On Caputo fractional elliptic equation with nonlocal condition

This paper is first study for considering nonlocal elliptic equation with Caputo derivative. We obtain the upper bound of the mild solution. The second contribution is to provide the lower bound of the solution at terminal time. We prove the non-correction of the problem in the sense of Hadamard. The main tool is the use of upper and lower bounds of the Mittag-Lefler function, combined with analysis in Hilbert scales space.