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In this study, we establish existence-uniqueness of a vector function in appropriate Sobolev-type space for a boundary value problem of a fifth-order operator differential equation. Proper conditions are obtained for the given problem to be well-posed. Much effort is devoted to develop the association between these conditions and the operator coefficients of the investigated equation. In this paper, accurate estimates of the norms of the intermediate derivatives operators are presented and used to determine the solvability conditions.

On this paper, for an arbitrary order operator-differential equation with the weight $e^{\frac{-\alpha t}{2}}, \alpha \in (-\infty ,+ \infty )$e−αt2,α∈(−∞,+∞), in the space $W^{n+m}_{2}(R_{+};H)$W2n+m(R+;H), we attain sufficient conditions for the well-posedness of a regular solvable of the boundary value problem. These conditions are provided only by the operator coefficients of the investigated equation where the leading part of the equation has multiple characteristics. We prove the connection between the lower bound of the spectrum of the higher-order differential operator in the main part and the exponential weight and also obtain estimations of the norms of operator intermediate derivatives. We apply the results of this paper to a mixed problem for higher-order partial differential equations (HOPDs).

On the whole real axis, we demonstrate sufficient conditions of regular solvability of third order operator-differential equations with complicated characteristics. These conditions were formulated only by the operator coefficients of the equation. In addition, by the principal part of the equation, the norms of the operators of intermediate derivative were estimated.

In this paper, a class of operator-differential equation of the first order with multiple characteristics is considered, for which the initial boundary value problem on the semi-axis is well-posed and uniquely solvable in the Sobolev space.

The paper studied the matters of solvability of the second order operator differential equations with multiple characteristics on the semi-axis. In the space of Sobolev type W22 (R+ ; H) identified conditions that ensured the unique and well-posed solvability are of the considered equation which is cleared by its operator coefficients