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Criteria of stability and boundedness of solutions of second-order and third-order equations in Hilbert spaces. (Russian) Zbl 0536.34037
The second-order equation (1) \(y''+A(t)y=0\); y(t),y”(t)\(\in H\); A(t)\(\in L(H,H)\); \(t\in [a,\omega)\), where H is a Hilbert space and L(H,H) is the space of linear operators acting from H into H, is investigated with respect to stability and boundedness of the solutions, the results are partially generalized to third-order equations. The first two theorems indicate boundedness, and stability resp., for the solution of (1) in case of a representation \(A(t)=a(t)E+A_ 1(t)\), where a(t) is a twice continuously differentiable positive function on \([a,\omega)\) bounded from below, and an integral limiting condition involving a differential invariant is valid for \(A_ 1\). Another necessary and sufficient criterion for boundedness refers to an operator T(t) transforming A(t) to a diagonal matrix. An analogous criterion is stated for third-order equations.
Reviewer: E.Ihle
34G10 Linear differential equations in abstract spaces
34D20 Stability of solutions to ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations