Quasilinearization for the Boundary Value Problem of Second-Order Singular Differential System

Abstract

We study the boundary value problems of second-order singular differential equations. At first, we reduce the BVPs to initial value problems of first-order singular integrodifferential equations and then we employ the quasilinearization method in studying the IVPs and obtain two monotone iterative sequences, which converge uniformly and quadratically to the unique solution of the IVPs. Finally, we get the similar result for the given BVPs.

1. Introduction

It is well known that quasilinearization method is a powerful tool for proving the existence of approximate solutions of nonlinear systems and the converge quadratically to the unique solution of the given problems (see [1]). Recently, Abd-Ellateef Kamar et al. investigated the first-order singular systems of differential equations with initial value problem [2]. In [3], Wang and Liu developed monotone iterative technique combined with the method of upper and lower solutions for studying the second-order singular systems with boundary value problems (BVPs).

In this paper, we extend quasilinearization method to study the second-order singular systems with the boundary conditions: 𝐴𝑋̈=𝑓𝑡,𝑋,𝑋̇,𝑋̇(0)=𝑋₀,𝑋(𝑏)=𝑋₁, (1) where 𝐴∈𝑅^𝑛×𝑛 is a singular matrix, 𝑓∈𝐶[𝐼×𝑅^𝑛×𝑅^𝑛,𝑅^𝑛], 𝐼=[0,𝑏], and 𝑋₀ and 𝑋₁ are two constant vectors. By using the existence result [4] for linear singular systems and the comparison result [5], we investigate two monotone iterative sequences which converge uniformly and quadratically to the solution of the problem.

2. Preliminaries

Consider the following initial value problem: −𝐴𝑈̇=𝑓(𝑡,𝑆𝑈,−𝑈),𝑈(0)=−𝑋₀, (2) where 𝐴 is a singular 𝑛×𝑛 matrix, 𝑓∈𝐶[𝐼×𝑅^𝑛×𝑅^𝑛,𝑅^𝑛], 𝐼=[0,𝑏], 𝑆𝑈=𝑋₁+∫^𝑏_𝑡𝑈(𝑡)𝑑𝑠 is an increasing operator, and 𝑋₀ is a constant vector.

In order to obtain two monotone sequences, we introduce an existence result for the corresponding linear singular systems and a comparison result.

The existence of the solution of the linear initial value problem of the form −𝐴𝑈̇+𝑀(𝑡)𝑈=𝑔(𝑡),𝑈(0)=−𝑋₀ (3) is well known and is given by the following lemma.

Lemma 1 (see [4]). Assume that the nonhomogeneous linear system (3) exists and if(i)there exists a 𝜆∈𝑅 such that (−𝜆𝐴+𝑀(𝑡))⁻¹ exists,(ii)𝑋₀ is the solution of??(𝐼−𝐴𝐴^𝐷)(−𝑋₀−𝑊(0))=0, where 𝐴=(−𝐴+𝑀(𝑡))⁻¹(),((𝑡−𝐴𝑀=−𝐴+𝑀))⁻¹𝑀(𝑡),𝑔(𝑡)=(−𝐴+𝑀(𝑡))⁻¹𝑔(𝑡),𝑊(𝑡)=𝐼−𝐴^𝐷𝐴𝑀^𝐷𝑔(𝑡), (4)and 𝐴^𝐷,𝑀^𝐷 are the Drazin inverses of 𝐴,𝑀(𝑡), respectively, then the solution is given by 𝑈(𝑡)=𝑒⁻𝐴^𝐷𝑀_𝑡𝐴^𝐷𝐴−𝑋₀+𝑒⁻𝐴^𝐷𝑀_𝑡^𝑡₀𝑒𝐴^𝐷𝑀_𝑠𝐴^𝐷𝑔(𝑠)+𝐴𝑑𝑠𝐼−^𝐷𝐴𝑀^𝐷𝑔(𝑡). (5)
Now one gives the following assumptions for convenience.(𝐻_2.1)Let 𝐴 and 𝑀(𝑡) be matrices such that (−𝜆𝐴+𝑀(𝑡))⁻¹ exists and is nonnegative for some 𝜆∈𝑅¹. Suppose further that 𝑇,𝑇⁻¹ exist and are nonnegative, such that 𝑇⁻¹𝐴𝑇=𝐶000,𝑇⁻¹𝐼𝑀𝑇=¹−𝜆𝐶00𝐼₂, (6) where 𝐴=(−𝜆𝐴+𝑀(𝑡))⁻¹𝐴, 𝑀=(−𝜆𝐴+𝑀(𝑡))⁻¹𝑀(𝑡), and 𝐶 is a diagonal square matrix with 𝐶⁻¹>0 and 𝐶⁻¹(𝐼₁−𝜆𝐶)>0.(𝐻_2.2)There exist 𝛼₀(𝑡),𝛽₀(𝑡)∈𝐶¹[𝐼,𝑅^𝑛] with 𝛼₀(𝑡)≤𝛽₀(𝑡) on 𝐼, such that −𝐴𝛼̇₀≤𝑓𝑡,𝑆𝛼₀,−𝛼₀,𝛼₀(0)≤−𝑋₀,−𝐴𝛽̇₀≥𝑓𝑡,𝑆𝛽₀,−𝛽₀,𝛽₀(0)≥−𝑋₀. (7)(𝐻_2.3)All second-order derivatives of 𝑓(𝑡,𝑋,𝑈) exist and are bounded, 𝑓(𝑡,𝑋,𝑈) is convex in 𝑋 for each (𝑡,𝑈), 𝑓(𝑡,𝑋,𝑈) is convex in 𝑈 for each (𝑡,𝑋), 𝑓_𝑥(𝑡,𝑋,𝑈) is nonincreasing in 𝑈 for each (𝑡,𝑋), and 𝑓_𝑦(𝑡,𝑋,𝑈) is nonincreasing in 𝑋 for each (𝑡,𝑈).(𝐻_2.4)Moreover 𝑀(𝑡)=−[𝑓_𝑥(𝑡,𝑆𝜂,−𝜂)𝑏−𝑓_𝑦(𝑡,𝑆𝜂,−𝜂)].
To obtain the results, one needs the following comparison theorem.

Lemma 2. Let −𝐴𝑃̇+𝑀(𝑡)𝑃≤0 such that 𝐴 and 𝑀(𝑡) satisfy assumptions (𝐻_2.1) and (𝐻_2.4). Then 𝑃(0)≤0 implies 𝑃(𝑡)≤0 on 𝐼.
The proof is similar to [5] and one omits it.

3. Main Results

Firstly, we develop the following result which is important for the final result.

Theorem 3. Suppose that assumptions (𝐻_2.1)--(𝐻_2.4) hold and, for 𝑋₀∈𝑅^𝑛,(i)(𝐼−𝐴𝐴^𝐷)(−𝑋₀−𝑊(0))=0, where 𝑊(𝑡)=(𝐼−𝐴^𝐷𝐴)𝑀^𝐷𝑔(𝑡).
Then there exist two monotone sequences {𝛼_𝑛} and {𝛽_𝑛}, which converge uniformly on 𝐼 to the unique solution of problem (2) and the convergence rate is quadratic.

Proof. From (𝐻_2.3) we find that 𝑓_𝑥𝑥(𝑡,𝑋,𝑈)≥0, and 𝑓𝑡,𝑋₁,𝑈≥𝑓𝑡,𝑋₂,𝑈+𝑓_𝑥𝑡,𝑋₂,𝑈𝑋₁−𝑋₂, (8) for all 𝑋₁,𝑋₂,𝑈∈𝑅^𝑛 and 𝑡∈𝐼. The convexity of 𝑓(𝑡,𝑋,𝑈) in 𝑈 implies that 𝑓𝑡,𝑋,𝑈₁≥𝑓𝑡,𝑋,𝑈₂+𝑓_𝑦𝑡,𝑋,𝑈₂𝑈₁−𝑈₂, (9)
for 𝑈₁,𝑈₂∈𝑅^𝑛, 𝑡∈𝐼, and 𝑋∈𝑅^𝑛.
Now consider the following linear problems. For 𝑘=0,1,2,3…, −𝐴𝛼̇_𝑘+1=𝑓𝑡,𝑆𝛼_𝑘,−𝛼_𝑘+𝑓_𝑥𝑡,𝑆𝛼_𝑘,−𝛼_𝑘𝑆𝛼_𝑘+1−𝑆𝛼_𝑘+𝑓_𝑦𝑡,𝑆𝛼_𝑘,−𝛼_𝑘−𝛼_𝑘+1+𝛼_𝑘,𝛼_𝑘+1(0)=−𝑋₀,−𝐴𝛽̇_𝑘+1=𝑓𝑡,𝑆𝛽_𝑘,−𝛽_𝑘+𝑓_𝑥𝑡,𝑆𝛼_𝑘,−𝛼_𝑘𝑆𝛽_𝑘+1−𝑆𝛽_𝑘+𝑓_𝑦𝑡,𝑆𝛼_𝑘,−𝛼_𝑘−𝛽_𝑘+1+𝛽_𝑘,𝛽_𝑘+1(0)=−𝑋₀. (10)(11) For (10), set 𝑔(𝑡,𝑆𝑥,−𝑥)≡𝑓𝑡,𝑆𝛼₀,−𝛼₀+𝑓_𝑥𝑡,𝑆𝛼₀,−𝛼₀𝑆𝑥−𝑆𝛼₀+𝑓_𝑦𝑡,𝑆𝛼₀,−𝛼₀−𝑥+𝛼₀. (12) Then, we obtain that −𝐴𝛼̇₀≤𝑓(𝑡,𝑆𝛼₀,−𝛼₀)=𝑔(𝑡,𝑆𝛼₀,−𝛼₀). Furthermore, we can get −𝐴𝛽̇₀≥𝑓𝑡,𝑆𝛽₀,−𝛽₀≥𝑓𝑡,𝑆𝛼₀,−𝛼₀+𝑓_𝑥𝑡,𝑆𝛼₀,−𝛼₀𝑆𝛽₀−𝑆𝛼₀+𝑓_𝑦𝑡,𝑆𝛼₀,−𝛼₀−𝛽₀+𝛼₀=𝑔𝑡,𝑆𝛽₀,−𝛽₀. (13) Hence 𝛼₀ and 𝛽₀ are lower and upper solutions of (12), respectively. Thus (12) has a solution 𝛼₁ on 𝐼, and we have 𝛼₀≤𝛼₁≤𝛽₀. Similarly, set 𝑔^∗(𝑡,𝑆𝑥,−𝑥)≡𝑓𝑡,𝑆𝛽₀,−𝛽₀+𝑓_𝑥𝑡,𝑆𝛼₀,−𝛼₀𝑆𝑥−𝑆𝛽₀+𝑓_𝑦𝑡,𝑆𝛼₀,−𝛼₀−𝑥+𝛽₀. (14) Then we obtain that (14) has a solution 𝛽₁ on 𝐼, and we have 𝛼₀≤𝛽₁≤𝛽₀.
Now we claim that 𝛼₀≤𝛼₁≤𝛽₁≤𝛽₀on𝐼. (15) Let 𝑝=𝛼₁−𝛽₁. We get that −𝐴𝑝̇=−𝐴𝛼̇₁−−𝐴𝛽̇₁=𝑓𝑡,𝑆𝛼₀,−𝛼₀+𝑓_𝑥𝑡,𝑆𝛼₀,−𝛼₀𝑆𝛼₁−𝑆𝛼₀+𝑓_𝑦𝑡,𝑆𝛼₀,−𝛼₀−𝛼₁+𝛼₀−𝑓𝑡,𝑆𝛽₀,−𝛽₀−𝑓_𝑥𝑡,𝑆𝛼₀,−𝛼₀𝑆𝛽₁−𝑆𝛽₀−𝑓_𝑦𝑡,𝑆𝛼₀,−𝛼₀−𝛽₁+𝛽₀=𝑓𝑡,𝑆𝛼₀,−𝛼₀−𝑓𝑡,𝑆𝛽₀,−𝛼₀+𝑓𝑡,𝑆𝛽₀,−𝛼₀−𝑓𝑡,𝑆𝛽₀,−𝛽₀+𝑓_𝑥𝑡,𝑆𝛼₀,−𝛼₀𝑆𝛼₁−𝑆𝛼₀+𝑓_𝑦𝑡,𝑆𝛼₀,−𝛼₀−𝛼₁+𝛼₀−𝑓_𝑥𝑡,𝑆𝛼₀,−𝛼₀𝑆𝛽₁−𝑆𝛽₀−𝑓_𝑦𝑡,𝑆𝛼₀,−𝛼₀−𝛽₁+𝛽₀≤𝑓_𝑥𝑡,𝑆𝛼₀,−𝛼₀𝑆𝛼₀−𝑆𝛽₀+𝑓_𝑦𝑡,𝑆𝛼₀,−𝛼₀−𝛼₀+𝛽₀+𝑓_𝑥𝑡,𝑆𝛼₀,−𝛼₀𝑆𝛼₁−𝑆𝛼₀+𝑓_𝑦𝑡,𝑆𝛼₀,−𝛼₀−𝛼₁+𝛼₀−𝑓_𝑥𝑡,𝑆𝛽₀,−𝛽₀𝑆𝛽₁−𝑆𝛽₀−𝑓_𝑦𝑡,𝑆𝛽₀,−𝛽₀−𝛽₁+𝛽₀≤𝑓_𝑥𝑡,𝑆𝛼₀,−𝛼₀×𝑆𝛼₀−𝑆𝛽₀+𝑆𝛼₁−𝑆𝛼₀−𝑆𝛽₁+𝑆𝛽₀+𝑓_𝑦𝑡,𝑆𝛼₀,−𝛼₀×−𝛼₀+𝛽₀−𝛼₁+𝛼₀+𝛽₁−𝛽₀=𝑓_𝑥𝑡,𝑆𝛼₀,−𝛼₀𝑆𝛼₁−𝑆𝛽₁−𝑓_𝑦𝑡,𝑆𝛼₀,−𝛼₀𝛼₁−𝛽₁≤−𝑀(𝑡)𝛼₁−𝛽₁. (16) Noticing that 𝑝(0)=𝛼₁(0)−𝛽₁(0)≤0, we get 𝛼₁(𝑡)≤𝛽₁(𝑡), on 𝐼.
Now, assume that, for 𝑛=0,1,2,…,𝑘, (10) and (11) admit solutions 𝛼_𝑛 and 𝛽_𝑛, respectively, such that 𝛼₀≤𝛼₁≤⋯≤𝛼_𝑘≤𝛽_𝑘≤⋯≤𝛽₁≤𝛽₀,on𝐼. (17) Then setting 𝑛=𝑘 in (10) and (11), we observe that the assumptions in Theorem 3 are satisfied. Thus, there exist solutions 𝛼_𝑘+1(𝑡) and 𝛽_𝑘+1(𝑡) for (10) and (11), respectively, and we now will show that the following relation 𝛼_𝑘(𝑡)≤𝛼_𝑘+1(𝑡)≤𝛽_𝑘+1(𝑡)≤𝛽_𝑘(𝑡),𝑡∈𝐼, (18) holds. Firstly, we can easily know that 𝛼_𝑘(𝑡)≤𝛼_𝑘+1(𝑡)≤𝛽_𝑘(𝑡) and 𝛼_𝑘(𝑡)≤𝛽_𝑘+1(𝑡)≤𝛽_𝑘(𝑡).
To prove that 𝛼_𝑘+1≤𝛽_𝑘+1, consider 𝑝=𝛼_𝑘+1−𝛽_𝑘+1. Then −𝐴𝑝̇=−𝐴𝛼̇_𝑘+1−−𝐴𝛽̇_𝑘+1=𝑓𝑡,𝑆𝛼_𝑘,−𝛼_𝑘+𝑓_𝑥𝑡,𝑆𝛼_𝑘,−𝛼_𝑘𝑆𝛼_𝑘+1−𝑆𝛼_𝑘+𝑓_𝑦𝑡,𝑆𝛼_𝑘,−𝛼_𝑘−𝛼_𝑘+1+𝛼_𝑘−𝑓𝑡,𝑆𝛽_𝑘,−𝛽_𝑘−𝑓_𝑥𝑡,𝑆𝛽_𝑘,−𝛽_𝑘𝑆𝛽_𝑘+1−𝑆𝛽_𝑘−𝑓_𝑦𝑡,𝑆𝛽_𝑘,−𝛽_𝑘−𝛽_𝑘+1+𝛽_𝑘=𝑓𝑡,𝑆𝛼_𝑘,−𝛼_𝑘−𝑓𝑡,𝑆𝛽_𝑘,−𝛼_𝑘+𝑓𝑡,𝑆𝛽_𝑘,−𝛼_𝑘−𝑓𝑡,𝑆𝛽_𝑘,−𝛽_𝑘+𝑓_𝑥𝑡,𝑆𝛼_𝑘,−𝛼_𝑘𝑆𝛼_𝑘+1−𝑆𝛼_𝑘+𝑓_𝑦𝑡,𝑆𝛼_𝑘,−𝛼_𝑘−𝛼_𝑘+1+𝛼_𝑘−𝑓_𝑥𝑡,𝑆𝛽_𝑘,−𝛽_𝑘𝑆𝛽_𝑘+1−𝑆𝛽_𝑘−𝑓_𝑦𝑡,𝑆𝛽_𝑘,−𝛽_𝑘−𝛽_𝑘+1+𝛽_𝑘≤𝑓_𝑥𝑡,𝑆𝛼_𝑘,−𝛼_𝑘𝑆𝛼_𝑘−𝑆𝛽_𝑘+𝑓_𝑦𝑡,𝑆𝛽_𝑘,−𝛼_𝑘−𝛼_𝑘+𝛽_𝑘+𝑓_𝑥𝑡,𝑆𝛼_𝑘,−𝛼_𝑘𝑆𝛼_𝑘+1−𝑆𝛼_𝑘+𝑓_𝑦𝑡,𝑆𝛼_𝑘,−𝛼_𝑘−𝛼_𝑘+1+𝛼_𝑘−𝑓_𝑥𝑡,𝑆𝛽_𝑘,−𝛽_𝑘𝑆𝛽_𝑘+1−𝑆𝛽_𝑘−𝑓_𝑦𝑡,𝑆𝛽_𝑘,−𝛽_𝑘−𝛽_𝑘+1+𝛽_𝑘≤𝑓_𝑥𝑡,𝑆𝛼_𝑘,−𝛼_𝑘×𝑆𝛼_𝑘−𝑆𝛽_𝑘+𝑆𝛼_𝑘+1−𝑆𝛼_𝑘−𝑆𝛽_𝑘+1+𝑆𝛽_𝑘+𝑓_𝑦𝑡,𝑆𝛼_𝑘,−𝛼_𝑘×−𝛼_𝑘+𝛽_𝑘−𝛼_𝑘+1+𝛼_𝑘−𝛽_𝑘+1+𝛽_𝑘=𝑓_𝑥𝑡,𝑆𝛼_𝑘,−𝛼_𝑘𝑆𝛼_𝑘+1−𝑆𝛽_𝑘+1+𝑓_𝑦𝑡,𝑆𝛼_𝑘,−𝛼_𝑘−𝛼_𝑘+1+𝛽_𝑘+1=−𝑀(𝑡)𝑝. (19) We know that 𝑝(0)=𝛼_𝑘+1(0)−𝛽_𝑘+1(0)≤0. Hence, from Lemma 2, we deduce that 𝛼_𝑘+1(𝑡)≤𝛽_𝑘+1(𝑡), on 𝐼. Thus, we have monotone sequences {𝛼_𝑛},{𝛽_𝑛} such that 𝛼₀≤𝛼₁≤⋯≤𝛼_𝑛≤𝛽_𝑛≤⋯≤𝛽₁≤𝛽₀,on𝐼. (20) Now, employing Ascoli-Arzela's theorem we conclude that the sequences converge uniformly and monotonically to the unique solution 𝑈(𝑡) of (2) on 𝐼.
To show that the convergence rate is quadratic, we begin with 𝑝_𝑛+1=𝑥−𝛼_𝑛+1. Then −𝐴𝑝̇_𝑛+1=−𝐴𝑥̇−−𝐴𝛼̇_𝑛+1=𝑓(𝑡,𝑆𝑥,−𝑥)−𝑓𝑡,𝑆𝛼_𝑛,−𝛼_𝑛−𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑆𝛼_𝑛+1−𝑆𝛼_𝑛−𝑓_𝑦𝑡,𝑆𝛼_𝑛,−𝛼_𝑛−𝛼_𝑛+1+𝛼_𝑛=𝑓(𝑡,𝑆𝑥,−𝑥)−𝑓𝑡,𝑆𝛼_𝑛,−𝛼_𝑛−𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑆𝛼_𝑛+1−𝑆𝑥+𝑆𝑥−𝑆𝛼_𝑛−𝑓_𝑦𝑡,𝑆𝛼_𝑛,−𝛼_𝑛−𝛼_𝑛+1+𝑥−𝑥+𝛼_𝑛=𝑓(𝑡,𝑆𝑥,−𝑥)−𝑓𝑡,𝑆𝛼_𝑛,−𝛼_𝑛+𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑆𝑝_𝑛+1−𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑆𝑝_𝑛−𝑓_𝑦𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑝_𝑛+1+𝑓_𝑦𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑝_𝑛=𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑆𝑝_𝑛+1−𝑓_𝑦𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑝_𝑛+1+𝑓(𝑡,𝑆𝑥,−𝑥)−𝑓𝑡,𝑆𝛼_𝑛,−𝛼_𝑛−𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑆𝑝_𝑛+𝑓_𝑦𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑝_𝑛≤−𝑀(𝑡)𝑝+𝑓(𝑡,𝑆𝑥,−𝑥)−𝑓𝑡,𝑆𝛼_𝑛,−𝛼_𝑛−𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑆𝑝_𝑛+𝑓_𝑦𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑝_𝑛=−𝑀(𝑡)𝑝+𝑓(𝑡,𝑆𝑥,−𝑥)−𝑓𝑡,𝑆𝛼_𝑛,−𝑥+𝑓𝑡,𝑆𝛼_𝑛,−𝑥−𝑓𝑡,𝑆𝛼_𝑛,−𝛼_𝑛−𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑆𝑝_𝑛+𝑓_𝑦𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑝_𝑛(𝑡)𝑝+=−𝑀₁₀𝑓_𝑥𝑡,𝜎𝑆𝑥+(1−𝜎)𝑆𝛼_𝑛,−𝑥𝑆𝑥−𝑆𝛼_𝑛+𝑑𝜎₁₀𝑓_𝑦𝑡,𝑆𝛼_𝑛,𝜎(−𝑥)+(1−𝜎)−𝛼_𝑛×−𝑥+𝛼_𝑛𝑑𝜎−𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑆𝑝_𝑛+𝑓_𝑦𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑝_𝑛=−𝑀(𝑡)𝑝+¹₀𝑓_𝑥𝑡,𝜎𝑆𝑥+(1−𝜎)𝑆𝛼_𝑛,−𝑥−𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑆𝑝_𝑛−𝑑𝜎₁₀𝑓_𝑦𝑡,𝑆𝛼_𝑛,𝜎(−𝑥)+(1−𝜎)−𝛼_𝑛−𝑓_𝑦𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑝_𝑛(𝑡)𝑑𝜎=−𝑀𝑝+𝐴₁+𝐵₁, (21) where 𝐴₁=¹₀𝑓_𝑥𝑡,𝜎𝑆𝑥+(1−𝜎)𝑆𝛼_𝑛,−𝑥−𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑆𝑝_𝑛𝐵𝑑𝜎,₁=−¹₀𝑓_𝑦𝑡,𝑆𝛼_𝑛,𝜎(−𝑥)+(1−𝜎)−𝛼_𝑛−𝑓_𝑦𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑝_𝑛𝑑𝜎. (22) Set 𝜉₁=𝜎𝑆𝑥+(1−𝜎)𝑆𝛼_𝑛. Then 𝐴₁=¹₀𝑓_𝑥𝑡,𝜉₁,−𝑥−𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑆𝑝_𝑛=𝑑𝜎₁₀𝑓_𝑥𝑡,𝜉₁,−𝑥−𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝑥+𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝑥−𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑆𝑝_𝑛=𝑑𝜎₁₀𝑓_𝑥𝑥𝑡,𝜏𝜉₁+(1−𝜏)𝑆𝛼_𝑛,−𝑥𝜉₁−𝑆𝛼_𝑛+𝑓_𝑥𝑦𝑡,𝑆𝛼_𝑛,𝜏(−𝑥)+(1−𝜏)−𝛼_𝑛×−𝑥+𝛼_𝑛𝑆𝑝_𝑛=𝑑𝜏𝑑𝜎₁₀𝑓_𝑥𝑥𝑡,𝜉₂,−𝑥𝜎𝑆𝑥−𝑆𝛼_𝑛−𝑓_𝑥𝑦𝑡,𝑆𝛼_𝑛,𝜉₃𝑥−𝛼_𝑛𝑆𝑝_𝑛=1𝑑𝜏𝑑𝜎2𝑁₁𝑆𝑝²_𝑛+𝑁₂𝑝_𝑛𝑆𝑝_𝑛≤12𝑁₁𝑏²+𝑁₂𝑏𝑝²_𝑛, (23) where 𝜉₂=𝜏𝜉₁+(1−𝜏)𝑆𝛼_𝑛, 𝜉₃=𝜏(−𝑥)+(1−𝜏)(−𝛼_𝑛), 𝑓_𝑥𝑥(𝑡,𝑈,𝑋)≤𝑁₁, and −𝑓_𝑥𝑦(𝑡,𝑈,𝑋)≤𝑁₂.
Next, set 𝜂₁=𝜎(−𝑥)+(1−𝜎)(−𝛼_𝑛). Then 𝐵₁=−¹₀𝑓_𝑦𝑡,𝑆𝛼_𝑛,𝜂₁−𝑓_𝑦𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑝_𝑛𝑑𝜎=−¹₀𝑓_𝑦𝑦𝑡,𝑆𝛼_𝑛,𝜏𝜂₁+(1−𝜏)−𝛼_𝑛×𝜂₁+𝛼_𝑛𝑝_𝑛=𝑑𝜎𝑑𝜏₁₀𝑓_𝑦𝑦𝑡,𝑆𝛼_𝑛,𝜂₂𝜎𝑝²_𝑛1𝑑𝜎𝑑𝜏≤2𝑁₃𝑝²_𝑛, (24) where 𝑓_𝑦𝑦(𝑡,𝑈,𝑋)≤𝑁₃ and 𝜂₂=𝜏𝜂₁+(1−𝜏)(−𝛼_𝑛).
Furthermore, we have that −𝐴𝑝̇_𝑛+1≤−𝑀(𝑡)𝑝_𝑛+1+𝑁₄𝑝²_𝑛, (25) where 𝑁₄=(1/2)𝑁₁𝑏²+𝑁₂𝑏+(1/2)𝑁₃.
Using Lemma 2, we show that 𝑝_𝑛+1(𝑡)≤𝑈(𝑡) on 𝐼, where 𝑈(𝑡) is the solution of −𝐴𝑈̇+𝑀(𝑡)𝑈=𝑁₄𝑝²_𝑛,𝑈(0)=0. (26) Thus, using Lemma 1, the solution of the previously mentioned equation is given as 𝑈(𝑡)=𝑒⁻𝐴^𝐷𝑀_𝑡^𝑡₀𝑒𝐴^𝐷𝑀_𝑠𝐴^𝐷(−𝜆𝐴+𝑀(𝑠))⁻¹𝑁₄𝑝²_𝑛(𝑠)+𝐴𝑑𝑠𝐼−^𝐷𝐴𝑀^𝐷(−𝜆𝐴+𝑀(𝑡))⁻¹𝑁₄𝑝²_𝑛(𝑡). (27) After taking suitable estimates, we obtain 𝑝_𝑛+1≤𝐾₁𝑝²_𝑛, (28) where 𝐾₁=𝑒⁻𝐴^𝐷𝑀_𝑡∫^𝑡₀𝑒𝐴^𝐷𝑀_𝑠𝐴^𝐷(−𝜆𝐴+𝑀(𝑠))⁻¹𝑁₄𝑑𝑠+(𝐼−𝐴^𝐷𝐴)𝑀^𝐷(−𝜆𝐴+𝑀(𝑡))⁻¹𝑁₄.
Set 𝑞_𝑛+1=𝛽_𝑛+1−𝑥. Then we can get −𝐴𝑞̇_𝑛+1=−𝐴𝛽̇_𝑛+1−(−𝐴𝑥̇)=𝑓𝑡,𝑆𝛽_𝑛,−𝛽_𝑛+𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑆𝛽_𝑛+1−𝑆𝛽_𝑛+𝑓_𝑦𝑡,𝑆𝛼_𝑛,−𝛼_𝑛−𝛽_𝑛+1+𝛽_𝑛−𝑓(𝑡,𝑆𝑥,−𝑥)=𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑆𝛽_𝑛+1−𝑆𝑥+𝑆𝑥−𝑆𝛽_𝑛+𝑓_𝑦𝑡,𝑆𝛼_𝑛,−𝛼_𝑛−𝛽_𝑛+1+𝑥−𝑥+𝛽_𝑛+𝑓𝑡,𝑆𝛽_𝑛,−𝛽_𝑛−𝑓(𝑡,𝑆𝑥,−𝑥)=𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑆𝛽_𝑛+1−𝑆𝑥−𝑓_𝑦𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝛽_𝑛+1−𝑥+𝑓𝑡,𝑆𝛽_𝑛,−𝛽_𝑛−𝑓(𝑡,𝑆𝑥,−𝑥)−𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑆𝛽_𝑛−𝑆𝑥+𝑓_𝑦𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝛽_𝑛(𝑡)𝑞−𝑥≤−𝑀_𝑛+1+𝑓𝑡,𝑆𝛽_𝑛,−𝛽_𝑛−𝑓𝑡,𝑆𝑥,−𝛽_𝑛+𝑓𝑡,𝑆𝑥,−𝛽_𝑛−𝑓(𝑡,𝑆𝑥,−𝑥)−𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑆𝛽_𝑛−𝑆𝑥+𝑓_𝑦𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝛽_𝑛(𝑡)𝑞−𝑥=−𝑀_𝑛+1+¹₀𝑓_𝑥𝑡,𝜎𝑆𝛽_𝑛+(1−𝜎)𝑆𝑥,−𝛽_𝑛𝑆𝛽_𝑛+−𝑆𝑥𝑑𝜎₁₀𝑓_𝑦𝑡,𝑆𝑥,𝜎−𝛽_𝑛+()×1−𝜎)(−𝑥−𝛽_𝑛+𝑥𝑑𝜎−𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑆𝛽_𝑛−𝑆𝑥+𝑓_𝑦𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝛽_𝑛(𝑡)𝑞−𝑥=−𝑀_𝑛+1+¹₀𝑓_𝑥𝑡,𝜎𝑆𝛽_𝑛+(1−𝜎)𝑆𝑥,−𝛽_𝑛−𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑆𝑞_𝑛−𝑑𝜎₁₀𝑓_𝑦𝑡,𝑆𝑥,𝜎−𝛽_𝑛+(1−𝜎)(−𝑥)−𝑓_𝑦𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑞_𝑛(𝑡)𝑞𝑑𝜎=−𝑀_𝑛+1+𝐴₂+𝐵₂, (29) in which 𝐴₂=¹₀𝑓_𝑥𝑡,𝜎𝑆𝛽_𝑛+(1−𝜎)𝑆𝑥,−𝛽_𝑛−𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑆𝑞_𝑛𝐵𝑑𝜎,₂=−¹₀𝑓_𝑦𝑡,𝑆𝑥,𝜎−𝛽_𝑛+(1−𝜎)(−𝑥)−𝑓_𝑦𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑞_𝑛𝑑𝜎. (30) Set 𝜃₁=𝜎𝑆𝛽_𝑛+(1−𝜎)𝑆𝑥. Then we get that 𝐴₂=¹₀𝑓_𝑥𝑡,𝜃₁,−𝛽_𝑛−𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑆𝑞_𝑛=𝑑𝜎₁₀𝑓_𝑥𝑡,𝜃₁,−𝛽_𝑛−𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝛽_𝑛+𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝛽−𝑓_𝑥𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑆𝑞_𝑛=𝑑𝜎₁₀𝑓_𝑥𝑥𝑡,𝜏𝜃₁+(1−𝜏)𝑆𝛼_𝑛,−𝛽_𝑛𝜃₁−𝑆𝛼_𝑛+𝑓_𝑥𝑦𝑡,𝑆𝛼_𝑛,𝜏−𝛽_𝑛+(1−𝜏)−𝛼_𝑛×−𝛽_𝑛+𝛼_𝑛𝑆𝑞_𝑛=𝑑𝜎𝑑𝜏₁₀𝑓_𝑥𝑥𝑡,𝜃₂,−𝛽_𝑛𝜎𝑞_𝑛+𝑝_𝑛−𝑓_𝑥𝑦𝑡,𝑆𝛼_𝑛,𝜃₃𝑞_𝑛+𝑝_𝑛𝑆𝑞_𝑛𝑑𝜎𝑑𝜏≤𝑁₁𝜎𝑞_𝑛𝑆𝑞_𝑛+𝑁₁𝑝_𝑛𝑆𝑞_𝑛+𝑁₂𝑞_𝑛𝑆𝑞_𝑛+𝑁₂𝑝_𝑛𝑆𝑞_𝑛≤12𝑁₁𝑞²_𝑛𝑏+𝑏𝑁₁12𝑝²_𝑛+𝑞²_𝑛+𝑁₂𝑏𝑞²_𝑛+𝑁₂𝑏12𝑝²_𝑛+𝑞²_𝑛=𝑁₁+32𝑁₂𝑏𝑞²_𝑛+12𝑏𝑁₁+𝑁₂𝑝²_𝑛, (31) where 𝜃₂=𝜏𝜃₁+(1−𝜏)𝑆𝛼_𝑛 and 𝜃₃=𝜏(−𝛽_𝑛)+(1−𝜏)(−𝛼_𝑛).
Similarly, set 𝜗₁=𝜎(−𝛽_𝑛)+(1−𝜎)(−𝑥). Then 𝐵₂=−¹₀𝑓_𝑦𝑡,𝑆𝑥,𝜗₁−𝑓_𝑦𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑞_𝑛𝑑𝜎=−¹₀𝑓_𝑦𝑡,𝑆𝑥,𝜗₁−𝑓_𝑦𝑡,𝑆𝛼_𝑛,𝜗₁+𝑓_𝑦𝑡,𝑆𝛼_𝑛,𝜗₁−𝑓_𝑦𝑡,𝑆𝛼_𝑛,−𝛼_𝑛𝑞_𝑛𝑑𝜎=−¹₀𝑓_𝑦𝑥𝑡,𝜏𝑆𝑥+(1−𝜏)𝑆𝛼_𝑛,𝜗₁𝑆𝑥−𝑆𝛼_𝑛+𝑓_𝑦𝑦𝑡,𝑆𝛼_𝑛,𝜏𝜗₁+(1−𝜏)−𝛼_𝑛×𝜗₁+𝛼_𝑛𝑞_𝑛𝑑𝜎𝑑𝜏=−¹₀𝑓_𝑦𝑥𝑡,𝜗₂,𝜗₁𝑆𝑝_𝑛+𝑓_𝑦𝑦𝑡,𝑆𝛼_𝑛,𝜗₃×𝜎−𝛽_𝑛+𝑥−𝑥−𝛼_𝑛𝑞_𝑛=𝑑𝜎𝑑𝜏₁₀−𝑓_𝑦𝑥𝑡,𝜗₂,𝜗₁𝑆𝑝_𝑛+𝑓_𝑦𝑦𝑡,𝑆𝛼_𝑛,𝜗₃𝜎𝑞_𝑛+𝑝_𝑛𝑞_𝑛𝑑𝜎𝑑𝜏≤𝑁₅𝑏𝑞_𝑛𝑝_𝑛+12𝑁₃𝑞²_𝑛+𝑁₃𝑝_𝑛𝑞_𝑛≤12𝑏𝑁₅+𝑁₃𝑞²_𝑛+12𝑏𝑁₅+𝑁₃𝑝²_𝑛, (32) where 𝜗₂=𝜏𝑆𝑥+(1−𝜏)𝑆𝛼_𝑛,𝜗₃=𝜏𝜗₁+(1−𝜏)(−𝛼_𝑛), and −𝑓_𝑦𝑥(𝑡,𝑋,𝑈)≤𝑁₅.
Then we get that −𝐴𝑞̇_𝑛+1≤−𝑀(𝑡)𝑞_𝑛+1+𝑁₆𝑞²_𝑛+𝑁₇𝑝²_𝑛, (33) where 𝑁₆=(𝑁₁+(3/2)𝑁₂)𝑏+((1/2)𝑏𝑁₅+𝑁₃) and 𝑁₇=(1/2)𝑏(𝑁₁+𝑁₂)+(1/2)(𝑏𝑁₅+𝑁₃). Thus we have that 𝑞_𝑛+1(𝑡)≤𝑈(𝑡)=𝑒⁻𝐴^𝐷𝑀_𝑡^𝑡₀𝑒𝐴^𝐷𝑀_𝑠𝐴^𝐷(−𝜆𝐴+𝑀(𝑠))⁻¹×𝑁₆𝑞_𝑛(𝑠)²+𝑁₇𝑝_𝑛(𝑠)²+𝐴𝑑𝑠𝐼−^𝐷𝐴𝑀^𝐷(−𝜆𝐴+𝑀(𝑡))⁻¹𝑁₆𝑞_𝑛(𝑡)²+𝑁₇𝑝_𝑛(𝑡)². (34) Furthermore, we obtain after taking suitable estimates 𝑝_𝑛+1≤𝐾₂𝑞²_𝑛+𝐾₃𝑝²_𝑛, (35) where 𝐾₂=𝑒⁻𝐴^𝐷𝑀_𝑡∫^𝑡₀𝑒𝐴^𝐷𝑀_𝑠𝐴^𝐷(−𝜆𝐴+𝑀(𝑠))⁻¹𝑁₆𝑑𝑠+(𝐼−𝐴^𝐷𝐴)𝑀^𝐷(−𝜆𝐴+𝑀(𝑡))⁻¹𝑁₆ and 𝐾₃=𝑒⁻𝐴^𝐷𝑀_𝑡∫^𝑡₀𝑒𝐴^𝐷𝑀_𝑠𝐴^𝐷(−𝜆𝐴+𝑀(𝑠))⁻¹𝑁₇𝑑𝑠+(𝐼−𝐴^𝐷𝐴)𝑀^𝐷(−𝜆𝐴+𝑀(𝑡))⁻¹𝑁₇.
Hence we proved that the convergence rate is quadratic.

Next, we consider singular differential systems BVPs and prove the following main result.

Theorem 4. Let assumptions (𝐻_2.1)--(𝐻_2.4) hold. Suppose further the following.(𝐻_3.1)There exist 𝑉₀(𝑡),𝑊₀(𝑡)∈𝐶²[𝐼,𝑅^𝑛] with 𝑉₀(𝑡)≤𝑊₀(𝑡) and 𝑉̇₀(𝑡)≥𝑊̇₀(𝑡) on 𝐼 and 𝐴𝑉̈₀≤𝑓𝑡,𝑉₀,𝑉̇₀,𝑉̇₀(0)≥𝑋₀,𝑉₀(𝑏)≤𝑋₁,𝐴𝑊̈₀≥𝑓𝑡,𝑊₀,𝑊̇₀,𝑊̇₀(0)≤𝑋₀,𝑊₀(𝑏)≥𝑋₁. (36)(𝐻_3.2)For 𝑋₀∈𝑅^𝑛, (𝐼−𝐴𝐴^𝐷)(−𝑋₀−𝑊(0))=0, where 𝑊(𝑡)=(𝐼−𝐴^𝐷𝐴)𝑀^𝐷𝑔(𝑡).Then there exist monotone sequences {𝑉_𝑛} and {𝑊_𝑛} which converge uniformly on 𝐼 to the unique solution of (1) and the convergence rate is quadratic.

Proof. Using the transformation 𝑈=−𝑋̇, we have that 𝑆𝑈(𝑡)=𝑋(𝑡)=𝑋₁+^𝑏_𝑡𝑈(𝑠)𝑑𝑠,𝑡∈𝐼. (37) BVPs (1) can be transformed into IVPs of singular system: −𝐴𝑈̇=𝑓(𝑡,𝑆𝑈,−𝑈),𝑈(0)=−𝑋₀. (38) Let 𝛼₀(0)=−𝑉̇₀(0) and 𝛽₀(0)=−𝑊̇₀(0). By using assumption (𝐻_3.1), we get that 𝛼₀(𝑡)≤−𝑉̇₀(𝑡)≤−𝑊̇₀(𝑡)=𝛽₀(𝑡) (39)
and 𝛼₀(0)=−𝑉̇₀(0)≤−𝑋₀≤−𝑊̇₀(0)=𝛽₀(0).
Noticing that 𝛼₀(0)=−𝑉̇₀(0), we have that 𝑉₀(𝑡)=𝑉₀(𝑏)+^𝑏_𝑡𝛼₀(𝑠)𝑑𝑠,𝑡∈𝐼,𝑆𝛼₀(𝑡)=𝑋₁+^𝑏_𝑡𝛼₀(𝑠)𝑑𝑠,𝑡∈𝐼. (40) Then 𝑉₀(𝑡)≤[𝑆𝛼₀](𝑡) from 𝑉₀(𝑏)≤𝑋₁, and −𝐴𝛼̇₀=𝐴𝑉̈₀≤𝑓𝑡,𝑉₀,𝑉̇₀≤𝑓𝑡,𝑆𝛼₀,𝑉̇₀=𝑓𝑡,𝑆𝛼₀,−𝛼₀. (41)
A similar argument shows that −𝐴𝛽̇₀=𝐴𝑊̈₀≥𝑓𝑡,𝑊₀,𝑊̇₀≥𝑓𝑡,𝑆𝛽₀,𝑊̇₀=𝑓𝑡,𝑆𝛽₀,−𝛽₀. (42)
By Theorem 3, there exist monotone sequences {𝛼_𝑛},{𝛽_𝑛} such that 𝛼_𝑛⟶𝑈⟵𝛽_𝑛 (43) and the convergence rate is quadratic. Again set 𝑉_𝑛=𝑆𝛼_𝑛 and 𝑊_𝑛=𝑆𝛽_𝑛. Then 𝑉₀≤𝑉₁≤𝑉₂≤⋯≤𝑉_𝑛≤𝑊_𝑛≤⋯≤𝑊₂≤𝑊₁≤𝑊₀,𝑉_𝑛⟶𝑈⟵𝑊_𝑛, (44) and also the convergence rate is quadratic. Noticing that 𝑥(𝑡)−𝑉_𝑛+1(𝑡)=𝑋₁+^𝑏_𝑡𝑈(𝑠)𝑑𝑠−𝑋₁+^𝑏₀𝛼_𝑛+1(𝑠)=𝑑𝑠_𝑏0𝑈(𝑠)−𝛼_𝑛+1(𝑠)≤𝑑𝑠_𝑏0𝐾₁𝑈(𝑠)−𝛼_𝑛(𝑠)²𝑑𝑠=𝐾₁𝑥(𝑡)−𝑉_𝑛(𝑡)², (45) we can obtain the similar result that 𝑊_𝑛 converges quadratically to the solution of (1). The proof is complete.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

All authors completed the paper together. All authors read and approved the final paper.

Acknowledgments

The authors would like to thank the reviewers for their valuable suggestions and comments. This paper is supported by the National Natural Science Foundation of China (11271106) and the Natural Science Foundation of Hebei Province of China (A2013201232).