# A Second-Order Boundary Value Problem with Nonlinear and Mixed Boundary Conditions: Existence, Uniqueness, and Approximation

Authors: Zheyan Zhou; Jianhe Shen
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

A second-order boundary value problem with nonlinear and mixed two-point boundary conditions is considered,Lx=f(t,x,x′), t∈(a,b), g(x(a),x(b),x′(a),x′(b))=0, x(b)=x(a) in which L is a formally self-adjoint second-order differential operator. Under appropriate assumptions on L, f, and g, existence and uniqueness of solutions is established by the method of upper and lower solutions and Leray-Schauder degree theory. The general quasilinearization method is then applied to this problem. Two monotone sequences converging quadratically to the unique solution are constructed.


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## Body

## 1. Introduction

The investigation of boundary value problems (denoted as BVPs for short) of ordinary differential equations is of great significance. On one hand, it makes a great impact on the studies of partial differential equations [1]. On the other hand, BVPs of ordinary differential equations can be used to describe a large number of mechanical, physical, biological, and chemical phenomena; see [2–5] for example. So far a lot of work has been carried out, including second-order, third-order, and higher-order BVPs with various boundary conditions.As far as we know, for a long term most of works focused on existence and uniqueness of solutions. The works relating to approximation of solutions are relatively rare. In recent years, some approximate methods, such as the shooting method [6], monotone iterative technique [7], homotopy analysis method [8], and general quasilinearization method have been applied to BVPs for obtaining approximations of solutions. Among these methods, the general quasilinearization method becomes more and more popular.The quasilinearization method was originally proposed by Bellman and Kalaba [9]. It is a very powerful approximation technique and unlike perturbation methods, is not dependent on the existence of a small or large parameter. The method, whose sequence of solutions of linear problems convergences to the solution of the original nonlinear problem, is quadratic and monotone, which is one of the reasons for the popularity of this technique. This method was generalized by Lakshmikantham and Vatsala [10] in which the convexity or concavity assumption on the nonlinear functions involved in the problems is relaxed.So far, the general quasilinearization method, coupled with the method of upper and lower solutions, has been applied to obtain approximation of solutions for a large number of nonlinear problems, for example, BVPs of ordinary differential equations, such as first-order BVP with nonlinear boundary condition [11] and second-order BVPs with Dirichlet boundary condition [12], periodic boundary condition [13], three-point boundary condition [14], four-point boundary condition [15], and m-point boundary condition [16]; BVPs of partial differential equations, such as parabolic initial-boundary value problem [17], elliptic problems with nonlinear boundary condition [18] and p-Laplacian equations with nonlinear boundary condition [19], and so forth; BVPs of impulsive differential equations [20] and impulse functional differential equations with anti-periodic boundary condition [21]; BVPs of some practically nonlinear problems, such as Duffing equation involving both integral and nonintegral forcing terms with Robin boundary condition [22] and forced Duffing equation with discontinuous-type integral boundary condition [23]; as well as some abstract problems such as fixed point theorems in ordered Banach space [24].Recently, El-Gebeily and O’Regan [25, 26] consider the singular and nonsingular second-order ordinary differential equations(1.1)l(u(t))=f(t,u(t)),t∈(a,b)
with nonlinear boundary conditions as follows:(1.2)g(u(a),u(b),pu′(a))=0,(1.3)h(u(a),u(b),pu′(b))=0,
in which lu=-(pu′)′+qu is a formally self-adjoint second-order differential operator. This type of BVPs arises in a variety of problems in applied mathematics and physics [27]. By defining the upper and lower solutions of BVP (1.1)–(1.3) suitably, Gebeily and Regan established the existence and uniqueness of solutions and constructed two monotonic iterative sequences converging to the unique solution quadratically. However, it can be noted that the u'(b) and u′(a) terms are not involved in (1.2) and (1.3), respectively. For second-order two-point BVPs, if all the terms u(a),u(b),u′(a) and u′(b) are involved in the boundary conditions, it seems quite difficult to deal with.In this paper, we consider a second-order BVP with nonlinear and mixed two-point boundary conditions as follows:(1.4)Lx=f(t,x,x′),t∈(a,b),(1.5)g(x(a),x(b),x′(a),x′(b))=0,(1.6)x(b)=x(a),
where Lx=-(px')'+qx, in which p∈C1[a,b], p>0, and q∈C[a,b], q≥0.In BVP (1.4)–(1.6), it can be found that the boundary condition (1.5) is dependent on all the x(a),x(b),x′(a), and x′(b) terms. First, existence and uniqueness of solutions of BVP (1.4)–(1.6) is established by combining the method of upper and lower solutions with Leray-Schauder degree theory. Then, the general quasilinearization method is applied to construct the approximations of the unique solution. Two monotone sequences of iterations converging to the unique solution quadratically are obtained.


## 2. Preliminaries

In this section, several definitions and lemmas needed to the main results are given first.Definition 2.1.
β(t),α(t)∈C2[a,b] are called the upper and lower solutions of BVP (1.4)–(1.6), respectively, if
(2.1)Lβ≥f(t,β,β′),t∈(a,b),g(β(a),β(b),β′(a),β′(b))≤0,β(b)=β(a),Lα≤f(t,α,α′),t∈(a,b),g(α(a),α(b),α′(a),α′(b))≥0,α(b)=α(a).Definition 2.2.
LetE be a subset of [a,b]×R2; it is said that the right-hand side function of (1.4) satisfies Nagumo condition on E if
(2.2)|f(t,x,x′)|≤h(|x′|)=O(|x′|2)
holds for (t,x,x′)∈E and |x′|→+∞.Lemma 2.3 (see [28]).
Letf:[a,b]×R2→R be a continuous function satisfying Nagumo  condition on
(2.3)E={(t,x,x′)∈[a,b]×R2:α(t)≤x(t)≤β(t)},
where α,β:[a,b]→R are continuous functions such that α(t)≤β(t) for all t∈[a,b]. Then there exists a constant N>0 such that every solution x(t) of second-order equations x′′=f(t,x,x′) with
(2.4)α(t)≤x(t)≤β(t),t∈[a,b]
satisfies ∥x'∥∞≤N, in which N is called the Nagumo constant.Lemma 2.4.
Boundary value problem as follows:(2.5)Lx=-x,t∈(a,b),(2.6)x(a)=0,x(b)=0
has only the trivial solution.Proof.
Assume thatx0(t) is an arbitrarily nontrivial solution of BVP (2.5)-(2.6). From the boundary conditions (2.6), it can be concluded that x0(t) can achieve its positive maximum or negative minimum in the interior of [a,b], suppose at t0, t0∈(a,b).
Ifx0(t) achieves its positive maximum, then
(2.7)x0(t0)>0,x0′(t0)=0,x0′′(t0)≤0
which means that
(2.8)Lx0(t0)=-p(t0)x0′′(t0)-p′(t0)x0′(t0)+q(t0)x(t0)≥0.
On the other hand, it can be derived from (2.5) that
(2.9)Lx0(t0)=-x0(t0)<0.
It is a contradiction.
Ifx0(t) achieves its negative minimum, similar arguments lead to a contradiction too. Hence, BVP (2.5)-(2.6) has only the trivial solution.Lemma 2.5 (see [26]).
Define a linear operator(2.10)l:C1[a,b]→C[a,b]×R×R
by
(2.11)lx(t)=(px′(a)-px′(t)+∫atq(s)x(s)ds,x(a),x(b)).
Then l-1 exists and is continuous.Lemma 2.6.
Assume that(1)
α(t),β(t) are the lower and upper solutions of BVP (1.4)–(1.6), respectively;(2)
f(t,x,x') is continuous on [a,b]×R2 and is strictly decreasing in x on
(2.12)D={(t,x,x′)∈[a,b]×R2:min{α(t),β(t)}≤x≤max{α(t),β(t)}};(3)
g is continuously differentiable on R4, strictly decreasing in the first and second variables, and nondecreasing and nonincreasing, respectively, in the third and forth variables.
Then,(2.13)α(t)≤β(t),t∈[a,b].Proof.
Suppose thatα(t)>β(t) for some t∈[a,b]. Then there exist some t0∈[a,b] such that
(2.14)maxt∈[a,b][α(t)-β(t)]=α(t0)-β(t0)>0.
Ift0∈(a,b), then α(t0)>β(t0), α'(t0)=β'(t0), and α′′(t0)≤β′′(t0) and consequently,
(2.15)L[α(t0)-β(t0)]=Lα(t0)-Lβ(t0)≥0.
However, it follows from Definition 2.1 and mean value theorem that
(2.16)Lα(t0)-Lβ(t0)≤f(t0,α(t0),α′(t0))-f(t0,β(t0),β′(t0))=fx′(t0,ξ,β′(t0))(α(t0)-β(t0))<0,
in which ξ∈(β(t0),α(t0)), and the last inequality follows from the strictly decreasing property of f in x. It is a contradiction.
Ift0=a, then α(a)>β(a) and
(2.17)α′(a)-β′(a)=limt→a+[α(t)-β(t)]-[α(a)-β(a)]t-a≤0.
By the definitions of the lower and upper solutions, we have
(2.18)α(b)=α(a)>β(a)=β(b).
Moreover,
(2.19)α'(b)-β'(b)=limt→b-(α(t)-β(t))-(α(b)-β(b))t-b=limt→b-(α(t)-β(t))-(α(a)-β(a))t-b≥0.
Consequently, in view of the monotonicity of g in its variables, it follows that
(2.20)0≥g(β(a),β(b),β′(a),β′(b))>g(α(a),α(b),α′(a),α′(b))≥0,
which is a contradiction.
Ift0=b, similar deductions lead to a contradiction too. Hence, Lemma 2.6 is proved.


## 3. Existence and Uniqueness of Solutions

Theorem 3.1.
Assume that(1)
α(t),β(t) are the lower and upper solutions of BVP (1.4)–(1.6), respectively;(2)
f(t,x,x′) is continuous on [a,b]×R2 and is strictly decreasing in x on
(3.1)D={(t,x,x′)∈[a,b]×R2:α(t)≤x(t)≤β(t)}
and satisfies Nagumo  condition on D;(3)
g is continuously differentiable, strictly decreasing in the first and second variables, nondecreasing and nonincreasing, respectively, in the third and forth variables on [α(a),β(a)]×[α(b),β(b)]×R2.Then there exists a unique solutionx(t)∈C2[a.b] of BVP (1.4)–(1.6) such that (3.2)α(t)≤x(t)≤β(t),t∈[a,b].Proof.
Define(3.3)w(t,x)={α(t),x(t)<α(t),x(t),α(t)≤x(t)≤β(t),β(t),x(t)>β(t).
Introduce the following auxiliary BVPs with homotopy character:
(3.4)Lx=λf(t,w(t,x),x′)-x+λw(t,x),x(a)=λ[g(w(a,x(a)),w(b,x(b)),x′(a),x′(b))+w(a,x(a))],x(b)=x(a)
in which λ∈[0,1] is called the embedded parameter.
By the continuity off and g, and by the boundedness of α(t) and β(t), we can select a sufficiently large constant M0>0 such that
(3.5)-M0<α(t)≤β(t)<M0,(3.6)f(t,β(t),0)+β(t)<M0,(3.7)g(β(a),β(b),0,0)+β(a)<M0,(3.8)-M0<f(t,α(t),0)+α(t),(3.9)-M0<g(α(a),α(b),0,0)+α(a).
In what follows, the proof of the existence of solutions is divided into four steps.
Step 1.
Show that, forλ∈[0,1], every solution x(t) of BVPs (3.4) satisfies
(3.10)|x(t)|<M0,t∈[a,b].
Suppose that the estimate|x(t)|<M0 is not true. Then, there must be some points in [a,b] such that either x(t)≥M0 or x(t)≤-M0. For the former case, x(t) has a positive maximum, suppose at t1, that is,
(3.11)maxt∈[a,b]x(t)=x(t1)(≥M0>0)
can be assumed. We have three cases to consider.
Case 1 (t1∈(a,b)).
In this case,x(t1)≥M0>β(t1), x'(t1)=0 and x′′(t1)≤0 which leads to Lx(t1)≥0. On the other hand, since the definition of w(t,x) yields w(t1,x(t1))=β(t1); hence, for λ∈(0,1], we have
(3.12)Lx(t1)=λf(t1,β(t1),0)-x(t1)+λβ(t1)=λ[f(t1,β(t1),0)+β(t1)]-x(t1)≤λ[f(t1,β(t1),0)+β(t1)-M0]<0,
in which the last inequality is obtained by the inequality (3.6). It is a contradiction.
Forλ=0, it can be derived that
(3.13)0≤Lx(t1)=-x(t1)≤-M0<0,
which is also a contradiction.Case 2 (t1=a).
In this case,x(a)≥M0>β(a), x′(a)≤0. Furthermore, we have
(3.14)x(b)=x(a)≥M0>β(b),x'(b)=limt→b-x(t)-x(b)t-b=limt→b-x(t)-x(a)t-b≥0.
The definition of w(t,x) means that w(a,x(a))=β(a) and w(b,x(b))=β(b). Hence, for λ∈(0,1], in view of x′(a)≤0 and x'(b)≥0, it follows from the monotonicity of g in its third and forth variables and the inequality (3.7) that
(3.15)M0≤x(a)=λ[g(w(a,x(a)),w(b,x(b)),x′(a),x′(b))+w(a,x(a))]=λ[β(a)+g(β(a),β(b),x′(a),x′(b))]≤λ[β(a)+g(β(a),β(b),0,0)]<λM0≤M0
which is a contradiction.
Forλ=0, we can obtain from the second equation in (3.4) that
(3.16)0<M0≤x(a)=0,
which is a contradiction too.Case 3 (t1=b).
In this case,x(b)≥M0>β(b), x′(b)≥0, and furthermore,
(3.17)x(a)=x(b)≥M0>β(a),x'(a)=limt→a+x(t)-x(a)t-a=limt→a+x(t)-x(b)t-a≤0.
The same deductions with those in Case 2 yield, for λ∈(0,1], that
(3.18)M0≤x(b)=x(a)≤λ[β(a)+g(β(a),β(b),0,0)]<λM0≤M0.
It is also a contradiction. For λ=0, it can be deduced from the boundary conditions of (3.4) that
(3.19)0<M0≤x(b)=x(a)=0,
which is a contradiction too.Step 2.
Show that there exists a positive constantM1 such that, for λ∈[0,1], every solution x(t) of BVPs (3.4) satisfies
(3.20)|x′(t)|<M1,t∈[a,b].
Let(3.21)E={(t,x,x′)∈[a,b]×R2:|x(t)|<M0}.
Define
(3.22)Fλ(t,x,x′)=1-p[p′x′-qx+λf(t,w(t,x),x′)-x+λw(t,x)],(t,x,x′)∈E.
Consequently, (1.4) can be rewritten as
(3.23)x′′=Fλ(t,x,x′).
Since the function f satisfies Nagumo condition on E,
(3.24)|Fλ(t,x,x′)|=1p|p′x′-qx+λf(t,w(t,x),x′)-x+λw(t,x)|=O(|x′|2),for(t,x,x′)∈E,|x′|→+∞
is obvious if the boundedness of x(t) and w(t,x), p∈C1[a,b], p>0, and q∈C[a,b] is kept in mind. That is, Fλ(t,x,x′) satisfies Nagumo condition on E. Hence, by Lemma 2.3, the estimate |x'(t)|<M1, t∈[a,b], can be obtained.Step 3.
Show that forλ=1, BVP (3.4) has at least one solution x1(t).
Define a nonlinear operator(3.25)Nλ:C1[a,b]→C[a,b]×R×R
by
(3.26)Nλx(t)=(∫at[λf(s,w(s,x),x′)-x+λw(s,x)]ds,Aλ,Bλ)
with
(3.27)Aλ=λ[g(w(a,x(a)),w(b,x(b)),x′(a),x′(b))+w(a,x(a))],Bλ=x(a).
Consequently, BVPs (3.4) are equivalent to the following operator equations:
(3.28)(I-l-1Nλ)x=0,
in which l is the linear operator defined in Lemma 2.5, and I is the unit operator.
Define the norm inC[a,b]×R×R as
(3.29)∥x(t)∥C[a,b]×R×R=∥x(t)∥C[a,b]+|R1|+|R2|
for x(t)∈C[a,b]×R×R, in which R1,R2 are two real numbers.
Let{xn(t)} be the bounded sequence on C1[a,b]. It then follows from Steps 1 and 2 that {Nλxn} and {Nλ′xn} are both uniformly bounded in the sense of the norm defined above. Thus, {Nλxn} is equicontinuous on C[a,b]×R×R. Consequently, Arzela-Ascoli theorem yields that {Nλxn} is compact on C[a,b]×R×R. Therefore, Nλx is a completely continuous operator.
Furthermore, the operator(3.30)Tλ:C1[a,b]→C1[a,b]
defined by
(3.31)Tλ=l-1Nλ
is also a completely continuous operator.
Define a bounded and open domain as follows:(3.32)Ω={x(t)∈C1[a,b]:|x(t)|<M0,|x′(t)|<M1}.
It follows from Steps 1 and 2 that
(3.33)(I-l-1Nλ)∂Ω≠0.
Therefore, the degree Deg(I-l-1Nλ,Ω,0) is well defined. Then the invariance of degree under homotopy yields
(3.34)Deg(I-l-1N1,Ω,0)=Deg(I-l-1N0,Ω,0).
Since the operator equation(I-l-1N0)x=0 is equivalent to  BVP (2.5)-(2.6) which has only the trivial solution, therefore
(3.35)Deg(I-l-1N1,Ω,0)=Deg(I-l-1N0,Ω,0)=±1.
Consequently, the operator equation (I-l-1N1)x=0 gives
(3.36)Lx=f(s,w(s,x),x′)-x+w(s,x),x(a)=g(w(a,x(a)),w(b,x(b)),x′(a),x′(b))+w(a,x(a)),x(b)=x(a),which has at least one solution x1(t)∈C2[a,b].Step 4.
Show that every solutionx(t)∈C2[a,b] of BVP (3.36) satisfies
(3.37)α(t)≤x(t)≤β(t),t∈[a,b].
The right-hand side of this inequality is first proved. For the sake of contradiction, suppose thatx(t)>β(t) for some t∈[a,b]. Define h(t)=x(t)-β(t), then h(t) has a positive maximum at some t1∈[a,b].
Case 1 (t1∈(a,b)).
In this case,h(t1)=x(t1)-β(t1)>0, h'(t1)=x'(t1)-β'(t1)=0, and h′′(t1)≤0 and consequently, Lh(t1)≥0.
On the other hand,(3.38)Lh(t1)=Lx(t1)-Lβ(t1)≤f(t1,w(t1,x(t1)),x′(t1))-x(t1)+w(t,x(t1))-f(t1,β(t1),β′(t1))=f(t1,β(t1),β′(t1))-x(t1)+β(t1)-f(t1,β(t1),β′(t1))=-x(t1)+β(t1)<0,
which is a contradiction.Case 2 (t1=a).
In this case,(3.39)h(a)=x(a)-β(a)>0,h'(a)=x'(a)-β′(a)≤0.
Furthermore, we have
(3.40)β(b)=β(a)<x(a)=x(b),x'(b)-β'(b)=h'(b)=limt→b-h(t)-h(b)t-b=limt→b-h(t)-h(a)t-b≥0.
The definition of w(t,x) yields w(a,x(a))=β(a) and w(b,x(b))=β(b). In view of x′(a)≤β′(a) and x'(b)≥β′(b), it then follows from the monotonicity of g in its third and forth variables that
(3.41)β(a)<x(a)=g(w(a,x(a)),w(b,x(b)),x′(a),x′(b))+w(a,x(a))≤g(β(a),β(b),β′(a),β′(b))+β(a)≤β(a).
It is also a contradiction.Case 3 (t1=b).
In this case,(3.42)h(b)=x(b)-β(b)>0,h′(b)=x′(b)-β′(b)≥0.
Furthermore,
(3.43)β(a)=β(b)<x(b)=x(a),x′(a)-β′(a)=h′(a)=limt→a+h(t)-h(a)t-a=limt→a+h(t)-h(b)t-a≤0.
Similar deductions lead to a contradiction too.
Therefore, the inequalityx(t)≤β(t) holds. In a similar way, α(t)≤x(t) can be proved.
Till now, by Steps1–4, the proof of the existence of solutions is completed. In what follows, we turn to the proof of the uniqueness of solutions.
We may assume thatx1(t) and x2(t) are two arbitrary solutions of BVP (1.4)–(1.6). Define
(3.44)y(t)=x1(t)-x2(t),t∈[a,b].
If y(t)≡0 for t∈[a,b], the uniqueness of solutions is obvious. Otherwise, there must be some points in [a,b] such that either y(t)>0 or y(t)<0. For the former case, we define
(3.45)maxt∈[a,b]y(t):=y(t1).
Similarly, we only need to consider the following three cases.
Case 1 (t1∈(a,b)).
In this case,(3.46)y(t1)=x1(t1)-x2(t1)>0,y′(t1)=x1′(t1)-x2′(t1)=0,y′′(t1)≤0.
Thus, 0≤Ly(t1). On the other hand, the mean value theorem and the strictly decreasing property of f(t,x,x') in x results that
(3.47)Ly(t1)=Lx1(t1)-Lx2(t1)=f(t1,x1(t1),x1′(t1))-f(t1,x2(t1),x2′(t1))=fx′(t1,θ,x1′(t1))(x1(t1)-x2(t1))<0,
which is a contradiction, where θ∈(x2(t1),x1(t1)).Case 2 (t1=a).
In this case,(3.48)y(a)=x1(a)-x2(a)>0,y′(a)=x1′(a)-x2′(a)≤0.
Moreover,
(3.49)y(b)=x1(b)-x2(b)=x1(a)-x2(a)=y(a)>0,x1′(b)-x2′(b)=y′(b)=limt→b-y(t)-y(b)t-b=limt→b-y(t)-y(a)t-b≥0.
Nevertheless, in view of the monotonicity of g in its variables, it follows from the mean value theorem that
(3.50)0=g(x1(a),x1(b),x1′(a),x1′(b))-g(x2(a),x2(b),x2′(a),x2′(b))=g1′[x1(a)-x2(a)]+g2′[x1(b)-x2(b)]+g3′[x1′(a)-x2′(a)]+g4′[x1′(b)-x2′(b)]<0,
which is a contradiction, in which gi′=gi′(ξ1,ξ2,ξ3,ξ4), i=1,2,3,4, ξ1∈(x2(a),x1(a)) and ξ2,ξ3,ξ4 can be located by analogy.Case 3 (t1=b).
For this case, in the same way, it can be proved that this case is also impossible.Consequently, the conclusion in Theorem 3.1 is proved.


## 4. Approximations of the Unique Solution

In this section, a series of boundary value problems whose sequence of solutions converging to the unique solution of  BVP (1.4)–(1.6) is constructed.Theorem 4.1.
Assume that(1)
there exist the upper and lower solutionsβ0(t),α0(t) of BVP (1.4)–(1.6);(2)
f(t,x,x') is continuous on [a,b]×R2 and satisfies Nagumo  condition on
(4.1)D={(t,x,x′)∈[a,b]×R2:α0(t)≤x(t)≤β0(t)};(3)
fxx(t,x,x′) exists and is continuous on [a,b]×R2 and fx(t,x,x′)<0, fxx(t,x,x′)≤0 on D×[-C,C], where
(4.2)C≥max{N,maxt∈[a,b]α0′(t),maxt∈[a,b]β0′(t)}
is chosen sufficiently large, and N is the Nagumo constant;(4)
g(u,v,z,w)∈C1([α0(a),β0(a)]×[α0(b),β0(b)]×R2,R), and it is strictly decreasing in u and v, no-decreasing and nonincreasing, respectively, in z and w.Then there exist a monotone nonincreasing sequence {βn(t)} and a monotone nondecreasing sequence {αn(t)} converging uniformly to the unique solution of BVP (1.4)–(1.6). Moreover, the rate of the convergence is quadratic.Proof.
Define a piecewise functionq(x')=max{-C,min{x′,C}} for x'∈R, and introduce f(t,x,q(x′)) on D×[-C,C] and
(4.3)F(t,x,x′;z)=f(t,z,q(x′))+fx(t,z,q(x′))(x-z)
for (t,x,x′;z)∈D×[-C,C]×[α0(t),β0(t)].
Obviously,F(t,x,x′;z) is linear in x if z is viewed as a known quantity. Moreover, f(t,x,q(x′)) and F(t,x,x′;z) both satisfy the Nagumo condition.
We begin by considering the following BVP:(4.4)Lx=F(t,x,x′;β0),g(x(a),x(b),x′(a),x′(b))=0,x(b)=x(a).
In view of the definition of the upper solution β0(t) of BVP (1.4)–(1.6), it can be verified directly that
(4.5)Lβ0≥f(t,β0,β0′)=F(t,β0,β0′;β0),g(β0(a),β0(b),β0′(a),β0′(b))≤0,β0(b)=β0(a).
Hence, β0(t) is the upper solution of BVP (4.4).
According to condition(3), it can be deduced that
(4.6)F(t,α0,α0′;β0)-f(t,α0,α0′)=f(t,β0,α0′)+fx(t,β0,α0′)(α0-β0)-f(t,α0,α0′)=f(t,β0,α0′)+fx(t,β0,α0′)(α0-β0)-[f(t,β0,α0′)+fx(t,β0,α0′)(α0-β0)+12fxx(t,ξ,α0′)(α0-β0)2]=-12fxx(t,ξ1,α0′)(α0-β0)2≥0,
in which |α0′|<C has been taken into account, and ξ1∈(α0,β0). Consequently, by taking the definition of the lower solution α0(t) of BVP (1.4)–(1.6) into account, we have
(4.7)Lα0≤f(t,α0,α0′)≤F(t,α0,α0′;β0),g(α0(a),α0(b),α0′(a),α0′(b))≥0,α0(b)=α0(a),
which indicates that α0(t) is the lower solution of BVP (4.4). Since BVP (4.4) satisfies all the requirements in Theorem 3.1, hence BVP (4.4) has a unique solution β1(t)∈C2[a,b] with α0(t)≤β1(t)≤β0(t) and |β1′(t)|<C1, t∈[a,b], where C1≤C is a constant.
Next, we consider(4.8)Lx=F(t,x,x′;β1),g(x(a),x(b),x′(a),x′(b))=0,x(b)=x(a).
It can be shown that
(4.9)Lβ1=F(t,β1,β1′;β0)=f(t,β0,β1′)+fx(t,β0,β1′)(β1-β0)≥f(t,β0,β1′)+fx(t,β0,β1′)(β1-β0)+12fxx(t,ξ2,β1′)(β1-β0)2=f(t,β1,β1′)=F(t,β1,β1′;β1)
in which ξ2∈(β1,β0). Thus, β1(t) is the upper solution of BVP (4.8) since
(4.10)g(β1(a),β1(b),β1′(a),β1′(b))≤0,β1(b)=β1(a)
are obvious if we keep that β1(t) is the solution of BVP (4.4) in mind.
Similarly, we can show thatα0(t) is the lower solution of BVP (4.8). Hence, according to Theorem 3.1, there exists a unique solution β2(t)∈C2[a,b] of BVP (4.8) such that α0(t)≤β2(t)≤β1(t) and |β2′(t)|<C1, t∈[a,b], where C1 is a constant.
By repeating the above procedure iteratively, we obtain a monotone nonincreasing sequence{βn(t)} with
(4.11)α0(t)≤⋯≤βn(t)≤⋯≤β1(t)≤β0(t),|βi′(t)|≤C1,t∈[a,b],
where {βn(t)} are, respectively, the unique solution of the  BVPs as follows:
(4.12)Lx=F(t,x,x′;βn-1),g(x(a),x(b),x′(a),x′(b))=0,x(b)=x(a).
Formula (4.11), the properties of the operator L, and the continuity of F yield the uniform boundedness of {βn′′(t)} for t∈[a,b]. Therefore, the sequences {βni(t)}(i=0,1) are uniformly bounded and equicontinuous on [a,b]. Hence, Ascoli-Arzela theorem guarantees the existence of subsequences {βnki(t)}⊂{βni(t)} and a function x(t)∈C1[a,b] with {βnki(t)} converging uniformly to xi(t)(i=0,1) on [a,b] as k→+∞.
Now let(4.13)fk=F(t,βnk,βnk′;βnk-1),gk=g(βnk(a),βnk(b),βnk′(a),βnk′(b)).
The continuity of the functions F and g imply that, as k→+∞,
(4.14)fk=F(t,βnk,βnk′;βnk-1)→f(t,x,x′),gk=g(βnk(a),βnk(b),βnk′(a),βnk′(b))→g(x(a),x(b),x′(a),x′(b)).
The Lebeague dominated convergence theorem then yields
(4.15)∫atfk(s)ds→∫atf(s,x(s),x′(s))ds
for each t∈[a,b] as k→+∞.
Integrating the equation(4.16)Lβnk=F(t,βnk,βnk′;βnk-1)
from a to t results in
(4.17)pβnk′(a)-pβnk′(t)+∫atq(s)βnk(s)ds=∫atfk(s)ds.
Passing to the limit in (4.17), we obtain
(4.18)px'(a)-px'(t)+∫atq(s)x(s)ds=∫atf(s,x(s),x′(s))ds.
Hence, x(t) is the unique solution of BVP (1.4)–(1.6).
To show the quadratic rate of convergence, we define the error function as follows:(4.19)en(t)=βn(t)-x(t).
Hence, en(t)≥0 for t∈[a,b].(4.20)Len=L(βn-x)=Lβn-Lx=F(t,βn,βn′;βn-1)-f(t,x,x′)=f(t,βn-1,βn′)+fx(t,βn-1,βn′)(βn-βn-1)-f(t,x,x′)=f(t,x,βn′)-f(t,x,x′)+f(t,βn-1,βn′)-f(t,x,βn′)+fx(t,βn-1,βn′)(βn-x)+fx(t,βn-1,βn′)(x-βn-1)=fx′(t,βn-1,η1)en′+fx(t,βn-1,βn′)en-12fxx(t,η2,βn′)(x-βn-1)2≤fx′(t,βn-1,η1)en′+K1en+K2∥en-1∥2,
in which η1 lies between βn′ and x', η2∈(x,βn-1), K1=maxt∈[a,b]fx(t,βn-1,βn′) is a negative constant, K2 is a constant with 0≤-(1/2)fxx(t,η2,βn′)≤K2 for (t,η2,βn′)∈D×[-C,C], and ∥en-1∥=maxt∈[a,b]|en-1(t)|.
Moreover, sinceβn(t),x(t) are the solutions of BVPs (4.12) and BVP (1.4)–(1.6), respectively, therefore, mean value theorem yields
(4.21)g1′en(a)+g2′en(b)+g3′en′(a)+g2′en′(b)=0,(4.22)en(a)=en(b),
in which gi′=gi′(η3,η4,η5,η6), i=1,2,3,4, where η3∈(x(a),βn(a)), and so forth.
Define a constant function by(4.23)W(t)=-K2K1∥en-1∥2≥0,t∈[a,b].
Then, W(t) satisfies
(4.24)LW≥fx′(t,βn-1,η1)W′+K1W+K2∥en-1∥2
and moreover, since W(t) is a nonnegative constant function, therefore, we have W(a)≥0,W(b)≥0 and W′(a)=0,  W′(b)=0. Consequently, the condition (4) guarantees that
(4.25)g1′W(a)+g2′W(b)+g3′W′(a)+g2′W′(b)≤0,(4.26)W(a)=W(b)
in which gi′, i=1,2,3,4, are defined in (4.21).
Ifen(t)≤W(t) holds uniformly for t∈[a,b], the quadratic convergence of the iteration has been proved. Otherwise, there must be some points t∈[a,b] such that en(t)>W(t). We then introduce a function
(4.27)qn(t)=en(t)-W(t),t∈[a,b]
and define
(4.28)maxt∈[a,b]qn(t):=qn(t1).Case 1 (t1∈(a,b)).
In this case,qn(t1)=en(t1)-W(t1)>0,  qn′(t1)=en′(t1)-W′(t1)=0, and qn′′(t1)=en′′(t1)-W′′(t1)≤0. Hence, Lqn(t1)≥0. On the other hand, it can be deduced that
(4.29)Lqn(t1)=Len(t1)-LW(t1)≤fx′(t1,βn-1,η1)(en′(t1)-W′(t1))+K1(en(t1)-W(t1))=K1qn(t1)<0
which is a contradiction.Case 2 (t1=a).
In this case,qn(a)>0,  qn′(a)≤0. Furthermore, we can obtain
(4.30)qn(b)=en(b)-W(b)=en(a)-W(a)=qn(a)>0,qn′(b)=limt→b-qn(t)-qn(b)t-b=limt→b-qn(t)-qn(a)t-b≥0.
Consequently, the monotonicity of g given in the condition (4) yields
(4.31)g1′qn(a)+g2′qn(b)+g3′qn′(a)+g2′qn′(b)<0.
However, equation (4.21) minus inequality (4.25) gives
(4.32)g1′qn(a)+g2′qn(b)+g3′qn′(a)+g2′qn′(b)≥0
which contradicts to (4.31).
Similarly, we can prove that Case3, that is, t1=b, is also impossible.
Hence, the inequality(4.33)en(t)≤W(t)=-K2K1∥en-1∥2
holds uniformly for t∈[a,b], that is,
(4.34)∥en(t)∥≤K∥en-1∥2,t∈[a,b]
holds uniformly, in which K=-K2/K1≥0 and ∥en(t)∥=maxt∈[a,b]|en(t)|.


## 5. An Example

Consider a nonlinear BVP as follows:(5.1)-((t2+1)x′)′+kx=-(t2+3)ex-2(x′)2+t2+1,t∈(-1,1),-x(-1)-x(1)+13(x′(-1))3-13(x′(1))3=0,x(-1)=x(1)
in which k is a constant to be determined, that is, (5.2)Lx=-((t2+1)x′)′+kx,f(t,x,x′)=-(t2+3)ex-2(x′)2+t2+1,g(x,y,z,w)=-x-y+13z3-13w3
in this BVP.It can be verified thatα0(t)=-2 and β0(t)=t2 are, respectively, the lower and upper solutions of BVP (5.1) if k≥(3-e2)/2e2 is chosen. Moreover, it is easy to show that all the assumptions in Theorem 4.1 are satisfied. Therefore, by Theorem 4.1, BVP (5.1) has a unique solution and it can be approximated quadratically by two monotone sequences.



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*Source: 287473-2010-08-31.xml*