Uniqueness properties in finite-continuous additive measurement

Let ? be a binary relation on A×X, and suppose that there are real valued functions f on A and g on X such that, for all ax, by ∈ A×X, ax ? by if and only if f (a)+g(x) ? f(b)+g(y). This paper establishes uniqueness properties for f and g when A is a finite set, X is a real interval with g increasing on X, and for any a, b and x there is a y for which f(a)+g(x)=f(b)+g(y). The resultant uniqueness properties occupy an intermediate position among uniqueness properties for other structural cases of two-factor additive measurement.It is shown that f is unique up to a positive affine transformation (αf+β₁ with α > 0), but that g is unique up to a similar positive affine transformation (αg+β₂) if and only if the ratio [f(a)?f(b)]/[f(a)?f(c)] is irrational for some a, b, c ∈ A. When the f ratios are rational for all cases where they are defined, there will be a half-open interval (x₀, x₁) in X such that the restriction of g on (x₀, x₁) can be any increasing function for which sup {g(x)?g(x₀): x₀ ? x < x₁} does not exceed a specified bound, and, when g is thus defines on (x₀, x₁), it will be uniquely determined on the rest of X. In general, g must be continuous only in the ‘irrational’ case.     

The pexiderized Go?a?b-Schinzel functional equation

Let X be a linear space over a commutative field K. We characterize a general solution f,g,h,k:X→K of the pexiderized Go?a?b-Schinzel equation f(x+g(x)y)=h(x)k(y), as well as real continuous solutions of the equation.     

Analysis of a singular functional differential equation that arises from stochastic modeling of population growth

The singular functional differential equation x(1 ? x)A(x)y′(x) + by(h(x)) ? by(x) = ?bg(x), x in (0, 1), is studied for initial data y = 0 on x ? a, y continuous on (a, 1) and y(1?) bounded. The singularity at x = 0+ is removable for a certain class of delayed arguments, h(x). The final end point at x = 1? is the most important singularity because it results in a genuine singular boundary value problem. A formal solution is constructed and is shown to be unique and bounded when g(x) is bounded. A singular decomposition transforms the problem into two nonsingular initial value problems. Singular FDEs of this type arise in the study of the persistence of populations undergoing large random fluctuations when modeled by compound Poisson processes superimposed on logistic-type growth.     

Structural Stability of Constrained Polynomial Systems

The structural stability of constrained polynomial differentialsystems of the form a(x, y)x'+b(x, y)y'=f(x, y), c(x, y)x'+d(x,y)y'=g(x, y), under small perturbations of the coefficientsof the polynomial functions a, b, c, d, f and g is studied.These systems differ from ordinary differential equations at‘impasse points’ defined by ad–bc=0. Extensionsto this case of results for smooth constrained differentialsystems [7] and for ordinary polynomial differential systems[5] are achieved here. 1991 Mathematics Subject Classification34C35, 34D30.     

A generalization of the cosine-sine functional equation on groups

The functional equation f(xy)=f(x)g(y)+g(x)f(y)+h(x)h(y) is solved where f, g, h are complex functions defined on a group.     

Binding number and minimum degree for the existence of (g,f,n)-critical graphs

Let G be a graph of order p. The binding number of G is defined as $\mbox{bind}(G):=\min\{\frac{|N_{G}(X)|}{|X|}\mid\emptyset\neq X\subseteq V(G)\,\,\mbox{and}\,\,N_{G}(X)\neq V(G)\}$ . Let g(x) and f(x) be two nonnegative integer-valued functions defined on V(G) with g(x)≤f(x) for any x∈V(G). A graph G is said to be (g,f,n)-critical if G?N has a (g,f)-factor for each N?V(G) with |N|=n. If g(x)≡a and f(x)≡b for all x∈V(G), then a (g,f,n)-critical graph is an (a,b,n)-critical graph. In this paper, several sufficient conditions on binding number and minimum degree for graphs to be (a,b,n)-critical or (g,f,n)-critical are given. Moreover, we show that the results in this paper are best possible in some sense.     

Continuous on rays solutions of an equation of the Go?a?b-Schinzel type

Let X be a real linear space. We characterize continuous on rays solutions f,g:X→R of the equation f(x+g(x)y)=f(x)f(y). Our result refers to papers of J. Chudziak (2006) [14] and J. Brzd?k (2003) [11].     

On the algebraic structure of Abelian integrals for a kind of perturbed cubic Hamiltonian systems

The finite generators of Abelian integral are obtained, where Γh is a family of closed ovals defined by H(x,y)=x²+y²+ax⁴+bx²y²+cy⁴=h, h∈Σ, ac(4ac−b²)≠0, Σ=(0,h₁) is the open interval on which Γh is defined, f(x,y), g(x,y) are real polynomials in x and y with degree 2n+1 (n?2). And an upper bound of the number of zeros of Abelian integral I(h) is given by its algebraic structure for a special case a>0, b=0, c=1.     

Asymmetry of twisted convolution operators

Let K be a distribution on $\text{R}$². We denote by λ(K) the twisted convolution operator f → K × f defined by the formula K × f(x, y) = ∝∝ dudvK(x ? u, y ? v) f(u, v) exp(ixv ? iyu). We show that there exists K such that the operator λ(K) is bounded on L^p(R)² for every p in (1, 2¦, but is unbounded on L^q(R)² for every q > 2.     

Skew Derivations of Prime Rings

Given a prime ring R, a skew g-derivation for g : R → R is an additive map f : R → R such that f(xy) = f(x)g(y) + xf(y) = f(x)y + g(x)f(y) and f(g(x)) = g(f(x)) for all x, y ∈ R. We generalize some properties of prime rings with derivations to the class of prime rings with skew derivations.     

New results on value functions and applications to MaxSup and MaxInf problems

In this paper, in the setting of sequential spaces, we consider the value functions v_S and v_I defined by v_S(x)=sup_y∈K(x)f(x,y) and v_I(x)=inf_y∈K(x)f(x,y), and we introduce sufficient conditions on the value functions and on the data, weaker than upper semicontinuity, which guarantee the existence of maximum for such functions. Various examples illustrate the results and the connections with previous results.     

Generalized weighted quasi-arithmetic means

Under some natural assumptions on real functions f and g defined on a real interval I, we show that a two variable function M _f,g : I ² → I defined by

On Olaleru’s open problem on Gregus fixed point theorem

Let (X, d) be a complete metric space and ${TX \longrightarrow X }$ be a mapping with the property d(Tx, Ty) ≤ ad(x, y) + bd(x, Tx) + cd(y, Ty) + ed(y, Tx) + fd(x, Ty) for all ${x, y \in X}$ , where 0 < a < 1, b, c, e, f ≥ 0, a + b + c + e + f = 1 and b + c > 0. We show that if e + f > 0 then T has a unique fixed point and also if e + f ≥ 0 and X is a closed convex subset of a complete metrizable topological vector space (Y, d), then T has a unique fixed point. These results extend the corresponding results which recently obtained in this field. Finally by using our main results we give an answer to the Olaleru’s open problem.     

A nonlinear three-point boundary value problem

We consider the differential equation ?(py′)′ + qy + λay + μby + f(x, y, y′) = 0, x? (α, γ) subject to the boundary conditions cos(α₁) y(α) ? sin(α₁) y′(α) = 0cos(β₁) y(β) ? sin(β₁) y′(β) = 0 β? (α, γ)cos(γ₁) y(γ) ? sin(γ₁) y′(γ) = 0. The functions p, g, a, b, and f are well-behaved functions of x; f is smooth and of “higher order” in y and y′; the scalars λ and μ are eigenparameters. With mild restrictions on a and b it is known that the linearized problem, f ≡ 0, has eigensolutions, ($\text{λ}{}^1\text{, μ}{}^1\text{, ψ}{}^1$). In this paper we use an Implicit Function Theorem argument to establish the existence of a local branch of solutions, bifurcating from ($\text{λ}{}^1\text{, μ}{}^1\text{, 0}$), to the above nonlinear two-parameter eigenvalue problem.     

Oscillation of partial difference equations with continuous variables

In this paper, we consider the partial difference equation with continuous variables of the form P₁z(x + a, y + b) + p₂z (x + a, y) + p₃z (x, y + b) − p₄z (x, y) + P (x, y) z (x − τ, y − σ) = 0, where P ϵ C(R⁺ × R⁺, R⁺ − {0}), a, b, τ, σ are real numbers and p_i (i = 1, 2, 3, 4) are nonnegative constants. Some sufficient conditions for all solutions of this equation to be oscillatory are obtained.     

A Priori Estimates for Solutions of Singular Perturbations with a Turning Point

A priori estimates are derived for the solutions of the boundary value problem εy″ + a(x)y′ + b(x)y = f(x), c ? x ? d, y(c) = α, y(d) = β. Here 0 < ε ? 1 is a small parameter and a(x) has a single simple zero in [c,d] (the turning point). It is shown that the solutions of this problem are uniformly bounded for ε→0 by the norms of f, α and β if and only if b(x)<0 at the turning point. However, in certain cases there are weak a priori estimates for the solutions even if this condition is not fulfilled.     

Uniformly continuous superposition operators in the space of bounded variation functions

Let I, J ? ? be intervals. The main result says that if a superposition operator H generated by a function of two variables h: I × J → ?, H (φ)(x) ? h (x, φ (x)), maps the set BV (I, J) of all bounded variation functions, φ: I → J into the Banach space BV (I, ?) and is uniformly continuous with respect to the BV ‐norm, then h (x, y) = a (x)y + b (x), x ∈ I, y ∈ J, for some a, b ∈ BV (I, ?) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Parameter Depending Differential Systems Under Carathéodory Conditions

For every λ in a complex domain G, consider on some interval I the initial value problem y′(λ,x) = A(λ,x)y(λ,x) + b(λ,x), y(λ,x₀) - y₀. If this problem satisfies the Carathéodory conditions for every A, then there exist locally absolutely continuous and almost everywhere differentiable solutions y(λ,· ) of the initial value problem. In general, the union N of the exceptional sets N _λ ? I where y(λ, ·) is not differentiate or does not fulfill the differential equation, is not of Lebesgue measure zero. It will be shown that N is of Lebesgue measure zero provided that A and b are holomorphic with respect to λ and their integrals with respect to x are locally bounded on G × I.     

Asymptotically stable sets and the stability of ω-limit sets

Let C be the collection of continuous self-maps of the unit interval I=[0,1] to itself. For f∈C and x∈I, let ω(x,f) be the ω-limit set of f generated by x, and following Block and Coppel, we take Q(x,f) to be the intersection of all the asymptotically stable sets of f containing ω(x,f). We show that Q(x,f) tells us quite a bit about the stability of ω(x,f) subject to perturbations of either x or f, or both. For example, a chain recurrent point y is contained in Q(x,f) if and only if there are arbitrarily small perturbations of f to a new function g that give us y as a point of ω(x,g). We also study the structure of the map Q taking (x,f)∈I×C to Q(x,f). We prove that Q is upper semicontinuous and a Baire 1 function, hence continuous on a residual subset of I×C. We also consider the map given by x?Q(x,f), and find that this map is continuous if and only if it is a constant map; that is, only when the set is a singleton.