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We consider an equation with left and right fractional derivatives and with the boundary condition y(0) = lim x→0+ y(x )=0 ,y(b) = lim x→by(x )=0 in the space L 1 (0,b) and in the subspace of tempered distributions. The asymptotic behavior of solutions in the end points 0 and b have been specially analyzed by using Karamata's regularly varying functions. In the last years differential equations of fractional orders have been used in many branches of mechanics and physics. Many results have been published with concrete problems solved in classical spaces of functions and in the spaces of gen- eralized functions. We cite only some of them, recently published or with a new approach: (2)-(4), (7), (8), (13), (15), (17), (19), (20), (22), (23) and with Karamata's regularly varying functions: (11), (24). In this paper we treat such an equation with the boundary condition y(0) = y(b) = 0 in the space L 1 (0 ,b ) and in a subspace of tempered distributions constructed for this problem. We specially discussed as- ymptotic behavior of solutions in the end points 0 and b using Karamata's regularly varying functions and quasi-asymptotics in the space of tempered distributions. As far as we are aware the equation treated in this paper has been solved only in (1) and (18) in some very special cases.

An equation with left and right fractional derivatives