Periodic singular problem with quasilinear differential operator

Irena Rachůnková; Milan Tvrdý

Displaying similar documents to “Periodic singular problem with quasilinear differential operator”

Strong singularities in mixed boundary value problems

Irena Rachůnková (2006)

Mathematica Bohemica

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We study singular boundary value problems with mixed boundary conditions of the form ${(p (t) u^{'})}^{'} + p (t) f (t, u, p (t) u^{'}) = 0, lim_{t \to 0 +} p (t) u^{'} (t) = 0, u (T) = 0,$ where $[0, T] \subset ℝ .$ We assume that $ℝ^{2},$ $f$ satisfies the Carathéodory conditions on $(0, T) \times$ $p \in C [0, T]$ and $1 / p$ need not be integrable on $[0, T] .$ Here $f$ can have time singularities at $t = 0$ and/or $t = T$ and a space singularity at $x = 0$ . Moreover, $f$ can change its sign. Provided $f$ is nonnegative it can have even a space singularity at $y = 0 .$ We present conditions for the existence of solutions positive on $[0, T) .$ ...

Periodic solutions of dissipative dynamical systems with singular potential and p-Laplacian

Bing Liu (2002)

Annales Polonici Mathematici

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By using the topological degree theory and some analytic methods, we consider the periodic boundary value problem for the singular dissipative dynamical systems with p-Laplacian: ${(ϕ_{p} (x^{'}))}^{'} + d / d t g r a d F (x) + g r a d G (x) = e (t)$ , x(0) = x(T), x’(0) = x’(T). Sufficient conditions to guarantee the existence of solutions are obtained under no restriction on the damping forces d/dt gradF(x).

An existence theorem of positive solutions to a singular nonlinear boundary value problem

Gabriele Bonanno (1995)

Commentationes Mathematicae Universitatis Carolinae

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In this note we consider the boundary value problem $y^{''} = f (x, y, y^{'})$ $(x \in [0, X]; X > 0)$ , $y (0) = 0$ , $y (X) = a > 0$ ; where $f$ is a real function which may be singular at $y = 0$ . We prove an existence theorem of positive solutions to the previous problem, under different hypotheses of Theorem 2 of L.E. Bobisud [J. Math. Anal. Appl. 173 (1993), 69–83], that extends and improves Theorem 3.2 of D. O’Regan [J. Differential Equations 84 (1990), 228–251].

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

Daqing Jiang, Junyu Wang (1997)

Annales Polonici Mathematici

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The generalized periodic boundary value problem -[g(u’)]’ = f(t,u,u’), a < t < b, with u(a) = ξu(b) + c and u’(b) = ηu’(a) is studied by using the generalized method of upper and lower solutions, where ξ,η ≥ 0, a, b, c are given real numbers, $g (s) = {| s |}^{p - 2} s$ , p > 1, and f is a Carathéodory function satisfying a Nagumo condition. The problem has a solution if and only if there exists a lower solution α and an upper solution β with α(t) ≤ β(t) for a ≤ t ≤ b.

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.

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.

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 $\pm 10^{- 7}$ .

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].