Variational problems with lags

In this paper, we study some questions concerning the minima of the functional $$J\left( y \right) = \int_{x_1 }^{x_2 } {f\left( {x,y\left( x \right),y\left( {x - r} \right),\dot y\left( x \right),\dot y\left( {x - r} \right)} \right)dx} $$ In Section 2, we obtain an analogue to the Jacobi condition to add to the list of previously obtained necessary conditions. A transversality condition is developed in Section 3. In Section 4, we obtain an existence theorem. The techniques used are modifications of those used in the classical problems. In Section 5, we show that the theory of fields for the classical problem fails to work for our problem.     

A New Proof of Spinks' Theorem

Skew lattices form a class of non-commutative lattices. Spinks' Theorem [Matthew Spinks, On middle distributivity for skew lattices, ] states that for symmetric skew lattices the two distributive identities and are equivalent. Up to now only computer proofs of this theorem have been known. In the present paper the author presents a direct proof of Spinks' Theorem. In addition, a new result is proved showing that the assumption of symmetry can be omitted for cancellative skew lattices.     

On the stability of Hosszú’s functional equation

Two stability results are proved. The first one states that Hosszú’s functional equation $$f(x+y-xy)+f(xy)=f(x)-f(y)=0\ \ \ \ \ (x,y \in \rm R)$$ is stable. The second is a local stability theorem for additive functions in a Banach space setting.     

Lower and upper solution method for the problem of elastic beam with hinged ends

We develop the method of lower and upper solutions for the fourth-order differential equation which models the stationary states of the deflection of an elastic beam, whose both ends simply supported

$$\begin{aligned}&y^{(4)}(x)+(k_1+k_2) y''(x)+k_1k_2 y(x)=f(x,y(x)), \ \ \ \ x\in (0,1),\\&y(0) = y(1) = y''(0) = y''(1) = 0\\ \end{aligned}$$

under the condition \(0<k_1<k_2<x_1^2\approx 4.11585\), where \(x_1\) is the first positive solution of the equation \(x\cos (x)+\sin (x)=0\). The main tools are Schauder fixed point theorem and the Elias inequality.

     

Multiplicative symmetry and related functional equations

Summary Let (G, *) be a commutative monoid. Following J. G. Dhombres, we shall say that a functionf: G G is multiplicative symmetric on (G, *) if it satisfies the functional equationf(x * f(y)) = f(y * f(x)) for allx, y inG. (1)Equivalently, iff: G G satisfies a functional equation of the following type:f(x * f(y)) = F(x, y) (x, y G), whereF: G × G G is a symmetric function (possibly depending onf), thenf is multiplicative symmetric on (G, *).In Section I, we recall the results obtained for various monoidsG by J. G. Dhombres and others concerning the functional equation (1) and some functional equations of the formf(x * f(y)) = F(x, y) (x, y G), (E) whereF: G × G G may depend onf. We complete these results, in particular in the case whereG is the field of complex numbers, and we generalize also some results by considering more general functionsF. In Section II, we consider some functional equations of the formf(x * f(y)) + f(y * f(x)) = 2F(x, y) (x, y K), where (K, +, ·) is a commutative field of characteristic zero, * is either + or · andF: K × K K is some symmetric function which has already been considered in Section I for the functional equation (E). We investigate here the following problem: which conditions guarantee that all solutionsf: K K of such equations are multiplicative symmetric either on (K, +) or on (K, ·)? Under such conditions, these equations are equivalent to some functional equations of the form (E) for which the solutions have been given in Section I. This is a partial answer to a question asked by J. G. Dhombres in 1973.     

Generic boundary behaviour of Taylor series in Hardy and Bergman spaces

We establish a general identity between the Mahler measures \(\mathrm {m}(Q_k(x,y))\) and \(\mathrm {m}(P_k(x,y))\) of two polynomial families, where \(Q_k(x,y)=0\) and \(P_k(x,y)=0\) are generically hyperelliptic and elliptic curves, respectively.     

The equivalence of two functional equations involving the arithmetic mean, the geometric mean and their Gauss composition

In this paper, the equivalence of the two functional equations $$f\left(\frac{x+y}{2} \right)+f\left(\sqrt{xy} \right)=f(x)+f(y)$$ and $$2f\left(\mathcal{G}(x,y)\right)=f(x)+f(y)$$ will be proved by showing that the solutions of either of these equations are constant functions. Here I is a nonvoid open interval of the positive real half-line and ${\mathcal{G}}$ is the Gauss composition of the arithmetic and geometric means.     

The undecidability of pseudo real closed fields

The aim of this note is to establish the following result: THEOREM: Let be a non-empty class of Boolean spaces and letPRC() be the class of pseudo real closed fields whose spaces of orderings belong to . Then the elementary theory ofPRC() is undecidable.Our proof appears to be an interesting application of the theory of Artin-Schreier structures, which has been initiated in [5] for the purpose of characterization of the absolute Galois groups of PRC fields. In Section 1 we define and investigate Frattini covers of Artin-Schreier structures, in analogy with [6], Section 2. In Section 2 we consider the analogues of proofs of [1] and [3], to attain the Theorem.This work corresponds to a part of the doctoral dissertation done under the supervision of Prof. Moshe Jarden at Tel-Aviv University     

Nonlinear wavelet methods for high-dimensional backward heat equation

The backward heat equation is a typical ill-posed problem. In this paper, we shall apply a dual least squares method connecting Shannon wavelet to the following equation ut (x, y, t) = u xx (x, y, t) + uyy (x, y, t), x ∈ R, y ∈ R, 0 ≤ t 1, u(x, y, 1) = (x, y), x ∈ R, y ∈ R. Motivated by Regińska's work, we shall give two nonlinear approximate methods to regularize the approximate solutions for high-dimensional backward heat equation, and prove that our methods are convergent.     

On a nonlinear Sturm-Liouville problem arising in the study of soliton

In this article,we consider the following nonlinear Sturm-Liouville problem(?)and prove the existence of the eigenvalue and the eigenfunction by using Schauder's fixed opinttheorem.This problem arises from finding the solutions of solitons and stationary states of thenonlinear Schr(?)dinger equation (NLS Eq.) with external fields.Using the result obtained,we provethe existence of solitons and stationary states of the NLS equation for the oscillater.     

Additivity and exponentiality are alien to each other

Let (S, +) be a (semi)group and let (R,+, ·) be an integral domain. We study the solutions of a Pexider type functional equation $$f(x+y) + g(x+y) = f(x) + f(y) + g(x)g(y)$$ for functions f and g mapping S into R. Our chief concern is to examine whether or not this functional equation is equivalent to the system of two Cauchy equations $$\left\{\begin{array}{@{}ll} f(x+y) = f(x) + f(y)\\ g(x+y) = g(x)g(y)\end{array}\right.$$ for every ${x,y \in S}$ .     

Fixed points and additive $${\rho}$$-functional equations

In this paper, we solve the additive \({\rho}\)-functional equations

$$\begin{aligned} f(x+y)-f(x)-f(y)= & {} \rho(2f(\frac{x+y}{2})-f(x)-f(y)), \\ 2f(\frac{x+y}{2})-f(x)-f(y)= & {} \rho(f(x+y)-f(x)-f(y)), \end{aligned}$$

where \({\rho}\) is a fixed non-Archimedean number or a fixed real or complex number with \({\rho \neq 1}\). Using the fixed point method, we prove the Hyers–Ulam stability of the above additive \({\rho}\)-functional equations in non-Archimedean Banach spaces and in Banach spaces.

     

On a Functional Equation Associated with Simpson’s Rule

In this paper, we determine the general solution of the functional equation $$f(x)-g(y)=(x-y)\lbrack h(x+y)+\psi (x)+\phi (y)\rbrack$$ for all real numbers x and y. This equation arises in connection with Simpson’s Rule for the numerical evaluation of definite integrals. The solution of this functional equation is achieved through the functional equation $$g(x)-g(y)=(x-y)f(x+y)+(x+y)f(x-y).$$      

Commutativity theorems for cancellative semigroups

Psomopoulos has proved that \([x^n, y] = [x, y^{n+1}]\) for a positive integer n implies commutativity in groups. Here we show that cancellative semigroups admitting commutators and satisfying the identity \([x^n, y] = [x, y^{n+k}]\) implies that the element \(y^k\) is central. The special case of \(k=1\) yields the above mentioned commutativity theorem. To accommodate negative exponents, we consider the functional equation \([f(x), y] = [x, g(y)f(y)] \) where f and g are unary functions satisfying certain formal syntactic rules and prove that in cancellative semigroups admitting commutators, the functional equation \([f(x), y] = [x, g(y)f(y)]\) implies that the element g(y) is central i.e. \(xg(y) = g(y)x\) for all x and y. By the way, these results are new even in group theory.     

On the equality and comparison problem of a class of mean values

Let \({I\subset \mathbb {R}}\) be a nonvoid open interval. A function \({K:I^2\to I}\) is called an M-conjugate mean if there exists \({(p,q)\in [0,1]^2}\) and a continuous strictly monotone real valued function \({\varphi}\) on I such that

$K(x,y)=\varphi^{-1}(p\varphi(x)+q\varphi(y)+(1-p-q)\varphi(M(x,y)))=:M_ \varphi^{(p,q)}(x,y)$

holds for all \({x,y\in I}\). In this paper, we investigate the equality and comparison problem in the class of M-conjugate means, in the case when

$M(x,y):=\min\{x,y\}\quad (x,y\in I)$

.

     

Blue sky catastrophe in relaxation systems with one fast and two slow variables

We establish conditions under which three-dimensional relaxational systems of the form

$$\dot x = f(x,y,\mu ),\varepsilon \dot y = g(x,y),x = (x_1 ,x_2 ) \in \mathbb{R}^2 ,y \in \mathbb{R},$$

where 0 ≤ ε ? 1, |µ| ? 1, and f, g ∈ C ^∞, exhibit the so-called blue sky catastrophe [the appearance of a stable relaxational cycle whose period and length tend to infinity as µ tends to some critical value µ_*(ε), µ_*(0) = 0].

     

On the Non-Existence of Periodic Orbits for a Class of Two-Dimensional Kolmogorov Systems

In this paper we introduce an explicit expression of first integral, then we prove the nonexistence of periodic orbits, then consequently the non-existence of limit cycles of two-dimensional Kolmogorov system, where R(x, y), S (x, y), P (x, y), Q(x, y),M (x, y), N (x, y) are homogeneous polynomials of degrees m, a, n, n, b, b, respectively. We introduce concrete example exhibiting the applicability of our result.     

The Variety of Commutative BCI-Algebras is 2-Based

In this note, we first solve the following open problem in [5]: Can the variety of commutative BCI-algebras be defined by two identities? An algebra of type (2, 0) is a commutative BCI-algebra if and only if it satisfies $u\ast\left(((x \ast y) \ast (x \ast y))(z \ast y)\right) = uIn this note, we first solve the following open problem in [5]: Can the variety of commutative BCI-algebras be defined by two identities? An algebra of type (2, 0) is a commutative BCI-algebra if and only if it satisfies and (see Theorem 2 below).Next, we prove that I-variety [2] is also 2-based. Finally, we show that I-variety is a proper subvariety of the variety of commutative BCI-algebras.AMS Subject Classification (2000): 03G25, 06A10, 06D99     

A functional equation with a symmetric binary operation

Let X be a nonempty set containing at least two elements and let \({\circ :X^2\to X}\) be a symmetric binary operation. Furthermore, let A, B, C be real parameters and let \({f,g:X\to\mathbb{R}_+}\) be unknown functions. We investigate the functional equation

$f(x\circ y)[Ag(y)-Bg(x)]=(A+C)f(x)g(y)-(B+C)f(y)g(x)\quad {\rm for\,\,all}\,x,y \in X.$

     

Free algebras of a unary variety with Mal’tsev’s operation that satisfies the Pixley conditions

In this paper we consider the variety V P of algebras with one unary and one ternary operation p that satisfies the Pixley identities, provided that operations are permutable. We describe the structure of a free algebra of the variety V P and study the structure of unary reducts of free algebras. We prove the solvability of the word problem in free algebras and the uniqueness of a free basis; we also describe groups of automorphisms of free algebras. Similar results are obtained for free algebras of a subvariety of the variety V P defined by the identities p(p(x, y, z), y, z) = p(x, y, z) and p(x, y, p(x, y, z)) = p(x, y, z).