On Stability of Vector Nonlinear Integrodifferential Equations

L 2 → L 2 satisfy conditions pointed out below, and u(⋅, ⋅) 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 u(t, ⋅) : [0,∞) → L having a measurable derivative bounded on each finite interval. It is assumed that under consideration F 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 m 0 ≥ 1, δ 0 > 0, and α > 0, such that ‖u(t)‖ L 2 ≤ m 0 ‖u(0)‖ L 2e −αt (t ≥ 0), provided ‖u(0)‖ L 2 ≤


Introduction and Statement of the Main Result
Throughout this paper, C  is the complex -dimensional Euclidean space with a scalar product (⋅, ⋅)  and norm ‖ ⋅ ‖  = √(⋅, ⋅)  ; C × is the set of  × -matrices;  is the unit operator in corresponding space; Ω is a bounded domain with a smooth boundary in a real Euclidean space;  2 (C  , Ω) =  2 is the Hilbert space of functions defined on Ω with values in C  , the scalar product and the norm ‖ ⋅ ‖  2 = √(⋅, ⋅)  2 .
Our main object in this paper is the equation where (⋅) and (⋅, ⋅) are matrix-valued functions defined on Ω and Ω × Ω, respectively, with values in C × , and (⋅) :  2 →  2 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][11][12][13][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][17][18][19][20][21][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.
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  0 ≥ 1,  0 > 0, and Suppose that, for a positive  ≤ ∞, International Journal of Engineering Mathematics For example, for an integer  > 1, let [(ℎ)]() = (ℎ())  . Here, with a matrix kernel (, ) satisfying Then, by the Schwarz inequality, Thus, Hence, for any  < ∞, we have condition (3) with  =  −1   .
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   (),  = 1, . . ., , are the eigenvalues of , counted with their multiplicities;  2 () = (Trace  * ) If  is a normal matrix,  * =  * , then () = 0. Furthermore, denote and assume that In addition, with the notation  0 = sup  (()), 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.
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 sup , ‖() − ()‖  is "small."If (2) is scalar, then  0 = 0, 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.

Auxiliary Results
Let H be a Hilbert space with a scalar product (⋅, ⋅) H and the norm ‖ ⋅ ‖ H = √(⋅, ⋅) H ; B(H) denotes the set of bounded linear operators in H and Then, the Lyapunov equation has a unique solution  ∈ B(H) and it can be represented as (cf. [25]).Denote where   = inf ().
Lemma 3.Under condition (19), one has Proof.Making use of ( 21), we can write We have This proves the lemma.

Equations in a Hilbert Space
In this section, for simplicity, we put where ,  ∈ B(H) and  continuously maps () into H 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).