The Solvability of the Discrete Boundary Value Problem on the Half-Line

This paper provides conditions for the existence of a solution to the second-order nonlinear boundary value problem on the half-line of the form Δa(n)Δx(n)=f(n+1,x(n+1),Δx(n+1)),n∈N∪{0}, with αx(0)+βa(0)Δx(0)=0,x(∞)=d, where d,α,β∈R, α2+β2>0. To achieve our goal, we use Schauder’s fixed-point theorem and the perturbation technique for a Fredholm operator of index 0. Moreover, we construct the necessary condition for the existence of a solution to the considered problem.


Introduction
In various physical areas, such as hydrodynamics or the unsteady flow of gas through a semi-infinite porous media, studying radially symmetric solutions leads to the Sturm-Liouville equation with boundary value conditions of the form x (0) = 0, x(∞) = C, C ∈ (0, 1); see for example [1,2]. Let us remind the reader of the classical Sturm-Liouville boundary value problem on the half-line: ds p(s) > 0. Many authors considered the above problem or its simplifications, see, for instance, [3][4][5][6][7] or slightly different boundary value problems on the half-line [8][9][10][11] and the references therein.
Difference equations represent the discrete counterpart of ordinary differential equations and are usually studied in connection with the numerical analysis. In this paper we consider the following discrete boundary value problem on the half-line: ∆(a(n)∆x(n)) = f (n + 1, x(n + 1), ∆x(n + 1)), n ∈ N ∪ {0}, where x(∞) = lim n→∞ x(n), d ∈ R, α, β ∈ R, α 2 + β 2 > 0. We want to construct sufficient conditions for the existence of a solution to (2) in dependence on the parameters α, β. First, we divide our consideration into two cases, when problem (2) is without resonance, which means that β = α ∑ ∞ l=0 1 a(l) , and with resonance. For the problem without resonance we use the fixed-point approach, which requires ∑ ∞ l=0 l a(l) < ∞ and the growth condition on a nonlinear continuous function f . In both cases, with and without resonance, we prove that ∑ ∞ l=0 l a(l) < ∞ is the necessary condition for the existence of a solution to (2). By the resonant case we mean the following problem: ∆(a(n)∆x(n)) = f (n + 1, x(n + 1), ∆x(n + 1)), n ∈ N ∪ {0}, This is called resonant, because for d = 0 we can write the above problem in an abstract form, Lx = Nx, where L is a linear, noninvertible operator, and N is a nonlinear operator. We notice that the noninvertible operator L is a Fredholm operator of index 0 and to obtain a solution to the above problem we use the Przeradzki perturbation method, see [12,13]. We construct Landesmann-Lazer type conditions for a bounded and continuous nonlinear function f . The used perturbation technique allows us to establish sufficient conditions for the existence of a solution to the above problem not only for d = 0, but for all d ∈ R. The Przeradzki perturbation method is one of the tools used to deal with boundary value problems in the resonant case. Another classical approach is Mawhin's coincidence degree, see for example [14]. More information about properties of Fredholm operators can be found in [13,15,16] and the references therein. Many authors have considered a discrete version of (1) or its generalizations using different tools, see for example [2,[17][18][19][20][21] and the references therein. Lian et al.,in [19], established the sufficient conditions for the existence of one and three solutions of the following problem: with a > 0, B, C ∈ R, using an upper and lower solutions method combined with the fixed-point approach and the degree theory. The method of upper and lower solutions on finite intervals with the degree theory was used by Tian et al. in [21] to prove the existence of three solutions to with l ∈ N, B ∈ R, p ≥ q > 0. For α, β ≥ 0, α 2 + β 2 > 0, p, q > 0, 1 + p > q, Tian and Ge in [20] searched for positive solutions to the following problem: To obtain the main results of this paper we need some auxiliary tools. Let us remind the reader of some of them. By c 0 we denote the Banach space of all sequences convergent to zero, whereas by c we denote the Banach space of all convergent sequences. We consider the supremum norm in both spaces. Proposition 1 ([23], p. 107). A set A ⊂ c 0 is relatively compact (with respect to norm topology) if and only if there is a sequence {λ(n)} ∈ c 0 such that |x(n)| ≤ λ(n) for any {x(n)} ∈ A and for any n ∈ N ∪ {0}.
From Lemma 3.1 in [24] or Lemma 5 in [25] we have: |w j | is convergent, then the second series is convergent and The plan of the paper is as follows: Section 2 is devoted to the study of Problem (2) with β = α ∑ ∞ l=0 1 a(l) , and in Section 3 the resonant case is presented.

Problem without Resonance
This section begins with the presentation of sufficient conditions for the existence of a solution to (2) Assumptions of this case allow us to look for a solution to (2) via a fixed point of an operator defined on some subset of c. . In this proof for {x(n)} n∈N∪{0} ∈ c we use the notation We define an operator T : B M (d) → c as follows: The above estimation yields Tx ∈ c. Now, we show that every fixed point of T is a solution to (2). Indeed, let x ∈ B M (d) be a fixed point of T; then: for n ∈ N ∪ {0}. Eventually, we obtain that ∆(a(n)∆x(n)) = f (n + 1, x(n + 1), ∆x(n + 1)) n ∈ N ∪ {0}. Moreover, Finally, passing to n → ∞ in (5) we obtain x(∞) = lim n→∞ x(n) = d, which ends the proof that every fixed point of T is a solution to (2). Now, we are in a position to check the assumptions of Schauder's theorem. We show that Let x ∈ B M (d) and n ∈ N ∪ {0}. By the definition of T and (4), we have: From the above we obtain that is a relatively compact subset of c 0 ; see Proposition 1. To prove that T is a compact operator in c, we have to prove its continuity. Let ε > 0. From assumption (H 2 ) we obtain the existence of n 0 ∈ N such that Moreover, there exists η > 0 such that From the uniform continuity of function f on {1, . . . , we obtain the existence of δ > 0 such that for any n ∈ {0, 1, . . . , From (6) and (7) above, we have By Schauder's theorem we obtain that there exists a fixed point x ∈ B M (d) of T, which is a solution to (2).
We will now present examples of classes of functions which satisfy (H 4 ). Example 1. Let d ∈ R and f : N × R 2 → R be a continuous function fulfilling with nonnegative sequences {b(n)}, {c(n)}, {e(n)} such that It is easy to see, that for assumption (H 4 ) of Theorem 2 is satisfied. Note that, for any L > 0, condition (10) with b(n) = c(n) = 0 and e(n) = L, n ∈ N ∪ {0} is satisfied. It means that this case includes a class of bounded functions. Moreover, (H 4 ) holds for a linear function with respect to second and third variables, i.e., f (n + 1, x, y) = b(n)(x − d) + c(n)(y) + e(n) for n ∈ N ∪ {0}, x, y ∈ R, where {b(n)}, {c(n)} satisfy (9) and {e(n)} is bounded.
The next example is a simple consequence of Example 1.
Let us remind the reader that in the classical approach we assume that ∑ ∞ l=0 1 a(l) < ∞. To see that our condition ∑ ∞ l=0 l a(l) < ∞ is not too strong we present the following necessary condition for the existence of a solution to (2). It is worth mentioning that the following necessary condition is true in both cases when Problem (2) is with and without resonance. Proof. If {x(n)} n∈N∪{0} is a nontrivial solution to (2), then there exists n 1 ∈ N such that |x(n) − d| ≤ K for any n ≥ n 1 . Then |∆x(n)| ≤ 2K for any n ≥ n 1 and summing up from n 2 := max{n 1 , n 0 } to n − 1 from the equation in (2) we obtain: for n ≥ n 2 . Hence, we have: . Summing up the above from n 2 to n − 1 and using (H 6 ), we obtain: for n > n 2 . Using the fact that x(+∞) = lim n→∞ x(n) = d and letting n → ∞ in the above, we have: ηl+a(n 2 )∆x(n 2 )−ηn 2 a(l) .
Dividing the equation and the first boundary condition in (13) Let k ∈ N. We consider the perturbed problem ∆(a(n)∆x(n)) = f (n + 1, x(n + 1), ∆x(n + 1)), n ∈ N ∪ {0}, under (14). Notice that after dividing (15) by ∑ ∞ l=0 ∑ ∞ i=l+1 1 a(i) , the nonlinear function f is still bounded. Hence, there exists L > 0 such that | f (n, x, y)| ≤ L for n ∈ N ∪ {0}, x, y ∈ R. It is clear that problem (15) satisfies assumptions of Theorem 2 with M k := kL ∑ ∞ l=0 l a(l) + |d|(k − 1) = kL + |d|(k − 1). Hence, there exists a solution x k to (15). We prove that {||x k ||} k∈N is bounded in c. On the contrary, suppose that {||x k ||} k∈N is unbounded. Passing to subsequence if necessary we assume that ||x k || → ∞, as k → ∞. Dividing (15) by ||x k || we obtain: for any k ∈ N. By the boundedness of f there exists M 1 > 0 such that for any n ∈ N ∪ {0}, k ∈ N. Summing the above from 0 to n − 1 we have for any n ∈ N ∪ {0}, k ∈ N. Summing the above from n to m − 1, (m − 1 > n) we obtain: for any n ∈ N ∪ {0}, k ∈ N. Passing to m → ∞ we obtain: It is easy to see that Let us assume that x 0 (n) = , n ∈ N ∪ {0}. Knowing that x k , k ∈ N is a fixed point of operator (3), we obtain from Theorem 4 that for any n ∈ N ∪ {0}, k ∈ N. By the boundedness of f and (H 2 ), Lemma 1 and (14) we have: Hence, there exists k 0 such that for any k ≥ k 0 . Using Fatou's lemma with summable lower bound We consider two cases.
We exclude (21) in the same way as in Case 1. On the other hand, for t = n 0 , we have that x k (n 0 + 1) ≥ 1 2 x 0 (n 0 + 1)||x k || for all large k. Hence, we exclude (22) by (H 8 ).

Conclusions
In this paper, we constructed sufficient conditions for the existence of a solution to the discrete boundary value problem on the half-line (2) in dependence on parameters α, β, d ∈ R. For α = 0, d ∈ (0, 1) the considered problem can be interpreted as a discrete version of some problem from hydrodynamics; see [2]. The fixed-point approach is used when problem (2) is without resonance. In the resonant case the considered problem can be solved via the perturbation technique for a Fredholm operator of index 0. We proved that the constructed assumptions are not too strong by providing the necessary condition for the existence of a solution to this problem.