Periodic solutions for equation with A(t) and B(t) changing signs

In this paper, we investigate the differential equation , where A,B,C∈C∞([0,1]), m>n>l and m,n,l∈Z+. A solution x(t) with x(1)=x(0) is called a periodic solution. Under some hypotheses which admit A(t) and B(t) without fixed sign, we obtain the upper bound (sometimes sharp) for the number of isolated periodic solutions of the equation. Applying these results for the Abel equation (i.e. m=3, n=2, l=1), we get that if there exists λ≠0 such that S(λ,t)⋅C(t)⋅λ<0 (resp. S(λ,t)⋅(A(t)λ+B(t))<0), then the equation has at most 2 (resp. 4) non-zero isolated periodic solutions. Furthermore, suppose that γ=(a(t),t) is a smooth curve which lies in (R\{0})×[0,1] with a(0)=a(1). We obtain that if vector fields (S(x,t),1) (resp. ) and (C(t)x,1) are transverse to (resp. (S(x,t),1)) on γ in opposite directions, then the number of non-zero isolated periodic solutions of this Abel equation is still no more than 2 (resp. 4). These conclusions generalize the known criteria about the Abel equation which only refer to the cases with either A(t) or B(t) keeping sign. Finally, as an application we study a kind of trigonometrical Abel equation.