QUASILINEARIZATION FOR THE PERIODIC BOUNDARY VALUE PROBLEM FOR SYSTEMS OF IMPULSIVE DIFFERENTIAL EQUATIONS

The method of quasilinearization of Bellman and Kalaba [2] has been extended, refined, and generalized when the forcing function is the sum of a convex and concave function using coupled lower and upper solutions. This method is now known as the method of generalized quasilinearization. It has all the advantages of the quasilinearization method such that the iterates are solutions of linear systems and the sequences simultaneously converge to the unique solution of the nonlinear problem. See [1–4, 6–8] for details. In this paper, we extend the method of generalized quasilinearization to system of nonlinear impulsive differential equations with periodic boundary conditions. For this purpose, we develop a linear comparison theorem for system of impulsive differential equations with periodic boundary conditions. We develop two iterates which are solutions of linear impulsive system with periodic boundary conditions which converge monotonically and quadratically to the unique solution of the nonlinear problem. Results related to different types of coupled lower and upper solutions are developed.We note that the results of [1] are a special case of our results where the forcing function is made to be convex and in addition they obtain semiquadratic convergence only. The results of [3] can be obtained as the scalar case of our result.


Introduction
The method of quasilinearization of Bellman and Kalaba [2] has been extended, refined, and generalized when the forcing function is the sum of a convex and concave function using coupled lower and upper solutions.This method is now known as the method of generalized quasilinearization.It has all the advantages of the quasilinearization method such that the iterates are solutions of linear systems and the sequences simultaneously converge to the unique solution of the nonlinear problem.See [1][2][3][4][6][7][8] for details.
In this paper, we extend the method of generalized quasilinearization to system of nonlinear impulsive differential equations with periodic boundary conditions.For this purpose, we develop a linear comparison theorem for system of impulsive differential equations with periodic boundary conditions.We develop two iterates which are solutions of linear impulsive system with periodic boundary conditions which converge monotonically and quadratically to the unique solution of the nonlinear problem.Results related to different types of coupled lower and upper solutions are developed.We note that the results of [1] are a special case of our results where the forcing function is made to be convex and in addition they obtain semiquadratic convergence only.The results of [3] can be obtained as the scalar case of our result.
2 Method impulsive systems with PB conditions Consider the system of nonlinear impulsive differential equations (PBVP) x = f t,x(t) + g t,x(t) for t ∈ [0,T], t = τ k , with periodic boundary conditions where

., p).
We consider the set PC(X,Y ) of all functions u : X → Y ,(X,Y ⊂ R n ) which are piecewise continuous in X with points of discontinuity of first kind at the points τ k ∈ X, that is, there exist the limits lim t↓τk u(t) = u(τ k + 0) < ∞ and lim t↑τk u(t) = u(τ k − 0) = u(τ k ).
We consider the set PC 1 (X,Y ) of all functions u ∈ PC(X,Y ) that are continuously differentiable for t ∈ X, t = τ k .
Consider the sets α 0 (0) ≤ α 0 (T). (2.4) If the inequalities are reversed, then α 0 (t) is called an upper solution.This is referred to as natural upper and lower solutions.
(2.5) S. G. Hristova (2.6) One can define other kinds of coupled lower and upper solutions of (2.1)-(2.2) on the same lines.
We will prove some preliminary results for linear systems of impulsive differential equations.
Let A = {a i j } N i, j=1 be a matrix, N a natural number.We will say that A > 0 if a i j > 0 for i, j = 1,2,...,N.
We will use the following notation: e = (1,1,...,1).We will note that the vector e is the unit vector according to the operation @.
For our main results, we need the following lemma for linear systems of impulsive differential inequalities.Lemma 2.6.Assume that (1) the matrix ) (2.9) Proof.Let τ 0 = 0, τ p+1 = T. Consider the numbers (2.11) The obtained contradiction proves that this case is not possible.Consider the case when for every k : 0 ≤ k ≤ p there exists a natural number j k such that lim t→τk+0 m jk (t) = k+1 (2.12) and m i (t) < k+1 for t ∈ (τ k ,τ k+1 ], i = 1,2,...,N.Then from the jump condition (2.8) we . The last inequality contradicts the condition (2.9).Therefore, this case is impossible.Case 2. There exists a natural number l : According to the jump condition (2.8), we obtain m(τ k + 0) ≤ B k m(τ k ) ≤ 0. Therefore, there exist a natural number j k and a point ξ k ∈ (τ k ,τ k+1 ] such that m jk (ξ k ) = k+1 and m jk (ξ k ) ≥ 0. From the inequality (2.7), we have (2.13) The obtained contradiction proves that k = p + 1.Therefore, m(T) ≤ 0 and from the boundary condition (2.9) it follows that m(0) ≤ 0. As in the proof above, we obtain that As an application of Lemma 2.6, the following corollary is implied.This will be useful in proving the existence and uniqueness of the linear nonhomogeneous impulsive system with periodic boundary condition.
Then the PBVP for the homogeneous linear system has only the trivial solution.
S. G. Hristova and A. S. Vatsala 5 We note that the solution of the linear system of impulsive equations (2.14), (2.15) with the initial condition m(0) = m 0 is m(t) = W(t,0)m 0 , where and U k (t,s) is the fundamental matrix of the linear system m = A(t)m(t), t ∈ (τ k ,τ k+1 ] (for more details, see [5,9]).
Consider the periodic boundary value problem for the nonhomogeneous linear systems of impulsive differential equations (2.20) Lemma 2.9 (see [ where In our main result, we will use the following integral mean-value theorem.

Main results
In this section, we develop the method of quasilinearization for the periodic boundary value problem for the system of nonlinear impulsive differential equations (2.1)-(2.2).We obtain two monotone sequences which are solutions of appropriately chosen linear impulsive differential systems with periodic boundary conditions.These monotone sequences converge quadratically to the unique solution of (2.1)-(2.2).
Theorem 3.1.Let the following conditions hold.
(2) The functions f x , g x exist and are continuous on where where The function f x (t,α 0 (t))x is quasimonotone nondecreasing in x and the function ( f x (t,β 0 ) − g x (t,β 0 ))e@x is strictly decreasing in x on [0,T].
Proof.Consider the periodic boundary value problem for the system of impulsive linear differential equations (3. 3) The PBVP (3.3) can be written in the form ) where Consider the matrices From conditions (2), ( 3), (4), and (5) of Theorem 3.1, it follows that C 0 (t) ∈ Ξ and Therefore, according to Lemma 2.9, the boundary value problem (3.4) has a unique solution, which can be written in the form (2.21)-(2.22).We denote the solution of (3.4) by α 1 (t), β 1 (t).
We can prove in the same way as for the function α 1 (t) and We will prove the convergence of the sequences {α m (t)} ∞ 0 and {β m (t)} ∞ 0 .
The next theorem is for the case when the lower and upper solutions are completely opposite to those in Theorem 3.1.Theorem 3.2.Let the following conditions hold.
( Proof.Consider the periodic boundary value problem for the system of impulsive linear differential equations (3.46) The PBVP (3.46) can be written in the form where (3.48) Consider the matrices of Theorem 3.2, it follows that C 0 (t) ∈ Ξ and Therefore, according to Lemma 2.9, the boundary value problem (3.46) has a unique solution α 1 (t),β 1 (t).
We will prove that α 0 (t) ≤ α 1 (t) and β 0 (t) ≥ β 1 (t) on [0,T].Set p(t) = α 0 (t) − α 1 (t), q(t) = β 1 (t) − β 0 (t).Then we have (3.50) The PBVP (3.50) can be written in the form where where the functions are the unique solution of the boundary value problem for the linear system of impulsive differential equations (3.53) The inequality α m (t) ≤ β m (t),t ∈ [0,T], holds and both sequences are uniformly convergent.Denote From the uniform convergence and the definition of the functions α m (t) and β m (t), the validity of the inequalities follows.
Since the functions α m (t) and β m (t) are solutions of the PBVP (3.53), we obtain that the functions u(t) and v(t) are solutions of the PBVP As in the proof of Theorem 3.1, we can obtain that u(t) = v(t) on [0,T].We will prove that the convergence is quadratic.
The next theorem is about the case when the PBVP (2.1)-(2.2) has a lower solution as well as an upper solution.
Theorem 3.3.Let the following conditions hold.
(2) The functions f x , g x exist and are continuous on Ω(α 0 ,β 0 ), f x (t,x) is nondecreasing in x, g x (t,x) is nonincreasing in x for t ∈ [0,T], and for x ≥ y, where S 1 > 0, S 2 > 0 are constant matrices.
As particular cases of the proved theorems, we can obtain some results for the PBVP for systems of nonlinear ordinary differential equations.