The strongly positive operator A defines the fractional spaces () consisting of all for which the following norms are finite:
We consider the initial value problem (1) for delay differential equations of parabolic type in the space of all continuous functions defined on the segment with values in a Banach space . First, we consider the problem (1) when and commute, i.e.
(5)
Theorem 2.1
Assume that the condition
(6)
holds for every , where M is the constant from (2). Then for every t, , , we have the following coercive stability estimate:
(7)
where does not depend on and . Here, we put when .
Proof It is clear that
where is the solution of the problem
(9)
and is the solution of the problem
(10)
First, we consider the problem (9). Using the formula
(11)
the semigroup property, condition (5), and the estimates (2), (6), we obtain
for every t, and λ, . This shows that
(12)
for every t, . Applying mathematical induction, one can easily show that it is true for every t. Namely, assume that the inequality
is true for t, , for some n. Letting , we have
Using the estimate (12), we obtain
for every t, , and λ, . This shows that
for every t, , . Therefore
(13)
is true for every . Applying (9), the triangle inequality, condition (5), and the estimates (6) and (13), we get
(14)
for every . Second, we consider the problem (10). To prove the theorem it suffices to establish the following stability inequality:
(15)
for the solution of the problem (10) for every t, , . Using the formula
(16)
the semigroup property, and the definition of the spaces , we obtain
for every t, and λ, . This shows that
(17)
for every t, . Applying mathematical induction, one can easily show that it is true for every t. Namely, assume that the inequality (15) is true for t, , , for some n. Using the formula
(18)
the semigroup property, the definition of the spaces , the estimate (2), and condition (6), we obtain
for every t, , and λ, . This shows that
(19)
for every t, , . Applying equation (9), the triangle inequality, and condition (5) and estimates (6) and (19), we get
for every . This result completes the proof of Theorem 2.1. □
Now, we consider the problem (1) when
for some . Note that A is a strongly positive operator in a Banach spaces E iff its spectrum lies in the interior of the sector of angle φ, , symmetric with respect to the real axis, and if on the edges of this sector, and and outside it the resolvent is the subject to the bound
(20)
for some . First of all let us give lemmas from the paper [18] that will be needed in the sequel.
Lemma 2.1 For any z on the edges of the sector,
and
and outside it the estimate
holds for any . Here and in the future M and are the same constants of the estimates (2) and (20).
Lemma 2.2 Let for all the operator with domain which coincide with admit a closure bounded in E. Then for all the following estimate holds:
Here .
Suppose that
(21)
holds for every . Here and in the future ε is some constant, .
The application of Lemmas 2.1 and 2.2 enables us to establish the following fact.
Theorem 2.2
Assume that the condition
(22)
holds for every . Then for every the coercive stability estimate (7) holds.
Proof In a similar manner as in the proof of Theorem 2.1 we establish estimates for the solution of the problems (9) and (10), separately. First, we consider the problem (9). Let and λ, . Then using (11), we have
where
Using the estimates (2), (20), and condition (22), we obtain
for every t, and λ, . Now let us estimate . By Lemma 2.1 and using the estimate (21), we obtain
for every t, and λ, . Using the triangle inequality, we obtain
for every t, and λ, . This shows that
for every t, . In a similar manner as with Theorem 2.1 applying mathematical induction, one can easily show that it is true for every t. Therefore, to prove the theorem it suffices to establish the coercive stability inequality (15) for the solution of the problem (10). Now, we consider the problem (10). Exactly in the same manner, using (16), the semigroup property, and the definition of the spaces , we obtain (15) for every t, . Applying mathematical induction, one can easily show that it is true for every t. Namely, assume that the inequality (15) is true for t, , for some n. Using (18) and the semigroup property, we write
where
Using the estimate (2) and condition (22), we obtain
for every t, , , and λ, . Now let us estimate . By Lemma 2.2 and using the estimate (2) and condition (21), we obtain
for every t, , and λ, . Using the triangle inequality and estimates for all , , we obtain
for every t, , and λ, . This shows that
for every t, , . This result completes the proof of Theorem 2.2. □
Note that these abstract results are applicable to the study of stability of various delay parabolic equations with local and nonlocal boundary conditions with respect to the space variables. However, it is important to study the structure of for space operators in Banach spaces. The structure of for some space differential and difference operators in Banach spaces has been investigated (see [20–30]). In Section 3, applications of Theorem 2.1 to the study of the coercive stability of initial-boundary value problem for delay parabolic equations are given.