On existence of periodic solutions for gradient systems with applications

On a generalization of fractional Langevin equation

“…Marasi et al [12] studied the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems by using new fixed point results. For some recent works on existence results, see [3,4,11,14,17,18] According to the applications of Langevin equation, on the existence theory of solutions for this class of equations is an important subject. Ahmad and Nieto [1] discussed the existence of solutions of nonlinear Langevin equation involving two fractional orders with Dirichlet boundary conditions as follows:…”