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

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 𝑛th 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 ℝ₊=[0,∞), ℤ be the set of all integers. For any 𝑐,𝑏∈ℤ∶𝑐<𝑏 we denote ℤ[𝑐,𝑏]={𝑧∈ℤ∶𝑐≤𝑧≤𝑏}.

Let 𝑎,𝑇∈ℤ∶𝑇>𝑎+1, 𝑟,𝑝∈ℤ[0,𝑇−𝑎], and ℎ∈ℤ∶ℎ>0 be fixed.

Consider the following nonlinear mixed boundary value problem for the generalized nonlinear delay-difference equation with maxima (BVP) ()(𝑘)𝜏Δ𝑢𝑘−1=𝑓𝑘,𝑢,𝑢₁(𝑘),𝑢𝜏₂(𝑘)𝑢𝜏,…,_𝑟(𝑘),max_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝑢(𝑠)[],𝑔𝑢(𝑎),for𝑘∈ℤ𝑎+1,𝑇,𝑢𝑇−𝜆₁,𝑢𝑇−𝜆₂,…,𝑢𝑇−𝜆_𝑝(𝑇)𝑢(𝑘)(𝑎)[],,𝑢=0,=𝑢for𝑘∈ℤ𝑎−ℎ+1,𝑎−1 (1)(2) where 𝑢∈ℝ, Δ𝑢(𝑘−1)=𝑢(𝑘)−𝑢(𝑘−1), 𝑓∶ℤ[𝑎+1,𝑇]×ℝ^𝑟+2→ℝ, 𝑔∶ℝ^𝑝+2→ℝ, 𝜏_𝑚(𝑘)∶ℤ[𝑎+1,𝑇]→ℤ[𝑎+1−ℎ,𝑇]∶𝑘−ℎ≤𝜏_𝑚(𝑘)≤𝑘 for 𝑚=1,2,…,𝑟.

Any solution of BVP (1)-(2) is a finite sequence of 𝑇−𝑎+ℎ 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 (𝑔(𝑥)≡𝑥+const for 𝑥∈ℝ) and the antiperiodic boundary value problem (𝑔(𝑥,𝑢)≡𝑥+𝑢 for 𝑥,𝑢∈ℝ).

The presence of the delays 𝜏_𝑚 generalizes the type of the considered difference equation since the function 𝑓 could depend on different delays at any point 𝑘. As a partial case 𝑟=ℎ and 𝜏_𝑗(𝑘)=𝑘−𝑗, 𝑗=1,…,𝑟 the right side of (1) is reduced to 𝑓(𝑘,𝑢(𝑘−ℎ),𝑢(𝑘−ℎ+1),…,𝑢(𝑘−1),𝑢(𝑘),max_{𝑠∈ℤ[𝑘−ℎ,𝑘]}𝑢(𝑠)). 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 𝑢(𝑘), 𝑘∈ℤ[𝑎+1,𝑇], 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, ∑^𝑚_𝑖=𝑛𝑎_𝑖=0 and ∏^𝑚_𝑖=𝑛𝑎_𝑖=1 where 𝑚<𝑛<∞ and 𝑎_𝑗∈ℝ, 𝑗∈𝑍[𝑚,𝑛].

For any function 𝑢∶ℤ[𝑎−ℎ+1,𝑇]→ℝ we introduce the following notations: 𝜒(𝑢(𝑇))=𝑢𝑇−𝜆₁,𝑢𝑇−𝜆₂,…,𝑢𝑇−𝜆_𝑝(𝑇,𝑈(𝑢(𝜏(𝑘=𝑢𝜏,𝑢))))₁(𝑘),𝑢𝜏₂(𝑘),…,𝑢𝜏_𝑟(𝑘),[],Ξ(𝑢(𝜏(𝑘=𝑢(𝑘)(𝑢(𝜏(𝑘𝑘∈ℤ𝑎+1,𝑇))),𝑈))),max_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝑢(𝑠),𝑘∈ℤ[𝑎+1,𝑇]. (3)

Using the previous notations the right side part of (1) could be written in a simpler way 𝑓(𝑘,Ξ(𝑢(𝜏(𝑘)))) and the nonlinear function in boundary condition (2) could be written as 𝑔(𝑢(𝑎),𝜒(𝑢(𝑇))).

Let 𝛼,𝛽∶ℤ[𝑎+1−ℎ,𝑇]→ℝ be given functions such that 𝛼(𝑘)≤𝛽(𝑘) for 𝑧∈ℤ[𝑎+1−ℎ,𝑇]. Introduce the following sets: 𝑆=[](𝑘)(𝑘)(𝑘),[,𝑊=𝛼,𝛽𝑢∶ℤ𝑎−ℎ+1,𝑇⟶ℝ∶𝛼≤𝑢≤𝛽𝑘∈ℤ𝑎−ℎ+1,𝑇]𝛼,𝛽𝑥∈ℝ^𝑝+1(𝛼(𝑇𝛽(𝑇),Ω∶𝜒))≤𝑥≤𝜒_𝑘𝛼,𝛽=𝑉∈ℝ^𝑟+2(𝛼(𝜏(𝑘𝛽(𝜏(𝑘,[].∶Ξ)))≤𝑉≤Ξ))𝑘∈ℤ𝑎+1,𝑇 (4)

Definition 1. The function 𝑔(𝑥,𝑦)∈𝐶([𝛼(𝑎),𝛽(𝑎)]×𝑊(𝛼,𝛽),ℝ) is said to be from the class Λ(𝛼,𝛽) if for 𝑥∈[𝛼(𝑎),𝛽(𝑎)] and for 𝑦₁,𝑦₂∈𝑊(𝛼,𝛽)∶𝑦₁≤𝑦₂ the inequality 𝑔(𝑥,𝑦₁)≤𝑔(𝑥,𝑦₂) holds and there exists a constant 𝛾>0 such that for 𝑢₁,𝑢₂∈[𝛼(𝑎),𝛽(𝑎)]∶𝑢₂≤𝑢₁, the inequality 𝑔(𝑢₁,𝑣)−𝑔(𝑢₂,𝑣)≤𝛾(𝑢₁−𝑢₂) holds for any 𝑣∈𝑊(𝛼,𝛽).

Remark 2. Note if the function 𝑔(𝑥,𝑦) is from the class Λ(𝛼,𝛽) 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 𝛼,𝛽∈ℤ[𝑎−ℎ+1,𝑇] form a couple of quasi-solutions of BVP (1)-(2), if they satisfy (1) and 𝑔𝛼(𝑎),𝛽𝑇−𝜆₁,𝛽𝑇−𝜆₂,…,𝛽𝑇−𝜆_𝑝(𝑇𝑔𝛽(𝑎),𝛽)=0,,𝛼𝑇−𝜆₁,𝛼𝑇−𝜆₂,…,𝛼𝑇−𝜆_𝑝(𝑇)𝛼(𝑘)(𝑎)(𝑘)(𝑎),[].,𝛼=0,=𝛼,𝛽=𝛽for𝑘∈ℤ𝑎−ℎ+1,𝑎−1 (5)

Definition 4. One will say that the functions 𝛼,𝛽∈ℤ[𝑎−ℎ+1,𝑇] form a couple of quasi lower and upper solutions of BVP (1)-(2), if ()(𝑘)𝜏Δ𝛼𝑘−1≤𝑓𝑘,𝛼,𝛼₁(𝑘),𝛼𝜏₂(𝑘)𝛼𝜏,…,_𝑟(𝑘),max_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝛼(𝑠)()(𝑘)𝜏Δ𝛽𝑘−1≥𝑓𝑘,𝛽,𝛽₁(𝑘),𝛽𝜏₂(𝑘)𝛽𝜏,…,_𝑟(𝑘),min_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝛽(𝑠)[],𝑔𝛼(𝑎)for𝑘∈ℤ𝑎+1,𝑇,𝛽𝑇−𝜆₁,𝛽𝑇−𝜆₂,…,𝛽𝑇−𝜆_𝑝(𝑇)𝑔𝛽(𝑎),𝛽≤0,,𝛼𝑇−𝜆₁,𝛼𝑇−𝜆₂,…,𝛼𝑇−𝜆_𝑝(𝑇)𝛼(𝑘)(𝑎)(𝑘)(𝑎),[].,𝛼≥0,=𝛼,𝛽=𝛽for𝑘∈ℤ𝑎−ℎ+1,𝑎−1 (6)

4. Linear Delay Difference Inequalities with Maxima

One will consider the following linear difference inequality with maxima Δ𝑢(𝑘−1)≤−𝑄(𝑘)𝑢(𝑘)−^𝑟_𝑗=1𝐶_𝑗(𝑘)𝑢𝜏_𝑗(𝑘(𝑘))−𝑞min_{𝑠∈ℤ[𝑘−ℎ,𝑘]}𝑢(𝑠)[],𝑢(𝑘)(𝑎)[].,𝑘∈ℤ𝑎+1,𝑇=𝑢≤0,𝑘∈ℤ𝑎−ℎ+1,𝑎−1 (7)

Lemma 5 (Comparison result). Let the function 𝑢∶ℤ[𝑎−ℎ+1,𝑇]→ℝ satisfy inequalities (7) and ^𝑇_𝑙=𝑎+1𝑄(𝑙)+^𝑟_𝑗=1𝐶_𝑗(𝑙)+𝑞(𝑙)≤1, (8) where 𝑄,𝑞,𝐶_𝑗∶ℤ[𝑎+1,𝑇]→ℝ₊ and 𝜏_𝑗(𝑘)∶ℤ[𝑎+1,𝑇]→ℤ[𝑎+1−ℎ,𝑇]∶𝑘−ℎ≤𝜏_𝑗(𝑘)≤𝑘 for 𝑗∈ℤ[1,𝑟], 𝑘∈ℤ[𝑎+1,𝑇].
Then 𝑢(𝑡)≤0 for 𝑡∈ℤ[𝑎−ℎ+1,𝑇].

Proof. Assume that the claim of Lemma 5 is not true. Consider the following two cases.
Case  1. Let 𝑢(𝑎)<0. Therefore, there exists 𝜂∈ℤ[𝑎+1,𝑇] such that 𝑢(𝑘)≤0 for 𝑘∈ℤ[𝑎−ℎ+1,𝜂−1] and 𝑢(𝜂)>0.
Denote min_{𝑠∈ℤ[𝑎−ℎ+1,𝜂]}𝑢(𝑠)=−𝜆<0. Let 𝜍∈ℤ[𝑎,𝜂], 𝜍<𝜂, be such that 𝑢(𝜍)=−𝜆. Then 𝑢(𝜏_𝑗(𝑙))≥−𝜆, 𝑗∈ℤ[1,𝑟], and min_{𝑠∈ℤ[𝑙−ℎ,𝑙]}𝑢(𝑠)≥−𝜆 for all 𝑙∈ℤ[𝜍+1,𝜂].
Therefore, 𝜂(𝜍)=𝑢𝜂+𝑢(𝑢()(𝜍(𝜍)≤𝜆<𝑢−𝑢−𝑢𝜂−1𝜂−1−𝑢𝜂−2+⋯+𝜍+1−𝑢))=Δ𝑢𝜂−1+Δ𝑢𝜂−2+⋯+Δ𝑢_{𝜂𝑙=𝜍+1}−𝑄(𝑙)𝑢(𝑙)−^𝑟_𝑗=1𝐶_𝑗(𝑙)𝑢𝜏_𝑗(𝑙(𝑙))−𝑞min_{[]𝑠∈ℤ𝑙−ℎ,𝑙}𝑢(𝑠)≤𝜆^𝜂_𝑙=𝜍+1𝑄(𝑙)+^𝑟_𝑗=1𝐶_𝑗(𝑙)+𝑞(𝑙)≤𝜆^𝑇_𝑙=𝑎+1𝑄(𝑙)+^𝑟_𝑗=1𝐶_𝑗(𝑙)+𝑞(𝑙). (9)
Inequality (9) contradicts (8).
Case  2. Let 𝑢(𝑎)=0. Then there exists 𝜂∈ℤ[𝑎,𝑇−1] such that 𝑢(𝑘)=0 for 𝑘∈ℤ[𝑎+1−ℎ,𝜂] and 𝑢(𝜂+1)≠0.
Let 𝑢(𝜂+1)<0. As in the proof of Case 1 for 𝑎=𝜂+1 we get a contradiction.
Let 𝑢(𝜂+1)>0. Introduce the following sets: Ω_𝜂=𝑗∈ℤ[1,𝑟]∶𝜏_𝑗𝜂+1=𝜂+1,Ω_𝜂=𝑗∶𝑗∈ℤ[1,𝑟],𝑗∉Ω_𝜂. (10)
Note that the set Ω_𝜂 could be empty. Therefore, 𝑢(𝜏_𝑗(𝜂+1))=𝑢(𝜂+1)>0 for 𝑗∈Ω_𝜂 and 𝑢(𝜏_𝑗(𝜂+1))=0 for 𝑗∈Ω_𝜂. Then from inequality (7) we get 1+𝑄𝜂+1+_{𝑗∈Ω𝜂}𝐶_𝑗𝜂+1𝑢𝜂+1≤−𝑞𝜂+1min0,𝑢𝜂+1=0. (11)
The obtained contradiction proves 𝑢(𝜂+1)<0.

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) ()(𝑘)𝑢(𝑘)(𝑘)Δ𝑢𝑘−1=−𝑄−𝑞max_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝑢(𝑠)−^𝑟_𝑗=1𝐶_𝑗(𝑘)𝑢𝜏_𝑗(𝑘(𝑘)[],𝑢(𝑘)(𝑎)[],)+𝑃,𝑘∈ℤ𝑎+1,𝑇=𝑢,𝑘∈ℤ𝑎−ℎ+1,𝑎−1 (12)(13) where 𝑃∶ℤ[𝑎+1,𝑇]→ℝ, 𝑄,𝑞,𝐶_𝑗∶ℤ[𝑎+1,𝑇]→ℝ₊, and 𝜏_𝑗(𝑘)∶ℤ[𝑎+1,𝑇]→ℤ[𝑎+1−ℎ,𝑇]∶𝑘−ℎ≤𝜏_𝑗(𝑘)≤𝑘 for 𝑗∈ℤ[1,𝑟], 𝑘∈ℤ[𝑎+1,𝑇].

Note that the unknown function 𝑢(𝑘) at any moment 𝑘∈ℤ[𝑎+1,𝑇] 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 𝜏_𝑗.

Case 1. Let the inequality 𝜏_𝑗(𝑘)<𝑘 hold for all 𝑗∈ℤ[1,𝑟], 𝑘∈ℤ[𝑎+1,𝑇].
Assume that the values 𝑢(𝑘) of the unknown solution are obtained for all 𝑘∈ℤ[𝑎−ℎ+1,𝑚], where 𝑚<𝑇. Now let 𝑘=𝑚+1.
Case  1.1. Let the following inequality be satisfied: 𝑃(𝑘)−∑^𝑟_𝑗=1𝐶_𝑗(𝑘)𝑢𝜏_𝑗(𝑘)+𝑢(𝑘−1)(𝑘)(𝑘){𝑢()}.1+𝑄+𝑞≥max𝑘−𝑙∶𝑙=1,2,…,ℎ (14)
Therefore, max_{𝑠∈ℤ[𝑘−ℎ,𝑘]}𝑢(𝑠)=𝑢(𝑘). Then the unique solution of IVP (12)-(13) is given by 𝑢(𝑘)=𝑃(𝑘)−∑^𝑟_𝑗=1𝐶_𝑗(𝑘)𝑢𝜏_𝑗(𝑘)+𝑢(𝑘−1)(𝑘)(𝑘).1+𝑄+𝑞 (15)Case  1.2. Let inequality (14) be not satisfied for all 𝑙∈ℤ[1,ℎ]. Therefore, there exists 𝑚∈ℤ[1,ℎ] such that max_{𝑠∈ℤ[𝑘−ℎ,𝑘]}𝑢(𝑠)=𝑢(𝑘−𝑚).
Then the unique solution of IVP (12)-(13) is 𝑢(𝑘)=𝑃(𝑘)−∑^𝑟_𝑗=1𝐶_𝑗(𝑘)𝑢𝜏_𝑗(𝑘)+𝑢(𝑘−1)−𝑞(𝑘)𝑢(𝑘−𝑚)(𝑘).1+𝑄 (16)

Case 2. Let there exists at least one 𝑗∈ℤ[1,𝑟] and 𝑘∈ℤ[𝑎+1,𝑇] such that 𝜏_𝑗(𝑘)=𝑘.
Assume that the values 𝑢(𝑘) of the unknown solution are obtained for all 𝑘∈ℤ[𝑎−ℎ+1,𝑚], where 𝑚<𝑇.
Now let 𝑘=𝑚+1. Introduce the following sets: Ω_𝑘=𝑗∈ℤ[1,𝑟]∶𝜏_𝑗(𝑘)=𝑘,Ω_𝑘=𝑗∶𝑗∈ℤ[1,𝑟],𝑗∉Ω_𝑘. (17)
Note that the set Ω_𝑘 could be empty. Therefore, 𝑢(𝜏_𝑗(𝑘))=𝑢(𝑘) for 𝑗∈Ω_𝑘 and 𝑢(𝜏_𝑗(𝑘))<𝑢(𝑘) for 𝑗∈Ω_𝑘. Equation (12) is reduced to 1+𝑄(𝑘)+_{𝑗∈Ω𝑘}𝐶_𝑗(𝑘)𝑢(𝑘)=𝑢(𝑘−1)−_{𝑗∈Ω𝑘}𝐶_𝑗(𝑘)𝑢𝜏_𝑗(𝑘)−𝑞(𝑘)max_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝑢(𝑠)+𝑃(𝑘). (18)Case  2.1. Let the following inequality be satisfied 𝑃(𝑘)+𝑢(𝑘−1)−∑_{𝑗∈Ω𝑘}𝐶_𝑗(𝑘)𝑢𝜏_𝑗(𝑘)1+𝑄(𝑘)+∑_{𝑗∈Ω𝑘}𝐶_𝑗(𝑘)(𝑘)()[].+𝑞≥𝑢𝑘−𝑠,𝑠∈ℤ1,ℎ (19)
Therefore, max_{𝑠∈ℤ[𝑘−ℎ,𝑘]}𝑢(𝑠)=𝑢(𝑘) and from (18) we obtain the solution 𝑢(𝑘)=𝑃(𝑘)−∑_{𝑗∈Ω𝑘}𝐶_𝑗(𝑘)𝑢𝜏_𝑗(𝑘)+𝑢(𝑘−1)1+𝑄(𝑘)+∑_{𝑗∈Ω𝑘}𝐶_𝑗(𝑘)(𝑘).+𝑞 (20)Case  2.2. Let inequality (19) be not satisfied for all 𝑠∈ℤ[1,ℎ].
Therefore, there exists 𝑚∈ℤ[1,ℎ] such that max_{𝑠∈ℤ[𝑘−ℎ,𝑘]}𝑢(𝑠)=𝑢(𝑘−𝑚).
Then the unique solution of problem (12)-(13) is given by 𝑢(𝑘)=𝑃(𝑘)−∑_{𝑗∈Ω𝑘}𝐶_𝑗(𝑘)𝑢𝜏_𝑗(𝑘)+𝑢(𝑘−1)−𝑞(𝑘)𝑢(𝑘−𝑚)1+𝑄(𝑘)+∑_{𝑗∈Ω𝑘}𝐶_𝑗(𝑘). (21)

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 𝛼₀,𝛽₀∶ℤ[𝑎−ℎ+1,𝑇]→ℝ form a couple of quasi lower and upper solutions of (1)-(2) such that 𝛼₀(𝑘)≤𝛽₀(𝑘) for 𝑘∈ℤ[𝑎−ℎ+1,𝑇].(2) The function 𝑓∶ℤ[𝑎+1,𝑇]×ℝ^𝑟+2→ℝ is continuous in Ω_𝑘(𝛼₀,𝛽₀) for any 𝑘∈ℤ[𝑎+1,𝑇] and for 𝑥,𝑦∈Ω_𝑘(𝛼₀,𝛽₀), 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑟+2), 𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑟+2) with 𝑥_𝑗≤𝑦_𝑗, 𝑗∈ℤ[1,𝑟+2] the inequality 𝑓𝑘,𝑥₁,𝑥₂,…,𝑥_𝑟+2−𝑓𝑘,𝑦₁,𝑦₂,…,𝑦_𝑟+2≤−𝑄(𝑘)𝑥₁−𝑦₁−^𝑟_𝑗=1𝐶_𝑗(𝑘)𝑥_𝑗+1−𝑦_𝑗+1−𝑞(𝑘)𝑥_𝑟+2−𝑦_𝑟+2 (22) holds, where the functions 𝑄,𝑞,𝐶_𝑗∶ℤ[𝑎+1,𝑇]→ℝ₊, 𝑗∈ℤ[1,𝑟], and inequality (8) is satisfied.(3) The function 𝑔∈Λ(𝛼₀,𝛽₀).(4) The functions 𝜏_𝑗(𝑘)∶ℤ[𝑎+1,𝑇]→ℤ[𝑎+1−ℎ,𝑇]∶𝑘−ℎ≤𝜏_𝑗(𝑘)≤𝑘 for 𝑗∈ℤ[1,𝑟], 𝑘∈ℤ[𝑎+1,𝑇].
Then there exist two sequences of functions {𝛼_𝑛}^∞_𝑛=0 and {𝛽_𝑛}^∞_𝑛=0 such that (a) The functions 𝛼_𝑛, 𝛽_𝑛∶ℤ[𝑎−ℎ+1,𝑇]→ℝ, (𝑛=1,2,…) form a couple of quasi lower and upper solutions of BVP (1)-(2). (b) For 𝑘∈ℤ[𝑎−ℎ+1,𝑇] the inequalities 𝛼₀(𝑘)≤𝛼₁(𝑘)≤⋯≤𝛼_𝑛(𝑘)≤⋯≤𝛽_𝑛(𝑘)≤⋯≤𝛽₁(𝑘)≤𝛽₀(𝑘) (23) hold. (c) Both sequences are convergent on ℤ[𝑎−ℎ+1,𝑇] and their limits 𝛼(𝑘)=lim_𝑛→∞𝛼_𝑛(𝑘) and 𝛽(𝑘)=lim_𝑛→∞𝛽_𝑛(𝑘) form a couple of quasi solutions of BVP (1)-(2) in 𝑆(𝛼₀,𝛽₀). In the case of uniqueness of the solution of BVP (1)-(2) in 𝑆(𝛼₀,𝛽₀) the equality 𝛼≡𝛽≡𝑈 holds, where 𝑈 is the unique solution of (1)-(2) in 𝑆(𝛼₀,𝛽₀).

Proof. Note if 𝑢∈𝑆(𝛼₀,𝛽₀), then 𝛼₀(𝜏_𝑗(𝑘))≤𝑢(𝜏_𝑗(𝑘))≤𝛽₀(𝜏_𝑗(𝑘)), 𝑗∈ℤ[1,𝑟], and max_{𝑠∈ℤ[𝑘−ℎ,𝑘]}𝛼₀(𝑠)≤max_{𝑠∈ℤ[𝑘−ℎ,𝑘]}𝑢(𝑠)≤max_{𝑠∈ℤ[𝑘−ℎ,𝑘]}𝛽₀(𝑠) for 𝑘∈ℤ[𝑎+1,𝑇]. Therefore, according to condition 2 of Theorem 6 if the functions 𝑢,𝑣∈𝑆(𝛼₀,𝛽₀) satisfy 𝑢(𝑘)≤𝑣(𝑘) for 𝑘∈ℤ[𝑎−ℎ+1,𝑇], then for any 𝑘∈ℤ[𝑎+1,𝑇] the inequality 𝑓((𝑢(𝜏(𝑘((𝑣(𝜏(𝑘(𝑘𝑢(𝑘)(𝑘−𝑘,Ξ))))−𝑓𝑘,Ξ))))≤−𝑄)[−𝑣)]_𝑟𝑗=1𝐶_𝑗(𝑘)𝑢𝜏_𝑗(𝑘)−𝑣𝜏_𝑗(𝑘)−𝑞(𝑘)max_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝑢(𝑠)−max_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝑣(𝑠) (24) holds.
We will give an algorithm for construction of successive approximations to the unknown exact solution of BVP (1)-(2).
Assume the functions 𝛼₁,𝛼₂,…,𝛼_𝑛∶ℤ[𝑎−ℎ+1,𝑇]→ℝ and 𝛽₁,𝛽₂,…,𝛽_𝑛∶ℤ[𝑎−ℎ+1,𝑇]→ℝ are constructed such that (H1)𝛼_𝑗−1(𝑘)≤𝛼_𝑗(𝑘), 𝛽_𝑗−1(𝑘)≥𝛽_𝑗(𝑘) for 𝑗∈ℤ[1,𝑛], 𝑘∈ℤ[𝑎−ℎ+1,𝑇]; (H2) the functions 𝛼_𝑗 and 𝛽_𝑗, 𝑗∈ℤ[1,𝑛] form a couple of quasi lower and upper solutions of BVP (1)-(2); (H3)𝛼_𝑗(𝑘)≤𝛽_𝑗(𝑘), 𝑘∈ℤ[𝑎−ℎ+1,𝑇], 𝑗∈ℤ[1,𝑛]; (H4) each function 𝛼_𝑗, 𝑗∈ℤ[1,𝑛], is the unique solution of the initial value problem for the linear difference equation with maxima Δ𝑥(𝑘−1)=𝑃𝑘,𝛼_𝑗−1(𝑘)−𝑄(𝑘)𝑥(𝑘)−^𝑟_𝑙=1𝐶_𝑙(𝑘)𝑥𝜏_𝑙(𝑘)−𝑞(𝑘)max_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝑥(𝑠)[],𝑥(𝑎),𝑘∈ℤ𝑎+1,𝑇=𝛼_𝑗−1(𝑎)−1𝛾𝑔𝛼_𝑗−1(𝑎),𝜒𝛽_𝑗(𝑇)𝑥(𝑘)=𝑥(𝑎),𝑘∈ℤ[𝑎−ℎ+1,𝑎−1], (25) where 𝑃∶ℤ[𝑎+1,𝑇]×𝑆(𝛼₀,𝛽₀)→ℝ is defined by 𝑃((𝑘((𝑢(𝜏(𝑘(𝑘)𝑢(𝑘)+𝑘,𝑢))=𝑓𝑘,Ξ))))+𝑄_𝑟𝑗=1𝐶_𝑗(𝑘)𝑢𝜏_𝑗(𝑘)+𝑞(𝑘)max_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝑢(𝑠); (26)(H5) each function 𝛽_𝑗, 𝑗∈ℤ[1,𝑛], is the unique solution of the initial value problem Δ𝑥(𝑘−1)=𝑃𝑘,𝛽_𝑗−1(𝑘)−Q(𝑘)𝑥(𝑘)−^𝑟_𝑙=1𝐶_𝑙(𝑘)𝑥𝜏_𝑙(𝑘)−𝑞(𝑘)max_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝑥(𝑠)[],𝑥(𝑎),𝑘∈ℤ𝑎+1,𝑇=𝛽_𝑗−1(𝑎)−1𝛾𝑔𝛽_𝑗−1(𝑎),𝜒𝛼_𝑗(𝑇),𝑥(𝑘)=𝑥(𝑎),𝑘∈ℤ[𝑎−ℎ+1,𝑎−1]. (27)
Now we will explain how to construct the next approximations 𝛼_𝑛+1 and 𝛽_𝑛+1.
Consider the initial value problem (25) for 𝑗=𝑛. According to Section 4 linear initial value problem (25) has a unique solution 𝛼_𝑛+1∶ℤ[𝑎−ℎ+1,𝑇]→ℝ.
Define a function 𝑝₁∶ℤ[𝑎−ℎ+1,𝑇]→ℝ by 𝑝₁(𝑘)=𝛼_𝑛(𝑘)−𝛼_𝑛+1(𝑘).
From claims (H2) and (H4) we get 𝑝₁(𝑘)=𝑝₁(𝑎) for 𝑘∈ℤ[𝑎−ℎ+1,𝑎−1] and 𝑝₁(𝑎)=(1/𝛾)𝑔(𝛼_𝑛(𝑎),𝜒(𝛽_𝑛(𝑇)))≤0.
Let 𝑘∈ℤ[𝑎+1,𝑇]. Then from claims (H1), (H4), and inequality max_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝛼_𝑛(𝑠)−max_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝛼_𝑛+1(𝑠)≥min_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝛼_𝑛(𝑠)−𝛼_𝑛+1(𝑠) (28) we get Δ𝑝₁(𝑘−1)=−𝑄(𝑘)𝛼_𝑛(𝑘)−𝛼_𝑛+1(𝑘−_𝑟𝑗=1)𝐶_𝑗(𝑘)𝛼_𝑛𝜏_𝑗(𝑘)−𝛼_𝑛+1𝜏_𝑗(𝑘)−𝑞(𝑘)max_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝛼_𝑛(𝑠)−max_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝛼_𝑛+1(𝑠𝛼)+𝑓𝑘,Ξ_𝑛−1(𝜏(𝑘))−𝑓𝑘,Ξ𝛼_𝑛(𝜏(𝑘(𝑘𝛼))+𝑄)_𝑛−1(𝑘)−𝛼_𝑛(𝑘+_𝑟𝑗=1)𝐶_𝑗(𝑘)𝛼_𝑛−1𝜏_𝑗(𝑘)−𝛼_𝑛𝜏_𝑗(𝑘)+𝑞(𝑘)max_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝛼_𝑛−1(𝑠)−max_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝛼_𝑛(𝑠(𝑘)𝑝)≤−𝑄₁(𝑘)−^𝑟_𝑗=1𝐶_𝑗(𝑘)𝑝₁𝜏_𝑗(𝑘)−𝑞(𝑘)min_{𝑠∈ℤ[𝑘−ℎ,𝑘]}𝑝₁(𝑠). (29)
According to Lemma 5  𝑝₁(𝑘)≤0 for 𝑘∈ℤ[𝑎−ℎ+1,𝑇]. Hence, 𝛼_𝑛(𝑘)≤𝛼_𝑛+1(𝑘) for 𝑘∈ℤ[𝑎−ℎ+1,𝑇].
Consider the linear initial value problem (27) for 𝑗=𝑛. According to Section 4 the linear initial value problems (27) has a unique solution 𝛽_𝑛+1∶ℤ[𝑎−ℎ+1,𝑇]→ℝ.
Similarly as previously mentioned we prove 𝛽_𝑛(𝑘)≥𝛽_𝑛+1(𝑘) for 𝑘∈ℤ[𝑎−ℎ+1,𝑇]; that is, claim (H1) is satisfied for 𝑗=𝑛+1.
Now we will prove (H3) for 𝑗=𝑛+1. Define a function 𝑝₂∶ℤ[𝑎−ℎ+1,𝑇]→ℝ by 𝑝₂(𝑘)=𝛼_𝑛+1(𝑘)−𝛽_𝑛+1(𝑘).
From claims (H2), (H3), and condition 3 of Theorem 6 we have 𝑝₂(𝑘)=𝑝₂(𝑎) for 𝑘∈ℤ[𝑎−ℎ+1,𝑎−1]. Also, 𝑝₂(𝑎)=𝛼_𝑛(𝑎)−𝛽_𝑛(𝑎)+1𝛾𝑔𝛽_𝑛(𝑎),𝜒𝛼_𝑛(𝑇)−𝑔𝛼_𝑛(𝑎),𝜒𝛽_𝑛(𝑇)≤𝛼_𝑛(𝑎)−𝛽_𝑛(𝑎)+1𝛾𝑔𝛽_𝑛(𝑎),𝜒𝛽_𝑛(𝑇)−𝑔𝛼_𝑛(𝑎),𝜒𝛽_𝑛(𝑇)≤0. (30) Let 𝑘∈ℤ[𝑎+1,𝑇]. From claims (H1), (H5), and inequality (24) we get Δ𝑝₂()𝛼𝑘−1=𝑓𝑘,Ξ_𝑛(𝜏(𝑘))−𝑓𝑘,Ξ𝛽_𝑛(𝜏(𝑘(𝑘𝛼))+𝑄)_𝑛(𝑘)−𝛽_𝑛(𝑘+_𝑟𝑗=1)𝐶_𝑗(𝑘)𝛼_𝑛𝜏_𝑗(𝑘)−𝛽_𝑛𝜏_𝑗(𝑘)+𝑞(𝑘)max_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝛼_𝑛(𝑠)−max_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝛽_𝑛(𝑠(𝑘)𝑝)−𝑄₃(𝑘)−^𝑟_𝑗=1𝐶_𝑗(𝑘)𝑝₃𝜏_𝑗(𝑘)−𝑞(𝑘)max_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝛼_𝑛+1(𝑠)−max_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝛽_𝑛+1(𝑠(𝑘)𝑝)≤−𝑄₂(𝑘)−^𝑟_𝑗=1𝐶_𝑗(𝑘)𝑝₂𝜏_𝑗(𝑘)−𝑞(𝑘)min_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝑝₂(𝑠). (31)
In view of Lemma 5 it follows 𝑝₂(𝑘)≤0 for 𝑘∈ℤ[𝑎−ℎ+1,𝑇]; that is, claim (H3) is satisfied for 𝑗=𝑛+1.
Therefore, 𝛼_𝑛+1,𝛽_𝑛+1∈𝑆(𝛼₀,𝛽₀).
We will prove (H2) for 𝑗=𝑛+1.
From condition 3 of Theorem 6, claim (H2) for 𝑗=𝑛 and the construction of functions 𝛼_𝑛+1 and 𝛽_𝑛+1 we obtain 1𝛾𝑔𝛼_𝑛+1(𝑎),𝜒𝛽_𝑛+1(𝑇≤1)𝛾𝑔𝛼_𝑛+1(𝑎),𝜒𝛽_𝑛(𝑇)=1𝛾𝑔𝛼_𝑛+1(𝑎),𝜒𝛽_𝑛(𝑇)−𝑔𝛼_𝑛(𝑎),𝜒𝛽_𝑛(𝑇)+1𝛾𝑔𝛼_𝑛(𝑎),𝜒𝛽_𝑛(𝑇)≤𝛼_𝑛+1(𝑎)−𝛼_𝑛(𝑎)+1𝛾𝑔𝛼_𝑛(𝑎),𝜒𝛽_𝑛(𝑇1)=0,𝛾𝑔𝛽_𝑛+1(𝑎),𝜒𝛼_𝑛+1(𝑇)≥1𝛾𝑔𝛽_𝑛+1(𝑎),𝜒𝛼_𝑛(𝑇)1=−𝛾𝑔𝛽_𝑛(𝑎),𝜒𝛼_𝑛(𝑇)−𝑔𝛽_𝑛+1(𝑎),𝜒𝛼_𝑛(𝑇)+1𝛾𝑔𝛽_𝑛(𝑎),𝜒𝛼_𝑛(𝑇)≥𝛽_𝑛+1(𝑎)−𝛽_𝑛(𝑎)+1𝛾𝑔𝛽_𝑛(𝑎),𝜒𝛼_𝑛(𝑇)=0. (32)
Let 𝑘∈ℤ[𝑎+1,𝑇]. From claims (H1), (H4), and inequality (24) we get Δ𝛼_𝑛+1()𝛼𝑘−1=𝑓𝑘,Ξ_𝑛(𝜏(𝑘))−𝑄(𝑘)𝛼_𝑛+1(𝑘)−𝛼_𝑛(𝑘−_𝑟𝑗=1)𝐶_𝑗(𝑘)𝛼_𝑛+1𝜏_𝑗(𝑘)−𝛼_𝑛𝜏_𝑗(𝑘)−𝑞(𝑘)max_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝛼_𝑛+1(𝑠)−max_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝛼_𝑛(𝑠𝛼)=𝑓𝑘,Ξ_𝑛+1(𝜏(𝑘+𝑓𝛼))𝑘,Ξ_𝑛(𝜏(𝑘))−𝑓𝑘,Ξ𝛼_𝑛+1(𝜏(𝑘(𝑘𝛼))−𝑄)_𝑛+1(𝑘)−𝛼_𝑛(𝑘−_𝑟𝑗=1)𝐶_𝑗(𝑘)𝛼_𝑛+1𝜏_𝑗(𝑘)−𝛼_𝑛𝜏_𝑗(𝑘(𝑘)−𝑞)max_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝛼_𝑛+1(𝑠)−max_{[]𝑠∈ℤ𝑘−ℎ,𝑘}𝛼_𝑛(𝑠𝛼)≤𝑓𝑘,Ξ_𝑛+1(𝜏(𝑘)). (33)
Similarly, we prove Δ𝛽_𝑛+1(𝑘−1)≥𝑓𝑘,Ξ𝛽_𝑛+1(𝜏(𝑘)). (34)
From inequalities (32), (33), and (34) it follows the functions 𝛼_𝑛+1 and 𝛽_𝑛+1 form a couple of quasi lower and upper solution of BVP (1)-(2) on ℤ[𝑎−ℎ+1,𝑇]; that is, claim (H2) is satisfied for 𝑗=𝑛+1.
Following the suggested previous scheme we can construct two sequences of functions {𝛼_𝑛}^∞_𝑛=1 and {𝛽_𝑛}^∞_𝑛=1 which satisfy claims (H1)–(H5).
For any fixed 𝑘∈ℤ[𝑎−ℎ+1,𝑇] the sequences {𝛼_𝑛}^∞_𝑛=0 and {𝛽_𝑛}^∞_𝑛=0 are monotone nondecreasing and monotone nonincreasing, respectively, and they are bounded by the functions 𝛼₀ and 𝛽₀. Therefore, they are convergent; that is, there exist functions 𝛼,𝛽∶ℤ[𝑎−ℎ+1,𝑇]→ℝ such that lim_𝑛→∞𝛼_𝑛(𝑘)=𝛼(𝑘),lim_𝑛→∞𝛽_𝑛(𝑘)(𝑘),[].=𝛽𝑘∈ℤ𝑎−ℎ+1,𝑇 (35)
From claim (H3) it is clear that 𝛼,𝛽∈𝑆(𝛼₀,𝛽₀).
Also for any 𝑘∈ℤ[𝑎+1,𝑇] the equalities lim_𝑛→∞max_{[]𝑚∈ℤ𝑘−ℎ,𝑘}𝛼_𝑛(𝑚)=max_{𝑚∈ℤ[𝑘−ℎ,𝑘]}lim_𝑛→∞𝛼_𝑛(𝑚),lim_𝑛→∞max_{[]𝑚∈ℤ𝑘−ℎ,𝑘}𝛽_𝑛(𝑚)=max_{[]𝑚∈ℤ𝑘−ℎ,𝑘}lim_𝑛→∞𝛽_𝑛(𝑚) (36) hold (for the proof see [19]).
We will prove that the limit 𝛼 is a solution of BVP (1)-(2) on ℤ[𝑎−ℎ+1,𝑇].
Taking a limit as 𝑛→∞ in the last equalities of (25) and (27) we get 𝛼(𝑘)=𝛼(𝑎) and 𝛽(𝑘)=𝛽(𝑎) for 𝑘∈ℤ[𝑎−ℎ+1,𝑎−1]. Also we obtain 𝛼(𝑎)=𝛼(𝑎)−(1/𝛾)𝑔(𝛼(𝑎),𝜒(𝛽(𝑇))) and 𝛽(𝑎)=𝛽(𝑎)−(1/𝛾)𝑔(𝛽(𝑎),𝜒(𝛼(𝑇))) or 𝑔(𝛼(𝑎),𝜒(𝛽(𝑇)))=0 and 𝑔(𝛽(𝑎),𝜒(𝛼(𝑇)))=0.
Taking a limit as 𝑛→∞ in the first equality of (25) and (27) we get the functions 𝛼 and 𝛽 satisfy equality (1) for 𝑘∈ℤ[𝑎+1,𝑇].
Therefore, functions 𝛼 and 𝛽 form a couple of quasi solutions of BVP (1)-(2) in 𝑆(𝛼₀,𝛽₀).
If nonlinear boundary value problem (1)-(2) has an unique solution 𝑈 in 𝑆(𝛼₀,𝛽₀), then 𝛼≡𝛽≡𝑈.

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 𝑢(𝑘)=(𝐶𝑢²(𝑘−1))/(𝑢²(𝑘−1)+𝐷) 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 ()=(𝑢(𝑘Δ𝑢𝑘−10.1875))²(𝑢(𝑘))²(𝜏(𝑘+16−0.05𝑢))−0.05max_{𝑠∈ℤ[𝑘−3,𝑘]}𝑢(𝑠),[],𝑢(0)(1)(2)(2)−(𝑢(2for𝑘∈ℤ3,10=𝑢=𝑢,𝑢))²+𝑒^𝑢(10)=0, (37)(38) where 𝜏(𝑘)=𝑘−[√𝑘], 𝑘∈ℤ[3,10], and [𝑠] denotes the integer part of the real number 𝑠; that is, 𝜏(𝑘)=𝑘−1𝑘=3,𝑘−2,𝑘=4,5,6,7,8,𝑘−3,𝑘=9,10. (39)
Boundary value problem for the nonlinear delay difference equation with maxima (37)-(38) is from type (1)-(2), where ℎ=3, 𝑎=2, 𝑇=10, 𝑔(𝑢,𝑣)=𝑢−𝑢²+𝑒^𝑣, 𝑓(𝑘,𝑥,𝑦,𝑧)=0.1875𝑥²/(𝑥²+16)−0.05𝑦−0.05𝑧.
The functions 𝛼₀(𝑘)=−1 and 𝛽₀(𝑘)=0.3, 𝑘∈ℤ[0,10] form a couple of quasi lower and upper solutions of (37), (38), respectively, because 0≥(0.1875∗0.3²)/(0.3²+16)−0.3/20−0.3/20=−0.028951212, 0≤0.1875/(1²+16)+0.05+0.05=0.111029, −1−(−1)²+𝑒^0.3=−0.65014119≤0, and 0.3−(0.3)²+𝑒⁻¹=0.5778794≥0.
Consider the function 𝑔(𝑢,𝑣)=𝑢−𝑢²+𝑒^𝑣 which is increasing in 𝑣. Let 𝑢₁,𝑢₂,𝑣∈[−1,0.3]∶𝑢₂≤𝑢₁. Then 𝑔(𝑢₁,𝑣)−𝑔(𝑢₂,𝑣)=(1−𝑢₁−𝑢₂)(𝑢₁−𝑢₂)≤3(𝑢₁−𝑢₂); that is, 𝛾=3 and the boundary condition (38) is of antiperiodic type.
Let 𝑥₁,𝑥₂,𝑦₁,𝑦₂,𝑧₁,𝑧₂∈[−1,0.3], 𝑥₁≤𝑥₂, 𝑦₁≤𝑦₂, 𝑧₁≤𝑧₂. Then 𝑓𝑘,𝑥₁,𝑦₁,𝑧₁−𝑓𝑘,𝑥₂,𝑦₂,𝑧₂=0.1875𝑥²₁𝑥²₁−+160.1875𝑥²₂𝑥²₂𝑦+16−0.05₁−𝑦₂−0.05𝑧₁−𝑧₂=3𝑥₁−𝑥₂𝑥₁+𝑥₂𝑥²₁+16𝑥²₂𝑦+16−0.05₁−𝑦₂−0.05𝑧₁−𝑧₂≤−6𝑥₁−𝑥₂𝑥²₁+16𝑥²₂𝑦+16−0.05₁−𝑦₂−0.05𝑧₁−𝑧₂≤−3𝑥₁−𝑥₂𝑦128−0.05₁−𝑦₂−0.05𝑧₁−𝑧₂; (40) that is, 𝑞(𝑘)=0.05, 𝐶(𝑘)=0.05, and 𝑄(𝑘)=3/128. Note that the inequality (8) is satisfied because ∑¹⁰_𝑙=3(3/128+0.05+0.05)=0.9875<1.
According to Theorem 6 the nonlinear boundary value problem (37)-(38) has a solution 𝑢(𝑘)∈[−1,0.3], 𝑘∈ℤ[0,10]. We will obtain 𝑢(𝑘) as a limit of two sequences of successive approximations.
The approximation 𝛼_𝑛, 𝑛=1,2,…, is a solution of the initial value problem for the linear delay difference equation with maxima (25) which is reduced to 𝛼_𝑛(𝑘)=𝛼_𝑛(𝑘−1)−0.0234375𝛼_𝑛(𝑘)−0.05max_{𝑠∈ℤ[𝑘−3,𝑘]}𝛼_𝑛(𝑠)−0.05𝛼_𝑛(𝜏(𝑘))+𝑃𝛼_𝑛−1(𝑘)[],𝛼,𝑘∈ℤ3,10_𝑛(0)=𝛼_𝑛(1)=𝛼_𝑛(2)=23𝛼_𝑛−1(2)+13𝛼_𝑛−1(2)²−13𝑒^{𝛽𝑛−1(10)}, (41) and the approximation 𝛽_𝑛, 𝑛=1,2,…, is a solution of IVP (27) which is reduced to 𝛽_𝑛(𝑘)=𝛽_𝑛(𝑘−1)−0.0234375𝛽_𝑛(𝑘)−0.05max_{𝑠∈ℤ[𝑘−3,𝑘]}𝛽_𝑛(𝑠)−0.05𝛽_𝑛(𝜏(𝑘))+𝑃𝛽_𝑛−1(𝑘)[],𝛽,𝑘∈ℤ3,10_𝑛(0)=𝛽_𝑛(1)=𝛽_𝑛(2)=23𝛽_𝑛−1(2)+13𝛽_𝑛−1(2)²−13𝑒^{𝛼𝑛−1(10)}, (42) where 𝑃(𝑢)=0.1875𝑢²/(𝑢²+16)+0.0234375𝑢.
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 {𝛼_𝑛} is increasing and the sequence {𝛽_𝑛} is decreasing and 𝛼_𝑛(𝑘)≤𝛽_𝑛(𝑘), 𝑘=ℤ[0,10]. The unknown exact solution 𝑢(𝑘)∈[𝛼₈(𝑘),𝛽₈(𝑘)], 𝑘∈ℤ[0,10] and its approximate values are written in Table 1.

Table 1 

Values of the successive lower/upper approximations 𝛼_𝑛(𝑘) and 𝛽_𝑛(𝑘), 𝑛=0,1,2,3,4,5,6,7,8.

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.