On The Periodic Solutions of a Generalized Smooth and Non-Smooth Perturbed Planar Double Pendulum with Small Oscillations

We provide sufficient conditions for the existence of periodic solutions of the smooth and non-smooth perturbed planar double pendulum with small oscillations having equations of motion\[\begin{array}{l}\ddot{\T}_{1}=-a\T_{1}+\T_{2}+\e \left(F_1(t,\T_1,\dot {\T}_1,\T_2, \dot{\T}_2)+F_2(t,\T_1,\dot {\T}_1,\T_2, \dot {\T}_2){\rmsgn}(\dot{\T_{1}})\right),\\\ddot{\T}_{2}=b\T_{1}-b\T_{2}+\e \left(F_3(t,\T_1,\dot {\T}_1,\T_2, \dot{\T}_2)+F_4(t,\T_1,\dot {\T}_1,\T_2, \dot {\T}_2){\rmsgn}(\dot{\T_{2}})\right),\end{array}\]where $a>1,b>0$ and $\e$ are real parameters. Here the parameter $\e$ is small and the smooth functions $F_i$ for $i=1,2,3,4$ define the perturbation which are periodic functions in $t$ and in resonance $p_{i}$:$q_{i}$ with some of the periodic solutions of the unperturbed double pendulum, being $p_{i}$ and $q_{i}$ relatively prime positive integers.