The article is devoted to the investigation of solutions of differential equations on an infinite interval, whose coefficients are unbounded linear operators in a Banach space $\mathfrak{B}$ over the field $\mathbb{C}$ of complex numbers. Namely, we consider equations of the form $y'(t) - Ay(t) = 0$ and $y'(t) + Ay(t) = 0$ on the whole real axis, where $A$ is the infinitesimal generator of a bounded analytic $C_{0}$-semigroup $\{e^{tA}\}_{t≥0}$ of linear operators in $\mathfrak{B}$, that is, a parabolic and inverse parabolic type, respectively, differential equations in a Banach space. Most the problems under consideration in the paper are related to the theory of abstract differential equations, one of the main directions of modern functional analysis which, as is well-known, covers a number of partial differential equations. For example, if $\mathfrak{B} = L^{p}(\mathbb{R}^{n}) \ (1 \leq p < \infty) $ and $Au(x) = \Delta u(x), x \in \mathbb{R}^{n}, (\Delta$ is the Laplacian), then the first equation above is none other than the classical heat one. But the study of such equations is useful not only because they cover a lot of partial differential equations, it also enables to look from a uniform point of view at ordinary as well as partial differential equations. The origin of the mentioned theory dates from the work of E.Hille and K.Yosida (1948), in which the first theorems on the existence of solutions to the Cauchy problem for the equation $y' = Ay$ with an unbounded operator $A$ in a Banach space, formulated in terms of the theory semigroups, were obtained. In the middle of the last century, P.Lax, R.S. Phillips, A.Milgram, V.Lyantce, and T.Kato applied the semigroup methods to the investigation of various classes of parabolic equations. These scientists laid the foundations of the theory of differential equations with unbounded operators, which thereafter became a field of independent interest, attracting the attention of many mathematicians including S.D. Eidelman. We describe all the solutions of the above abstract differential equations and find the conditions which are necessary and sufficient for a solution to admit an extension to an entire vector-valued function with given order of growth and type. Moreover, the criterions for such classes of solutions to be dense in the set of all solutions are presented. So, the conditions are established under which for each solution $y(z)$ of the corresponding equation, there exists a sequence $y_{n}(z)$ of a certain order and type entire solutions converging uniformly to $y(z)$ on every compact set $K \subset \mathbb{C}$. It should be noted that similar problems for equations on $[0, \infty)$ were considered by M.L.Gorbachuk.

[1] Arendt W., Batty C.J.K., Hieber M., and Neubrander F. Vector-valued Laplace transforms and Cauchy problems. Birkhauser Verlag, Basel-Boston-Berlin, 1999.

[2] Engel K.-J. and Nagel R. One-parameter semigroups for linear evolution equations. Springer-Verlag, Berlin - New York, 2000.

[3] Gorbachuk M.L.On analytic solutions of differential-operator equations. Ukrain. Mat.Zh. 2000, 52 (5), 596–607. (in Ukrainian)

[4] Gorbachuk M.L., Gorbachuk V.I. On approximation of smooth vectors of a closed operator by entire vectors of exponential type. Ukrain. Mat.Zh. 1995, 47 (5), 616–628. (in Ukrainian)

[5] Gorbachuk M.L., Gorbachuk V.I. On the well-posed solvability in some classes of entire functions of the Cauchy problem for differential equations in a Banach space. Methods Funct. Anal. Topology 2005, 11 (2), 113–125.

[6] Gorbachuk M.L., Mokrousov Yu.G. On density of some sets of infinitely differentiable vectors of a closed operator on a Banach space. Methods Funct. Anal. Topology 2002, 8 (1), 23–29.

[7] Gorbachuk V.I., Gorbachuk M.L. Boundary value problems for operator differential equations. Kluwer Academic Publishers., Dordrecht-Boston-London, 1991.

[8] Gorbachuk V.M. On solutions of parabolic and elliptic type differential equations on $(−∞, ∞)$ in a Banach space. Methods Funct. Anal. Topology 2008, 14 (2), 177–183.

[9] Gorbachuk V.M. On the structure of solutions of operator-differential equations on the whole real axis. Methods Funct. Anal. Topology 2015, 21 (2), 170–178.

[10] Gorbachuk V. On solutions of parabolic type differential equations in a Banach space. In: Proc. of the Intern. Conf. “Modern Problems of Differential Equations and Applications”, Chernivtsi, Ukraina: National University of Chernivtsi, September 16–19, 2020, 108–109. (in Ukrainian)

[11] Hille E. and Phillips R.S. Functional analysis and semi-groups. Amer. Math. Soc., Providence, RI, 1957. [12] Krein S.G. Linear differential equations in Banach space. Nauka, Moscow, 1967 . (in Russian)

[13] Vasilyev V.V., Krein S.G., and Piskariov S.I. Semigroups of operators, cosine operator functions, and linear differential equations. Itogi Nauki i Techniki, Ser. Math., Math. Anal. 1990, 28, 87–202. (in Russian)

[14] Yosida K. Functional Analysis. Springer-Verlag, Berlin-Gottingen-Heidelberg, 1965.