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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
TL;DR: In this paper, the existence results of positive ω-periodic solutions are obtained for the second-order differential equation with delays, where is a ωperiodic function, is a continuous function, which is ω -periodic in, and are positive constants.
Abstract: The 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.

12 citations

Journal ArticleDOI
TL;DR: In this article, the existence results of positive ω-periodic solutions are obtained for the second-order functional differential equation with multiple delays, where is a positive ε -periodic function, is a continuous function which is ωperiodic in t, and are ω periodic functions, and existence conditions concern the first eigenvalue of the associated linear periodic boundary problem.
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
TL;DR: In this paper, Banaś, Mohammad Mursaleen, Beata Rzepka, and Kishin Sadarangani discuss Compactness Conditions in the Theory of Nonlinear Differential and Integral Equations.
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
TL;DR: In this article, 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.
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
TL;DR: In this article, the existence of positive ω-periodic solutions for nth-order ODEs with delays in Banach space was studied and the strong positivity estimation was established.
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), ....

    [...]

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

    [...]

  • ...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
TL;DR: A survey of the main ideas, concepts, and methods that constitute nonlinear functional analysis can be found in this article, with extensive commentary, many examples, and interesting, challenging exercises.
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.

4,910 citations

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2,292 citations

Journal ArticleDOI
TL;DR: In this article, the existence of periodic solutions of the second-order Caratheodory problem is studied, by combining some new properties of Green's function together with Krasnoselskii fixed point theorem on compression and expansion of cones.

290 citations

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TL;DR: In this paper, the boundary value problem was considered in the context of boundary value maximization, where the authors considered the problem of finding a boundary value for a given set of variables.

99 citations

Journal ArticleDOI
TL;DR: In this paper, the existence of positive solutions of second-order nonlinear differential equations on a finite interval with periodic boundary conditions was proved and upper and lower bounds for these positive solutions were given.

86 citations