Differential Equations and Computational Simulations III J. Graef, R. Shivaji, B. Soni J. & Zhu (Editors) Electronic Journal of Differential Equations, Conference 01, 1997, pp. 81–95. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp 147.26.103.110 or 129.120.3.113 (login: ftp)

Quadratic Convergence of Approximate Solutions of Two-Point Boundary Value Problems with

Impulse ^{∗}

Vidya Doddaballapur, Paul W. Eloe, & Yongzhi Zhang

Abstract

The method of quasilinearization, coupled with the method of upper and lower solutions, is applied to a boundary value problem for an ordinary differential equation with impulse that has a unique solution. The method generates sequences of approximate solutions which converge monotonically and quadratically to the unique solution. In this work, we allow nonlinear terms with respect to velocity; in particular, Nagumo conditions are employed.

1 Introduction

Let 0 = t_{0} < t_{1} < … < t_{m} < t_{m+1} = 1 be given. In this paper, we shall apply the method of quasilinearization to the two-point conjugate boundary value problem (BVP) with impulse,

x′′(t) = f(t, x(t), x′(t)), t_{k} < t < t_{k+1}, k = 0, …, m,  (1)
x(0) = a, x(1) = b,  (2)

and for k = 1, …, m,

Δx(t_{k}) = u_{k}  (3)
Δx′(t_{k}) = v_{k}(x(t_{k}), x′(t_{k})) ,

where f : [0, 1] × ℝ^{2} → ℝ is continuous, u_{k} ∈ ℝ, v_{k} : ℝ^{2} → ℝ is continuous, k = 1, …, m. Define the impulse, Δx(t_{k}) = x(t_{k}^{+})−x(t_{k}^{−}), and by convention, let x(t_{k}) = x(t_{k}^{−}), k = 1, …, m. We shall employ the method of upper and lower solutions and the method of quasilinearization to obtain a bilateral iteration scheme in which the iterates converge quadratically to the unique solution of the BVP with impulse, (1), (2), (3).

∗1991 Mathematics Subject Classifications: 34A37, 34B15.

Key words and phrases: Quasilinearization, boundary value problem with impulse,

quadratic convergence, Nagumo conditions.

\begin{matrix}c \\ ◯ \end{matrix}1998 Southwest Texas State University and University of North Texas.

Published November 12, 1998.

The method of quasilinearization is described by Bellman [4, 5], and has recently been generalized by Lakshmikantham, Leela and various co-authors to apply to a wide variety of problems. See, for example, [14, 15], and references therein. The method generates sequences of approximate solutions which converge monotonically and quadratically to the problem of interest. Recently, Vatsala, et. al., [16], [17, 18], have applied the method of quasilinearization to families of two-point BVPs related to (1), in the case that f is independent of x′, and the boundary conditions are more general than (2).

More recently, Eloe and Zhang [9] extended the work of Vatsala, et. al. [16, 17, 18] to the BVP, (1), (2), in the case where f depends on x′. As pointed out in [9], Knobloch [13] and Jackson and Schrader [12] have obtained conditions such that there exists a sequence of solutions of (1) converging monotonically and in C^{1}[0, 1] to a solution of the BVP, (1), (2). Neither Knoblach [13] nor Jackson and Schrader [12] considered the rate of convergence.

Also, recently, Devi, Chandrakala and Vatsala [6] applied the method of quasilinearization to initial value problems for scalar ordinary differential equations with impulse. Doddaballapur and Eloe [7] have extended the work of Vatsala, et. al. [16, 17], [18] to the BVP with impulse, (1), (2), (3), in the case that f and each v_{k} are independent of x′. Thus, the primary contribution of this paper then is that we extend the work in [16], [17], [18], [9] and [7] to the BVP with impulse, (1), (2), (3), when f and each v_{k} depend on x′.

The paper is organized in the following manner. We shall obtain a preliminary result in Theorem 1 concerning the properties of upper and lower solutions of the BVP with impulse, (1), (2), (3). In Theorem 2, we shall obtain a fundamental existence of solutions result for the BVP with impulse, (1), (2), (3). The proof of this result employs the Schauder fixed point theorem. Due to the dependence on x′, technical difficulties arise which require the assumption of Nagumo type conditions and extensions of the Kamke convergence theorem [10, 11]. In Theorem 3, we shall state a uniqueness of solutions result for the BVP with impulse, (1), (2), (3). We shall state and prove the main result of this paper in Theorem 4. The proof of Theorem 4 employs a clever manipulation of Theorems 1 and 2. The iterative details in the proof of Theorem 4 are completely analogous to those found in [7, 9, 16, 17, 9] once Theorems 1 and 2 are obtained. Hence, we consider these details to be standard and only highlight those details in the proof of Theorem 4 that are particular to the BVP with impulse, (1), (2), (3).

2 Results

We begin with the definition of an appropriate Banach space, B. Let P C[0, 1] denote the piecewise continuous functions on [0, 1] and let P C^{1}[0, 1] denote the functions, x, such that x ∈ P C[0, 1] and x′ ∈ P C[0, 1]. Define

B = {x ∈ P C^{1}[0, 1] : x^{(i)}|_{[t_{k}, t_{k+1}]} ∈ C^{i}[t_{k}, t_{k+1}], k = 0, …, m, i = 0, 1},

with ∥x∥_{B} = max_{k=0,...,m} ∥x∥_{k} and ∥x∥_{k} = max_{i=0,1} sup_{t_{k}≤t≤t_{k+1}} |x^{(i)}(t)|. We shall say that α ∈ B is a lower solution of the BVP with impulse, (1), (2), (3), if

α′′(t) ≥ f(t, α(t), α′(t)), t_{k} < t < t_{k+1}, k = 0, …, m,
α(0) ≤ a, α(1) ≤ b,

and for k = 1, …, m,

Δα(t_{k}) = u_{k}
Δα′(t_{k}) ≥ v_{k}(α(t_{k}), α′(t_{k})) .

We shall say that β ∈ B is an upper solution of the BVP with impulse, (1), (2), (3), if

β′′(t) ≤ f(t, β(t), β′(t)), t_{k} < t < t_{k+1}, k = 0, …, m,
β(0) ≥ a, β(1) ≥ b ,

and for k = 1, …, m,

Δβ(t_{k}) = u_{k}
Δβ′(t_{k}) ≤ v_{k}(β(t_{k}), β′(t_{k})) .

For the remainder of this paper, we shall assume that

f ∈ C([0, 1] × ℝ^{2}), (∂f/∂x) = f_{x} ∈ C([0, 1] × ℝ^{2}),  (4)
f_{x}(t, x, y) > 0, (t, x, y) ∈ [0, 1] × ℝ^{2},  (5)
v_{k} ∈ C^{1}(ℝ^{2}),  (6)

and for k = 1, …, m,

v_{kx}(x, y) > 0, (x, y) ∈ ℝ^{2}, v_{ky}(x, y) > 0, (x, y) ∈ ℝ^{2}.  (7)

In order to obtain Theorem 2, we shall define an appropriate fixed point operator, T. For x ∈ B, define an operator T on x by

T x(t) = p(t) + I(t, x) + ∫_{0}^{1} G(t, s)f(s, x(s), x′(s)) ds,  (8)

where p(t) = a(1 − t) + bt, I(t, x) = ∑_{k=1}^{m} I_{k}(t, x). For k = 1, …, m, let

I_{k}(t, x) = \{ \begin{matrix}t(−u_{k} − (1 − t_{k})v_{k}(x(t_{k}), x′(t_{k}))) & , 0 ≤ t ≤ t_{k}, \\ (1 − t)(u_{k} − t_{k}v_{k}(x(t_{k}), x′(t_{k}))) & , t_{k} ≤ t ≤ 1 . \end{matrix}

Let

G(t, s) = \{ \begin{matrix}t(s − 1) & , 0 ≤ t < s ≤ 1, \\ s(t − 1) & , 0 ≤ s < t ≤ 1, \end{matrix}

denote the Green’s function for the BVP, x′′(t) = 0, 0 ≤ t ≤ 1, x(0) = 0, x(1) = 0. Eloe and Henderson [8] have argued that x is a solution of the BVP with impulse, (1), (2), (3), if, and only if, x ∈ B and T x = x. Finally, we shall define a partial order on B as follows: for α, β ∈ B, we say that α ≤ β if, and only if,

α|_{[t_{k},t_{k+1}]}(t) ≤ β|_{[t_{k},t_{k+1}]}(t), t_{k} ≤ t ≤ t_{k+1}, k = 0, …, m.

Theorem 1 Assume (4), (5), (6), and (7) hold. Let α, β be lower and upper solutions of the BVP with impulse, (1), (2), (3), respectively. Then α ≤ β.

Proof. Set w(t) = α(t) − β(t) and note that w is continuous on [0, 1] by (3). Assume, for the sake of contradiction, that w is positive on [0, 1]. Since w(0) ≤ 0, w(1) ≤ 0, w has a positive maximum at some τ ∈ (0, 1). Assume τ ∈ ∪_{k=0}^{m}(t_{k}, t_{k+1}). Then w′′(τ) ≤ 0 and α′(τ) = β′(τ). However, employing that α and β are lower and upper solutions of the BVP with impulse, (1), (2), (3), respectively, and employing (5), it follows that

w′′(τ) = α′′(τ) − β′′(τ) ≥ f(τ, α(τ), α′(τ)) − f(τ, β(τ), β′(τ)) > 0.

This provides a contradiction and so, τ ∉ ∪_{k=0}^{m}(t_{k}, t_{k+1}). Now, assume that τ = t_{k} for some k ∈ {1, …, m}. By Taylor’s theorem, w′(t_{k}^{−}) ≥ 0 and w′(t_{k}^{+}) ≤ 0, or Δw′(t_{k}) ≤ 0 and

α′(t_{k}^{−}) = α′(t_{k}) ≥ β′(t_{k}) = β′(t_{k}^{−}).

But

Δw′(t_{k}) = Δα′(t_{k}) − Δβ′(t_{k}) ≥ v_{k}(α(t_{k}), α′(t_{k})) − v_{k}(β(t_{k}), β′(t_{k})) > 0

by (7). Thus, τ ∉ {t_{1}, … t_{m}}, and w(t) ≤ 0, 0 ≤ t ≤ 1.

Theorem 2 Assume g ∈ C([0, 1] × ℝ^{2}), z_{k} ∈ C(ℝ^{2}), k = 1, …, m, and assume that each z_{k}(x, y) is monotone increasing in y for fixed x. Assume that each solution of x′′(t) = g(t, x(t), x′(t)) extends to [0, 1], or becomes unbounded on its maximal interval of convergence. Let α, β be lower and upper solutions of the BVP,

x′′(t) = g(t, x(t), x′(t)), t_{k} < t < t_{k+1},  (9)
Δx(t_{k}) = u_{k}
Δx′(t_{k}) = z_{k}(x(t_{k}), x′(t_{k})) ,  (10)

with k = 1, …, m and boundary conditions given by (2), respectively, such that

α ≤ β.

Then, there exists a solution, x, of the BVP with impulse, (9), (2), (10), satisfying

α ≤ x ≤ β.

Proof. Define

f̂(t, x, y) = \{ \begin{matrix}g(t, β(t), y) + (x − β(t))/[1 + (x − β(t))], & x > β(t), \\ g(t, x, y), & α(t) ≤ x ≤ β(t), \\ g(t, α(t), y) + (x − α(t))/[1 + |x − α(t)|], & x < α(t), \end{matrix}

and for k = 1, …, m, define

v̂_{k}(x, y) = \{ \begin{matrix}z_{k}(β(t_{k}), y) + (x − β(t_{k}))/[1 + (x − β(t_{k}))], & x > β(t_{k}), \\ z_{k}(x, y), & α(t_{k}) ≤ x ≤ β(t_{k}), \\ z_{k}(α(t_{k}), y) + (x − α(t_{k}))/[1 + |x − α(t_{k})|], & x < α(t_{k}). \end{matrix}

Let N > 0 be such that |α′(t)| ≤ N, |β′(t)| ≤ N, t ∈ [t_{k}, t_{k+1}], k = 0, …, m. For each positive integer, l, define

f_{l}(t, x, y) = \{ \begin{matrix}f̂(t, x, N + l), & y > N + l, \\ f̂(t, x, y), & |y| ≤ N + l, \\ f̂(t, x, −(N + l)), & y < −(N + l), \end{matrix}

and

v_{kl}(t, x, y) = \{ \begin{matrix}v̂_{k}(x, N + l), & y > N + l, \\ v̂_{k}(x, y), & |y| ≤ N + l, \\ v̂_{k}(x, −(N + l)), & y < −(N + l). \end{matrix}

Notice that f_{l} and each v_{kl} are bounded and continuous. With a standard application of the Schauder fixed point theorem to the operator T, defined by (8), one obtains a solution, x_{l} ∈ B, to the BVP with impulse, (1), (2), (3), with f = f_{l} and each v_{k} = v_{kl} bounded and continuous.

We now argue that each solution, x_{l}, satisfies α ≤ x_{l} ≤ β. We shall show that x_{l} ≤ β. As in the proof of Theorem 1, assume for the sake of contradiction that x_{l} − β has a positive maximum at τ. As in the proof of Theorem 1, τ ∈ (0, 1). If τ ∈ ∪_{k=0}^{m}(t_{k}, t_{k+1}), then x′′_{l}(τ) ≤ β′′(τ), and |x′_{l}(τ)| = |β′(τ)| ≤ N < N + l. Thus,

(x_{l} − β)′′(τ) ≥ (x_{l} − β)(τ)/[1 + (x − β)(τ)] > 0,

which is a contradiction. If τ = t_{k}, for some k ∈ {1, …, m}, then x′_{l}(t_{k}) ≥ β′(t_{k}). Since each z_{k}(x, y) is monotone increasing in y for fixed x, it follows that each v_{kl}(x, y) is monotone increasing in y for fixed x. Moreover, note that v_{kl}(β(t_{k}), β′(t_{k})) =z_{k}(β(t_{k}), β′(t_{k})). Thus,

Δ(x_{l} − β)′(t_{k}) ≥ v_{kl}(β(t_{k}), x′_{l}(t_{k})) − v_{kl}(β(t_{k}), β′(t_{k}))
+(x_{l} − β)(t_{k})/[1 + (x_{l} − β(t_{k}))]
≥ (x_{l} − β)(t_{k})/[1 + (x_{l} − β(t_{k}))] > 0

which is also a contradiction. Therefore, x_{l} ≤ β. To show that α ≤ x_{l} we follow a similar procedure.

For each l there exists t_{l} ∈ [0, t_{1}] such that

t_{1}|xkl^{′}(t_{l})| = |x_{kl}(t_{1}) − a| ≤ max{|β(0) − α(t_{1})|, |β(t_{1}) − α(0)|}.

Thus, each of the sequences {x_{kl}(t_{l})} and {xkl^{′}(t_{l})} are bounded. One can now apply the Kamke convergence theorem (see [11]) for solutions of initial value problems and obtain a subsequence of {x_{kl}} which converges to a solution of x′′(t) = f̂(t, x(t), x′(t)) on a maximal subinterval of [0, t_{1}]. Clearly, α(t) ≤ x(t) ≤ β(t) and solutions of x′′(t) = g(t, x(t), x′(t)) extend to all of [0, 1] or become unbounded; thus, x′′(t) = f̂(t, x(t), x′(t)) on [0, t_{1}].

Now, apply the impulse defined by (10) at t_{1}. Apply the Kamke theorem to the subsequence that was extracted in the preceding paragraph. Because of (10) one can employ t_{1} = t_{l} for each l. Thus, one obtains a further subsequence which converges to a solution, x, of x′′(t) = f̂(t, x(t), x′(t)) on (0, t_{1}) ∪ (t_{1}, t_{2}) such that x satisfies (10) at t_{1}.

Continue inductively, first applying (10) at each t_{j} and then applying the Kamke convergence theorem on that subinterval (t_{j}, t_{j+1}). Finally, since α ≤ x ≤ β, f̂(t, x(t), x′(t)) = f(t, x(t), x′(t)) and the proof of Theorem 2 is complete.

Remark. For simplicity, we can assume that g satisfies a Nagumo condition in x′ ([10], [11]). That is, assume that for each M > 0 there exists a positive continuous function, h_{M}(s), defined on [0, ∞) such that

|g(t, x, x′)| ≤ h_{M}(|x′|)

for all (t, x, x′) ∈ [0, 1] × [−M, M] × ℝ and such that

∫_{0}^{∞}(s/h_{M}(s))ds = +∞.

The assumption that g satisfies a Nagumo condition implies that each solution of the differential equation, x′′(t) = g(t, x(t), x′(t)), either extends to [0, 1] or becomes unbounded on its maximal interval of existence ([10], [11]). In our main result, Theorem 4, g will represent a modification of f. Thus, we shall assume in Theorem 4 that f satisfies a Nagumo condition in x′.

Theorem 3 Assume that (4), (5), (6), and (7) hold. Then, solutions of the BVP with impulse, (1), (2), (3), are unique.

Proof. The uniqueness of solutions result follows immediately from Theorem 1 and the observation that solutions are respectively upper and lower solutions.

Theorem 4 Assume that (4), (5), (6), and (7) hold, and assume that

(∂^{2}/∂x^{2})f ∈ C([0, 1] × ℝ^{2}), v′′_{k} ∈ C(ℝ^{2}), k = 1, …, m .

Assume that f satisfies a Nagumo condition in x′. Assume that α_{0} and β_{0} are lower and upper solutions of the BVP with impulse, (1), (2), (3), respectively. Then there exist monotone sequences, {α_{n}(t)} and {β_{n}(t)}, which converge in B to the unique solution, x(t), of the BVP with impulse, (1), (2), (3), and the convergence is quadratic.

Proof. Let F (t, x) : [0, 1] × ℝ → ℝ be such that F, F_{x}, F_{xx} are continuous on [0, 1] × ℝ and

F_{xx}(t, x) ≥ 0, (t, x) ∈ [0, 1] × ℝ .  (11)

Set ϕ_{1}(t, x_{1}, x_{2}) = F (t, x_{1})−f(t, x_{1}, x_{2}) on [0, 1]×ℝ^{2}. From (11) it follows that, if x_{1}, y_{1} ∈ ℝ, then F (t, x_{1}) ≥ F (t, y_{1}) + F_{x}(t, y_{1})(x_{1} − y_{1}). In particular, for x_{1}, y_{1}, x_{2}, y_{2} ∈ ℝ,

f(t, x_{1}, x_{2}) ≥ f(t, y_{1}, y_{2}) + F_{x}(t, y_{1})(x_{1} − y_{1}) − ϕ_{1}(t, x_{1}, x_{2}) + ϕ_{1}(t, y_{1}, y_{2}). (12)

For each k = 1, …, m, let V_{k}(x) : ℝ → ℝ be such that V_{k}, V′_{k}, V′′_{k} are continuous on ℝ and

V′′_{k}(x) ≥ 0, x ∈ ℝ .  (13)

Set ϕ_{2k}(x_{1}, x_{2}) = V_{k}(x_{1})−v_{k}(x_{1}, x_{2}) on ℝ^{2}. From (13) it follows that, if x_{1}, y_{1} ∈ ℝ, then V_{k}(x_{1}) ≥ V_{k}(y_{1}) + V′_{k}(y_{1})(x_{1} − y_{1}). In particular, for x_{1}, y_{1}, x_{2}, y_{2} ∈ ℝ,

v_{k}(x_{1}, x_{2}) ≥ v_{k}(y_{1}, y_{2}) + V′_{k}(y_{1})(x_{1} − y_{1}) − (ϕ_{2k}(x_{1}, x_{2}) − ϕ_{2k}(y_{1}, y_{2})). (14)

Define

g(t, x_{1}, x_{2}; α_{0}, β_{0}, α′_{0}) = f(t, α_{0}(t), α′_{0}(t)) + F_{x}(t, β_{0}(t))(x_{1} − α_{0}(t))
−ϕ_{1}(t, x_{1}, x_{2}) + ϕ_{1}(t, α_{0}(t), α′_{0}(t)) ,
G(t, x_{1}, x_{2}; β_{0}, β′_{0}) = f(t, β_{0}(t), β′_{0}(t)) + F_{x}(t, β_{0}(t))(x_{1} − β_{0}(t))
−ϕ_{1}(t, x_{1}, x_{2}) + ϕ_{1}(t, β_{0}(t), β′_{0}(t)) ,
h_{k}(x_{1}, x_{2}; α_{0}, β_{0}, α′_{0}) = v_{k}(α_{0}(t_{k}), α′_{0}(t_{k})) + V′_{k}(β_{0}(t_{k}))(x_{1} − α_{0}(t_{k}))
−(ϕ_{2k}(x_{1}, x_{2}) − ϕ_{2k}(α_{0}(t_{k}), α′_{0}(t_{k}))) ,
H_{k}(x_{1}, x_{2}; β_{0}, β′_{0}) = v_{k}(β_{0}(t_{k}), β′_{0}(t_{k})) + V′_{k}(β_{0}(t_{k}))(x_{1} − β_{0}(t_{k}))
−(ϕ_{2k}(x_{1}, x_{2}) − ϕ_{2k}(β_{0}(t_{k}), β′_{0}(t_{k}))) .

First consider the BVP with impulse,

x′′(t) = g(t, x(t), x′(t); α_{0}, β_{0}, α′_{0}), t_{k} < t < t_{k+1}, k = 0, …, m, (15)

and for k = 1, …, m,

Δx(t_{k}) = u_{k}  (16)
Δx′(t_{k}) = h_{k}(x(t_{k}), x′(t_{k}); α_{0}, β_{0}, α′_{0}) ,

with boundary conditions given by (2). Each h_{k} readily satisfies the hypotheses of Theorem 2. A limit comparison implies that g satisfies a Nagumo condition in x′.

We now show that α_{0} and β_{0} are lower and upper solutions, respectively, of the BVP with impulse, (15), (2), (16); thus, by Theorem 2, there exists a solution α_{1}(t) of the BVP with impulse, (15), (2), (16), satisfying

α_{0} ≤ α_{1} ≤ β_{0}.

To this end, note that for t_{k} < t < t_{k+1}, k = 0, …, m,

α′′_{0}(t) ≥ f(t, α_{0}(t), α′_{0}(t)) = g(t, α_{0}(t), α′_{0}(t); α_{0}, β_{0}, α′_{0}),

and, for k = 1, …, m,

Δα′_{0}(t_{k}) ≥ v_{k}(α_{0}(t_{k}), α′_{0}(t_{k})) = h_{k}(α_{0}(t_{k}), α′_{0}(t_{k}); α_{0}, β_{0}, α′_{0}).

Moreover, from (12) and (14), it follows that for t_{k} < t < t_{k+1}, k = 0, …, m,

β′′_{0}(t) ≤ f(t, β_{0}(t), β′_{0}(t)) ≤ f(t, α_{0}(t), α′_{0}(t)) − F_{x}(t, β_{0}(t))(α_{0}(t) − β_{0}(t))
+ϕ_{1}(t, α_{0}(t), α′_{0}(t)) − ϕ_{1}(t, β_{0}(t), β′_{0}(t))
= g(t, β_{0}(t), β′_{0}(t); α_{0}, β_{0}, α′_{0}),

and for k = 1, …, m,

Δβ′_{0}(t_{k}) ≤ v_{k}(β_{0}(t_{k}), β′_{0}(t_{k}))
≤ v_{k}(α_{0}(t), α′_{0}(t)) − V′_{k}(β_{0}(t_{k}))(α_{0}(t_{k}) − β_{0}(t_{k}))
+(ϕ_{2k}(α_{0}(t_{k}), α′_{0}(t)) − ϕ_{2k}(β_{0}(t_{k}), β′_{0}(t_{k}))
= h_{k}(β_{0}(t_{k}), β′_{0}(t_{k}); α_{0}, β_{0}, β′_{0}).

Since α_{0} and β_{0} satisfy (2), α_{0} and β_{0} are lower and upper solutions, respectively, of the BVP with impulse, (15), (2), (16), and thus, by Theorem 2, there exists a solution α_{1}(t) of the BVP with impulse, (15), (2), (16), such that

α_{0} ≤ α_{1} ≤ β_{0} .

Now, consider the BVP with impulse,

x′′(t) = G(t, x(t), x′(t); β_{0}, β′_{0}), t_{k} < t < t_{k+1}, k = 0, …, m,
x(0) = a, x(1) = b,  (17)

and for k = 1, …, m,

Δx(t_{k}) = u_{k}  (18)
Δx′(t_{k}) = H_{k}(x(t_{k}), x′(t_{k}); α_{0}, β_{0}, β′_{0}) .

Again, G and each H_{k} satisfy the hypotheses of Theorem 2, and, again, (12) and (14) are employed to show that α_{0} and β_{0} are lower and upper solutions, respectively, of the BVP with impulse, (17), (2), (18); thus, there exists a solution β_{1}(t) of the BVP with impulse, (17), (2), (18), such that

α_{0} ≤ β_{1} ≤ β_{0} .

We now show that α_{1} and β_{1} are lower and upper solutions, respectively, of the BVP with impulse, (1), (2), (3). Thus, it will follow by Theorem 1 that

α_{0} ≤ α_{1} ≤ β_{1} ≤ β_{0} .

V. Doddaballapur, P. W. Eloe & Y. Zhang 89 Employ (12) and (11) to see that for t ∈ ∪_{k=0}^{m}(t_{k}, t_{k+1}),

α′′_{1}(t) = g(t, α_{1}(t), α′_{1}(t); α_{0}, β_{0}, α′_{0})
= f(t, α_{0}(t), α′_{0}(t)) + F_{x}(t, β_{0}(t))(α_{1}(t) − α_{0}(t))
−(ϕ(t, α_{1}(t), α′_{1}(t)) − ϕ(t, α_{0}(t), α′_{0}(t)))
≥ f(t, α_{1}(t), α′_{1}(t)) + F_{x}(t, α_{1}(t))(α_{0}(t) − α_{1}(t)) + ϕ(t, α_{1}(t), α′_{1}(t))
−ϕ(t, α_{0}(t), α′_{0}(t)) + F_{x}(t, β_{0}(t))(α_{1}(t) − α_{0}(t))
−(ϕ(t, α_{1}(t), α′_{1}(t)) − ϕ(t, α_{0}(t), α′_{0}(t)))
= f(t, α_{1}(t), α′_{1}(t)) + (F_{x}(t, β_{0}(t)) − F_{x}(t, α_{1}(t)))(α_{1}(t) − α_{0}(t))
≥ f(t, α_{1}(t), α′_{1}(t)).

Similarly, for k = 1, …, m, employ (14) and (13) to see that

Δα′_{1}(t_{k}) = h_{k}(α_{1}(t_{k}), α′_{1}(t_{k})); α_{0}, β_{0}, α′_{0})
= v_{k}(α_{0}(t_{k}), α′_{0}(t_{k})) + V′_{k}(β_{0}(t_{k}))(α_{1}(t_{k}) − α_{0}(t_{k}))
−(ϕ_{2k}(α_{1}(t_{k}), α′_{1}(t_{k})) − ϕ_{2k}(α_{0}(t_{k}), α′_{0}(t_{k})))
≥ v_{k}(α_{1}(t_{k}), α′_{1}(t_{k})) + V′_{k}(α_{1}(t_{k}))(α_{0}(t_{k}) − α_{1}(t_{k}))
+ϕ_{2k}(α_{1}(t_{k}), α′_{1}(t_{k})) − ϕ_{2k}(α_{0}(t_{k}), α′_{0}(t_{k}))
+V′_{k}(β_{0}(t_{k}))(α_{1}(t_{k}) − α_{0}(t_{k})) − (ϕ_{2k}(α_{1}(t_{k}), α′_{1}(t_{k}))
−ϕ_{2k}(α_{0}(t_{k}), α′_{0}(t_{k})))
= v_{k}(α_{1}(t_{k})) + (V′_{k}(β_{0}(t_{k})) − V′_{k}(α_{1}(t_{k})))(α_{1}(t_{k}) − α_{0}(t_{k}))
≥ v_{k}(α_{1}(t_{k})).

Similarly, it follows by (11)-(14) that for t ∈ ∪_{k=0}^{m}(t_{k}, t_{k+1}),

β′′_{1}(t) ≤ f(t, β_{1}(t), β′_{1}(t)),

and for k ∈ {1, …, m},

Δβ′_{1}(t_{k}) ≤ v_{k}(β_{1}(t_{k}), β′_{1}(t_{k})).

In particular, α_{1} and β_{1} are lower and upper solutions, respectively, of the BVP with impulse, (1), (2), (3), and by Theorem 1,

α_{0} ≤ α_{1} ≤ β_{1} ≤ β_{0}.

Inductively, define sequences of functions {g_{l}}, {G_{l}}, {h_{kl}}, and {H_{kl}} by

g_{l}(t, x_{1}, x_{2}) = g(t, x_{1}, x_{2}; α_{l}, β_{l}, α′_{l})
= f(t, α_{l}(t), α′_{l}(t)) + F_{x}(t, β_{l}(t))(x_{1} − α_{l}(t))
−ϕ_{1}(t, x_{1}, x_{2}) + ϕ_{1}(t, α_{l}(t), α′_{l}(t))
G_{l}(t, x_{1}, x_{2}) = G(t, x_{1}, x_{2}; β_{l}, β′_{l})
= f(t, β_{l}(t), β′_{l}(t)) + F_{x}(t, β_{l}(t))(x_{1} − β_{l}(t))

−ϕ_{1}(t, x_{1}, x_{2}) + ϕ_{1}(t, β_{l}(t), β′_{l}(t)) ,
h_{kl} = h_{k}(x_{1}, x_{2}; α_{l}, β_{l}, α′_{l})
= v_{k}(α_{l}(t_{k}), α′_{l}(t_{k})) + V′_{k}(β_{l}(t_{k}))(x_{1} − α_{l}(t_{k}))
−(ϕ_{2k}(x_{1}, x_{2}) − ϕ_{2k}(α_{l}(t_{k}))) ,
H_{kl} = H_{k}(x_{1}, x_{2}; β_{l}, β′_{l})
= v_{k}(β_{l}(t_{k}), β′_{l}(t_{k})) + V′_{k}(β_{l}(t_{k}))(x_{1} − β_{l}(t_{k}))
−(ϕ_{2k}(x_{1}, x_{2}) − ϕ_{2k}(β_{l}(t_{k}), β′_{l}(t_{k}))) .

Inductively, Theorem 2 implies there exists a solution α_{l+1}(t) of the BVP with impulse, (15), (2), (16), with g = g_{l} and each h_{k} = h_{kl} satisfying

α_{0} ≤ … ≤ α_{l} ≤ α_{l+1} ≤ β_{l} ≤ … ≤ β_{0} .

Similarly, there exists a solution β_{l+1}(t) of the BVP with impulse, (17), (2), (18), with G = G_{l} and each H_{k} = H_{kl} satisfying

α_{0} ≤ … ≤ α_{l} ≤ β_{l+1} ≤ β_{l} ≤ … ≤ β_{0}.

Finally, inductively, α_{l+1} and β_{l+1} are lower and upper solutions, respectively, of the BVP with impulse, (1), (2), (3), and by Theorem 1,

α_{0} ≤ … ≤ α_{l} ≤ α_{l+1} ≤ β_{l+1} ≤ β_{l} ≤ … ≤ β_{0}.

We now show that each sequence {α_{l}} and {β_{l}} converge in B to x, the unique solution of the BVP with impulse, (1), (2), (3). Recall

B = {x ∈ P C^{1}[0, 1] : x^{(i)}|_{[t_{k},t_{k+1}]} ∈ C^{i}[t_{k}, t_{k+1}], k = 0, …, m, i = 0, 1},

with ∥x∥_{B} = max_{k=0,...,m} ∥x∥_{k} and ∥x∥_{k} = max_{i=0,1} sup_{t_{k}≤t≤t_{k+1}} |x^{(i)}(t)|. The Kamke convergence theorem does not apply directly to either sequence, {α_{l}} or {β_{l}} since neither g_{l} nor G_{l} converge uniformly on compact sets to f. To see this, note that

g_{l}(t, x_{1}, x_{2}) = f(t, x_{1}, x_{2}) + F_{x}(t, β_{l}(t))(x_{1} − α_{l}(t)) + F (t, α_{l}(t)) − F (t, x_{1})

and

G_{l}(t, x_{1}, x_{2}) = f(t, x_{1}, x_{2}) + F_{x}(t, β_{l}(t))(x_{1} − β_{l}(t)) + F (t, β_{l}(t)) − F (t, x_{1}).

Define

ĝ_{l}(t, x_{1}, x_{2}) = f(t, x_{1}, x_{2}) + F_{x}(t, β_{l}(t))(α_{l+1} − α_{l})(t) + F (t, α_{l}(t)) − F (t, α_{l+1}(t))

and

Ĝ_{l}(t, x_{1}, x_{2}) = f(t, x_{1}, x_{2})+F_{x}(t, β_{l}(t))(β_{l+1} −β_{l})(t)+F (t, β_{l}(t))−F (t, β_{l+1}(t)).

Theorem 3 applies to the BVP with impulse, (1), (2), (3), with f = ĝ_{l} and each v_{k} = h_{kl} and note that α_{k+1} is the unique solution. The Kamke convergence theorem now does apply and, with omitted details that are similar to those given in the proof of Theorem 2, {α_{l}} converges in B to x, the unique solution of the BVP with impulse, (1), (2), (3). Similarly, {β_{l}} converges in B to x, the unique solution of the BVP with impulse, (1), (2), (3).

We now argue that the convergence is quadratic. Let q_{n}(t) = β_{n}(t) − x(t) and p_{n}(t) = x(t) − α_{n}(t), where x(t) denotes the unique solution of the BVP with impulse, (1), (2), (3). Set

e_{n} = max{∥q_{n}∥_{B}, ∥p_{n}∥_{B}}.

First, consider q_{n+1}(t) and note that q_{n+1} ≥ 0. For t ∈ ∪_{k=0}^{m}(t_{k}, t_{k+1}),

q′′_{n+1}(t) = F (t, β_{n}(t)) + F_{x}(t, β_{n}(t))(β_{n+1} − β_{n})(t)
−ϕ_{1}(t, β_{n+1}(t), β′_{n+1}(t)) − F (t, x(t)) + ϕ_{1}(t, x(t), x′(t))
= F_{x}(t, c_{1}(t))q_{n}(t) − F_{x}(t, β_{n}(t))q_{n}(t) + F_{x}(t, β_{n}(t))q_{n+1}(t)
−ϕ_{1x}(t, c_{2}(t), c_{3}(t))q_{n+1}(t) − ϕ_{1x′}(t, c_{2}(t), c_{3}(t))q′_{n+1}(t) ,

where x(t) ≤ c_{1}(t) ≤ β_{n}(t), x(t) ≤ c_{2}(t) ≤ β_{n+1}(t), and c_{3}(t) is between x′(t) and β′_{n+1}(t). Thus, there exists c_{1}(t) ≤ c_{4}(t) ≤ β_{n}(t) such that

q′′_{n+1}(t)

= F_{xx}(t, c_{4}(t))q_{n}(t)(c_{1}(t) − β_{n}(t))

+(F_{x}(t, β_{n}(t)) − ϕ_{1x}(t, c_{2}(t), c_{3}(t)))q_{n+1}(t) − ϕ_{1x′}(t, c_{2}(t), c_{3}(t))q′_{n+1}(t)

≥ −F_{xx}(t, c_{4}(t))q_{n}^{2}(t) + f_{x′}(t, c_{2}(t), c_{3}(t))q′_{n+1}(t) .

Note that to obtain this inequality, we have employed the monotonicity of F_{x} in the second component. In particular, there exists M > 0, such that

q′′_{n+1}(t) − f_{x′}(t, c_{2}(t), c_{3}(t))q′_{n+1}(t) ≥ −Me_{n}^{2},  (19)

where M > max_{i} max_{(t,x)∈D_{i}} F_{xx}(t, x), and for i = 0, … m,

D_{i} = {(t, x) : t_{i} ≤ t ≤ t_{i+1}, α_{0}(t) ≤ x ≤ β_{0}(t)} .

Similarly, there exist appropriate c_{4} and c_{5} such that for k = 1, …, m,

Δq′_{n+1}(t_{k}) − v_{ky}(c_{4}, c_{5})q′_{n+1}(t_{k}) ≥ −Me_{n}^{2} .  (20)

Let m(t) = exp ( − ∫_{0}^{t} f_{x′}(s, c_{2}(s), c_{3}(s))ds) denote the integrating factor asso-

ciated with (19). Then

(q′_{n+1}(t)m(t))′ ≥ −Mm(t)e_{n}^{2} .  (21)

Thus, for t_{m} ≤ t ≤ 1,

q′_{n+1}(1)m(1) − q′_{n+1}(t)m(t) ≥ −Me_{n}^{2} ∫_{t}^{1} m(s)ds .

Since, q′_{n+1}(1) ≤ 0, it follows that

q′_{n+1}(t) ≤ Me_{n}^{2} ∫_{t}^{1} m(s)ds/m(t) .

Since q_{n+1} converges to 0 in B, eventually (s, c_{2}(s), c_{3}(s)) belongs to

D̂ = {(s, x_{1}, x_{2}) : t_{m} ≤ s ≤ 1, α_{0}(s) ≤ x_{1} ≤ β_{0}(s), x′(s) − 1 ≤ x_{2} ≤ x′(s) + 1}.

Thus, we can bound m(t) away from both 0 and ∞ for n sufficiently large; in particular, there exists N_{1} > 0 such that for t_{m} ≤ t ≤ 1 and n sufficiently large,

q′_{n+1}(t) ≤ N_{1}e_{n}^{2} .  (22)

Apply (20) at t_{m}. Then

q′_{n+1}(t_{m}^{+}) − q′_{n+1}(t_{m}) − v_{my}(c_{4}, c_{5})q′_{n+1}(t_{m}) ≥ −Me_{n}^{2} .

Employ (7) and also bound v_{my} away from both 0 and ∞ to obtain some M̂ > 0 such that

q′_{n+1}(t_{m}^{−}) ≥ −M̂e_{n}^{2} .  (23)

Now, employ (21) and (23) to obtain (22) for t_{m−1} ≤ t ≤ t_{m} for some N_{2} > 0. Again, apply (20) to obtain a suitable (23) at t_{m−1}. Proceed inductively and obtain that there exists N > 0 such that for t ∈ ∪_{k=0}^{m}[t_{k}, t_{k+1}] and n sufficiently large,

q′_{n+1}(t) ≤ Ne_{n}^{2}.  (24)

Recall that q_{n+1}(t) ≥ 0, and that q_{n+1} ∈ C[0, 1]. Integrate (24) from 0 to t; then for n sufficiently large,

0 ≤ q_{n+1} ≤ Ne_{n}^{2} .  (25)

Beginning again at (21), integrate from 0 to t ≤ t_{1} to obtain

q′_{n+1}(t)m(t) − q′_{n+1}(0) ≥ −Me_{n}^{2} ∫_{0}^{t} m(s) ds .

Since, q′_{n+1}(0) ≥ 0, it follows that for 0 ≤ t ≤ t_{1}, there exists N_{1} > 0, such that

q′_{n+1}(t) ≥ −Me_{n}^{2} ∫_{0}^{t} m(s) ds/m(t) ≥ −N_{1}e_{n}^{2} ,

for n sufficiently large. This is analogous to (22). Proceed analogously, then, and choose N large enough such that for t ∈ ∪_{k=0}^{m}[t_{k}, t_{k+1}] for n sufficiently large,

q′_{n+1}(t) ≥ −Ne_{n}^{2} .  (26)

It now follows from (24), (25), and (26) that β_{n} converges to x quadratically in B.

The argument that {α_{n}} converges quadratically to x in B is similar and we provide some details.

p′′_{n+1}(t)

= F (t, x(t)) − ϕ_{1}(t, x(t), x′(t))

−(F (t, α_{n}(t)) + F_{x}(t, β_{n}(t))(α_{n+1} − α_{n})(t) − ϕ_{1}(t, α_{n+1}(t), α′_{n+1}(t)))

= F_{x}(t, c_{1}(t))p_{n}(t) − F_{x}(t, β_{n}(t))p_{n}(t) + F_{x}(t, β_{n}(t))p_{n+1}(t)
−ϕ_{1x}(t, c_{2}(t), c_{3}(t))p_{n+1}(t) − ϕ_{1x′}(t, c_{2}(t), c_{3}(t))p′_{n+1}(t)
= F_{xx}(t, c_{4}(t))p_{n}(t)(c_{1}(t) − β_{n}(t))

+(F_{x}(t, β_{n}(t)) − ϕ_{1x}(t, c_{2}(t), c_{3}(t)))p_{n+1}(t) − ϕ_{1x′}(t, c_{2}(t), c_{3}(t))p′_{n+1}(t)

≥ −F_{xx}(t, c_{4}(t))p_{n}(t)(p_{n}(t) + q_{n}(t)) + f_{x′}(t, c_{2}(t), c_{3}(t))p′_{n+1}(t) .

In particular,

p′′_{n+1}(t) − f_{x′}(t, c_{2}(t), c_{3}(t))p′_{n+1}(t) ≥ −2Me_{n}^{2}

on an appropriate set and for sufficiently large n. A similar inequality is obtained with respect to the impulse and the details for quadratic convergence follow as above.

Corollary 5 The sequence {β′′_{n}(t) − f(t, β_{n}(t), β′_{n}(t))} converges quadratically to 0 in B.

Proof: There exist β_{n} ≥ c_{2} ≥ c_{1} ≥ β_{n+1} such that

f(t, β_{n+1}(t), β′_{n+1}(t)) ≥ β′′_{n+1}(t)
= f(t, β_{n}(t), β′_{n}(t)) + F_{x}(t, β_{n}(t))(β_{n+1}(t) − β_{n}(t))
−(ϕ_{1}(t, β_{n+1}(t), β′_{n+1}(t)) − ϕ_{1}(t, β_{n}(t), β′_{n}(t)))
= f(t, β_{n+1}(t), β′_{n+1}(t))
+F_{xx}(t, c_{2}(t))(β_{n+1}(t) − β_{n}(t))(β_{n}(t) − c_{1}(t)) .

Thus,

0 ≤ f(t, β_{n+1}(t), β′_{n+1}(t)) − β′′_{n+1}(t)
≤ F_{xx}(t, c_{2}(t))(β_{n+1}(t) − β_{n}(t))^{2}
≤ F_{xx}(t, c_{2}(t))e_{n}^{2} .

Similar inequalities are obtained for the impulse. Quadratic convergence can also be obtained for the sequence

{f(t, α_{n}(t), α′_{n}(t)) − α′′_{n}(t)}.

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Vidya Doddaballapur, Paul W. Eloe, & Yongzhi Zhang Department of Mathematics University of Dayton Dayton, Ohio 45469-2316, USA Email address: eloe@@saber.udayton.edu