# V. I. Gorbachuk

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### On behavior at infinity of solutions of parabolic differential equations in a Banach space

M. L. Gorbachuk, V. I. Gorbachuk

MFAT **20** (2014), no. 3, 274-283

274-283

For a differential equation of the form $y'(t) + Ay(t) = 0, \ t \in (0, \infty)$, where $A$ is the generating operator of a $C_{0}$-semigroup of linear operators on a Banach space $\mathfrak{B}$, we give conditions on the operator $A$, under which this equation is uniformly (uniformly exponentially) stable, that is, every its weak solution defined on the open semiaxis $(0, \infty)$ tends (tends exponentially) to 0 as $t \to \infty$. As distinguished from the previous works dealing only with solutions continuous at 0, in this paper no conditions on the behavior of a solution near 0 are imposed. In the case where the equation is parabolic, there always exist weak solutions which have singularities of any order. The criterions below not only generalize, but make more precise a number of earlier results in this direction.

### The Dirichlet problem for differential equations in a Banach space

M. L. Gorbachuk, V. I. Gorbachuk

MFAT **18** (2012), no. 2, 140-151

140-151

In the paper, we consider an abstract differential equation of the form $\left(\frac{\partial^{2}}{\partial t^{2}}- B \right)^{m}y(t) = 0$, where $B$ is a positive operator in a Banach space $\mathfrak{B}$. For solutions of this equation on $(0, \infty)$, it is established the analogue of the Phragmen-Lindelof principle on the basis of which we show that the Dirichlet problem for the above equation is uniquely solvable in the class of vector-valued functions admitting an exponential estimate at infinity. The Dirichlet data may be both usual and generalized with respect to the operator $-B^{1/2}$.The formula for the solution is given, and some applications to partial differential equations are adduced.

### On the approximation to solutions of operator equations by the least squares method

Myroslav L. Gorbachuk, Valentyna I. Gorbachuk

MFAT **14** (2008), no. 3, 229-241

229-241

We consider the equation $Au = f$, where $A$ is a linear operator with compact inverse in a Hilbert space. For the approximate solution $u_n$ of this equation by the least squares method in a coordinate system that is an orthonormal basis of eigenvectors of a self-adjoint operator $B$ similar to $A \ ({\mathcal{D}} (A) = {\mathcal{D}} (B))$, we give a priori estimates for the asymptotic behavior of the expression $R_n = \|Au_n - f\|$ as $n \to \infty$. A relationship between the order of smallness of this expression and the degree of smoothness of the solution $u$ with respect to the operator $B$ (direct and converse theorems) is established.

### On completeness of the set of root vectors for unbounded operators

Myroslav L. Gorbachuk, Valentyna I. Gorbachuk

MFAT **12** (2006), no. 4, 353-362

353-362

For a closed linear operator $A$ in a Banach space, the notion of a vector accessible in the resolvent sense at infinity is introduced. It is shown that the set of such vectors coincides with the space of exponential type entire vectors of this operator and the linear span of root vectors if, in addition, the resolvent of $A$ is meromorphic. In the latter case, the completeness criteria for the set of root vectors are given in terms of behavior of the resolvent at infinity.

### On well-posed solvability in some classes of entire functions of the Cauchy problem for differential equations in a Banach space

Myroslav L. Gorbachuk, Valentyna I. Gorbachuk

MFAT **11** (2005), no. 2, 113-125

113-125

### Damir Zyamovich Arov (to the 70th anniversary of his birth)

V. M. Adamyan, Yu. M. Berezansky, M. L. Gorbachuk, V. I. Gorbachuk, G. M. Gubreev, A. N. Kochubei, M. M. Malamud

MFAT **10** (2004), no. 2, 1-3

1-3

### On the Cauchy problem for differential equations in a Banach space over the field of p-adic numbers

Myroslav L. Gorbachuk, Valentyna I. Gorbachuk

MFAT **9** (2003), no. 3, 207-212

207-212