Positive periodic solutions for high-order differential equations with multiple delays in Banach spaces
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.
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Journal Article•
TL;DR: In this paper, the authors considered nonlinear singular third order periodic boundary value problem and obtained a positive solution under some conditions concern to the eigenvalue of relevant linear operator by fixed index theory on cone.
Abstract: We considered nonlinear singular third order periodic boundary value problem{u″'+ρ3u=f(t,u),t∈I=(0,2π),ρ∈(0,1/3~(1/2))u(i)(0)=u(i)(2π),i=0,1,2 and obtained positive solution under some conditions concern to the eigenvalue of relevant linear operator by fixed index theory on cone.
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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
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
TL;DR: In this article, the existence of positive solutions for non-linear third-order periodic boundary value problem was proved for the case where ρ ∈ ( 0, 1 3 ) is a positive constant and the nonlinearity f ( t, u ) may be singular at u = 0.
Abstract: In this paper, we are concerned with the problem of the existence of positive solutions for non-linear third-order periodic boundary value problem u ‴ + ρ 3 u = f ( t , u ) , 0 ⩽ t ⩽ 2 π , u ( i ) ( 0 ) = u ( i ) ( 2 π ) , i = 0 , 1 , 2 . Here, ρ ∈ ( 0 , 1 3 ) is a positive constant and our non-linearity f ( t , u ) may be singular at u = 0 . The proof relies on a non-linear alternative of Leray–Schauder type and on Krasnoselskii fixed point theorem on compression and expansion of cones.
43 citations
TL;DR: In this paper, the authors considered a type of second-order neutral functional differential equations and obtained existence results of multiplicity and nonexistence of positive periodic solutions based on a fixed point theorem in cones.
42 citations