In paper [1] there is given an abstract scheme of investigation of mixed problems for hyperbolic equations with non stationary boundary conditions. In this direction, some results were obtained in [2].
In this paper, we offer the analogues abstract model of investigation of mixed problem with non stationary boundary and transmission conditions for impulsive linear and semilinear hyperbolic equations.
1.1. Statement of the Problem and Main Theorem
Let (; ; ; ) be Hilbert Spaces. Consider the following abstract initialboundary value problem:
where , , , are the linear closed operators in ; are the linear operators from to ; are the linear operators from to ; are the linear operators from to; , , , .
We will investigate this problem under the following conditions.

(i)
Let , and let be densely in and continuously imbedded into it, .
In the Hilbert space , it was defined the system of the inner products , which generate uniform equivalent norms, that is,
For each , the function is continuously differentiable, .
In the Hilbert space , it was defined the system of the inner products , which generate uniform equivalent norms, that is,
For each , the function is continuously differentiable.

(ii)
For each and is a linear closed operator in whose domain is ; acts boundedly from to ; is strongly continuously differentiable.

(iii)
The linear operators , that act from to , bounded, where is interpolation space between and of order (see [3]).

(iv)
For each , the linear operators , that act from to , are bounded; is strongly continuously differentiable , ; .

(v)
The linear operators , from into , act boundedly , ; .
Let us introduce the following designations:
From condition (v), it follows that the space with the norm
is a subspace of

(vi)
Let the linear manifold be dense in , and let linear manifold be dense in .

(vii)
(Green's Identity). For arbitrary and , the following identity is valid:

(viii)
For all , the following inequality is fulfilled:
where .

(ix)
For each
, an operator pencil
which acts boundedly from to , has a regular point , where

(x)

(xi)
,
Definition 1.1.
The function is called a solution of problem (1.1)(1.2) if the function from to is continuous, and the function
from to is twice continuously differentiable and (1.1)(1.2) are satisfied.
Theorem 1.2.
Let conditions (i)–(xi) are satisfied, then the problem (1.1)(1.2) has a unique solution.
Proof.
We define the operator in the Hilbert space in the following way:
Then the problem (1.1)(1.2) is represented as the Cauchy problem
where ,
It is obvious that if is the solution of problem (1.1)(1.2), then is the solution of the problem (1.16). On the contrary, if
is the solution of problem (1.16), then , and is the solution of problem (1.1)(1.2).
Let us define the system of inner product in Hilbert space in the following way:
where ,.
We denote space with inner product (1.19) by .
We will prove later the following auxiliary results.
Statement 1.3.
There exists such , that
and the function is continuously differentiable, where = .
Statement 1.4.
is a symmetric operator in for each .
Statement 1.5.
has a regular point for each in .
is symmetric and , for some ; therefore, for each is a selfadjoint operator in (see [4, chapter x]).
Taking into account (viii) and Statement 1.3, we get
that is, is a lower semibounded selfadjoint operator in .
Thus, the operator is selfadjoint and positive definite, where .
Problem (1.16) can be rewritten as
It is known that if and , then the problem (1.22) has a unique solution (see [5, 6]).
To complete the proof of the theorem, we need to show that and .
By conditions of the theorem ; , and are bounded operators from to . Therefore,
On the other hand, and , therefore, . Consequently,
From the definition of interpolation spaces (see [3, chapter 1], [7, chapter 1]), we get the following inclusion:
By virtue of definition, the powers of positive selfadjoint operator (see [8, chapter 2], [7, chapter 1]), we have that and
Assume that , then
By virtue of conditions (ii), (viii), (1.26), and (1.27), we get
Let . By virtue of condition (vi), is dense in ; therefore, there exists a sequence , such that and
Hence it follows, that
Then from (1.28) and (1.30) it follows that is fundamental in , that is,
where ,….
Thus, there exists such that
On the other hand, , therefore,
Hence,
where . From this, by virtue of (1.29), , that is,
Thus, . The theorem is proved.
1.2. Proof of Auxiliary Results
Validity of Statement 1.3 follows from condition (i), the Statement 1.4 from condition (vii).
Proof.
Consider in Hilbert space the equation
where .
Equation (1.36) is equivalent to the following system of differentialoperator equations:
By virtue of (ix), problem (1.37) has a solution for some . Thus, for each ,
where is an identity operator in , that is, has a regular point.