Open Access

# The Dirichlet problem for differential equations in a Banach space

### Abstract

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.

### Article Information

 Title The Dirichlet problem for differential equations in a Banach space Source Methods Funct. Anal. Topology, Vol. 18 (2012), no. 2, 140-151 MathSciNet MR2978190 zbMATH 1265.34214 Copyright The Author(s) 2012 (CC BY-SA)

### Authors Information

M. L. Gorbachuk
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine

V. I. Gorbachuk
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine

### Citation Example

M. L. Gorbachuk and V. I. Gorbachuk, The Dirichlet problem for differential equations in a Banach space, Methods Funct. Anal. Topology 18 (2012), no. 2, 140-151.

### BibTex

@article {MFAT628,
AUTHOR = {Gorbachuk, M. L. and Gorbachuk, V. I.},
TITLE = {The Dirichlet problem for differential equations in a Banach space},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {18},
YEAR = {2012},
NUMBER = {2},
PAGES = {140-151},
ISSN = {1029-3531},
MRNUMBER = {MR2978190},
ZBLNUMBER = {1265.34214},
URL = {http://mfat.imath.kiev.ua/article/?id=628},
}