|
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）(∈Lloc1（R）) is 2πperiodic. Assume that the following conditions are satisfied,and g（x）,F（x） (F（x=∫0x f（u）du） are bounded. Moreover, the limits （?） =G±, （?） exist and are finite, where G（x） =∫0x g（u）du,（?）=∫0x 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）,
|