Asymptotic Behavior of Solutions of Delayed Difference Equations

Abstract

This contribution is devoted to the investigation of the asymptotic behavior of delayed difference equations with an integer delay. We prove that under appropriate conditions there exists at least one solution with its graph staying in a prescribed domain. This is achieved by the application of a more general theorem which deals with systems of first-order difference equations. In the proof of this theorem we show that a good way is to connect two techniques—the so-called retract-type technique and Liapunov-type approach. In the end, we study a special class of delayed discrete equations and we show that there exists a positive and vanishing solution of such equations.

1. Introduction

Throughout this paper, we use the following notation: for an integer 𝑞, we defineℤ^∞𝑞∶={𝑞,𝑞+1,…}.(1.1) We investigate the asymptotic behavior for 𝑛→∞ of the solutions of the discrete delayed equation of the (𝑘+1)-th orderΔ𝑣(𝑛)=𝑓(𝑛,𝑣(𝑛),𝑣(𝑛−1),…,𝑣(𝑛−𝑘)),(1.2) where 𝑛 is the independent variable assuming values from the set ℤ^∞𝑎 with a fixed 𝑎∈ℕ. The number 𝑘∈ℕ, 𝑘≥1 is the fixed delay, Δ𝑣(𝑛)=𝑣(𝑛+1)−𝑣(𝑛), and 𝑓∶ℤ^∞𝑎×ℝ^𝑘+1→ℝ.

A function 𝑣∶ℤ^{∞𝑎−𝑘}→ℝ is a solution of (1.2) if it satisfies (1.2) for every 𝑛∈ℤ^∞𝑎.

We will study (1.2) together with 𝑘+1 initial conditions𝑣(𝑎+𝑠−𝑘)=𝑣^{𝑎+𝑠−𝑘}∈ℝ,𝑠=0,1,…,𝑘.(1.3) Initial problem (1.2), (1.3) obviously has a unique solution, defined for every 𝑛∈ℤ^{∞𝑎−𝑘}. If the function 𝑓 is continuous with respect to its last 𝑘+1 arguments, then the solution of (1.2) continuously depends on initial conditions (1.3).

Now we give a general description of the problem solved in this paper.

Problem 1. Let 𝑏,𝑐∶ℤ^{∞𝑎−𝑘}→ℝ be functions such that 𝑏(𝑛)<𝑐(𝑛) for every 𝑛∈ℤ^{∞𝑎−𝑘}. The problem under consideration is to find sufficient conditions for the right-hand side of (1.2) that will guarantee the existence of a solution 𝑣=𝑣^∗(𝑛) of initial problem (1.2), (1.3) such that 𝑏(𝑛)<𝑣^∗(𝑛)<𝑐(𝑛),𝑛∈ℤ^{∞𝑎−𝑘}.(1.4)

This problem can be solved with help of a result which is valid for systems of first-order difference equations and which will be presented in the next section. This is possible because the considered equation (1.2) can be rewritten as a system of 𝑘+1 first-order difference equations, similarly as a differential equation of a higher order can be transformed to a special system of first-order differential equations. Although the process of transforming a (𝑘+1)-st order difference equation to a system of first order equations is simple and well-known (it is described in Section 3), the determination of the asymptotic properties of the solutions of the resulting system using either Liapunov approach or retract-type method is not trivial. These analogies of classical approaches, known from the qualitative theory of differential equations, were developed for difference systems in [1] (where an approach based on Liapunov method was formulated) and in [2–5] (where retract-type analysis was modified for discrete equations). It occurs that for the mentioned analysis of asymptotic problems of system (1.2), neither the ideas of Liapunov, nor the retract-type technique can be applied directly. However, in spite of the fact that each of the two mentioned methods fails when used independently, it appears that the combination of both these techniques works for this type of systems. Therefore, in Section 2 we prove the relevant result suitable for the asymptotic analysis of systems arising by transformation of (1.2) to a system of first-order differential equations (Theorem 2.1), where the assumptions put to the right-hand side of the system are of both types: those caused by the application of the Liapunov approach and those which are typical for the retract-type technique. Such an idea was applied in a particular case of investigation of asymptotic properties of solutions of the discrete analogue of the Emden-Fowler equation in [6, 7]. The approach is demonstrated in Section 3 where, moreover, its usefulness is illustrated on the problem of detecting the existence of positive solutions of linear equations with a single delay (in Section 3.4) and asymptotic estimation of solutions (in Section 3.3).

Advantages of our approach can be summarized as follows. We give a general method of analysis which is different from the well-known comparison method (see, e.g., [8, 9]). Comparing our approach with the scheme of investigation in [10, 11] which is based on a result from [12], we can see that the presented method is more general because it unifies the investigation of systems of discrete equations and delayed discrete equations thanks to the Liapunov-retract-type technique.

For related results concerning positive solutions and the asymptotics of solutions of discrete equations, the reader is referred also to [13–25].

2. The Result for Systems of First-Order Equations

Consider the system of 𝑚 difference equationsΔ𝑢(𝑛)=𝐹(𝑛,𝑢(𝑛)),(2.1) where 𝑛∈ℤ^∞𝑎, 𝑢=(𝑢₁,…,𝑢_𝑚), and 𝐹∶ℤ^∞𝑎×ℝ^𝑚→ℝ^𝑚, 𝐹=(𝐹₁,…,𝐹_𝑚). The solution of system (2.1) is defined as a vector function 𝑢∶ℤ^∞𝑎→ℝ^𝑚 such that for every 𝑛∈ℤ^∞𝑎, (2.1) is fulfilled. Again, if we prescribe initial conditions𝑢_𝑖(𝑎)=𝑢^𝑎𝑖∈ℝ,𝑖=1,…,𝑚(2.2) the initial problem (2.1), (2.2) has a unique solution. Let us define a set Ω⊂ℤ^∞𝑎×ℝ^𝑚 asΩ∶=^∞_𝑛=𝑎Ω(𝑛),(2.3) whereΩ(𝑛)∶=(𝑛,𝑢)∶𝑛∈ℤ^∞𝑎,𝑢_𝑖∈ℝ,𝑏_𝑖(𝑛)<𝑢_𝑖<𝑐_𝑖(𝑛),𝑖=1,…,𝑚(2.4) with 𝑏_𝑖,𝑐_𝑖∶ℤ^∞𝑎→ℝ, 𝑖=1,…,𝑚, being auxiliary functions such that 𝑏_𝑖(𝑛)<𝑐_𝑖(𝑛) for each 𝑛∈ℤ^∞𝑎. Such set Ω is called a polyfacial set.

Our aim (in this part) is to solve, in correspondence with formulated Problem 1, the following similar problem for systems of difference equations.

Problem 2. Derive sufficient conditions with respect to the right-hand sides of system (2.1) which guarantee the existence of at least one solution 𝑢(𝑛)=(𝑢^∗1(𝑛),…,𝑢^∗𝑚(𝑛)), 𝑛∈ℤ^∞𝑎, satisfying 𝑛,𝑢^∗1(𝑛),…,𝑢^∗𝑚(𝑛)∈Ω(𝑛)(2.5) for every 𝑛∈ℤ^∞𝑎.

As we mentioned above, in [1] the above described problem is solved via a Liapunov-type technique. Here we will combine this technique with the retract-type technique which was used in [2–5] so as the result can be applied easily to the system arising after transformation of (1.2). This brings a significant increase in the range of systems we are able to investigate. Before we start, we recall some basic notions that will be used.

2.1. Consequent Point

Define the mapping 𝒞∶ℤ^∞𝑎×ℝ^𝑚→ℤ^∞𝑎×ℝ^𝑚 as𝒞∶(𝑛,𝑢)⟼(𝑛+1,𝑢+𝐹(𝑛,𝑢)).(2.6) For any point 𝑀=(𝑛,𝑢)∈ℤ^∞𝑎×ℝ^𝑚, the point 𝒞(𝑀) is called the first consequent point of the point 𝑀. The geometrical meaning is that if a point 𝑀 lies on the graph of some solution of system (2.1), then its first consequent point 𝒞(𝑀) is the next point on this graph.

2.2. Liapunov-Type Polyfacial Set

We say that a polyfacial set Ω is Liapunov-type with respect to discrete system (2.1) if𝑏_𝑖(𝑛+1)<𝑢_𝑖+𝐹_𝑖(𝑛,𝑢)<𝑐_𝑖(𝑛+1)(2.7) for every 𝑖=1,…,𝑚 and every (𝑛,𝑢)∈Ω. The geometrical meaning of this property is this: if a point 𝑀=(𝑛,𝑢) lies inside the set Ω(𝑛), then its first consequent point 𝒞(𝑀) stays inside Ω(𝑛+1).

In this contribution we will deal with sets that need not be of Liapunov-type, but they will have, in a certain sense, a similar property. We say that a polyfacial set Ω is Liapunov-type with respect to the 𝑗th variable (𝑗∈{1,…,𝑚}) and to discrete system (2.1) if(𝑛,𝑢)∈Ω⟹𝑏_𝑗(𝑛+1)<𝑢_𝑗+𝐹_𝑗(𝑛,𝑢)<𝑐_𝑗(𝑛+1).(2.8) The geometrical meaning is that if 𝑀=(𝑛,𝑢)∈Ω(𝑛), then the 𝑢_𝑗-coordinate of its first consequent point stays between 𝑏_𝑗(𝑛+1) and 𝑐_𝑗(𝑛+1), meanwhile the other coordinates of 𝒞(𝑀) may be arbitrary.

2.3. Points of Strict Egress and Their Geometrical Sense

An important role in the application of the retract-type technique is played by the so called strict egress points. Before we define these points, let us describe the boundaries of the sets Ω(𝑛), 𝑛∈ℤ^∞𝑎, in detail. As one can easily see, _𝑛_{∈ℤ∞𝑎}𝜕Ω(𝑛)=^𝑚_𝑗=1Ω^𝑗𝐵∪^𝑚_𝑗=1Ω^𝑗𝐶(2.9) withΩΩ^𝑗𝐵∶=(𝑛,𝑢)∶𝑛∈ℤ^∞𝑎,𝑢_𝑗=𝑏_𝑗(𝑛),𝑏_𝑖(𝑛)≤𝑢_𝑖≤𝑐_𝑖(𝑛),𝑖=1,…,𝑚,𝑖≠𝑗_𝑗𝐶,∶=(𝑛,𝑢)∶𝑛∈ℤ^∞𝑎,𝑢_𝑗=𝑐_𝑗(𝑛),𝑏_𝑖(𝑛)≤𝑢_𝑖≤𝑐_𝑖(𝑛),𝑖=1,…,𝑚,𝑖≠𝑗.(2.10) In accordance with [3, Lemmas 1 and 2], a point (𝑛,𝑢)∈𝜕Ω(𝑛) is a point of the type of strict egress for the polyfacial set Ω with respect to discrete system (2.1) if and only if for some 𝑗∈{1,…,𝑚}𝑢_𝑗=𝑏_𝑗(𝑛),𝐹_𝑗(𝑛,𝑢)<𝑏_𝑗(𝑛+1)−𝑏_𝑗(𝑛),(2.11) or𝑢_𝑗=𝑐_𝑗(𝑛),𝐹_𝑗(𝑛,𝑢)>𝑐_𝑗(𝑛+1)−𝑐_𝑗(𝑛).(2.12) Geometrically these inequalities mean the following: if a point 𝑀=(𝑛,𝑢)∈𝜕Ω(𝑛) is a point of the type of strict egress, then the first consequent point 𝒞(𝑀)∉Ω(𝑛+1).

2.4. Retract and Retraction

If 𝐴⊂𝐵 are any two sets in a topological space and 𝜋∶𝐵→𝐴 is a continuous mapping from 𝐵 onto 𝐴 such that 𝜋(𝑝)=𝑝 for every 𝑝∈𝐴, then 𝜋 is said to be a retraction of 𝐵 onto 𝐴. If there exists a retraction of 𝐵 onto 𝐴, then 𝐴 is called a retract of 𝐵.

2.5. The Existence Theorem for the System of First-Order Equations (Solution of Problem 2)

The following result, solving Problem 2, gives sufficient conditions with respect to the right-hand sides of (2.1) which guarantee the existence of at least one solution satisfying (2.5) for every 𝑛∈ℤ^∞𝑎.

Theorem 2.1. Let 𝑏_𝑖(𝑛),𝑐_𝑖(𝑛),𝑏_𝑖(𝑛)<𝑐_𝑖(𝑛), 𝑖=1,…,𝑚, be real functions defined on ℤ^∞𝑎 and let 𝐹_𝑖∶ℤ^∞𝑎×ℝ^𝑚→ℝ, 𝑖=1,…,𝑚, be continuous functions. Suppose that for one fixed 𝑗∈{1,…,𝑚} all the points of the sets Ω^𝑗𝐵, Ω^𝑗𝐶 are points of strict egress, that is, if (𝑛,𝑢)∈Ω^𝑗𝐵, then 𝐹_𝑗(𝑛,𝑢)<𝑏_𝑗(𝑛+1)−𝑏_𝑗(𝑛),(2.13) and if (𝑛,𝑢)∈Ω^𝑗𝐶, then 𝐹_𝑗(𝑛,𝑢)>𝑐_𝑗(𝑛+1)−𝑐_𝑗(𝑛).(2.14) Further suppose that the set Ω is of Liapunov-type with respect to the 𝑖th variable for every 𝑖∈{1,…,𝑚}, 𝑖≠𝑗, that is, that for every (𝑛,𝑢)∈Ω 𝑏_𝑖(𝑛+1)<𝑢_𝑖+𝐹_𝑖(𝑛,𝑢)<𝑐_𝑖(𝑛+1).(2.15) Then there exists a solution 𝑢=(𝑢^∗1(𝑛),…,𝑢^∗𝑚(𝑛)) of system (2.1) satisfying the inequalities 𝑏_𝑖(𝑛)<𝑢^∗𝑖(𝑛)<𝑐_𝑖(𝑛),𝑖=1,…,𝑚,(2.16) for every 𝑛∈ℤ^∞𝑎.

Proof. The proof will be by contradiction. We will suppose that there exists no solution satisfying inequalities (2.16) for every 𝑛∈ℤ^∞𝑎. Under this supposition we prove that there exists a continuous mapping (a retraction) of a closed interval onto both its endpoints which is, by the intermediate value theorem of calculus, impossible.
Without the loss of generality we may suppose that the index 𝑗 in Theorem 2.1 is equal to 1, that is, all the points of the sets Ω^1𝐵 and Ω^1𝐶 are strict egress points. Each solution of system (2.1) is uniquely determined by the chosen initial condition𝑢(𝑎)=𝑢₁(𝑎),…,𝑢_𝑚(𝑎)=𝑢^𝑎1,…,𝑢^𝑎𝑚=𝑢^𝑎.(2.17) For the following considerations, let 𝑢^𝑎𝑖 with 𝑢^𝑎𝑖∈(𝑏_𝑖(𝑎),𝑐_𝑖(𝑎)), 𝑖=2,…,𝑚, be chosen arbitrarily but fixed. Now the solution of (2.1) is given just by the choice of 𝑢^𝑎1, we can write 𝑢=𝑢𝑛,𝑢^𝑎1=𝑢₁𝑛,𝑢^𝑎1,…,𝑢_𝑚𝑛,𝑢^𝑎1.(2.18) Define the closed interval 𝐼∶=[𝑏₁(𝑎),𝑐₁(𝑎)]. Hereafter we show that, under the supposition that there exists no solution satisfying inequalities (2.16), there exists a retraction ℛ (which will be a composition of two auxiliary mappings ℛ₁ and ℛ₂ defined below) of the set 𝐵∶=𝐼 onto the set 𝐴∶=𝜕𝐼={𝑏₁(𝑎),𝑐₁(𝑎)}. This contradiction will prove our result. To arrive at such a contradiction, we divide the remaining part of the proof into several steps.Construction of the Leaving Value 𝑛^∗
Let a point 𝑢₁̃∈𝐼 be fixed. The initial condition 𝑢₁(𝑎)=𝑢₁̃ defines a solution 𝑢=𝑢(𝑛,𝑢₁̃)=(𝑢₁(𝑛,𝑢₁̃),…,𝑢_𝑚(𝑛,𝑢₁̃)). According to our supposition, this solution does not satisfy inequalities (2.16) for every 𝑛∈ℤ^∞𝑎. We will study the moment the solution leaves the domain Ω for the first time. The first value of 𝑛 for which inequalities (2.16) are not valid will be denoted as 𝑠.
(I) First consider the case 𝑢₁̃∈int𝐼. Then there exists a value 𝑠>1 in ℤ^∞𝑎+1 such that𝑠,𝑢𝑠,𝑢₁̃∉Ω(𝑠)(2.19) while 𝑟,𝑢𝑟,𝑢₁̃∈Ω(𝑟)for𝑎≤𝑟≤𝑠−1.(2.20) As the set Ω is of the Liapunov-type with respect to all variables except the first one and (𝑠−1,𝑢(𝑠−1,𝑢₁̃))∈Ω(𝑠−1), then 𝑏_𝑖(𝑠)<𝑢_𝑖𝑠,𝑢₁̃<𝑐_𝑖(𝑠),𝑖=2,…,𝑚.(2.21) Because 𝑗=1 was assumed, and Ω is of Liapunov-type for each variable 𝑢_𝑖, 𝑖≠𝑗, then the validity of inequalities (2.16) has to be violated in the 𝑢₁-coordinate. The geometrical meaning was explained in Section 2.2.
Now, two cases are possible: either (𝑠,𝑢(𝑠,𝑢₁̃))∉Ω(𝑠) or (𝑠,𝑢(𝑠,𝑢₁̃))∈𝜕Ω(𝑠). In the first case 𝑢₁(𝑠,𝑢₁̃)<𝑏₁(𝑠) or 𝑢₁(𝑠,𝑢₁̃)>𝑐₁(𝑠). In the second case 𝑢₁(𝑠,𝑢₁̃)=𝑏₁(𝑠) or 𝑢₁(𝑠,𝑢₁̃)=𝑐₁(𝑠) and, due to (2.13) and (2.14), 𝑢₁(𝑠+1,𝑢₁̃)<𝑏₁(𝑠+1) or 𝑢₁(𝑠+1,𝑢₁̃)>𝑐₁(𝑠+1), respectively.
(II) If 𝑢₁̃∈𝜕𝐼, then (𝑎,𝑢(𝑎,𝑢₁̃))∉Ω(𝑎). Thus, for this case, we could put 𝑠=𝑎. Further, because of the strict egress property of Ω^1𝐵 and Ω^1𝐶, either 𝑢₁(𝑎+1,𝑢₁̃)<𝑏₁(𝑎+1) (if 𝑢₁̃=𝑏₁(𝑎)) or 𝑢₁(𝑎+1,𝑢₁̃)>𝑐₁(𝑎+1) (if 𝑢₁̃=𝑐₁(𝑎)) and thus (𝑎+1,𝑢(𝑎+1,𝑢₁̃))∉Ω(𝑎+1).
Unfortunately, for the next consideration the value 𝑠 (the first value of the independent variable for which the graph of the solution is out of Ω) would be of little use. What we will need is the last value for which the graph of the solution stays in Ω. We will denote this value as 𝑛^∗ and will call it the leaving value. We can define 𝑛^∗ as𝑛𝑛^∗=𝑠−1if𝑠,𝑢𝑠,𝑢₁̃∉Ω(𝑠),_∗=𝑠if𝑠,𝑢𝑠,𝑢₁̃∈𝜕Ω(𝑠).(2.22) As the value of 𝑛^∗ depends on the chosen initial point 𝑢₁̃, we could write 𝑛^∗=𝑛^∗(𝑢₁̃) but we will mostly omit the argument 𝑢₁̃, unless it is necessary. From the above considerations it follows that 𝑢₁𝑏₁𝑛^∗≤𝑢₁𝑛^∗,𝑢₁̃≤𝑐₁𝑛^∗,𝑛^∗+1,𝑢₁̃<𝑏₁𝑛^∗+1or𝑢₁𝑛^∗+1,𝑢₁̃>𝑐₁𝑛^∗+1.(2.23)Auxiliary Mapping ℛ₁
Now we construct the auxiliary mapping ℛ₁∶𝐼→ℝ×ℝ. First extend the discrete functions 𝑏₁,𝑐₁ onto the whole interval [𝑎,∞): 𝑏𝑐₁(𝑡)∶=𝑏₁(⌊𝑡⌋)+𝑏₁(⌊𝑡⌋+1)−𝑏₁(⌊𝑡⌋)(𝑡−⌊𝑡⌋),₁(𝑡)∶=𝑐₁(⌊𝑡⌋)+𝑐₁(⌊𝑡⌋+1)−𝑐₁(⌊𝑡⌋)(𝑡−⌊𝑡⌋),(2.24)⌊𝑡⌋ being the integer part of 𝑡 (the floor function). Note that 𝑏₁,𝑐₁ are now piecewise linear continuous functions of a real variable 𝑡 such that 𝑏₁(𝑡)<𝑐₁(𝑡) for every 𝑡 and that the original values of 𝑏₁(𝑛),𝑐₁(𝑛) for 𝑛∈ℤ^∞𝑎 are preserved. This means that the graphs of these functions connect the points (𝑛,𝑏₁(𝑛)) or (𝑛,𝑐₁(𝑛)) for 𝑛∈ℤ^∞𝑎, respectively. Denote 𝑉 the set 𝑉∶=𝑡,𝑢₁∶𝑡∈[𝑎,∞),𝑏₁(𝑡)≤𝑢₁≤𝑐₁(𝑡).(2.25) The boundary of 𝑉 consists of three mutually disjoint parts 𝑉_𝑎, 𝑉_𝑏, and 𝑉_𝑐: 𝜕𝑉=𝑉_𝑎∪𝑉_𝑏∪𝑉_𝑐,(2.26) where 𝑉𝑉_𝑎∶=𝑎,𝑢₁∶𝑏₁(𝑎)<𝑢₁<𝑐₁(𝑎)_𝑏,𝑉∶=𝑡,𝑢₁∶𝑡∈[𝑎,∞),𝑢₁=𝑏₁(𝑡)_𝑐,∶=𝑡,𝑢₁∶𝑡∈[𝑎,∞),𝑢₁=𝑐₁(𝑡).(2.27) Define the mapping ℛ₁∶𝐼→𝑉_𝑏∪𝑉_𝑐 as follows: let ℛ₁(𝑢₁̃) be the point of intersection of the line segment defined by its end points (𝑛^∗,𝑢₁(𝑛^∗,𝑢₁̃)), (𝑛^∗+1,𝑢₁(𝑛^∗+1,𝑢₁̃)) with 𝑉_𝑏∪𝑉_𝑐 (see Figure 1). The mapping ℛ₁ is obviously well defined on 𝐼 and ℛ₁(𝑏₁(𝑎))=(𝑎,𝑏₁(𝑎)), ℛ₁(𝑐₁(𝑎))=(𝑎,𝑐₁(𝑎)).
Prove that the mapping ℛ₁ is continuous. The point ℛ₁(𝑢₁̃)=(𝑡(𝑢₁̃),𝑢₁(𝑢₁̃)) lies either on 𝑉_𝑏 or on 𝑉_𝑐. Without the loss of generality, consider the second case (the first one is analogical). The relevant boundary line segment for 𝑡∈[𝑛^∗,𝑛^∗+1], which is a part of 𝑉_𝑐, is described by (see (2.24))𝑢₁=𝑐𝑛^∗+𝑐𝑛^∗+1−𝑐𝑛^∗𝑡−𝑛^∗,(2.28) and the line segment joining the points (𝑛^∗,𝑢₁(𝑛^∗,𝑢₁̃)), (𝑛^∗+1,𝑢₁(𝑛^∗+1,𝑢₁̃)) by the equation 𝑢₁=𝑢₁𝑛^∗,𝑢₁̃+𝑢₁𝑛^∗+1,𝑢₁̃−𝑢₁𝑛^∗,𝑢₁̃𝑡−𝑛^∗,𝑡∈𝑛^∗,𝑛^∗+1.(2.29) The coordinates of the point ℛ₁(𝑢₁̃)=(𝑡(𝑢₁̃),𝑢₁(𝑢₁̃)), which is the intersection of both these line segments, can be obtained as the solution of the system consisting of (2.28) and (2.29). Solving this system with respect to 𝑡 and 𝑢₁, we get 𝑡𝑢₁̃=𝑛^∗+𝑢¹𝑛^∗,𝑢₁̃−𝑐₁𝑛^∗𝑐₁(𝑛^∗+1)−𝑢₁𝑛^∗+1,𝑢₁̃+𝑢₁𝑛^∗,𝑢₁̃−𝑐₁(𝑛^∗),(2.30)𝑢₁𝑢₁̃=𝑐₁𝑛^∗+𝑢¹𝑛^∗,𝑢₁̃−𝑐₁𝑛^∗𝑐₁𝑛^∗+1−𝑐₁𝑛^∗𝑐₁(𝑛^∗+1)−𝑢₁𝑛^∗+1,𝑢₁̃+𝑢₁𝑛^∗,𝑢₁̃−𝑐₁(𝑛^∗).(2.31) Let {𝑣_𝑘}^∞𝑘=1 be any sequence with 𝑣_𝑘∈𝐼 such that 𝑣_𝑘→𝑢₁̃. We will show that ℛ₁(𝑣_𝑘)→ℛ₁(𝑢₁̃). Because of the continuity of the functions 𝐹_𝑖, 𝑖=1,…,𝑚, 𝑢₁𝑛,𝑣_𝑘→𝑢₁𝑛,𝑢₁̃foreveryfixed𝑛∈ℤ^∞𝑎.(2.32) We have to consider two cases: (I)(𝑛^∗,𝑢(𝑛^∗,𝑢₁̃))∈Ω(𝑛^∗), that is, 𝑏₁(𝑛^∗)<𝑢₁(𝑛^∗,𝑢₁̃)<𝑐₁(𝑛^∗), (II)(𝑛^∗,𝑢(𝑛^∗,𝑢₁̃))∈𝜕Ω(𝑛^∗), that is, 𝑢₁(𝑛^∗,𝑢₁̃)=𝑐₁(𝑛^∗).
Recall that (due to our agreement) in both cases 𝑢₁(𝑛^∗+1,𝑢₁̃)>𝑐₁(𝑛^∗+1).
(I)In this case also 𝑢₁(𝑛^∗,𝑣_𝑘)<𝑐₁(𝑛^∗) and 𝑢₁(𝑛^∗+1,𝑣_𝑘)>𝑐₁(𝑛^∗+1) for 𝑘 sufficiently large. That means that the leaving value 𝑛^∗(𝑣_𝑘) is the same as 𝑛^∗ given by 𝑢₁̃ and thus the point ℛ₁(𝑣_𝑘)=(𝑡(𝑣_𝑘),𝑢₁(𝑣_𝑘)) is given by 𝑡𝑣_𝑘=𝑛^∗+𝑢¹𝑛^∗,𝑣_𝑘−𝑐₁𝑛^∗𝑢𝑐₁(𝑛^∗+1)−𝑢₁𝑛^∗+1,𝑣_𝑘+𝑢₁𝑛^∗,𝑣_𝑘−𝑐₁(𝑛^∗),(2.33)₁𝑣_𝑘=𝑐₁𝑛^∗+𝑢¹𝑛^∗,𝑣_𝑘−𝑐₁𝑛^∗𝑐₁𝑛^∗+1−𝑐₁𝑛^∗𝑐₁(𝑛^∗+1)−𝑢₁𝑛^∗+1,𝑣_𝑘+𝑢₁𝑛^∗,𝑣_𝑘−𝑐₁(𝑛^∗).(2.34) The desired convergence ℛ₁(𝑣_𝑘)→ℛ₁(𝑢₁̃) is implied by equations (2.30) to (2.34).(II)Suppose 𝑛^∗=𝑎. Then 𝑢₁̃=𝑐₁(𝑎), 𝑣_𝑘=𝑢₁(𝑎,𝑣_𝑘)<𝑐₁(𝑎) for all 𝑘 and as 𝑘→∞, 𝑢₁(𝑎+1,𝑣_𝑘)>𝑐₁(𝑎+1). A minor edit of the text in the case (I) proof provides the continuity proof. Suppose 𝑛^∗>𝑎. In this case there can be 𝑢₁(𝑛^∗,𝑣_𝑘)≤𝑐₁(𝑛^∗) for some members of the sequence {𝑣_𝑘} and 𝑢₁(𝑛^∗,𝑣_𝑘)>𝑐₁(𝑛^∗) for the others. Without the loss of generality, we can suppose that {𝑣_𝑘} splits into two infinite subsequences {𝑣_𝑞𝑘} and {𝑣_𝑟𝑘} such that 𝑢₁𝑛^∗,𝑣_𝑞𝑘≤𝑐₁𝑢𝑛^∗,𝑢₁𝑛^∗+1,𝑣_𝑞𝑘>𝑐₁𝑛^∗+1₁𝑛^∗,𝑣_𝑟𝑘>𝑐₁𝑛^∗.(2.35)For the subsequence {𝑣_𝑞𝑘}, the text of the proof of (I) can be subjected to a minor edit to provide the proof of continuity. As for the subsequence {𝑣_𝑟𝑘}, the leaving value 𝑛^∗(𝑣_𝑟𝑘) is different from 𝑛^∗ given by 𝑢₁̃ because (𝑛^∗,𝑢₁(𝑛^∗,𝑣_𝑟𝑘)) is already out of Ω. For 𝑘 sufficiently large,𝑛^∗𝑣_𝑟𝑘=𝑛^∗−1(2.36)because 𝑢₁(𝑛^∗−1,𝑢₁̃)<𝑐₁(𝑛^∗−1) and thus, as 𝑘→∞, 𝑢₁(𝑛^∗−1,𝑣_𝑟𝑘)<𝑐₁(𝑛^∗−1).Hence, the value of the mapping ℛ₁ for 𝑣_𝑟𝑘 is (in (2.33), (2.34) we replace 𝑛^∗ by 𝑛^∗−1) 𝑡𝑣_𝑟𝑘=𝑛^∗−1+𝑢¹𝑛^∗−1,𝑣_𝑟𝑘−𝑐₁𝑛^∗−1𝑢𝑐₁(𝑛^∗)−𝑢₁𝑛^∗,𝑣_𝑟𝑘+𝑢₁𝑛^∗−1,𝑣_𝑟𝑘−𝑐₁(𝑛^∗−1)₁,𝑣_𝑟𝑘=𝑐𝑛^∗−1+𝑢¹𝑛^∗−1,𝑣_𝑟𝑘−𝑐₁𝑛^∗−1𝑐₁𝑛^∗−𝑐₁𝑛^∗−1𝑐₁(𝑛^∗)−𝑢₁𝑛^∗,𝑣_𝑟𝑘+𝑢₁𝑛^∗−1,𝑣_𝑟𝑘−𝑐₁(𝑛^∗−1).(2.37)Due to (2.32), 𝑢₁(𝑛^∗,𝑣_𝑟𝑘)→𝑢₁(𝑛^∗,𝑢₁̃)=𝑐₁(𝑛^∗) and thus 𝑡𝑣_𝑟𝑘⟶𝑛^∗−1+𝑢¹𝑛^∗−1,𝑣_𝑟𝑘−𝑐₁𝑛^∗−1𝑢₁𝑛^∗−1,𝑣_𝑟𝑘−𝑐₁(𝑛^∗−1)=𝑛^∗𝑢₁,𝑣_𝑟𝑘⟶𝑐𝑛^∗−1+𝑢¹𝑛^∗−1,𝑣_𝑟𝑘−𝑐₁𝑛^∗−1𝑐₁𝑛^∗−𝑐₁𝑛^∗−1𝑢₁𝑛^∗−1,𝑣_𝑟𝑘−𝑐₁(𝑛^∗−1)=𝑐¹ℛ₁𝑛^∗,𝑣_𝑟𝑘=𝑡𝑣_𝑟𝑘,𝑢₁𝑣_𝑟𝑘⟶𝑛^∗,𝑐₁𝑛^∗=ℛ₁𝑢₁̃.(2.38)We have shown that ℛ₁(𝑣_𝑞𝑘)→ℛ₁(𝑢₁̃) and ℛ₁(𝑣_𝑟𝑘)→ℛ₁(𝑢₁̃) and thus ℛ₁(𝑣_𝑘)→ℛ₁(𝑢₁̃).Auxiliary Mapping ℛ₂
Define ℛ₂∶𝑉_𝑏∪𝑉_𝑐→{𝑏₁(𝑎),𝑐₁(𝑎)} as ℛ₂(𝑃)=𝑏¹𝑐₁(𝑎)if𝑃∈𝑉_𝑏,(𝑎)if𝑃∈𝑉_𝑐.(2.39) The mapping ℛ₂ is obviously continuous.Resulting Mapping ℛ and Its Properties
Define ℛ∶=ℛ₂∘ℛ₁. Due to construction we have ℛ𝑏₁(𝑎)=𝑏₁(𝑎),ℛ𝑐₁(𝑎)=𝑐₁(𝑎),(2.40) and ℛ(𝐼)=𝜕𝐼. The mapping ℛ is continuous because of the continuity of the two mappings ℛ₁ and ℛ₂. Hence, it is the sought retraction of 𝐼 onto 𝜕𝐼. But such a retraction cannot exist and thus we get a contradiction and the proof is complete.

Figure 1 

Construction of the mapping ℛ1.

3. Application of Theorem 2.1 to the Delayed Discrete Equation

Now, let us return to the original delayed discrete equation (1.2), that is, Δ𝑣(𝑛)=𝑓(𝑛,𝑣(𝑛),𝑣(𝑛−1),…,𝑣(𝑛−𝑘)).(3.1) As it was said in Section 1, this equation will be transformed to a system of 𝑘+1 first-order discrete equations. Then we will apply Theorem 2.1 to this system and prove that under certain conditions there exists a solution of delayed equation (1.2) that stays in the prescribed domain. In the end, we will study a special case of (1.2).

3.1. Transformation of (1.2) to the System of First-Order Equations

We will proceed in accordance with the well-known scheme similarly as when constructing the system of first-order differential equations from a differential equation of a higher order. Put 𝑢𝑢₁(𝑛)∶=𝑣(𝑛),₂𝑢(𝑛)∶=⋯𝑣(𝑛−1),_𝑘+1(𝑛)∶=𝑣(𝑛−𝑘),(3.2) where 𝑢₁,𝑢₂,…,𝑢_𝑘+1 are new unknown functions. From (1.2) we get Δ𝑢₁(𝑛)=𝑓(𝑛,𝑢₁(𝑛),𝑢₂(𝑛),…,𝑢_𝑘+1(𝑛)). Obviously 𝑢₂(𝑛+1)=𝑢₁(𝑛),…,𝑢_𝑘+1(𝑛+1)=𝑢_𝑘(𝑛). Rewriting these equalities in terms of differences, we have Δ𝑢₂(𝑛)=𝑢₁(𝑛)−𝑢₂(𝑛),…,Δ𝑢_𝑘+1(𝑛)=𝑢_𝑘(𝑛)−𝑢_𝑘+1(𝑛). Altogether, we get the systemΔ𝑢₁(𝑛Δ)𝑢=₂𝑓𝑛,𝑢₁(𝑛),…,𝑢_𝑘+1(𝑛),Δ𝑢_𝑘+1(𝑛)=𝑢⋯₁(𝑛)−𝑢₂(𝑛),(𝑛)=𝑢_𝑘(𝑛)−𝑢_𝑘+1(𝑛)(3.3) which is equivalent to (1.2).

3.2. The Existence Theorem for the Delayed Equation (1.2) (Solution of Problem 1)

The following theorem is a consequence of Theorem 2.1. In fact, this theorem has been already proved in [12]. There, the proof is based upon a modification of the retract method for delayed equations. Our method (rearranging a delayed equation to a system of first-order equations) is, by its principle, more general than that used in [12].

Theorem 3.1. Let 𝑏(𝑛), 𝑐(𝑛), 𝑏(𝑛)<𝑐(𝑛), be real functions defined on ℤ^{∞𝑎−𝑘}. Further, let 𝑓∶ℤ^∞𝑎×ℝ^𝑘+1→ℝ be a continuous function and let the inequalities 𝑏𝑐((𝑛𝑛))++𝑓𝑓𝑛𝑛,,𝑏𝑐((𝑛𝑛)),,𝑣𝑣₂₂,…,𝑣_𝑘+1<𝑏(𝑛+1),(3.4),…,𝑣_𝑘+1>𝑐(𝑛+1)(3.5) hold for every 𝑛∈ℤ^∞𝑎 and every 𝑣₂,…,𝑣_𝑘+1 such that 𝑏(𝑛−𝑖+1)<𝑣_𝑖<𝑐(𝑛−𝑖+1),𝑖=2,…,𝑘+1.(3.6) Then there exists a solution 𝑣=𝑣^∗(𝑛) of (1.2) satisfying the inequalities 𝑏(𝑛)<𝑣^∗(𝑛)<𝑐(𝑛)(3.7) for every 𝑛∈ℤ^{∞𝑎−𝑘}.

Proof. We have shown that (1.2) is equivalent to system (3.3) which can be seen as a special case of system (2.1) with 𝑚=𝑘+1 and 𝐹=(𝐹₁,…𝐹_𝑘+1) where 𝐹₁𝑛,𝐹𝑢₁₂,…,𝑢_𝑘+1∶=𝑓𝑛,𝑢₁,…,𝑢_𝑘+1,𝐹_𝑘𝑛,𝑢₁,…,𝑢⋯_𝑘+1∶=𝑢₁−𝑢₂,𝐹_𝑘+1𝑛,𝑢₁,…,𝑢_𝑘+1∶=𝑢_𝑘−1−𝑢_𝑘,𝑛,𝑢₁,…,𝑢_𝑘+1∶=𝑢_𝑘−𝑢_𝑘+1.(3.8) Define the polyfacial set Ω as Ω∶=(𝑛,𝑢)∶𝑛∈ℤ^∞𝑎,𝑏_𝑖(𝑛)<𝑢_𝑖<𝑐_𝑖(𝑛),𝑖=1,…,𝑘+1(3.9) with 𝑏_𝑖(𝑛)∶=𝑏(𝑛−𝑖+1),𝑐_𝑖(𝑛)∶=𝑐(𝑛−𝑖+1),𝑖=1,…,𝑘+1.(3.10) We will show that for system (3.3) and the set Ω, all the assumptions of Theorem 2.1 are satisfied.
As the function 𝑓 is supposed to be continuous, the mapping 𝐹 is continuous, too. Put the index 𝑗 from Theorem 2.1, characterizing the points of egress, equal to 1. We will verify that the set Ω is of Liapunov-type with respect to the 𝑖th variable for any 𝑖=2,…,𝑘+1, that is, (see (2.8)) that for every (𝑛,𝑢)∈Ω𝑏_𝑖(𝑛+1)<𝑢_𝑖+𝐹_𝑖(𝑛,𝑢)<𝑐_𝑖(𝑛+1)for𝑖=2,…,𝑘+1.(3.11) First, we compute 𝑢_𝑖+𝐹_𝑖(𝑛,𝑢)=𝑢_𝑖+𝑢_𝑖−1−𝑢_𝑖=𝑢_𝑖−1for𝑖=2,…,𝑘+1.(3.12) Thus we have to show that for 𝑖=2,…,𝑘+1𝑏_𝑖(𝑛+1)<𝑢_𝑖−1<𝑐_𝑖(𝑛+1).(3.13) Because (𝑛,𝑢)∈Ω, then 𝑏_𝑝(𝑛)<𝑢_𝑝<𝑐_𝑝(𝑛) for any 𝑝∈{1,…,𝑘+1}, and therefore 𝑏_𝑖−1(𝑛)<𝑢_𝑖−1<𝑐_𝑖−1(𝑛)for𝑖=2,…,𝑘+1.(3.14) But, by (3.10), we have 𝑏_𝑖−1(𝑛)=𝑏(𝑛−𝑖+1+1)=𝑏(𝑛−𝑖+2),(3.15) meanwhile 𝑏_𝑖(𝑛+1)=𝑏(𝑛+1−𝑖+1)=𝑏(𝑛−𝑖+2),(3.16) and thus 𝑏_𝑖−1(𝑛)=𝑏_𝑖(𝑛+1). Analogously we get that 𝑐_𝑖−1(𝑛)=𝑐_𝑖(𝑛+1). Thus inequalities (3.11) are fulfilled.
Further we will show that all the boundary points 𝑀∈Ω^1𝐵∪Ω^1𝐶 are points of strict egress for the set Ω with respect to system (3.3). According to (2.11), we have to show that if 𝑢₁=𝑏₁(𝑛) and 𝑏_𝑖(𝑛)<𝑢_𝑖<𝑐_𝑖(𝑛) for 𝑖=2,…,𝑘+1, then𝑏₁(𝑛)+𝐹₁(𝑛,𝑢)<𝑏₁(𝑛+1),(3.17) that is, 𝑏₁(𝑛)+𝑓𝑛,𝑏₁(𝑛),𝑢₂,…,𝑢_𝑘+1<𝑏₁(𝑛+1).(3.18) Notice that the condition 𝑏_𝑖(𝑛)<𝑢_𝑖<𝑐_𝑖(𝑛) for 𝑖=2,…,𝑘+1 is equivalent with condition 𝑏(𝑛−𝑖+1)<𝑢_𝑖<𝑐(𝑛−𝑖+1) (see (3.10)). Looking at the supposed inequality (3.4) and realizing that 𝑏₁(𝑛)=𝑏(𝑛) and 𝑏₁(𝑛+1)=𝑏(𝑛+1), we can see that inequality (3.18) is fulfilled.
Analogously, according to (2.12), we have to prove that for 𝑢₁=𝑐₁(𝑛) and 𝑏_𝑖(𝑛)<𝑢_𝑖<𝑐_𝑖(𝑛) for 𝑖=2,…,𝑘+1 the inequality𝑐₁(𝑛)+𝐹₁(𝑛,𝑢)>𝑐₁(𝑛+1),(3.19) that is, 𝑐₁(𝑛)+𝑓𝑛,𝑐₁(𝑛),𝑢₂,…,𝑢_𝑘+1>𝑐₁(𝑛+1)(3.20) holds.
Again, considering (3.5) and the fact that 𝑐₁(𝑛)=𝑐(𝑛) and 𝑐₁(𝑛+1)=𝑐(𝑛+1), we can see that this inequality really holds.
Thus, by the assertion of Theorem 2.1, there exists a solution 𝑢=𝑢^∗(𝑛) of system (3.3) such that for every 𝑛∈ℤ^∞𝑎𝑏_𝑖(𝑛)<𝑢^∗𝑖(𝑛)<𝑐_𝑖(𝑛)for𝑖=1,…,𝑘+1.(3.21)
In our case, 𝑣=𝑣^∗(𝑛)=𝑢^∗1(𝑛) is the solution of the original equation (1.2). Further, 𝑏₁(𝑛)=𝑏(𝑛) and 𝑐₁(𝑛)=𝑐(𝑛), and thus the existence of a solution of the delayed equation (1.2) such that inequalities (3.7) are satisfied is guaranteed.

3.3. Asymptotic Solution Estimates for Delayed Difference Equations

Let us suppose that two functions 𝑢,𝑤∶ℤ^{∞𝑎−𝑘}→ℝ are given such that Δ𝑢(𝑛)≥𝑓(𝑛,𝑢𝑢((𝑛𝑛)),<𝑢(𝑤𝑛−(𝑛1)),,…𝑛∈,𝑢ℤ(𝑛−^{∞𝑎−𝑘},𝑘)),𝑛∈ℤ_∞𝑎(3.22)Δ𝑤(𝑛)≤𝑓(𝑛,𝑤(𝑛),𝑤(𝑛−1),…,𝑤(𝑛−𝑘)),𝑛∈ℤ_∞𝑎,(3.23).(3.24) Consider the problem of whether there exists a solution 𝑣=𝑣^∗(𝑛), 𝑛∈ℤ^{∞𝑎−𝑘} of (1.2) such that 𝑢(𝑛)<𝑣^∗(𝑛)<𝑤(𝑛),𝑛∈ℤ^{∞𝑎−𝑘}.(3.25) The following corollary of Theorem 3.1 presents sufficient conditions for the existence of a solution of this problem.

Corollary 3.2. Let functions 𝑢,𝑤∶ℤ^{∞𝑎−𝑘}→ℝ satisfy inequalities (3.22)–(3.24). Let 𝑓∶ℤ^∞𝑎×ℝ^𝑘+1→ℝ be a continuous function such that 𝑓𝑓𝑛𝑛,,𝑤𝑢((𝑛𝑛)),,𝑦𝑦₂₂,…,𝑦_𝑘+1>𝑓𝑛,𝑢(𝑛),𝑧₂,…,𝑧_𝑘+1,(3.26),…,𝑦_𝑘+1>𝑓𝑛,𝑤(𝑛),𝑧₂,…,𝑧_𝑘+1(3.27) for every 𝑛∈ℤ^∞𝑎 and every 𝑦₂,…,𝑦_𝑘+1,𝑧₂,…,𝑧_𝑘+1∈ℝ such that 𝑦_𝑖<𝑧_𝑖,𝑖=2,…,𝑘+1.(3.28) Then there exists a solution 𝑣=𝑣^∗(𝑛) of (1.2) satisfying inequalities (3.25) for every 𝑛∈ℤ^{∞𝑎−𝑘}.

Proof. This assertion is an easy consequence of Theorem 3.1.
Put 𝑏(𝑛)∶=𝑢(𝑛), 𝑐(𝑛)∶=𝑤(𝑛). Considering inequalities (3.23) and (3.26), we can see thatΔ𝑢(𝑛)>𝑓𝑛,𝑢(𝑛),𝑣₂,…,𝑣_𝑘+1(3.29) for every 𝑛∈ℤ^∞𝑎 and every 𝑣₂,…,𝑣_𝑘+1 such that 𝑏(𝑛−𝑖+1)<𝑣_𝑖<𝑐(𝑛−𝑖+1),𝑖=2,…,𝑘+1.(3.30) Similarly, Δ𝑤(𝑛)<𝑓𝑛,𝑤(𝑛),𝑣₂,…,𝑣_𝑘+1(3.31) for every 𝑛∈ℤ^∞𝑎 and every 𝑏(𝑛−𝑖+1)<𝑣_𝑖<𝑐(𝑛−𝑖+1), 𝑖=2,…,𝑘+1.
Obviously, inequalities (3.29) and (3.31) are equivalent with inequalities (3.4) and (3.5), respectively. Thus, all the assumptions of Theorem 3.1 are satisfied and there exists a solution 𝑣=𝑣^∗(𝑛) of (1.2) satisfying inequalities (3.25) for every 𝑛∈ℤ^{∞𝑎−𝑘}.

Example 3.3. Consider the equation Δ𝑣(𝑛)=𝑣²(𝑛)−𝑣(𝑛−1)(3.32) for 𝑛∈ℤ^∞3 which is a second-order delayed discrete equation with delay 𝑘=1. We will show that there exists a solution 𝑣=𝑣^∗(𝑛) of (3.32) that satisfies the inequalities 1<𝑣^∗(𝑛)<𝑛(3.33) for 𝑛∈ℤ^∞2.

We will prove that for the functions𝑢(𝑛)∶=1,𝑤(𝑛)∶=𝑛,𝑓𝑛,𝑣₁,𝑣₂∶=𝑣²¹−𝑣₂(3.34) all the assumptions of Corollary 3.2 are satisfied. Inequality (3.22) is obviously fulfilled for 𝑛∈ℤ^∞2. Inequality (3.23) can be also proved very easily:Δ𝑢(𝑛)=0,𝑓(𝑛,𝑢(𝑛),𝑢(𝑛−1))=1²−1=0,(3.35) and thus for every 𝑛∈ℤ^∞3, Δ𝑢(𝑛)≥𝑓(𝑛,𝑢(𝑛),𝑢(𝑛−1)).

As for inequality (3.24), we getΔ𝑤(𝑛)=1,𝑓(𝑛,𝑤(𝑛),𝑤(𝑛−1))=𝑛²−𝑛+1(3.36) and thus Δ𝑤(𝑛)≤𝑓(𝑛,𝑤(𝑛),𝑤(𝑛−1)) for 𝑛∈ℤ^∞3.

Finally, the functions𝑓𝑛,𝑢(𝑛),𝑣₂=1−𝑣₂,𝑓𝑛,𝑤(𝑛),𝑣₂=𝑛²−𝑣₂(3.37) are decreasing with respect to 𝑣₂. Therefore, conditions (3.26) and (3.27) are satisfied, too. Hence, due to Corollary 3.2, there exists a solution of (3.32) satisfying (3.33).

3.4. Positive Solutions of a Linear Equation with a Single Delay

We will apply the result of Theorem 3.1 to the investigation of a simple linear difference equation of the (𝑘+1)-st order with only one delay, namely, the equationΔ𝑣(𝑛)=−𝑝(𝑛)𝑣(𝑛−𝑘),(3.38) where, again, 𝑛∈ℤ^∞𝑎 is the independent variable and 𝑘∈ℕ, 𝑘≥1, is the fixed delay. The function 𝑝∶ℤ^∞𝑎→ℝ is assumed to be positive. Our goal is to give sharp sufficient conditions for the existence of positive solutions. The existence of such solutions is very often substantial for a concrete model considered. For example, in biology, when a model of population dynamics is described by an equation, the positivity of a solution may mean that the studied biological species can survive in the supposed environment.

For its simple form, (3.38) often serves for testing new results and is very frequently investigated. It was analyzed, for example, in papers [10, 11, 26]. A sharp result on existence of positive solutions given in [26] is proved by a comparison method [8, 9]. Here we will use Theorem 3.1 to generalize this result.

For the purposes of this section, define the expression ln_𝑞𝑡, where 𝑞∈ℕ, as ln_𝑞𝑡∶ln=₀lnln_𝑞−1𝑡𝑡∶=𝑡.(3.39) We will write only ln𝑡 instead of ln₁𝑡. Further, for a fixed integer ℓ≥0 define auxiliary functions𝜇_ℓ(𝑛)∶=18𝑛²+18(𝑛ln𝑛)²+⋯+18𝑛ln𝑛⋯ln_ℓ𝑛²𝑝_ℓ,(𝑛)∶=𝑘𝑘+1^𝑘⋅1𝑘+1+𝑘𝜇^ℓ𝜈(𝑛),(3.40)_ℓ(𝑛)∶=𝑘𝑘+1^𝑛⋅√𝑛ln𝑛ln₂𝑛⋯ln_ℓ𝑛.(3.41) In [26], it was proved that if 𝑝(𝑛) in (3.38) is a positive function bounded by 𝑝_ℓ(𝑛) for some ℓ≥0, then there exists a positive solution of (3.38) bounded by the function 𝜈_ℓ(𝑛) for 𝑛 sufficiently large. Since lim_𝑛→∞𝜈_ℓ(𝑛)=0, such solution will vanish for 𝑛→∞. Here we show that (3.38) has a positive solution bounded by 𝜈_ℓ(𝑛) even if the coefficient 𝑝(𝑛) satisfies a less restrictive inequality (see inequality (3.58) below). The proof of this statement will be based on the following four lemmas. The symbols “𝑜” and “𝑂” stand for the Landau order symbols and are used for 𝑛→∞.

Lemma 3.4. The formula ln(𝑦−𝑧)=ln𝑦−^∞_𝑖=1𝑧^𝑖𝑖𝑦^𝑖(3.42) holds for any numbers 𝑦,𝑧∈ℝ such that 𝑦>0 and |𝑧|<𝑦.

Proof. The assertion is a simple consequence of the well-known Maclaurin expansion ln(1−𝑥)=−^∞_𝑖=11𝑖𝑥^𝑖for−1≤𝑥<1.(3.43) As ln(𝑦−𝑧)−ln𝑦=ln(1−𝑧/𝑦), substituting 𝑥=𝑧/𝑦 we get ln(𝑦−𝑧)−ln𝑦=−^∞_𝑖=1𝑧^𝑖𝑖𝑦^𝑖for−𝑦≤𝑧<𝑦(3.44) and adding ln𝑦 to both sides of this equality, we get (3.42).

Lemma 3.5. For fixed 𝑟∈ℝ⧵{0} and fixed 𝑞∈ℕ, the asymptotic representation ln_𝑞(𝑛−𝑟)=ln_𝑞𝑛−𝑟−𝑛ln𝑛𝑟⋯₂ln_𝑞−1𝑛−2𝑛²ln𝑛𝑟⋯²ln_𝑞−1𝑛2(𝑛ln𝑛)²ln₂𝑛⋯ln_𝑞−1𝑛−⋯−2𝑛ln𝑛⋯𝑟²ln_𝑞−1𝑛²−𝑟³(1+𝑜(1))3𝑛³ln𝑛⋯ln_𝑞−1𝑛(3.45) holds for 𝑛→∞.

Proof. We will prove relation (3.45) by induction with respect to 𝑞. For 𝑞=1, (3.45) reduces to ln(𝑛−𝑟)=ln𝑛−𝑟𝑛−2𝑟𝑛²²−𝑟³(1+𝑜(1))3𝑛³(3.46) which holds due to Lemma 3.4. Suppose that relation (3.45) holds for some 𝑞. We can write ln_𝑞(𝑛−𝑟)=𝑦−𝑧 with 𝑦=ln_𝑞𝑛 and 𝑧=𝑟+𝑛⋯ln𝑛+⋯ln_𝑞−1𝑛+𝑟₂2𝑛²ln𝑛𝑟⋯²ln_𝑞−1𝑛+2(𝑛ln𝑛)²ln𝑟²₂𝑛⋯ln_𝑞−1𝑛2𝑛ln𝑛⋯ln_𝑞−1𝑛²+𝑟³(1+𝑜(1))3𝑛³ln𝑛⋯ln_𝑞−1𝑛.(3.47) Now we will show that (3.45) holds for 𝑞+1. Notice that in our case, the condition |𝑧|<𝑦 from Lemma 3.4 is fulfilled for 𝑛 sufficiently large because 𝑧→0 for 𝑛→∞, meanwhile 𝑦→∞ for 𝑛→∞. Thus we are justified to use Lemma 3.4 and doing so, we get ln_𝑞+1(𝑛−𝑟)==llnn(𝑦ln−_𝑞𝑧(𝑛)−=𝑟ln)𝑦−1𝑦𝑧−21𝑦²=lnln_𝑞𝑧²−⋯𝑛−1ln_𝑞𝑛+⋅𝑛ln𝑛⋯𝑟𝑟ln_𝑞−1𝑛+2𝑛²ln𝑛𝑟⋯²ln_𝑞−1𝑛+⋯₂2𝑛ln𝑛⋯ln_𝑞−1𝑛²+𝑟³(1+𝑜(1))−13𝑛³ln𝑛⋯ln_𝑞−1𝑛2ln_𝑞𝑛²⋅𝑟²𝑛ln𝑛⋯ln_𝑞−1𝑛²+𝑂1+𝑂1𝑛³ln𝑛⋯ln_𝑞−1𝑛²=ln_𝑞+1𝑛ln𝑛⋯ln_𝑞𝑛³𝑛−𝑟−⋯−𝑛ln𝑛𝑟⋯₂ln_𝑞𝑛−2𝑛²ln𝑛𝑟²⋯ln_𝑞𝑛−2(𝑛ln𝑛)²l𝑟n²₂𝑛⋯ln_𝑞𝑛2𝑛ln𝑛⋯ln_𝑞𝑛²−𝑟³(1+𝑜(1))3𝑛³ln𝑛⋯ln_𝑞𝑛.(3.48) Thus, formula (3.45) holds for 𝑞+1, too, which ends the proof.

Lemma 3.6. For fixed 𝑟∈ℝ⧵{0} and fixed 𝑞∈ℕ, the asymptotic representations ln_𝑞(𝑛−𝑟)ln_𝑞𝑛=−1−2𝑛ln𝑛𝑟⋯𝑟ln_𝑞𝑛−4𝑛²ln𝑛𝑟²⋯ln_𝑞𝑛−4(𝑛ln𝑛)²l𝑟n²₂𝑛⋯ln_𝑞𝑛−⋯₂4𝑛ln𝑛⋯ln_𝑞−1𝑛²ln_𝑞𝑛−8𝑛ln𝑛𝑟⋯²ln_𝑞𝑛²−𝑟³(1+𝑜(1))6𝑛³ln𝑛⋯ln_𝑞𝑛,(3.49)𝑛−𝑟𝑛=1−2𝑟𝑛−8𝑟𝑛²²−𝑟³16𝑛³+𝑜1𝑛³(3.50) hold for 𝑛→∞.

Proof. Both these relations are simple consequences of the asymptotic formula √1−𝑥=1−12𝑥−81𝑥²−116𝑥³+𝑜𝑥³for𝑥⟶0(3.51) and of Lemma 3.5 (for formula (3.49)). In the case of relation (3.49), we put 𝑥=𝑟𝑛ln𝑛⋯ln_𝑞𝑛+2𝑛²ln𝑛𝑟²⋯ln_𝑞𝑛+⋯+2𝑛ln𝑛⋯l𝑟n²_𝑞−1𝑛²ln_𝑞𝑛+𝑟³(1+𝑜(1))3𝑛³ln𝑛⋯ln_𝑞𝑛(3.52) and in the case of relation (3.50), we put 𝑥=𝑟/𝑛.

Lemma 3.7. For fixed 𝑟∈ℝ⧵{0} and fixed 𝑞∈ℕ, the asymptotic representation (𝑛−𝑟)𝑛ln(l𝑛n𝑛−𝑟)⋯ln^𝑞(𝑛−𝑟)=1−𝑟1ln_𝑞𝑛2𝑛+2𝑛1ln𝑛+⋯+2𝑛ln𝑛1⋯ln_𝑞𝑛−𝑟²𝜇_𝑞(𝑛)−𝑟³16𝑛³+𝑜1𝑛³(3.53) holds for 𝑛→∞.

Proof. We will prove relation (3.53) by induction with respect to 𝑞. For 𝑞=1, (3.53) reduces to (𝑛−𝑟)𝑛ln(l𝑛n𝑛−𝑟)=1−𝑟21𝑛+2𝑛1ln𝑛−𝑟²𝜇₁(𝑛)−𝑟³16𝑛³+𝑜1=1−𝑟1𝑛³2𝑛+2𝑛1ln𝑛−𝑟²81𝑛²+18(𝑛ln𝑛)²−16𝑟³𝑛³+𝑜1𝑛³.(3.54) On the other hand, using Lemma 3.6, we get (𝑛−𝑟)=𝑛1l−n(l𝑛n𝑟𝑛−𝑟)2𝑛−8𝑟𝑛²²−𝑟³16𝑛³+𝑜1×1−𝑟𝑛³2𝑛ln𝑛−4𝑛²𝑟²ln𝑛−8(𝑛𝑟ln²𝑛)²−𝑟³(1+𝑜(1))=1−𝑟6𝑛³ln𝑛2𝑛ln𝑛−4𝑛²𝑟²ln𝑛−8(𝑛𝑟ln²𝑛)²−𝑟2𝑛+4𝑛²𝑟²ln𝑛−8𝑟𝑛²²−𝑟³16𝑛³+𝑜1=1−𝑟1𝑛³2𝑛+2𝑛1ln𝑛−𝑟²81𝑛²+18(𝑛ln𝑛)²−16𝑟³𝑛³+𝑜1𝑛³.(3.55) Thus, for 𝑞=1, relation (3.53) holds. Now suppose that (3.53) holds for some 𝑞 and prove that it holds for 𝑞+1. In the following calculations, we use Lemma 3.6 and we skip some tedious expressions handling. (𝑛−𝑟)𝑛ln(l𝑛n𝑛−𝑟)⋯ln^𝑞+1(𝑛−𝑟)=ln_𝑞+1𝑛(𝑛−𝑟)𝑛ln(l𝑛n𝑛−𝑟)⋯ln^𝑞(𝑛−𝑟)ln_𝑞𝑛⋅ln_𝑞+1(𝑛−𝑟)=1−𝑟1ln_𝑞+1𝑛2𝑛+2𝑛1ln𝑛+⋯+2𝑛ln𝑛1⋯ln_𝑞𝑛−𝑟²𝜇_𝑞(𝑛)−𝑟³16𝑛³+𝑜1×1−𝑟𝑛³−2𝑛ln𝑛⋯𝑟ln_𝑞+1𝑛−4𝑛²ln𝑛𝑟⋯²ln_𝑞+1𝑛−⋯₂4𝑛ln𝑛⋯ln_𝑞𝑛²ln_𝑞+1𝑛−8𝑛ln𝑛⋯𝑟²ln_𝑞+1𝑛²+𝑜1=1−𝑟1𝑛³2𝑛+2𝑛1ln𝑛+⋯+2𝑛ln𝑛⋯1ln_𝑞+1𝑛−𝑟²𝜇_𝑞+1(𝑛)−𝑟³16𝑛³+𝑜1𝑛³.(3.56) We can see that formula (3.53) holds for 𝑞+1, too, which ends the proof.

Now we are ready to prove that there exists a bounded positive solution of (3.38). Remind that functions 𝑝_ℓ and 𝜈_ℓ were defined by (3.40) and (3.41), respectively.

Theorem 3.8. Let 𝜔∶ℤ^∞𝑎→ℝ satisfy the inequality ||𝜔(𝑛)||≤𝜀𝑘𝑘+1^𝑘⋅𝑘2𝑘²+𝑘−116𝑛³(𝑘+1),𝑛∈ℤ^∞𝑎,(3.57) for a fixed 𝜀∈(0,1). Suppose that there exists an integer ℓ≥0 such that the function 𝑝 satisfies the inequalities 0<𝑝(𝑛)≤𝑝_ℓ(𝑛)+𝜔(𝑛)(3.58) for every 𝑛∈ℤ^∞𝑎. Then there exists a solution 𝑣=𝑣^∗(𝑛), 𝑛∈ℤ^{∞𝑎−𝑘} of (3.38) such that for 𝑛 sufficiently large the inequalities 0<𝑣^∗(𝑛)<𝜈_ℓ(𝑛)(3.59) hold.

Proof. Show that all the assumptions of Theorem 3.1 are fulfilled. For (3.38), 𝑓(𝑛,𝑣₁,…,𝑣_𝑘+1)=−𝑝(𝑛)𝑣_𝑘+1. This is a continuous function. Put 𝑏(𝑛)∶=0,𝑐(𝑛)∶=𝜈_ℓ(𝑛).(3.60) We have to prove that for every 𝑣₂,…,𝑣_𝑘+1 such that 𝑏(𝑛−𝑖+1)<𝑣_𝑖<𝑐(𝑛−𝑖+1), 𝑖=2,…,𝑘+1, the inequalities (3.4) and (3.5) hold for 𝑛 sufficiently large. Start with (3.4). That gives that for 𝑏(𝑛−𝑘)<𝑣_𝑘+1<𝑐(𝑛−𝑘), it has to be 0−𝑝(𝑛)⋅𝑣_𝑘+1<0.(3.61) This certainly holds, because the function 𝑝 is positive and so is 𝑣_𝑘+1.
Next, according to (3.5), we have to prove that𝜈_ℓ(𝑛)−𝑝(𝑛)𝑣_𝑘+1>𝜈_ℓ(𝑛+1)(3.62) which is equivalent to the inequality −𝑝(𝑛)𝑣_𝑘+1>𝜈_ℓ(𝑛+1)−𝜈_ℓ(𝑛).(3.63) Denote the left-hand side of (3.63) as 𝐿_(3.63). As 𝑣_𝑘+1<𝑐(𝑛−𝑘)=𝜈_ℓ(𝑛−𝑘) and as by (3.40), (3.58), and (3.57) 𝑝(𝑛)≤𝑘𝑘+1^𝑘⋅1𝑘+1+𝑘𝜇^ℓ(𝑛)+𝜀𝑘𝑘+1^𝑘⋅𝑘2𝑘²+𝑘−116𝑛³(𝑘+1),(3.64) we have 𝐿_(3.63)>−𝑘𝑘+1^𝑘𝑘+11+𝑘𝜇^ℓ(𝑛)+𝜀⋅𝑘2𝑘²+𝑘−1×𝑘16𝑛³(𝑘+1)𝑘+1^𝑛−𝑘=−𝑘√(𝑛−𝑘)ln(𝑛−𝑘)⋯ln_ℓ(𝑛−𝑘)𝑘+1^𝑛𝑘+11+𝑘𝜇^ℓ(𝑛)+𝜀⋅𝑘2𝑘²+𝑘−116𝑛³(𝑘+1)⋅√(𝑛−𝑘)ln(𝑛−𝑘)⋯ln_ℓ(𝑛−𝑘).(3.65) Further, we can easily see that 𝜈_ℓ(𝑛+1)−𝜈_ℓ(𝑛)=𝑘𝑘+1^𝑛√𝑛ln𝑛⋯ln_ℓ𝑛⎛⎜⎜⎝𝑘𝑘+1(𝑛+1)𝑛ln(l𝑛n+𝑛1)⋯ln^ℓ(𝑛+1)ln_ℓ𝑛−1⎞⎟⎟⎠.(3.66) Thus, to prove (3.63), it suffices to show that for 𝑛 sufficiently large, −1𝑘+1+𝑘𝜇^ℓ(𝑛)+𝜀⋅𝑘2𝑘²+𝑘−116𝑛³(𝑘+1)(𝑛−𝑘)𝑛ln(l𝑛n−𝑛𝑘)⋯ln^ℓ(𝑛−𝑘)>𝑘ln_ℓ𝑛𝑘+1(𝑛+1)𝑛ln(l𝑛n+𝑛1)⋯ln^ℓ(𝑛+1)ln_ℓ𝑛−1.(3.67) Denote the left-hand side of inequality (3.67) as 𝐿_(3.67) and the right-hand side as 𝑅_(3.67). Using Lemma 3.7 with 𝑟=𝑘 and 𝑞=ℓ, we can write 𝐿_(3.67)=−1𝑘+1+𝑘𝜇^ℓ(𝑛)+𝜀⋅𝑘2𝑘²+𝑘−1×1−𝑘116𝑛³(𝑘+1)2𝑛+2𝑛1ln𝑛+⋯+2𝑛ln𝑛1⋯ln_ℓ𝑛−𝑘²𝜇_ℓ(𝑛)−𝑘³16𝑛³+𝑜1=−1𝑛³+𝑘𝑘+₂1+𝑘+𝑘121𝑛+2𝑛1ln𝑛+⋯+2𝑛ln𝑛1⋯ln_ℓ𝑛𝑘+1𝜇^ℓ(𝑛)+𝑘³16𝑛³(𝑘+1)−𝑘𝜇^ℓ(𝑛)+𝑘²16𝑛³−𝜀⋅𝑘2𝑘²+𝑘−1=−116𝑛³(𝑘+1)+𝑜𝑛1³−𝑘+𝑘1+𝑘+𝑘121𝑛+2𝑛1ln𝑛+⋯+2𝑛ln𝑛1⋯ln_ℓ𝑛𝑘+1𝜇^ℓ(𝑛)+2𝑘³(1−𝜀)+𝑘²(1−𝜀)+𝑘𝜀16𝑛³(𝑘+1)+𝑜𝑛1³.(3.68) Using Lemma 3.7 with 𝑟=−1 and 𝑞=ℓ, we get for 𝑅_(3.67)𝑅_(3.67)=𝑘𝑘+11+21𝑛+2𝑛1ln𝑛+⋯+2𝑛ln𝑛1⋯ln_ℓ𝑛−𝜇^ℓ(𝑛)+116𝑛³+𝑜1=−1𝑛³−1−𝑘+𝑘1+𝑘+𝑘121𝑛+2𝑛1ln𝑛+⋯+2𝑛ln𝑛1⋯ln_ℓ𝑛𝑘+1⋅𝜇^ℓ(𝑛)+𝑘16𝑛³(𝑘+1)+𝑜𝑛1³.(3.69) It is easy to see that the inequality (3.67) reduces to 2𝑘³(1−𝜀)+𝑘²(1−𝜀)+𝑘𝜀16𝑛³(𝑘+1)+𝑜𝑛1³>16𝑛³(𝑘𝑘+1)+𝑜𝑛1³.(3.70) This inequality is equivalent to 𝑘2𝑘²(1−𝜀)+𝑘(1−𝜀)−(1−𝜀)16𝑛³(𝑘+1)+𝑜𝑛1³>0.(3.71) The last inequality holds for 𝑛 sufficiently large because 𝑘≥1 and 1−𝜀∈(0,1). We have proved that all the assumptions of Theorem 3.1 are fulfilled and hence there exists a solution of (3.38) satisfying conditions (3.59).

Acknowledgment

This paper was supported by Grant P201/10/1032 of the Czech Grant Agency (Prague), by the Council of Czech Government MSM 00216 30529, and by the Grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology.