On the number of solutions for the forced pendulum equation (Q1262436)

The problem considered is the problem of the number of T-periodic solutions for the forced pendulum equation: (1) ẍ\(+\lambda \dot x+k \sin x=e(t)+c\), where \(\lambda\),k,c\(\in R\), \(k>0\), \(e=e(t)\in C[0,T]\) T- periodic function with zero average value. The main result of the article is following: There exists a positive \(\lambda_ 0=\lambda_ 0(k,e)\) depending on k,e such that for every \(\lambda:| \lambda | >\lambda_ 0\) there exist two numbers \(d=d(e,\lambda)\), \(D=D(e,\lambda)\) \((d<0<D)\) such that equation (1) has 1) no T -periodic solution if \(c\not\in [d,D];\) 2) exactly one T-periodic solution if \(c=d\) or \(c=D;\) 3) exactly two T-periodic solution if \(c\in (d,D).\) The corresponding result is also obtained for the equation (1) without friction (i.e., \(\lambda =0)\) under some restrictions imposed on e(t),k,T.