The existence of subharmonic solutions with prescribed minimal period for forced pendulum equations with impulses (Q358065)

The authors study the existence of subharmonic solutions with prescribed minimal period for a forced pendulum equation with constant impulses NEWLINE\[NEWLINE u''(t)+A \sin u(t)=f(t) \text{ for a.e. } t \in [0,pT]\backslash \{t_k\}_{k=1}^{m}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE\Delta u'(t_k)=d_k, ~~~k=1,2,\ldots,m,NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(0)=u(pT),NEWLINE\]NEWLINE where \(0=t_0<t_1<\cdots<t_m<t_{m+1}=pT\), \(f\) is a \(T\)-periodic function regarded as an external force.NEWLINENEWLINEMotivated by the paper due to \textit{J. J. Nieto} and \textit{D. O'Regan} [Nonlinear Anal., Real World Appl. 10, No. 2, 680--690 (2009; Zbl 1167.34318)], the authors mainly apply the variational methods and critical point theory [\textit{P. H. Rabinowitz}, Reg. Conf. Ser. Math. 65, vii, 100 p. (1986; Zbl 0609.58002)] to obtain the existence of subharmonic solutions of the above impulsive problem.NEWLINENEWLINETheir results do not require that \(f\) is odd, which, in the case of forced pendulum equation without impulses, improve the results by \textit{Qi Wang} et al. [Nonlinear Anal., Theory Methods Appl. 28, No. 7, 1273--1282 (1997; Zbl 0872.34022)] and \textit{J. Yu} [J. Dyn. Differ. Equations 20, No. 4, 787--796 (2008; Zbl 1361.34041); J. Differ. Equations 247, No. 2, 672--684 (2009; Zbl 1179.34038)].NEWLINENEWLINESeveral examples are provided to show the applications of their results and advantages in comparison to previous works in the literature.