Abstract and Applied Analysis

Volume 2013 (2013), Article ID 571954, 9 pages

http://dx.doi.org/10.1155/2013/571954

Research Article

Monotone-Iterative Method for Solving Antiperiodic Nonlinear Boundary Value Problems for Generalized Delay Difference Equations with Maxima

Angel Golev,¹ Snezhana Hristova,¹ and Svetoslav Nenov²

¹Faculty of Mathematics and Informatics, Plovdiv University, Tzar Asen 24, 4000 Plovdiv, Bulgaria

²Department of Mathematics, University of Chemical Technology and Metallurgy, Kl. Ohridski 8, 1756 Sofia, Bulgaria

Received 26 February 2013; Revised 30 June 2013; Accepted 17 July 2013

Academic Editor: Ferhan M. Atici

Copyright © 2013 Angel Golev 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.

A nonlinear generalized difference equation with both delays and the maximum value of the unknown function over a discrete past time interval are studied. A nonlinear boundary value problem of antiperiodic type for the given difference equation is set up. One of the main characteristics of the considered difference equation is the presence of the unknown function in both sides of the equation. It leads to impossibility for using the step method for explicit solving of the nonlinear difference equation. In this paper, an approximate method, namely, the monotone iterative technique, is applied to solve the problem. An important feature of the given algorithm is that each successive approximation of the unknown solution is equal to the unique solution of an appropriately constructed initial value problem for a linear difference equation with “maxima,” and an algorithm for its explicit solving is given. Also, each approximation is a lower/upper solution of the given nonlinear boundary value problem. The suggested scheme for approximate solving is computer realized, and it is applied to a particular example, which is a generalization of a model in population dynamics.

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 these problems are modeled discretely the corresponding equations are called difference equations with maxima. The presence of the maximum function in equations mostly requires new techniques for investigation of the qualitative behavior of their solutions. Note also differential equations with maxima are not well studied and still not widely applied as models (see, e.g., [1–5], the monograph [6], and cited therein references). The character of the maximum function leads to a variety of different types of difference equations. The properties of solutions of some special types of difference equations with maxima have been studied in [7–9]. However, such nonlinear difference equations can hardly be solved in explicit forms, and hence we need to develop some approximate methods for their solutions.

Note in several papers various types of boundary value problems for difference equations have been studied, and monotone iterative method has been applied. For example, in [10–13] first order difference equations have been studied in [14, 15] some criteria for existence and uniqueness results for th order antiperiodic difference equations have been developed; in [16] a generalized delay difference equation is studied by lower and upper solutions, but the problem consists only of a boundary condition, which does not get uniqueness of the solution; the global boundary value problem for difference equations without any kind of delay is well studied by Cabada in [17]; in [18] a nonlinear boundary value problem for a delay difference equation with one delay is studied by monotone iterative method; in [19] an approximate method with a rapid convergence is applied to an initial value problem for difference equations with maxima, in [20] the monotone-iterative technique for nonlinear boundary value problem for difference equations with various delays as well as maxima is applied. Also, in [21], monotone iterative technique is applied to a periodic boundary value problem for difference equations with maxima but the successive approximation are solutions of periodic boundary value problems, which is practically difficult to be obtained.

However, in connection with the applications of difference equations as models of real world problems it is necessary to study difference equations with both delays and maximum over a discrete past time interval.

In this paper the authors consider a class of nonlinear difference equations with both delays and maximum of the unknown function over a discrete past time interval. A nonlinear boundary value problem of antiperiodic type is studied. An algorithm for constructing approximate solutions of this type of boundary value problem is developed. Any successive approximation is a solution of an initial value problem for a linear difference equation, and a practical algorithm for its obtaining is given and computer realized.

2. Statement of the Problem

Let , be the set of all integers. For any we denote .

Let , , and be fixed.

Consider the following nonlinear mixed boundary value problem for the generalized nonlinear delay-difference equation with maxima (BVP)

where , , , , for .

Any solution of BVP (1)-(2) is a finite sequence of real numbers, and we will consider it as a real valued function with a discrete domain.

In this paper we will study boundary condition (2) in the case when it is from antiperiodic type. The studied problem includes as partial cases the initial value problem ( for ) and the antiperiodic boundary value problem ( for ).

The presence of the delays generalizes the type of the considered difference equation since the function could depend on different delays at any point . As a partial case and , the right side of (1) is reduced to . For some other particular cases see Example 7 of the paper.

Note the explicit solving of BVP (1)-(2) is often impossible because of the nonlinearity and the presence of the unknown function , , in both sides of (1). It requires an application of approximate methods, one of which is an object of investigation in the paper.

3. Preliminary Notes, Basic Notations, and Definitions

Note, and where and , .

For any function we introduce the following notations:

Using the previous notations the right side part of (1) could be written in a simpler way and the nonlinear function in boundary condition (2) could be written as .

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

Definition 1. The function is said to be from the class if for and for the inequality holds and there exists a constant such that for , the inequality holds for any .

Remark 2. Note if the function is from the class then BVP (1)-(2) is from antiperiodic type.

In connection with the construction of successive approximations we will introduce a couple of quasi-solutions of BVP (1)-(2):

Definition 3. One will say that the functions form a couple of quasi-solutions of BVP (1)-(2), if they satisfy (1) and

Definition 4. One will say that the functions form a couple of quasi lower and upper solutions of BVP (1)-(2), if

4. Linear Delay Difference Inequalities with Maxima

One will consider the following linear difference inequality with maxima

Lemma 5 (Comparison result). Let the function satisfy inequalities (7) and

where and for , .

Then for .

Proof. Assume that the claim of Lemma 5 is not true. Consider the following two cases.

Case  1. Let . Therefore, there exists such that for and .

Denote . Let , , be such that . Then , , and for all .

Therefore,

Inequality (9) contradicts (8).

Case  2. Let . Then there exists such that for and .

Let . As in the proof of Case 1 for we get a contradiction.

Let . Introduce the following sets:

Note that the set could be empty. Therefore, for and for . Then from inequality (7) we get

The obtained contradiction proves .

5. Linear Delay Difference Equations with Maxima

In connection with the computer realization of the suggested method, we will give an algorithm for exact solving of the initial value problem for the scalar linear delay-difference equation with maxima (IVP)

where , , and for , .

Note that the unknown function at any moment is included in the maximum function of the right side part of (12) and it is not obvious to solve (12). We will give the algorithm for obtaining the solution.

We will consider two cases with respect to the type of delays .

Case 1. Let the inequality hold for all , .

Assume that the values of the unknown solution are obtained for all , where . Now let .

Case  1.1. Let the following inequality be satisfied:

Therefore, . Then the unique solution of IVP (12)-(13) is given by

Case  1.2. Let inequality (14) be not satisfied for all . Therefore, there exists such that .

Then the unique solution of IVP (12)-(13) is

Case 2. Let there exists at least one and such that .

Assume that the values of the unknown solution are obtained for all , where .

Now let . Introduce the following sets:

Note that the set could be empty. Therefore, for and for . Equation (12) is reduced to

Case  2.1. Let the following inequality be satisfied

Therefore, and from (18) we obtain the solution

Case  2.2. Let inequality (19) be not satisfied for all .

Therefore, there exists such that .

Then the unique solution of problem (12)-(13) is given by

6. Monotone-Iterative Method

We will give an algorithm for approximatly obtaining the solution of the mixed boundary value problem for the nonlinear delay-difference equation with maxima (1)-(2) as a limit of two monotone sequences.

Theorem 6. Let the following conditions be fulfilled:

(1) The functions form a couple of quasi lower and upper solutions of (1)-(2) such that for .(2) The function is continuous in for any and for , , with , the inequality

 holds, where the functions , , and inequality (8) is satisfied.(3) The function .(4) The functions for , .

Then there exist two sequences of functions and such that

(a) The functions , , () form a couple of quasi lower and upper solutions of BVP (1)-(2). (b) For the inequalities

 hold. (c) Both sequences are convergent on and their limits and form a couple of quasi solutions of BVP (1)-(2) in . In the case of uniqueness of the solution of BVP (1)-(2) in the equality holds, where is the unique solution of (1)-(2) in .

Proof. Note if , then , , and for . Therefore, according to condition 2 of Theorem 6 if the functions satisfy for , then for any the inequality

holds.

We will give an algorithm for construction of successive approximations to the unknown exact solution of BVP (1)-(2).

Assume the functions and are constructed such that

(H1), for , ; (H2) the functions and , form a couple of quasi lower and upper solutions of BVP (1)-(2); (H3), , ; (H4) each function , , is the unique solution of the initial value problem for the linear difference equation with maxima

 where is defined by

(H5) each function , , is the unique solution of the initial value problem

Now we will explain how to construct the next approximations and .

Consider the initial value problem (25) for . According to Section 4 linear initial value problem (25) has a unique solution .

Define a function by .

From claims (H2) and (H4) we get for and .

Let . Then from claims (H1), (H4), and inequality

we get

According to Lemma 5   for . Hence, for .

Consider the linear initial value problem (27) for . According to Section 4 the linear initial value problems (27) has a unique solution .

Similarly as previously mentioned we prove for ; that is, claim (H1) is satisfied for .

Now we will prove (H3) for . Define a function by .

From claims (H2), (H3), and condition 3 of Theorem 6 we have for . Also,

Let . From claims (H1), (H5), and inequality (24) we get

In view of Lemma 5 it follows for ; that is, claim (H3) is satisfied for .

Therefore, .

We will prove (H2) for .

From condition 3 of Theorem 6, claim (H2) for and the construction of functions and we obtain

Let . From claims (H1), (H4), and inequality (24) we get

Similarly, we prove

From inequalities (32), (33), and (34) it follows the functions and form a couple of quasi lower and upper solution of BVP (1)-(2) on ; that is, claim (H2) is satisfied for .

Following the suggested previous scheme we can construct two sequences of functions and which satisfy claims (H1)–(H5).

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

From claim (H3) it is clear that .

Also for any the equalities

hold (for the proof see [19]).

We will prove that the limit is a solution of BVP (1)-(2) on .

Taking a limit as in the last equalities of (25) and (27) we get and for . Also we obtain and or and .

Taking a limit as in the first equality of (25) and (27) we get the functions and satisfy equality (1) for .

Therefore, functions and form a couple of quasi solutions of BVP (1)-(2) in .

If nonlinear boundary value problem (1)-(2) has an unique solution in , then .

7. Applications

Now we will give an example which exact solution will be obtained by a computer realization of the suggested previous algorithm. The example is a modification of the equation which is used in population biology to model a situation where the species survive if the initial size of population is greater than a certain threshold.

Example 7. Consider the nonlinear boundary value problem for the nonlinear difference equation with maxima

where , , and denotes the integer part of the real number ; that is,

Boundary value problem for the nonlinear delay difference equation with maxima (37)-(38) is from type (1)-(2), where , , , , .

The functions and , form a couple of quasi lower and upper solutions of (37), (38), respectively, because , , , and .

Consider the function which is increasing in . Let . Then ; that is, and the boundary condition (38) is of antiperiodic type.

Let , , , . Then

that is, , , and . Note that the inequality (8) is satisfied because .

According to Theorem 6 the nonlinear boundary value problem (37)-(38) has a solution , . We will obtain as a limit of two sequences of successive approximations.

The approximation , , is a solution of the initial value problem for the linear delay difference equation with maxima (25) which is reduced to

and the approximation , , is a solution of IVP (27) which is reduced to

where .

The algorithm for solving linear problems (41) and (42) is given in Section 4. It is computer realized, and some of the results are given in Table 1.

From Table 1 it could be seen that the sequence is increasing and the sequence is decreasing and , . The unknown exact solution , and its approximate values are written in Table 1.

Table 1: Values of the successive lower/upper approximations and , .

0
3
4
5

9
10

0.300000
0.300000
0.300000
0.300000

0.300000
0.300000

0.107373
0.102318
0.097378
0.092799

0.079779
0.077504

−0.161693
−0.140751
−0.121361
−0.104382

−0.054887
−0.045805

−0.350522
−0.313072
−0.277815
−0.246387

−0.150852
−0.132489

−0.450828
−0.404758
−0.361295
−0.322444

−0.203309
−0.180134

−0.495330
−0.445438
−0.398367
−0.356278

−0.226976
−0.201742

−0.513731
−0.462242
−0.413672
−0.370244

−0.236787
−0.210719

−0.521346
−0.469185
−0.419987
−0.376000

−0.240816
−0.214404

−0.524570
−0.472122
−0.422654
−0.378427

−0.242506
−0.215947

−0.526
−0.474
−0.424
−0.380

−0.244
−0.217

−0.529472
−0.476579
−0.426694
−0.382097

−0.245037
−0.218253

−0.532445
−0.479278
−0.429136
−0.384311

−0.246548
−0.219624

−0.538843
−0.485075
−0.434371
−0.389046

−0.249751
−0.222525

−0.551920
−0.496893
−0.445013
−0.398643

−0.256153
−0.228299

−0.576447
−0.518984
−0.464830
−0.416445

−0.267835
−0.238790

−0.617154
−0.555535
−0.497512
−0.445709

−0.286810
−0.255788

−0.677874
−0.610052
−0.546283
−0.489435

−0.315694
−0.281933

−0.783286
−0.704773
−0.631632
−0.567151

−0.376223
−0.340584

−1.000000
−1.000000
−1.000000
−1.000000

−1.000000
−1.000000

Acknowledgments

Research was partially supported by Project BG051PO001/3.3-05-001 Science and Business, financed by the Operative Program “Development of Human Resources,” European Social Fund.

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