scholarly journals Multiple periodic solutions for a class of second-order nonlinear neutral delay equations

2006 ◽  
Vol 2006 ◽  
pp. 1-9 ◽  
Author(s):  
Xiao-Bao Shu ◽  
Yuan-Tong Xu
Keyword(s):  
Periodic Solutions ◽  
Index Theory ◽  
Second Order ◽  
Delay Equations ◽  
Group Index ◽  
Neutral Delay ◽  
Variational Structure ◽  
Multiple Periodic Solutions ◽  
T Tau

By means of a variational structure andZ2-group index theory, we obtain multiple periodic solutions to a class of second-order nonlinear neutral delay equations of the formx″(t−τ)+λ(t)f(t,x(t),x(t−τ),x(t−2τ))=x(t),λ(t)>0,τ>0.

scholarly journals Multiple periodic solutions to a class of second-order nonlinear mixed-type functional differential equations

2005 ◽  
Vol 2005 (17) ◽  
pp. 2689-2702
Author(s):  
Xiao-Bao Shu ◽  
Yuan-Tong Xu
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Mixed Type ◽  
Functional Differential Equations ◽  
Index Theory ◽  
Second Order ◽  
Group Index ◽  
Variational Structure ◽  
Multiple Periodic Solutions ◽  
T Tau

By means of variational structure andZ2group index theory, we obtain multiple periodic solutions to a class of second-order mixed-type differential equationsx''(t−τ)+f(t,x(t),x(t−τ),x(t−2τ))=0andx''(t−τ)+λ(t)f1(t,x(t),x(t−τ),x(t−2τ))=x(t−τ).

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>

scholarly journals Multiple solutions of boundary value problem

2006 ◽  
Vol 37 (2) ◽  
pp. 149-154
Author(s):  
Yongjin Li ◽  
Xiaobao Shu ◽  
Yuantong Xu
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Multiple Solutions ◽  
Boundary Value ◽  
Index Theory ◽  
Second Order ◽  
Group Index ◽  
Variational Structure

By means of variational structure and $ Z_2 $ group index theory, we obtain multiple solutions of boundary value problems for second-order ordinary differential equations$ \begin{cases} & - (ru')' + qu = \lambda f(t, u),\qquad 0 < t < 1 \\ & u'(0) = 0 = \gamma u(1)+ u'(1), \qquad \text{ where } \gamma \geq 0. \end{cases} $

scholarly journals On the Existence of Positive Periodic Solutions for Second-Order Functional Differential Equations with Multiple Delays

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Qiang Li ◽  
Yongxiang Li
Keyword(s):  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Index Theory ◽  
Second Order ◽  
Multiple Delays ◽  
First Eigenvalue ◽  
Fixed Point Index Theory ◽  
Existence Conditions ◽  
Functional Differential ◽  
Periodic Boundary Problem

The existence results of positiveω-periodic solutions are obtained for the second-order functional differential equation with multiple delaysu″(t)+a(t)u(t)=f(t,u(t),u(t−τ1(t)),…,u(t−τn(t))), wherea(t)∈C(ℝ)is a positiveω-periodic function,f:ℝ×[0,+∞)n+1→[0,+∞)is a continuous function which isω-periodic int, andτ1(t),…,τn(t)∈C(ℝ,[0,+∞))areω-periodic functions. The existence conditions concern the first eigenvalue of the associated linear periodic boundary problem. Our discussion is based on the fixed-point index theory in cones.

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