Homoclinic solutions for a class of nonlinear second-order differential equations with time-varying delays (Q395648)

The paper studies the following second-order differential equation with time-varying delays NEWLINE\[NEWLINE\begin{aligned} & x''(t)+a(t)g(x'(t))\\ & +\sum_{i=1}^n b_i(t)h_i (x(t-\tau_1(t)),x(t-\tau_2(t)),\ldots,x(t-\tau_n(t)))=f(t), \end{aligned}NEWLINE\]NEWLINE where the functions \(a,b_i,\tau_i,f,g,h_i\), are continuous on \(\mathbb R\), \(a,b_i\) are \(2T\)-periodic, positive, and bounded, \(\tau_i\)'s are positive for \(i=1,\ldots, n\).NEWLINENEWLINEThe paper mainly applies Mawhin's continuation theorem of coincidence degree theory to obtain some sufficient criteria for the existence of homoclinic solutions of the above equation and then provides an example.