Solvability of a Class of Operator-Differential Equations of Third Order with Complicated Characteristic on the Whole Real Axis

Abstract

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.

Sufficient conditions for regular solvability of an arbitrary order operator-differential equation with initial-boundary conditions

“…n+1 j=1 c j (α, γ 0 )b(α, γ 0 ) A j A -j H→H < 1,where the numbers b(α, γ 0 ) and c j (α, γ 0 ), j = 1, n + 1, are calculated in Theorem 11. Then the initial-boundary value problem (4)-(5) is regularly solvable[1].…”

On the Solvability of Higher-Order Operator-Differential Equations in a Weighted Sobolev Space

“…The principal part of the investigated equation has mixed multiple-complicated characteristics with m, n order (m, n ≥ 1). It should be noted that for specific values of m, n, the solvability problem for second (m = 0, n = 2), third (m = 1, n = 2), fourth (m = 1, n = 3) and fifth (m = 5, n = 0)− order operator differential equations have been studied elsewhere (see [1]- [7], [16]). This study differs from the study in work [5].…”

“…is true, then u(t) is called a regular solution of equation (1) and equation ( 1) is called regularly solvable (see [1], [2], [12], [16]).…”