A note on the nonexistence of limit cycles (Q1801737)

Many results on the non-existence of limit cycles for the differential system \[ dx/dt= y- F(x),\quad dy/dt= -g(x)\tag{1} \] depend on the inconsistency of the simultaneous equations (2) \(F(x)= F(y)\), \(G(x)= G(y)\), \(y< 0<x\), where \(G(x)= \int^ x_ 0 g(u)du\). In some cases, it is easier to examine the inconsistency of the equations (3) \(F(x)= F(y)\), \(f(x)/g(x)= f(y)/g(y)\), \(y<0<x\) than to do that of (2). Here, \(f(x)= F'(x)\). The following theorem is established. Theorem 1. Let \(xg(x)>0\) for \(x\neq 0\). Suppose one of the following two conditions holds: (i) \(\lim_{x\to 0^ +}| f(x)/g(x)|=\lim_{x\to 0^ -}| f(x)/g(x)|=\infty\), (ii) \(xf(x)\) keeps sign and \(\not\equiv 0\) in a neighborhood of \(x=0\). Then the inconsistency of (3) implies the inconsistency of (2). The author also proves the following result on the non-existence of separatrix cycles. Theorem 2. Let \(F\in C^ 1(x_ 2,x_ 1)\) and \(g\in C^ 0(x_ 2,x_ 1)\) where \(x_ 2<0<x_ 1\) and let (i) \(F(0)=0\), and \(g(x_ 1)=0\) or \(g(x_ 2)=0\); (ii) \(xg(x)>0\) for \(x\in (x_ 2,x_ 1)\), \(x\neq 0\); (iii) \(F_ 1(Z)- F_ 2(Z)\geq 0\) (or \(\leq 0)\), \(\not\equiv 0\) for \(0<Z<Z_ 0\), where \(F_ i(Z)= F_ i(x_ i(Z))\), \(i=1,2\), \(x_ 1(Z)>0>x_ 2(Z)\) are the inverse functions of the function \(Z=G(x)\), \(Z_ 0=\min\{G(x_ 2+0),\;G(x_ 1-0)\}\). Then (1) has no separatrix cycles which surround the origin and pass through \(A\) (when \(x_ 2\) is finite and \(g(x_ 2)=0\)) or \(B\) (when \(x_ 1\) is finite and \(g(x_ 1)=0\)) (or simultaneously through both \(A\) and \(B\)).