On nonlinear implicit fractional differential equations without compactness

The main purpose of this research paper is to develop some sufficient conditions for the existence of solution of a nonlinear problem of implicit fractional differential equations (IFDEs) with boundary conditions, using prior estimate method. The distinction of the method applied here is, it does not require compactness of the operator. This idea is the result of motivation from the book of O’Regan and other [R. P. Agarwal, D. O’Regan, Y. J. Cho, Y.-Q. Chen, Taylor Francis Group, New York, (2006)]. Devising the respective conditions, we also developed some conditions for Hyers-Ulam type stability to the solution of the said problem. To justify the relevant results a suitable example is provided.