Periodic solutions of $x^{\prime \prime } + f(x)x^{\prime 2n} + g(x) = μp(t)$

S. Sędziwy

Displaying similar documents to “Periodic solutions of $x^{''} + f (x) x^{' 2 n} + g (x) = μ p (t)$ ”

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.

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.

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 $X^{1 + 10^{- 52}}$ .

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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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.

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 $\omega$ .

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 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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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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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.

Periodic solutions of $x " + f (x) x^{' m} + g (x) = 0$

W. R. Utz (1971)

Annales Polonici Mathematici

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Periodic solutions of $x " + f (x) x^{' 2 n} + g (x) = 0$ with arbitrarily large period

J. W. Heidel (1971)

Annales Polonici Mathematici

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Addenda to “Periodic solutions of $x " + f (x) x^{' 2 n} + g (x) = 0$ with arbitrarily large period”

J. W. Heidel (1973)

Annales Polonici Mathematici

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$E_{1}$ -degeneration and $d^{'} d^{''}$ -lemma

Tai-Wei Chen, Chung-I Ho, Jyh-Haur Teh (2016)

Commentationes Mathematicae Universitatis Carolinae

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For a double complex $(A, d^{'}, d^{''})$ , we show that if it satisfies the $d^{'} d^{''}$ -lemma and the spectral sequence ${E_{r}^{p, q}}$ induced by $A$ does not degenerate at $E_{0}$ , then it degenerates at $E_{1}$ . We apply this result to prove the degeneration at $E_{1}$ of a Hodge-de Rham spectral sequence on compact bi-generalized Hermitian manifolds that satisfy a version of $d^{'} d^{''}$ -lemma.

Boundedness criteria for a class of second order nonlinear differential equations with delay

Daniel O. Adams, Mathew Omonigho Omeike, Idowu A. Osinuga, Biodun S. Badmus (2023)

Mathematica Bohemica

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We consider certain class of second order nonlinear nonautonomous delay differential equations of the form $a (t) x^{''} + b (t) g (x, x^{'}) + c (t) h (x (t - r)) m (x^{'}) = p (t, x, x^{'})$ and ${(a (t) x^{'})}^{'} + b (t) g (x, x^{'}) + c (t) h (x (t - r)) m (x^{'}) = p (t, x, x^{'}),$ where $a$ , $b$ , $c$ , $g$ , $h$ , $m$ and $p$ are real valued functions which depend at most on the arguments displayed explicitly and $r$ is a positive constant. Different forms of the integral inequality method were used to investigate the boundedness of all solutions and their derivatives. Here, we do not require construction of the Lyapunov-Krasovski functional to establish our results....

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 $\alpha$ .

The periodic Ambrosetti-Prodi problem for nonlinear perturbations of the p-Laplacian

Jean Mawhin (2006)

Journal of the European Mathematical Society

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We prove an Ambrosetti–Prodi type result for the periodic solutions of the equation $(| u^{'} |^{p - 2} u^{'} {))}^{'} + f (u) u^{'} + g (x, u) = t$ , when $f$ is arbitrary and $g (x, u) \to + \infty$ or $g (x, u) \to - \infty$ when $| u | \to \infty$ . The proof uses upper and lower solutions and the Leray–Schauder degree.

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].

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...

Region of variability for spiral-like functions with respect to a boundary point

S. Ponnusamy, A. Vasudevarao, M. Vuorinen (2009)

Colloquium Mathematicae

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For μ ∈ ℂ such that Re μ > 0 let $ℱ_{μ}$ denote the class of all non-vanishing analytic functions f in the unit disk with f(0) = 1 and $R e (2 π / μ z f^{'} (z) / f (z) + (1 + z) / (1 - z)) > 0$ in . For any fixed z₀ in the unit disk, a ∈ ℂ with |a| ≤ 1 and λ ∈ ̅, we shall determine the region of variability V(z₀,λ) for log f(z₀) when f ranges over the class $ℱ_{μ} (λ) = f \in ℱ_{μ} : f^{'} (0) = (μ / π) (λ - 1) a n d f^{''} (0) = (μ / π) (a (1 - | λ | ²) + (μ / π) (λ - 1) ² - (1 - λ ²))$ . In the final section we graphically illustrate the region of variability for several sets of parameters.

Displaying similar documents to “Periodic solutions of x ' ' + f ( x ) x ' 2 n + g ( x ) = μ p ( t ) ”

Displaying similar documents to “Periodic solutions of $x^{''} + f (x) x^{' 2 n} + g (x) = μ p (t)$ ”