Existence of uncountably many asymptotically constant solutions to discrete nonlinear three-dimensional system with $p$-Laplacian

This work is devoted to the study of the existence of uncountably many asymptotically constant solutions to discrete nonlinear three-dimensional system with p -Laplacian.

1. Introduction and preliminaries. Equations involving the discrete p-Laplacian operator have been widely studied by many authors and several approaches, see for example [4], [5] and [6]. Agarwal, Perera and O'Regan obtain multiple positive solutions of singular discrete p-Laplacian problems via variational methods in [1]. Paper [2] is devoted to the study of the existence of at least one (non-zero) solution to a problem involving the discrete p-Laplacian. In [8], He by means of a fixed point theorem in a cone, studies the existence of positive solutions of p-Laplacian difference equations. In [9], Lee and Lee consider p-Laplacian systems with singular weights. By exploiting Amann type three solutions theorem for a singular system, the authors prove the existence, nonexistence, and multiplicity of positive solutions when nonlinear terms have a combined sublinear effect at infinity.
Three-dimensional systems were considered in [10], [13] and [14]. Here we continue such a study by considering the following three-dimensional system of difference equations with p-Laplacian operator where n ≥ n 0 = max {k, l, m} ∈ N 0 = {0, 1, 2, . . . } and ∆ is the forward difference operator, (a(n)), (b(n)), (c(n)) are sequences of real numbers, l, k, m are nonnegative integers, functions f, g, h : R → R are continuous functions. Moreover, for p ∈ R, p > 1 the p-Laplacian operator Φ p : R → R is defined by formula In system (1), (x(n)), (y(n)) and (z(n)) are real sequences defined for n ∈ N. Throughout this paper, X(n) denotes a vector [x(n), y(n), z(n)] T . By a solution of system (1) we mean a sequence X = (X(n)) which fulfils system (1) for n ≥ n 0 . Notice that by taking p = 2 system (1) reduces to the system which was considered in [11] and [12].
For the elements of R 3 the symbol | · | stands for the maximum norm. By B we denote the Banach space of all bounded sequences in R 3 with the supremum norm, i.e., It is well known that Φ p given by (2) is a homeomorphism on R, with (Φ p ) −1 = Φ p , where 1 p + 1 p = 1 (or equivalently p = p p−1 ). Under continuity of functions f, g, h and assumptions we prove the sufficient conditions under which for any real constants d 1 , d 2 , d 3 there exists a solution to the considered system convergent to [d 1 , d 2 , d 3 ] T . That connects the subject of this paper with so-called permanence of solutions, which is important for example in mathematical modelling of problems arising in biology.
The following theorems will be used in the sequel. By c 0 we denote the Banach space of all convergent to zero sequences with the supremum norm.
for any x = (x(n)) ∈ A and for any n ∈ N 0 .

Main results.
In this section, we present conditions under which there exists asymptotically constant solution to system (1). In the first step we prove the following existence result.
then for any real constants d 1 , d 2 , d 3 there exists a sequence (X(n)) which fulfils system (1) for sufficiently large n such that Proof. Taking into consideration that Φ p : R → R is a homeomorphism, then the considered system (1) by substitutionx(n) = Φ p (x(n)) is equivalent to the following problem Let It is easy to see that Ω is a closed and convex subset of Banach space B. Taking into consideration that c 0 is a closed subspace of space B, to prove that Ω is a compact subset of B we use the relative compactness criterion in c 0 . From (3) we get that From Theorem 1.2 and above we get that for any sequences ( and (F X)(n) = D for 0 < n < n 2 . Firstly, we will show that F : Ω → Ω. For s ≥ n 2 and X ∈ Ω we get from (9) |y(s − l) − d 2 | ≤ M g ∞ s=n |b(s)| ≤ δ, and from (6) |f (y(s − l))| ≤ M f . Hence, For s ≥ n 2 and X ∈ Ω we get from (9) and from (7) |g(z(s − m))| ≤ M g . Hence, For s ≥ n 2 and X ∈ Ω we get from (9) and from (8) |h(x(s − k))| ≤ Mh. Hence, Therefore, F (Ω) ⊂ Ω. Now we prove the continuity of F . Let X = (X(n)) = [x(n), y(n), z(n)] T ∈ Ω, X j = (X j (n)) = [x j (n), y j (n), z j (n)] T ∈ Ω such that X j → X as j → ∞. Let ε > 0.
The functions f, g,h are uniformly continuous on any compact interval, so then there exists δ 1 > 0 such that for any j ≥ j 0 , s ≥ n 0 . Then from (13), (16) we get for any n ≥ n 2 , j ≥ j 0 We get conditions |(F 2 X j )(n) − (F 2 X)(n)| ≤ ε, |(F 3 X j )(n) − (F 3 X)(n)| ≤ ε for n ≥ n 2 , j ≥ j 0 in an analogous way from (15). Because (F X)(n) = (F X j )(n) for n < n 2 and j ∈ N 0 we get that for j ≥ j 0 which means that F is continuous on Ω.
All assumptions of Theorem 2.1 are satisfied. It is easy to check that T is the solution of the above system having the property lim Proof. On virtue of Theorem 2.1, there exists a sequence (X(n)), satisfying condition (4), which fulfils system (1) for sufficiently large n. Since (A 3 ) and (A 4 ) hold, we can rewrite system (1) in the following form , n ≥ n 2 .
Next, we present the necessary condition for existence of asymptotically constant solution to system (1).