In this research paper, the nonlinear fractional relaxation equation involving the generalized Caputo derivative is reduced to an equivalent integral equation via the generalized Laplace transform. Moreover, the upper and lower solutions method combined with some xed point theorems, and the properties of the Mittag-Leer function are applied to investigate the existence and uniqueness of positive solutions for the problem at hand. At the end, to illustrate our results, we give an example.

In this paper, a version modied of contraction Hardy-Rogers type in a metric space and is proved. Moreover, we apply this modied version to investigate the existence of unique solution of boundary value problems for the dierential equations and generalized fractional dierential equations through help of the properties of Green function. We also provide an example in support of acquired results. These results extend various comparable results from literature.