METHOD OF THE QUASILINEARIZATION

FOR NONLINEAR IMPULSIVE DIFFERENTIAL EQUATIONS WITH LINEAR

BOUNDARY CONDITIONS

P. Eloe ^{1} S. G. Hristova^{2}

Abstract

The method of quasilinearization for nonlinear impulsive differential 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 Classifications: 34A37, 34E05.

Key words: impulsive differential equations,linear boundary value problem, quasilinearization, quadratic convergence.

1 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.

1University of Dayton, Dayton, OH, Paul.Eloe@notes.udayton.edu

2University of Louisiana at Lafayette, Lafayette, LA, snehri@hotmail.com

2 Preliminary notes and definitions

Let the points τ_{k} ∈ (0, T), k = 1, 2, ..., p be such that

τ_{k+1} > τ_{k}, k = 1, 2, …, p − 1.

Consider the nonlinear impulsive differential equation (BVP)

x′ = f(t, x(t)) for t ∈ [0, T], t ≠ τ_{k},  (1)
x(τ_{k} + 0) = I_{k}(x(τ_{k})), k = 1, 2, …, p  (2)

with the linear boundary value condition

Mx(0) − Nx(T) = c,  (3)

where x ∈ R, f : [0, T]×R → R, I_{k} : R → R, (k = 1, 2, …, p), c, M, N are constants.

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_{t↓τ_{k}} u(t) = u(τ_{k} + 0) < ∞ and lim_{t↑τ_{k}} u(t) = u(τ_{k} − 0) = u(τ_{k}).

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:

α′(t) ≤ f(t, α(t)) for t ∈ [0, T], t ≠ τ_{k},  (4)
α(τ_{k} + 0) ≤ I_{k}(α(τ_{k})), k = 1, 2, …, p  (5)
Mα(0) − Nα(T) ≤ c.  (6)

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.

Let the functions α, β ∈ P C([0, T], R) be such that α(t) ≤ β(t).

Consider the sets:

S(α, β) = {u ∈ P C([0, T], R) : α(t) ≤ u(t) ≤ β(t) for t ∈ [0, T]},
Ω(α, β) = {(t, x) ∈ [0, T] × R : α(t) ≤ x ≤ β(t)},
D_{k}(α, β) = {x ∈ R : α(τ_{k}) ≤ x ≤ β(τ_{k})}, k = 1, 2, …, p.

Consider the linear boundary value problem for the linear impulsive differential equation (LBVP)

u′(t) = g(t)u(t) + σ(t), t ∈ [0, T], t ≠ τ_{k}  (7)
u(τ_{k} + 0) = b_{k}u(τ_{k}) + γ_{k}, k = 1, 2, …, p  (8)
Mu(0) − Nu(T) = 0.  (9)

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.

Lemma 2.1 . Let the functions g, σ ∈ P C([0, T], R) and M, N, b_{k}, γ_{k}, (k = 1, 2, … p) be constants such that N(∏_{k=1}^{p} b_{k})exp( ∫_{0}^{T} g(s)ds) ≠ M.

Then the LBVP (7), (8), (9) has a unique solution u(t) on the interval [0, T], where

u(t) = u(0)( ∏_{0<τ_{k}<t} b_{j})exp(∫_{0}^{t} g(τ)dτ)

+ ∑_{0<τ_{k}<t} γ_{k}( ∏_{τ_{k}<τ_{j}<t} b_{j})exp(∫_{τ_{k}}^{t} g(τ)dτ)ds

+ ∫_{0}^{t} σ(s)( ∏_{s<τ_{k}<t} b_{k})exp(∫_{s}^{t} g(τ)dτ)ds,

τ_{0} = 0, b_{0} = 1, ∏_{j=k}^{n} f(j) = 1 for k > n,

u(0) = [1 − \frac{N}{M} ( ∏_{k=1}^{p} b_{k})exp( ∫_{0}^{T} g(s)ds)]^{−1}

×{ ∑_{i=1}^{p} γ_{i}( ∏_{j=i+1}^{p} b_{j})exp(∫_{τ_{i}}^{T} g(τ)dτ)ds

+ ∫_{0}^{T} σ(s)( ∏_{s<τ_{j}<T} b_{j})exp(∫_{s}^{T} g(τ)dτ)}.

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). 2.The function m ∈ P C^{1}[R_{+}, R] and

m′(t) ≤ p(t)m(t) + q(t), t ∈ [0, T], t ≠ τ_{k}
m(τ_{k} + 0) ≤ d_{k}u(τ_{k}) + b_{k}, k = 1, 2, …, p,

where d_{k}, b_{k} (k = 1, 2, …, p) are constants, d_{k} ≥ 0, p, q ∈ C[R_{+}, R].

Then for t ≥ 0 the inequality

m(t) ≤ m(0)( ∏_{0<τ_{k}<t} d_{j})exp(∫_{0}^{t} p(τ)dτ)

+ ∑_{0<τ_{k}<t} b_{k}( ∏_{τ_{k}<τ_{j}<t} d_{j})exp(∫_{τ_{k}}^{t} p(τ)dτ)ds

+ ∫_{0}^{t} ( ∏_{s<τ_{k}<t} d_{k})exp(∫_{s}^{t} p(τ)dτ)q(s)ds

is valid.

In the proof of the main results we will use the following comparison result. Lemma 2.3 . Assume that the function m ∈ P C^{1}([0, T], R) satisfies the inequalities

m′(t) ≤ ϕ(t)m(t), t ∈ [0, T], t ≠ t_{k},  (10)
m(t_{k} + 0) ≤ α_{k}m(t_{k}), k = 1, 2, …, p  (11)
Mm(0) − Nm(T) ≤ 0,  (12)

where α_{k} ≥ 0, M > 0, N ≥ 0 are constants such that

M − N( ∏_{k=1}^{p} α_{k})exp( ∫_{0}^{T} ϕ(s)ds) > 0.  (13)

Then m(t) ≤ 0 for t ∈ [0, T]. Proof:According to Lemma 2.2 the function m(t) satisfies the inequality

m(t) ≤ m(0)( ∏_{k:0<t_{k}<t} α_{k})exp( ∫_{0}^{t} ϕ(s)ds) for t ∈ [0, T].  (14)

From inequality (12) we have

m(0) ≤ \frac{N}{M} m(T)

and therefore

m(0) ≤ \frac{N}{M} m(0)( ∏_{k=1}^{p} α_{k})exp(∫_{0}^{T} ϕ(s)ds).  (15)

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].

As a partial case of Lemma 2.3 we obtain the following result:

EJQTDE, 2002 No. 10, p. 4 Corollary 1 . Let the function m ∈ P C^{1}([0, T], R) satisfies the inequalities (10) - (12), where ∫_{0}^{T} ϕ(s)ds ≤ 0, 0 ≤ α_{k} < 1 and M > 0, N ≥ 0, M ≥ N.

Then the function m(t) is nonpositive on the interval [0, T].

3 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.

Theorem 3.1 . Let the following conditions be fulfilled:

1. The functions α, β ∈ P C^{1}([0, T], R) are lower and upper solutions of the BVP (1)-(3) and α(t) ≤ β(t) for t ∈ [0, T].

2. The function f ∈ C(Ω(α, β), R).

3. The functions I_{k} : D_{k}(α, β) → R, (k = 1, 2, …) are nondecreasing in D_{k}(α, β).

4. The constants M > 0, N ≥ 0.

Then the BVP (1)-(3) has a solution u ∈ S(α, β).

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

F (t, x) = \{ \begin{matrix}f(t, β(t)) + \frac{β(t)−x}{1+|x|} & for x > β(t) \\ f(t, x) & for α(t) ≤ x ≤ β(t) \\ f(t, α(t)) + \frac{α(t)−x}{1+|x|} & for x < α(t). \end{matrix}

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 μ ∈ C([0, T], [0, ∞)) such that sup{|F (t, x)| : x ∈ R} ≤ μ(t) for t ∈ [0, T].

Therefore, the initial value problem for the ordinary differential equation x′ = F (t, x), x(0) = x_{0} has a solution X(t; x_{0}) for t ∈ [0, t_{1}].

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

m′(t^{∗}) ≤ f(t^{∗}, β(t^{∗})) + \frac{β(t^{∗}) − X(t^{∗}; x_{0})}{1 + |X(t^{∗}; x_{0})|} − f(t^{∗}, β(t^{∗})) = \frac{−m(t^{∗})}{1 + |X(t^{∗}; x_{0})|} < 0.

According to the obtained contradiction, the assumption is not true. Therefore,

X(t; x_{0}) ≤ β(t), t ∈ [0, t_{1}].

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

α(t_{1} + 0) ≤ I_{1}(α(t_{1})) ≤ I_{1}(X(t_{1}; x_{0})) ≤ I_{1}(β(t_{1})) ≤ β(t_{1} + 0),

i.e.

α(t_{1} + 0) ≤ y_{0} ≤ β(t_{1} + 0).

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 α(t) ≤ Y (t; y_{0}) ≤ β(t) for t ∈ [t_{1}, T] hold. At the same time Y (t_{1}; y_{0}) = I_{1}(X(t_{1}; x_{0})).

Define the function

x(t; x_{0}) = \{ \begin{matrix}X(t; x_{0}) & for t ∈ [0, t_{1}] \\ Y (t; y_{0}) & for t ∈ (t_{1}, T]. \end{matrix}

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}.

From the inequality α(t) ≤ β(t) for t ∈ [0, T] it follows that the following two cases are possible:

Case 1. Let α(0) = β(0). Then x_{0} = α(0) = β(0). Therefore

Mx(0; x_{0}) − Nx(T; x_{0}) = Mx_{0} − Nx(T; x_{0}) ≤ Mα(0) − Nα(T) ≤ c

and

Mx(0; x_{0}) − Nx(T; x_{0}) ≥ Mx_{0} − Nβ(T) ≥ c.

Therefore Mx(0; x_{0}) − Nx(T; x_{0})) = c, i.e. the function x(t; x_{0}) is a solution of the BVP (1)-(3).

Case 2. Let α(0) < β(0). We will prove that there exists a point x_{0} ∈ [α(0), β(0)] such that the solution x(t; x_{0}) of the impulsive differential equation (1), (2) with initial condition x(0) = x_{0} satisfies the boundary condition (3). Assume the opposite, i.e. for every point x_{0} ∈ [α(0), β(0)] the inequality Mx(0; x_{0}) − Nx(T; x_{0})) ≠ c holds, where x(t; x_{0}) is the solution of the impulsive equation (1),(2).

If x_{0} = β(0) then from the relation x(t; x_{0}) ∈ S(α, β) we obtain that

Mx(0; x_{0}) − Nx(T; x_{0})) = Mβ(0) − Nx(T; x_{0}) ≥ Mβ(0) − Nβ(T) ≥ c.

According to the assumption and the above inequality we obtain

Mx(0; x_{0}) − Nx(T; x_{0}) > c.  (16)

Then there exists a number δ : 0 < δ < β(0) − α(0), such that for x_{0} : 0 ≤ β(0)−x_{0} < δ the corresponding solution x(t; x_{0}) of the impulsive differential equation (1), (2) satisfies the inequality

Mx(0; x_{0}) − Nx(T; x_{0}) > c.  (17)

Indeed, assume that for every natural number n there exists a point z_{n} : 0 ≤ β(0)−z_{n} < \frac{1}{n} such that the corresponding solution x^{(n)}(t; z_{n}) of the impulsive equation (1),(2) with the initial condition x(0) = z_{n} satisfies the inequality

Mx^{(n)}(0; z_{n}) − Nx^{(n)}(T; z_{n}) < c.

Let {z_{n_{j}}} is a subsequence such that lim_{j→∞}z_{n_{j}} = β(0) and lim_{j→∞}x^{(n_{j})}(t; z_{n_{j}}) = x(t) uniformly on the intervals [0, t_{1}] and (t_{1}, T]. The function x(t) is a solution of the impulsive differential equation (1), (2) such that x(0) = β(0), x(t) ∈ S(α, β) and

Mx(0) − Nx(T) ≤ c.  (18)

The inequality (18) contradicts the inequality (16) and therefore the assumption is not true.

Let

δ^{∗} = sup{δ ∈ (0, β(0) − α(0)] : for which there exists a point x_{0} ∈ (β(0) − δ, β(0)]

such that the solution x(t; x_{0}) satisfies the inequality (17)}.

Choose a sequence of points x_{n} ∈ (α(0), β(0) − δ^{∗}) such that lim_{n→∞} x_{n} = β(0) − δ^{∗}. From the choice of δ^{∗} and the assumption it follows that the corresponding solutions x^{(n)}(t; x_{n}) satisfy the inequality

Mx^{(n)}(0; x_{n}) − Nx^{(n)}(T; x_{n}) < c.
There exists a subsequence {x_{n_{j}}}_{0}^{∞} of the sequence {x_{n}}_{0}^{∞} such that

lim_{j→∞} x^{(n_{j})}(t; x_{n_{j}}) = x^{∗}(t) uniformly on the intervals [0, t_{1}] and (t_{1}, T]. The function x^{∗}(t) ∈ S(α, β) is a solution of the impulsive equation (1), (2) with the initial condition x(0) = β(0) − δ^{∗} and satisfies the inequality Mx^{∗}(0) − Nx^{∗}(T) ≤ c. The last inequality contradicts the choice of δ^{∗}.

Therefore, there exists a point x_{0} ∈ [α(0), β(0)] such that the solution x(t; x_{0}) of the impulsive differential equation (1), (2) satisfies the condition (3), i.e. the function x(t; x_{0}) is a solution of the BVP (1), (2), (3). This completes the proof of Theorem 3.1.

We will construct the method of quasilinearization to approximate the solution of the BVP (1), (2), (3). We will prove that the convergence of the successive approximations is quadratic.

Theorem 3.2 Let the following conditions hold:

1. The functions α_{0}(t), β_{0}(t) are lower and upper solutions of the BVP (1), (2), (3) and α_{0}(t) ≤ β_{0}(t) for t ∈ [0, T].

2. The function f ∈ C^{0,2}(Ω(α_{0}, β_{0}), R) and there exist two functions F, g ∈ C^{0,2}(Ω(α_{0}, β_{0}), R) such that F (t, x) = f(t, x) + g(t, x), F′′_{xx}(t, x) ≥ 0,

g′′_{xx}(t, x) ≥ 0,

∫_{0}^{T} [F′_{x}(s, β_{0}(s)) − g′_{x}(s, α_{0}(s))]ds < 0.

3. The functions I_{k} ∈ C^{2}(D_{k}(α_{0}, β_{0}), R), k = 1, 2, …, p, and there exist functions G_{k}, J_{k} ∈ C^{2}(D_{k}(α_{0}, β_{0}), R) such that G_{k}(x) = I_{k}(x) + J_{k}(x), G′′_{k}(x) ≥ 0, J′′_{k}(x) ≥ 0,

G′_{k}(β_{0}(τ_{k})) − J′_{k}(α_{0}(τ_{k})) < 1,
G′_{k}(α_{0}(τ_{k})) − J′_{k}(β_{0}(τ_{k})) ≥ 0.

4. The constants M > 0, N ≥ 0, M ≥ N.

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).

c. Both sequences are uniformly convergent on the intervals (τ_{k}, τ_{k+1}], for k = 0, 1, 2, …, p, to the unique solution of the BVP (1), (2), (3) in S(α_{0}, β_{0}).

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 x_{1} ≥ x_{2} then

f(t, x_{1}) ≥ f(t, x_{2}) + F′_{x}(t, x_{2})(x_{1} − x_{2}) + g(t, x_{2}) − g(t, x_{1}),  (19)
g(t, x_{1}) ≥ g(t, x_{2}) + g′_{x}(t, x_{2})(x_{1} − x_{2}).  (20)

From the condition 3 of Theorem 3.2 it follows that if x_{1} ≥ x_{2}, x_{1}, x_{2} ∈ D_{k}(α_{0}, β_{0}), then

I_{k}(x_{1}) ≥ I_{k}(x_{2}) + G′_{k}(x_{2})(x_{1} − x_{2}) + J_{k}(x_{2}) − J_{k}(x_{1}),  (21)

and

G_{k}(x_{1}) ≥ G_{k}(x_{2}) + G′_{k}(x_{2})(x_{1} − x_{2}).  (22)

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 I′_{k}(x) = G′_{k}(x)−J′_{k}(x) ≥ G′_{k}(α_{0}(τ_{k})) − J′_{k}(β_{0}(τ_{k})) ≥ 0 holds, which proves that the functions I_{k}(x) are nondecreasing, k = 1, 2, …, p.

According to Theorem 3.1 the BVP (1), (2), (3) has a solution in S(α_{0}, β_{0}). We consider the linear boundary value problem for the impulsive linear differential

equation (LBVP)

x′(t) = f(t, α_{0}(t)) + Q_{0}(t)(x − α_{0}(t)) for t ∈ [0, T], t ≠ τ_{k},  (23)
x(τ_{k} + 0) = I_{k}(α_{0}(τ_{k})) + B_{k}^{0}[x(τ_{k}) − α_{0}(τ_{0})],  (24)
Mx(0) − Nx(T) = c,  (25)

where

Q_{0}(t) = F′_{x}(t, α_{0}(t)) − g′_{x}(t, β_{0}(t)),
B_{k}^{0} = G′_{k}(α_{0}(τ_{k})) − J′_{k}(β_{0}(τ_{k})), k = 1, 2, …, p.

It is easy to verify that the function α_{0}(t) is a lower solution of the LBVP (23), (24), (25).

According to the condition 1 of Theorem 3.2, inequalities (19) and (21) we obtain the inequalities

β′_{0}(t) ≥ f(t, α_{0}(t)) + Q_{0}(t)(β_{0}(t) − α_{0}(t))

−[F (t, α_{0}(t)) − F (t, β_{0}(t)) + F′_{x}(t, α_{0}(t))(β_{0}(t) − α_{0}(t))]

+g(t, α_{0}(t)) − g(t, β_{0}(t)) + gx^{′}(t, β_{0}(t))(α_{0}(t) − β_{0}(t))

≥ f(t, α_{0}(t)) + Q_{0}(t)(β_{0}(t) − α_{0}(t)) for t ∈ [0, T], t ≠ τ_{k}, (26)
β_{0}(τ_{k} + 0) ≥ I_{k}(α_{0}(τ_{k})) + [I_{k}(β_{0}(τ_{k})) − I_{k}(α_{0}(τ_{k}))]

≥ I_{k}(α_{0}(τ_{k})) + [G′_{k}(α_{0}(τ_{k})) − J′_{k}(β_{0}(τ_{k}))](β_{0}(τ_{k}) − α_{0}(τ_{k}))

≥ I_{k}(α_{0}(τ_{k})) + B_{k}^{0}(β_{0}(τ_{k}) − α_{0}(τ_{k})).  (27)

From the inequalities (26), (27) it follows that the function β_{0}(t) is an upper solution of the LBVP (23), (24), (25).

According to the Lemma 2.1 the LBVP (23), (24), (25) has a unique solution α_{1}(t) ∈ S(α_{0}, β_{0}).

We consider the linear boundary value problem for the impulsive linear differential equation (LBVP)

x′(t) = f(t, β_{0}(t)) + Q_{0}(t)(x(t) − β_{0}(t)) for t ∈ [0, T], t ≠ τ_{k},  (28)
x(τ_{k} + 0) = I_{k}(β_{0}(τ_{k})) + B_{k}^{0}(x(τ_{k}) − β_{0}(τ_{k})),  (29)
Mx(0) − Mx(T) = c.  (30)

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}).

We will prove that α_{1}(t) ≤ β_{1}(t) for t ∈ [0, T].

Define the function u(t) = α_{1}(t) − β_{1}(t) for t ∈ [0, T]. 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

u′ = f(t, α_{0}(t)) − f(t, β_{0}(t)) + Q_{0}(t)u(t) + Q_{0}(t)(β_{0}(t) − α_{0}(t))
≤ Q_{0}(t)u(t) for t ∈ [0, T], t ≠ τ_{k}.  (31)

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

u(τ_{k} + 0) ≤ I_{k}(α_{0}(τ_{k})) − I_{k}(β_{0}(τ_{k})) + B_{k}^{0}u(τ_{k}) + B_{k}^{0}[β_{0}(τ_{k}) − α_{0}(τ_{k})]
≤ B_{k}^{0}u(τ_{k}).  (32)

From the boundary value condition for the functions α_{1}, β_{1} and the condition 4 we obtain the inequality

Mu(0) − Nu(T) = Mα_{1}(0) − Nα_{1}(T) − (Mβ_{1}(0) − Nβ_{1}(T)) = c − c = 0. (33)

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).

The function α_{1}(t) is a lower solution of the BVP (1), (2), (3). Indeed, for t ∈ [0, T], t ≠ τ_{k},

α′_{1} ≤ f(t, α_{1}(t)) + F′_{x}(t, α_{0}(t))(α_{0}(t) − α_{1}(t))
−g(t, α_{0}(t)) + g(t, α_{1}(t)) + Q_{0}(t)(α_{1}(t) − α_{0}(t))  (34)
≤ f(t, α_{1}(t)).

From the inequality (21) and the choice of the function α_{1}(t) we obtain the inequalities

α_{1}(τ_{k} + 0) ≤ I_{k}(α_{1}(τ_{k})) + [G′_{k}(α_{0}(τ_{k})) − J′_{k}(α_{1}(τ_{k}))
−B_{k}^{0}](α_{0}(τ_{k}) − α_{1}(τ_{k}))
≤ I_{k}(α_{1}(τ_{k})) − [J′_{k}(α_{0}(τ_{k})) − J′_{k}(β_{0}(τ_{k}))](α_{0}(τ_{k}) − α_{1}(τ_{k}))
≤ I_{k}(α_{1}(τ_{k})), k = 1, 2, …, p.  (35)

From the inequalities (34), (35) and the boundary condition for the function α_{1}(t) it follows that the function α_{1}(t) is a lower solution of the BVP (1), (2), (3).

Analogously, it can be proved that the function β_{1}(t) is an upper solution of the BVP (1), (2), (3).

By this way we can construct two sequences of functions {α_{n}(t)}_{0}^{∞} and {β_{n}(t)}_{0}^{∞}, α_{n}, β_{n} ∈ S(α_{n−1}, β_{n−1}). The function α_{n+1}(t) is the unique solution of the linear boundary value problem for the impulsive linear differential equation (LBVP)

x′(t) = f(t, α_{n}(t)) + Q_{n}(t)(x − α_{n}(t)) for t ∈ [0, T], t ≠ τ_{k},  (36)
x(τ_{k} + 0) = I_{k}(α_{n}(τ_{k})) + B_{k}^{n}(x(τ_{k}) − α_{n}(τ_{k})),  (37)
Mx(0) − Nx(T) = c  (38)

and the function β_{n+1}(t) is the unique solution of the linear boundary value problem for the impulsive linear differential equation (LBVP)

x′(t) = f(t, β_{n}(t)) + Q_{n}(t)(x − β_{n}(t)) for t ∈ [0, T], t ≠ τ_{k},  (39)
x(τ_{k} + 0) = I_{k}(β_{n}(τ_{k})) + B_{k}^{n}(x(τ_{k}) − β_{n}(τ_{k})),  (40)
Mx(0) − Nx(T) = c,  (41)

where

Q_{n}(t) = F′_{x}(t, α_{n}(t)) − g′_{x}(t, β_{n}(t)),
B_{k}^{n} = G′_{k}(α_{n}(τ_{k})) − J′_{k}(β_{n}(τ_{k})).

As in the case n = 0 it can be proved that the functions α_{n+1}(t) and β_{n+1}(t) are lower and upper solutions of the BVP (1), (2), (3) and the inequalities

α_{0}(t) ≤ α_{1}(t) ≤ … ≤ α_{n}(t) ≤ β_{n}(t) ≤ … ≤ β_{0}(t)  (42)

hold.

Therefore, the sequences {α_{n}(t)}_{0}^{∞} and {β_{n}(t)}_{0}^{∞} are uniformly bounded and equi-continuous on the intervals (τ_{k}, τ_{k+1}], k = 0, 1, 2, …, p and they are uniformly convergent.

Denote

lim_{n→∞} α_{n}(t) = u(t), lim_{n→∞} β_{n}(t) = v(t).

From the uniform convergence and the definition of the functions α_{n}(t) and β_{n}(t) it follows that

α_{0}(t) ≤ u(t) ≤ v(t) ≤ β_{0}(t).  (43)

From the LBPVPs (36)-(38) and (39)-(41) we obtain that the functions u(t) and v(t) are solutions of the BVP (1), (2), (3) in S(α_{0}, β_{0}) and therefore u(t) = v(t).

We will prove the convergence is quadratic.

Define the functions a_{n+1}(t) = u(t) − α_{n+1}(t) and b_{n+1}(t) = β_{n+1}(t) − u(t), t ∈ [0, T]. For t ∈ [0, T], t ≠ τ_{k} we obtain the inequalities

a′_{n+1} ≤ Q_{n}(t)a_{n+1}(t) + [F′_{x}(t, u(t)) − g′_{x}(t, α_{n}(t)) − Q_{n}(t)]a_{n}(t)
= Q_{n}(t)a_{n+1}(t) + F′′_{xx}(t, ξ_{1})a_{n}^{2}(t)
+g′′_{xx}(t, η_{1})a_{n}(t)(β_{n}(t) − α_{n}(t))  (44)

where u(t) ≤ ξ_{1} ≤ α_{n}(t), α_{n}(t) ≤ η_{1} ≤ β_{n}(t).

It is easy to verify that the inequality

a_{n}(t)(β_{n}(t) − α_{n}(t)) = a_{n}(t)(b_{n}(t) + a_{n}(t)) ≤ \frac{1}{2}b_{n}^{2}(t) + \frac{3}{2}a_{n}^{2}(t).  (45)

From the inequalities (44) and (45) it follows that for t ∈ [0, T], t ≠ τ_{k} the inequality

a′_{n+1}(t) ≤ Q_{n}(t)a_{n+1}(t) + σ_{n}(t),  (46)

holds, where

σ_{n}(t) = [F′′_{xx}(t, ξ_{1}) + \frac{3}{2}g′′_{xx}(t, η_{1})]a_{n}^{2} + \frac{1}{2}g′′_{xx}(t, η_{1})b_{n}^{2}.

Analogously, it can be proved that

a_{n+1}(τ_{k} + 0) ≤ B_{k}^{n}a_{n+1}(τ_{k}) + γ_{k},  (47)

where

γ_{k} = [G′′_{k}(ω_{k}) + \frac{3}{2}J′′_{k}(ν_{k})]a_{n}^{2}(τ_{k}) + \frac{1}{2}J′′_{k}(ν_{k})b_{n}^{2}(τ_{k}),

α_{n}(τ_{k}) ≤ ω_{k} ≤ u(τ_{k}), α_{n}(τ_{k}) ≤ κ_{k} ≤ β_{n}(τ_{k}), k = 1, 2, …, p.

From the boundary conditions for the functions u(t) and α_{n}(t) we obtain the equality

Ma_{n+1}(0) − Na_{n+1}(T) = 0.  (48)

From the inequalities (46), (47) according to Lemma 2.2 it follows that the function a_{n+1}(t) satisfies the estimate

a_{n+1}(t) ≤ a_{n+1}(0)( ∏_{0<τ_{k}<t} B_{k}^{n})exp(∫_{0}^{t} Q_{n}(τ)dτ)

+ ∑_{0<τ_{k}<t} γ_{k}( ∏_{τ_{k}<τ_{j}<t} B_{j}^{n})exp(∫_{τ_{k}}^{t} Q_{n}(τ)dτ)ds

+ ∫_{0}^{t} σ_{n}(s)( ∏_{s<τ_{k}<t} B_{k}^{n})exp(∫_{s}^{t} Q_{n}(τ)dτ)ds.  (49)

From the boundary condition (48) we have a_{n+1}(0) = \frac{N}{M} a_{n+1}(T) and therefore

a_{n+1}(0) ≤ [1 − \frac{N}{M} ( ∏_{k=1}^{p} B_{k}^{n})exp( ∫_{0}^{T} Q_{n}(s)ds)]^{−1}

×{ ∑_{i=0}^{p} γ_{i}( ∏_{j=i+1}^{p} B_{j}^{n}exp(∫_{τ_{i}}^{T} Q_{n}(τ)dτ)ds

+ ∫_{0}^{T} σ_{n}(s)( ∏_{s<τ_{j}<T} B_{j}^{n})exp(∫_{s}^{T} Q_{n}(τ)dτ).  (50)

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

||a_{n+1}|| ≤ λ_{1}||a_{n}||^{2} + λ_{2}||b_{n}||^{2}.  (51)

Analogously, it can be proved that there exists constants μ_{1} > 0 and μ_{2} > 0 such that

||b_{n+1}|| ≤ μ_{1}||b_{n}||^{2} + μ_{2}||a_{n}||^{2}.  (52)

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].

In the case when M = 1, N = 1, c = 0 the BVP (1), (2), (3) is reduced to the periodic boundary value problem for an impulsive differential equations.

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.

Acknowledgments. The present investigation was partially supported by the Fund NIMP of Plovdiv University.

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