In this paper, we use the Banach fixed point theorem to obtain the existence, interval of existence and uniqueness of solutions for nonlinear hybrid implicit Caputo-Hadamard fractional differential equations. We also use the generalization of Gronwall's inequality to show the estimate of the solutions.

This paper is devoted to the study of nonlinear fractional Langevin integro differential equations with boundary conditions. Some effective results concerning the existence and uniqueness are obtained by applying the Banach contraction mapping principle and the Schauder fixed point theorem. An example is presented illustrating the effectiveness of the theoretical results.

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In this paper, we use the fixed point theory to obtain the existence and uniqueness of solutions for nonlinear implicit Riemann-Liouville fractional differential equations with nonlocal conditions. An example is given to illustrate this work.

This paper introduces a new methodology for fault diagnosis of rotating machinery in complex industrial process. In this fault diagnosis system, multi-scale principal component analysis (MSPCA) is used. The main contribution of this paper is the use of the MSPCA technique in the fault diagnosis based on vibration analysis in rotating machinery. Their maintenance as well as their diagnosis then became an economic state. It is important to early detect the faults likely to appear in those motors and to implement a preventive maintenance.