Stability of periodic solutions of first-order difference equations lying between lower and upper solutions

STABILITY OF PERIODIC SOLUTIONS OF FIRST-ORDER DIFFERENCE EQUATIONS LYING BETWEEN LOWER AND UPPER SOLUTIONS ALBERTO CABADA, VICTORIA OTERO-ESPINAR, AND DOLORES RODR´ IGUEZ-VIVERO Received 8 January 2004 and in revised form 2 September 2004 We prove that if there exists α≤β, a pair of lower and upper solutions of the first-order discrete periodic problem ∆u(n)=f(n,u(n)); n∈IN≡{0,...,N−1},u(0) =u(N), with facontinuousN-periodic function in its first variable and such that x+f(n,x)isstrictly increasing in x,foreveryn∈IN, then, this problem has at least one solution such that its N-periodic extension to Nis stable. In several particular situations, we may claim that this solution is asymptotically stable. 1. Introduction It is well known that one of the most important concepts in the qualitative theory of differential and difference equations is the stability of the solutions of the treated problems. Classical tools, as approximation by linear equations or Lyapunov functions, have been developed for both type of equations, see [7] for ordinary differential equations and [8] for difference ones. More recently, some authors as, among others, de Coster and Habets [6], Nieto [9], or Ortega [10], have proved the stability of solutions of adequate ordinary differential equations that lie between a pair of lower and upper solutions. In this case, fixed points theorems and degree and index theory are the fundamental arguments to deduce the mentioned stability results. Stability for order-preserving operators defined on Banach spaces have been obtained by Dancer in [4] and Dancer and Hess in [5]. On these papers, the authors describe the assymptotic behavior of the iterates that lie between a lower and an upper solution of suitable operators. Our purpose is to ensure the stability of at least one periodic solution of a first-order difference equation. We will prove such result by using a monotone nondecreasing operator. In this case, the defined operator does not verify the conditions imposed in [5]. The so-obtained results are in the same direction as the ones proved by the authors in [3]for the first-order implicit difference equation ∆u(i)=f(i,u(i+ 1)) coupled with periodic boundary conditions. In that situation, we give some optimal conditions on function f and on the number of the possible periodic solutions of the considered problem, that warrant the existence of at least one stable solution. The arguments there are different Copyright ©2005 Hindawi Publishing Corporation Advances in Difference Equations 2005:3 (2005) 333–343 DOI: 10.1155/ADE.2005.333 334 Stability of periodic solutions from the ones used in this paper because in that situation, due to the lack of uniqueness of solutions of the initial problem, the discrete operator considered here cannot be defined. The paper is organized as follows. In Section 2, we present some fundamental properties of the set of solutions of initial and periodic problems. In Section 3,weprovethe existence of at least one N-periodic stable solution. Finally, Section 4 is devoted to give some examples that point out the, in some sense, optimality of the obtained results. 2. Preliminaries This paper is devoted to study the stability, by using the method of lower and upper solutions, of the N-periodic solutions of the following first-order difference equation: (P) ∆u(n)=fn,u(n);n∈N, (2.1) where, for every n∈N,∆u(n)=u(n+1)−u(n), f:N×R→Ris a continuous function, N-periodic in its first variable for some N∈N∗=N\{0}given. Throughout the paper, having x=(x(0),...,x(N)) and y=(y(0),...,y(N)), we will say that x≤yin JN≡{0,...,N}if x(j)≤y(j)forallj∈JN,wewillsaythatx<yin JN when x≤yin JNand there is at least one j0∈JNsuch that x(j0)<y(j0), moreover xy in JNwhen x(j)<y(j)forallj∈JN. We will denote [x,y]=z=z(0),...,z(N);x≤z≤yin JN.(2.2) We say that u:N→Ris an N-periodic solution of problem (P) if it satisfies equation (P)inNand u(n)=u(N+n)foralln∈N. From the periodicity of fin its first variable, it is obvious that to look for an N-periodic solution of (P) is equivalent to solve equation (PN) ∆u(i)=fi,u(i);i∈IN≡{0,...,N−1},u(0) =u(N).(2.3) Below, we will denote u:JN→Rand ¯ u:N→Ras a solution of (PN)anditsN-periodic extension to N, respectively. We define the concept of lower and upper solutions for problem (PN)asfollows. Definition 2.1. Let N∈N∗be given. Say that α=(α(0),...,α(N)) is a lower solution of problem (PN) if it satisfies ∆α(i)≤fi,α(i);i∈IN,α(0) ≤α(N).(2.4) The concept of upper solution is given by reversing the previous inequalities. It is important to note, see [1,2], that the existence of αand β,apairoflowerand upper solutions of problem (PN), such that α≤βin JN, does not imply the existence of a solution of this problem. Alberto Cabada et al. 335 Now, by defining for each n∈N,hn:R→Ras hn(x):=x+f(n,x)forallx∈R,and g:R→Ras g:=hN−1◦···◦h1◦h0,wehavethatproblem(PN) has a solution if and only if ghas a fixed point. If for every n∈IN,hnis a strictly increasing function on [α(n),β(n)], with α≤βapair of lower and upper solutions of problem (PN), then it turns out that for every n∈IN, α(n+1)≤hnα(n)≤hnβ(n)≤β(n+ 1), (2.5) which implies that g([α(0),β(0)]) ⊂[α(0),β(0)]. Now, we assume the following properties. (H) There exist αand βa pair of lower and upper solutions that are no solutions of problem (PN), such that α≤βin JN. (Hf)f:N×R→Ris N-periodic in its first variable and for all n∈IN, the function f(n,·)iscontinuouson[α(n),β(n)], with αand βgivenin(H). Moreover, hn(x):=x+f(n,x) is strictly increasing on [α(n),β(n)]. As a consequence of the ideas exposed above, we deduce the following existence result for problem (P). Lemma 2.2. Assume that conditions (H)and (Hf)are fulfilled, then problem (P)has at least one N-periodic solution ¯ usuch that u(≡¯ u|JN)∈[α,β]. Remark 2.3. One can see in [1] that the previous property is optimal in the sense that if hnis not monotone increasing on [α(n),β(n)] for some n∈IN, then the existence result is not guaranteed. On the other hand, plainly for every ξ∈R, the initial problem (Pξ) ∆u(n)=fn,u(n);n∈N,u(0) =ξ, (2.6) has a unique solution which will be denoted throughout the paper as uξ. If we denote its restriction to the interval INas (PN ξ) ∆u(n)=fn,u(n);n∈IN,u(0) =ξ, (2.7) then, one can see in [2, Example 2.2] that the existence of α≤β,apairoflowerandupper solutions of this problem, that is, ∆α(i)−fi,α(i)≤0≤∆β(i)−fi,β(i);i∈IN,α(0) ≤ξ≤β(0), (2.8) is not sufficient to ensure that the unique solution of problem (PN ξ)liesin[α,β]. However, with an analogous argument to the periodic case, whenever hnis a strictly increasing function for all n∈IN, we derive that this solution belongs to the sector formed by α and β. Moreover, the strict monotony of hnallows to ensure that the solutions starting at different initial conditions are not equal at any point. 336 Stability of periodic solutions From this arguments, one can prove the following result. Lemma 2.4. If conditions (H)and (Hf)hold, then for all ξ∈[α(0),β(0)], the unique solution of the initial problem (PN ξ)belongs to the sector [α,β]. Moreover, the unique solution uξof problem (Pξ)is such that uξ|JN∈[α,β]. If ξ,η∈[α(0),β(0)] are such that ξ<η,then,α≤uξ|JNuη|JN≤βin JN. Below, we give a more precise description of the set of solutions of problem (PN). Lemma 2.5. Assume conditions (H)and (Hf).Letu,v∈[α,β]be two solutions of problem (PN). Then, one of the following statements is true: (i) αu≡vβin JN; (ii) αuvβin JN; (iii) αvuβin JN. Proof. First we show that αuin JN. Suppose that there exists n0∈JNsuch that α(n0)=u(n0), by using inequalities (2.5), we obtain that α(n)=u(n)foralln∈{0,...,n0}. Therefore, α(N)=u(N), and so α(n)= u(n)foralln∈JN, which implies that αis a solution of (PN) and it contradicts hypothesis (H). Hence, αuin JN. Inequality vβin JNcan be proven in a similar way. Now, if u(0) =v(0), then, from the uniqueness of solutions of the initial problem, we conclude that u≡vin JN, and so assertion (i) holds. However, if u(0) <v(0), then, from Lemma 2.4 we may assert that u(n)<v(n)forall n∈JN, so that claim (ii) is proved. Claim (iii) is fulfilled whenever u(0) >v(0).  Previous result establishes that the set of solutions in [α,β]ofproblem(PN) (and their N-periodic extensions of problem (P)) is totally ordered and bounded. From the continuity of function f, we know that it is closed. Thus, we conclude that there exist ψand φthe minimum and the maximum of the aforementioned set and, clearly, they match up the minimal and the maximal solutions, respectively, in [α,β]ofproblem(PN). 3. Stability In this section, we prove the stability of at least one N-periodic solution ¯ uof problem (P) such that ubelongs to the sector [α,β]. Here, we say that u¯ ξ:N→R, the unique solution of the initial problem (P¯ ξ), is stable if and only if for all ε>0, there exists δ=δ(ε)∈(0,ε)suchthat|uξ(n)−u¯ ξ(n)|<εfor all n∈N∗,wheneverξ∈(¯ ξ−δ,¯ ξ+δ). It is asymptotically stable if and only if it is stable and there exists µ>0suchthat limn→∞ (uξ−u¯ ξ)(n)=0forallξ∈(¯ ξ−µ,¯ ξ+µ). We will say that it is stable from above if the interval ( ¯ ξ−δ,¯ ξ+δ)isreplacedby(¯ ξ,¯ ξ+ δ). Similar comment is valid for stable from below and from asymptotically stable from above and from below. We will call attractivity set of u¯ ξto the biggest interval V¯ ξsuch that ¯ ξ∈V¯ ξand limn→∞ (uξ−u¯ ξ)(n)=0forallξ∈V¯ ξ. Alberto Cabada et al. 337 Now, for every u:JN→R,letT(u) denote the unique solution of problem (PN u(N)). Note that, from Lemma 2.4,weknowthatT([α,β]) ⊂[α,β]. Following properties for operator T,carryover. Proposition 3.1. Let ξ∈Rbe given, then uξ:N→Ris the solution of the initial problem (Pξ)if and only if uξ(n)=Tknv(in),withv:JN→Rthe unique solution of the initial problem (PN ξ),kn∈N,in∈IN,andn=knN+in. Proof. Let uξand vbe the unique solutions of (Pξ)and(PN ξ), respectively. By definition, uξ=vin JN.Since fis N-periodic in its first variable, for every k∈N,thevaluesofuξ on the intervals {kN,...,kN +N−1}match up the values of the unique solution of the initial problem (PN uξ(kN))onIN, that is, with the values of Tkvon IN. As a straightforward consequence of the previous result, we obtain the following characterization of the set of solutions of problem (PN). Corollary 3.2. Let u:JN→R, then u is a solution of problem (PN)if and only if uis a fixed point of operator T. Proposition 3.3. Assume that conditions (H)and (Hf)are satisfied, then operator Tis monotone nondecreasing on [α,β]. Moreover, if u,v∈[α,β]are such that u(N)<v(N), then TuTv in JN. Proof. Let u,v∈[α,β]besuchthatu≤vin JN.Thus,(Tu)(0) =u(N)≤v(N)=(Tv)(0). The monotonicity of operator Tis a direct consequence of condition (Hf). Last part can be easily obtained from Lemma 2.4. Remark 3.4. Note that, despite Proposition 3.3 is valid, operator Tis not strictly increasing on [α,β], that is, if u,v∈[α,β]aresuchthatu<vin JN,thenTu <Tv in JN. Indeed, it is enough to consider a pair of functions u,v∈[α,β]suchthatu(N)=v(N). It is obvious that Tu≡Tv. This property guarantees that the results given in [5] for strictly increasing operators, defined on Banach spaces, cannot be applied to the operator Tdefined above. Proposition 3.5. Assume that conditions (H)and (Hf)are fulfilled, then α<TαTβ< βin JN. Proof. Let u=Tα,bydefinition u(n+1)=u(n)+ fn,u(n),n∈IN;u(0) =α(N)≥α(0).(3.1) Condition (Hf) ensures that α≤Tαin JN. If the equality holds, then we have that αis a solution of the periodic problem (PN) which contradicts hypothesis (H). Hence, α<Tα in JNand the proof of the first inequality is complete. One can prove in a similar way the fact that Tβ<β in JN. As we have proved Lemma 2.5,αβin JN, so that the second inequality follows from Proposition 3.3. 338 Stability of periodic solutions Proposition 3.6. Assume that conditions (H)and (Hf)are fulfilled. Let α0=α,β0=β and for all m≥1,αm=Tαm−1and βm=Tβm−1. Then, the two following properties hold. (1) {αm}m∈Nis a strictly increasing sequence which converges uniformly in JNto ψ,the minimal solution in [α,β]of problem (PN). (2) {βm}m∈Nis a strictly decreasing sequence which converges uniformly in JNto φ,the maximal solution in [α,β]of problem (PN). Proof. We only prove the first assertion; the second one holds similarly. If α1(0) =α1(N), then α1is a solution of problem (PN). Thus, equality α1(0) =α(N) establishes that this case is not possible because it contradicts Lemma 2.5. Moreover, inequality α1(0) >α 1(N) does not hold, so that α1(0) <α 1(N). Inequality α1(0) =α0(N)<ψ(N)=ψ(0), Proposition 3.3,andLemma 2.4 yield α1=Tα0Tα1 ψ=Tψ in JN. Inductively by using Proposition 3.3,wededucethatαmTαmTψ =ψ in JN. The conclusion follows from the boundness from above by ψof the sequence {αm}m∈N and the definition of ψ. This property allows us to deduce the one-sided asymptotic stability of the N-periodic extensions of the extremal solutions of the periodic problem (PN). The obtained result is the following theorem. Theorem 3.7. If assumptions (H)and (Hf)hold, then ¯ φis asymptotically stable from above and ¯ ψis asymptotically stable from below. Moreover, sets [φ(0),β(0)] and [α(0),ψ(0)] are contained in the attractivity sets of ¯ φand ¯ ψ,respectively. Proof. We only prove the claim for ¯ φ. The other one can be proven by similar arguments. Let ξ∈(φ(0),β(0)]; we know, from Lemma 2.5,that(φ(0),β(0)] is not empty. There exists, by Proposition 3.6,i0≥1suchthatξ∈(βi0(0),βi0−1(0)]. Let v:JN→Rbe the unique solution of the initial problem (PN ξ); monotony properties of operator Tensure that the sequence {Tmv}m∈Nis strictly decreasing in JNand it converges uniformly in JNto φ. Proposition 3.1 leads to the desired result.  Plainly from this result, we may establish the asymptotic stability by assuming uniqueness of solutions in [α,β]ofproblem(PN). Corollary 3.8. If assumptions (H)and (Hf)hold and there exists a unique solution uof problem (PN)in [α,β], then ¯ uis asymptotically stable. Moreover, set [α(0),β(0)] is contained in the attractivity set of ¯ u. Whenever fis a strictly decreasing function in its second variable, we achieve the following result. Corollary 3.9. Suppose that conditions (H)and (Hf)hold and that for every n∈IN, f(n,·)is strictly decreasing on [α(n),β(n)]. Then, problem (PN)has a unique solution uin [α,β]and ¯ uis asymptotically stable. Moreover, set [α(0),β(0)] is contained in the attractivity set of ¯ u. Proof. Let uand vbe two different solutions of problem (PN)in[α,β]. From Lemma 2.5, we may assume, without loss of generality, that uvin JN. As a consequence, we obtain Alberto Cabada et al. 339 the following contradiction: 0=u(N)−u(0) = N−1  n=0 fn,u(n)> N−1  n=0 fn,v(n)=v(N)−v(0) =0.(3.2)  Remark 3.10. One could intend to prove the previous result by replacing decreasing with increasing. However, under this assumption, we have that α(N)−α(0) ≤ N−1  n=0 fn,α(n)< N−1  n=0 fn,β(n)≤β(N)−β(0), (3.3) which contradicts hypothesis (H). Below, for every ξ∈(ψ(0),φ(0)), we analyse the behavior of the solution of the initial problem (Pξ). In order to do this, we study the orbits of the operator Tin [ψ,φ]. We achieve similar properties to the ones proved by Dancer and Hess in [5]forstrictly increasing operators defined on an arbitrary Banach space. However, as we have noted in the previous section, we cannot deduce the stability results as a consequence of the proved results in that reference, because we are not in the presence of a strictly increasing operator. Moreover, Theorem 3.7 allows us to prove the following property of one-sided asymptotic stability. Proposition 3.11. Assume that conditions (H)and (Hf)hold. Let u,vbe two solutions of the periodic problem (PN)such that uvin JNand there is no solution of this problem lying between both functions. Then one of the two following assertions holds. (1) The N-periodic solution ¯ vis asymptotically stable from below. Moreover, set (u(0), v(0)] is contained in the attractivity set of ¯ v. (2) The N-periodic solution ¯ uis asymptotically stable from above. Moreover, set [u(0), v(0)) is contained in the attractivity set of ¯ u. Proof. Suppose that there exists ξ∈(u(0),v(0)) such that uξ(0) <u ξ(N). By defining of α0:JN→Ras the unique solution of the initial problem (PN ξ), it turns out that α0and β verify condition (H)andvis the minimal solution in [α0,β]ofproblem(PN). Hence, the first part of assertion (1) follows from Theorem 3.7. Now, for all m≥1, let αmbe the unique solution of the final problem ∆u(n)=fn,u(n);n∈IN,u(N)=αm−1(0); (3.4) it follows from condition (Hf)that{αm}m∈Nis a strictly decreasing sequence of lower solutions of problem (PN) and it converges uniformly in JNto u. Now, given η∈(u(0),ξ], there exists m0≥0suchthatη∈(αm0(0),αm0−1(0)] and so uη|JN∈(αm0,αm0−1]. The second part of claim (1) follows from Theorem 3.7. Statement (2) is true provided that there exists ξ∈(u(0),v(0)) such that uξ(0)>uξ(N).  340 Stability of periodic solutions As a consequence of this result, we can ensure asymptotic stability by assuming a finite number of solutions of problem (PN)in[α,β]. Theorem 3.12. If assumptions (H)and (Hf)are fulfilled and problem (PN)has a finite number of solutions in [α,β], then at least one N-periodic solution of (P)is asymptotically stable. Proof. Define C:=u∈[α,β]:¯ uis an N-periodic solution of (P)anda.s.b. , (3.5) where a.s.b. means asymptotically stable from below. From Theorem 3.7, we know that this set is not empty (ψ∈C), moreover it is bounded from above by φfinite, and by Lemma 2.5 well ordered. Proposition 3.11 establishes that function maxCis asymptotically stable.  Lastly, we consider the opposite case to the previous one, that is, there are not finite number of solutions of problem (PN)in[α,β]. In this situation, we only guarantee stability, not asymptotic stability. Theorem 3.13. If assumptions (H)and (Hf)hold and problem (PN)does not have a finite number of solutions in [α,β],thenatleastoneN-periodic solution of (P)is stable. Proof. Consider set Cdefined in (3.5); given that it is not empty, bounded from above, there exists function uS:=supC. Due to the fact that the set of solutions of (PN)isclosed and well ordered, we conclude that uSis a solution of problem (PN). If uSis isolated from below, it is clear that uS∈C, and so it is asymptotically stable from below. Suppose that uSis not isolated from below, by supremum’s definition and Lemma 2.5, there exists a strictly increasing monotone sequence {um}m∈N⊂Cwhich converges uniformly in JNto uS. Therefore, given >0, we know that there exists m0∈Nsuch that 0 <u S(n)−um(n)< for all n∈JNand m≥m0. Hence, since ¯ umis asymptotically stable from below for every m∈N,Proposition 3.1 together with the nondecreasing properties of operator T guarantees that ¯ uSis stable from below. If uSis the limit of a decreasing sequence of solutions of (PN), then it is stable from above. Otherwise, if it is isolated from above, then it is asymptotic stable from above and we conclude the proof.  4. Examples and counterexamples In this section, we present two examples which illustrate the results obtained in the previous section. In the first one, we consider a problem with a unique N-periodic solution in the whole space; we show that this solution is asymptotically stable. In the second example, we show that Theorem 3.13 cannot be improved, in the sense that it is possible to find a nontrivial function fsuch that the set of N-periodic solutions of problem (P) is not finite and none of these solutions is asymptotically stable. Alberto Cabada et al. 341 Example 4.1. Let N∈N∗be fixed, consider the following problem ∆u(n)=−arctanu(n)−in,n∈[kN,kN +N−1], k∈N, (4.1) where for every k∈N,in∈INis given as n=kN +in. Obviously f(n,x)=−arctan(x−in) is a continuous and strictly decreasing function in its second variable and N-periodic in the first one. Let D>0, it is clear that, for Dlarge enough, αDand βD,definedforeveryn∈JN as αD(n)=n−Dand βD(n)=−n+D, is a pair of lower and upper solutions that are no solutions of problem (PN)(with fdefined above) and they are such that α≤βin JN, that is, condition (H) is fulfilled. Since fsatisfies condition (Hf), we deduce, from Corollary 3.9, that this problem has a unique solution such that its restriction to JNbelongs to [αD,βD] and it is asymptotically stable. DuetothefactthatαD→−∞and βD→+∞whenever D→+∞, we claim that the restriction to JNof problem (4.1)hasonlyasolutioninRN+1,andsoproblem(4.1)has auniqueN-periodic solution defined in N. Moreover, the attractive set of this solution is the whole space R. Remark 4.2. Note that in the previous example, we may ensure the character not only local as in the obtained results above, but global of the asymptotic stability of the considered solution. Example 4.3. Let f:[−1,2] →Rbe defined as follows: f(x)=              −x 2,ifx∈[−1,0], limi→+∞fi(x), if x∈[0,1], −x−1 2,ifx∈[1,2]. (4.2) Here, the functional sequence fi: [0,1] →Ris defined in the following way: f0(x)=0, and for i≥1, consider Di=2i−1 j=1(ai j,bi j) the union of the 2(i−1) open intervals dropped from [0,1] in the ith step of the construction of the classical ternary Cantor set. We define fi(x)=fi−1(x)forallx∈ Diand, for any j∈{1,...,2i−1}, fi(x)=            aj−x 2,ifx∈aj,aj+bj 2, x−bj 2,ifx∈aj+bj 2,bj. (4.3) It is clear that this function is continuous and nonpositive on [0,1], moreover the set of zeros of fis the ternary Cantor set.