Periodic solutions for a second-order differential equation with indefinite weak singularity

2019 ◽  
Vol 149 (5) ◽  
pp. 1135-1152 ◽  
Author(s):  
José Godoy ◽  
Manuel Zamora
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Second Order ◽  
Constant Sign ◽  
Piecewise Constant ◽  
Weak Singularity ◽  
Second Order Differential Equation ◽  
Image Position

AbstractAs a consequence of the main result of this paper efficient conditions guaranteeing the existence of a T −periodic solution to the second-order differential equation $${u}^{\prime \prime} = \displaystyle{{h(t)} \over {u^\lambda }}$$are established. Here, h ∈ L(ℝ/Tℤ) is a piecewise-constant sign-changing function and the non-linear term presents a weak singularity at 0 (i.e. λ ∈ (0, 1)).

scholarly journals Periodic solutions of a second order nonlinear differential equation

1989 ◽  
Vol 40 (3) ◽  
pp. 357-361 ◽  
Author(s):  
Bahman Mehri
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Fixed Point ◽  
Periodic Solutions ◽  
Nonlinear Differential Equation ◽  
Order Differential Equation ◽  
Second Order ◽  
Fixed Point Technique ◽  
Image Position ◽  
Schauder Fixed Point

We consider the following non-linear nonautonomous second order differential equationwhere h(x) is continuous, f, p are continuous and periodic with respect to t of period w. Using the Leray-Schauder fixed point technique we prove that the above equation possesses at least one non-trivial periodic solution of period w.

Periodic Solutions of a Singular Equation With Indefinite Weight

2010 ◽  
Vol 10 (4) ◽  
Author(s):  
José Luis Bravo ◽  
Pedro J. Torres
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Periodic Solutions ◽  
Existence And Uniqueness ◽  
Order Differential Equation ◽  
Singular Equation ◽  
Indefinite Weight ◽  
Piecewise Constant ◽  
Second Order Differential Equation ◽  
The Stability

AbstractMotivated by some relevant physical applications, we study the existence and uniqueness of T-periodic solutions for a second order differential equation with a piecewise constant singularity which changes sign. Other questions like the stability and robustness of the periodic solution are considered.

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

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.

On the Forced Lienard Equation

1969 ◽  
Vol 12 (1) ◽  
pp. 79-84 ◽  
Author(s):  
R.R. Stevens
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Second Order ◽  
Liénard Equation ◽  
Second Order Differential Equation ◽  
Image Position ◽  
Lienard Equation

We consider the second order differential equation(1)with the assumptions that(2) f(x) is continuous (- ∞ < x < ∞) and p(t) is continuous and bounded: |p(t)| ≤ E, - ∞ < t < ∞.Also, throughout this paper, F(x) denotes an antiderivative of f(x).

scholarly journals Existence and Uniqueness of Periodic Solutions for a Second-Order Nonlinear Differential Equation with Piecewise Constant Argument

2009 ◽  
Vol 2009 ◽  
pp. 1-14 ◽  
Author(s):  
Gen-qiang Wang ◽  
Sui Sun Cheng
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Periodic Solutions ◽  
Nonlinear Differential Equation ◽  
Existence And Uniqueness ◽  
Second Order ◽  
Piecewise Constant ◽  
Piecewise Constant Argument ◽  
Order Nonlinear Differential Equation ◽  
Constant Argument

Based on a continuation theorem of Mawhin, a unique periodic solution is found for a second-order nonlinear differential equation with piecewise constant argument.

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