The Laplace derivative

Ralph E. Svetic

Displaying similar documents to “The Laplace derivative”

Asymptotic behavior of solutions of a $2 n^{t h}$ order nonlinear differential equation

C. S. Lin (2002)

Czechoslovak Mathematical Journal

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In this paper we prove two results. The first is an extension of the result of G. D. Jones [4]: (A) Every nontrivial solution for $\}\begin{matrix} {(- 1)}^{n} u^{(2 n)} + f (t, u) = 0, in (α, \infty), u^{(i)} (ξ) = 0, i = 0, 1, \dots, n - 1, and ξ \in (α, \infty), \end{matrix}$ must be unbounded, provided $f (t, z) z \geq 0$ , in $E \times ℝ$ and for every bounded subset $I$ , $f (t, z)$ is bounded in $E \times I$ . (B) Every bounded solution for ${(- 1)}^{n} u^{(2 n)} + f (t, u) = 0$ , in $ℝ$ , must be constant, provided $f (t, z) z \geq 0$ in $ℝ \times ℝ$ and for every bounded subset $I$ , $f (t, z)$ is bounded in $ℝ \times I$ .

On uniqueness and successive approximations in the Darboux problem for the equation $u_{x y} = f (x, y, u, u_{x}, u_{y}, \int_{0}^{x} \int_{0}^{y} g (x, y, s, t, u (s, t), u_{s} (s, t), u_{t} (s, t)) d s d t)$

B. Palczewski (1965)

Annales Polonici Mathematici

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The inverse Laplace transform of the product of two modified Bessel functions $K_{n v} [a^{1 / 2 n} p^{1 / 2 n}] K_{n μ} [a^{1 / 2 n} p^{1 / 2 n}]$ where n=1, 2, 3,...

F. M. Ragab (1963)

Annales Polonici Mathematici

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On the existence of solutions for some nondegenerate nonlinear wave equations of Kirchhoff type

Jong Yeoul Park, Jeong Ja Bae (2002)

Czechoslovak Mathematical Journal

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Let $Ω$ be a bounded domain in $ℝ^{n}$ with a smooth boundary $Γ$ . In this work we study the existence of solutions for the following boundary value problem: $\frac{\partial^{2} y}{\partial t^{2}} - M (\int_{Ω} {| \nabla y |}^{2} d x) Δ y - \frac{\partial}{\partial t} Δ y = f (y) in Q = Ω \times (0, \infty), . 1 y = 0 in Σ_{1} = Γ_{1} \times (0, \infty), M (\int_{Ω} {| \nabla y |}^{2} d x) \frac{\partial y}{\partial ν} + \frac{\partial}{\partial t} (\frac{\partial y}{\partial ν}) = g in Σ_{0} = Γ_{0} \times (0, \infty), y (0) = y_{0}, \frac{\partial y}{\partial t} (0) = y_{1} in Ω, (1)$ where $M$ is a $C^{1}$ -function such that $M (λ) \geq λ_{0} > 0$ for every $λ \geq 0$ and $f (y) = {| y |}^{α} y$ for $α \geq 0$ .

Covariant differential operators and Green's functions

Miroslav Engliš, Jaak Peetre (1997)

Annales Polonici Mathematici

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The basic idea of this paper is to use the covariance of a partial differential operator under a suitable group action to determine suitable associated Green’s functions. For instance, we offer a new proof of a formula for Green’s function of the mth power $Δ^{m}$ of the ordinary Laplace’s operator Δ in the unit disk found in a recent paper (Hayman-Korenblum, J. Anal. Math. 60 (1993), 113-133). We also study Green’s functions associated with mth powers of the Poincaré invariant Laplace operator...

On the spectrum and eigenfunctions of the operator $(V f) (x) = \int_{0}^{x^{α}} f (t) d t$

I. Yu. Domanov (2007)

Banach Center Publications

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General Dirichlet series, arithmetic convolution equations and Laplace transforms

Helge Glöckner, Lutz G. Lucht, Štefan Porubský (2009)

Studia Mathematica

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In the earlier paper [Proc. Amer. Math. Soc. 135 (2007)], we studied solutions g: ℕ → ℂ to convolution equations of the form $a_{d} * g^{* d} + a_{d - 1} * g^{* (d - 1)} + \dots + a ₁ * g + a ₀ = 0$ , where $a ₀, . . ., a_{d} : ℕ \to ℂ$ are given arithmetic functions associated with Dirichlet series which converge on some right half plane, and also g is required to be such a function. In this article, we extend our previous results to multidimensional general Dirichlet series of the form $\sum_{x \in X} f (x) e^{- s x}$ ( $s \in ℂ^{k}$ ), where $X \subseteq {[0, \infty)}^{k}$ is an additive subsemigroup. If X is discrete and a certain solvability criterion...

Differentiation of n-convex functions

H. Fejzić, R. E. Svetic, C. E. Weil (2010)

Fundamenta Mathematicae

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The main result of this paper is that if f is n-convex on a measurable subset E of ℝ, then f is n-2 times differentiable, n-2 times Peano differentiable and the corresponding derivatives are equal, and $f^{(n - 1)} = f_{(n - 1)}$ except on a countable set. Moreover $f_{(n - 1)}$ is approximately differentiable with approximate derivative equal to the nth approximate Peano derivative of f almost everywhere.

A class of retracts in $L^{p}$ with some applications to differential inclusion

Grzegorz Bartuzel, Andrzej Fryszkowski (2002)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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Generalized α-variation and Lebesgue equivalence to differentiable functions

Jakub Duda (2009)

Fundamenta Mathematicae

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We find conditions on a real function f:[a,b] → ℝ equivalent to being Lebesgue equivalent to an n-times differentiable function (n ≥ 2); a simple solution in the case n = 2 appeared in an earlier paper. For that purpose, we introduce the notions of $C B V G_{1 / n}$ and $S B V G_{1 / n}$ functions, which play analogous rôles for the nth order differentiability to the classical notion of a VBG⁎ function for the first order differentiability, and the classes $C B V_{1 / n}$ and $S B V_{1 / n}$ (introduced by Preiss and Laczkovich) for Cⁿ smoothness....

The spectrum of the Laplace operator for a spherical space form

Gr. Tsagas (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Si determina lo spettro di un operatore di Laplace di una «spherical space form» $(M,g)$ e si studia l’influenza di tale spettro su $(M,g)$ .

Continuity in the Alexiewicz norm

Erik Talvila (2006)

Mathematica Bohemica

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If $f$ is a Henstock-Kurzweil integrable function on the real line, the Alexiewicz norm of $f$ is $∥ f ∥ = {sup}_{I} | \int_{I} f |$ where the supremum is taken over all intervals $I \subset ℝ$ . Define the translation $τ_{x}$ by $τ_{x} f (y) = f (y - x)$ . Then $∥ τ_{x} f - f ∥$ tends to $0$ as $x$ tends to $0$ , i.e., $f$ is continuous in the Alexiewicz norm. For particular functions, $∥ τ_{x} f - f ∥$ can tend to 0 arbitrarily slowly. In general, $∥ τ_{x} f - f ∥ \geq osc f | x |$ as $x \to 0$ , where $osc f$ is the oscillation of $f$ . It is shown that if $F$ is a primitive of $f$ then $∥ τ_{x} F - F ∥ \leq ∥ f ∥ | x |$ . An example shows that the function $y \mapsto τ_{x} F (y) - F (y)$ need not be in $L^{1}$ . However, if...