# Approximation of Solutions for Second-Order -Point Nonlocal Boundary Value Problems via the Method of Generalized Quasilinearization

- Ahmed Alsaedi
^{1}Email author

**2011**:929061

**DOI: **10.1155/2011/929061

© Ahmed Alsaedi. 2011

**Received: **11 May 2010

**Accepted: **2 October 2010

**Published: **4 October 2010

## Abstract

We discuss the existence and uniqueness of the solutions of a second-order -point nonlocal boundary value problem by applying a generalized quasilinearization technique. A monotone sequence of solutions converging uniformly and quadratically to a unique solution of the problem is presented.

## 1. Introduction

The monotone iterative technique coupled with the method of upper and lower solutions [1–7] manifests itself as an effective and flexible mechanism that offers theoretical as well as constructive existence results in a closed set, generated by the lower and upper solutions. In general, the convergence of the sequence of approximate solutions given by the monotone iterative technique is at most linear [8, 9]. To obtain a sequence of approximate solutions converging quadratically, we use the method of quasilinearization [10]. This method has been developed for a variety of problems [11–20]. In view of its diverse applications, this approach is quite an elegant and easier for application algorithms.

The subject of multipoint nonlocal boundary conditions, initiated by Bicadze and Samarskiĭ [21], has been addressed by many authors, for instance, [22–32]. The multipoint boundary conditions appear in certain problems of thermodynamics, elasticity and wave propagation, see [23] and the references therein. The multipoint boundary conditions may be understood in the sense that the controllers at the endpoints dissipate or add energy according to censors located at intermediate positions.

where is continuous and are nonnegative real constants such that , and with

where is given by (1.5). Thus, (1.6) represents the correct form of the solution for the problem (1.1) and (1.3).

## 2. Preliminaries

Note that on

Definition 2.1.

for . We say that satisfies a Nagumo condition on relative to if for every and there exists a Nagumo function such that

We need the following result [33] to establish the main result.

Theorem 2.2.

Let be a continuous function satisfying the Nagumo condition on where are continuous functions such that for all Then there exists a constant (depending only on the Nagumo function ) such that every solution of (1.1)-(1.2) with , satisfies

If are assumed to be lower and upper solutions of (1.1)-(1.2), respectively, in the statement of Theorem 2.2, then there exists a solution, of (1.1) and (1.2) such that ,

Theorem 2.3.

Assume that are, respectively, lower and upper solutions of (1.1)-(1.2). If is decreasing in for each then on

Proof.

which implies that Now as , , , therefore we obtain a contradiction. We have a similar contradiction at Thus, we conclude that ,

## 3. Main Results

Theorem 3.1.

Assume that

the functions are, respectively, lower and upper solutions of (1.1)-(1.2) such that on

Then, there exists a monotone sequence of approximate solutions converging uniformly to a unique solution of the problems (1.1)-(1.2).

Proof.

where . Thus, any solution of (3.2) with , satisfies on and hence it is a solution of (1.1)-(1.2).

Thus, are lower and upper solutions of (3.14), respectively. Since satisfies a Nagumo condition, there exists a constant (depending on and a Nagumo function) such that any solution of (3.14) with satisfies on

This proves that is a solution of (3.2).

Theorem 3.2.

Assume that and hold. Further, one assumes that

the function satisfies for where and

Then, the convergence of the sequence of approximate solutions (obtained in Theorem 3.1) is quadratic.

Proof.

where and ,

This proves the quadratic convergence in norm.

Example 3.3.

Let and be, respectively, lower and upper solutions of (3.60). Clearly and are not the solutions of (3.60) and Also, the assumptions of Theorem 3.1 are satisfied. Thus, the conclusion of Theorem 3.1 applies to the problem (3.60).

## Declarations

### Acknowledgment

The author is grateful to the referees and professor G. Infante for their valuable suggestions and comments that led to the improvement of the original paper.

## Authors’ Affiliations

## References

- Ladde GS, Lakshmikantham V, Vatsala AS:
*Monotone Iterative Techniques for Nonlinear Differential Equations, Monographs, Advanced Texts and Surveys in Pure and Applied Mathematics, 27*. Pitman, Boston, Mass, USA; 1985:x+236. - Nieto JJ, Jiang Y, Jurang Y: Monotone iterative method for functional-differential equations.
*Nonlinear Analysis: Theory, Methods & Applications*1998, 32(6):741–747. 10.1016/S0362-546X(97)00524-5View ArticleMathSciNetMATH - Vatsala AS, Yang J: Monotone iterative technique for semilinear elliptic systems.
*Boundary Value Problems*2005, 2005(2):93–106. 10.1155/BVP.2005.93View ArticleMathSciNetMATH - Drici Z, McRae FA, Devi JV: Monotone iterative technique for periodic boundary value problems with causal operators.
*Nonlinear Analysis: Theory, Methods & Applications*2006, 64(6):1271–1277. 10.1016/j.na.2005.06.033View ArticleMathSciNetMATH - Jiang D, Nieto JJ, Zuo W: On monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations.
*Journal of Mathematical Analysis and Applications*2004, 289(2):691–699. 10.1016/j.jmaa.2003.09.020View ArticleMathSciNetMATH - Nieto JJ, Rodríguez-López R: Monotone method for first-order functional differential equations.
*Computers & Mathematics with Applications*2006, 52(3–4):471–484. 10.1016/j.camwa.2006.01.012View ArticleMathSciNetMATH - Ahmad B, Sivasundaram S: The monotone iterative technique for impulsive hybrid set valued integro-differential equations.
*Nonlinear Analysis: Theory, Methods & Applications*2006, 65(12):2260–2276. 10.1016/j.na.2006.01.033View ArticleMathSciNetMATH - Cabada A, Nieto JJ: Rapid convergence of the iterative technique for first order initial value problems.
*Applied Mathematics and Computation*1997, 87(2–3):217–226. 10.1016/S0096-3003(96)00285-8View ArticleMathSciNetMATH - Lakshmikantham V, Nieto JJ: Generalized quasilinearization for nonlinear first order ordinary differential equations.
*Nonlinear Times and Digest*1995, 2(1):1–9.MathSciNetMATH - Bellman RE, Kalaba RE:
*Quasilinearization and Nonlinear Boundary-Value Problems, Modern Analytic and Computional Methods in Science and Mathematics*.*Volume 3*. American Elsevier, New York, NY, USA; 1965:ix+206. - Lakshmikantham V, Vatsala AS:
*Generalized Quasilinearization for Nonlinear Problems, Mathematics and Its Applications*.*Volume 440*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1998:x+276.View Article - Cabada A, Nieto JJ: Quasilinearization and rate of convergence for higher-order nonlinear periodic boundary-value problems.
*Journal of Optimization Theory and Applications*2001, 108(1):97–107. 10.1023/A:1026413921997View ArticleMathSciNetMATH - Ahmad B, Nieto JJ, Shahzad N: The Bellman-Kalaba-Lakshmikantham quasilinearization method for Neumann problems.
*Journal of Mathematical Analysis and Applications*2001, 257(2):356–363. 10.1006/jmaa.2000.7352View ArticleMathSciNetMATH - Eloe PW, Gao Y: The method of quasilinearization and a three-point boundary value problem.
*Journal of the Korean Mathematical Society*2002, 39(2):319–330.View ArticleMathSciNetMATH - Akyildiz FT, Vajravelu K: Existence, uniqueness, and quasilinearization results for nonlinear differential equations arising in viscoelastic fluid flow.
*Differential Equations & Nonlinear Mechanics*2006, 2006:-9. - Ahmad B: A quasilinearization method for a class of integro-differential equations with mixed nonlinearities.
*Nonlinear Analysis: Real World Applications*2006, 7(5):997–1004. 10.1016/j.nonrwa.2005.09.003View ArticleMathSciNetMATH - Krivec R, Mandelzweig VB: Quasilinearization method and WKB.
*Computer Physics Communications*2006, 174(2):119–126. 10.1016/j.cpc.2004.12.017View ArticleMathSciNetMATH - Amster P, De Nápoli P: A quasilinearization method for elliptic problems with a nonlinear boundary condition.
*Nonlinear Analysis: Theory, Methods & Applications*2007, 66(10):2255–2263. 10.1016/j.na.2006.03.016View ArticleMathSciNetMATH - Ahmad B, Alsaedi A, Alghamdi BS: Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions.
*Nonlinear Analysis: Real World Applications*2008, 9(4):1727–1740. 10.1016/j.nonrwa.2007.05.005View ArticleMathSciNetMATH - Ahmad B, Nieto JJ: Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions.
*Nonlinear Analysis: Theory, Methods & Applications*2008, 69(10):3291–3298. 10.1016/j.na.2007.09.018View ArticleMathSciNetMATH - Bicadze AV, Samarskiĭ AA: Some elementary generalizations of linear elliptic boundary value problems.
*Doklady Akademii Nauk SSSR*1969, 185: 739–740.MathSciNet - Gupta CP: A second order
-point boundary value problem at resonance.
*Nonlinear Analysis: Theory, Methods & Applications*1995, 24(10):1483–1489. 10.1016/0362-546X(94)00204-UView ArticleMathSciNetMATH - Ma R: Multiple positive solutions for nonlinear
-point boundary value problems.
*Applied Mathematics and Computation*2004, 148(1):249–262. 10.1016/S0096-3003(02)00843-3View ArticleMathSciNetMATH - Eloe PW, Ahmad B: Positive solutions of a nonlinear
th order boundary value problem with nonlocal conditions.
*Applied Mathematics Letters*2005, 18(5):521–527. 10.1016/j.aml.2004.05.009View ArticleMathSciNetMATH - Webb JRL, Infante G: Positive solutions of nonlocal boundary value problems: a unified approach.
*Journal of the London Mathematical Society*2006, 74(3):673–693. 10.1112/S0024610706023179View ArticleMathSciNetMATH - Khan RA: Generalized approximations and rapid convergence of solutions of
-point boundary value problems.
*Applied Mathematics and Computation*2007, 188(2):1878–1890. 10.1016/j.amc.2006.11.138View ArticleMathSciNetMATH - Pei M, Chang SK: The generalized quasilinearization method for second-order three-point boundary value problems.
*Nonlinear Analysis: Theory, Methods & Applications*2008, 68(9):2779–2790. 10.1016/j.na.2007.02.025View ArticleMathSciNetMATH - Pei M, Chang SK: A quasilinearization method for second-order four-point boundary value problems.
*Applied Mathematics and Computation*2008, 202(1):54–66. 10.1016/j.amc.2008.01.026View ArticleMathSciNetMATH - Ahmad B: Approximation of solutions of the forced Duffing equation with
-point boundary conditions.
*Communications in Applied Analysis*2009, 13(1):11–20.MathSciNetMATH - Wang L, Pei M, Ge W: Existence and approximation of solutions for nonlinear second-order four-point boundary value problems.
*Mathematical and Computer Modelling*2009, 50(9–10):1348–1359. 10.1016/j.mcm.2008.11.018View ArticleMathSciNetMATH - Chang Y-K, Nieto JJ, Li W-S: On impulsive hyperbolic differential inclusions with nonlocal initial conditions.
*Journal of Optimization Theory and Applications*2009, 140(3):431–442. 10.1007/s10957-008-9468-1View ArticleMathSciNetMATH - Graef JR, Webb JRL: Third order boundary value problems with nonlocal boundary conditions.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 71(5–6):1542–1551. 10.1016/j.na.2008.12.047View ArticleMathSciNetMATH - Grossinho M, Minhós FM: Upper and lower solutions for higher order boundary value problems.
*Nonlinear Studies*2005, 12(2):165–176.MathSciNetMATH

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.