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Journal ArticleDOI

On Positive Periodic Solutions to Impulsive Differential Equations with Delays

01 Mar 2004-Results in Mathematics (Birkhäuser-Verlag)-Vol. 45, Iss: 1, pp 67-78
TL;DR: In this article, the existence of positive periodic solutions of differential equations with impulses and delays was studied and some new results were obtained by using the Krasnoselskii fixed point theorem on cone compression and expansion.
Abstract: In this paper, by using the Krasnoselskii fixed point theorem on cone compression and expansion, we study the existence of positive periodic solutions of differential equations with impulses and delays, and obtain some new results.
Citations
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Journal ArticleDOI
01 Jan 2019-Filomat
TL;DR: In this article, four functionals fixed point theorem is used to investigate the existence of positive solutions for second-order time-scale boundary value problem of impulsive dynamic equations on the half-line.
Abstract: In this paper, four functionals fixed point theorem is used to investigate the existence of positive solutions for second-order time-scale boundary value problem of impulsive dynamic equations on the half-line.

4 citations

Posted Content
TL;DR: In this article, a new representation of periodic solution to the impulsive logistic equation was provided, which is based on the one presented in [7] and [8].
Abstract: In this paper is provided a new representation of periodic solution to the impulsive Logistic equation considered in [7].

3 citations

Journal ArticleDOI
TL;DR: In this article, the authors employ the well-known Krasnoselskii fixed point theorem to study the existence and n-multiplicity of positive periodic solutions for the periodic impulsive functional differential equations with two parameters.
Abstract: In this paper, we employ the well-known Krasnoselskii fixed point theorem to study the existence and n-multiplicity of positive periodic solutions for the periodic impulsive functional differential equations with two parameters. The form including an impulsive term of the equations in this paper is rather general and incorporates as special cases various problems which have been studied extensively in the literature. Easily verifiable sufficient criteria are obtained for the existence and n-multiplicity of positive periodic solutions of the impulsive functional differential equations.

2 citations

Journal ArticleDOI
TL;DR: In this article, the existence of three positive solutions to a class of nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales with parameter are obtained by using the Leggett-Williams fixed point theorem.
Abstract: By using the Leggett-Williams fixed point theorem, the existence of three positive solutions to a class of nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales with parameter are obtained. An example is given to illustrate the main results in this paper.
References
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Book
14 Oct 1993
TL;DR: The present book builds upon the earlier work of J. Hale, "Theory of Functional Differential Equations" published in 1977 and attempts to maintain the spirit of that book and have retained approximately one-third of the material intact.
Abstract: The present book builds upon the earlier work of J. Hale, "Theory of Functional Differential Equations" published in 1977. The authors have attempted to maintain the spirit of that book and have retained approximately one-third of the material intact. One major change was a completely new presentation of linear systems (Chapter 6-9) for retarded and neutral functional differential equations. The theory of dissipative systems (Chapter 4) and global attractors was thoroughly revamped as well as the invariant manifold theory (Chapter 10) near equilibrium points and periodic orbits. A more complete theory of neutral equations is presented (Chapters 1,2,3,9,10). Chapter 12 is also entirely new and contains a guide to active topics of research. In the sections on supplementary remarks, the authors have included many references to recent literature, but, of course, not nearly all, because the subject is so extensive.

6,844 citations

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

Book
01 May 1989
TL;DR: Impulsive differential equations, that is, differential equations involving impulse effects, appear as a natural description of observed evolution phenomena of several real world problems.
Abstract: Many evolution processes are characterized by the fact that at certain moments of time they experience a change of state abruptly. These processes are subject to short-term perturbations whose duration is negligible in comparison with the duration of the process. Consequently, it is natural to assume that these perturbations act instantaneously, that is, in the form of impulses. It is known, for example, that many biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, pharmacokinetics and frequency modulated systems, do exhibit impulsive effects. Thus impulsive differential equations, that is, differential equations involving impulse effects, appear as a natural description of observed evolution phenomena of several real world problems.

3,856 citations

Journal ArticleDOI
08 Mar 1940-Science

521 citations