Solution of nonlinear Langevin equations involving Hilfer-Hadamard fractional order derivatives and variable coefficients

This study innovates a novel technique of the nonlinear fractional Langevin equation of Hilfer-Hadamard type, incorporating an initial condition. The research demonstrates that this problem can be reformulated as an integral equation featuring a Mittag-Leffler function within the kernel. Through rigorous analysis, we establish the existence and uniqueness of solutions for this problem without imposing a contractive assumption. Furthermore, the study extends these findings to encompass two specific types of fractional differential equations characterized by two fractional derivatives and a variable coefficient. Theoretical conclusions are supported by the provision of two illustrative examples.