In a separable Hilbert space H, we have the following equation:
A is a self-adjoint positive-definite operator, and
are generally linear unbounded operators. All derivatives are understood in the sense of distributions theory.
(see   ), and
, which are determined as follows:
With the norm
See  .
Notice that the principal part of the investigated equation possesses complicated characteristic, not multiple characteristics as in  .
Definition 1. If for any
there exists a vector function
that satisfies Equation (1) almost everywhere in R, then it is called a regular solution of Equation (1)
Definition 2. If for any
there exists a regular solution of Equation (1), and satisfies the inequality
then Equation (1) is called regularly solvable.
It is known that if
And the following inequalities are valid (see  ).
Definition 3. Parseval’s equality
2. Main Results
Theorem 1. The operator
is an isomorphism from the space
to the space
Proof. From (2), it is easy to prove that the operator
be bounded. Using Fourier transforms for the equation
, we obtain
(E is the unit operator), where
are Fourier transform for the functions
, respectively. The operator pencil
is invertible and moreover
We show that
. By using the Parseval equality and (3), we obtain:
is a spectrum of the operator A, then we consider
Taking into account (5) and (6) into (4) we obtain:
Applying Banach theorem on the inverse operator, we get that the operator
is an isomorphism from
Now, we estimate the norms of intermediate derivative operators participating in the main part of the Equation (1) for finding exact conditions on regular solvability of the given equation, expressed only by its operator coefficients.
From theorem 1, we have that the norms
are equivalent in the space
. Therefore by the norm
the theorem on intermediate derivatives is valid as well.
Theorem 2. Let
. Then there hold the following inequalities:
Proof. To establish the validity of inequality (11) we make change
and apply the Fourier transformation. We get
we estimate the following norms:
Finally, from (12), we have
Lemma. The operator
continuously acts from
provided that the operators
are bounded in H.
Taking into account the results found up  to now we get possibility to establish regular solvability conditions of Equation (1).
Theorem 3. Let the operators
be bounded in H and it holds the inequality
, where the numbers
are determined in theorem 2. Then the Equation (1) is regularly solvable.
Proof. By theorem 1, provided that the operator
has a bounded inverse operator
, then after replacing
in Equation (1) can be written as
Now we prove under the theorem conditions (see  ), that the norm
By theorem (2), we have:
Thus, the operator
is invertible in
can be determined by
The theorem is proved.
We formulated exact conditions on regular solvability of Equation (1), expressed only by its operator coefficients. We estimated the norms of intermediate derivative operators participating in the principle part of the given equation. In the case when in the perturbed part of Equation (1), the participant variable operator coefficients, i.e.
are linear operators, which determined for all
, are investigated in a similar way.