Causal difference equations with upper and lower solutions in the reverse order

This paper is devoted to studying the existence conditions for difference equations involving causal operators in the presence of upper and lower solutions in the reverse order. To this end, we prove some new comparison theorems and develop the upper and lower solutions method. Our results improve and extend some relevant results in difference equations. Two examples are given to illustrate the obtained results.


Introduction
In this paper, we are concerned with the existence of solutions for the following difference equations with causal operators: where x(k) = x(k + 1)x(k), E 0 = C(Z[0, T -1], R), Q ∈ C(E 0 , E 0 ) is a causal operator, g ∈ C(R × R, R), and the following type of equations: where x(k -1) = x(k)x(k -1), E 1 = C(Z [1, T], R), Q ∈ C(E 1 , E 1 ) is a causal operator, and g ∈ C(R × R, R).
With the development of boundary value problems (BVPs) for differential equations and for difference equations [18,19,25,26], and the theory of causal differential equations [6-9, 14, 21, 23], many authors have focused their attention on BVPs for causal difference equations [11,12,24]. In particular, in 2011, Jankowski [11] investigated first-order BVPs of difference equations with causal operators and developed the monotone iterative technique. In 2006, Atici, Cabada, and Ferreiro [2] considered the difference equations with functional boundary value conditions. Inspired by this paper, in 2015, Wang and Tian [24] established some existence criteria for the following difference equations involving causal operators with nonlinear boundary conditions: To obtain existence results of causal difference equations for problem (1) and (2), we use the method of lower and upper solutions coupled with the monotone iterative technique. This method is well known not only for the continuous case but also for the discrete case, see [1,10,13,15,17,20,22]. However, in the above papers, the definition of lower and upper solutions is not perfect, for example, in [2], and most results only discuss the case when lower solution is less than upper solution. In fact, in many cases, the lower and upper solutions often occur in the reverse order, which is a fundamentally different situation. So far only a few papers have investigated the existence results for the non-ordered case [3-5, 16, 27]. In this paper, we shall consider the causal difference equations with nonlinear periodic boundary conditions under the assumption of the existing upper and lower solutions for the reverse case.
We shall divide the results of this paper into six sections. First, some comparison principles are established. Next, by using the notion of lower and upper solutions v(k), w(k) and the monotone iterative technique, we testify the existence of the extremal solutions for (1) and (2) with v(k) ≥ w(k). Then, by using the definition of coupled lower and upper solutions v(k), w(k), we obtain the existence of the coupled quasi-solutions of (1) and (2) with lower and upper solutions in the reverse order. Finally, two examples are given to illustrate the results.

Lemmas
Let R be a real numbers set, Z denote the set of nonnegative integer numbers, Z[m, n] = {m, m + 1, . . . , n}, E = C(Z[m, n], R), where m, n ∈ Z and m < n. We define x = max k∈Z[m,n] |x(k)|. Moreover, in the paper, we only consider the discrete topology for the set Z[0, T].
A function x ∈ C(Z[0, T], R) is said to be a solution of problem (1) if it satisfies (1). Similarly the solution of problem (2) is defined analogously above. Definition 2.1 Assume that Q ∈ C(E, E), then Q is said to be a causal operator if the following property holds: if u, v ∈ E are such that u(s) = v(s) for m ≤ s ≤ k < n, k ∈ Z[m, n] arbitrary, then (Qu)(s) = (Qv)(s) for m ≤ s ≤ k. where L ∈ C(E 0 , E 0 ) is a positive linear operator, that is, Lm ≥ 0 whenever m ≥ 0, and Proof Suppose that the conclusion is not true, then p(k) ≥ 0 for some k ∈ Z[0, T]. We have two cases as follows. Thus, so λ > 1, this is in contradiction with (4).
where L ∈ C(E 1 , E 1 ) is a positive linear operator and The proof is analogous to Lemma 2.2, so it is omitted.

Lemma 3.1 A function x ∈ E 0 is a solution of (6) if and only if x is a solution of the summation equation below:
By applying (7), one arrives at Let k = T in (8). Then one has From the boundary condition y(T) = Substituting (9) into (8) and using y( We see that x is a solution of (6) and the proof is complete.
In the remainder of the paper, Then problem (6) has a unique solution.
Proof Define an operator F : For any x 1 , x 2 ∈ E 0 , we have Hence, by the Banach contraction principle, F has a unique fixed point and (6) has only one solution. We complete the proof.
Next, we give the following definitions which help us to testify our main results.

Definition 3.3 A function w is called an upper solution of (1) if
and a lower solution of (1) is defined similarly by reversing the inequalities above.
Proof First, we define the sequences {v n (k)}, {w n (k)} as follows: for n = 1, 2, . . . , where v 0 = v, w 0 = w. It follows from Lemma 3.2 that both (11) and (12) have unique solutions, respectively. We have four steps to complete the proof.
Step 1. We demonstrate that w n-1 ≤ w n and v n ≤ v n-1 , n = 1, 2, . . . . and From Lemma 2.2 and M 2 ≥ M 1 > 0, we get p ≤ 0, so v 1 ≤ v. Employing mathematical induction, it is readily seen that v n is a nonincreasing sequence. Analogously, we can show w n is a nondecreasing sequence.
Step 3. By the first two steps, we get and each v n , w n satisfies (10) and (11). It is easy to see that sequences {v n (k)}, {w n (k)} are monotonously and bounded, passing to the limit when n → ∞, we have lim n→∞ v n (k) = ρ(k) and lim n→∞ w n (k) = r(k) uniformly on Z[0, T]. Clearly, ρ(k), r(k) satisfy problem (1).
Step 4. We show that ρ and r are extremal solutions of (1) in [w, v]. Let x(k) be any solution of (1) such that w(k) ≤ x(k) ≤ v(k). Assume that there exists a positive integer n such that w n (k) ≤ x(k) ≤ v n (k). Then, setting p = w n+1x, we have By Lemma 2.2, p ≤ 0, i.e., w n+1 ≤ x. Similarly, we may get that x ≤ v n+1 on Z[0, T]. Since w 0 (k) ≤ x(k) ≤ v 0 (k), by induction we obtain w n (k) ≤ x(k) ≤ v n (k) for every n ∈ N, which implies r(k) ≤ x(k) ≤ ρ(k), and the proof is complete.

Existence results to (1.2)
In this section, to avoid repetition, we merely state the next lemmas and theorems without proofs since they are similar to those in Sect. 3.

Definition 4.1 Function w is called an upper solution of (2) if
and a lower solution of (2) is defined similarly by reversing the inequalities above.

Lemma 4.2 Let C u = -g(u(0), u(T)) + M 1 u(0) -M 2 u(T). A function x ∈ E 1 is a solution of (13) iff x is a solution of the following summation equation:
In the remainder of the paper, we denote ξ = H(k, i) = max{| Then problem (13) has a unique solution. (14) is satisfied, further (A 1 ) w, v are upper and lower solutions of problem (2) and

Coupled lower and upper solutions
In this section, we shall prove the existence of the coupled quasi-solutions for problems (1) and (2).

Definition 5.1
Functions v, w are called coupled lower and upper solutions of (1) if

Definition 5.2 A pair (U, V ) is said to be a coupled quasi-solution of problem (1) if
The definitions of coupled lower and upper solutions and coupled quasi-solution for (2) are similar to above.
In regard to Lemma 3.1 and Lemma 3.2, it is easy to obtain that v, w are well defined. First we prove that v 0 ≤ v 1 ≤ w 1 ≤ w 0 .