Existence of positive solutions of linear delay difference equations with continuous time

Consider the delay difference equation with continuous time of the form x(t)− x(t− 1) + m ∑ i=1 Pi(t)x(t− ki(t)) = 0, t ≥ t0, where Pi : [t0, ∞) 7→ R, ki : [t0, ∞) 7→ {2, 3, 4, . . . } and limt→∞(t − ki(t)) = ∞, for i = 1, 2, . . . , m. We introduce the generalized characteristic equation and its importance in oscillation of all solutions of the considered difference equations. Some results for the existence of positive solutions of considered difference equations are presented as the application of the generalized characteristic equation.


Introduction
Difference equations with continuous time are difference equations in which the unknown function is a function of a continuous variable. Equations of this type appear as natural descriptions of observed evolution phenomena in many branches of the natural sciences and therefore appear in various mathematical models. This is the main reason why they have been studied in many papers recently. See, for example, the papers of Domshlak [1], Ferreira and Pinelas [2,3], Golda and Werbowski [4], Korenevskii and Kaizer [7], Ladas et al. [8], Medina and Pituk [9], Meng et al. [10], Nowakowska and Werbowski [11,12,13,14], Shaikhet [17], Shen et al. [18,19,20,21], Zhang et al. [22,23,24,25], and the references cited therein. In this paper, we introduce the generalized characteristic equation and its importance in oscillation of all solutions of linear delay difference equations with continuous time. Some results regarding the existence of positive solutions of the considered difference equations are presented as the application of the generalized characteristic equation.
The investigated equation is Let t 0 be a positive real number such that It is clear that t −1 (t 0 ) ≤ t 0 − 2 < t 0 − 1.
A motivating example is the equation

5)
Existence of positive solutions 3 where s ≥ 2 is a given integer and P : [t 0 , ∞) → R. (1.6) In this case t −1 (t 0 ) = t 0 − s and hence the initial condition is The generalized characteristic equation is with the initial condition assuming that the function φ is defined by (1.7) and We can formulate the following statement. , and x may be written in the form Proof. Let x be a positive solution of the initial value problem (1.5) and (1.7). By dividing both sides of equation (1.5) with x(t − 1) we get Define the function , From the definition is obvious that the function λ is positive and follows that 13) and so λ satisfies the initial value problem (1.8) and (1.9) on [t 0 − s + 1, ∞).
On the other hand, let λ be a positive solution of the initial value problem (1.8) and (1.9) on [t 0 − s + 1, ∞) with positive function φ defined by (1.7). Then, function x defined by (1.10) is positive. From the definition it follows also that it is equal to the initial function (1.7) for t 0 − s ≤ t < t 0 . For n = 1 it follows that x(t) = λ(t)x(t − 1) and so the equalities (1.13) hold. That means that the characteristic equation (1.8) may be written in the form (1.11) and the function x defined by (1.10) satisfying the difference equation (1.5). The proof is complete. The goal is to find necessary and sufficient conditions for the solutions of the initial value problem (1.5) and (1.7) to be positive on [t 0 − s, ∞). The simplest case is P(t) ≤ 0, t ≥ t 0 , since for every initial function φ(t) > 0, t 0 − s ≤ t < t 0 , the solution of the initial value problem (1.5) and (1.7) is positive. When P(t) ≥ 0, t ≥ t 0 , then the existence of a positive solution is more delicate, while the most difficult case being whenever the coefficient P(t) is oscillatory on [t 0 , ∞). Theorem 1.2. Assume that (1.6) holds. Let P(t) ≥ 0 for t ≥ t 0 , and assume that there are two positive functions α, β : (1.14) Then there exists a solution λ : [t 0 − s + 1, ∞) → (0, ∞) of the initial value problem (1.8) and (1.9) with In this case we can prove that and hence the limit function λ of the sequence of functions {λ r (t)} r∈N exists for t ≥ t 0 − s + 1.

Preliminaries
In the work of Győri and Ladas [6] some results, such as Theorem 3.1.1, are shown related to the generalized characteristic equation of linear delay differential equation with an initial condition of the form In Theorem 3.1.1, a condition for the existence of a positive solution of the initial value problem (2.1) and (2.2) is formulated. The unique solution of the initial value problem (2.1) and (2.2) is denoted with x(ϕ)(t) and exists for t 0 ≤ t ≤ T.
Győri and Ladas have also obtained some results for the existence of positive solutions of the considered differential equation.
Theorem A ([6, Theorem 3.3.2]). Assume that (H * 1 ) holds and that there exists a positive number µ such that Papers [15] and [16] deal with the discrete analogues of the generalized characteristic equation and Theorem A. Consider the linear retarded difference equation where N * = {n ∈ N : n 0 ≤ n < M, n 0 < M ≤ ∞} and N is the set of positive integers. Let Associated with equation (2.3), we define the initial condition a n = φ n , for n = n −1 , n −1 + 1, ..., n 0 , φ n ∈ R, (2.4) where The unique solution of the initial value problem (2.3) and (2.4) is denoted with a(φ) n and exists for n ∈ N * .
The papers of Golda and Werbowski [4], Shen and Stavroulakis [21], Zhang and Choi [25] deal with the functional equation with variable coefficients of the form for large t, then equation (2.5) has a non-oscillatory solution. Shen and Stavroulakis [21] studied the linear functional equation of the form then equation (2.9) has a non-oscillatory solution.
Zhang and Choi [25] have studied also the functional equation of the form then equation (2.11) has a positive solution.

Main results
The following lemma can be easily proved by mathematical induction.
two given functions and consider the difference equation with the initial condition Then, the initial value problem (3.1) and (3.2) has a solution which is given in the form (b) x satisfies (3.1) and (3.2) or equivalently it is given by (3.3).
Proof. Let us assume that equation (1.1) has a positive solution, say x : [t −1 (t 0 ), ∞) → R. Then, one can show that On the other hand, if (a) and (b) hold then one can get that x is a positive solution of equation (1.1).
The following lemma will be useful in proving the main results.  Proof. The left side of the equality we can rewrite in the form Using the above transformation we get The following theorem is the discrete analogue of Theorem 3.1.1 [6] and simultaneously the generalization of the Theorem 1.1 [15] for continuous time.
Then the following statements are equivalent: where the positive function ψ is defined by (1.4), such that β(t) ≤ δ(t) ≤ γ(t) for t ≥ t 0 + 1, the following inequalities hold: , ∞) → R be the solution of the initial value problem (1.1) and (1.3) and suppose that x(t) > 0 for t ≥ t −1 (t 0 ). Our aim is to show that the positive function , is a solution of the characteristic equation (1.2) with the initial condition (1.4). From definition (3.6) denotes the integer part of the real number t).
By dividing both sides of equation (1.1) with x(t − 1) we get Because of (3.7) we have Thus, this part of the proof is complete.
By condition (3.4) and using induction, it follows that and so λ r : [t −1 (t 0 ) + 1, ∞) → R + . Next, we show that the sequence {λ r (t)} r∈N converges uniformly on any subinterval [t 0 + 1, Then from (3.8) it follows that Thus for all r = 1, 2, . . . and t 0 + 1 ≤ t ≤ T 1 the inequality holds. By induction, we can show that for all r = 0, 1, 2, . . . and t 0 For r = 0 we have Suppose that the inequality is true for r = q, i.e.
Existence of positive solutions 11 We will show that the inequality is true also for r = q + 1.
because t −1 (t 0 ) > 0 and so t − k + 1 > 0. For given n ∈ N, t ∈ R + and a function f : R → R we use the standard notation

It follows by the Weierstrass M-test that the series
converges uniformly on every compact interval [t 0 + 1, T 1 ] and therefore the sequence also converges uniformly. Thus, the limit function is positive for t 0 + 1 ≤ t ≤ T 1 . Because of the convergence, and λ(t) = λ 0 (t) for t −1 (t 0 ) + 1 ≤ t < t 0 + 1, which shows that λ, as defined by   (3.10) and the generalized characteristic equation Then, for every φ ∈ F C , equation (3.10) has a unique piecewise continuous solution x : We can formulate the following corollary.
Then the following statements are equivalent. (c) There exist functions β, for t ≥ t 0 + 1, the following inequalities hold:

Comparison results
Consider, now, the delay functional equation and the delay functional inequalities Existence of positive solutions

13
The oscillatory behavior of delay differential equations and inequalities has been the subject of many investigations. For a result we refer to [6,5] and the references therein. The next result is a discrete analogue of Theorem 3.2.1 [6] formulated for differential equations and inequalities and the generalization of the Theorem 1.2 [16] in the continuous time domain.

Existence of positive solutions
Our aim in this section is to derive results on the existence of positive solutions of equation (1.1) by applying statement (c) of Theorem 3.4. To that end, we will postulate first the Theorem which is the discrete analogue of the Theorem 3.3.2 [6] and at the same time the generalization of Theorem 3.2 [16] in the continuous time domain.
The next theorem is a generalization of Theorem 5.1.
Proof. Consider the functional equation It is possible to show that the statement (c) of Theorem 3.5 is true for any function δ Because of (H 3 ) and (5.5), it follows that Combining (5.3), (5.5) and (5.6), we obtain Therefore, the solution y(φ)(t) of (5.4) is positive for t ≥ t 0 . Since the solution x(φ)(t) of (1.1) is also a solution of inequality and by using Theorem 4.1, it follows that x(φ)(t) ≥ y(φ)(t) > 0 for t ≥ t 0 , the proof is complete.
Then equation (1.1) has positive increasing solution for t ≥ t 0 .
Proof. Let φ(t) ≡ 1 for t −1 (t 0 ) ≤ t < t 0 . The statement (c) of Theorem 3.5 will be true for any function δ (5.7) and (5.8), it follows that Therefore, by Theorem 3.5, the solution where λ is a positive solution of the characteristic equation associated with equation (1.1), greater than 1 for t ≥ t 0 + 1.
Hence, x(φ)(t) is an increasing solution of equation (1.1) and the proof is complete.