Electronic Journal of Qualitative Theory of Differential Equations Spec. Ed. I, 2009 No. 17, 1–13; http://www.math.u-szeged.hu/ejqtde/

QUASILINEARIZATION METHOD AND NONLOCAL

SINGULAR THREE POINT BOUNDARY VALUE PROBLEMS

Rahmat Ali Khan^{1, 2}

1 Department of Mathematics, University of Dayton, Dayton, Ohio 45469-2316 USA 2 Centre for Advanced Mathematics and Physics, National University of Sciences and Technology(NUST), Campus of College of Electrical and Mechanical Engineering,

Peshawar Road, Rawalpindi, Pakistan e-mail: rahmat¯ alipk@yahoo.com

Honoring the Career of John Graef on the Occasion of His Sixty-Seventh Birthday

Abstract

The method of upper and lower solutions and quasilinearization for nonlinear singular equations of the type

−x′′(t) + λx′(t) = f(t, x(t)), t ∈ (0, 1),

subject to nonlocal three-point boundary conditions

x(0) = δx(η), x(1) = 0, 0 < η < 1,

are developed. Existence of a C^{1} positive solution is established. A monotone sequence of solutions of linear problems converging uniformly and rapidly to a solution of the nonlinear problem is obtained.

Key words and phrases: Nonlinear singular equation, nonlocal three-point conditions, quasilinearization, rapid convergence. 2 Permanent address. Acknowledgement: Department of Mathematics University of Dayton for Hospitality. Research is supported by HEC, Pakistan, 2-3(50)/PDFP/HEC/2008/1.

AMS (MOS) Subject Classifications: 34A45, 34B15

1 Introduction

Nonlocal singular boundary value problems (BVPs) have various applications in chemical engineering, underground water flow and population dynamics. These problems arise in many areas of applied mathematics such as gas dynamics, Newtonian fluid mechanics, the theory of shallow membrane caps, the theory of boundary layer and so on; see for example, [2, 7, 12, 13, 16] and the references therein. An excellent resource with an extensive bibliography was produced by Agarwal and O’Regan [1]. Existence theory for nonlinear multi-point singular boundary value problems has attracted the attention of many researchers; see for example, [3, 4, 5, 14, 15, 17, 18] and the references therein.

In this paper, we study existence and approximation of C^{1}-positive solution of a nonlinear forced Duffing equation with three-point boundary conditions of the type

\begin{matrix}− & x′′(t) + λx′(t) = f(t, x(t)), t ∈ (0, 1), \\ x(0) = δx(η), x(1) = 0, 0 < η < 1, 0 < δ < \frac{e^{λ} − 1}{e^{λ} − e^{λη}} , \end{matrix}  (1)

where the nonlinearity f : (0, 1) × ℝ \ {0} → ℝ is continuous and may be singular at x = 0, t = 0 and/or t = 1. By singularity we mean the function f(t, x) is allowed to be unbounded at x = 0, t = 0 and/or t = 1 and by a C^{1}-positive solution x we mean that x ∈ C[0, 1] ∩ C^{2}(0, 1) satisfies (1), x(t) > 0 for t ∈ (0, 1) and both x′(0+) and x′(1−) exist.

For the existence theory, we develop the method of upper and lower solutions and to approximate the solution of the BVP (1), we develop the quasilinearization technique [5, 8, 9, 10, 11]. We obtain a monotone sequence of solutions of linear problems and show that, under suitable conditions on f, the sequence converges uniformly and quadratically to a solution of the original nonlinear problem (1).

2 Some basic results

For u ∈ C[0, 1] we write ∥u∥ = max{|u(t)| : t ∈ [0, 1]}. For any λ ∈ ℝ \ {0}, consider the singular boundary value problem

\begin{matrix}− x′′(t) + λx′(t) = f(t, x(t)), t ∈ (0, 1), \\ x(0) = δx(η), x(1) = 0, 0 < η < 1, 0 < δ < \frac{e^{λ} − 1}{e^{λ} − e^{λη}} . \end{matrix}  (2)

We seek a solution x via the singular integral equation

x(t) = ∫_{0}^{1} G(t, s)f(s, x(s))ds + \frac{(e^{λ} − e^{λt})δ}{(e^{λ} − 1) − δ(e^{λ} − e^{λη})} ∫_{0}^{1} G(η, s)f(s, x(s))ds, (3)

where

G(t, s) = \frac{1}{λe^{λs}(e^{λ} − 1)} \begin{cases}(e^{λt} − 1)(e^{λ} − e^{λs}), & 0 < t < s < 1, \\ (e^{λs} − 1)(e^{λ} − e^{λt}), & 0 < s < t < 1, \end{cases}

is the Green’s function corresponding to the homogeneous two-point BVP

\begin{matrix}− x′′(t) + λx′(t) = 0, t ∈ (0, 1), \\ x(0) = 0, x(1) = 0. \end{matrix}

EJQTDE Spec. Ed. I, 2009 No. 17

Clearly, G(t, s) > 0 on (0, 1) × (0, 1). From (3), x ≥ 0 on [0, 1] provided f ≥ 0. Hence for a positive solution we assume f ≥ 0 on [0, 1] × ℝ.

We recall the concept of upper and lower solutions for the BVP (2).

Definition 2.1. A function α is called a lower solution of the BVP (2) if α ∈ C[0, 1]∩ C^{2}(0, 1) and satisfies

\begin{matrix}−α′′(t) + λα′(t) ≤ f(t, α(t)), t ∈ (0, 1), \\ α(0) ≤ δα(η), α(1) ≤ 0. \end{matrix}

An upper solution β ∈ C[0, 1]∩C^{2}(0, 1) of the BVP (2) is defined similarly by reversing the inequalities.

Choose b > η, a finite positive number, such that δ < \frac{e^{λb}−1}{e^{λb}−e^{λη}}. Since the homogeneous linear problem

\begin{matrix}− x′′(t) + λx′(t) = 0, t ∈ [0, b], \\ x(0) = 0, x(b) = 0, \end{matrix}

has only the trivial solution, hence, for any σ ∈ C[0, b] and ρ, τ ∈ ℝ, the corresponding nonhomogeneous linear three point problem

\begin{matrix}− x′′(t) + λx′(t) = σ(t), t ∈ [0, b], \\ x(0) − δx(η) = τ, x(b) = ρ, 0 < η < b, 0 < δ < \frac{e^{λb} − 1}{e^{λb} − e^{λη}} , \end{matrix}  (4)

has a unique solution

x(t) = ∫_{0}^{b} G_{b}(t, s)σ(s)ds + \frac{(e^{λb} − e^{λt})}{D}\{δ ∫_{0}^{b} G_{b}(η, s)σ(s)ds + τ\} + \frac{ρψ(t)}{D}, (5)

where ψ(t) = (e^{λt} − 1) + δ(e^{λη} − e^{λt}), D = (e^{λb} − 1) − δ(e^{λb} − e^{λη}) and

G_{b}(t, s) = \frac{1}{λe^{λs}(e^{λb} − 1)} \begin{cases}(e^{λt} − 1)(e^{λb} − e^{λs}), & 0 ≤ t ≤ s ≤ b, \\ (e^{λs} − 1)(e^{λb} − e^{λt}), & 0 ≤ s ≤ t ≤ b. \end{cases}

We note that ψ(t) ≥ 0 on [0, b] and if τ ≥ 0, ρ ≥ 0 and σ ≥ 0 on [0, b], then x ≥ 0 on [0, b]. Thus, we have the following comparison result (maximum principle): Maximum Principle: Let δ, η ∈ ℝ such that 0 < δ < \frac{e^{λb}−1}{e^{λb}−e^{λη}} and 0 < η < b. For any x ∈ C^{1}[0, b] such that

−x′′(t) + λx′(t) ≥ 0, t ∈ (0, b), x(0) − δx(η) ≥ 0 and x(b) ≥ 0,

we have x(t) ≥ 0, t ∈ [0, b].

In the following theorem, we prove existence of a C^{1}[0, 1] positive solution of the singular BVP (2). We generate a sequence of C^{1}[0, 1] positive solutions of nonsingular problems that has a convergent subsequence converging to a solution of the original problem.

Theorem 2.1. Assume that there exist lower and upper solutions α, β ∈ C[0, 1] ∩ C^{2}(0, 1) of the BVP (2) such that α(1) = β(1), and 0 < α ≤ β on [0, 1), and α(0) − δα(η) < β(0) − δβ(η). Assume that f : (0, 1) × ℝ \ {0} → (0, ∞) is continuous and there exists h(t) such that e^{−λt}h(t) ∈ L^{1}[0, 1] and

|f(t, x)| ≤ h(t) if x ∈ [ᾱ, β̄],  (6)

where ᾱ = min{α(t) : t ∈ [0, 1]} = 0 and β̄ = max{β(t) : t ∈ [0, 1]}. Then the BVP (2) has a C^{1}[0, 1] positive solution x such that α(t) ≤ x(t) ≤ β(t), t ∈ [0, 1].

Proof. Let {a_{n}}, {b_{n}} be two monotone sequences satisfying

0 < ⋯ < a_{n} < ⋯ < a_{1} < η < b_{1} < ⋯ < b_{n} < ⋯ < 1

and are such that {a_{n}} converges to 0, {b_{n}} converges to 1. Clearly, ∪_{n=1}^{∞}[a_{n}, b_{n}] = (0, 1). Let α(a_{n}) − δα(η) ≤ β(a_{n}) − δβ(η) for sufficiently large n, and choose two null sequences {τ_{n}} and {ρ_{n}} [that is, {τ_{n}} and {ρ_{n}} both converge to 0] such that

\begin{matrix}α(a_{n}) − δα(η) ≤ τ_{n} ≤ β(a_{n}) − δβ(η), \\ α(b_{n}) ≤ ρ_{n} ≤ β(b_{n}), n = 1, 2, 3, …. \end{matrix}  (7)

Define a partial order in C[0, 1] ∩ C^{2}(0, 1) by x ≤ y if and only if x(t) ≤ y(t), t ∈ [0, 1]. Define a modification F of f with respect to α, β as follows:

F (t, x) = \begin{cases}f(t, β(t)) + \frac{x−β(t)}{1+|x−β(t)|}, & if x ≥ β(t), \\ f(t, x(t)), & if α(t) ≤ x ≤ β(t), \\ f(t, α(t)) + \frac{α(t)−x}{1+|α(t)−x|}, & if x ≤ α(t). \end{cases}  (8)

Clearly, F is continuous and bounded on (0, 1) × C[0, 1]. For each n ∈ ℕ, consider the nonsingular modified problems

\begin{matrix}− x′′(t) + λx′(t) = F (t, x), t ∈ [a_{n}, b_{n}], \\ x(a_{n}) − δx(η) = τ_{n}, x(b_{n}) = ρ_{n}. \end{matrix}  (9)

We write the BVP (9) as an equivalent integral equation

\begin{matrix}x(t) = ∫_{a_{n}}^{b_{n}} G_{n}(t, s)F (s, x)ds + \frac{(e^{λb_{n}} − e^{λt})}{D_{n}}\{δ ∫_{a_{n}}^{b_{n}} G_{n}(η, s)F (s, x)ds + τ_{n}\} \\ +\frac{ρ_{n}ψ_{n}(t)}{D_{n}}, t ∈ [a_{n}, b_{n}], \end{matrix} (10)

where D_{n} = (e^{λb_{n}} − e^{λa_{n}}) − δ(e^{λb_{n}} − e^{λη}), ψ_{n}(t) = (e^{λt} − e^{λa_{n}}) + δ(e^{λη} − e^{λt}) and

G_{n}(t, s) = \frac{1}{λe^{λs}(e^{λb_{n}} − e^{λa_{n}})} \begin{cases}(e^{λt} − e^{λa_{n}})(e^{λb_{n}} − e^{λs}), & a_{n} ≤ t ≤ s ≤ b_{n}, \\ (e^{λs} − e^{λa_{n}})(e^{λb_{n}} − e^{λt}), & a_{n} ≤ s ≤ t ≤ b_{n}. \end{cases}

EJQTDE Spec. Ed. I, 2009 No. 17

Clearly, G_{n}(t, s) → G(t, s) as n → ∞. By a solution of (10), we mean a solution of the operator equation

(I − T_{n})x = 0, that is, a fixed point of T_{n},

where I is the identity and for each x ∈ C[a_{n}, b_{n}], the operator T_{n} : C[a_{n}, b_{n}] → C[a_{n}, b_{n}] is defined by

\begin{matrix}T_{n}(x)(t) = ∫_{a_{n}}^{b_{n}} G_{n}(t, s)F (s, x)ds + \frac{(e^{λb_{n}} − e^{λt})}{D_{n}}\{δ ∫_{a_{n}}^{b_{n}} G_{n}(η, s)F (s, x)ds + τ_{n}\} \\ +\frac{ρ_{n}ψ_{n}(t)}{D_{n}}, t ∈ [a_{n}, b_{n}]. \end{matrix} (11)

Since F is continuous and bounded on [a_{n}, b_{n}] × C[a_{n}, b_{n}] for each n ∈ ℕ, hence T_{n} is compact for each n ∈ ℕ. By Schauder’s fixed point theorem, T_{n} has a fixed point (say) x_{n} ∈ C[a_{n}, b_{n}] for each n ∈ ℕ.

Now, we show that

α ≤ x_{n} ≤ β on [a_{n}, b_{n}], n ∈ ℕ

and consequently, x_{n} is a solution of the BVP

\begin{matrix}− x′′(t) + λx′(t) = f(t, x(t)), t ∈ [a_{n}, b_{n}], \\ x(a_{n}) − δx(η) = τ_{n}, x(b_{n}) = ρ_{n}. \end{matrix}  (12)

Firstly, we show that α ≤ x_{n} on [a_{n}, b_{n}], n ∈ ℕ. Assume that α ≰ x_{n} on [a_{n}, b_{n}]. Set z(t) = x_{n}(t) − α(t), t ∈ [a_{n}, b_{n}], then

z ∈ C^{1}[a_{n}, b_{n}] and z ≱ 0 on [a_{n}, b_{n}].  (13)

Hence, z has a negative minimum at some point t_{0} ∈ [a_{n}, b_{n}]. From the boundary conditions, it follows that

\begin{matrix}z(a_{n}) & − δz(η) = [x_{n}(a_{n}) − δx_{n}(η)] − [α(a_{n}) − δα(η)] ≥ τ_{n} − τ_{n} = 0, \\ z(b_{n}) = x_{n}(b_{n}) − α(b_{n}) ≥ ρ_{n} − ρ_{n} ≥ 0. \end{matrix}  (14)

Hence, t_{0} ≠ b_{n}. If t_{0} ≠ a_{n}, then

z(t_{0}) < 0, z′(t_{0}) = 0, z′′(t_{0}) ≥ 0.

However, in view of the definition of F and that of lower solution, we obtain

−z′′(t_{0}) = −z′′(t_{0}) + λz′(t_{0}) ≥ −\frac{z(t_{0})}{1 + |z(t_{0})|} > 0,

a contradiction. Hence z has no negative local minimum.

If t_{0} = a_{n}, then z(a_{n}) < 0 and z′(a_{n}) ≥ 0. From the boundary condition (14), we have z(η) ≤ \frac{1}{δ}z(a_{n}) < 0. Let [t_{1}, t_{2}] be the maximal interval containing η such that z(t) ≤ 0, t ∈ [t_{1}, t_{2}]. Clearly, t_{1} ≥ a_{n}, t_{2} ≤ b_{n} and z(t_{1}) ≥ z(a_{n}) ≥ δz(η). Further, for t ∈ [t_{1}, t_{2}], we have

−z′′(t) + λz′(t) ≥ f(t, α(t)) − \frac{z(t)}{1 + |z(t)|} − f(t, α(t)) > 0.

Hence, by comparison result, z > 0 on [t_{1}, t_{2}], again a contradiction. Thus, α ≤ x_{n} on [a_{n}, b_{n}].

Similarly, we can show that x_{n} ≤ β on [a_{n}, b_{n}].

Now, define

u_{n}(t) = \begin{cases}δx_{n}(η) + τ_{n}, if 0 ≤ t ≤ a_{n} \\ x_{n}(t), if a_{n} ≤ t ≤ b_{n} \\ ρ_{n}, if b_{n} ≤ t ≤ 1. \end{cases}

Clearly, u_{n} is continuous extension of x_{n} to [0, 1] and α ≤ u_{n} ≤ β on [a_{n}, b_{n}]. Since,

\begin{matrix}u_{n}(t) = δx_{n}(η) + τ_{n} = x_{n}(a_{n}), t ∈ [0, a_{n}], \\ u_{n}(t) = ρ_{n} = x_{n}(b_{n}), t ∈ [b_{n}, 1]. \end{matrix}

Hence,

α ≤ u_{n} ≤ β on [0, 1], n ∈ ℕ.

Since [a_{1}, b_{1}] ⊂ [a_{n}, b_{n}], for each n there must exist t_{n} ∈ (a_{1}, b_{1}) such that

|u_{n}(t_{n})| ≤ M; |u′_{n}(t_{n})| = |\frac{u_{n}(b_{1}) − u_{n}(a_{1})}{b_{1} − a_{1}}| ≤ N,

where M = max_{t∈[a_{1},b_{1}]}\{|α(t)|, |β(t)|, N = \frac{2M}{b_{1}−a_{1}} . We can assume that

t_{n} → t_{0} ∈ [a_{1}, b_{1}],
u_{n}(t_{n}) → x_{0} ∈ [α(t_{0}), β(t_{0})],
u′_{n}(t_{n}) → x′_{0} ∈ [−N, N], as n → ∞

By standard arguments [6], (also see [1, 3, 14]), there is a C[0, 1] positive solution x(t) of (2) such that α ≤ x ≤ β on [0, 1], x(t_{0}) = x_{0}, x′(t_{0}) = x′_{0} and a subsequence {u_{nj}(t)} of {u_{n}(t)} such that u_{nj}(t), u′_{nj}(t) converges uniformly to x(t), x′(t) respectively, on any compact subinterval of (0, 1).

Now, using (6), we obtain

| − (x′(t)e^{−λt})′| = e^{−λt}|f(t, x(t))| ≤ e^{−λt}h(t) ∈ L^{1}[0, 1],

which implies that x ∈ C^{1}[0, 1].

3 Approximation of solution

We develop the approximation technique (quasilinearization) and show that under suitable conditions on f, there exists a bounded monotone sequence of solutions of linear problems that converges uniformly and quadratically to a solution of the nonlinear original problem. Choose a function Φ(t, x) such that Φ, Φ_{x}, Φ_{xx} ∈ C([0, 1] × ℝ),

Φ_{xx}(t, x) ≥ 0 for every t ∈ [0, 1] and x ∈ [0, β̄]

and

\frac{∂^{2}}{∂x^{2}} [f(t, x) + Φ(t, x)] ≥ 0 on (0, 1) × (0, β̄].  (15)

Here, we do not require the condition that \frac{∂^{2}}{∂x^{2}} f(t, x) ≥ 0 on (0, 1) × (0, β̄]. Define F : (0, 1) × ℝ → ℝ by F (t, x) = f(t, x) + Φ(t, x). Note that F ∈ C((0, 1) × ℝ) and

\frac{∂^{2}}{∂x^{2}} F (t, x) ≥ 0 on (0, 1) × (0, β̄],  (16)

where β̄ = max{β(t) : t ∈ [0, 1]}.

Theorem 3.1. Assume that

(A_{1}) α, β are lower and upper solutions of the BVP (1) satisfying the hypotheses of Theorem 2.1.

(A_{2}) f, f_{x}, f_{xx} ∈ C((0, 1) × ℝ) and there exist h_{1}, h_{2}, h_{3} such that e^{−λt}h_{i} ∈ L^{1}[0, 1]

and

|\frac{∂^{i}}{∂x^{i}} f(t, x)| ≤ h_{i}(t) for |x| ≤ β̄, t ∈ (0, 1), i = 0, 1, 2.

Moreover, f is non-increasing in x for each t ∈ (0, 1).

Then, there exists a monotone sequence {w_{n}} of solutions of linear problems converging uniformly and quadratically to a unique solution of the BVP (2).

Proof. The conditions (A_{1}) and (A_{2}) ensure the existence of a C^{1} positive solution x of the BVP (2) such that

α(t) ≤ x(t) ≤ β(t), t ∈ [0, 1].

For t ∈ (0, 1), using (16), we obtain

f(t, x) ≥ f(t, y) + F_{x}(t, y)(x − y) − [Φ(t, x) − Φ(t, y)],  (17)

where x, y ∈ (0, β̄]. The mean value theorem and the fact that Φ_{x} is increasing in x on [0, β̄] for each t ∈ [0, 1], yields

Φ(t, x) − Φ(t, y) = Φ_{x}(t, c)(x − y) ≤ Φ_{x}(t, β̄)(x − y) for x ≥ y,  (18)

where x, y ∈ [0, β̄] such that y ≤ c ≤ x. Substituting in (17), we have

f(t, x) ≥ f(t, y) + [F_{x}(t, y) − Φ_{x}(t, β̄)](x − y), for x ≥ y  (19)

on (0, 1) × (0, β̄]. Define g : (0, 1) × ℝ × ℝ \ {0} → ℝ by

g(t, x, y) = f(t, y) + [F_{x}(t, y) − Φ_{x}(t, β̄)](x − y).  (20)

We note that g(t, x, y) is continuous on (0, 1) × ℝ × ℝ \ {0}. Moreover, for every t ∈ (0, 1) and x, y ∈ (0, β̄], g satisfies the following relations

g_{x}(t, x, y) = F_{x}(t, y) − Φ_{x}(t, β̄) ≤ F_{x}(t, y) − Φ_{x}(t, y) = f_{x}(t, y) ≤ 0 and

\begin{cases}f(t, x) ≥ g(t, x, y), for x ≥ y, \\ f(t, x) = g(t, x, x). \end{cases}  (21)

Moreover, for every t ∈ (0, 1) and x, y ∈ (0, β̄], using mean value theorem, we have

g(t, x, y) = f(t, y) + f_{x}(t, y)(x − y) − Φ_{xx}(t, c)(β̄ − y)(x − y),

where y < c < β̄. Consequently, in view of (A_{2}), we obtain

|g(t, x, y)| ≤ |f(t, y)| + |f_{x}(t, y)||(x − y)| + |Φ_{xx}(t, c)||β̄ − y||x − y|

≤ h_{1}(t) + h_{2}(t)β̄ + M = H(t) (say), for every t ∈ (0, 1) and x, y ∈ (0, β̄], (22)

where M = max{|Φ_{xx}(t, c)||β̄ − y||x − y| : t ∈ [0, 1], x, y ∈ [0, β̄]}. Hence

e^{−λt}H(t) = e^{−λt}h_{1}(t) + e^{−λt}h_{2}(t)β̄ + e^{−λt}M ∈ L^{1}[0, 1].

Now, we develop the iterative scheme to approximate the solution. As an initial approximation, we choose w_{0} = α and consider the linear problem

\begin{aligned}−x′′ & (t) + λx′(t) = g(t, x(t), w_{0}(t)), t ∈ (0, 1) \\ & x(0) = δx(η), x(1) = 0. \end{aligned}  (23)

Using (21) and the definition of lower and upper solutions, we get

\begin{matrix}g(t, w_{0}(t) & , w_{0}(t)) = f(t, w_{0}(t)) ≥ −w′′_{0}(t) + λw′_{0}(t), t ∈ (0, 1), \\ w_{0}(0) ≤ δ(w_{0}(η)), w_{0}(1) ≤ 0, \end{matrix}

\begin{matrix}g(t, β(t) & , w_{0}(t)) ≤ f(t, β(t)) ≤ −β′′(t) + λβ′(t), t ∈ (0, 1), \\ β(0) ≥ δβ(η), β(1) ≥ 0, \end{matrix}

which imply that w_{0} and β are lower and upper solutions of (23) respectively. Hence by Theorem 2.1, there exists a C^{1} positive solution w_{1} ∈ C[0, 1] ∩ C^{2}(0, 1) of (23) such that

w_{0} ≤ w_{1} ≤ β on [0, 1].

Using (21) and the fact that w_{1} is a solution of (23), we obtain

\begin{matrix}−w′′_{1}(t) & + λw′_{1}(t) = g(t, w_{1}(t), w_{0}(t)) ≤ f(t, w_{1}(t)), t ∈ (0, 1) \\ w_{1}(0) = δw_{1}(η), w_{1}(1) = 0, \end{matrix}  (24)

which implies that w_{1} is a lower solution of (2). Similarly, in view of (A_{1}), (21) and (24), we can show that w_{1} and β are lower and upper solutions of

\begin{aligned}−x′′ & (t) + λx′(t) = g(t, x(t), w_{1}(t)), t ∈ (0, 1) \\ & x(0) = δx(η), x(1) = 0. \end{aligned}  (25)

Hence by Theorem 2.1, there exists a C^{1} positive solution w_{1} ∈ C[0, 1] ∩ C^{2}(0, 1) of

(25) such that

w_{1} ≤ w_{2} ≤ β on [0, 1].

Continuing in the above fashion, we obtain a bounded monotone sequence {w_{n}} of C^{1}[0, 1] positive solutions of the linear problems satisfying

w_{0} ≤ w_{1} ≤ w_{2} ≤ w_{3} ≤ ... ≤ w_{n} ≤ β on [0, 1],  (26)

where the element w_{n} of the sequence is a solution of the linear problem

−x′′(t) + λx′(t) = g(t, x(t), w_{n−1}(t)), t ∈ (0, 1)
x(0) = δx(η), x(1) = 0

and for each t ∈ (0, 1), is given by

\begin{aligned}w_{n}(t) & = ∫_{0}^{1} G(t, s)g(s, w_{n}(s), w_{n−1}(s))ds+ \\ & \frac{(e^{λ} − e^{λt})δ}{(e^{λ} − 1) − δ(e^{λ} − e^{λη})} ∫_{0}^{1} G(η, s)g(s, w_{n}(s), w_{n−1}(s))ds. \end{aligned}  (27)

The monotonicity and uniform boundedness of the sequence {w_{n}} implies the existence of a pointwise limit w on [0, 1]. From the boundary conditions, we have

0 = w_{n}(0) − δw_{n}(η) → w(0) − δw(η) and 0 = w_{n}(1) → w(1).

Hence w satisfy the boundary conditions. Moreover, from (22), the sequence {g(t, w_{n}, w_{n−1})} is uniformly bounded by h_{3}(t) ∈ L^{1}[0, 1] on (0, 1). Hence, the continuity of the function g on (0, 1) × (0, β̄] × (0, β̄] and the uniform boundedness of the sequence {g(t, w_{n}, w_{n−1})} implies that the sequence {g(t, w_{n}, w_{n−1})} converges pointwise to the function g(t, w, w) = f(t, w). By Lebesgue dominated convergence theorem, for any t ∈ (0, 1),

∫_{0}^{1} G(t, s)g(s, w_{n}(s), w_{n−1}(s))ds → ∫_{0}^{1} G(t, s)f(s, w(s))ds.

Passing to the limit as n → ∞, we obtain

\begin{matrix}w(t) = ∫_{0}^{1} G(t, s)g(s, w(s), w(s))ds + \frac{(e^{λ} − e^{λt})δ}{(e^{λ} − 1) − δ(e^{λ} − e^{λη})} ∫_{0}^{1} G(η, s)g(s, w(s), w(s))ds \\ = ∫_{0}^{1} G(t, s)f(s, w(s))ds + \frac{(e^{λ} − e^{λt})δ}{(e^{λ} − 1) − δ(e^{λ} − e^{λη})} ∫_{0}^{1} G(η, s)f(s, w(s))ds, t ∈ (0, 1); \end{matrix}

that is, w is a solution of (2).

Now, we show that the convergence is quadratic. Set v_{n}(t) = w(t)−w_{n}(t), t ∈ [0, 1], where w is a solution of (2). Then, v_{n}(t) ≥ 0 on [0, 1] and the boundary conditions imply that v_{n}(0) = δv_{n}(η) and v_{n}(1) = 0. Now, in view of the definitions of F and g, we obtain

\begin{aligned}−v′′_{n} & (t) + λv′_{n}(t) = f(t, w(t)) − g(t, w_{n}(t), w_{n−1}(t)) \\ & = [F (t, w(t)) − Φ(t, w(t))] \\ & − [f(t, w_{n−1}(t)) + (F_{x}(t, w_{n−1}(t)) − Φ_{x}(t, β̄))(w_{n}(t) − w_{n−1}(t))] \\ & = [F (t, w(t)) − F (t, w_{n−1}(t)) − F_{x}(t, w_{n−1}(t))(w_{n}(t) − w_{n−1}(t))] \\ & − [Φ(t, w(t)) − Φ(t, w_{n−1}(t)) − Φ_{x}(t, β̄))(w_{n}(t) − w_{n−1}(t))], t ∈ (0, 1). \end{aligned} (28)

Using the mean value theorem repeatedly and the fact that Φ_{xx} ≥ 0 on [0, 1] × [0, β̄], we obtain, Φ(t, w(t)) − Φ(t, w_{n−1}(t)) ≥ Φ_{x}(t, w_{n−1}(t))(w(t) − w_{n−1}(t)) and

F (t, w(t)) − F (t, w_{n−1}(t)) − F_{x}(t, w_{n−1}(t))(w_{n}(t) − w_{n−1}(t))

= F_{x}(t, w_{n−1}(t))(w(t) − w_{n−1}(t)) + \frac{F_{xx}(t, ξ_{1})}{2}(w(t) − w_{n−1}(t))^{2}

− F_{x}(t, w_{n−1}(t))(w_{n}(t) − w_{n−1}(t))

= F_{x}(t, w_{n−1}(t))(w(t) − w_{n}(t)) + \frac{F_{xx}(t, ξ_{1})}{2}(w(t) − w_{n−1}(t))^{2}

≤ F_{x}(t, w_{n−1}(t))(w(t) − w_{n}(t)) + \frac{F_{xx}(t, ξ_{1})}{2}∥v_{n−1}∥^{2}, t ∈ (0, 1),

where w_{n−1}(t) ≤ ξ_{1} ≤ w(t) and ∥v∥ = max{v(t) : t ∈ [0, 1]}. Hence the equation (28) can be rewritten as

\begin{aligned}−v′′_{n}(t) & + λv′_{n}(t) ≤ F_{x}(t, w_{n−1}(t))(w(t) − w_{n}(t)) + \frac{F_{xx}(t, ξ_{1})}{2}∥v_{n−1}∥^{2} \\ & − Φ_{x}(t, w_{n−1}(t))(w(t) − w_{n−1}(t)) + Φ_{x}(t, β̄))(w_{n}(t) − w_{n−1}(t)) \\ & = f_{x}(t, w_{n−1}(t))(w(t) − w_{n}(t)) + \frac{F_{xx}(t, ξ_{1})}{2}∥v_{n−1}∥^{2} \\ & + [Φ_{x}(t, β̄) − Φ_{x}(t, w_{n−1}(t))](w_{n}(t) − w_{n−1}(t)) \\ & ≤ \frac{F_{xx}(t, ξ_{1})}{2}∥v_{n−1}∥^{2} + Φ_{xx}(t, ξ_{2})(β̄ − w_{n−1}(t))(w_{n}(t) − w_{n−1}(t)) \\ & ≤ \frac{f_{xx}(t, ξ_{1}) + Φ_{xx}(t, ξ_{1})}{2}∥v_{n−1}∥^{2} + Φ_{xx}(t, ξ_{2})(β̄ − w_{n−1}(t))(w(t) − w_{n−1}(t)) \\ & ≤ \frac{f_{xx}(t, ξ_{1})}{2}∥v_{n−1}∥^{2} + d_{1}(\frac{∥v_{n−1}∥^{2}}{2} + |β̄ − w_{n−1}(t)||w(t) − w_{n−1}(t)|), t ∈ (0, 1) \end{aligned} ; (29)

where w_{n−1}(t) ≤ ξ_{2} ≤ w_{n}(t), d_{1} = max{|Φ_{xx}| : (t, x) ∈ [0, 1] × [0, β̄]} and we used the fact that f_{x} ≤ 0 on (0, 1) × (0, β̄]. Choose r > 1 such that

|β(t) − w_{n−1}(t)| ≤ r|w(t) − w_{n−1}(t)| on [0, 1].

We obtain

−v′′_{n}(t) + λv′_{n}(t) ≤ (\frac{f_{xx}(t, ξ_{1})}{2} + d_{1}(r + 1/2))∥v_{n−1}∥^{2} ≤ (\frac{h_{3}(t)}{2} + d_{2})∥v_{n−1}∥^{2}, t ∈ (0, 1),

(30)

where e^{−λt}h_{3}(t) ∈ L^{1}[0, 1] and d_{2} = d_{1}(r + 1/2).

By the comparison result, v_{n}(t) ≤ z(t), t ∈ [0, 1], where z(t) is the unique solution of the linear BVP

\begin{matrix}−z′′(t) & + λz′(t) = (\frac{h_{3}(t)}{2} + d_{2})∥v_{n−1}∥^{2}, \\ z(0) = δz(η), z(1) = 0. \end{matrix}  (31)

Thus,

\begin{aligned}v_{n}(t) ≤ z(t) & = [ ∫_{0}^{1} G(t, s)(\frac{h_{3}(s)}{2} + d_{2})ds+ \\ & \frac{(e^{λ} − e^{λt})δ}{(e^{λ} − 1) − δ(e^{λ} − e^{λη})} ∫_{0}^{1} G(η, s)(\frac{h_{3}(s)}{2} + d_{2})ds]∥v_{n−1}∥^{2} \\ & ≤ A∥v_{n−1}∥^{2}, \end{aligned}  (32)

where A denotes

max_{t∈[0,1]}\{∫_{0}^{1} G(t, s)(\frac{h_{3}(s)}{2} + d_{2})ds + \frac{(e^{λ} − e^{λt})δ}{(e^{λ} − 1) − δ(e^{λ} − e^{λη})} ∫_{0}^{1} G(η, s)(\frac{h_{3}(s)}{2} + d_{2})ds\}.

(32) gives quadratic convergence.

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