Well-posedness of a class of fractional Langevin equations (Q6554399)

The authors studied the existence, uniqueness and stability of solutions for the following nonlinear fractional Langevin equation in a real Banach space\N\[\N^CD^{\beta}_{0^+} (^CD^{\alpha}_{0^+}+\lambda) u(\tau)=f(\tau, u(\tau), I^{\gamma}_{0^+} u(\tau) ^CD^{\rho}_{0^+} u(\tau)),\N\]\N\[\Nu^{(k)}(0) =u_k\\\N\]\N\[\Nu^{(k+\alpha)}(0) =v_k\N\]\Nwhere all the terms are well defined. They have worked in a special space defined by \(X=\{ u| u(\tau) \in C(J,\mathbb R), ^CD^{\rho}_{0^+} u(\tau) \in C(J,\mathbb R)\}\)\N\Nwith norm defined by \(||u||_X = \max\{ ||u||,||^CD^{\rho}_{0^+} u||\)\N\Nwhere \(||u|| =\sup\{ |u(\tau)|,\tau \in J\},\; ||^CD^{\rho}_{0^+} u||=\sup\{ |^CD^{\rho}_{0^+}u(\tau)|,\tau \in J\} \) With the developed the necessary infrastructure and showed that \(X\) ia Banach Space.\N\NThey transformed the given problem to a fixed point problem and using Diaz-Margolis fixed point alternative, Boyd and Wang fixed point theorem and the Nonlinear alternative for single valued maps they studied he existence and uniqueness of solutions for the considered problem. They also, studied the stability of Ulam-Hyers, Ulam-Hyers-Rassias and semi-Ulam-Hyers-Rassias for the considered nonlinear fractional Langevin equation. They illustrated their work with examples.