Existence and uniqueness of periodic solutions for odd-order ordinary differential equations

Yongxiang Li; He Yang

Displaying similar documents to “Existence and uniqueness of periodic solutions for odd-order ordinary differential equations”

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

W. R. Utz (1971)

Annales Polonici Mathematici

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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 the uniqueness of periodic decomposition

Viktor Harangi (2011)

Fundamenta Mathematicae

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Let $a ₁, . . ., a_{k}$ be arbitrary nonzero real numbers. An $(a ₁, . . ., a_{k})$ -decomposition of a function f:ℝ → ℝ is a sum $f ₁ + \dots + f_{k} = f$ where $f_{i} : ℝ \to ℝ$ is an $a_{i}$ -periodic function. Such a decomposition is not unique because there are several solutions of the equation $h ₁ + \dots + h_{k} = 0$ with $h_{i} : ℝ \to ℝ a_{i}$ -periodic. We will give solutions of this equation with a certain simple structure (trivial solutions) and study whether there exist other solutions or not. If not, we say that the $(a ₁, . . ., a_{k})$ -decomposition is essentially unique. We characterize those periods for which essential...

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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Existence of nonnegative periodic solutions in neutral integro-differential equations with functional delay

Imene Soulahia, Abdelouaheb Ardjouni, Ahcene Djoudi (2015)

Commentationes Mathematicae Universitatis Carolinae

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The fixed point theorem of Krasnoselskii and the concept of large contractions are employed to show the existence of a periodic solution of a nonlinear integro-differential equation with variable delay $x^{'} (t) = - \int_{t - τ (t)}^{t} a (t, s) g (x (s)) d s + \frac{d}{d t} Q (t, x (t - τ (t))) + G (t, x (t), x (t - τ (t))) .$ We transform this equation and then invert it to obtain a sum of two mappings one of which is completely continuous and the other is a large contraction. We choose suitable conditions for $τ$ , $g$ , $a$ , $Q$ and $G$ to show that this sum of mappings fits into the framework of a modification of...

Three periodic solutions for a class of higher-dimensional functional differential equations with impulses

Yongkun Li, Changzhao Li, Juan Zhang (2010)

Annales Polonici Mathematici

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By using the well-known Leggett–Williams multiple fixed point theorem for cones, some new criteria are established for the existence of three positive periodic solutions for a class of n-dimensional functional differential equations with impulses of the form ⎧y’(t) = A(t)y(t) + g(t,yt), $t \neq t_{j}$ , j ∈ ℤ, ⎨ ⎩ $y (t ⁺_{j}) = y (t ¯_{j}) + I_{j} (y (t_{j}))$ , where $A (t) = {(a_{i j} (t))}_{n \times n}$ is a nonsingular matrix with continuous real-valued entries.

A class of $I_{0}$ -sets

David Grow (1987)

Colloquium Mathematicae

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Stability of periodic stationary solutions of scalar conservation laws with space-periodic flux

Anne-Laure Dalibard (2011)

Journal of the European Mathematical Society

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This article investigates the long-time behaviour of parabolic scalar conservation laws of the type $\partial_{t} u + {div}_{y} A (y, u) - Δ_{y} u = 0$ , where $y \in ℝ^{N}$ and the flux $A$ is periodic in $y$ . More specifically, we consider the case when the initial data is an $L^{1}$ disturbance of a stationary periodic solution. We show, under polynomial growth assumptions on the flux, that the difference between u and the stationary solution behaves in $L^{1}$ norm like a self-similar profile for large times. The proof uses a time and space change of variables...

Multiple positive solutions of a nonlinear fourth order periodic boundary value problem

Lingbin Kong, Daqing Jiang (1998)

Annales Polonici Mathematici

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The fourth order periodic boundary value problem $u^{(4)} - m ⁴ u + F (t, u) = 0$ , 0 < t < 2π, with $u^{(i)} (0) = u^{(i)} (2 π)$ , i = 0,1,2,3, is studied by using the fixed point index of mappings in cones, where F is a nonnegative continuous function and 0 < m < 1. Under suitable conditions on F, it is proved that the problem has at least two positive solutions if m ∈ (0,M), where M is the smallest positive root of the equation tan mπ = -tanh mπ, which takes the value 0.7528094 with an error of $\pm 10^{- 7}$ .

The Lebesgue constant for the periodic Franklin system

Markus Passenbrunner (2011)

Studia Mathematica

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We identify the torus with the unit interval [0,1) and let n,ν ∈ ℕ with 0 ≤ ν ≤ n-1 and N:= n+ν. Then we define the (partially equally spaced) knots $t_{j}$ = ⎧ j/(2n) for j = 0,…,2ν, ⎨ ⎩ (j-ν)/n for for j = 2ν+1,…,N-1. Furthermore, given n,ν we let $V_{n, ν}$ be the space of piecewise linear continuous functions on the torus with knots $t_{j} : 0 \leq j \leq N - 1$ . Finally, let $P_{n, ν}$ be the orthogonal projection operator from L²([0,1)) onto $V_{n, ν}$ . The main result is $l i m_{n \to \infty, ν = 1} | | P_{n, ν} : L^{\infty} \to L^{\infty} | | = s u p_{n \in ℕ, 0 \leq ν \leq n} | | P_{n, ν} : L^{\infty} \to L^{\infty} | | = 2 + (33 - 18 \sqrt 3) / 13$ . This shows in particular that the Lebesgue constant of the classical...

Well-posedness of second order degenerate differential equations in vector-valued function spaces

Shangquan Bu (2013)

Studia Mathematica

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Using known results on operator-valued Fourier multipliers on vector-valued function spaces, we give necessary or sufficient conditions for the well-posedness of the second order degenerate equations (P₂): d/dt (Mu’)(t) = Au(t) + f(t) (0 ≤ t ≤ 2π) with periodic boundary conditions u(0) = u(2π), (Mu’)(0) = (Mu’)(2π), in Lebesgue-Bochner spaces $L^{p} (, X)$ , periodic Besov spaces $B_{p, q}^{s} (, X)$ and periodic Triebel-Lizorkin spaces $F_{p, q}^{s} (, X)$ , where A and M are closed operators in a Banach space X satisfying D(A) ⊂ D(M)....

One-dimensional symmetry of periodic minimizers for a mean field equation

Chang-Shou Lin, Marcello Lucia (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider on a two-dimensional flat torus $T$ defined by a rectangular periodic cell the following equation $Δ u + ρ (\frac{e^{u}}{\int_{T} e^{u}} - \frac{1}{| T |}) = 0, \int_{T} u = 0 .$ It is well-known that the associated energy functional admits a minimizer for each $ρ \leq 8 π$ . The present paper shows that these minimizers depend actually only on one variable. As a consequence, setting $λ_{1} (T)$ to be the first eigenvalue of the Laplacian on the torus, the minimizers are identically zero whenever $ρ \leq min {8 π, λ_{1} (T) | T |}$ . Our results hold more generally for solutions that...

Functionals on Banach Algebras with Scattered Spectra

H. S. Mustafayev (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let A be a complex, commutative Banach algebra and let $M_{A}$ be the structure space of A. Assume that there exists a continuous homomorphism h:L¹(G) → A with dense range, where L¹(G) is a group algebra of the locally compact abelian group G. The main results of this note can be summarized as follows: (a) If every weakly almost periodic functional on A with compact spectra is almost periodic, then the space $M_{A}$ is scattered (i.e., $M_{A}$ has no nonempty perfect subset). (b) Weakly almost periodic...

Cobham's theorem for substitutions

Fabien Durand (2011)

Journal of the European Mathematical Society

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The seminal theorem of Cobham has given rise during the last 40 years to a lot of work about non-standard numeration systems and has been extended to many contexts. In this paper, as a result of fifteen years of improvements, we obtain a complete and general version for the so-called substitutive sequences. Let $α$ and $β$ be two multiplicatively independent Perron numbers. Then a sequence $x \in A^{ℕ}$ , where $A$ is a finite alphabet, is both $α$ -substitutive and $β$ -substitutive if and only if $x$ is ultimately...

Characterising weakly almost periodic functionals on the measure algebra

Matthew Daws (2011)

Studia Mathematica

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Let G be a locally compact group, and consider the weakly almost periodic functionals on M(G), the measure algebra of G, denoted by WAP(M(G)). This is a C*-subalgebra of the commutative C*-algebra M(G)*, and so has character space, say $K_{W A P}$ . In this paper, we investigate properties of $K_{W A P}$ . We present a short proof that $K_{W A P}$ can naturally be turned into a semigroup whose product is separately continuous; at the Banach algebra level, this product is simply the natural one induced by the Arens...

On a certain class of arithmetic functions

Antonio M. Oller-Marcén (2017)

Mathematica Bohemica

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A homothetic arithmetic function of ratio $K$ is a function $f : ℕ \to R$ such that $f (K n) = f (n)$ for every $n \in ℕ$ . Periodic arithmetic funtions are always homothetic, while the converse is not true in general. In this paper we study homothetic and periodic arithmetic functions. In particular we give an upper bound for the number of elements of $f (ℕ)$ in terms of the period and the ratio of $f$ .

$C^{1}$ self-maps on closed manifolds with finitely many periodic points all of them hyperbolic

Jaume Llibre, Víctor F. Sirvent (2016)

Mathematica Bohemica

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Let $X$ be a connected closed manifold and $f$ a self-map on $X$ . We say that $f$ is almost quasi-unipotent if every eigenvalue $λ$ of the map $f_{* k}$ (the induced map on the $k$ -th homology group of $X$ ) which is neither a root of unity, nor a zero, satisfies that the sum of the multiplicities of $λ$ as eigenvalue of all the maps $f_{* k}$ with $k$ odd is equal to the sum of the multiplicities of $λ$ as eigenvalue of all the maps $f_{* k}$ with $k$ even. We prove that if $f$ is $C^{1}$ having finitely many periodic points all of them...

On asymptotic behavior of solutions to Emden-Fowler type higher-order differential equations

Irina Astashova (2015)

Mathematica Bohemica

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For the equation $y^{(n)} + {| y |}^{k} sgn y = 0, k > 1, n = 3, 4,$ existence of oscillatory solutions $y = {(x^{*} - x)}^{- α} h (log (x^{*} - x)), α = \frac{n}{k - 1}, x < x^{*},$ is proved, where $x^{*}$ is an arbitrary point and $h$ is a periodic non-constant function on $ℝ$ . The result on existence of such solutions with a positive periodic non-constant function $h$ on $ℝ$ is formulated for the equation $y^{(n)} = {| y |}^{k} sgn y, k > 1, n = 12, 13, 14 .$