A generalized periodic boundary value problem for the one-dimensional p-Laplacian

Daqing Jiang; Junyu Wang

Displaying similar documents to “A generalized periodic boundary value problem for the one-dimensional p-Laplacian”

Periodic solutions to a non-linear differential equation of the order $2n+1$

Monika Kubicova (1989)

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

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A criterion for the existance of periodic solutions of an ordinary differential equation of order k proved by J. Andres and J. Vorâcek for k = 3 is extended to an arbitrary odd k.

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 $\pm 10^{- 7}$ .

Periodic solutions to a non-linear differential equation of the order $2n+1$

Monika Kubicova (1989)

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

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A criterion for the existance of periodic solutions of an ordinary differential equation of order k proved by J. Andres and J. Vorâcek for k = 3 is extended to an arbitrary odd k.

Anti-periodic solutions to a parabolic hemivariational inequality

Jong Yeoul Park, Hyun Min Kim, Sun Hye Park (2004)

Kybernetika

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In this paper we deal with the anti-periodic boundary value problems with nonlinearity of the form $b (u)$ , where $b \in L_{loc}^{\infty} (R) .$ Extending $b$ to be multivalued we obtain the existence of solutions to hemivariational inequality and variational-hemivariational inequality.

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

Yongxiang Li, He Yang (2011)

Annales Polonici Mathematici

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The paper deals with the existence and uniqueness of 2π-periodic solutions for the odd-order ordinary differential equation $u^{(2 n + 1)} = f (t, u, u^{'}, . . ., u^{(2 n)})$ , where $f : ℝ \times ℝ^{2 n + 1} \to ℝ$ is continuous and 2π-periodic with respect to t. Some new conditions on the nonlinearity $f (t, x ₀, x ₁, . . ., x_{2 n})$ to guarantee the existence and uniqueness are presented. These conditions extend and improve the ones presented by Cong [Appl. Math. Lett. 17 (2004), 727-732].

Stieltjes differential problems with general boundary value conditions. Existence and bounds of solutions

Valeria Marraffa, Bianca Satco (2025)

Czechoslovak Mathematical Journal

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We are concerned with first order set-valued problems with very general boundary value conditions $\{\begin{matrix} u_{g}^{'} (t) \in F (t, u (t)), μ_{g} -a.e. \in [0, T], \\ L (u (0), T)) = 0 \end{matrix}$ involving the Stieltjes derivative with respect to a left-continuous nondecreasing function $g : [0, T] \to ℝ$ , a Carathéodory multifunction $F : [0, T] \times ℝ \to 𝒫 (ℝ)$ and a continuous $L : ℝ^{2} \to ℝ$ . Using appropriate notions of lower and upper solutions, we prove the existence of solutions via a fixed point result for condensing mappings. In the periodic single-valued case, combining an existence theory for the linear case with a recent...

On the solution set of the nonconvex sweeping process

Andrea Gavioli (1999)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We prove that the solutions of a sweeping process make up an $R_{δ}$ -set under the following assumptions: the moving set C(t) has a lipschitzian retraction and, in the neighbourhood of each point x of its boundary, it can be seen as the epigraph of a lipschitzian function, in such a way that the diameter of the neighbourhood and the related Lipschitz constant do not depend on x and t. An application to the existence of periodic solutions is given.

Periodic solutions of evolution problem associated with moving convex sets

Charles Castaing, Manuel D.P. Monteiro Marques (1995)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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This paper is concerned with periodic solutions for perturbations of the sweeping process introduced by J.J. Moreau in 1971. The perturbed equation has the form $- D u \in N_{C (t)} (u (t)) + f (t, u (t))$ where C is a T-periodic multifunction from [0,T] into the set of nonempty convex weakly compact subsets of a separable Hilbert space H, $N_{C (t)} (u (t))$ is the normal cone of C(t) at u(t), f:[0,T] × H∪H is a Carathéodory function and Du is the differential measure of the periodic BV solution u. Several existence results of periodic solutions...

Generalized $c$ -almost periodic type functions in $ℝ^{n}$

M. Kostić (2021)

Archivum Mathematicum

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In this paper, we analyze multi-dimensional quasi-asymptotically $c$ -almost periodic functions and their Stepanov generalizations as well as multi-dimensional Weyl $c$ -almost periodic type functions. We also analyze several important subclasses of the class of multi-dimensional quasi-asymptotically $c$ -almost periodic functions and reconsider the notion of semi- $c$ -periodicity in the multi-dimensional setting, working in the general framework of Lebesgue spaces with variable exponent. We provide...

Abstract inclusions in Banach spaces with boundary conditions of periodic type

Lahcene Guedda, Ahmed Hallouz (2014)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We study in the space of continuous functions defined on [0,T] with values in a real Banach space E the periodic boundary value problem for abstract inclusions of the form ⎧ $x \in S (x (0), s e l_{F} (x))$ ⎨ ⎩ x (T) = x(0), where, $F : [0, T] \times \to 2^{E}$ is a multivalued map with convex compact values, ⊂ E, $s e l_{F}$ is the superposition operator generated by F, and S: × L¹([0,T];E) → C([0,T]; ) an abstract operator. As an application, some results are given to the periodic boundary value problem for nonlinear differential inclusions governed...

Stable periodic solutions in scalar periodic differential delay equations

Anatoli Ivanov, Sergiy Shelyag (2023)

Archivum Mathematicum

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A class of nonlinear simple form differential delay equations with a $T$ -periodic coefficient and a constant delay $τ > 0$ is considered. It is shown that for an arbitrary value of the period $T > 4 τ - d_{0}$ , for some $d_{0} > 0$ , there is an equation in the class such that it possesses an asymptotically stable $T$ -period solution. The periodic solutions are constructed explicitly for the piecewise constant nonlinearities and the periodic coefficients involved, by reduction of the problem to one-dimensional maps. The...

Global Attractor for the Convective Cahn-Hilliard Equation in $H^{k}$

Xiaopeng Zhao, Ning Duan (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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We consider the convective Cahn-Hilliard equation with periodic boundary conditions. Based on the iteration technique for regularity estimates and the classical theorem on existence of a global attractor, we prove that the convective Cahn-Hilliard equation has a global attractor in $H^{k}$ .

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.

Existence and uniqueness of positive periodic solutions for a class of integral equations with parameters

Shu-Gui Kang, Bao Shi, Sui Sun Cheng (2009)

Annales Polonici Mathematici

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Existence of periodic solutions of functional differential equations with parameters such as Nicholson’s blowflies model call for the investigation of integral equations with parameters defined over spaces with periodic structures. In this paper, we study one such equation $ϕ (x) = λ \int_{[x, x + ω] \cap Ω} K (x, y) h (y) f (y, ϕ (y - τ (y))) d y$ , x ∈ Ω, by means of the proper value theory of operators in Banach spaces with cones. Existence, uniqueness and continuous dependence of proper solutions are established.

A class of $I_{0}$ -sets

David Grow (1987)

Colloquium Mathematicae

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

W. R. Utz (1971)

Annales Polonici Mathematici

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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.

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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Weak Wecken's theorem for periodic points in dimension 3

Jerzy Jezierski (2003)

Fundamenta Mathematicae

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We prove that a self-map f: M → M of a compact PL-manifold of dimension ≥ 3 is homotopic to a map with no periodic points of period n iff the Nielsen numbers $N (f^{k})$ for k dividing n all vanish. This generalizes the result from [Je] to dimension 3.

On solutions of a fourth-order Lidstone boundary value problem at resonance

Mariusz Jurkiewicz (2009)

Annales Polonici Mathematici

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We consider a Lidstone boundary value problem in $ℝ^{k}$ at resonance. We prove the existence of a solution under the assumption that the nonlinear part is a Carathéodory map and conditions similar to those of Landesman-Lazer are satisfied.

On the Converse of Caristi's Fixed Point Theorem

Szymon Głąb (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let X be a nonempty set of cardinality at most $2^{ℵ ₀}$ and T be a selfmap of X. Our main theorem says that if each periodic point of T is a fixed point under T, and T has a fixed point, then there exist a metric d on X and a lower semicontinuous map ϕ :X→ ℝ ₊ such that d(x,Tx) ≤ ϕ(x) - ϕ(Tx) for all x∈ X, and (X,d) is separable. Assuming CH (the Continuum Hypothesis), we deduce that (X,d) is compact.