The Hukuhara-Kneser property for parabolic systems with nonlinear boundary conditions

We analyze the asymptotic behavior as x → ∞ of the product integral Π_x0^xe^A(s)ds, where A(s) is a perturbation of a diagonal matrix function by an integrable function on [x₀,∞). Our results give information concerning the asymptotic behavior of solutions of certain linear ordinary differential equations, e.g., the second order equation y″ = a(x)y.     

Linear Differential Equations with Variable Coefficients in Regular and Some Singular Cases

In this paper the asymptotic properties as t → + ∞ for a single linear differential equation of the form x⁽ⁿ⁾ + a₁ (t)x^(n?1)+…. + a_n(t)x = 0, where the coefficients aj (z) are supposed to be of the power order of growth, are considered. The results obtained in the previous publications of the author were related to the so called regular case when a complete set of roots {λ,(t)}, j = 1, 2, …, n of the characteristic polynomial yⁿ + a₁ (t)y^n?1 + … + a_n(t) possesses the property of asymptotic separability. One of the main restrictions of the regular case consists of the demand that the roots of the set {λ,(t)} have not to be equivalent in pairs for t → + ∞. In this paper we consider the some more general case when the set of characteristic roots possesses the property of asymptotic independence which includes the case when the roots may be equivdent in pairs. But some restrictions on the asymptotic behaviour of their differences λ_i(t)→ λ_j(t) are preserved. This case demands more complicated technique of investigation. For this purpose the so called asymptotic spaces were introduced. The theory of asymptotic spaces is used for formal solution of an operator equation of the form x = A(x) and has the analogous meaning as the classical theory of solving this equation in Band spaces. For the considered differential equation, the main asymptotic terms of a fundamental system of solution is given in a simple explicit form and the asymptotic fundamental system is represented in the form of asymptotic Emits for several iterate sequences.     

On diophantine equations of the form {({\boldsymbol x}-{\boldsymbol a}_{\bf 1})({\boldsymbol x}-{\boldsymbol a}_{\bf 2}) \ldots ({\boldsymbol x}-{\boldsymbol a}_{\boldsymbol k}) + {\boldsymbol r} = {\boldsymbol y}^{ \boldsymbol n}}

Erd?s and Selfridge [3] proved that a product of consecutive integers can never be a perfect power. That is, the equation x(x?+?1)(x?+?2)...(x?+?(m???1))?=?y ⁿ has no solutions in positive integers x,m,n where m, n?>?1 and y?∈?Q. We consider the equation $$ (x-a_1)(x-a_2) \ldots (x-a_k) + r = y^n $$ where 0?≤?a ₁?<?a ₂?<???<?a _k are integers and, with r?∈?Q, n?≥?3 and we prove a finiteness theorem for the number of solutions x in Z, y in Q. Following that, we show that, more interestingly, for every nonzero integer n?>?2 and for any nonzero integer r which is not a perfect n-th power for which the equation admits solutions, k is bounded by an effective bound.     

The Connection between the Characteristic Roots and the Corresponding Solutions of a Single Linear Differential Equation with Comparable Coefficients

Given a linear differential equation of the form x (ⁿ) + a₁ (t) x (n-1) + …+ a_n (t) x = 0 with variable coefficients defined on the positive semi -axis for t ? 1. We denote its fundamental set of solutions (FSS) by {exp [∫ ri (t) dt] } (i = 1, 2,…,n). In this paper we look for the asymptotic connection (as t → ∞) between the logarithmic derivatives ri (t) of an FSS and of the roots of the characteristic equation yⁿ + a₁ (t) y^n-1 +… + a_n (t) = 0. We mainly consider the case when the coefficients of the equation and the characteristic roots are comparable and have the power order of growth for t → ∞. We discuss the conditions when the functions λ_ii(t) are equivalent to the corresponding roots λ_ii(t) of the characteristic equation as t → ∞.     

Exact solution to a nonlinear Klein-Gordon equation

The nonlinear Klein-Gordon equation ?^μ?_μΦ + M²Φ + λ₁Φ^1?m + λ₂Φ^1?2m = 0 has the exact formal solution Φ = [u^2m ?λ₁u^m/(m ? 2)M²+λ₁²/(m?2)²M⁴?λ₂/4(m ? 1)M²]^1/mu^?1, m ≠ 0, 1, 2, where u and v^?1 are solutions of the linear Klein-Gordon equation. This equation is a simple generalization of the ordinary second order differential equation satisfied by the homogeneous function y = [au^m + b(uv)^m/2 + cv^m]^k/m, where u and v are linearly independent solutions of y″ + r(x) y′ + q(x) y = 0.     

Asymptotic forms of solutions of certain linear ordinary differential equations

We analyze the asymptotic behavior as x → ∞ of the product integral Π_x0^xe^A(s)ds, where A(s) is a perturbation of a diagonal matrix function by an integrable function on [x₀,∞). Our results give information concerning the asymptotic behavior of solutions of certain linear ordinary differential equations, e.g., the second order equation y″ = a(x)y.     

On Heuvers’ Logarithmic Functional Equation

?(x + y) - ?(x) - ?(y) = ?(x ^?1 + y ^?l) are identical to those of the Cauchy equation ?(xy) = ?(x) + ?(y) when ? is a function from the positive real numbers into the reals. In the present article, we prove this equivalence for functions mapping the set of nonzero elements of a field (excluding ?₂) .     

Note on the Dočev-Grosswald asymptotic series for generalized Bessel polynomials

The Do?ev-Grosswald asymptotic series for the generalized Bessel polynomials y_n(z; a, b) is extended to O(1/n⁴) relative accuracy. The differential equation of the asymptotic factor, derived from the differential equation for y_n(z; a, b), is the basis of a different and easier method that employs simple recurrence relations and much less algebra for obtaining the same series. This is applied to the important special case of a = 1 to obtain the asymptotic series to O(1/n¹¹) relative accuracy.     

The convolution method for fourier expansions. The case of generalized transmission conditions of crack (screen) type in piecewise inhomogeneous media

We consider boundary value problems for the equation ? _x (K ? _x ?) + KL[?] = 0 in the space R ⁿ with generalized transmission conditions of the type of a strongly permeable crack or a weakly permeable screen on the surface x = 0. (Here L is an arbitrary linear differential operator with respect to the variables y ₁, …, y _n?1.) The coefficients K(x) > 0 are monotone functions of certain classes in the regions separated by the screen x = 0. The desired solution has arbitrary given singular points and satisfies a sufficiently weak condition at infinity.We derive formulas expressing the solutions of the above-mentioned problems in the form of simple quadratures via the solutions of the considered equation with a constant coefficient K for given singular points in the absence of a crack or a screen. In particular, the obtained formulas permit one to solve boundary value problems with generalized transmission conditions for equations with functional piecewise continuous coefficients in the framework of the theory of harmonic functions.     

Oscillation criteria and growth of nonoscillatory solutions of higher order differential equations

We look for conditions under which all solutions of the nonlinear ordinary differential equation y⁽ⁿ⁾ + f(t, y) = 0, t ? 0, ?∞ < y < ∞, are oscillatory, as well as consider the asymptotic behaviour of the nonoscillatory solutions.     

Elliptic equation with a singular potential in a domain with a conic point

This paper deals with the behavior of the nonnegative solutions of the problem $$- \Delta u = V(x)u, \left. u \right|\partial \Omega = \varphi (x)$$ in a conical domain Ω ? ? ⁿ , n ≥ 3, where 0 ≤ V (x) ∈ L₁(Ω), 0 ≤ ?(x) ∈ L₁(?Ω) and ?(x) is continuous on the boundary ?Ω. It is proved that there exists a constant C _*(n) = (n ? 2)²/4 such that if V ₀(x) = (c + λ ₁)|x|^?2, then, for 0 ≤ c ≤ C _*(n) and V(x) ≤ V ₀(x) in the domain Ω, this problem has a nonnegative solution for any nonnegative boundary function ?(x) ∈ L ₁(?Ω); for c > C _*(n) and V(x) ≥ V ₀(x) in Ω, this problem has no nonnegative solutions if ?(x) > 0.     

A Functional Equation Of J.A. Baker Involving Integral Means

The functional equation $$f(x)={1\over 2}\int^{x+1}_{x-1}f(t)\ dt\ \ \ {\rm for}\ \ \ x\ \in\ {\rm R}$$ has the linear functions ?(x) = a + bx (a, b ∈ ?) as trivial solutions. It is shown that there are two kinds of nontrivial solutions, (i) ?(x) = eλ_i ^x (i = 1, 2, …), where the λ_i∈ ? are the fixed points of the map z ? sinh z, and (ii) C^∞-solutions ? for which the values in the interval [?1,1] can be prescribed arbitrarily, but with the provision that ?^(j)(? 1) = ?^(j)(0) = ?^(j)(1) = 0 for all j = 0, 1, 2 …     

Asymptotic properties of solutions of the emden-fowler equation at infinity

We prove a number of theorems on asymptotic properties of solutions of the equation y″+x ^a y ^σ = 0, σ < 0. First, we prove the absence of solutions on (x ₁, +∞) for some values of the parameters a and σ; after that, we obtain asymptotic formulas for solutions defined on (x ₀, +∞).     

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.     

Positive integer solutions of the diophantine equation x 2 − L n xy + (−1) n y 2 = ±5 r

In this paper, we consider the equation x ²?L _n x y+(?1) ⁿ y ² = ±5 ^r and determine the values of n for which the equation has positive integer solutions x and y. Moreover, we give all positive integer solutions of the equation x ²?L _n x y+(?1) ⁿ y ² = ±5 ^r when the equation has positive integer solutions.     

Construction of the asymptotics of the solutions of the one-dimensional Schrödinger equation with rapidly oscillating potential

We obtain asymptotic formulas for the solutions of the one-dimensional Schrödinger equation ? y″ +q(x)y = 0 with oscillating potential q(x)=x ^β P(x ^1+α)+cx ^?2 as x→ +∞. The real parameters α and β satisfy the inequalities β ? α ≥ ?1, 2α ? β > 0 and c is an arbitrary real constant. The real function P(x) is either periodic with period T, or a trigonometric polynomial. To construct the asymptotics, we apply the ideas of the averaging method and use Levinson’s fundamental theorem.     

Short cubic exponential sums over primes

For y ≥ x ^4/5 L ^8B+151 (where L = log(xq) and B is an absolute constant), a nontrivial estimate is obtained for short cubic exponential sums over primes of the form S ₃(α; x, y) = ∑ _x?y<n≤x Λ(n)e(αn ³), where α = a/q + θ/q ², (a, q) = 1, L ^32(B+20) < q ≤ y ⁵ x ^?2 L ^?32(B+20), |θ| ≤ 1, Λ is the von Mangoldt function, and e(t) = e ^2πit.     

Solution of the Szili type conjugate problem on the roots of the laguerre polynomials

Let ?1<α≤0 and let $$L_n^{(\alpha )} (x) = \frac{1}{{n!}}x^{ - \alpha } e^x \frac{{d^n }}{{dx^n }}(x^{\alpha + n} e^{ - x} )$$ be the generalizednth Laguerre polynomial,n=1,2,… Letx ₁,x ₂,…,x _n andx*₁,x*₂,…,x* _n?1 denote the roots ofL _n ^(α) (x) andL _n ^(α)′ (x) respectively and putx*₀=0. In this paper we prove the following theorem: Ify ₀,y ₁,…,y _n ?1 andy ₁ ^′ ,…,y _n ^′ are two systems of arbitrary real numbers, then there exists a unique polynomialP(x) of degree 2n?1 satisfying the conditions $$\begin{gathered} P\left( {x_k^* } \right) = y_k (k = 0,...,n - 1) \hfill \\ P'\left( {x_k } \right) = y_k^\prime (k = 1,...,n). \hfill \\ \end{gathered} $$ .     

The diophantine equation x3 + 3y3 = 2n

It is proved that the equation of the title has a finite number of integral solutions (x, y, n) and necessary conditions are given for (x, y, n) in order that it can be a solution (Theorem 2). It is also proved that for a given odd x₀ there is at most one integral solution (y, n), n ≥ 3, to x₀³ + 3y³ = 2ⁿ and for a given odd y₀ there is at most one integral solution (x, n), n ≥ 3, to x³ + 3y₀³ = 2ⁿ.     

A lower bound for the first eigenvalue of an elliptic operator

Oscillatory behavior of the solutions of the nth-order delay differential equation L_nx(t) + q(t)f(x[g(t)]) = 0 is discussed. The results obtained are extensions of some of the results by Kim (Proc. Amer. Math. Soc.62 (1977), 77–82) for y⁽ⁿ⁾ + py = 0.