Abstract singular hyperbolic equations

Giuseppe Coppoletta

Displaying similar documents to “Abstract singular hyperbolic equations”

Abstract singular hyperbolic equations

Giuseppe Coppoletta (1982)

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

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Si annunziano alcuni risultati di esistenza e unicità per l’equazione astratta singolare $\varphi(t){u^{\prime}}(t)=A(t)\,u(t)+f(t)$ nel caso iperbolico.

Periodic solutions to a non-linear parametric differential equation of the third order

Jan Andres, Jan Vorácek (1984)

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

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Si dimostra un teorema di esistenza di soluzioni periodiche dell'equazione differenziale ordinaria del terzo ordine $x^{\prime\prime\prime}+a(t,x,x^{\prime},x^{\prime\prime})x^{\prime\prime}+b(t,% x,x^{\prime},x^{\prime\prime})x^{\prime}+h(x)=e(t,x,x^{\prime},x^{\prime\prime})$ con le funzioni $a$ , $b$ , $e$ periodiche in $t$ di periodo $\omega$ .

Periodic solutions to a non-linear parametric differential equation of the third order

Jan Andres, Jan Vorácek (1984)

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

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On the nonconvergence of successive approximations in the theory of the equation $z "_{x y} = f (x, y, z, z_{x}^{'}, z_{y}^{'})$

Michał Kisielewicz (1979)

Annales Polonici Mathematici

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Boundedness results of solutions to the equation $x^{\prime\prime\prime}+ax^{\prime\prime}+g(x)x^{\prime}+h(x)=p(t)$ without the hypothesis $h(x)\,\operatorname{sgn}x\geq 0$ for $|x|>R$ .

Ján Andres (1986)

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

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Per l'equazione differenziale ordinaria non lineare del 3° ordine indicata nel titolo, studiata da numerosi autori sotto l'ipotesi $h(x)\,\text{sgn}x\geq 0$ $for|x|>R$ , si dimostra l'esistenza di almeno una soluzione limitata sopprimendo l'ipotesi suddetta.

Boundedness results of solutions to the equation $x^{\prime\prime\prime}+ax^{\prime\prime}+g(x)x^{\prime}+h(x)=p(t)$ without the hypothesis $h(x)\,\text{sgn}x\geq 0$ $for|x|>R$ .

Ján Andres (1986)

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

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On the behaviour of solutions of the differential equations ${(r (t) y^{''})}^{'} + q (t) {(y^{'})}^{β} + p (t) y^{α} = f (t)$

N. Parhi, S. Parhi (1986)

Annales Polonici Mathematici

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The largest prime factor of X³ + 2

A. J. Irving (2015)

Acta Arithmetica

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Improving on a theorem of Heath-Brown, we show that if X is sufficiently large then a positive proportion of the values n³ + 2: n ∈ (X,2X] have a prime factor larger than $X^{1 + 10^{- 52}}$ .

Periodic solutions of $x^{''} + f (x) x^{' 2 n} + g (x) = μ p (t)$

S. Sędziwy (1969)

Annales Polonici Mathematici

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Positive periodic solutions to super-linear second-order ODEs

Jiří Šremr (2025)

Czechoslovak Mathematical Journal

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We study the existence and uniqueness of a positive solution to the problem $u^{''} = p (t) u + q (t, u) u + f (t); u (0) = u (ω), u^{'} (0) = u^{'} (ω)$ with a super-linear nonlinearity and a nontrivial forcing term $f$ . To prove our main results, we combine maximum and anti-maximum principles together with the lower/upper functions method. We also show a possible physical motivation for the study of such a kind of periodic problems and we compare the results obtained with the facts well known for the corresponding autonomous case.

Positive solutions of a fourth-order differential equation with integral boundary conditions

Seshadev Padhi, John R. Graef (2023)

Mathematica Bohemica

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We study the existence of positive solutions to the fourth-order two-point boundary value problem $\{\begin{matrix} u^{''''} (t) + f (t, u (t)) = 0, & 0 < t < 1, \\ u^{'} (0) = u^{'} (1) = u^{''} (0) = 0, & u (0) = α [u], \end{matrix}$ where $α [u] = \int_{0}^{1} u (t) d A (t)$ is a Riemann-Stieltjes integral with $A \geq 0$ being a nondecreasing function of bounded variation and $f \in 𝒞 ([0, 1] \times ℝ_{+}, ℝ_{+})$ . The sufficient conditions obtained are new and easy to apply. Their approach is based on Krasnoselskii’s fixed point theorem and the Avery-Peterson fixed point theorem.

Analytic solutions of a second-order iterative functional differential equation near resonance

Houyu Zhao, Jianguo Si (2009)

Annales Polonici Mathematici

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We study existence of analytic solutions of a second-order iterative functional differential equation $x^{''} (z) = \sum_{j = 0}^{k} \sum_{t = 1}^{\infty} C_{t, j} (z) {(x^{[j]} (z))}^{t} + G (z)$ in the complex field ℂ. By constructing an invertible analytic solution y(z) of an auxiliary equation of the form $α ² y^{''} (α z) y^{'} (z) = α y^{'} (α z) y^{''} (z) + [y^{'} (z)] ³ [\sum_{j = 0}^{k} \sum_{t = 1}^{\infty} C_{t, j} (y (z)) {(y (α^{j} z))}^{t} + G (y (z))]$ invertible analytic solutions of the form $y (α y^{- 1} (z))$ for the original equation are obtained. Besides the hyperbolic case 0 < |α| < 1, we focus on α on the unit circle S¹, i.e., |α|=1. We discuss not only those α at resonance, i.e. at a root of unity, but also near resonance...

On the non-convergence of successive approximations in the Darboux problem for the equation $z_{x y}^{''} = f (x, y, z)$

M. Kisielewicz (1977)

Colloquium Mathematicae

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Shifted values of the largest prime factor function and its average value in short intervals

Jean-Marie De Koninck, Imre Kátai (2016)

Colloquium Mathematicae

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We obtain estimates for the average value of the largest prime factor P(n) in short intervals [x,x+y] and of h(P(n)+1), where h is a complex-valued additive function or multiplicative function satisfying certain conditions. Letting $s_{q} (n)$ stand for the sum of the digits of n in base q ≥ 2, we show that if α is an irrational number, then the sequence ${(α s_{q} (P (n)))}_{n \in ℕ}$ is uniformly distributed modulo 1.

On non-oscillation on semi-axis of solutions of second order deviating differential equations

Sergey Labovskiy, Manuel Alves (2018)

Mathematica Bohemica

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We obtain conditions for existence and (almost) non-oscillation of solutions of a second order linear homogeneous functional differential equations $u^{''} (x) + \sum_{i} p_{i} (x) u^{'} (h_{i} (x)) + \sum_{i} q_{i} (x) u (g_{i} (x)) = 0$ without the delay conditions $h_{i} (x), g_{i} (x) \leq x$ , $i = 1, 2, ...$ , and $u^{''} (x) + \int_{0}^{\infty} u^{'} (s) d_{s} r_{1} (x, s) + \int_{0}^{\infty} u (s) d_{s} r_{0} (x, s) = 0 .$

On hyperbolic partial differential equations in Banach spaces

Bogdan Rzepecki (1986)

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

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Viene dimostrata l'esistenza di soluzioni del problema di Darboux per l'equazione iperbolica $z^{\prime\prime}_{xy}=f(x,y,z,Z^{\prime}_{x},z_{y})$ sul planiquarto $x\geq 0$ , $y\geq 0$ . Qui, $f$ è una funzione continua, con valori in uno spazio Banach che soddisfano alcune condizioni di regolarità espresse in termini della misura di non-compattezza $\alpha$ .

On hyperbolic partial differential equations in Banach spaces

Bogdan Rzepecki (1986)

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

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Anatomy of a macro (tutorial)

Denis Roegel (2010)

Zpravodaj Československého sdružení uživatelů TeXu

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In this article, we explain in detail a TeX macro computing prime numbers. This gives us an opportunity to illustrate technical aspects often ignored by beginners in the TeX world. The source codes are included as small parts in the article commented in detail. You may find the original English version of the article in TUGboat, see http://www.tug.org/TUGboat/Articles/tb22-1-2/tb70roeg.pdf.

Existence results and iterative method for fully third order nonlinear integral boundary value problems

Quang A Dang, Quang Long Dang (2021)

Applications of Mathematics

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We consider the boundary value problem $\begin{matrix} u^{'''} (t) = f (t, u (t), u^{'} (t), u^{''} (t)), 0 < t < 1, \\ u (0) = u^{'} (0) = 0, u (1) = \int_{0}^{1} g (s) u (s) d s, \end{matrix}$ where $f : [0, 1] \times ℝ^{3} \to ℝ^{+}$ , $g : [0, 1] \to ℝ^{+}$ are continuous functions. The case when $f = f (u (t))$ was studied in 2018 by Guendouz et al. Using the fixed-point theory on cones they established the existence of positive solutions. Here, by the method developed by ourselves very recently, we establish the existence, uniqueness and positivity of the solution under easily verified conditions and propose an iterative method for finding the solution. Some examples demonstrate the validity of the...

Positive solutions for one-dimensional singular p-Laplacian boundary value problems

Huijuan Song, Jingxue Yin, Rui Huang (2012)

Annales Polonici Mathematici

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We consider the existence of positive solutions of the equation $1 / λ (t) {(λ (t) φ_{p} (x^{'} (t)))}^{'} + μ f (t, x (t), x^{'} (t)) = 0$ , where $φ_{p} (s) = {| s |}^{p - 2} s$ , p > 1, subject to some singular Sturm-Liouville boundary conditions. Using the Krasnosel’skiĭ fixed point theorem for operators on cones, we prove the existence of positive solutions under some structure conditions.

Truncatable primes and unavoidable sets of divisors

Artūras Dubickas (2006)

Acta Mathematica Universitatis Ostraviensis

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We are interested whether there is a nonnegative integer $u_{0}$ and an infinite sequence of digits $u_{1}, u_{2}, u_{3}, \dots$ in base $b$ such that the numbers $u_{0} b^{n} + u_{1} b^{n - 1} + \dots + u_{n - 1} b + u_{n},$ where $n = 0, 1, 2, \dots,$ are all prime or at least do not have prime divisors in a finite set of prime numbers $S .$ If any such sequence contains infinitely many elements divisible by at least one prime number $p \in S,$ then we call the set $S$ unavoidable with respect to $b$ . It was proved earlier that unavoidable sets in base $b$ exist if $b \in {2, 3, 4, 6},$ and that no unavoidable set exists in base $b = 5 .$ Now,...

On sets which contain a qth power residue for almost all prime modules

Mariusz Ska/lba (2005)

Colloquium Mathematicae

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A classical theorem of M. Fried [2] asserts that if non-zero integers $β ₁, . . ., β_{l}$ have the property that for each prime number p there exists a quadratic residue $β_{j}$ mod p then a certain product of an odd number of them is a square. We provide generalizations for power residues of degree n in two cases: 1) n is a prime, 2) n is a power of an odd prime. The proofs involve some combinatorial properties of finite Abelian groups and arithmetic results of [3].