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

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.

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.

In this paper, we develop the method of generalized quasilinearization to obtain a sequence of approximate solutions converging monotonically and quadratically to a unique solution of the following second-order point nonlocal boundary value problem

(1.1)

(1.2)

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

Here we remark that [26] studies (1.1) with the boundary conditions of the form

(1.3)

A perturbed integral equation equivalent to the problem (1.1) and (1.3) considered in [26] is

(1.4)

where

(1.5)

It can readily be verified that the solution given by (1.4) does not satisfy (1.1). On the other hand, by Green's function method, a unique solution of the problem (1.1) and (1.3) is

(1.6)

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

For we define where It can easily be verified that the homogeneous problem associated with (1.1)-(1.2) has only the trivial solution. Therefore, by Green's function method, the solution of (1.1)-(1.2) can be written as

(2.1)

where is the Green's function and is given by

(2.2)

Note that on

We say that is a lower solution of the boundary value problem (1.1) and (1.2) if

(2.3)

and is an upper solution of (1.1) and (1.2) if

(2.4)

Definition 2.1.

A continuous function is called a Nagumo function if

(2.5)

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.

Let us define so that and satisfies the boundary conditions

(2.6)

For the sake of contradiction, let have a positive maximum at some . If , then and On the other hand, in view of the decreasing property of in we have

(2.7)

which is a contradiction. If we suppose that has a positive maximum at , then it follows from the first of boundary conditions (2.6) that

(2.8)

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

Theorem 3.1.

Assume that

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

the function satisfies a Nagumo condition relative to and on where is a positive constant depending on and the Nagumo function . Further, there exists a function such that with on where

(3.1)

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

Proof.

For we define and consider the following modified -point BVP

(3.2)

We note that are, respectively, lower and upper solutions of (3.2) and for every we have

(3.3)

where As

(3.4)

so is a Nagumo function. Furthermore, there exists a constant depending on , and Nagumo function such that

(3.5)

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

Let us define a function by

(3.6)

In view of the assumption it follows that and satisfies on Therefore, by Taylor's theorem, we obtain

(3.7)

We set

(3.8)

and observe that

(3.9)

By the mean value theorem, we can find and ( depend on , resp.), such that

(3.10)

Letting

(3.11)

we note that

(3.12)

Let us define as

(3.13)

Clearly is continuous and bounded on and satisfies a Nagumo condition relative to . For every and , we consider the -point BVP

(3.14)

Using (3.9), (3.12) and (3.13), we have

(3.15)

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

Now, we choose and consider the problem

(3.16)

Using , (3.9), (3.12) and (3.13), we obtain

(3.17)

which imply that and are lower and upper solutions of (3.16). Hence by Theorems 2.2 and 2.3, there exists a unique solution of (3.16) such that

(3.18)

Note that the uniqueness of the solution follows by Theorem 2.3. Using (3.9) and (3.13) together with the fact that is solution of (3.16), we find that is a lower solution of (3.2), that is,

(3.19)

In a similar manner, it can be shown by using , (3.12), (3.13), and (3.19) that and are lower and upper solutions of the following -point BVP

(3.20)

Again, by Theorems 2.2 and 2.3, there exists a unique solution of (3.20) such that

(3.21)

Continuing this process successively, we obtain a bounded monotone sequence of solutions satisfying

(3.22)

where is a solution of the problem

(3.23)

and is given by

(3.24)

Since is bounded on , , therefore it follows that the sequences are uniformly bounded and equicontinuous on Hence, by Ascoli-Arzela theorem, there exist the subsequences and a function such that uniformly on as Taking the limit we find that which consequently yields

(3.25)

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.

Let us set so that satisfies the boundary conditions

(3.26)

In view of the assumption for every it follows that

(3.27)

Now, by Taylor's theorem, we have

(3.28)

where , , , on , and , with satisfying on Also, in view of (3.13), we have

(3.29)

where and ,

Now we show that By the mean value theorem, for every and we obtain

(3.30)

Let for some Then and (3.30) becomes

(3.31)

In particular, taking and using (3.27), we have

(3.32)

which contradicts that Similarly, letting for some we get a contradiction. Thus, it follows that for every , which implies that and consequently, (3.28) and (3.29) take the form

(3.33)

where and

(3.34)

Now, by a comparison principle, we can obtain on , where is a solution of the problem

(3.35)

Since is continuous and bounded on , there exist (independent of ) such that on Since on so we can rewrite (3.35) as

(3.36)

whose solution is given by

(3.37)

where

(3.38)

Introducing the integrating factor such that (3.34) takes the form

(3.39)

Integrating (3.39) from to and using we obtain

(3.40)

which can alternatively be written as

(3.41)

where , . Using the fact that together with (3.41) yields

(3.42)

which, on substitutingin (3.37), yields

(3.43)

where

(3.44)

Taking the maximum over and then solving (3.43) for we obtain

(3.45)

Also, it follows from (3.33) that

(3.46)

Integrating (3.46) from to and using (from the boundary condition we obtain

(3.47)

which, in view of the fact and (3.45), yields

(3.48)

where

(3.49)

As , there exists such that

(3.50)

Integrating (3.46) from to () and using (3.50), we have

(3.51)

Using (3.45) in (3.34), we obtain

(3.52)

where . Since is bounded on , we can choose such that on , and so that (3.52) takes the form

(3.53)

Integrating (3.53) from to (), and using (3.51), we find that

(3.54)

Letting

(3.55)

it follows from (3.51) and (3.54) that

(3.56)

Hence, from (3.48) and (3.56), it follows that

(3.57)

where From (3.45) and (3.57) with

(3.58)

we obtain

(3.59)

This proves the quadratic convergence in norm.

Example 3.3.

Consider the boundary value problem

(3.60)

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).

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.

Affiliations

1. Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

   • Ahmed Alsaedi

Corresponding author

Correspondence to Ahmed Alsaedi.

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