existence and ulam−hyers stability analysis for nonlinear langevin equation featuring two fractional orders involving anti-periodic boundary conditions

This paper presents an investigation of the existence and the unique feature of solutions for nonlinear langevin equations involving two fractional orders with anti-periodic boundary conditions (apbcs). as a result of employing some fixed point theorems like  schauder and contraction mapping principles, the existence and uniqueness of solutions are examined. on top of this,   the stability within the scope of ulam–hyers of solutions to this problem is also considered. the distinctive features of the present study are its similar variant and the existence of derivatives of caputo and riemann in the problem structure. finally, to illustrate the result of the study, an example is presented.