Journal of Applied Mathematics

Volume 2012 (2012), Article ID 159031, 17 pages

doi:10.1155/2012/159031

Research Article

Quasilinearization of the Initial Value Problem for Difference Equations with “Maxima”

S. Hristova, A. Golev, and K. Stefanova

Faculty of Mathematics and Informatics, Plovdiv University, Tzar Asen 24, 4000 Plovdiv, Bulgaria

Received 26 May 2012; Accepted 6 July 2012

Academic Editor: Mehmet Sezer

Copyright © 2012 S. Hristova et al. 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.

Abstract.

The object of investigation of the paper is a special type of difference equations containing the maximum value of the unknown function over a past time interval. These equations are adequate models of real processes which present state depends significantly on their maximal value over a past time interval. An algorithm based on the quasilinearization method is suggested to solve approximately the initial value problem for the given difference equation. Every successive approximation of the unknown solution is the unique solution of an appropriately constructed initial value problem for a linear difference equation with “maxima,” and a formula for its explicit form is given. Also, each approximation is a lower/upper solution of the given mixed problem. It is proved the quadratic convergence of the successive approximations. The suggested algorithm is realized as a computer program, and it is applied to an example, illustrating the advantages of the suggested scheme.

1. Introduction

In the last few decades, great attention has been paid to automatic control systems and their applications to computational mathematics and modeling. Many problems in the control theory correspond to the maximal deviation of the regulated quantity. In the case when the dynamic of these problems is modeled discretely, the corresponding equations are called difference equations with “maxima”. The presence of the maximum function in the equation requires not only more complicated calculations but also a development of new methods for qualitative investigations of the behavior of their solutions(see, e.g., the monograph [1]). The character of the maximum function leads to variety of the types of difference equations. Some special types of difference equations are studied in [2–18]. At the same time, when the unknown function at any point is presented in both sides of the equation nonlinearly as well as it is involved in the maximum function, the given equation is not possible to be solved in an explicit form. It requires development of some approximate methods for their solving.

In the present paper, a nonlinear difference equation of delayed type is considered. The equation contains the maximum of the unknown function over a discrete past time interval. The main goal of the paper is suggesting an algorithm for an approximate solving of an initial value problem for the given difference equation.

2. Preliminary Notes

Let , be the set of all integers, let be a given fixed integer and be such that . Denote by .

Note that for any function , , the equalities and hold.

Consider the following nonlinear difference equation with “maxima”:

with an initial condition

where the functions , , , and , the points are such that .

Definition 2.1. One will say that the function is a lower (upper) solution of the problem (2.1), (2.2), if

Let be given functions such that for . Define the following sets:

3. Comparison Results

In our further investigations, we will use the following results for difference inequalities with “maxima”.

Lemma 3.1 (existence and uniqueness). Let the following conditions be fulfilled:

(1)The function .(2)The functions , are such that

Then the initial value problem for linear difference equation with “maxima”

has an unique solution on the interval .

Proof. We will use the step method to solve the initial value problem (3.2). Assume for a fixed all values , are known. Then from (3.2), we obtain .

Consider the following two possible cases.

Case  1. Let , or for .

Therefore, . Then applying the inequality (3.1), we obtain the unique solution of the problem (3.2) given by the equality

Case  2. Let where .

If we assume then from (3.2), we obtain . The obtained contradiction proves and .

From the inequalities (3.1) and , it follows that and therefore, the unique solution of problem (3.2) is given by the equality

Thus, we receive the value , .

Lemma 3.2. Let the following conditions be fulfilled:

(1) The functions satisfy the inequality

(2) The function satisfies the inequalities

Then for .

Proof. Assume the claim of Lemma 3.2 is not true. Then there exists such that and for . Therefore, and according to inequality (3.6), we get

The above inequality contradicts (3.5).

Remark 3.3. Note if both for , then inequality (3.5) is satisfied.

Now, we will prove a linear difference inequality in which at any point both the unknown function and its maximum over a past time interval are involved also in the right side of the inequality.

In the proof of our preliminary results, we will need the following lemma.

Lemma 3.4 (see [2], Theorem  4.1.1). Let , and

Then for all , the following inequality is valid:

Now, we will solve a generalized linear difference inequality with “maxima.”

Lemma 3.5. Let the following conditions be fulfilled:

(1) The functions and

(2) The function and satisfies the inequalities

where .

Then,

Proof. Define a function by the equalities

It is easy to check that for any , the inequality holds. Also, from the definition of , it follows that , and for . Therefore, for any , we obtain

From inequality (3.14), it follows

According to Lemma 3.4 from inequality (3.15), we get for

Inequality (3.16) implies the validity of the required inequality (3.12).

4. Quasilinearization

We will apply the method of quasilinearization to obtain approximate solution of the IVP for the nonlinear difference equation with “maxima’’ (2.1), (2.2). We will prove the convergence of the sequence of successive approximations is quadratic.

Theorem 4.1. Let the following conditions be fulfilled:

(1) The functions are a lower and an upper solutions of the problem (2.1), (2.2), respectively, and such that for .(2) The function satisfies for the equality

 where the functions are continuous and twice continuously differentiable with respect to their second and third arguments and the following inequalities are valid for , :

 where

(3) The function . Then there exist two sequences and , , such that:

(a) The functions , are lower solutions of IVP (2.1), (2.2).(b) The functions , are upper solutions of IVP (2.1), (2.2).(c) The following inequalities hold for

(d) Both sequences are convergent on and their limits and are the minimal and the maximal solutions of IVP (2.1), (2.2) in . In the case, IVP (2.1), (2.2) has an unique solution in both limits coincide, that is, .(e) The convergence is quadratic, that is, there exist constants such that for the solution of IVP (2.1), (2.2) in , the inequalities

 hold, where for any function .

Proof. From Taylor formula and condition (2) of Theorem 4.1 for , , the following inequalities are valid:

Consider the initial value problem for the linear difference equation with “maxima”

where the functions are defined by the equalities

From inequality (4.4), it follows that and from inequalities (4.2), (4.5), we get for . According to Lemma 3.1 the IVP (4.10), (4.11) has an unique solution , defined on the interval .

Define a function by the equality . Then we get for .

Let . From the choice of the function and (4.10) for the function , we get

According to Lemma 3.2 for the function , it follows that for . Therefore, for .

Consider the linear difference equation with “maxima”

where the functions and are defined by equalities (4.12) and

According to Lemma 3.1, the linear initial value problem (4.14), (4.15) has a unique solution , defined on the interval .

Define a function by the equality . Then for .

Now, let . From the choice of the function and (4.14) for the function , we get

Inequality (4.17) proves the function satisfies inequality (3.6). According to Lemma 3.2, it follows that for . Therefore, for .

Define a function by the equality . Then for .

Let . Then for the function , we get

Inequality (4.18) proves the function satisfies inequality (3.6). According to Lemma 3.2, it follows that for . Therefore, for .

Furthermore, the functions and .

Now, we will prove that the function is a lower solution of (2.1), (2.2) on the interval .

Let . From the inequalities for , for , inequalities (4.9), definitions (4.12), and inequalities (4.2) which prove the monotonic property of the first derivatives of the functions and we get

Thus, the function is a lower solution of (2.1), (2.2) on .

In a similar way, we can prove that the function is an upper solution of (2.1), (2.2) on the interval .

Analogously, we can construct two sequences of functions and . If the functions and , , are obtained such that and the claims (a), (b), (c) of Theorem 4.1 are satisfied, then we consider the initial value problem for the linear difference equation with “maxima”

and the initial value problem for the linear difference equation with “maxima’’

where

Since , the first derivatives of the function and are nondecreasing in , and inequalities (4.7) hold, we obtain , and , that is, . Therefore, according to Lemma 3.1, the initial value problems (4.20), (4.21), and (4.22) have unique solutions and , , correspondingly.

The proof that the functions , , and they are lower/upper solutions of (2.1), (2.2) on the interval is the same as in the case of and we omit it.

For any fixed , the sequences and are monotone nondecreasing and monotone nonincreasing, respectively, and they are bounded by and . Therefore, they are convergent on , that is, there exist functions such that

From inequalities (4.7), it follows that .

Now, we will prove that for any the following equality holds:

Let be fixed. We denote . From inequalities (4.7) for every , the inequalities hold and thus, , that is, the sequence is monotone nondecreasing and bounded from above by . Therefore, there exists the limit .

From the monotonicity of the sequence of the lower solutions , we get that for it is fulfilled . Let be such that . From the inequalities for every it follows . Assume that . Then there exists a natural number such that the inequalities hold. Therefore, there exists such that or . The obtained contradiction proves the validity of the required inequality (4.25).

Analogously, we can show that the functions also satisfy (4.25).

Now, we will prove that the function is a solution of the IVP (2.1), (2.2) on .

Let . Take a limit as in (4.21) and get .

Therefore, the function satisfies equality (2.2) for .

Let . Taking a limit in (4.20) as and applying (4.25), we obtain the function satisfies equality (2.1) for .

In a similar way, we can prove that is a solution of the IVP (2.1), (2.2).

Therefore, we obtain two solutions of (2.1), (2.2) in .

In the case of uniqueness of the solution of (2.1), (2.2) in , we have for . In the case of nonuniqueness, let be another solution of (2.1), (2.2). Then it is easy to prove that , that is, is the minimal solution and is the maximal solution of (2.1), (2.2) in .

We will prove that the convergence of the sequences , is quadratic. Let be a solution of (2.1), (2.2) in .

Define the functions , by the equalities

It is obvious that for .

Let . According to the definitions of the functions , and the condition (2) of Theorem 4.1, we get

According to the mean value theorem, there exist points and such that

Then the following inequalities are valid:

From inequalities (4.29) and (4.31), we obtain

From inequalities (4.30), (4.31), we get

Since the second derivatives of the functions are continuous and bounded in , it follows from inequalities in (4.27), (4.32), and (4.33), there exist positive constants such that

Therefore,

From inequalities in (4.35), we obtain

where .

According to Lemma 3.5 from inequalities in (4.36), it follows

From (4.37) and the condition (2) of Theorem 4.1, it follows that there exist positive constants , where , such that

In a similar way, we can prove that there exist positive constants , where , such that

Inequalities (4.38), (4.39) and the definitions of the functions , imply the validity of (4.8), that is, the convergence of the monotone sequences and is quadratic.

5. Application

Now, we will give an example to illustrate the suggested above scheme for approximate obtaining of a solution.

Consider the following nonlinear difference equation with “maxima”:

with an initial condition

The function , , is a lower solution of (5.1), (5.2) because the inequality holds.

The function , , is an upper solution of (5.1), (5.2) because the inequality holds.

The conditions of Theorem 4.1 are satisfied since and for . Also, the inequality (4.5) holds, because in this case , and .

According to Theorem 4.1, the initial value problem (5.1), (5.2) has a solution which is between and . It is obviously the problem (5.1), (5.2) has a zero solution. This solution also could be obtained by constructing two sequences of successive approximations.

The successive approximation is a solution of (4.20), (4.21) which is reduced to the following initial value problem:

and the successive approximation is a solution of (4.22) which is reduced to the following initial value problem:

where

Initial value problems (5.3) and (5.4) are solved by a computer program, using the algorithm given in the proof of Lemma 3.1 and the results are written in Table 1.

Table 1: Values of successive approximations and , .

1
2
3

0.126984126984127
0.126984126984127
0.126984126984127

0.00245312888643079
0.00245312888643079
0.0025469207078195

5.93137225955726E-07
5.93137225955726E-07
9.79796674723704E-07

2.96900520626677E-14
2.96900520626677E-14
9.99338575141566E-14

2.13083324289133E-17
2.13083324289133E-17
1.43434637320051E-16

−1.16621056078124E-17
−1.16621056078124E-17
−6.48937233257787E-18

−4.08173696273438E-17
−4.08173696273438E-17
−2.74428886272319E-15

−7.22563191957548E-09
−7.22563191957548E-09
−2.43398432836776E-07

−0.000449855202856569
−0.000449855202856569
−0.00207365792650451

−0.115942028985507
−0.115942028985507
−0.126023944549464

Table 1 demonstrates both sequences monotonically approach the exact zero solution. This illustrates the application of the proved above procedure for approximately obtaining of the solution.

Acknowledgments

The research in this paper was partially supported by Fund Scientific Research MU11FMI005/29.05.2011, Plovdiv University, Bulgaria and BG051PO001/3.3-05-001 Science and Business, financed by the Operative Program “Development of Human Resources”, European Social Fund.

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