Periodic Solutions, Such As When The System Under The Resonance Condition

In this paper, we study the existence of periodic solutions of the isochronous systemx" + f(x)x' + V'(x) + g(x)=p(t).under resonant conditions. Assume that all the solutions of x" + V'(x) = 0 are (?) periodic and V(x) is a strict convex potential function, whose derivative satisfies the local Lipschitz conditions and p(t)(∈L_loc¹(R)) is 2πperiodic. Assume that the following conditions are satisfied,and g(x),F(x) (F(x=∫₀^x f(u)du) are bounded. Moreover, the limits (?) =G^±, (?) exist and are finite, where G(x) =∫₀^x g(u)du,(?)=∫₀^x F(u)du.Define functions:Assume that one of the following conditions holds,(1)∏₁ has constant sign, or∏₂ has constant sign;(2)∏₁ and∏₂ don't take value 0 simultaneously, and∏₂ has constant sign at the zeros of∏₁;(3)∏₁ and∏₂ don't take value 0 simultaneously, and∏₂ change signs more than twice at the zero point of∏₁ in (?).We prove that the given equation has at least one 2πperiodic solution. On the other hand, we also deal with the existence of periodic solutions of equations as follows,x" + f(x') + V'(x)+g(x)=p(t),...