Electronic Journal of Qualitative Theory of Differential Equations 2007, No. 1, 1 - 19; http://www.math.u-szeged.hu/ejqtde/

An Extended Method of Quasilinearization for Nonlinear

Impulsive Differential Equations with a Nonlinear Three-Point Boundary Condition

Bashir Ahmad^{1} and Ahmed Alsaedi Department of Mathematics, Faculty of science, King Abdulaziz University, P.O. Box. 80257, Jeddah 21589, Saudi Arabia

Abstract

In this paper, we discuss an extended form of generalized quasilinearization technique for first order nonlinear impulsive differential equations with a nonlinear three-point boundary condition. In fact, we obtain monotone sequences of upper and lower solutions converging uniformly and quadratically to the unique solution of the problem.

Key words: Impulsive differential equations, three-point boundary conditions, quasilinearization, quadratic convergence. AMS(MOS) Subject Classification: 34A37, 34B10.

1 Introduction

The method of quasilinearizaion provides an adequate approach for obtaining approximate solutions of nonlinear problems. The origin of the quasilinearizaion lies in the theory of dynamic programming [1-3]. This method applies to semilinear equations with convex (concave) nonlinearities and generates a monotone scheme whose iterates converge quadratically to the solution of the problem at hand. The assumption of convexity proved to be a stumbling block for the further development of the method. The nineties brought new dimensions to this technique. The most interesting new idea was introduced by Lakshmikantham [4-5] who generalized the method of quasilinearizaion by relaxing the convexity assumption. This development proved to be quite significant and the method was studied and applied to a wide range of initial and boundary value problems for different types of differential equations, see [6-17] and references therein. Some real-world applications of the quasilinearizaion technique can be found in [18-20]. Many evolution processes are subject to short term perturbations which act instantaneously in the form of impulses. Examples include biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics and frequency modulated systems. Thus, impulsive differential equations provide a natural description of observed evolution processes of several real world problems. Moreover, the theory of impulsive differential equations is much richer than the corresponding theory of ordinary differential equations without impulse effects since a simple impulsive differential equation may exhibit several new phenomena such as rhythmical beating, merging of solutions and noncontinuability of solutions. Thus, the theory of impulsive differential equations is quite interesting and has attracted the attention of many scientists, for example, see [21-24]. In particular, Eloe and Hristova [23] discussed the method of quasilinearization for first order nonlinear impulsive differential equations with linear boundary conditions. Multi-point nonlinear boundary value problems, which refer to a different family of boundary conditions in the study of disconjugacy theory [25], have been addressed by many authors, for instance, see [26-27] and the references therein. In this paper, we develop an extended method of quasilinearization for a class of first order nonlinear impulsive differential equations involving a mixed type of nonlinearity with a nonlinear three-point boundary condition

1Corresponding author bashir_{−}qau@yahoo.com (B. Ahmad), aalsaedi@hotmail.com (A. Alsaedi)

x′(t) = F (t, x(t)) for t ∈ [0, T], t ≠ τ_{k}, τ_{k} ∈ (0, T),  (1)
x(τ_{k} + 0) = I_{k}(x(τ_{k})), k = 1, 2, ..., p,  (2)
γ_{1}x(0) − γ_{2}x(T) = h(x(\frac{T}{2} )),  (3)

where F ∈ C[[0, T] × R, R] and F (t, x(t)) = f(t, x(t)) + g(t, x(t)), γ_{1}, γ_{2} are constants with γ_{1} ≥ γ_{2} > 0, τ_{k} < τ_{k+1}, k = 1, 2, ..., p and the nonlinearity h : R ⟶ R is continuous. Here, it is worthmentioning that the convexity assumption on f(t, x) has been relaxed and instead f(t, x) + M_{1}x^{2} is taken to be convex for some M_{1} > 0 while a less restrictive condition is demanded on g(t, x), namely, [g(t, x) + M_{2}x^{1+ϵ}] satisfies a nondecreasing condition for some ϵ > 0 and M_{2} > 0. Moreover, we also relax the concavity assumption (h′′(x) ≤ 0) on the nonlinearity h(x) in the boundary condition (3) by requiring h′′(x) + ψ′′(x) ≤ 0 for some continuous function ψ(x) satisfying ψ′′ ≤ 0 on R. We construct two monotone sequences of upper and lower solutions converging uniformly and quadratically to the unique solution of the problem. Some special cases of our main result have also been recorded.

2 Some Basic Results

For A ⊂ R, B ⊂ R, let P C(A, B) denotes the set of all functions v : A → B which are piecewise continuous in A with points of discontinuity of first kind at the points τ_{k} ∈ A, that is, there exist the limits lim_{t↓τ_{k}} v(t) = v(τ_{k} + 0) < ∞ and lim_{t↑τ_{k}} v(t) = v(τ_{k} − 0) = v(τ_{k}). The set P C^{1}(A, B) consists of all functions v ∈ P C(A, B) that are continuously differentiable for t ∈ A, 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

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)
γ_{1}α(0) − γ_{2}α(T) ≤ h(α(\frac{T}{2} )).  (6)

The function β(t) ∈ P C^{1}([0, T], R) is called an upper solution of the BVP (1)-(3) if the inequalities are reversed in (4)-(6).

Let us set the following notations for the sequel.

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

Theorem 1. (Comparison Result) Let α, β ∈ P C^{1}([0, T], R) be lower and upper solutions of (1)-(3) respectively. Further, F (t, x) ∈ C(Ω(α, β), R) is quasimonotone nondecreasing in x for each t ∈ [0, T] and satisfies F (t, x) − F (t, y) ≤ L(x − y), L ≥ 0 whenever y ≤ x. Moreover, h is nondecreasing on R and I_{k} : D_{k}(α, β) → R are nondecreasing in D_{k}(α, β) for each k = 1, 2, ..., p and satisfies I_{k}(x) − I_{k}(y) ≤ M(x − y), M ≥ 0. Then α(t) ≤ β(t) on [0, T].

Proof. The method of proof is similar to the one used in proving Theorem 2.6.1 (page 87 [21]), so we omit the proof.

Theorem 2. (Existence of solution) Assume that F is continuous on Ω(α, β) and h is nondecreasing on R. Further, we assume that I_{k} : D_{k}(α, β) → R are nondecreasing in D_{k}(α, β) for each k = 1, 2, ..., p and α, β are respectively lower and upper solutions of (1)-(3) such that α(t) ≤ β(t) on [0, T]. Then there exists a solution x(t) of (1)-(3) such that x(t) ∈ S(α, β).

Proof. There is no loss of generality if we consider the case p = 1, that is, 0 < t_{1} < T. Let x_{0} be an arbitrary point such that α(0) ≤ x_{0} ≤ β(0). Define \overline{F} and H by

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

H(x) = \{ \begin{matrix}h(β(\frac{T}{2} )), & if x > β(\frac{T}{2} ), \\ h(x), & if α(\frac{T}{2} ) ≤ x ≤ β(\frac{T}{2} ), \\ h(α(\frac{T}{2} )), & if x < α(\frac{T}{2} ). \end{matrix}

Since \overline{F} (t, x) and H(x) are continuous and bounded, therefore, there exists a function μ ∈ ([0, T], [0, ∞)) such that sup{|F (t, x)| : x ∈ R} ≤ μ(t) for t ∈ [0, T]. Thus, the initial value problem x′(t) = \overline{F} (t, x), x(0) = x_{0} has a solution X(t; x_{0}) for t ∈ [0, t_{1}]. We define u(t) = X(t; x_{0}) − β(t) and prove that the function u(t) is non-positive on [0, t_{1}]. For the sake of contradiction, assume that u(t) > 0, that is, sup{u(t) : t ∈ [0, t_{1}]} > 0. Therefore, there exists a point t_{0} ∈ (0, t_{1}) such that u(t_{0}) > 0 and u′(t_{0}) ≥ 0. On the other hand, we have

u′(t_{0}) = X′(t_{0}; x_{0}) − β′(t_{0})
≤ \overline{F} (t_{0}, x) − F (t_{0}, β(t_{0}))
= F (t_{0}, β(t_{0})) + \frac{β(t_{0}) − X(t_{0}; x_{0})}{1 + |X(t_{0}; x_{0})|} − F (t_{0}, β(t_{0}))
= \frac{−u(t_{0})}{1 + |X(t_{0}; x_{0})|} < 0,

which is a contradiction. Hence we conclude that X(t; x_{0}) ≤ β(t), t ∈ [0, t_{1}). Similarly, it can be shown that X(t; x_{0}) ≥ α(t), t ∈ [0, t_{1}]. Now we set y_{0} = I_{1}(X(t_{1}; x_{0})) and note that y_{0} depends on x_{0}. From the nondecreasing property of 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),

that is, α(t_{1} + 0) ≤ y_{0} ≤ β(t_{1} + 0). Consider the initial value problem x′ = \overline{F} (t, x), x(t_{1}) = y_{0} for t ∈ [t_{1}, T] which has a solution Y (t; y_{0}) for t ∈ [t_{1}, T]. Employing the earlier arguments, It is not hard to prove that α(t) ≤ Y (t; y_{0}) ≤ β(t) for t ∈ [t_{1}, T]. Also, we notice that Y (t_{1}, y_{0}) = y_{0} = I_{1}(X(t_{1}, x_{0})). Let us define

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}

Obviously the function x(t; x_{0}) such that α(t) ≤ x(t; x_{0}) ≤ β(t) is a solution of the impulsive differential equation (1)-(2) with the initial condition x(0) = x_{0}. In view of the inequality α(t) ≤ β(t) for t ∈ [0, T], there are following two possible cases: Case 1. Let α(0) = β(0). Then x_{0} = α(0) = β(0) and

γ_{1}x(0; x_{0}) − γ_{2}x(T; x_{0}) = γ_{1}x_{0} − γ_{2}x(T; x_{0}) ≤ γ_{1}α(0) − γ_{2}α(T) ≤ h(α(\frac{T}{2} )),

γ_{1}x(0; x_{0}) − γ_{2}x(T; x_{0}) ≥ γ_{1}β(0) − γ_{2}β(T) ≥ h(β(\frac{T}{2} )).

Thus,

h(β(\frac{T}{2} )) ≤ γ_{1}x(0; x_{0}) − γ_{2}x(T; x_{0}) ≤ h(α(\frac{T}{2} )).  (10)

Using the nondecreasing property of h(t) together with the fact that α(t) ≤ x(t; x_{0}) ≤ β(t), t ∈ [0, T], we find that h(α(t)) ≤ h(x(t; x_{0})) ≤ h(β(t)), t ∈ [0, T]. In particular, for t = \frac{T}{2} , we have

h(α(\frac{T}{2} )) ≤ h(x(\frac{T}{2} ; x_{0})) ≤ h(β(\frac{T}{2} )).  (11)

Combining (10) and (11), we obtain

γ_{1}x(0; x_{0}) − γ_{2}x(T; x_{0}) = h(x(\frac{T}{2} ; x_{0})).

This shows that 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 the initial condition x(0) = x_{0} satisfies the boundary condition (3). For the sake of contradiction, let us assume that γ_{1}x(0; x_{0}) − γ_{2}x(T; x_{0}) ≠ h(β(\frac{T}{2} )), where x(t; x_{0}) is the solution of (1)-(2). Letting x_{0} = β(0) together with the relation α(t) ≤ x(t; x_{0}) ≤ β(t), we obtain

γ_{1}x(0; x_{0}) − γ_{2}x(T; x_{0}) = γ_{1}β(0) − γ_{2}x(T; x_{0})
≥ γ_{1}β(0) − γ_{2}β(T) ≥ h(β(\frac{T}{2} )) ≥ h(x(\frac{T}{2} ; x_{0})),

which, according to the above assumption, reduces to

γ_{1}x(0; x_{0}) − γ_{2}x(T; x_{0}) > h(x(\frac{T}{2} ; x_{0})).

Then there exists a number δ satisfying 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 (1)-(2) satisfies the inequality

γ_{1}x(0; x_{0}) − γ_{2}x(T; x_{0}) > h(x(\frac{T}{2} ; x_{0})).  (12)

We can indeed assume that for every natural number n there exists a point ν_{n} satisfying 0 ≤ β(0) − ν_{n} < \frac{1}{n} such that the corresponding solution x^{(n)}(t; ν_{n}) of (1)-(2) with the initial condition x(0) = ν_{n} satisfies the inequality

γ_{1}x^{(n)}(0; ν_{n}) − γ_{2}x^{(n)}(T; ν_{n}) < h(x(\frac{T}{2} ; ν_{n})).

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

γ_{1}x(0) − γ_{2}x(T) ≤ h(x(\frac{T}{2} )),  (13)

which contradicts the inequality (12) and consequently our assumption is false. Now, we define

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 (12) }.

Select 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

γ_{1}x^{(n)}(0; x_{n}) − γ_{2}x^{(n)}(T; x_{n}) < h(x(\frac{T}{2} ; x_{n})).

Thus, 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) satisfying α(t) ≤ x^{∗}(t) ≤ β(t), is a solution of(1)-(2) with the initial condition x(0) = β(0) − δ^{∗} and satisfies the inequality γ_{1}x^{∗}(0) − γ_{2}x^{∗}(T) ≤ h(x(\frac{T}{2} )). This 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 (1)-(2) satisfies (3). Thus, the function x(t, x_{0}) is a solution of (1)-(3). This completes the proof of the theorem.

Theorem 3. Let g, η ∈ P C([0, T], R) and γ_{1}, γ_{2}, ζ, b_{k}, δ_{k}(k = 1, 2, ..., p) be constants such that [γ_{1}/γ_{2} − (Π_{k=1}^{p}b_{k}) exp(∫_{0}^{T} g(m)dm)] ≠ 0. Then the following linear BVP

x′(t) = g(t)x(t) + η(t), for t ∈ [0, T], t ≠ τ_{k},
x(τ_{k} + 0) = b_{k}x(τ_{k}) + δ_{k}, , k = 1, 2, ..., p,
γ_{1}x(0) − γ_{2}x(T) = ζ,

has a unique solution u(t) on the interval [0, T] given by

u(t) = u(0)( ∏_{0<τ_{k}<t} b_{k}) exp(∫_{0}^{t} g(m)dm) + ∑_{0<τ_{k}<t} δ_{k}( ∏_{τ_{k}<τ_{j}<t} b_{j}) exp(∫_{τ_{k}}^{t} g(m)dm)
+ ∫_{0}^{t} η(s)( ∏_{s<τ_{k}<t} b_{k}) exp(∫_{s}^{t} g(m)dm)ds,

where

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

u(0) = [γ_{1}/γ_{2} − ( ∏_{k=1}^{p} b_{k}) exp(∫_{0}^{T} g(m)dm)]^{−1}{∑_{i=1}^{p} δ_{i}( ∏_{j=i+1}^{p} b_{j}) exp(∫_{τ_{i}}^{T} g(m)dm)
+ ∫_{0}^{T} η(s)( ∏_{s<τ_{j}<T} b_{j}) exp(∫_{s}^{T} g(m)dm)ds + ζ/γ_{2}}.

We need the following known theorem (Theorem 1.4.1, page 32 [21]) to prove our main result.

Theorem 4. Let the function m ∈ P C^{1}[R_{+}, R] be such that

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

where σ, q ∈ C[R_{+}, R], d_{k} ≥ 0 and b_{k} are constants. Then

m(t) = m(0)( ∏_{0<τ_{k}<t} d_{k}) exp(∫_{0}^{t} σ(ξ)dξ) + ∑_{0<τ_{k}<t}( ∏_{τ_{k}<τ_{j}<t} d_{j}) exp(∫_{τ_{k}}^{t} σ(ξ)dξ)b_{k}
+ ∫_{0}^{t}( ∏_{s<τ_{k}<t} d_{k}) exp(∫_{s}^{t} σ(ξ)dξ)q(s)ds, t ≥ 0.

3 Extended Method of Quasilinearization

Theorem 5. Assume that

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

(A_{2}) f_{x}, f_{xx} exist, are continuous and (f(t, x)+M_{1}x^{2})_{xx} ≥ 0 for (t, x) ∈ Ω, M_{1} > 0. For some ϵ > 0, M_{2} > 0, [g(t, x)+M_{2}x^{1+ϵ}] satisfies a nondecreasing condition. Further, g_{x} satisfies Lipschitz condition and

{[g_{x}(t, x) + (1 + ϵ)M_{2}x^{ϵ}] − [g_{x}(t, y) + (1 + ϵ)M_{2}y^{ϵ}]}(x − y) ≥ 0, ϵ > 0.

Moreover, ∫_{0}^{T} [F_{x}(s, β_{0}(s)) − 2M_{1}α_{0}(s)]ds < 0.

(A_{3}) For k = 1, 2, ..., p, the functions I_{k} ∈ C^{2}(D_{k}(α_{0}, β_{0}), R) and there exists 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.

(A_{4}) h(x), h′(x), h′′(x) exist, are continuous on R with 0 ≤ h′ and h′′(x)+ψ′′(x) ≤ 0 for some continuous function ψ(x) satisfying ψ′′ ≤ 0 on R.

Then there exist monotone sequences {α_{n}(t)}_{0}^{∞} and {β_{n}(t)}_{0}^{∞} of lower and upper solutions respectively that converge uniformly and quadratically on the intervals (τ_{k}, τ_{k+1}] for k = 1, 2, ..., p to the unique solution of the BVP (1)-(3) in S(α_{0}, β0).

Proof. For (t, x_{1}), (t, x_{2}) ∈ Ω(α_{0}, β_{0}) with x_{1} ≥ x_{2}, it follows from (A_{2}) that

f(t, x_{1}) ≥ f(t, x_{2}) + (f_{x}(t, x_{2}) + 2M_{1}x_{2})(x_{1} − x_{2}) − M_{1}(x_{1}^{2} − x_{2}^{2}),  (14)
g(t, x_{1}) ≥ g(t, x_{2}) + (g_{x}(t, x_{2}) + (1 + ϵ)M_{2}x_{2}^{ϵ})(x_{1} − x_{2}) − M_{2}(x_{1}^{1+ϵ} − x_{2}^{1+ϵ}). (15)

Define

Q(t, x_{1}, x_{2}) = f(t, x_{2}) + (f_{x}(t, x_{2}) + 2M_{1}x_{2})(x_{1} − x_{2}) − M_{1}(x_{1}^{2} − x_{2}^{2})
+ g(t, x_{2}) + [g_{x}(t, x_{2}) + (1 + ϵ)M_{2}x_{2}^{ϵ}](x_{1} − x_{2}) − M_{2}(x_{1}^{1+ϵ} − x_{2}^{1+ϵ}),

and observe that

f(t, x_{1}) + g(t, x_{1}) ≥ Q(t, x_{1}, x_{2}),  f(t, x_{1}) + g(t, x_{1}) = Q(t, x_{1}, x_{1}). (16)

Furthermore, for α_{0}(t) ≤ y ≤ x ≤ β_{0}(t), we have

[f(t, x) + g(t, x)] − [f(t, y) + g(t, y)] ≤ L(x − y), L > 0,  (17)
Q(t, x, x_{2}) − Q(t, y, x_{2}) ≤ N(x − y), N > 0.  (18)

From (A_{3}), we obtain

I_{k}(x_{1}) ≥ I_{k}(x_{2}) + G′_{k}(x_{2})(x_{1} − x_{2}) + J_{k}(x_{2}) − J_{k}(x_{1}),  (19)
G_{k}(x_{1}) ≥ G_{k}(x_{2}) + G′_{k}(x_{2})(x_{1} − x_{2}),  (20)

where x_{1}, x_{2} ∈ D_{k}(α_{0}, β_{0}) with x_{1} ≥ x_{2}. Since

I′_{k}(x) = G′_{k}(x) − J′_{k}(x) = G′_{k}(α_{0}τ_{k}) − J′_{k}(β_{0}τ_{k}) ≥ 0,

it follows that the functions I_{k}(x) are nondecreasing for k = 1, 2, ..., p. Now, we define H : R → R by H(x) = h(x) + ψ(x). Using the mean value theorem and (A_{4}), we obtain

h(x) ≤ h(y)+H′(y)(x−y)+ψ(y)−ψ(x) = E(x, y), h(x) = E(x, x), x, y ∈ R. (21)

Hence, by Theorem 2, the BVP (1)-(3) has a solution in S(α_{0}, β_{0}). We set

Q(t, x, α_{0}) = f(t, α_{0}) + [f_{x}(t, α_{0}) + 2M_{1}α_{0}](x − α_{0}) − M_{1}(x^{2} − α_{0}^{2})
+ g(t, α_{0}) + [g_{x}(t, α_{0}) + (1 + ϵ)M_{2}α_{0}^{ϵ}](x − α_{0}) − M_{2}(x^{1+ϵ} − α_{0}^{1+ϵ}),
Q̂(t, x, β_{0}) = f(t, β_{0}) + [f_{x}(t, α_{0}) + 2M_{1}α_{0}](x − β_{0}) − M_{1}(x^{2} − β_{0}^{2})
+ g(t, β_{0}) + [g_{x}(t, α_{0}) + (1 + ϵ)M_{2}α_{0}^{ϵ}](x − β_{0}) − M_{2}(x^{1+ϵ} − β_{0}^{1+ϵ}),
C_{k}(x(τ_{k}), α_{0}(τ_{k})) = I_{k}(α_{0}(τ_{k})) + B_{k}^{0}(x(τ_{k}) − α_{0}(τ_{k})),
B_{k}^{0} = G′_{k}(α_{0}(τ_{k})) − J′_{k}(β_{0}(τ_{k})),
Ĉ_{k}(x(τ_{k}), β_{0}(τ_{k})) = I_{k}(β_{0}(τ_{k})) + B_{k}^{0}(x(τ_{k}) − β_{0}(τ_{k})),
E(x(\frac{T}{2} ); α_{0}, β_{0}) = h(α_{0}(\frac{T}{2} )) + H′(β_{0}(\frac{T}{2} ))(x(\frac{T}{2} ) − α_{0}(\frac{T}{2} )) + ψ(α_{0}(\frac{T}{2} )) − ψ(x(\frac{T}{2} )),
e(x(\frac{T}{2} ); β_{0}) = h(β_{0}(\frac{T}{2} )) + H′(β_{0}(\frac{T}{2} ))(x(\frac{T}{2} ) − β_{0}(\frac{T}{2} )) + ψ(β_{0}(\frac{T}{2} )) − ψ(x(\frac{T}{2} )).

Obviously

Q(t, α_{0}, α_{0}) = f(t, α_{0})+g(t, α_{0}) = F (t, α_{0}), Q̂(t, β_{0}, β_{0}) = f(t, β_{0})+g(t, β_{0}) = F (t, β_{0}),

C_{k}(α_{0}(τ_{k}), α_{0}(τ_{k})) = I_{k}(α_{0}(τ_{k})), Ĉ_{k}(β_{0}(τ_{k}), β_{0}(τ_{k})) = I_{k}(β_{0}(τ_{k})),

E(α_{0}(\frac{T}{2} ); α_{0}, β_{0}) = h(α_{0}(\frac{T}{2} )), e(β_{0}(\frac{T}{2} ); β_{0}) = h(β_{0}(\frac{T}{2} )).

Now, we consider the following three-point impulsive boundary value problem

x′ = Q(t, x, α_{0}), for t ∈ [0, T], t ≠ τ_{k},  (22)
x(τ_{k} + 0) = C_{k}(x(τ_{k}), α_{0}(τ_{k})),  (23)
γ_{1}x(0) − γ_{2}x(T) = E(x(\frac{T}{2} ); α_{0}, β_{0}),  (24)

and show that α_{0} and β_{0} are its lower and upper solutions respectively. From (A_{1}) and (14)-(15), we obtain

α′_{0} ≤ f(t, α_{0}) + g(t, α_{0}) = Q(t, α_{0}, α_{0}),

α_{0}(τ_{k} + 0) ≤ I_{k}(α_{0}(τ_{k})) = C_{k}(α_{0}(τ_{k}), α_{0}(τ_{k})),

γ_{1}α_{0}(0) − γ_{2}α_{0}(T) ≤ h(α_{0}(\frac{T}{2} )) = E(α_{0}(\frac{T}{2} ); α_{0}, β_{0}),

which implies that α_{0} is a lower solution of the BVP (22)-(24) and

β′_{0}(t) ≥ F (t, β_{0}) = f(t, β_{0}) + g(t, β_{0})
≥ f(t, α_{0}) + [f_{x}(t, α_{0}) + 2M_{1}α_{0}](β_{0} − α_{0}) − M_{1}(β_{0}^{2} − α_{0}^{2})
+ g(t, α_{0}) + [g_{x}(t, α_{0}) + (1 + ϵ)M_{2}α_{0}^{ϵ}](β_{0} − α_{0}) − M_{2}(β_{0}^{1+ϵ} − α_{0}^{1+ϵ})
= Q(t, β_{0}, α_{0}).

Using (A_{1}), (19) and the nondecreasing property of J′_{k}, we get

β_{0}(τ_{k} + 0) ≥ I_{k}(β_{0}(τ_{k}))
≥ I_{k}(α_{0}(τ_{k})) + G′_{k}(α_{0}(τ_{k}))(β_{0}(τ_{k}) − α_{0}(τ_{k})) + J_{k}(α_{0}(τ_{k})) − J_{k}(β_{0}(τ_{k}))
= I_{k}(α_{0}(τ_{k})) + G′_{k}(α_{0}(τ_{k}))(β_{0}(τ_{k}) − α_{0}(τ_{k})) − J′_{k}(η_{0})(β_{0}(τ_{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})) = C_{k}(β_{0}(τ_{k}), α_{0}(τ_{k})),

where α_{0}(τ_{k}) ≤ η_{0} ≤ β_{0}(τ_{k}). In view of (A_{1}) and (A_{4}), for α_{0}(\frac{T}{2} ) ≤ c_{0} ≤ β_{0}(\frac{T}{2} ), we find that

h(β_{0}(\frac{T}{2} )) − E(β_{0}(\frac{T}{2} ); α_{0}, β_{0})

= h(β_{0}(\frac{T}{2} )) − h(α_{0}(\frac{T}{2} )) − H′(β_{0}(\frac{T}{2} ))(β_{0}(\frac{T}{2} ) − α_{0}(\frac{T}{2} ))

− ψ(α_{0}(\frac{T}{2} )) + ψ(β_{0}(\frac{T}{2} ))
= H(β_{0}(\frac{T}{2} )) − H(α_{0}(\frac{T}{2} )) − H′(β_{0}(\frac{T}{2} ))(β_{0}(\frac{T}{2} ) − α_{0}(\frac{T}{2} ))
= H′(c_{0})(β_{0}(\frac{T}{2} ) − α_{0}(\frac{T}{2} )) − H′(β_{0}(\frac{T}{2} ))(β_{0}(\frac{T}{2} ) − α_{0}(\frac{T}{2} ))
= (H′(c_{0}) − H′(β_{0}(\frac{T}{2} )))(β_{0}(\frac{T}{2} ) − α_{0}(\frac{T}{2} )) ≥ 0.

Thus, γ_{1}β_{0}(0) − γ_{2}β_{0}(T) ≥ β_{0}(\frac{T}{2} ) ≥ E(β_{0}(\frac{T}{2} ); α_{0}, β_{0}). Hence β_{0}(t) is an upper solution of the BVP (22)-(24). Then, by Theorem 2, there exists a unique solution α_{1}(t) ∈ S(α_{0}, β_{0}) of the BVP (22)-(24) such that α_{0}(t) ≤ α_{1}(t) ≤ β_{0}(t), t ∈ [0, T].

Next, consider the problem

x′ = Q̂(t, x, β_{0}), for t ∈ [0, T], t ≠ τ_{k},  (25)
x(τ_{k} + 0) = Ĉ_{k}(x(τ_{k}), β_{0}(τ_{k})),  (26)
γ_{1}x(0) − γ_{2}x(T) = e(x(\frac{T}{2} ); β_{0}).  (27)

From (A_{1}), it follows that

β′_{0}(t) ≥ f(t, β_{0}) + g(t, β_{0}) = Q̂(t, β_{0}, β_{0}),

β_{0}(τ_{k} + 0) ≥ I_{k}(β_{0}(τ_{k})) = Ĉ_{k}(β_{0}(τ_{k}), β_{0}(τ_{k})),

γ_{1}β_{0}(0) − γ_{2}β_{0}(T) ≥ h(β_{0}(\frac{T}{2} )) = e(β_{0}(\frac{T}{2} ); β_{0}),

which implies that β_{0} is an upper solution (25)-(27). Using (A_{1}) and (14)-(15) again, we obtain

α′_{0}(t) ≤ f(t, α_{0}) + g(t, α_{0})
≤ f(t, β_{0}) + g(t, β_{0}) − (f_{x}(t, α_{0}) + 2M_{1}α_{0})(β_{0} − α_{0}) + M_{1}(β_{0}^{2} − α_{0}^{2})
− (g_{x}(t, α_{0}) + (1 + ϵ)M_{2}α_{0}^{ϵ})(β_{0} − α_{0}) + M_{2}(β_{0}^{1+ϵ} − α_{0}^{1+ϵ})
= f(t, β_{0}) + (f_{x}(t, α_{0}) + 2M_{1}α_{0})(α_{0} − β_{0}) − M_{1}(α_{0}^{2} − β_{0}^{2})
+ g(t, β_{0}) + (g_{x}(t, α_{0}) + (1 + ϵ)M_{2}α_{0}^{ϵ})(α_{0} − β_{0}) − M_{2}(α_{0}^{1+ϵ} − β_{0}^{1+ϵ})
= Q̂(t, α_{0}, β_{0}).

In a similar manner, it can be shown that

α_{0}(τ_{k} + 0) ≤ Ĉ_{k}(α_{0}(τ_{k}), β_{0}(τ_{k})).

γ_{1}α_{0}(0) − γ_{2}α_{0}(T) ≤ e(α_{0}(\frac{T}{2} ), β_{0}).

Hence α_{0} is a lower solution of the BVP (25)-(27). Again, by Theorem 2, there exists a unique solution β_{1}(t) ∈ S(α_{0}, β_{0}) of the BVP (25)-(27) such that α_{0}(t) ≤ β_{1}(t) ≤ β_{0}(t), t ∈ [0, T]. Now, we show that α_{1}(t) ≤ β_{1}(t), for t ∈ [0, T]. For that, we will prove that α_{1}(t) and β_{1}(t) are lower and upper solution of the BVP (1)-(3) respectively. Using the fact that α_{1} is a solution of (22)-(24) together with (14)-(15), we obtain

α′_{1}(t) = Q(t, α_{1}, α_{0})
= f(t, α_{0}) + [f_{x}(t, α_{0}) + 2M_{1}α_{0}](α_{1} − α_{0}) − M_{1}(α_{1}^{2} − α_{0}^{2})
+ g(t, α_{0}) + [g_{x}(t, α_{0}) + (1 + ϵ)M_{2}α_{0}^{ϵ}](α_{1} − α_{0}) − M_{2}(α_{1}^{1+ϵ} − α_{0}^{1+ϵ})
≤ f(t, α_{1}) + g(t, α_{1}) − [f_{x}(t, α_{0}) + 2M_{1}α_{0}](α_{1} − α_{0}) + M_{1}(α_{1}^{2} − α_{0}^{2})
− [g_{x}(t, α_{0}) + (1 + ϵ)M_{2}α_{0}^{ϵ}](α_{1} − α_{0}) + M_{2}(α_{1}^{1+ϵ} − α_{0}^{1+ϵ})
+ [f_{x}(t, α_{0}) + 2M_{1}α_{0}](α_{1} − α_{0}) − M_{1}(α_{1}^{2} − α_{0}^{2})
+ [g_{x}(t, α_{0}) + (1 + ϵ)M_{2}α_{0}^{ϵ}](α_{1} − α_{0}) − M_{2}(α_{1}^{1+ϵ} − α_{0}^{1+ϵ})
= f(t, α_{1}) + g(t, α_{1}) = F (t, α_{1}(t)), t ∈ [0, T], t ≠ τ_{k}.

In view of (19) and the nonincreasing property of J′_{k}, we have

α_{1}(τ_{k} + 0) = I_{k}(α_{0}(τ_{k})) + B_{k}^{0}[α_{1}(τ_{k}) − α_{0}(τ_{k})]
≤ I_{k}(α_{1}(τ_{k})) − G′_{k}(α_{0}(τ_{k}))(α_{1}(τ_{k}) − α_{0}(τ_{k})) − J_{k}(α_{0}(τ_{k})) + J_{k}(α_{1}(τ_{k}))
+ B_{k}^{0}[α_{1}(τ_{k}) − α_{0}(τ_{k})]
= I_{k}(α_{1}(τ_{k})) + [G′_{k}(α_{0}(τ_{k})) − J′_{k}(η_{1}) − B_{k}^{0}](α_{0}(τ_{k}) − α_{1}(τ_{k}))
≤ I_{k}(α_{1}(τ_{k})) + [G′_{k}(α_{0}(τ_{k})) − J′_{k}(β_{0}(τ_{k})) − G′_{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})),

where α_{0} ≤ η_{1} ≤ α_{1} ≤ β_{0}. Utilizing the nonincreasing property of H′ together with (21) yields

Mα_{1}(0) − Nα_{1}(T)
= h(α_{0}(\frac{T}{2} )) + H′(β_{0}(\frac{T}{2} ))(α_{1}(\frac{T}{2} ) − α_{0}(\frac{T}{2} )) + ψ(α_{0}(\frac{T}{2} )) − ψ(α_{1}(\frac{T}{2} ))
≤ h(α_{1}(\frac{T}{2} )) + H′(α_{1}(\frac{T}{2} ))(α_{0}(\frac{T}{2} ) − α_{1}(\frac{T}{2} )) + ψ(α_{1}(\frac{T}{2} )) − ψ(α_{0}(\frac{T}{2} ))
+ H′(α_{1}(\frac{T}{2} ))(α_{1}(\frac{T}{2} ) − α_{0}(\frac{T}{2} )) + ψ(α_{0}(\frac{T}{2} )) − ψ(α_{1}(\frac{T}{2} ))
= h(α_{1}(\frac{T}{2} )).

Thus, α_{1}(t) is a lower solution of the BVP (1)-(3). Similarly, we can show that β_{1}(t) is an upper solution of (1)-(3). Thus, by Theorem 1, α_{1}(t) ≤ β_{1}(t) and consequently, we get

α_{0}(t) ≤ α_{1}(t) ≤ β_{1}(t) ≤ β_{0}(t), t ∈ [0, T].

Continuing this process, by induction, one can construct monotone sequences {α_{n}(t)}_{0}^{∞} and {β_{n}(t)}_{0}^{∞}, α_{n}, β_{n} ∈ S(α_{n−1}, β_{n−1}) such that

α_{0}(t) ≤ α_{1}(t) ≤ ... ≤ α_{n}(t) ≤ β_{n}(t) ≤ ... ≤ β_{1}(t) ≤ β_{0}(t), t ∈ [0, T],

where α_{n+1}(t) is the unique solution of the BVP

x′ = Q(t, x, α_{n}), for t ∈ [0, T], t ≠ τ_{k},  (28)
x(τ_{k} + 0) = C_{k}(x(τ_{k}), α_{n}(τ_{k})),  (29)
γ_{1}x(0) − γ_{2}x(T) = E(x(\frac{T}{2} ); α_{n}, β_{n}),  (30)

and β_{n+1}(t) is the unique solution of

x′ = Q̂(t, x, β_{n}), for t ∈ [0, T], t ≠ τ_{k},  (31)
x(τ_{k} + 0) = Ĉ_{k}(x(τ_{k}), β_{n}(τ_{k})),  (32)
γ_{1}x(0) − γ_{2}x(T) = e(x(\frac{T}{2} ); β_{n}).  (33)

Since the sequences {α_{n}(t)}_{0}^{∞} and {β_{n}(t)}_{0}^{∞} are uniformly bounded and equicontinuous on (τ_{k}, τ_{k+1}], k = 0, 1, ..., p, it follows that they are uniformly convergent [23] with

lim_{n→∞} α_{n}(t) = x(t), lim_{n→∞} β_{n}(t) = y(t).

Hence we conclude that

α_{0}(t) ≤ x(t) ≤ y(t) ≤ β_{0}(t).

Taking the limit n → ∞, we find that

Q(t, α_{n+1}, α_{n}) → f(t, x(t)) + g(t, x(t)), C_{k}(α_{n+1}(τ_{k}), α_{n}(τ_{k})) → I_{k}(x(t_{k})),

E(α_{n+1}(\frac{T}{2} ); α_{n}, β_{n}) → h(x(\frac{T}{2} )).

Now applying Theorem 3 to the BVP (28)-(30) together with Lebesgue dominated convergence theorem, it follows that x(t) is the solution of the BVP (1)-(3) in S(α_{0}, β_{0}). Similarly, applying Theorem 3 to the BVP (31)-(32), it can be shown that y(t) is the solution of the BVP (1)-(3) in S(α_{0}, β_{0}). Therefore, by the uniqueness of the solution, x(t) = y(t). Now, we prove that the convergence of each of the two sequences is quadratic. For that, we set a_{n+1}(t) = x(t) − α_{n+1}(t), b_{n+1}(t) = β_{n+1}(t) − x(t), t ∈ [0, T] and note that a_{n+1}(t) ≥ 0 and b_{n+1}(t) ≥ 0. We will only prove the quadratic convergence of the sequence {a_{n}(t)}_{0}^{∞} as that of the sequence {b_{n}(t)}_{0}^{∞} is similar one. Setting P (t, x) = f(t, x) + g(t, x) + M_{1}x^{2} + M_{2}x^{1+ϵ}, t ∈ [0, T], t ≠ τ_{k} and using the mean value theorem repeatedly, we obtain

a′_{n+1}(t) = x′(t) − α′_{n+1}(t)
= f(t, x) + g(t, x) − Q(t, α_{n+1}, α_{n})
= P (t, x) − P (t, α_{n}) − P_{x}(t, α_{n})(a_{n}(t) − a_{n+1}(t)) − M_{1}(x^{2} − α_{n+1}^{2})

− M_{2}(x^{1+ϵ} − α_{n+1}^{1+ϵ})
= P_{x}(t, c_{1})a_{n}(t) − P_{x}(t, α_{n})a_{n}(t) + P_{x}(t, α_{n})a_{n+1}(t)
− M_{1}(x^{2} − α_{n+1}^{2}) − M_{2}(x^{1+ϵ} − α_{n+1}^{1+ϵ})
= [P_{x}(t, c_{1}) − P_{x}(t, α_{n})]a_{n}(t)
+ [P_{x}(t, α_{n}) − M_{1}(x + α_{n+1}) − M_{2}σ(x, α_{n+1})]a_{n+1}(t)
= [f_{x}(t, c_{1}) + g_{x}(t, c_{1}) + 2M_{1}c_{1} + (1 + ϵ)M_{2}c_{1}^{ϵ}
− f_{x}(t, α_{n}) − g_{x}(t, α_{n}) − 2M_{1}α_{n} − (1 + ϵ)M_{2}α_{n}^{ϵ}]a_{n}(t)
+ [P_{x}(t, α_{n}) − M_{1}(x + α_{n+1}) − M_{2}σ(x, α_{n+1})]a_{n+1}(t)
= [f_{xx}(t, c_{2})(c_{1} − α_{n}) + g_{x}(t, c_{1}) − g_{x}(t, α_{n}) + (1 + ϵ)M_{2}(c_{1}^{ϵ} − α_{n}^{ϵ})
+ 2M_{1}(c_{1} − α_{n})]a_{n}(t) + [P_{x}(t, α_{n}) − M_{1}(x + α_{n+1}) − M_{2}σ(x, α_{n+1})]a_{n+1}(t)
≤ Q_{n}(t)a_{n+1}(t) + ρ_{n},  (34)

where L_{1} is Lipschitz constant (g_{x} satisfies the Lipschitz condition), α_{n} ≤ c_{1} ≤ c_{2} ≤ x ≤ β_{n} and

Q_{n} = P_{x}(t, α_{n}) − M_{1}(x + α_{n+1}) − M_{2}ω(x, α_{n+1}),
ρ_{n} = [f_{xx}(t, c_{2}) + L_{1} + (1 + ϵ)M_{2}ω(c_{1}, α_{n}) + 2M_{1}]a_{n}^{2}(t),
ω(x, α_{n+1}) = (x^{ϵ} + x^{ϵ−1}α_{n+1} + x^{ϵ−2}α_{n+1}^{2} + ... + x^{1}α_{n+1}^{ϵ−1} + α_{n}^{ϵ}) > 0.

Similarly it can be shown that

a_{n+1}(τ_{k} + 0) ≤ B_{k}^{n}a_{n+1}(τ_{k}) + σ_{k},  (35)

where σ_{k} = [G′′_{k}(ω_{k}) + \frac{3}{2}J′′_{k}(χ_{k})]a_{n}^{2}(τ_{k}) + \frac{1}{2}J′′_{k}(χ_{k})b_{n}^{2}(τ_{k}), α_{n}(τ_{k}) ≤ ω_{k} ≤ x(τ_{k}) and α_{n}(τ_{k}) ≤ χ_{k} ≤ β_{n}(τ_{k}), k = 1, 2, ..., p. Applying Theorem 1.4.1 (page 32 [21]) on (34)-(35), 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} ( ∏_{τ_{k}<τ_{j}<t} B_{j}^{n} exp (∫_{τ_{k}}^{t} Q_{n}(τ)dτ )) σ_{k}
+ ∫_{0}^{t} ∏_{s<τ_{k}<t} B_{k}^{n} exp (∫_{s}^{t} Q_{n}(τ)dτ ) ρ_{n}(s)ds, t ≥ t_{0}.  (36)

In view of (21), we have

γ_{1}a_{n+1}(0) − γ_{2}a_{n+1}(T)
= [γ_{1}x(0) − γ_{2}x(T)] − [γ_{1}α_{n+1}(0) − γ_{2}α_{n+1}(T)]
= h(x(\frac{T}{2} )) − h(α_{n}(\frac{T}{2} )) − H′(β_{n}(\frac{T}{2} ))(α_{n+1}(\frac{T}{2} ) − α_{n}(\frac{T}{2} ))

− ψ(α_{n}(\frac{T}{2} )) + ψ(α_{n+1}(\frac{T}{2} ))
≤ h(α_{n}(\frac{T}{2} )) + H′(α_{n}(\frac{T}{2} ))(x(\frac{T}{2} ) − α_{n}(\frac{T}{2} )) + ψ(α_{n}(\frac{T}{2} )) − ψ(x(\frac{T}{2} ))
− h(α_{n}(\frac{T}{2} )) − H′(β_{n}(\frac{T}{2} ))(α_{n+1}(\frac{T}{2} ) − α_{n}(\frac{T}{2} ))
− ψ(α_{n}(\frac{T}{2} )) + ψ(α_{n+1}(\frac{T}{2} ))
= H′(α_{n}(\frac{T}{2} ))a_{n}(\frac{T}{2} ) − H′(β_{n}(\frac{T}{2} ))a_{n}(\frac{T}{2} ) + H′(β_{n}(\frac{T}{2} ))a_{n+1}(\frac{T}{2} )
− ψ′(c_{3})a_{n+1}(\frac{T}{2} )
≤ −H′′(c_{4})(β_{n}(\frac{T}{2} ) − α_{n}(\frac{T}{2} ))a_{n}(\frac{T}{2} ) + H′(β_{n}(\frac{T}{2} ))a_{n+1}(\frac{T}{2} )
− ψ′(β_{n}(\frac{T}{2} ))a_{n+1}(\frac{T}{2} )
= −H′′(c_{4})(b_{n}(\frac{T}{2} ) + a_{n}(\frac{T}{2} ))a_{n}(\frac{T}{2} ) + h′(β_{n}(\frac{T}{2} ))a_{n+1}(\frac{T}{2} )
≤ −H′′(c_{4})(\frac{3}{2}a_{n}^{2}(\frac{T}{2} ) + \frac{1}{2}b_{n}^{2}(\frac{T}{2} )) + h′(β_{n}(\frac{T}{2} ))a_{n+1}(\frac{T}{2} )

where α_{n+1}(\frac{T}{2} ) ≤ c_{3} ≤ x(\frac{T}{2} ) ≤ β_{n}(\frac{T}{2} ) and x(\frac{T}{2} ) ≤ c_{4} ≤ β_{n}(\frac{T}{2} ). Thus, we have

a_{n+1}(0) ≤ \frac{γ_{2}}{γ_{1}} a_{n+1}(T) + \frac{1}{γ_{1}} [−H′′(c_{4})(\frac{3}{2}a_{n}^{2}(\frac{T}{2} ) + \frac{1}{2}b_{n}^{2}(\frac{T}{2} )) + h′(β_{n}(\frac{T}{2} ))a_{n+1}(\frac{T}{2} )]. (37)

Combining (36) and (37) yields

a_{n+1}(0) ≤ \frac{γ_{2}}{γ_{1}} a_{n+1}(T) + \frac{1}{γ_{1}} (−H′′(c_{4})b_{n}(\frac{T}{2} )a_{n}(\frac{T}{2} ) + h′(β_{n}(\frac{T}{2} ))a_{n+1}(\frac{T}{2} ))
≤ \frac{γ_{2}}{γ_{1}} [a_{n+1}(0) ( ∏_{0<τ_{k}<T} B_{k}^{n}) exp (∫_{0}^{T} Q_{n}(τ)dτ )

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

+ ∫_{0}^{T} ∏_{s<τ_{k}<T} B_{k}^{n} exp (∫_{s}^{T} Q_{n}(τ)dτ ) ρ_{n}(s)ds]
− \frac{1}{γ_{1}} H′′(c_{4})(\frac{3}{2}a_{n}^{2}(\frac{T}{2} ) + \frac{1}{2}b_{n}^{2}(\frac{T}{2} ))

+ \frac{1}{γ_{1}} h′(β_{n}(\frac{T}{2} ))[a_{n+1}(0) ( ∏_{0<τ_{k}< \frac{T}{2}} B_{k}^{n}) exp (∫0^{\frac{T}{2}} Q_{n}(τ)dτ )

+ ∑_{0<τ_{k}< \frac{T}{2}} ( ∏_{τ_{k}<τ_{j}< \frac{T}{2}} B_{j}^{n} exp (∫τ_{k}^{\frac{T}{2}} Q_{n}(τ)dτ )) σ_{k}

+ ∫0^{\frac{T}{2}} ∏_{s<τ_{k}< \frac{T}{2}} B_{k}^{n} exp (∫s^{\frac{T}{2}} Q_{n}(τ)dτ ) ρ_{n}(s)ds].

Solving for a_{n+1}(0), we get

a_{n+1}(0) ≤ [1 − \frac{γ_{2}}{γ_{1}} ( ∏_{0<τ_{k}<T} B_{k}^{n}) exp (∫_{0}^{T} Q_{n}(τ)dτ )

− \frac{1}{γ_{1}} h′(β_{n}(\frac{T}{2} )) ( ∏_{0<τ_{k}< \frac{T}{2}} B_{k}^{n}) exp (∫0^{\frac{T}{2}} Q_{n}(τ)dτ )]^{−1}

× \{\frac{γ_{2}}{γ_{1}} [ ∑_{0<τ_{k}<T} ( ∏_{τ_{k}<τ_{j}<T} B_{j}^{n} exp (∫_{τ_{k}}^{T} Q_{n}(τ)dτ )) σ_{k}

+ ∫_{0}^{T} ∏_{s<τ_{k}<T} B_{k}^{n} exp (∫_{s}^{T} Q_{n}(τ)dτ ) ρ_{n}(s)ds]

+ \frac{1}{γ_{1}} h′(β_{n}(\frac{T}{2} ))[ ∑_{0<τ_{k}< \frac{T}{2}} ( ∏_{τ_{k}<τ_{j}< \frac{T}{2}} B_{j}^{n} exp (∫τ_{k}^{\frac{T}{2}} Q_{n}(τ)dτ )) σ_{k}

+ ∫0^{\frac{T}{2}} ∏_{s<τ_{k}< \frac{T}{2}} B_{k}^{n} exp (∫s^{\frac{T}{2}} Q_{n}(τ)dτ ) ρ_{n}(s)ds]

− \frac{1}{γ_{1}} H′′(c_{4})(\frac{3}{2}a_{n}^{2}(\frac{T}{2} ) + \frac{1}{2}b_{n}^{2}(\frac{T}{2} ))\}.  (38)

Substituting (38) into (36) yields

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

× \{\frac{γ_{2}}{γ_{1}} [ ∑_{0<τ_{k}<T} ( ∏_{τ_{k}<τ_{j}<T} B_{j}^{n} exp (∫_{τ_{k}}^{T} Q_{n}(τ)dτ )) (λa_{n}^{2}(τ_{k}) + \frac{1}{2}δ_{2}b_{n}^{2}(τ_{k}))

+ ∫_{0}^{T} ∏_{s<τ_{k}<T} B_{k}^{n} exp (∫_{s}^{T} Q_{n}(τ)dτ ) [(δ_{3} + δ_{4})a_{n}^{2}(s)]ds

+ \frac{1}{γ_{1}} δ_{5}(\frac{T}{2} ))[ ∑_{0<τ_{k}< \frac{T}{2}} ( ∏_{τ_{k}<τ_{j}< \frac{T}{2}} B_{j}^{n} exp (∫τ_{k}^{\frac{T}{2}} Q_{n}(τ)dτ )) (λa_{n}^{2}(τ_{k}) + \frac{1}{2}δ_{2}b_{n}^{2}(τ_{k}))

+ ∫0^{\frac{T}{2}} ∏_{s<τ_{k}< \frac{T}{2}} B_{k}^{n} exp (∫s^{\frac{T}{2}} Q_{n}(τ)dτ ) [(δ_{3} + δ_{4})a_{n}^{2}(s)]ds

+ \frac{1}{γ_{1}} δ_{6}(\frac{3}{2}a_{n}^{2}(\frac{T}{2} ) + \frac{1}{2}b_{n}^{2}(\frac{T}{2} ))\}
+ ∑_{0<τ_{k}<t} ( ∏_{τ_{k}<τ_{j}<t} B_{j}^{n} exp (∫_{τ_{k}}^{t} Q_{n}(τ)dτ )) (λa_{n}^{2}(τ_{k}) + \frac{1}{2}δ_{2}b_{n}^{2}(τ_{k}))

+ ∫_{0}^{t} ∏_{s<τ_{k}<t} B_{k}^{n} exp (∫_{s}^{t} Q_{n}(τ)dτ ) [(δ_{3} + δ_{4})a_{n}^{2}(s)]ds

where |G′′_{k}| ≤ δ_{1}, |J′′_{k}| ≤ δ_{2}, |f_{xx}| ≤ δ_{3}, |L_{1} + (1 + ϵ)M_{2}ω(c_{1}, α_{n}) + 2M_{1}| ≤ δ_{4}, |h′| ≤ δ_{5}, |H′′| ≤ δ_{6}, λ = δ_{1} + \frac{3}{2}δ_{2} and

Φ = 1−\frac{γ_{2}}{γ_{1}} ∏_{0<τ_{k}<T} B_{k}^{n} exp (∫_{0}^{T} Q_{n}(τ)dτ )−\frac{1}{γ_{1}} h′(β_{n}(\frac{T}{2} )) ∏_{0<τ_{k}< \frac{T}{2}} B_{k}^{n} exp (∫0^{\frac{T}{2}} Q_{n}(τ)dτ ) .

Taking the maximum on [0, T], it follows that there exist positive constants η_{1} and η_{2} such that

∥a_{n+1}(t)∥ ≤ η_{1}∥a_{n}∥^{2} + η_{2}∥b_{n}∥^{2}.

On the same pattern, it can be proved that

∥b_{n+1}(t)∥ ≤ ζ_{1}∥b_{n}∥^{2} + ζ_{2}∥a_{n}∥^{2},

where ζ_{1} and ζ_{2} are positive constants. This establishes the quadratic convergence of the sequences.

Example. The impulsive BVP

x′(t) = \frac{1}{15} ln((x(t))^{3} + 1) + \frac{1}{300}((x(t)) + t)^{5} for t ∈ [0, 1], t ≠ \frac{1}{3},

x(\frac{1}{3} + 0) = x(\frac{1}{3}),

γ_{1}x(0) − γ_{2}x(1) = x(\frac{1}{2}), γ_{2} ≤ \frac{1}{6}γ_{1} − \frac{3}{4},

admits the minimal solution α_{0}(t) = 0, t ∈ [0, 1] and the maximal solution β_{0}(t) given by

β_{0}(t) = \{ \begin{matrix}t + \frac{1}{3}, & if t ∈ [0, \frac{1}{3}], \\ t + 1, & if t ∈ (\frac{1}{3}, 1]. \end{matrix}

Clearly α_{0}(t) and β_{0}(t) are not the solutions of the BVP and α_{0}(t) ≤ β_{0}(t), t ∈ [0, 1].

4 Concluding Remarks

This paper addresses a quasilinearization method for a nonlinear impulsive first order ordinary differential equation dealing with a nonlinear function F (t, x(t)) which is a sum of two functions of different nature together with a nonlinear three-point boundary condition in contrast to a problem containing a single function and a linear boundary condition considered in [23]. The condition on g(t, x(t)) in assumption (A_{3}) of Theorem 3 is motivated by the well known fact that χ(t) = t^{p} is convex for p > 1. The following results can be recorded as a special case of this problem:

(i) If we take g(t, x(t)) = 0, h(x(\frac{T}{2} )) = c (constant), we obtain the generalized quasilinearization for first order impulsive differential equations with linear boundary conditions [23].

(ii) By taking h(x(\frac{T}{2} )) = u_{0}, γ_{1} = 1, γ_{2} = 0, we can record the results of usual initial value problems with impulse. In reference [28], the authors have developed an extension of generalized quasilinearization for initial value problems without impulse. Thus our problem generalizes the results of [28] in the sense that impulsive effects have been taken into account along with a three-point nonlinear boundary condition.

(iii) The extension of generalized quasilinearization technique for periodic boundary value problems involving impulsive differential equations follows if we take γ_{1} = 1, γ_{2} = 1 and h(x(\frac{T}{2} )) = 0.

Acknowledgement. The authors thank the reviewer for his/her valuable suggestions and comments for the improvement of the original manuscript.

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(Received September 19, 2006)