chua and Wheeden, we obtain existence and uniqueness theorems and ‘continuous dependence of average condition’ for average value problem:

y^′=F(x,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)     

The stability of Cauchy's gamma-beta functional equation

We obtain the super stability of Cauchy's gamma-beta functional equation

B(x,y)f(x+y)=f(x)f(y),     

Fixed points of asymptotic contractions

Let be a contractive gauge function in the sense that φ is continuous, φ(s)<s for s>0, and if f:M→M satisfies d(f(x),f(y))?φ(d(x,y)) for all x,y in a complete metric space (M,d), then f always has a unique fixed point. It is proved that if T:M→M satisfies

     

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 solutions of a common generalization of the Go?a?b-Schinzel equation and of the addition formulae

Under some additional assumptions we determine solutions of the equation

f(x+M(f(x))y)=f(x)○f(y),     

Invariance of quasi-arithmetic means with function weights

Let I⊂R be a non-trivial interval and let . We present some results concerning the following functional equation, generalizing the Matkowski-Sutô equation,

λ(x,y)φ⁻¹(μ(x,y)φ(x)+(1−μ(x,y))φ(y))+(1−λ(x,y))ψ⁻¹(ν(x,y)ψ(x)+(1−ν(x,y))ψ(y))=λ(x,y)x+(1−λ(x,y))y,     

On Lie’s problem and differential invariants of ODEs <Emphasis Type="Italic">y</Emphasis>″ = <Emphasis Type="Italic">F</Emphasis>(<Emphasis Type="Italic">x</Emphasis>, <Emphasis Type="Italic">y</Emphasis>)

A point classification of ordinary differential equations of the form y″ = F(x, y) is considered. The algebra of differential invariants of the action of the point symmetry pseudogroup on the right-hand sides of equations of the form y″ = F(x, y) is calculated, and Lie’s problem on the point equivalence of such equations is solved.     

Existence theorems for a class of nonlocal differential equations

A class of nonlocal second-order ordinary differential equations of the form

y^″(x)=f(x,y(x),(y○λ)(x),y^′(x))     

Functional equations with exotic addition

We deal with the following “exotic” addition:

x⊕y:=xf(y)+yf(x)