Asymptotic properties of solutions to difference equations of Sturm–Liouville type
Introduction
Let denote the set of positive integers and the set of real numbers, respectively. In this paper we consider the second order nonlinear difference equation of the form where rn > 0, for any and with σ(n) → ∞. By a solution of (E) we mean a sequence satisfying (E) for large n.
The classical Sturm–Liouville equation is the second order linear differential equation given by the formula Many authors studied the above equation and its generalizations of the form or with deviating argument of the form where σ(t) → ∞ as t → ∞ with σ(t) ≤ t or σ(t) ≥ t for large t, see e.g. [5], [8], [9], [16], and the references therein. Since by the numerical point of view difference equations can be treated as discrete version of differential equations, many authors have considered discrete version of the above equations. We recall that the discretization of Sturm–Liouville equation has the form see, e.g. [4], [13], [14]. Therefore our Eq. (E) generalizes discrete Sturm–Liouville equation (3) in two directions. First, we consider the nonlinear function anf(xσ(n)) instead of . Secondly, we extend (3) by adding the sequence b which is independent of variable x. Moreover, equation (E) generalizes discrete version of (1) or (2) mainly in the second direction mentioned previously.
In this study we are concerned with two problems. First, we establish conditions under which for a given sequence y of the form and a given number there exists a solution x of (E) such that If (5) is satisfied, then x is called a solution with prescribed asymptotic behavior, and y is called an approximative solution of (E).
Secondly, we establish conditions under which for a given solution x of (E) and a given number there exist such that In recent years, in the cycle of papers which contains among others [18], [24], the first author presented a new theory of the study of asymptotic properties of solutions of difference equations. This paper is a continuation of these investigations. Approach presented here is similar, in spirit, to the idea used to an equation of the form If the function f is “sufficiently regular” and the sequences a, b are “sufficiently small”, then some solutions of the equation are approximative solutions of (6), (see [18], [19]).
Now, we notice that any solution of is of the form (4). It is worth mentioning that every nonconstant solution of is unbounded. On the other hand, there exist positive sequences r such that any solution of is not only bounded, but even convergent. We believe that this feature of kernel of linear part of equation (E) makes this equation interesting to study of its asymptotic behavior. Assuming y is a solution of (7), ε > 0, s ≤ 0, τ ∈ [s, ∞), we prove, in Theorem 1, that there exists a solution x of (E) such that To achieve our goal we use the nontrivial corollary from the Schauder fixed point theorem from [18]. Moreover, we present two examples which prove that assumption (8) is essential. Next, in Theorem 2, assuming we establish conditions under which for a given solution x of (E) there exist real constants c, d such that Prescribed asymptotic behavior to difference equations were discussed, for example, in [18], [19], [20], [21], [22], [23], [24] or in [28]. Prescribed asymptotic behavior to differential or dynamic equations were considered, for example, in [3], [6], [7], [11], [17], [25], [29]. Additionally, the paper [3] is an excellent survey of the literature concerning in asymptotic integration theory of ordinary differential equations. Mentioned previously papers included the class of second order equations with Bihari-like nonlinearity of the form If g is a nonnegative, continuous function and h1, h2 are nonnegative, continuous functions such that then for any there exists solution of (10) defined on [T, ∞) for some T ≥ 1 such that see [17]. Moreover, assuming (11), (12) and any solution of (10) is of the form (13), see [17]. It is worth mentioning that for our assumption (9) is a discrete counterpart of (12). Notice that, the case τ ≥ 0 includes sequence r constantly equal to 1. If the function g in (11) is bounded, results presented in this paper are discrete version of results from [17] for and the sequence r ≡ 1. Otherwise assumptions (8) and (11) are not comparable. Moreover, if the function g is not bounded, assumptions and conclusions of main theorems presented here depend on given solution of (7) which differs our consideration from the study in the continuous case. Prescribed asymptotic behavior in slightly different sense from presented in this study was considered in [6] for the dynamic equation on time scales of the form with g satisfying a counterpart of (9) with p(t) ≥ δ > 0 and a function f fulfilling a condition |f(x)| ≤ L|x|α, α ∈ (0, 1), L > 0 and rd-continuous function q. Additionally, different types of asymptotic properties of solutions of nonlinear difference, differential or dynamic equations were considered for example in [1], [2], [10], [12], [15], [26], [27], [30], [31], [32] and the references therein.
The paper is organized as follows. In Section 2, we introduce notation and terminology. In Section 3, we present some preliminary lemmas. Next, in Section 4, we prove our first main result and some consequences of it. Section 5 is devoted to our second main result Theorem 2. In Section 6 we present some additional results.
Section snippets
Notation and terminology
The space of all sequences we denote by SQ. If x, y in SQ, then xy and |x| denote the sequences defined by and respectively. Moreover, For an index we define the set by We will use the following notation: Note that S is a linear subspace of SQ. Moreover, D and R are linear operators. If
Preliminary lemmas
In this section we present properties of operators D and R which are useful in the proofs of main results.
Lemma 1 Proof Assume x ∈ S. Then
Hence
Let . Obviously K ⊂ Ker D. Assume x ∈ Ker D. Then . Hence there exists a constant c such that for any n. Therefore
Thus Ker D ⊂ K. □ Lemma 2 Assume x
Approximative solutions
In this section we establish various conditions under which a given sequence y ∈ Ker D is an approximative solution of (E). We say that a sequence y ∈ SQ is f-regular if there exist an index q and a uniform neighborhood U of the set such that f is continuous and bounded on U.
Theorem 1 Assume τ ∈ [s, ∞), y ∈ Ker D is f-regular,
Then there exists a solution x of (E) such that . Proof For and x ∈ SQ let
There exist
Approximations of solutions
In this section we establish conditions under which a given solution x of (E) can be approximated by sequences y from Ker D. More precisely, assuming we present conditions under which for a given solution x of (E) there exists a sequence y ∈ Ker D such that Note that, by Lemma 1, the condition (21) means that there exist and such that We say that a sequence x ∈ SQ is (f, σ)-bounded if the sequence (f(xσ(n))) is bounded. Note
Remarks
In our investigations the condition
plays an important role. In practice, this condition can be difficult to verify. In the following lemma we present the condition a little stronger but easier to check.
Lemma 7 Assume and . Then
Proof Let . Choose a positive constant L such that |an| ≤ Lnλ for any n. Then and
□
We can apply Lemma 7 to our results and receive many new
Acknowledgments
Authors wish to express their thanks to referees for insightful remarks improving quality of the paper.
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