ψ$$ \psi $$‐Caputo type time‐delay Langevin equations with two general fractional orders

Abstract

In the present paper, first, a ‐delayed Mittag–Leffler type function is introduced, which generalizes the existing delayed Mittag–Leffler type function. Second, by means of ‐delayed Mittag–Leffler type function, an exact analytical solution formula to non‐homogeneous linear delayed Langevin equations involving two distinct ‐Caputo type fractional derivatives of general orders is obtained. Moreover, existence and uniqueness, stability of solution to nonlinear delayed Langevin fractional differential equations is obtained in the weighted space. Numerical and simulated examples are shared to exemplify the theoretical findings.