scholarly journals NONTRIVIAL PERIODIC SOLUTIONS FOR SECOND-ORDER DIFFERENTIAL DELAY EQUATIONS

2017 ◽  
Vol 7 (3) ◽  
pp. 931-941
Author(s):  
Qi Wang ◽  
◽  
Wenjie Liu ◽  
Mei Wang ◽  
Keyword(s):  
Periodic Solutions ◽  
Second Order ◽  
Delay Equations ◽  
Differential Delay ◽  
Differential Delay Equations ◽  
Nontrivial Periodic Solutions

scholarly journals Periodic solutions for a class of second-order differential delay equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xuan Wu ◽  
Huafeng Xiao
Keyword(s):  
Periodic Solutions ◽  
Index Theory ◽  
Second Order ◽  
Delay Equations ◽  
Asymptotically Linear ◽  
Differential Delay ◽  
Differential Delay Equations ◽  
Existence Of Periodic Solutions ◽  
The Difference

<p style='text-indent:20px;'>In this paper, we study the existence of periodic solutions of the following differential delay equations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} z^{\prime\prime}(t) = \sum\limits_{k = 1}^{M-1}(-1)^kf(z(t-k)), \notag \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ f\in C(\mathbf{R}^N, \mathbf{R}^N) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ M,N\in \mathbf{N} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ M $\end{document}</tex-math></inline-formula> is odd. By making use of <inline-formula><tex-math id="M4">\begin{document}$ S^1 $\end{document}</tex-math></inline-formula>-geometrical index theory, we obtain an estimation about the number of periodic solutions in term of the difference between eigenvalues of asymptotically linear matrices at the origin and at infinity.</p>

Wright's equation has no solutions of period four

1989 ◽  
Vol 113 (3-4) ◽  
pp. 281-288 ◽  
Author(s):  
Roger D. Nussbaum
Keyword(s):  
Periodic Solutions ◽  
Fourth Order ◽  
Delay Equations ◽  
Order System ◽  
Minimal Period ◽  
Differential Delay ◽  
Differential Delay Equations

SynopsisLet N:ℝ→ℝ be a locally Lipschitzian map such that (y + l)N(y)>0 for all y ≠ –1 and such that N(y)=1 + y for – 1 ≦ y ≦ 3. For any positive number α the equation y'(t) αy(t–1)N(y(t)) has, aside from the constantsolutions y(t) ≡ –1, and y(t) ≡–1 solution y(t) such that y(t + 4) = y(t) for all real t If N(y) = 1 + y for all y, one obtains Wright's equation, which isknown to have periodic solutions of minimal period p (depending on α) arbitrarily close to 4. Some results concerning nonexistence of periodic solutions of period 4 of other differential-delay equations are also proved. In all cases the method of proof consists in analysing an associated fourth-order system of ordinary differential equationsand showing that this system has no nonconstant periodic solutions.

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