There is a huge number of phenomena

There is a huge number of phenomena that are modeled by different equations involving fractional order derivatives. On the monographs [1, 2, 3] the reader can found examples that appear in different fields as physics, chemistry, aerodynamics or electrodynamics. The main difference with the usual derivative is that the definition of the fractional one involves all the values of the function up to the point t in which it is valued. This fact makes it more useful for problems with some kind of memory, [4]. As it can be seen in recent published articles, the study of this subject is growing up in the last years, see [5]-[12], [13]-[19] and the references therein.We make special mention to the recent papers [20, 21] in which, among other results, it has beenproved the existence and uniqueness of solution of different types of nonlinear fractional Cauchyproblems. Furthermore, in [22] is deduced the existence of solutions for impulsive fractionalstochastic differential equations with infinite delay. In these three papers the fixed point theory plays a fundamental role in the proofs of the obtained results.This paper considers a kind of integral boundary conditions. As it has been stated in [23], this type of conditions appear in different real phenomena as, among others, blood flow problems,chemical engineering, thermo-elasticity or population dynamics, see [24]-[33] and the references therein.