$_{z}$′ = A ( t ) z + r ( t, z ) , Δ z | t = θ$_{i}$ = D ( j ) i z + q$_{i}$ ( z ) , j = 1 , 2 , . . . , m, (4.35) where r ( t, z ) = [ f (Ψ( t )+ z ) − f (Ψ( t ))] − A ( t ) z and q$_{i}$ ( z ) = W$_{i}$ (Ψ( θ$_{i}$ )+ z ) − W$_{i}$ (Ψ( θ$_{i}$ )) − D ( j ) i z, are contin- uous functions, and matrices D ( j ) i satisfy condition ( A 4) . The functions are continuously differentiable with respect to z. One can verify that r ( t, 0) ≡ q$_{i}$ (0) ≡ 0 and r ( t + ω, z ) = r ( t, z ) for t ∈ $^{R}$. Moreover, the derivatives satisfy r ′ ( t, 0) ≡ q ′ $_{iz}$(0) ≡ 0 and the functions r ( t, z ) → 0 , q$_{i}$ ( z ) → 0 , r ′ $_{z}$( t, z ) → 0 and q ′ $_{iz}$( z ) → 0 , as z → 0 uniformly in t ∈ [0 , ∞ ) , i ≥ 0 . Each system (4.35) for j = 1 , 2 , . . . , m, corresponds to a region adjoint to initial value, x$_{0}$ such that these regions cover a neighborhood of x$_{0}$. Fix a number j and denote Y$_{j}$ ( t ) the fundamental matrix of adjoint to (4.35) linear homogeneous system of the form (4.34). One can verify that for −∞ < t, s < ∞ . We can write where $^{O$_{n}$}$− 1 is the ( n − 1) × ( n − 1) zero matrix. Then it can be driven where $_{y}$′ = A ( t ) y, Δ y | t = θ$_{i}$ = D ( j ) i y, (4.36) Y$_{j}$ ( t ) Y − 1 j ( s ) = P$_{j}$ ( t )    1 0 0 exp( H$_{j}$ ( t − s ))    $_{P}$− 1 j ( s ) , (4.37)    1 0 0 exp( H$_{j}$ ( t − s ))    =    0 0 0 exp( H$_{j}$ ( t − s ))    +    1 0 0 $^{O$_{n}$}$− 1    , Y$_{j}$ ( t ) Y − 1 j ( s ) = G ( j ) 1 ( t, s ) + G ( j ) 2 ( t, s ) = G ( j $^{)}$( t, s ) , G ( j ) 1 ( t, s ) = P$_{j}$ ( t )    0 0 0 exp( H$_{j}$ ( t − s ))    $_{P}$− 1 j ( s ) , G ( j ) 2 ( t, s ) = P$_{j}$ ( t )    1 0 0 $^{O$_{n}$}$− 1    $_{P}$− 1 j ( s ) . 23