The extensive application of fractional differential equations and boundary-value problems for these equations promotes the development of the theory and the appearance of numerous publications in this field. One of the types of such equations are equations containing more than one differential operator of fractional order.

This paper deals with the study of linear boundary-value problem for the multi-term fractional differential equation with the Caputo derivative. We considered the left fractional Caputo derivative, which is convenient for the description of systems with memory. The boundary-value problem is specified by linear vector functional such that the number of it components does not coincide with the number of the orders of the derivative. Assume that the coefficients of the equation are continuous functions and the orders of the derivative are commensurate. A multi-term fractional differential equation is reduced to an equivalent system of differential equations containing only one fractional operator. The general solution of the system of fractional differential equations consisting of a general solution of the associated homogeneous system and the arbitrary particular solution of the inhomogeneous system is considered. The particular solution we found, which is also a solution of the system of linear Volterra integral equations of the second kind with square summable kernels. The question of the solvability of the boundary-value problem for the multi-term fractional differential equations was studied. We considered the critical case, i.e. case when the homogeneous problem has nontrivial solutions. By using the theory of pseudo-inverse matrices, the necessary and sufficient conditions for solvability of the given problem are established. Moreover, a family of linearly independent solutions of this boundary-value problem is constructed.

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