Method of the quasilinearization for nonlinear impulsive differential equations with linear boundary conditions

The method of quasilinearization for nonlinear impulsive dieren tial equations with linear boundary conditions is studied. The boundary conditions include periodic boundary conditions. It is proved that the convergence is quadratic. AMS ( MOS ) Subject Classic ations: 34A37, 34E05. In this paper a boundary value problem (BVP) for impulsive dieren tial equations with a family of linear two point boundary conditions is studied. An existence theo- rem is proved. An algorithm, based on methods of quasilinearization, for constructing successive approximations of the solution of the considered problem is given. The quadratic convergence of the iterates is proved. The obtained results are general- izations of the known results for initial value problems as well as boundary value problems for ordinary dieren tial equations and impulsive dieren tial equations. The method of quasilinearization has recently been studied and extended exten- sively. It is generating a rich history beginning with the works by Bellman and Kalaba (1). Lakshmikantham and Vatsala, and many co-authors have extensively developed the method and have applied the method to a wide range of problems. We refer the reader to the recent work by Lakshmikantham and Vatsala (9) and the extensive bibliography found there. The method has been applied to two-point boundary value problems for ordinary dieren tial equations and we refer the reader to the papers, (2, 3, 4, 8, 10, 11, 12), for example. Likewise impulsive equations have been generating a rich history. We refer the reader to the monograph by Lakshmikantham, Bainov, and Simeonov (6) for a thor- ough introduction to the material and an introduction to the literature. Methods of quasilinearization have been applied to impulsive dieren tial equations with various initial or boundary conditions. We refer the reader to (9) for references and we refer the reader to (2, 3, 13) in our bibliography. In this paper, we consider a family of


Introduction
In this paper a boundary value problem (BVP) for impulsive differential equations with a family of linear two point boundary conditions is studied.An existence theorem is proved.An algorithm, based on methods of quasilinearization, for constructing successive approximations of the solution of the considered problem is given.The quadratic convergence of the iterates is proved.The obtained results are generalizations of the known results for initial value problems as well as boundary value problems for ordinary differential equations and impulsive differential equations.
The method of quasilinearization has recently been studied and extended extensively.It is generating a rich history beginning with the works by Bellman and Kalaba [1].Lakshmikantham and Vatsala, and many co-authors have extensively developed the method and have applied the method to a wide range of problems.We refer the reader to the recent work by Lakshmikantham and Vatsala [9] and the extensive bibliography found there.The method has been applied to two-point boundary value problems for ordinary differential equations and we refer the reader to the papers, [2,3,4,8,10,11,12], for example.
Likewise impulsive equations have been generating a rich history.We refer the reader to the monograph by Lakshmikantham, Bainov, and Simeonov [6] for a thorough introduction to the material and an introduction to the literature.Methods of quasilinearization have been applied to impulsive differential equations with various initial or boundary conditions.We refer the reader to [9] for references and we refer the reader to [2,3,13] in our bibliography.In this paper, we consider a family of boundary value conditions that contain periodic boundary conditions.A quasilinearization method has been applied to problems with periodic boundary conditions, [8]; to our knowledge, this is the first application to impulsive problems with periodic boundary conditions.

Preliminary notes and definitions
Let the points τ k ∈ (0, T ), k = 1, 2, ..., p be such that Consider the nonlinear impulsive differential equation (BVP) with the linear boundary value condition where We consider the set P C(X, Y ) of all functions u : X → Y, (X ⊂ R, Y ⊂ R) which are piecewise continuous in X with points of discontinuity of first kind at the points τ k ∈ X, i.e. there exist the limits lim We consider the set P C 1 (X, Y ) of all functions u ∈ P C(X, Y ) that are continuously differentiable for t ∈ X, t = τ k .Definition 1 .The function α(t) ∈ P C 1 ([0, T ], R) is called a lower solution of the BVP (1)-(3), if the following inequalities are satisfied: Definition 2 .The function β(t) ∈ P C 1 ([0, T ], R) is called an upper solution of the BVP (1)-( 3), if the inequalities (4), ( 5), (6) are satisfied in the opposite direction.
Consider the sets: M u(0 Using the results for the initial value problem for the linear impulsive differential equation ( 7),(8) (Corollary 1.6.1 [6] ) we can easily prove the following existence result for the LBVP ( 7), ( 8), ( 9) and obtain the formula for the solution.7), ( 8), ( 9) has a unique solution u(t) on the interval [0, T ], where We will need the following results for differential inequalities.
Lemma 2.2 (Theorem 1.4.1 [6]).Let the following conditions be satisfied: 1. u, g, σ ∈ P C([0, T ], R).In the proof of the main results we will use the following comparison result.
Proof:According to Lemma 2.2 the function m(t) satisfies the inequality From inequality (12) we have and therefore From the inequalities ( 13) and (15) it follows that m(0) ≤ 0. Therefore according to (14) the inequality m(t) ≤ 0 holds for t ∈ [0, T ].

Main Results
We will obtain sufficient conditions for existence of a solution of the BVP (1)-( 3).The obtained result will be useful not only for the proof of the method of quasilinearization but for different qualitative investigation of nonlinear boundary value problem for impulsive differential equations.
Proof:Without loss of generality we will consider the case when p = 1, i.e. 0 < t 1 < T .Let x 0 be an arbitrary point such that α(0) ≤ x 0 ≤ β(0).Define a function F : [0, T ] × R → R by the equality From the condition 2 of the Theorem 3.1 it follows that the function f (t, x) is bounded on S(α, β) and therefore there exists a function Therefore, the initial value problem for the ordinary differential equation Consider the function m(t) = X(t; x 0 ) − β(t).We will prove that the function m(t) is non-positive on [0, t 1 ].Assume the opposite, i.e. sup{m(t) : t ∈ [0, t 1 ]} > 0. Therefore, there exists a point t * ∈ (0, t 1 ) such that m(t * ) > 0 and m (t * ) ≥ 0. From the definition of the function X(t; x 0 ) it also follows that EJQTDE, 2002 No. 10, p. 5 According to the obtained contradiction, the assumption is not true.Therefore, Analogously, we can prove that X(t; x 0 ) ≥ α(t), t ∈ [0, t 1 ].Let y 0 = I 1 (X(t 1 ; x 0 )).We note that y 0 depends on x 0 .From the monotonicity of the function I 1 (x) we obtain Consider the initial value problem for the ordinary differential equation x = F (t, x), x(t 1 ) = y 0 for t ∈ [t 1 , T ].This initial value problem has a solution Y (t; y 0 ) for t ∈ [t 1 , T ].Using the same ideas as above we can prove that the inequalities Define the function The function x(t; x 0 ) ∈ S(α, β) is a solution of the impulsive differential equation ( 1), (2) with the initial condition x(0) = x 0 .

, p, and there exist functions
Then there exist two sequences of functions {α n (t)} ∞ 0 and {β n (t)} ∞ 0 such that: a.The sequences are increasing and decreasing respectively.b.The functions α n (t) are lower solutions and the functions β n (t) are upper solutions of the BVP (1), ( 2), (3).
d.The convergence is quadratic.
Proof:From the condition 2 of Theorem 3.2 it follows that if (t, x 1 ), (t, x 2 ) ∈ Ω(α 0 , β 0 ) and From the condition 3 of Theorem 3.2 it follows that if and From the condition 3 it follows that the functions G k (x) and J k (x) are nondecreasing in D k (α o , β 0 ).Therefore for x ∈ D k (α 0 , β 0 ) the inequality ≥ 0 holds, which proves that the functions I k (x) are nondecreasing, k = 1, 2, . . ., p.
We consider the linear boundary value problem for the impulsive linear differential equation (LBVP) The functions α 0 (t) and β 0 (t) are lower and upper solutions of the LBVP (28), (29), (30) and according to Lemma 2.1 there exists a unique solution β 1 (t) ∈ S(α 0 , β 0 ).EJQTDE, 2002 No. 10, p. 9 We will prove that α From the choice of the functions α 1 (t) and β 1 (t) and the inequality (20) we obtain that the function u(t) satisfies the inequalities According to the inequality (21) for x 2 = β 0 (t k ) and x 1 = α 0 (t k ) and the definition of the functions α 1 , β 1 we obtain From the boundary value condition for the functions α 1 , β 1 and the condition 4 we obtain the inequality From the inequalities (31), (32) and boundary condition (33), according to Lemma 2.3, the function u(t) is non-positive, i.e. α 1 (t) ≤ β 1 (t).
We will prove the convergence is quadratic.Define the functions a n+1 where EJQTDE, 2002 No. 10, p. 11 It is easy to verify that the inequality From the inequalities (44) and (45) it follows that for t ∈ holds, where Analogously, it can be proved that where  From the properties of the functions F (t, x) and g(t, x), the definition of σ n (t) and the inequalities (49), (50) it follows that there exist constants λ 1 > 0 and λ 2 > 0 such that Analogously, it can be proved that there exists constants µ 1 > 0 and µ 2 > 0 such that The inequalities (51) and (52) prove that the convergence is quadratic.
Remark 1 In the case when N = 0 the BVP (1), ( 2), ( 3) is reduced to an initial value problem for impulsive differential equations for which the quasilinearization is applied in [9].
We also note that some of the results for ordinary differential equations, obtained in [5,7,8,9] are partial cases of the obtained results when I k (x) = x.