Elsevier

Applied Mathematics and Computation

Volume 340, 1 January 2019, Pages 126-137
Applied Mathematics and Computation

Asymptotic properties of solutions to difference equations of Sturm–Liouville type

https://doi.org/10.1016/j.amc.2018.08.001Get rights and content

Abstract

We consider the discrete Sturm–Liouville type equation of the form Δ(rnΔxn)=anf(xσ(n))+bn.Assume s is a given nonpositive real number. We present sufficient conditions for the existence of solution x with the asymptotic behavior xn=c(r11++rn11)+d+o(ns)where c, d are given real numbers. Moreover, we establish conditions under which for a given solution x there exist real numbers c, d such that x has the above asymptotic behavior.

Introduction

Let N, R 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 Δ(rnΔxn)=anf(xσ(n))+bn,where an,bnR, rn > 0, for any nN, f:RR, and σ:NN, with σ(n) → ∞. By a solution of (E) we mean a sequence x:NR satisfying (E) for large n.

The classical Sturm–Liouville equation is the second order linear differential equation given by the formula (rx)(t)+q(t)x(t)=λw(t)x(t).Many authors studied the above equation and its generalizations of the form (rx)(t)=a(t)f(x(t))or with deviating argument of the form (rx)(t)=a(t)f(x(σ(t)))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 Δ(rnΔxn)+gnxn+1=λwnxn+1,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 (λwngn)xn+1. 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 yn=c(r11++rn11)+d,c,dR,and a given number s(,0] there exists a solution x of (E) such that xn=yn+o(ns).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 s(,0] there exist c,dR such that xn=c(r11++rn11)+d+o(ns).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 Δ2xn=anf(xσ(n))+bn.If the function f is “sufficiently regular” and the sequences a, b are “sufficiently small”, then some solutions of the equation Δ2yn=0 are approximative solutions of (6), (see [18], [19]).

Now, we notice that any solution of Δ(rnΔyn)=0is of the form (4). It is worth mentioning that every nonconstant solution of Δ2yn=0 is unbounded. On the other hand, there exist positive sequences r such that any solution of Δ(rnΔyn)=0 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), qN, ε > 0, fiscontinuousandboundedonthesetn=q[ynɛ,yn+ɛ],s ≤ 0, τ ∈ [s, ∞), rn1=O(nτ),n=1n1+τs|an|<,andn=1n1+τs|bn|<,we prove, in Theorem 1, that there exists a solution x of (E) such that xn=yn+o(ns).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 s(,0], we establish conditions under which for a given solution x of (E) there exist real constants c, d such that xn=c(r11++rn11)+d+o(ns).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 u=f(t,u)with Bihari-like nonlinearity of the form |f(t,u)|h1(t)g(|u|t)+h2(t).If g is a nonnegative, continuous function and h1, h2 are nonnegative, continuous functions such that 1vhi(v)dv<+,i=1,2,then for any a,bR there exists solution of (10) defined on [T, ∞) for some T ≥ 1 such that u(t)=at+b+o(1),see [17]. Moreover, assuming (11), (12) and 1dvg(v)= any solution of (10) is of the form (13), see [17]. It is worth mentioning that for τ=0, s=0 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 τ=0, s=0, 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 (p(t)xΔ)Δ+q(t)(fxσ)=q(t)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 x:NR we denote by SQ. If x, y in SQ, then xy and |x| denote the sequences defined by (xy)n=xnyn and |x|n=|xn|, respectively. Moreover, x=sup{|xn|:nN}.For an index qN we define the set Nq by Nq={q,q+1,q+2,}.We will use the following notation: r*SQ,r1*=0,rn*=i=1n11ri,forn>1,D:SQSQ,D(x)(n)=Δ(rnΔxn),S={xSQ:theseriesj=11rji=jxiisconvergent},R:SSQ,R(x)(n)=j=n1rji=jxi.Note that S is a linear subspace of SQ. Moreover, D and R are linear operators. If s(,

Preliminary lemmas

In this section we present properties of operators D and R which are useful in the proofs of main results.

Lemma 1

D(R(x))=xforxS,KerD={cr*+d:c,dR}.

Proof

Assume x ∈ S. Then ΔR(x)(n)=R(x)(n+1)R(x)(n)=1rni=nxi,rnΔR(x)(n)=i=nxi.Hence D(R(x))(n)=i=n+1xi+i=nxi=xn.Let K={cr*+d:c,dR}. Obviously K ⊂ Ker D. Assume x ∈ Ker D. Then Δ(rnΔxn)=0. Hence there exists a constant c such that rnΔxn=c for any n. Therefore Δxn=crnandxn=d+cn=1n11rn=crn*+dforcertaindRandanynN.Thus Ker D ⊂ K. 

Lemma 2

Assume s(,0], 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 y(Nq) such that f is continuous and bounded on U.

Theorem 1

Assume s(,0], τ ∈ [s, ∞), y ∈ Ker D is f-regular, rn1=O(nτ),n=1n1+τs|an|<,n=1n1+τs|bn|<.Then there exists a solution x of (E) such that xn=yn+o(ns).

Proof

For nN and x ∈ SQ let xn*=anf(xσ(n))+bn.There exist c,d

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 s(,0], we present conditions under which for a given solution x of (E) there exists a sequence y ∈ Ker D such that xn=yn+o(ns).Note that, by Lemma 1, the condition (21) means that there exist c,dR and zn=o(ns) such that xn=crn*+d+zn=ck=1n11rk+d+zn.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 n=1n1+τs|an|<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 s(,0], τR, λ(,sτ2), and an=O(nλ). Then n=1n1+τs|an|<.

Proof

Let ɛ=sτ2λ. Choose a positive constant L such that |an| ≤ Lnλ for any n. Then λ=sτ2ɛ and n=1n1+τs|an|Ln=1n1+τsnλ=n=11n1+ɛ<. 

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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