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Journal ArticleDOI

Positive Periodic Solutions of Second-Order Differential Equations with Delays

03 Jul 2012-Abstract and Applied Analysis (Hindawi)-Vol. 2012, pp 1-13

AbstractThe existence results of positive ω-periodic solutions are obtained for the second-order differential equation with delays , where is a ω-periodic function, is a continuous function, which is ω-periodic in , and are positive constants. Our discussion is based on the fixed point index theory in cones.

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Citations
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Journal ArticleDOI
TL;DR: This paper builds a new maximum principle for the ω-periodic solutions of the corresponding linear equation with delay and studies the existence of the minimal and maximal periodic solutions for abstract delayed equation by combining perturbation method and monotone iterative technique of the lower and upper solutions.
Abstract: In this paper, we deal with the existence of ω-periodic solutions for second-order functional differential equation with delay in E - u ' ' ( t ) = f ( t , u ( t ) , u ( t - ? ) ) , t ? R , where E is an ordered Banach space, f : R × E × E ? E is a continuous function which is ω-periodic in t and ? ? 0 is a constant. We first build a new maximum principle for the ω-periodic solutions of the corresponding linear equation with delay. With the aid of this maximum principle, under the assumption that the nonlinear function is quasi-monotonicity, we study the existence of the minimal and maximal periodic solutions for abstract delayed equation by combining perturbation method and monotone iterative technique of the lower and upper solutions.

10 citations


Journal ArticleDOI
Abstract: The existence results of positive ω-periodic solutions are obtained for the second-order functional differential equation with multiple delays , where is a positive ω-periodic function, is a continuous function which is ω-periodic in t, and are ω-periodic functions. The existence conditions concern the first eigenvalue of the associated linear periodic boundary problem. Our discussion is based on the fixed-point index theory in cones.

6 citations


Journal ArticleDOI
Abstract: and Applied Analysis Compactness Conditions in the Theory of Nonlinear Differential and Integral Equations Guest Editors: Jozef Banaś, Mohammad Mursaleen, Beata Rzepka, and Kishin Sadarangani

2 citations


Journal ArticleDOI
Abstract: The existence results of positive -periodic solutions are obtained for the third-order ordinary differential equation with delays where is -periodic function and is a continuous function which is -periodic in are positive constants. The discussion is based on the fixed-point index theory in cones.

1 citations


Cites methods from "Positive Periodic Solutions of Seco..."

  • ...In [3, 8, 9, 11, 12], the authors obtained the existence of positive periodic solutions for some first-order and second-order delay differential equations by using Krasnoselskii’s fixed-point theorem of cone mapping....

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Journal ArticleDOI
Abstract: This paper deals with the existence of positive ω-periodic solutions for nth-order ordinary differential equation with delays in Banach space E of the form $$L_{n}u(t)=f\bigl(t,u(t-\tau_{1}),\ldots,u(t- \tau_{m})\bigr),\quad t\in\mathbb{R}, $$ where $L_{n}u(t)=u^{(n)}(t)+\sum_{i=0}^{n-1}a_{i} u^{(i)}(t)$ is the nth-order linear differential operator, $a_{i}\in\mathbb {R}$ ($i=0,1,\ldots,n-1$) are constants, $f: \mathbb{R}\times E^{m}\rightarrow E$ is a continuous function which is ω-periodic with respect to t, and $\tau_{i}>0$ ($i=1,2,\ldots,m$) are constants which denote the time delays. We first prove the existence of ω-periodic solutions of the corresponding linear problem. Then the strong positivity estimation is established. Finally, two existence theorems of positive ω-periodic solutions are proved. Our discussion is based on the theory of fixed point index in cones.

1 citations


Cites background from "Positive Periodic Solutions of Seco..."

  • ...Recently, Li [8] discussed the existence of positive ω-periodic solutions of the second-order differential equation with delays of the form –u′′(t) + a(t)u(t) = f ( t, u(t – τ1), ....

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  • ...The results obtained in [8] can deal with the case of second-order differential equations, but for high-order differential equations, for example,...

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  • ...Lnu(t) = 1 2 u2(t – τ1) + 1 4 u2(t – τ2) + 1 8 u2(t – τ3), t ∈ R, the results of [8] are not valid....

    [...]


References
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Book
01 Oct 1984
Abstract: This graduate-level text offers a survey of the main ideas, concepts, and methods that constitute nonlinear functional analysis. It features extensive commentary, many examples, and interesting, challenging exercises. Topics include degree mappings for infinite dimensional spaces, the inverse function theory, the implicit function theory, Newton's methods, and many other subjects. 1985 edition.

4,727 citations


Book
01 Jan 1988

2,241 citations


Journal ArticleDOI
Abstract: This paper is devoted to study the existence of periodic solutions of the second-order equation x 00 ¼ f ðt; xÞ; where f is a Caratheodory function, by combining some new properties of Green's function together with Krasnoselskii fixed point theorem on compression and expansion of cones. As applications, we get new existence results for equations with jumping nonlinearities as well as equations with a repulsive or attractive singularity. In this latter case, our results cover equations with weak singularities and are compared with some recent results by I. Rachunkova´ , M. Tvrdyand I. Vrkoc˘ . r 2002 Elsevier Science (USA). All rights reserved.

275 citations


Journal ArticleDOI
Abstract: The authors consider the boundary value problem � y �� − ρ 2 y + λg(t)f (y) = 0, 0 t 2π, y(0) = y(2 π) , y � (0) = y � (2 π) .

96 citations


Journal ArticleDOI
Abstract: We prove the existence of positive solutions of second-order nonlinear differential equations on a finite interval with periodic boundary conditions and give upper and lower bounds for these positive solutions. Obtained results yield positive periodic solutions of the equation on the whole real axis, provided that the coefficients are periodic.

84 citations