On the behaviour of solutions of the differential equations $(r(t)y^{\prime \prime })^{\prime } + q(t)(y^{\prime })^β + p(t)y^α = f(t)$

N. Parhi; S. Parhi

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

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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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 Lincei. Matematica e Applicazioni

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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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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.

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.

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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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 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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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.

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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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...

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...

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.

Products of factorials modulo p

Florian Luca, Pantelimon Stănică (2003)

Colloquium Mathematicae

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We show that if p ≠ 5 is a prime, then the numbers $1 / p (\binom{p}{m ₁, . . ., m_{t}}) | t \geq 1, m_{i} \geq 0 f o r i = 1, . . ., t a n d \sum_{i = 1}^{t} m_{i} = p$ cover all the nonzero residue classes modulo p.

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].

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 $u^{''} (x) + \int_{0}^{\infty} u^{'} (s) d_{s} r_{1} (x, s) + \int_{0}^{\infty} u (s) d_{s} r_{0} (x, s) = 0 .$

To the theory of central dispersions for the linear differential equations $y^{''} = q (t) y$ of a finite type-special

Eva Tesaříková (1987)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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Displaying similar documents to “On the behaviour of solutions of the differential equations ( r ( t ) y ' ' ) ' + q ( t ) ( y ' ) β + p ( t ) y α = f ( t ) ”

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