# 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

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