Solvability of a functional integral equation of fractional order in the class of functions having limits at infinity

A functional integral equation of fractional order is investigated in the space of real functions which are defined, continuous and bounded on the real half-axis. Using the technique of measures of noncompactness and the fixed point theorem of Darbo type we prove that the equation in question has solutions in the mentioned function space. Moreover, a suitable choice of a measure of noncompactness enables us to prove that those solutions tend to limits at infinity. An example illustrating our result is also given.