The main object of our discussion is the following system, In order to define a solution of (2.1), we need the following function and spaces. A function φ ( t ) : R → R $^{n}$, n ∈ $^{N}$, θ is a B − sequence, is from the set P C ( $^{R}$, θ ) if it : ( i ) is left continuous, ( ii ) is continuous, except, possibly, points of θ, where it has discontinuities of the first kind. A function φ ( t ) is from the set P C $^{1}$( $^{R}$, θ ) if φ ( t ) , φ ′ ( t ) ∈ P C ( $^{R}$, θ ) , where the derivative at points of θ is assumed to be the left derivative. If φ ( t ) is a solution of (2.1), then it is required that it belongs to P C $^{1}$( $^{R}$, θ ) [1]. We say that x ( t ) : I → R $^{n}$, I ⊂ $^{R}$, is a solution of (2.1) on I if there exists an extension ˜ x ( t ) of the function on R such that ˜ x ( t ) ∈ P C $^{1}$( $^{R}$, θ ) , the equality x ′ ( t ) = f ( x ( t )) , t ∈ I , is true if x ( t ) / ∈ Γ , x ( θ$_{i}$ +) = J ( x ( θ$_{i}$ )) for x ( θ$_{i}$ ) ∈ Γ and x ( θ$_{i}$ +) ∈ ˜ Γ , θ$_{i}$ ∈ I . If θ$_{i}$ is a discontinuity moment of x ( t ) , then x ( θ$_{i}$ ) ∈ Γ , for θ$_{i}$ > 0 and x ( θ$_{i}$ ) ∈ ˜ Γ , for θ$_{i}$ < 0 . If x ( θ$_{i}$ ) ∈ ∂ Γ or x ( θ$_{i}$ ) ∈ ∂ ˜ Γ , then x ( θ$_{i}$ ) is a point of discontinuity with zero jump. Definition 2.1 A point x ∗ from ∂ Γ or ∂ ˜ Γ is a grazing point of system (2.1) if 〈∇ Φ( x ∗ ) , f ( x ∗ ) 〉 = 0 or 〈∇ ˜ Φ( x ∗ ) , f ( x ∗ ) 〉 = 0 , respectively. If at least one of coordinates of ∇ ˜ Φ( x ∗ ) is zero then the grazing is axial, otherwise it is non-axial. Definition 2.2 An orbit γ ( x ∗ ) = { x ( t, 0 , x ∗ ) | x ∗ ∈ D, t ∈ $^{R}$} of (2.1) is grazing if there exists at least one grazing point on the orbit. Consider a solution x ( t ) : $^{R}$→ R n and { θ$_{i}$ } be the moments of the discontinuity, they are the moments where solution x ( t ) intersects Γ as time increases and the moments when the solution it intersects ˜ Γ as time decreases. A solution x ( t ) = x ( t, 0 , x$_{0}$ ) , x$_{0}$ ∈ D of (2.1) locally exists and is unique if the conditions ( C 1) − ( C 3) are valid [1]. In what follows, let ‖ · ‖ be the Euclidean norm, that is for a vector x = ( x$_{1}$, x$_{2}$, . . . , x$_{n}$ ) in R $^{n}$, the norm is equal to √ x 2 1 + x 2 + . . . + x 2 $_{n}$. The following condition for (2.1) guarantees that any set of discontinuity moments of the system constitutes a B − sequence and we call the condition B − sequence condition . In [1], some other B − sequence conditions are provided. We will request for discontinuous dynamical systems that any sequence of discontinuity moments to be a B − sequence. (C8) sup$_{D}$ ‖ f ( x ) ‖ < + ∞ , and $^{inf$_{x}$$_{0}$}$∈ ˜ $_{Γ}$( x$_{0}$, y ( ζ, 0 , x$_{0}$ )) > 0 . 4 x ′ = f ( x ) , Δ x | $^{x}$∈ Γ = I ( x ) . (2.1)