Multiple Positive Solutions of Second-Order Nonlinear Difference Systems with Repulsive Singularities

We study the existence of positive solutions for second-order nonlinear repulsive singular difference systems with periodic boundary conditions. Our nonlinearity may be singular in its dependent variable. The proof of the main result relies on a fixed point theorem in cones and a nonlinear alternative principle of Leray-Schauder; the result is applicable to the case of a weak singularity as well as the case of a strong singularity. An example is given; some recent results in the literature are improved and generalized.

Such repulsive singularity appears in many problems of applications such as the Brillouin focusing systems and nonlinear elasticity [2].
System (1) can be viewed as a discretization of the following more general class of the Sturm singular second-order differential system: Such systems, even in case p ≡ 1, where they are referred to as being of Klein-Gordon or Schrödinger type, appear in many scientific areas including fluid mechanics, gas dynamics, and quantum field theory. During the last few decades, the study of the existence of periodic solutions for singular differential equations has deserved the attention of many researchers [3][4][5][6][7][8][9][10][11]. Tracing back to 1987, Lazer and Solimini [5] investigated the singular model: where λ > 0, h, g are T-periodic functions and the mean value of g is negative, g < 0. One of the common conditions to guarantee the existence of positive periodic solution is a so-called strong force condition (corresponds to the case λ ≥ 1 in (5)) [11,12]. For example, if we consider the system: with Vðt, xÞ = 1/jxj a ; the strong force condition holds for α ≥ 2. On the other hand, the existence of positive periodic solutions of the singular differential equations has been established with a weak force condition (corresponds to the case 0 < λ < 1 in (5)) [13][14][15]. From then on, some classical tools have been used to study singular differential equations in the literature, including the degree theory [6,11,16], the method of the upper and lower solutions [8,17], Schauder's fixed point theorem [14], some fixed point theorems in cones for completely continuous operators [13,18], and a nonlinear Leray-Schauder alternative principle [19].
For the existence of periodic solutions of difference equations, some results have been obtained using the variational methods or the topological methods [1,[20][21][22][23][24][25]. For example, by minimax principle, Guo and Yu [23] discussed the existence of periodic solutions for difference equation: where the nonlinearity f is of superlinear or sublinear growth at infinity. Based on the method of the upper and lower solutions, Atici and Cabada [21] studied the existence of periodic solutions for difference equation: In [26], Zhou and Liu investigated the following autonomous difference equations: By Conley index theory, the author showed that the suitable assumptions of asymptotically linear nonlinear are enough to guarantee the existence of periodic solutions.
In this paper, we establish two different existence results of positive periodic solutions for (1) and (2) and proof of the existence of positive solutions; the first one is based on an application of a nonlinear alternative of Leray-Schauder, which has been used by many authors [19,27,28] and references therein; the second one is based on a fixed point theorem in cones. Our main motivation is to obtain new existence results for positive periodic solutions of the system: Here, we emphasize that the new results are applicable to the case of a strong singularity as well as the case of a weak singularity and that e does not need to be positive.
The rest of this paper is organized as follows. In Section 2, some preliminary results will be given. In Section 3, we will state and prove the main results. We will use the notation We say that a function φ : ℝ N ⟶ ℝ is nondecreasing if φðxÞ ≥ φðyÞ for x, y ∈ ℝ N with x ≥ y. For a given function p defined on ℤ½0, T, we denote its maximum and minimum by p * and p * , respectively.

Journal of Function Spaces
Lemma 2 (see [29]). Assume (H) holds. For the solution of the problem: the formula holds, where is the Green's function; the number D i is defined by (13).

Main Results
In this section, we state and prove the new existence results for (1). In order to prove our main results, the following nonlinear alternative of Leray-Schauder is needed, which can be found in [30].

Lemma 5.
Assume Ω is a relatively compact subset of a convex set E in a normed space X. Let A : Ω ⟶ E be a compact map with 0 ∈ Ω: Then, one of the following two conclusions holds: (i) T has at least one fixed point in Ω (ii) There exist u ∈ ∂Ω and 0 < λ < 1 such that u = λAu Then, X 1 is a Banach space with the norm We take with the norm Define which corresponds to the unique solution of (14), and the 3 Journal of Function Spaces Now, we present the first existence result of the positive solution to problem (1).
and g i ðxÞ > 0 is nonincreasing and h i ðxÞ/g i ðxÞ is nondecreasing in x (H 3 ) There exists a positive number r such that σr + γ * > 0 and for all i = 1, 2, ⋯, N. Here, Then, (1) and (2) has at least one positive periodic solution x with xðnÞ > γðnÞ for all n ∈ Z½0, T and 0 < jx − γj < r.

Journal of Function Spaces
We claim that any fixed point x of (34) for any λ ∈ ½0, 1 must satisfy jxj ≠ r. Otherwise, assume that x is a fixed point of (34) for some λ ∈ ½0, 1 such that jxj = r. Without loss of generality, we assume that jx l j = r for some l = 1, 2, ⋯, N.
Thus, we have Hence, for all n ∈ Z½1, T, we have Therefore, Using (34), we have from condition (H 2 ), for all n ∈ Z½1 , T, Therefore, This is a contradiction to the choice of j 0 , and the claim is proved.
From this claim, the nonlinear alternative of Leray-Schauder guarantees that has a fixed point, denoted by x j ðnÞ, in B r = fx ∈ X : jxj < rg, i.e., has a periodic solution x j with jx j j < r.
In more general, we can obtain the following result.
Then, for each e with γ * ≥ 0, we have (i) if β < 1, then (10) has at least one positive periodic solution for each μ > 0 (ii) if β ≥ 1, then (10) has at least one positive periodic solution for each 0 < μ < μ 2 , where μ 2 is some positive constant By using a fixed point theorem for compact maps on conical shells [31], we established the second positive periodic solution for (1). Recall that a compact operator means an operator which transforms every bounded set into a relatively compact set and introducing the definition of a cone. Definition 9. Let X be a Banach space and let K be a closed, nonempty subset of X. K is a cone if (i) αu + βv ∈ K for all u, v ∈ K and all α, β > 0 (ii) u, −u ∈ K implies u = 0 Lemma 10 (see [31]). Let X be a Banach space and K a cone in X. Assume Ω 1 , Ω 2 are open subsets of X with 0 ∈ Ω 1 , be a continuous and completely continuous operator such that (ii) There exist ψ ∈ K \ f0g such that x ≠ Φx + λψ for x ∈ K ∩ ∂Ω 2 and λ > 0