Decaying positive global solutions of second order difference equations with mean curvature operator

A boundary value problem on an unbounded domain, associated to difference equations with the Euclidean mean curvature operator is considered. The existence of solutions which are positive on the whole domain and decaying at infinity is examined by proving new Sturm comparison theorems for linear difference equations and using a fixed point approach based on a linearization device. %The process from the continuous problem to discrete one is examined, too. The process of discretization of the boundary value problem on the unbounded domain is examined, and some discrepancies between the discrete and the continuous case are pointed out, too.


Introduction
In this paper we study the boundary value problem (BVP) on the half-line for difference equation with the Euclidean mean curvature operator subject to the conditions where m ∈ Z + = N ∪ {0}, k ∈ Z m := {k ∈ Z : k ≥ m} and c ∈ (0, ∞). Throughout the paper the following conditions are assumed: (H 1 ) The sequence a satisfies a k > 0 for k ∈ Z m and ∞ j=m 1 a j < ∞.
(H 3 ) The function F is continuous on R, F (u)u > 0 for u = 0, and When modeling real life phenomena, boundary value problems for second order differential equations play important role. The BVP (1)-(2) originates from the discretisation process for searching radial solutions, which are globally positive and decaying, for PDE with Euclidean mean curvature operator. By globally positive solutions we mean solutions which are positive on the whole domain Z m . The Euclidean mean curvature operator arises in the study of some fluid mechanics problems, in particular capillarity-type phenomena for compressible and incompressible fluids.
Recently, discrete BVPs, associated to equation (1), have been widely studied, both in bounded and unbounded domains, see, e.g., [2] and references therein. Many of these papers can be seen as a finite dimensional variant of results established in the continuous case. For instance, we refer to [5,6,7,23] for BVPs involving mean curvature operators in Euclidean and Minkowski spaces, both in the continuous and in the discrete case. Other results in this direction are in [8,9], in which the multiplicity of solutions of certain BVPs involving the p-Laplacian is examined. Finally, in [13,15] for second order equations with p-Laplacian the existence of globally positive decaying Kneser solutions, that is solutions x such that x n > 0, ∆x n < 0 for n ≥ 1 and lim n→∞ x n = 0, is examined.
Several approaches have been used in literature for treating the above problems. Especially, we refer to variational methods [10], the critical point theory [9] and fixed point theorems on cones [25,26].
Here, we extend to second order difference equations with Euclidean mean curvature some results on globally positive decaying Kneser solutions stated in [13] for equations with p-Laplacian and b n < 0.
This paper is motivated also by [14], in which BVPs for differential equation with the Euclidean mean curvature operator on the half-line [1, ∞) have been studied subjected to the boundary conditions x(1) = 1 and lim t→∞ x(t) = 0. The study in [14] is accomplished by using a linearization device and some properties of principal solutions of certain disconjugate second-order linear differential equations. Here, we consider the discrete setting of the problem studied in [14]. However, the discrete analogue presented here requires different technique. This is caused by a different behavior of decaying solutions as well as by peculiarities of the discrete setting which lead to a modified fixed point approach. Jointly with this, we prove new Sturm comparison theorems and new properties of recessive solutions for linear difference equations. Our existence result is based on a fixed point theorem for operators defined in a Fréchet space by a Schauder's linearization device. This method is originated in [11], later extended to the discrete case in [21], and recently developed in [16]. This tool does not require the explicit form of the fixed point operator T and simplifies the check of the topological properties of T on the unbounded domain, since these properties become an immediate consequence of apriori bounds for an associated linear equation. These bounds are obtained in an implicit form by means of the concepts of recessive solutions for second order linear equations. The main properties and results which are needed in our arguments, are presented in Sections 2 and 3. In Section 4 the solvability of the BVP (1)-(2) is given, by assuming some implicit conditions on sequences a and b. Several effective criteria are given, too. These criteria are obtained by considering suitable linear equations which can be viewed as Sturm majorants of the auxiliary linearized equation. In Section 5 we compare our results with those stated in the continuous case in [14]. Throughout the paper we emphasis some discrepancies, which arise between the continuous case and the discrete one.

Discrete versus continuous decay
Several properties in the discrete setting have no continuous analogue. For instance, for a positive sequence x we always have In the continuous case, obviously, this does not occur in general, and the decay can be completely different. For example, if x(t) = e −2t then x ′ (t)/x(t) = −2 for all t. Further, the ratio x ′ /x can be also unbounded from below, as the function x(t) = e −e t shows. Another interesting observation is the following. If two positive continuous functions x, y satisfy the inequality then there exists K > 0 such that x(t) ≤ Ky M (t) for t ≥ t 0 . This is not true in the discrete case, as the following example illustrates.
Example 1. Consider the sequences x, y given by On the other hand, the inequality x k ≤ Ky 1/2 k is false for every value of K > 0. Indeed, which is clearly unbounded.
The situation in the discrete case is described in the following two lemmas.
Lemma 1. Let x, y be positive sequences on Z m such that M ∈ (0, 1) exists, satisfying for k ∈ Z m . Then 1 + M∆y k /y k > 0 for k ∈ Z m , and Proof. First of all note that, from M ∈ (0, 1) and the positivity of y, we have and taking the product from m to k − 1, k > m, we obtain From the classical theory of infinite products (see for instance [20]) the infinite product P = ∞ k=m (1 + q k ) of real numbers is said to converge if there is N ∈ Z m such that 1 + q k = 0 for k ≥ N and P n = n k=N (1 + q k ) has a finite and nonzero limit as n → ∞.
In case −1 < q k ≤ 0, {P n } is a positive nonincreasing sequence, thus P being divergent (not converging to a nonzero number) means that lim n→∞ n k=N (1 + q k ) = 0.
Moreover, the convergence of P is equivalent to the convergence of the series ∞ k=N ln(1+ q k ) and this is equivalent to the convergence of the series ∞ k=N q k . Indeed, if ∞ k=m q k is convergent, then lim k→∞ q k = 0 and hence, i.e., ln(1 + q k ) ∼ q k as k → ∞. Since summing preserves asymptotic equivalence, we get that ∞ k=m ln(1 + q k ) converges. Similarly, we obtain the opposite direction. Therefore, in case −1 < q k ≤ 0, (5) holds if and only if ∞ k=N q k diverges to −∞. The following holds.
Lemma 2. Let y be a positive nonincreasing sequence on Z m such that lim k→∞ y k = 0. Then, for any M ∈ (0, 1), Proof. From the theory of infinite products it is sufficient to show that We distinguish two cases: As for the former case, from the Lagrange mean value theorem, we have where ξ k is such that y k+1 ≤ ξ k ≤ y k for k ∈ Z m . Summing the above inequality from m to n − 1, n > m, we get Since lim n→∞ y n = 0, letting n → ∞ we get (6). Next we deal with the case inf k∈Zm y k+1 /y k = 0. This is equivalent to which implies (6), since k j=m ∆y j /y j is negative nonincreasing.

A Sturm-type comparison theorem for linear equations
The main idea of our approach is based on an application of a fixed point theorem and on global monotonicity properties of recessive solutions of linear equations. To this goal, in this section we prove a new Sturm-type comparison theorem for linear difference equations.
Consider the linear equation where p k ≥ 0 and r k > 0 on Z m . We say that a solution y of equation (7) has a generalized zero in n if either y n = 0 or y n−1 y n < 0, see e.g. [1,3]. A (nontrivial) solution y of (7) is said to be nonoscillatory if y k y k+1 > 0 for all large k. Equation (7) is said to be nonoscillatory if all its nontrivial solutions are nonoscillatory. It is well known that, by the Sturm type separation theorem, the nonoscillation of (7) is equivalent to the existence of a nonoscillatory solution see e.g. [2, Theorem 1.4.4], [3]. If (7) is nonoscillatory, then there exists a nontrivial solution u, uniquely determined up to a constant factor, such that where y denotes an arbitrary nontrivial solution of (7), linearly independent of u. Solution u is called recessive solution and y a dominant solution, see e.g. [4]. Recessive solutions can be characterized in the following ways (both these properties are proved in [4]): (7) is recessive if and only if (ii) For a recessive solution u of (7) and any linearly independent solution y (i.e. dominant solution) of (7), one has ∆u k u k < ∆y k y k eventually.
Along with equation (7) consider the equation where P k ≥ p k ≥ 0 and 0 < R k ≤ r k on Z m ; equation (9) is said to be a Sturm majorant of (7). From [2, Lemma 1.7.2], it follows that if (9) is nonoscillatory, then (7) is nonoscillatory as well. In this section we always assume that (9) is nonoscillatory.
The following two propositions are slight modifications of results in [17]. They are preparatory to the main comparison result.
. Let x be a positive solution of (9) on Z m and y be a solution of (7) such that y m > 0 and r m ∆y m /y m ≥ R m ∆x m /x m . Then Moreover, if y,ȳ are solutions of (7) such that y k > 0, k ∈ Z m , andȳ m > 0, ∆ȳ m /ȳ m > ∆y m /y m , thenȳ . If a recessive solution v of (7) has a generalized zero in N ∈ Z m and has no generalized zero in (N, ∞), then any solution of (9) has a generalized zero in (N − 1, ∞).
The following lemma is an improved version of [17, Theorem 1].
Lemma 3. Let u, v be recessive solutions of (7) and (9), respectively, satisfying Proof. By contradiction, assume that there exists N ∈ Z m such that r N ∆u N /u N > R N ∆v N /v N . Let y be a solution of (7) satisfying y N > 0 and r N ∆y N /y N = R N ∆v N /v N . Then r N ∆y N /y N < r N ∆u N /u N , (which implies that y is linearly independent with u) and from Proposition 1 we get y k > 0, ∆y k /y k < ∆u k /u k for k ∈ Z N , which contradicts (8).
Lemma 4. Let x be a positive solution of (9) on Z m . Then there exists a recessive solution u of (7), which is positive on Z m .
Proof. Let u be a recessive solution of (7), whose existence is guaranteed by nonoscillation of majorant equation (9). By contradiction, assume that there exists N ∈ Z m such that Then u cannot have a generalized zero in (N + 1, ∞). Theorem 1. Let x be a positive solution of (9) on Z m . Then there is a recessive solution u of (7), which is positive on Z m and satisfies In addition, if x is decreasing (nonincreasing) on Z m , then u is decreasing (nonincreasing) on Z m .
Proof. Let x be a positive solution of (9) on Z m . From Lemma 4, there exist a recessive solution u of (7) and a recessive solution v of (9), which are both positive on Z m . We claim that Indeed, suppose by contradiction that there is N ∈ Z m such that ∆x N /x N < ∆v N /v N . Then, in view of Proposition 1, ∆x k /x k < ∆v k /v k for k ∈ Z N , which contradicts (8).
Taking p = P and r = R in Theorem 1, we get the following corollary.
Corollary 1. If (9) has a positive decreasing (nonincreasing) solution on Z m , then there exists a recessive solution of (9) which is positive decreasing (nonincreasing) on Z m .
We close this section by the following characterization of the asymptotic behavior of recessive solutions which will be used later.
Then (7) is nonoscillatory. Moreover, for every d = 0, (7) has an eventually positive, nonincreasing recessive solution u, tending to zero and satisfying If the linear difference equation has a positive decreasing solution on Z m , then BVP (1)-(2) has at least one solution.
Effective criteria, ensuring the existence of a positive decreasing solution of (14), are given at the end of this section.
From this theorem and its proof we get the following. To prove Theorem 2, we use a fixed point approach, based on the Schauder-Tychonoff theorem on the Fréchet space of all sequences defined on Z m , endowed with the topology of pointwise convergence on Z m . The use of the Fréchet space X, instead of a suitable Banach space, is advantageous especially for the compactness test. Even if this is true also in the continuous case, in the discrete case the situation is even more simple, since any bounded set in X is relatively compact from the discrete Arzelà-Ascoli theorem. We recall that a set Ω ⊂ X is bounded if the sequences in Ω are equibounded on every compact subset of Z m . The compactness test is therefore very simple just owing to the topology of X, while in discrete Banach spaces can require some checks which are not always immediate. Notice that, if Ω ⊂ X is bounded, then Ω ∆ = {∆u, u ∈ Ω} is bounded, too. This is a significant discrepancy between the continuous and the discrete case; such a property can simplify the solvability of discrete boundary value problems associated to equations of order two or higher with respect to the continuous counterpart because a-priori bounds for the first difference ∆x n = x n+1 − x n are a direct consequences of a-priori bounds for x n , and similarly for higher order differences.
In [ where J is a continuous function on R such that lim v→0 J(v) = 1. This suggests the form of an auxiliary linearized equation. Using the same arguments as in the proof of [21, Theorem 2.1], with minor changes, we have the following.

Theorem 3. Consider the (functional) BVP
∆(a n ∆x n J(∆x n )) = g(n, x), n ∈ Z m , where J : R → R and g : Z m × X → R are continuous maps, and S is a subset of X.
Let G : Z m × X 2 → R be a continuous map such that G(k, q, q) = g(k, q) for all (k, q) ∈ Z m × X. If there exists a nonempty, closed, convex and bounded set Ω ⊂ X such that: a) for any u ∈ Ω the problem ∆(a n J(∆u n )∆y n ) = G(n, y, u), n ∈ Z m , y ∈ S, has a unique solution y = T (u); b) T (Ω) ⊂ Ω; c) T (Ω) ⊂ S, then (15) has at least one solution.
Proof. We briefly summarize the main arguments, for reader's convenience, which are a minor modification of the ones in [21, Theorem 2.1]. Let us show that the operator T : Ω → Ω is continuous with relatively compact image. The relatively compactness of T (Ω) follows immediately from b), since Ω is bounded. To prove the continuity of T in Ω, let {u j } be a sequence in Ω, u j → u ∞ ∈ Ω, and let v j = T (u j ). Since T (Ω) is relatively compact, {v j } admits a subsequence (still indicated with {v j }) which is convergent to v ∞ , with v ∞ ∈ S from c). Since J, G are continuous on their domains, we obtain as j → ∞. The uniqueness of the solution of (16) implies v ∞ = T (u ∞ ), and therefore T is continuous. By the Schauder-Tychonoff fixed point theorem, T has at least one fixed point in Ω, which is clearly a solution of (15).
Proof of Theorem 2. Let z be the recessive solution of (14) such that z m = c, z k > 0, ∆z k ≤ 0, k ∈ Z m ; the existence of a recessive solution with these properties follows from Corollary 1. Further, from Lemma 5, we have lim k→∞ z k = 0.
Define the set Ω by where X is the Fréchet space of all real sequences defined on Z m , endowed with the topology of pointwise convergence on Z m , and M = 1/ √ 1 + c 2 ∈ (0, 1). Clearly Ω is a closed, bounded and convex subset of X.
For any u ∈ Ω, consider the following BVP whereF Since 0 ≤ u k ≤ c, for every u ∈ Ω, we have −c ≤ ∆u k ≤ c, and so (∆u k ) 2 ≤ c 2 . Therefore, for every u ∈ Ω and k ∈ Z m . FurtherF (u k+1 ) ≤ L c for u ∈ Ω, and hence (14) is Sturm majorant for the linear equation in (17). Let y = y(u) be the recessive solution of the equation in (17) such that y m = c. Then y is positive nonincreasing on Z m by Theorem 1, and, in view of y m = c and the uniqueness of recessive solutions up to the constant factor, y is the unique solution of (17). Define the operator T : Ω → X by (T u) k = y k for u ∈ Ω. From Theorem 1, we get which implies ∆ y k / y k ≤ M∆z k /z k , k ∈ Z m . By Lemma 1, which yields T (Ω) ⊆ Ω. Next we show that T (Ω) ⊆ S. Let y ∈ T (Ω). Then there exists {u j } ⊂ Ω such that {T u j } converges to y (in the topology of X). It is not restrictive to assume {u j } →ū ∈ Ω since Ω is compact. Since T u j =: y j is the (unique) solution of (17), we have y j m = c, y j k > 0 and ∆ y j k ≤ 0 on Z m for every j ∈ N. Consequently, y m = c, y k ≥ 0, ∆y k ≤ 0 for k ∈ Z m . Further, sinceF is continuous, y is a solution of the equation in (17) for u = u. Suppose now that there is T ∈ Z m such that y T = 0. Then clearly ∆y T = 0 and by the global existence and uniqueness of the initial value problem associated to any linear equation, we get y ≡ 0 on Z m , which contradicts to y m = c > 0. Thus y k > 0 for all k ∈ Z m . We have just to prove that ∞ j=m (a j y j y j+1 ) −1 = ∞. In view of Lemma 5, there exists Thus y ∈ S, i.e., T (Ω) ⊆ S. By applying Theorem 3, we obtain that the problem x ∈ S has at least a solutionx ∈ Ω. From the definition of the set Ω, Proof of Corollary 2. Assume that (14) has a positive decreasing solution for c = c 0 > 0, and let c 1 ∈ (0, c 0 ). Then equation (14) with c = c 0 is a Sturm majorant of (14) with c = c 1 , and from Theorem 1, equation (14) with c = c 1 has a positive decreasing solution.
The application of Theorem 2 leads to the existence of a solution of (1)-(2) for c = c 1 .
Effective criteria for the solvability of BVP (1)-(2) can be obtained by considering as a Sturm majorant of (14) any linear equation that is known to have a global positive solution.
In the continuous case, a typical approach to obtaining global positivity of solutions for equation (t 2 y ′ ) ′ + γy = 0, t ≥ 1, where 0 < γ ≤ 1/4, is based on the Sturm theory. In virtue of the transformation x = t 2 y ′ , this equation is equivalent to the Euler equation whose general solutions are well-known.
In the discrete case, various types of Euler equations are considered in the literature, see, e.g. [19,24] and references therein. It is somehow problematic to find a solution for some natural forms of discrete Euler equations in the self-adjoint form (7).
Here our aim is to deal with solutions of Euler type equations.
Lemma 6. The equation has a recessive solution which is positive decreasing on N.
Equation (20) can be understand as the reciprocal equation to the Euler difference equation i.e., these equations are related by the substitution relation u k = d∆x k , d ∈ R, where u satisfies (23) provided x is a solution of (20). The form of (23) perfectly fits the discretization of the differential equation (19) with γ = 1/4, using the usual central difference scheme.
Corollary 3. Let (H i ), i=1,2,3, be satisfied and L c be defined by (13). The BVP (1)- (2) has at least one solution if there exists λ > 0 such that for k ≥ 1 Proof. Consider the equation (20). By Lemma 6, it has a positive decreasing solution on N. The same trivially holds for the equivalent equation Since (24) holds, (25) is a Sturm majorant of (14), and by Theorem 1, equation (14) has a positive decreasing solution on N. Now the conclusion follows from Theorem 2.
Remark. Note that the sequence b does not need to be bounded. For example, consider as a Sturm majorant of (14) the equation One can check that this equation has the solution x k = 2 −k . This leads to the conditions ensuring the solvability of the BVP (1)- (2). Another criteria can be obtained by considering the equation having the solution x k = 1/k. This comparison with (14) leads to the conditions The following example illustrates our result.
It is interesting to compare our discrete BVP with the continuous one investigated in [14].
Here the BVP for the differential equation with the Euclidean mean curvature operator has been considered. Sometimes solutions of differential equations satisfying the condition are called Kneser solutions and the problem to find such solution is called Kneser problem.
The problem (P) has been studied under the following conditions: The function a is continuous on [1, ∞), a(t) > 0 in [1, ∞), and The main result for solvability of (P) is the following. Note that the principal solution for linear differential equation is defined similarly as the recessive solution, see e.g. [14,18]. If the principal solution z 0 of the linear equation is positive and nonincreasing on [1, ∞), then the BVP (P) has at least one solution.
It is worth to note that the method used in [14] does not allow that α = 1 and thus Theorem 4 is not immediately applicable when a(t) = t 2 . In [14] there are given several effective criteria for the solvability of the BVP (P) which are similar to Corollary 3. An example, which can be viewed as a discrete counterpart, is the above Example 2.
Open problems.
(1) The comparison between Theorem 2 for the discrete BVP and Theorem 4 for the continuous one, suggests to investigate the BVP (1)-(2) on times scales.
(2) In [14], the solvability of the continuous BVP has been proved under the weaker assumption (27) posed on F . This is due to the fact that the set Ω is defined using a precise lower bound which is different from zero. It is an open problem if a similar estimation from below can be used in the discrete case and assumption (3) can be replaced by (27).
(3) Similar BVPs concerning the existence of Kneser solutions for difference equations with p-Laplacian operator are considered in [13] when b k < 0 for k ∈ Z + . It should be interesting to extend the solvability of the BVP (1)-(2) to the case in which the sequence b is negative and in the more general situation when the sequence b is of indefinite sign.