The authors generalise the results about the number of closed solutions for the equation in general form: \(dx/dt=f(t,x)\), \((t,x)\in I\times J\), where I and J are open intervals, \([0,1]\in I\) and \(f(t,x),(t,x)\in I\times J\) is a real valued continuous function having continuous derivatives \(f_{x^ j}(t,x)\) with respect to x of orders \(j\leq n\) and \(n=1,2,3\). The essential requirement is to satisfy the condition: \[ \forall t\in [0,1]\forall x\in J:\;f_{x^ n}(t,x)\geq 0,\exists t_ 0\in [0,1]\forall x\in J:\;f_{x^ n}(t_ 0,x)>0. \]
They also investigate the stability properties of closed solutions. Finally, the authors have shown that, under appropriate conditions, there is a relation between the two equations \(\dot x=f(t,x)\) and \(\dot x=f_ x(t,x)\) in the case \(n=2\), related to the number of closed solution (the derivative of function \(x(t)\) is denoted \(\dot x(t)\)). The results are valid for any other interval [0,T], \(T>0\). Hence the derived results applied for an equation \(dx/dt=p(t,x),\) \((t,x)\in R\times J,\) where \(p(t,x)\) is a real valued periodic function with period T, one can obtain theorems on the maximal number of T-periodic solutions.