Correct Solvability of Integrodifferential Equations in Spaces of Vector Functions Holomorphic in an Angular Domain

Integrodifferential equations with unbounded operator coefficients in a Hilbert space are studied. The main part of an equation of this kind is an abstract parabolic equation perturbed by a Volterra integral operator. The fundamental difference between this work and the other ones is that integrodifferential equations are considered and studied in this paper for vector functions the arguments of which take values in an angular domain on the complex plane.

In [1,2], the class of functions holomorphic in an angular domain of the form such that is studied. In [1,2], it is established that equipped with the corresponding norm is a Hilbert space and a theorem of Paley-Wiener type is proved for it.
Let H be a separable Hilbert space, and let A be a self-adjoint positive operator ( ) acting in H with a compact inverse. Let (⋅, ⋅) and ||⋅|| denote the scalar product and the norm in H, respectively.
This paper studies the classes and of functions with values in H that are holomorphic in the domain . The class consists of vector functions such that  We give a theorem that is an analogue of the Paley-Wiener theorem for the space .

Theorem 2. Let
The following assertions are true: 1 0 . The class of functions coincides with the set of functions admitting the representation

The function in representation (1) is unique for each fixed function
, and the following inversion formula is valid: Now we consider and study analogues of the Sobolev spaces of functions holomorphic in the angular domain .
In what follows, let denote the derivative of a function u(τ) in the sense of functions of a complex variable. The class of functions coincides with the class of functions holomorphic in the angular domain such that The following lemma is an analogue of the intermediate derivative theorem, which is well known for (see [3] [4]. Complete detailed proofs of the above assertions concerning the spaces and are given in [5].

CORRECT SOLVABILITY OF THE INITIAL VALUE PROBLEM FOR AN INTEGRODIFFERENTIAL EQUATION IN THE SPACE
We consider the initial value problem for an integrodifferential equation of the form where A is a self-adjoint positive operator acting in a separable Hilbert space H with a compact inverse. The kernel K(τ) belongs to the Hardy space (see [7]), , and . Note that the integration in the integral term in (4) is carried out over the interval connecting the origin and the point . However, due to the regularity of the functions K(τ) and u(τ), the integral can be taken over any recti-  (4), (5) such that (7) where the constant d is independent of the vector function f(τ) and the vector .

Corollary 1. Assume that the assumptions of Theorem 5 hold and
Then there exists a unique solution of problem (4), (5) satisfying inequality (6).
The idea of proving Theorem 5 can be described as follows. We consider the Laplace transform of the strong solution u(τ) of (4) with zero initial data , which has the form Here, the operator function is the symbol of Eq. (4) and it can be represented as where is the Laplace transform of the kernel and I is the identity operator in the space H. To prove Theorem 5, due to Theorem 2 (Paley-Wiener theorem), it suffices to show that the vector functions and are in the space and to derive their estimates. With this aim in view, we establish that the operator function is holomorphic and admits the following estimates in the domain = : (8) In turn, to deduce (8), we use the representation the estimates (11) and the inequality which is implied by (6), (10), (11), and (9). Based on (8), (10), and the fact that the vector function is in the space , we find that the vector functions and belong to the space . Thus, using Theorem 2 (Paley-Wiener theorem), we conclude that the vector functions and belong to and satisfy the inequalities Finally, it follows from (12) and (13) that (14) where . Furthermore, the case of inhomogeneous initial data is standardly reduced to a problem with homogeneous initial data and a new right-hand side of (4) of the form , where and the vector function is then estimated. Remark 1. It is well known that the solution of the initial boundary value problem for the homogeneous heat equation under natural assumptions on the initial data admits an analytic extension with respect to the time variable t to an angular domain on the complex plane. The theory of analytic semigroups of operators is closely related to this fact (see, for example, [6,[8][9][10]).
In Corollary 1 to Theorem 5, not only the analyticity of the solution of an abstract parabolic equation is established, but also estimate (7) is derived in the Hilbert spaces and , which is a much more profound result than the analyticity (holomorphicity) of the solution.
Theorem 5 also immediately yields the corresponding assertion for θ = 0, that is, for the case when , S H R problem (4), (5) is considered on the half-line , rather than in the angular domain . In this case, the space is replaced by , while is replaced by the Sobolev space . It is worth noting that Theorem 5 is closely related to Theorem 2.1 in [11] and Theorem 3.2.3 in [12]. These theorems establish the correct solvability of the initial value problem for an integrodifferential equation similar to (4) in the weighted Sobolev spaces . Remark 2. A significant difference of Theorem 5 from the solvability results deduced in [11,12] for the traditional Sobolev spaces is that we give estimates of the Laplace transform of the solution in the domain , rather than in the right halfplane. Our results rely heavily on Theorem 2 (an analogue of the Paley-Wiener theorem) for the angular domains and , whereas the previous works use the conventional Paley-Wiener theorem.
Note that a number of solvability results for integrodifferential equations with operator coefficients in the Sobolev spaces are given in [13]. Some profound results on the solvability of elliptic functional differential equations in Sobolev spaces are described in [14]. bution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Com-