Abstract
Let be a bounded domain in a real Euclidean space. We consider the equation , where and are matrix-valued functions and is a nonlinear mapping. Conditions for the exponential stability of the steady state are established. Our approach is based on a norm estimate for operator commutators.
1. Introduction and Statement of the Main Result
Throughout this paper, is the complex -dimensional Euclidean space with a scalar product and norm ; is the set of -matrices; is the unit operator in corresponding space; is a bounded domain with a smooth boundary in a real Euclidean space; is the Hilbert space of functions defined on with values in , the scalar product and the norm .
Our main object in this paper is the equation where and are matrix-valued functions defined on and , respectively, with values in , and satisfy conditions pointed out below, and is unknown.
Traditionally, (2) is called the Barbashin type integrodifferential equation or simply the Barbashin equation. It plays an essential role in numerous applications, in particular, in kinetic theory [1], transport theory [2], continuous mechanics [3], control theory [4], radiation theory [5, 6], and the dynamics of populations [7]. Regarding other applications, see [8]. The classical results on the Barbashin equation are represented in the well-known book [9]. The recent results about various aspects of the theory of the Barbashin equation can be found, for instance, in [10–14] and the references given therein. In particular, in [11], the author investigates the solvability conditions for the Cauchy problem for a Barbashin equation in the space of bounded continuous functions and in the space of continuous vector-valued functions with the values in an ideal Banach space. The stability and boundedness of solutions to a linear scalar nonautonomous Barbashin equation have been investigated in [15].
The literature on the asymptotic properties of integrodifferential equations is rather rich (cf. [16–22] and the references given therein), but the stability of nonlinear vector integrodifferential equations is almost not investigated. It is at an early stage of development.
A solution of (2) is a function having a measurable derivative bounded on each finite interval.
It is assumed that under consideration provides the existence and uniqueness of solutions (e.g., it is Lipschitz continuous). The zero solution of (2) is said to be exponentially stable, if there are constants , and , such that , provided . It is globally exponentially stable if .
Suppose that, for a positive , For example, for an integer , let . Here, with a matrix kernel satisfying Then, by the Schwarz inequality, Thus, Hence, for any , we have condition (3) with .
The following notations are introduced: for a linear operator , is the adjoint operator, is the operator norm, and is the spectrum. For -matrix , put where , are the eigenvalues of , counted with their multiplicities; is the Frobenius (Hilbert-Schmidt) norm of . The following relations are checked in [23, Section ]: , If is a normal matrix, , then . Furthermore, denote and assume that In addition, with the notation , put This integral is simply calculated. If is a normal matrix for all , then
Now, we are in a position to formulate our main result.
Theorem 1. Let conditions (3), (11), and hold. Then, the zero solution to (2) is exponentially stable. If, in addition, in (3), then the zero solution is globally exponentially stable.
This theorem is proved in the next 3 sections. It gives us “good” results when is “small,” that is, if matrices and “almost commute” and is “small.” If (2) is scalar, then , So, in the scalar case, condition (14) takes the form This condition is similar to the stability test derived in [24] for scalar integrodifferential equations.
2. Auxiliary Results
Let be a Hilbert space with a scalar product and the norm ; denotes the set of bounded linear operators in and is the commutator of .
Lemma 2. Let and . Then,
Proof. Put . Then, . On the other hand, So, , as claimed.
Let Then, the Lyapunov equation has a unique solution and it can be represented as (cf. [25]). Denote , where .
Lemma 3. Under condition (19), one has
Proof. Making use of (21), we can write But . So , where We have If , then . If , then . So .
In addition, by Lemma 2, This proves the lemma.
3. Equations in a Hilbert Space
In this section, for simplicity, we put . Put . Consider in the equation where and continuously maps into and satisfies The solution and stability are defined as in Section 1. The existence and uniqueness of solutions are assumed. Recall that is a solution of (20).
Lemma 4. Let conditions (19) and (29) with hold. Then, any solution of (28) satisfies the inequality
Proof. For brevity, we write . Multiplying (28) by and doing the scalar product, we get Since , due to (20) and Lemma 3, it can be written that Taking into account the fact that due to (29) we get From this inequality, we have . Hence, as claimed.
Lemma 5. Let conditions (29) and hold. Then, the zero solution to (28) is exponentially stable. If in (29), then the zero solution to (28) is globally exponentially stable.
Proof. If , then the required result is due to the previous lemma. If , then, taking due to the previous lemma, . Hence, we easily obtain the required result.
4. Proof of Theorem 1
Take Then, and So Due to [23, Example ], where . Hence, since . Consequently, . In addition, Now, the required result is due to Lemma 5.
Competing Interests
The author declares that there are no competing interests regarding the publication of this paper.