scholarly journals Note on the existence of periodic solutions of a second order differential equation

1969 ◽  
Vol 21 (3) ◽  
pp. 305-307
Author(s):  
F. H. Szafraniec
2018 ◽  
Vol 24 (2) ◽  
pp. 127-137
Author(s):  
Jaume Llibre ◽  
Ammar Makhlouf

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.


2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Zaihong Wang ◽  
Jin Li ◽  
Tiantian Ma

We study the existence of periodic solutions of the second-order differential equationx′′+ax+-bx-+g(x(t-τ))=p(t), wherea,bare two constants satisfying1/a+1/b=2/n,n∈N,τis a constant satisfying0≤τ<2π,g,p:R→Rare continuous, andpis2π-periodic. When the limitslimx→±∞g(x)=g(±∞)exist and are finite, we give some sufficient conditions for the existence of2π-periodic solutions of the given equation.


2003 ◽  
Vol 2003 (4) ◽  
pp. 209-228 ◽  
Author(s):  
O. Rabiei Motlagh ◽  
Z. Afsharnezhad

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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