scholarly journals Existence of Periodic Solutions for the Duffing Equation with Impulses

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Xuxin Yang ◽  
Weibing Wang ◽  
Jianhua Shen
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Solutions ◽  
Point Theorem ◽  
Existence Of Solutions ◽  
Duffing Equation ◽  
Infinitely Many Solutions ◽  
Sufficient Condition ◽  
Existence Of Periodic Solutions

We study the existence of solutions to the Duffing equation with impulses. By means of the Poincaré-Birkhoff fixed point theorem under given conditions, we obtain the sufficient condition of existence of infinitely many solutions. Our results generalize those of T. R. Ding. An example is presented to demonstrate applications of our main result.

scholarly journals Periodic Solutions and Nontrivial Periodic Solutions for a Class of Rayleigh-Type Equation with Two Deviating Arguments

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Meiqiang Feng
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Solutions ◽  
Point Theorem ◽  
Type Equation ◽  
Index Theorem ◽  
Rayleigh Equation ◽  
Deviating Arguments ◽  
Nontrivial Periodic Solutions ◽  
Existence Of Periodic Solutions

The Rayleigh equation with two deviating argumentsx′′(t)+f(x'(t))+g1(t,x(t-τ1(t)))+g2(t,x(t-τ2(t)))=e(t)is studied. By using Leray-Schauder index theorem and Leray-Schauder fixed point theorem, we obtain some new results on the existence of periodic solutions, especially for the existence of nontrivial periodic solutions to this equation. The results are illustrated with two examples, which cannot be handled using the existing results.

scholarly journals The Solvability and Optimal Controls for Some Fractional Impulsive Equations of Order1< α<2

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Xianghu Liu ◽  
Zhenhai Liu ◽  
Maojun Bin
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Existence Of Solutions ◽  
Schauder’S Fixed Point Theorem ◽  
Sufficient Condition ◽  
Optimal Controls ◽  
Schauder's Fixed Point Theorem ◽  
Gronwall’S Inequality ◽  
Impulsive Equations

We study the existence of solutions and optimal controls for some fractional impulsive equations of order1< α<2. By means of Gronwall’s inequality and Leray-Schauder’s fixed point theorem, the sufficient condition for the existence of solutions and optimal controls is presented. Finally, an example is given to illustrate our main results.

NONDEGENERACY OF THE PERIODICALLY FORCED LIÉNARD DIFFERENTIAL EQUATION WITH ϕ-LAPLACIAN

2011 ◽  
Vol 13 (02) ◽  
pp. 283-292 ◽  
Author(s):  
P. J. TORRES
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Solutions ◽  
Point Theorem ◽  
Multiplicity Results ◽  
Related Literature ◽  
Pendulum Equation ◽  
Existence Of Periodic Solutions ◽  
Schäuder Fixed Point Theorem

New results on the existence of periodic solutions of a forced Liénard differential equation with ϕ-Laplacian are provided. The method of proof relies on the Schauder fixed point theorem, so some information on the location of the solutions is also obtained, leading to multiplicity results. The flexibility of this approach is tested by comparing our results with some examples taken from the related literature, including the classical pendulum equation.

Periodic Solutions of Almost Linear Volterra Integro-dynamic Equations on Periodic Time Scales

2015 ◽  
Vol 58 (1) ◽  
pp. 174-181 ◽  
Author(s):  
Youssef N. Raffoul
Keyword(s):  
Fixed Point ◽  
Nonlinear System ◽  
Fixed Point Theorem ◽  
Periodic Solutions ◽  
Time Scales ◽  
Point Theorem ◽  
Discrete Time ◽  
Dynamic Equations ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Existence Of Periodic Solutions

AbstractUsing Krasnoselskii’s fixed point theorem, we deduce the existence of periodic solutions of nonlinear system of integro-dynamic equations on periodic time scales. These equations are studied under a set of assumptions on the functions involved in the equations. The equations will be called almost linear when these assumptions hold. The results of this paper are new for the continuous and discrete time scales.

scholarly journals A Note on Periodic Solutions of Second Order Nonautonomous Singular Coupled Systems

2010 ◽  
Vol 2010 ◽  
pp. 1-15 ◽  
Author(s):  
Zhongwei Cao ◽  
Chengjun Yuan ◽  
Daqing Jiang ◽  
Xiaowei Wang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Solutions ◽  
Point Theorem ◽  
Second Order ◽  
Coupled Systems ◽  
Schauder’S Fixed Point Theorem ◽  
Schauder's Fixed Point Theorem ◽  
Existence Of Periodic Solutions

We establish the existence of periodic solutions of the second order nonautonomous singular coupled systemsx′′+a1(t)x=f1(t,y(t))+e1(t)for a.e.t∈[0,T],y′′+a2(t)y=f2(t,x(t))+e2(t)for a.e.t∈[0,T]. The proof relies on Schauder's fixed point theorem.

scholarly journals Applications of Schauder’s Fixed Point Theorem to Semipositone Singular Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Zhongwei Cao ◽  
Chengjun Yuan ◽  
Xiuling Li
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Point Theorem ◽  
Schauder’S Fixed Point Theorem ◽  
Singular Differential Equations ◽  
Schauder's Fixed Point Theorem ◽  
Existence Of Periodic Solutions ◽  
Weak Singularities

We study the existence of positive periodic solutions of second-order singular differential equations. The proof relies on Schauder’s fixed point theorem. Our results generalized and extended those results contained in the studies by Chu and Torres (2007) and Torres (2007) . In some suitable weak singularities, the existence of periodic solutions may help.

scholarly journals Existence of Solutions of Abstract Nonlinear Mixed Functional Integrodifferential equation with nonlocal conditions

2017 ◽  
Vol 4 (1) ◽  
pp. 1-15
Author(s):  
Machindra B. Dhakne ◽  
Poonam S. Bora
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Integrodifferential Equation ◽  
Existence Of Solutions ◽  
Nonlocal Condition ◽  
Nonlocal Conditions ◽  
Fractional Power ◽  
Strong Solutions ◽  
Fractional Power Of Operators

Abstract In this paper we discuss the existence of mild and strong solutions of abstract nonlinear mixed functional integrodifferential equation with nonlocal condition by using Sadovskii’s fixed point theorem and theory of fractional power of operators.

scholarly journals Existence of Positive Periodic Solutions for a Class ofn-Species Competition Systems with Impulses

2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
Peilian Guo ◽  
Yansheng Liu
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Solutions ◽  
Point Theorem ◽  
Sufficient Conditions ◽  
Species Competition ◽  
Positive Periodic Solutions ◽  
The Fixed Point Theorem

By using the fixed point theorem on cone, some sufficient conditions are obtained on the existence of positive periodic solutions for a class ofn-species competition systems with impulses. Meanwhile, we point out that the conclusion of (Yan, 2009) is incorrect.

scholarly journals Multiplicity of positive periodic solutions to second order differential equations

2006 ◽  
Vol 73 (2) ◽  
pp. 175-182 ◽  
Author(s):  
Jifeng Chu ◽  
Xiaoning Lin ◽  
Daqing Jiang ◽  
Donal O'Regan ◽  
R. P. Agarwal
Keyword(s):  
Periodic Solution ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Point Theorem ◽  
Positive Periodic Solution ◽  
Second Order ◽  
Positive Periodic Solutions ◽  
Second Order Differential Equations

In this paper, we study the existence of positive periodic solutions to the equation x″ = f (t, x). It is proved that such a equation has more than one positive periodic solution when the nonlinearity changes sign. The proof relies on a fixed point theorem in cones.

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