Abstract
The main objective of this paper is to introduce the (alpha,beta)-type J-contraction, (alpha, beta)-type rational J-contraction, and cyclic (alpha-theta) contraction. Based on these definitions we prove fixed point theorems in the complete metric spaces. These results extend and improve some known results in the literature. As an application of the proved fixed point Theorems, we study the existence of solutions of an integral boundary value problem for scalar nonlinear Caputo fractional differential equations with a fractional order in (1, 2).