Periodic solutions of superlinear and sublinear state-dependent discontinuous differential equations

A classical, second-order differential equation is considered with state-dependent impulses at both the position and its derivative. This means that the instants of impulsive effects depend on the solutions and they are not fixed beforehand, making the study of this problem more difficult and interesting from the real applications point of view. The existence of periodic solutions follows from a transformation of the problem into a planar system followed by a study of the Poincaré map and the use of some fixed point theorems in the plane. Some examples are presented to illustrate the main results.

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A new Ф-generalized quasi metric space with some fixed point results and applications

Filomat ◽

10.2298/fil1711157a ◽

2017 ◽

Vol 31 (11) ◽

pp. 3157-3172

Author(s):

Mujahid Abbas ◽

Bahru Leyew ◽

Safeer Khan

Keyword(s):

Fixed Point ◽

Differential Equations ◽

Fixed Points ◽

Periodic Solutions ◽

Metric Space ◽

Delay Differential Equations ◽

Fixed Point Theorems ◽

Metric Spaces ◽

Existence Of Periodic Solutions ◽

Delay Differential

In this paper, the concept of a new ?-generalized quasi metric space is introduced. A number of well-known quasi metric spaces are retrieved from ?-generalized quasi metric space. Some general fixed point theorems in a ?-generalized quasi metric spaces are proved, which generalize, modify and unify some existing fixed point theorems in the literature. We also give applications of our results to obtain fixed points for contraction mappings in the domain of words and to prove the existence of periodic solutions of delay differential equations.

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Existence and Lyapunov Stability of Positive Periodic Solutions for a Third-Order Neutral Differential Equation

ISRN Applied Mathematics ◽

10.5402/2011/698529 ◽

2011 ◽

Vol 2011 ◽

pp. 1-28 ◽

Cited By ~ 1

Author(s):

Jingli Ren ◽

Zhibo Cheng ◽

Yueli Chen

Keyword(s):

Differential Equation ◽

Fixed Point ◽

Fixed Point Theorem ◽

Periodic Solutions ◽

Lyapunov Stability ◽

Order Differential Equation ◽

Sufficient Conditions ◽

Neutral Differential Equation ◽

Positive Periodic Solutions ◽

Third Order

By applying Green's function of third-order differential equation and a fixed point theorem in cones, we obtain some sufficient conditions for existence, nonexistence, multiplicity, and Lyapunov stability of positive periodic solutions for a third-order neutral differential equation.

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Periodic solutions for periodic second-order differential equations with variable potentials

Journal of Applied Analysis ◽

10.1515/jaa-2018-0013 ◽

2018 ◽

Vol 24 (2) ◽

pp. 127-137

Author(s):

Jaume Llibre ◽

Ammar Makhlouf

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Periodic Solutions ◽

Order Differential Equation ◽

Sufficient Conditions ◽

Second Order ◽

Second Order Differential Equations ◽

Second Order Differential Equation ◽

Existence Of Periodic Solutions

Abstract We provide sufficient conditions for the existence of periodic solutions of the second-order differential equation with variable potentials {-(px^{\prime})^{\prime}(t)-r(t)p(t)x^{\prime}(t)+q(t)x(t)=f(t,x(t))} , where the functions {p(t)>0} , {q(t)} , {r(t)} and {f(t,x)} are {\mathcal{C}^{2}} and T-periodic in the variable t.

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Periodic Solutions for a Class of Functional Differential Equations with State-Dependent Delay Close to Zero

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202503002738 ◽

2003 ◽

Vol 13 (06) ◽

pp. 807-841 ◽

Cited By ~ 6

Author(s):

R. Ouifki ◽

M. L. Hbid

Keyword(s):

Differential Equation ◽

Hopf Bifurcation ◽

Periodic Solutions ◽

Functional Differential Equations ◽

Delay Equation ◽

Constant Delay ◽

State Dependent ◽

State Dependent Delay ◽

Existence Of Periodic Solutions ◽

Functional Differential

The purpose of the paper is to prove the existence of periodic solutions for a functional differential equation with state-dependent delay, of the type [Formula: see text] Transforming this equation into a perturbed constant delay equation and using the Hopf bifurcation result and the Poincaré procedure for this last equation, we prove the existence of a branch of periodic solutions for the state-dependent delay equation, bifurcating from r ≡ 0.

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The Integrability and Existence of Periodic Solutions on a First-Order Nonlinear Differential Equation with a Polynomial Nonlinear Term

Mathematical Problems in Engineering ◽

10.1155/2012/703474 ◽

2012 ◽

Vol 2012 ◽

pp. 1-26

Author(s):

Ni Hua ◽

Tian Li-Xin

Keyword(s):

Differential Equation ◽

Periodic Solutions ◽

Nonlinear Differential Equation ◽

Nonlinear Term ◽

Order Differential Equation ◽

First Order ◽

Order Nonlinear Differential Equation ◽

Existence Of Periodic Solutions ◽

The Stability

This paper deals with a first-order differential equation with a polynomial nonlinear term. The integrability and existence of periodic solutions of the equation are obtained, and the stability of periodic solutions of the equation is derived.

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On the existence of periodic solutions of a certain third-order differential equation

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100034678 ◽

1960 ◽

Vol 56 (4) ◽

pp. 381-389 ◽

Cited By ~ 15

Author(s):

J. O. C. Ezeilo

Keyword(s):

Differential Equation ◽

Periodic Function ◽

Periodic Solutions ◽

Order Differential Equation ◽

Initial Conditions ◽

Continuous Periodic Function ◽

Third Order ◽

Third Order Differential Equation ◽

Existence Of Periodic Solutions ◽

Image Position

In this paper we shall be concerned with the differential equationin which a and b are constants, p(t) is a continuous periodic function of t with a least period ω, and dots indicate differentiation with respect to t. The function h(x) is assumed continuous for all x considered, so that solutions of (1) exist satisfying any assigned initial conditions. In an earlier paper (2) explicit hypotheses on (1) were established, in the two distinct cases:under which every solution x(t) of (1) satisfieswhere t0 depends on the particular x chosen, and D is a constant depending only on a, b, h and p. These hypotheses are, in the case (2),or, in the case (3),In what follows here we shall refer to (2) and (H1) collectively as the (boundedness) hypotheses (BH1), and to (3) and (H2) as the hypotheses (BH2). Our object is to examine whether periodic solutions of (1) exist under the hypotheses (BH1), (BH2).

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New Existence of Fixed Point Results in Generalized Pseudodistance Functions with Its Application to Differential Equations

Mathematics ◽

10.3390/math6120324 ◽

2018 ◽

Vol 6 (12) ◽

pp. 324 ◽

Cited By ~ 2

Author(s):

Sujitra Sanhan ◽

Winate Sanhan ◽

Chirasak Mongkolkeha

Keyword(s):

Differential Equation ◽

Fixed Point ◽

Differential Equations ◽

Contraction Mapping ◽

Fixed Point Theorems ◽

Metric Spaces ◽

Order Differential Equation ◽

Second Order ◽

Second Order Differential Equation ◽

Generalized Pseudodistance

The purpose of this article is to prove some existences of fixed point theorems for generalized F -contraction mapping in metric spaces by using the concept of generalized pseudodistance. In addition, we give some examples to illustrate our main results. As the application, the existence of the solution of the second order differential equation is given.

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Existence of periodic solutions and homoclinic orbits for third-order nonlinear differential equations

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171203107089 ◽

2003 ◽

Vol 2003 (4) ◽

pp. 209-228 ◽

Cited By ~ 3

Author(s):

O. Rabiei Motlagh ◽

Z. Afsharnezhad

Keyword(s):

Differential Equation ◽

Periodic Solutions ◽

Order Differential Equation ◽

Nonlinear Differential Equations ◽

Homoclinic Orbits ◽

Order Equation ◽

Second Order ◽

Third Order ◽

Bifurcation Theorem ◽

Existence Of Periodic Solutions

The existence of periodic solutions for the third-order differential equationx¨˙+ω2x˙=μF(x,x˙,x¨)is studied. We give some conditions for this equation in order to reduce it to a second-order nonlinear differential equation. We show that the existence of periodic solutions for the second-order equation implies the existence of periodic solutions for the above equation. Then we use the Hopf bifurcation theorem for the second-order equation and obtain many periodic solutions for it. Also we show that the above equation has many homoclinic solutions ifF(x,x˙,x¨)has a quadratic form. Finally, we compare our result to that of Mehri and Niksirat (2001).

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