Existence of nonoscillatory solutions of the third order quasilinear neutral differential equations (Q2756998)

The authors consider third-order quasilinear neutral differential equations of the form NEWLINE\[NEWLINE (r_2(t)(r_1(t)\varphi (L_0'x(t)))')'+f(t, x(g(t)))=0,\quad t\geq a\geq 0, NEWLINE\]NEWLINE with \(L_0x(t)=x(t)-p(t)x(h(t))\). First, they classify the nonoscillatory solutions in the set \(N^+=\{x\) is a nonoscillatory solution to the above equation: \(x(t)L_0x(t)>0\) for all large \(t\}\) into three types. Then, by using the Schauder-Tikhonov fixed-point theorem, they establish sufficient and necessary conditions for the existence of nonoscillatory solutions to the above equation for each type with specified asymptotic behavior as \(t\rightarrow \infty\).