V. M. Gorbachuk

For a differential equation of the form $y''(t) - By(t) = 0, \ t \in (0, \infty)$, where $B$ is a weakly positive linear operator in a Banach space $\mathfrak{B}$, the conditions on the operator $B$, under which this equation is uniformly or uniformly exponentially stable are given. As distinguished from earlier works dealing only with continuous at 0 solutions, in this paper no conditions on behavior of a solution near 0 are imposed.

We consider an equation of the form $y'(t) + Ay(t) = 0, \ t \in [0, \infty)$, where $A$ is a nonnegative self-adjoint operator in a Hilbert space. We give direct and inverse theorems on approximation of solutions of this equation with its entire solutions of exponential type. This establishes a one-to-one correspondence between the order of convergence to $0$ of the best approximation of a solution and its smoothness degree. The results are illustrated with an example, where the operator $A$ is generated by a second order elliptic differential expression in the space $L_{2}(\Omega)$ (the domain $\Omega \subset \mathbb{R}^{n}$ is bounded with smooth boundary) and a certain boundary condition.

We consider differential equations of the form $\left(\frac{d^{2}}{dt^{2}} - B\right)^{m}y(t) = f(t)$, $m \in \mathbb{N}, \ t \in (-\infty, \infty)$, where $B$ is a positive operator in a Banach space $\mathfrak{B}, \ f(t)$ is a bounded continuous vector-valued function on $(-\infty, \infty)$ with values in $\mathfrak{B}$, and describe all their solutions. In the case, where $f(t) \equiv 0$, we prove that every solution of such an equation can be extended to an entire $\mathfrak{B}$-valued function for which the Phragmen-Lindel\"{o}f principle is fulfilled. It is also shown that there always exists a unique bounded on $\mathbb{R}^{1}$ solution, and if $f(t)$ is periodic or almost periodic, then this solution is the same as $f(t)$.

We show that every classical solution of a parabolic or elliptic type homogeneous differential equation on $(-\infty, \infty)$ in a Banach space may be extended to an entire vector-valued function. The description of all the solutions is given, and necessary and sufficient conditions for a solution to be continued to a finite order and finite type entire vector-valued function are presented.