- Research Article
- Open access
- Published:
Approximation of Solutions for Second-Order
-Point Nonlocal Boundary Value Problems via the Method of Generalized Quasilinearization
Boundary Value Problems volume 2011, Article number: 929061 (2011)
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 SamarskiÄ [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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ1_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ3_HTML.gif)
A perturbed integral equation equivalent to the problem (1.1) and (1.3) considered in [26] is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ4_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ6_HTML.gif)
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
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ7_HTML.gif)
where is the Green's function and is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ8_HTML.gif)
Note that on
We say that is a lower solution of the boundary value problem (1.1) and (1.2) if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ9_HTML.gif)
and is an upper solution of (1.1) and (1.2) if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ10_HTML.gif)
Definition 2.1.
A continuous function is called a Nagumo function if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ11_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ12_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ13_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ14_HTML.gif)
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
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ15_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ16_HTML.gif)
We note that are, respectively, lower and upper solutions of (3.2) and for every
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ17_HTML.gif)
where As
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ18_HTML.gif)
so is a Nagumo function. Furthermore, there exists a constant
depending on
, and Nagumo function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ19_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ20_HTML.gif)
In view of the assumption it follows that
and satisfies
on
Therefore, by Taylor's theorem, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ21_HTML.gif)
We set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ22_HTML.gif)
and observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ23_HTML.gif)
By the mean value theorem, we can find and
(
depend on
, resp.), such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ24_HTML.gif)
Letting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ25_HTML.gif)
we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ26_HTML.gif)
Let us define as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ27_HTML.gif)
Clearly is continuous and bounded on
and satisfies a Nagumo condition relative to
. For every
and
, we consider the
-point BVP
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ28_HTML.gif)
Using (3.9), (3.12) and (3.13), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ29_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ30_HTML.gif)
Using , (3.9), (3.12) and (3.13), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ31_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ32_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ33_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ34_HTML.gif)
Again, by Theorems 2.2 and 2.3, there exists a unique solution of (3.20) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ35_HTML.gif)
Continuing this process successively, we obtain a bounded monotone sequence of solutions satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ36_HTML.gif)
where is a solution of the problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ37_HTML.gif)
and is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ38_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ39_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ40_HTML.gif)
In view of the assumption for every
it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ41_HTML.gif)
Now, by Taylor's theorem, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ42_HTML.gif)
where ,
,
,
on
,
and
,
with
satisfying
on
Also, in view of (3.13), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ43_HTML.gif)
where and
,
Now we show that By the mean value theorem, for every
and
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ44_HTML.gif)
Let for some
Then
and (3.30) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ45_HTML.gif)
In particular, taking and using (3.27), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ46_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ47_HTML.gif)
where and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ48_HTML.gif)
Now, by a comparison principle, we can obtain on
, where
is a solution of the problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ49_HTML.gif)
Since is continuous and bounded on
, there exist
(independent of
) such that
on
Since
on
so we can rewrite (3.35) as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ50_HTML.gif)
whose solution is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ51_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ52_HTML.gif)
Introducing the integrating factor such that
(3.34) takes the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ53_HTML.gif)
Integrating (3.39) from to
and using
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ54_HTML.gif)
which can alternatively be written as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ55_HTML.gif)
where ,
. Using the fact that
together with (3.41) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ56_HTML.gif)
which, on substitutingin (3.37), yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ57_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ58_HTML.gif)
Taking the maximum over and then solving (3.43) for
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ59_HTML.gif)
Also, it follows from (3.33) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ60_HTML.gif)
Integrating (3.46) from to
and using
(from the boundary condition
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ61_HTML.gif)
which, in view of the fact and (3.45), yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ62_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ63_HTML.gif)
As , there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ64_HTML.gif)
Integrating (3.46) from to
(
) and using (3.50), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ65_HTML.gif)
Using (3.45) in (3.34), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ66_HTML.gif)
where . Since
is bounded on
,
we can choose
such that
on
,
and
so that (3.52) takes the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ67_HTML.gif)
Integrating (3.53) from to
(
), and using (3.51), we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ68_HTML.gif)
Letting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ69_HTML.gif)
it follows from (3.51) and (3.54) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ70_HTML.gif)
Hence, from (3.48) and (3.56), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ71_HTML.gif)
where From (3.45) and (3.57) with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ72_HTML.gif)
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ73_HTML.gif)
This proves the quadratic convergence in norm.
Example 3.3.
Consider the boundary value problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ74_HTML.gif)
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).
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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.
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Alsaedi, A. Approximation of Solutions for Second-Order -Point Nonlocal Boundary Value Problems via the Method of Generalized Quasilinearization.
Bound Value Probl 2011, 929061 (2011). https://doi.org/10.1155/2011/929061
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DOI: https://doi.org/10.1155/2011/929061