Existence and uniqueness of solutions of higher-order antiperiodic dynamic equations

EXISTENCE AND UNIQUENESS OF SOLUTIONS OF HIGHER-ORDER ANTIPERIODIC DYNAMIC EQUATIONS ALBERTO CABADA AND DOLORES R. VIVERO Received 8 October 2003 and in revised form 9 February 2004 We prove existence and uniqueness results in the presence of coupled lower and upper solutions for the general nth problem in time scales with linear dependence on the ith ∆- derivatives for i=1,2,...,n, together with antiperiodic boundary value conditions. Here the nonlinear right-hand side of the equation is defined by a function f(t,x) which is rd-continuous in tand continuous in xuniformly in t. To do that, we obtain the expression of the Green’s function of a related linear operator in the space of the antiperiodic functions. 1. Introduction The theory of dynamic equations has been introduced by Stefan Hilger in his Ph.D. thesis [12]. This new theory unifies difference and differential equations and has experienced an important growth in the last years. Recently, many papers devoted to the study of this kind of problems have been presented. In the monographs of Bohner and Peterson [5,6] there are the fundamental tools to work with this type of equations. Surveys on this theory given by Agarwal et al. [2]andAgarwaletal.[1]giveusanideaoftheimportance of this new field. In this paper, we study the existence and uniqueness of solutions of the following nthorder dynamic equation with antiperiodic boundary value conditions: (Ln) u∆n(t)+ n−1  j=1 Mju∆j(t)=ft,u(t),∀t∈I=[a,b], u∆i(a)=−u∆iσ(b),0≤i≤n−1. (1.1) Here, n≥1, Mj∈Rare given constants for j∈{1,...,n−1},[a,b]=Tκn,withT⊂Ran arbitrary bounded time scale and f:I×R→Rsatisfies the following condition: Copyright ©2004 Hindawi Publishing Corporation Advances in Difference Equations 2004:4 (2004) 291–310 2000 Mathematics Subject Classification: 39A10 URL: http://dx.doi.org/10.1155/S1687183904310022 292 Higher-order antiperiodic dynamic equations (Hf)forallx∈R,f(·,x)∈Crd(I)and f(t,·)∈C(R)uniformlyatt∈I, that is, for all >0, there exists δ>0suchthat |x−y|<δ=⇒  f(t,x)−f(t,y) <,∀t∈I. (1.2) Asolutionofproblem(Ln) will be a function u:T→Rsuch that u∈Cn rd(I) and satisfies both equalities. Here, we denote by Cn rd(I) the set of all functions u:T→Rsuch that the ith derivative is continuous in Tκi,i=0,...,n−1, and the nth derivative is rdcontinuous in I. It is clear that for any given constant M∈R,problem(Ln)canberewrittenas u∆n(t)+ n−1  j=1 Mju∆j(t)+Mu(t)=ft,u(t)+Mu(t), ∀t∈I, u∆i(a)=−u∆iσ(b),0≤i≤n−1. (1.3) Defining the linear operator Tn[M]:Cn rd(I)→Crd(I)foreveryu∈Cn rd(I)as Tn[M]u(t):=u∆n(t)+ n−1  j=1 Mju∆j(t)+Mu(t), for every t∈I, (1.4) and the set Wn:=u∈Cn rd(I):u∆i(a)=−u∆iσ(b),0≤i≤n−1, (1.5) we can rewrite the dynamic equation (Ln)as Tn[M]u(t)=ft,u(t)+Mu(t), t∈I,u∈Wn.(1.6) From this fact, we deduce that to ensure the existence and uniqueness of solutions of the dynamic equation (Ln), we must determine the real values M,M1,...,Mn−1for which the operator Tn[M] is invertible on the set Wn, that is, the values for which Green’s function associated with the operator T−1 n[M]inWncan be defined. In Section 2,wepresent the expression of Green’s function associated to the operator T−1in Wn,whereTis a general nth-order linear operator that is invertible on that set. This formula is analogous to the one given in [9]fornth-order dynamic equations with periodic boundary value conditions. In Section 3,weproveasufficient condition for the existence and uniqueness of solutions of the dynamic equation (Ln). For this, we take as reference the results obtained in [3,4], where the existence and uniqueness of solutions of problem (Ln)isstudiedinthe particular case T={0,1,...,P+n}and so (Ln)isadifference equation with antiperiodic boundary conditions. In this case, the classical iterative methods based on the existence of a lower and an upper solution and on comparison principles of some adequate linear A. Cabada and D. R. Vivero 293 operators, cannot be applied and, as a consequence, extremal solutions do not exist in a given function’s set. Hence, to study the existence and uniqueness of solutions of problem (Ln) in an arbitrary bounded time scale T⊂R, we use the technique developed in [3,4], based on the concept of coupled lower and upper solutions, similar to the definition given in [10] for operators defined in abstract spaces and in [11] for antiperiodic boundary first-order differential equations. A survey of those results for difference equations can be founded in [8]. Using the results proved in Sections 2and 3,wewillobtaininSections4and 5the expression of Green’s function and a sufficient condition for the existence and uniqueness of solutions of the dynamic equations of first- and second-order, respectively; likewise, we will give details about the continuous case where a dynamic equation is a differential equation and the discrete case, in which either a difference equation or a q-difference equation are treated. 2. Expression of Green’s function In this section, we obtain the expression of Green’s function associated with the operator T−1in Wn,whereTis a general linear operator of nth-order that is invertible on the mentioned set. First, we introduce the concept of nth-order regressive operator, see [5, Definition 5.89 and Theorem 5.91]. Definition 2.1. Let Mi∈R,0≤i≤n−1 be given constants, the operator T:Cn rd(I)→ Crd(I), defined for every u∈Cn rd(I)as Tu(t):=u∆n(t)+ n−1  i=0 Miu∆i(t), for every t∈I, (2.1) is regressive on Iif and only if 1 + n i=1(−µ(t))iMn−i= 0forallt∈I. Theorem 2.2. Let Mi∈R,0≤i≤n−1be given constants such that the operator Tdefined in (2.1) is regressive on I(see Definition 2.1). If the operator Tis invertible on Wn, then Green’s function associated to the operator T−1in Wn,G:T×I→Ris given by the following expression: G(t,s)=   u(t,s)+v(t,s), if a≤σ(s)≤t≤σn(b), u(t,s), if a≤t<σ(s)≤σ(b), (2.2) where, for every s∈[a,b]fixed, v(·,s)is the unique solution of the problem (Qs) Txs(t)=0, t∈σ(s),b, x∆i sσ(s)=0, i=0,1,...,n−2, x∆n−1 sσ(s)=1, (2.3) and for every s∈[a,b]fixed, u(·,s)is given as the unique solution of the problem 294 Higher-order antiperiodic dynamic equations (Rs) Tys(t)=0, t∈[a,b], y∆i s(a)+y∆i sσ(b)=− v∆iσ(b),s,i=0,1,...,n−1.(2.4) Proof. First, we see that the function Gis well defined, that is, for every s∈[a,b]fixed, problems (Qs)and(Rs) have a unique solution. Since the operator Tis regressive on I,wehave,see[5, Corollary 5.90 and Theorem 5.91], that for every s∈[a,b] fixed, the initial value problem (Qs) has a unique solution. To verify that the periodic boundary problem (Rs) is uniquely solvable, we consider the following boundary value problem: (Pλ) w∆n(t)+ n−1  i=0 Miw∆i(t)=h(t), t∈I, w∆i(a)+w∆iσ(b)=λi,i=0,1,...,n−1, (2.5) with h∈Crd(I)andλi∈R,0≤i≤n−1fixed. We know that w∈Cn rd(I)isasolutionofproblem(Pλ)ifandonlyifW(t)= (w(t),w∆(t),...,w∆n−1(t))Tis a solution of the matrix equation W∆(t)=AW(t)+H(t), t∈I,W(a)+Wσ(b)=λ, (2.6) where H(t)=(0,...,0,h(t))T,λ=(λ0,...,λn−1)T,and A=             010··· 0 001··· 0 . . .. . .. . ..... . . 000 ··· 1 −M0−M1−M2··· −Mn−1             .(2.7) Since the operator Tis regressive on I,wehave,by[5, Definitions 5.5 and 5.89], that the matrix Ais regressive on Itooandso,itfollowsfrom[5, Theorem 5.24] that the initial value problem W∆(t)=AW(t)+H(t), t∈I,W(a)=Wa, (2.8) has a unique solution that is given by the following expression: W(t)=eA(t,a)Wa+t aeAt,σ(s)H(s)∆s. (2.9) A. Cabada and D. R. Vivero 295 If we denote the n×nidentity matrix by In, then we obtain, from the boundary conditions, that problem (2.6) has a unique solution if and only if there exists a unique Wa=W(a)∈Rnsuch that In+eAσ(b),aWa=λ−σ(b) aeAσ(b),σ(s)H(s)∆s, (2.10) or equivalently, if and only if the matrix In+eA(σ(b),a)isinvertible. Now, since the operator Tis invertible on Wn,wehavethatproblem(P0)hasaunique solution and then there exists the inverse of such matrix. As a consequence, problem (Rs) has a unique solution. Now, let z:T→Rbe defined for every t∈Tas z(t)=σ(b) aG(t,s)h(s)∆s. (2.11) It is not difficult to prove, by using [5, Theorem 1.117], that zis the unique solution of the problem (P0).  Now, we prove the following properties of Green’s function associated to the operator T−1in Wn. Proposition 2.3. Let Mi∈R,0≤i≤n−1be given constants such that the operator Tdefined in (2.1) is regressive on I.IfG:T×I→Ris Green’s function associated to the operator T−1in Wn,definedin(2.2), then the following conditions are satisfied. (1) There exists k>0such that |G(t,s)|≤kfor all (t,s)∈T×I. (2) If n=1,thenforeverys∈I,thefunctionG(·,s)is continuous at t∈Texcept at t=s=σ(s). (3) If n>1,thenforeverys∈I,thefunctionG(·,s)is continuous in T. (4) If n=1, then for every t∈T,thefunctionG(t,·)is rd-continuous at s∈Iexcept when s=t=σ(t). (5) If n>1,thenforeveryt∈T,thefunctionG(t,·)is rd-continuous in I. Proof. As we have seen in the proof of Theorem 2.2, we know that Green’s function associated to the operator T−1in Wnis given as the 1 ×nterm of the matrix function F(t,s)=     eAt,σ(s)−eA(t,a)In+eAσ(b),a−1eAσ(b),σ(s),σ(s)≤t, −eA(t,a)In+eAσ(b),a−1eAσ(b),σ(s),t<σ(s), (2.12) where Ais the matrix given in (2.7). From [5, Definition 5.18 and Theorem 5.23], we know that the matrix exponential function is continuous in both variables and so the function Gis bounded in the compact set T×I. Now, since eA(t,t)=In,ift=σ(s)=s, then the diagonal terms of F(·,s) are not continuous at t. It is clear that in any other situation, the function F(·,s)iscontinuousatt. 296 Higher-order antiperiodic dynamic equations On the other hand, given t0∈T,foreverys0∈Isuch that s0= t0,itfollows,from the continuity of the exponential function, that if s→s0and σ(s)→σ(s0), then F(t0,s)→ F(t0,s0). Hence, since G(t,s)(≡F1,n(t,s)) belongs to the diagonal of F(t,s)onlywhenn=1, the properties (2), (3), (4), and (5) of the statement hold.  3. Existence and uniqueness results In this section, we prove existence and uniqueness results for the nth-order nonlinear dynamic equation with antiperiodic boundary conditions (Ln). Suppose that the function f:I×R→Rsatisfies condition (Hf), the operator Tn[M] is regressive on Iand invertible on Wnand Gis Green’s function associated to the operator Tn−1[M]inWn,definedin(2.2). We define the functions G+,G−:T×I→Ras G+:=max{G,0}≥0, G−:=−min{G,0}≥0, (3.1) and so, G=G+−G−on T×I. (3.2) Considering the operators A+ n[M],A− n[M]:C(T)→C(T)definedforeveryη∈C(T) as A+ n[M]η(t):=σ(b) aG+(t,s)fs,η(s)+Mη(s)∆s,t∈T, A− n[M]η(t):=σ(b) aG−(t,s)fs,η(s)+Mη(s)∆s,t∈T, (3.3) the solutions of the dynamic equation (Ln) are the fixed points of the operator An[M]:=A+ n[M]−A− n[M].(3.4) Note that if condition (Hf) holds, then the operators A+ n[M]andA− n[M]arewell defined. To deduce the existence and uniqueness of solutions of the dynamic equation (Ln), we introduce the concept of coupled lower and upper solutions for such problem. Definition 3.1. Given M∈Rsuch that the operator Tn[M] is regressive on Iand invertible on Wn, a pair of functions α,β∈Cn rd(I)suchthatα≤βin Tis a pair of coupled lower and upper solutions of the dynamic equation (Ln) if the inequalities α(t)≤A+ n[M]α(t)−A− n[M]β(t), ∀t∈T, β(t)≥A+ n[M]β(t)−A− n[M]α(t), ∀t∈T,(3.5) hold. A. Cabada and D. R. Vivero 297 Under the conditions of the previous definition, if αand βare a pair of coupled lower and upper solutions for the dynamic equation (Ln), then defining the operator B[M]:[α,β]×[α,β]−→ C(T) (3.6) as B[M](η,ξ):=A+ n[M]η−A− n[M]ξ, (3.7) and considering the hypothesis (H)foreveryt∈Iand α(t)≤u≤v≤β(t), it is satisfied that f(t,u)+Mu≤f(t,v)+Mv, (3.8) we prove the following monotonicity property. Lemma 3.2. Suppose that M∈Ris a given constant such that the operator Tn[M]is regressive on Iand invertible on Wn,αand βare a pair of coupled lower and upper solutions of the dynamic equation (Ln)and the function f:I×R→Rsatisfies hypotheses (Hf) and (H).Then,B[M](η,ξ)∈[α,β]for all η,ξ∈[α,β].Moreover,ifα≤η1≤η2≤βand α≤ξ2≤ξ1≤β, then B[M]η1,ξ1≤B[M]η2,ξ2in T.(3.9) Proof. Let α≤η1≤η2≤βand α≤ξ2≤ξ1≤β. It follows, from the definitions of A+ n[M] and A− n[M], that A+ n[M]α≤A+ n[M]η1≤A+ n[M]η2≤A+ n[M]βin T, A− n[M]α≤A− n[M]ξ2≤A− n[M]ξ1≤A− n[M]βin T.(3.10) From the definitions of αand β,weobtainthat α≤A+ n[M]α−A− n[M]β≤A+ n[M]η1−A− n[M]ξ1 ≤A+ n[M]η2−A− n[M]ξ2≤A+ n[M]β−A− n[M]α≤βin T.(3.11) This completes the proof.  Now, we obtain a result which gives us a region where all the solutions in [α,β]ofthe dynamic equation (Ln)lie. Proposition 3.3. Suppose that M∈Ris a given constant such that the operator Tn[M]is regressive on Iand invertible on Wn,αand βare a pair of coupled lower and upper solutions of the dynamic equation (Ln)and the function f:I×R→Rsatisfies hypotheses (Hf)and (H). Then, there exist two monotone sequences in C(T),{ϕm}m∈N,and{ψm}m∈N,withα= ϕ0≤ϕm≤ψl≤ψ0=βin T,m,l∈Nwhich converge uniformly to the functions ϕand ψ that satisfy ϕ=A+ n[M]ϕ−A− n[M]ψ,ψ=A+ n[M]ψ−A− n[M]ϕin T.(3.12) 298 Higher-order antiperiodic dynamic equations Moreover, any solution u∈[α,β]of (Ln)belongs to the sector [ϕ,ψ]. If, in addition, ϕ=ψ, then ϕis the unique solution of (Ln)in [α,β]. Proof. The sequences {ϕm}m∈Nand {ψm}m∈Nare obtained recursively as ϕ0:=α,ψ0:=β and for every m≥1, ϕm:=B[M]ϕm−1,ψm−1,ψm:=B[M]ψm−1,ϕm−1.(3.13) From Lemma 3.2,weknowthatα=:ϕ0≤ϕ1≤ψ1≤ψ0:=βin T. By induction, we conclude that the sequence {ϕm}m∈Nis monotone increasing, the sequence {ψm}m∈Nis monotone decreasing, and ϕm≤ψlin Tfor every m,l∈N. As a consequence, for every t∈T, there exist ϕ(t):=limm→∞ ϕm(t)andψ(t):= limm→∞ ψm(t). From hypothesis (Hf)andProposition 2.3, we know that both sequences are uniformly equicontinuous on Iand so, Ascoli-Arzel` a’s theorem, (see [7, page 72], [14,page 735]), implies that such convergence is uniform in T.Now,[13, Theorem 1.4.3] shows that ϕ=A+ n[M]ϕ−A− n[M]ψ,ψ=A+ n[M]ψ−A− n[M]ϕin T.(3.14) Let ube a solution of the dynamic equation (Ln)suchthatu∈[α,β]. From Lemma 3.2,weknowthat ϕ1:=B[M](α,β)≤B[M](u,u)=u≤B[M](β,α)=:ψ1in T.(3.15) By recurrence, we arrive at ϕm≤u≤ψlin Tfor all m,l∈N. Thus, passing to the limit, we obtain that ϕ≤u≤ψin T. Finally, if ϕ=ψ,thenwehavethatϕ=A+ n[M]ϕ−A− n[M]ϕ=:An[M]ϕ, that is, ϕ=ψ is a solution of the dynamic equation (Ln)in[α,β]. Since all solutions of (Ln)thatbelong to [α,β] lie in the sector [ϕ,ψ], we conclude that ϕis the unique solution of (Ln)in [α,β].  Now, let ·be the supremum norm in C(T). We prove the following existence result, that gives us a sufficient condition to assure that the dynamic equation (Ln) has a unique solution lying between a pair of coupled lower and upper solutions of (Ln). Theorem 3.4. Assume that M∈Ris a given constant such that the operator Tn[M]is regressive on Iand invertible on Wn,αand βare a pair of coupled lower and upper solutions of the dynamic equation (Ln)and the function f:I×R→Rsatisfies hypothesis (Hf). If for every t∈Iand α(t)≤u≤v≤β(t)the inequalities −M(v−u)≤f(t,v)−f(t,u)≤(K−M)(v−u) (3.16) A. Cabada and D. R. Vivero 299 are satisfied for some K≥0such that K·    σ(b) a G(t,s) ∆s     <1, (3.17) then the dynamic equation (Ln)has a unique solution in [α,β]. Proof. Since the first part of the inequality (3.16) is hypothesis (H), we know, by Proposition 3.3, that there exists a pair of functions ϕ,ψ∈C(T)suchthatforeveryt∈T we have 0≤(ψ−ϕ)(t) =A+ n[M]ψ(t)−A− n[M]ϕ(t)−A+ n[M]ϕ(t)+A− n[M]ψ(t) =σ(b) aG+(t,s)fs,ψ(s)−fs,ϕ(s)+Mψ(s)−ϕ(s)∆s +σ(b) aG−(t,s)fs,ψ(s)−fs,ϕ(s)+Mψ(s)−ϕ(s)∆s =σ(b) a G(t,s) fs,ψ(s)−fs,ϕ(s)+Mψ(s)−ϕ(s)∆s ≤σ(b) a G(t,s) ·K·ψ(s)−ϕ(s)∆s ≤ ψ−ϕ ·K·    σ(b) a G(t,s) ∆s     . (3.18) Thus, it follows from the inequality (3.17)thatϕ=ψin Tand Proposition 3.3 allows us to conclude that the dynamic equation (Ln) has a unique solution in [α,β].  Remark 3.5. One can check, following the proofs given in these sections, that we can develop an analogous theory for problem (¯ Ln) −u∆n(t)+ n−1  j=1 Mju∆j(t)=ft,u(t),∀t∈I=[a,b], u∆i(a)=−u∆iσ(b),0≤i≤n−1. (3.19) In this case, we must study the operator ¯ Tn[M]u≡−u∆n+ n−1  j=1 Mju∆j+Mu (3.20) in the space Wn. The functions αand βare given as in Definition 3.1,withGGreen’s function related with operator ¯ Tn[M]inWn. 306 Higher-order antiperiodic dynamic equations One can verify that the operator T2[M]u(t):=−u(t)+M2u(t), t∈I, (5.12) is regressive on Iand invertible on W2for all M= 0. Moreover, Green’s function associated to the operator T−1 2[M]inW2is given by the following expression. If M=0, then G(t,s)=         1 2T 2−t+s,if0≤s<t≤T, 1 2T 2−s+t,if0≤t≤s≤T. (5.13) If M= 0, then G(t,s)=             1 2Me−M(t−s) 1+e−MT −eM(t−s) 1+eMT ,if0≤s<t≤T, 1 2MeM(T+t−s) 1+eMT −e−M(T+t−s) 1+e−MT ,if0≤t≤s≤T. (5.14) From the previous expressions we obtain that if M=0, then     T 0 G(t,s) ds     ≤5T2 4=:K0>0, (5.15) and if M= 0, then     T 0 G(t,s) ds     ≤1 2|M|MeMT −1 1+eMT +1−e−MT 1+e−MT =:KM>0.(5.16) Therefore, from Corollary 5.1 we obtain the following result. Corollary 5.2. Assume that M∈Ris a given constant, αand βare a pair of coupled loweranduppersolutionsofthedifferential equation (¯ L2),and f:I×R→Ris a continuous function. Assuming condition (3.16)(withM2instead of M)forsomeK≥0such that K<1/KM,withK0and KMdefined in (5.15)and(5.16), respectively, then the differential equation (¯ L2)has a unique solution in [α,β]. Difference equations. Let h>0, P∈N;P≥2, and T={0,h,...,hP}⊂R. Given u:T→R,itfollowsfrom[5, Theorem 1.16] that for every t∈Tκ2,uis twice ∆-differentiable at t, moreover, we have that u∆2(t)=u(t+2h)−2u(t+h)+u(h) h2.(5.17) A. Cabada and D. R. Vivero 307 In this case, one can verify that for all M∈Rsuch that |M| = 1/h,1+(1+Mh)P−1= 0, and 1 + (1 −Mh)P−1= 0, the operator ¯ T2[M]u(t):=−u(t+2h)+2u(t+h)−1−M2h2u(h) h2,t∈I, (5.18) is regressive on Iand invertible on W2. Now, the function Gis given by the following expression. If M=0, then G(t,s)=         1 2h(P−1) 2−t+s+h,if0≤s+h≤t≤hP, 1 2h(P−1) 2−s−h+t,if0≤t≤s≤h(P−2). (5.19) If M= 0, |M| = 1/h,1+(1+Mh)P−1= 0, and 1 + (1 −Mh)P−1= 0, then G(t,s)=           1 2M(1 −Mh)(t−s−h)/h 1+(1−Mh)P−1−(1 + Mh)(t−s−h)/h 1+(1+Mh)P−1,if0≤s+h≤t≤hP, 1 2M(1 + Mh)(Ph+t−s−2h)/h 1+(1+Mh)P−1−(1 −Mh)(Ph+t−s−2h)/h 1+(1−Mh)P−1,if0≤t≤s≤h(P−2). (5.20) From these expressions, we deduce the following estimates. If M=0, then we have that      h· h(P−2)  s=0 G(t,s)      ≤5h(P−1)2 4=:K0>0.(5.21) If |M|<1/h,M= 0, then we arrive at      h· h(P−2)  s=0 G(t,s)      ≤1 2|M|M(1 + Mh)P−1−1 1+(1+Mh)P−1+1−(1 −Mh)P−1 1+(1−Mh)P−1 =:KM>0. (5.22) As a consequence, we arrive at the following result. Corollary 5.3. Assume that M∈Ris a given constant such that |M|<1/h,αand βare a pair of coupled lower and upper solutions of the dynamic equation (¯ L2),andthefunction f:I×R→Rverifies that for every t∈I,f(t,·)∈C(R).If(3.16)(withM2instead of M) holds for some K≥0such that K<1/KM,withK0and KMdefined in (5.21)and(5.22), respectively, then the dynamic equation (¯ L2)has a unique solution in [α,β]. q-difference equations. Given q∈R,q>1, and N∈N,N≥2, let T={1,q,...,qN}⊂R. 308 Higher-order antiperiodic dynamic equations If u:T→R,thenweknow,by[5, Theorem 1.16], that for every t∈Tκ2,uis twice ∆-differentiable at tand u∆2(t)=uq2t−(q+1)u(qt)+qu(t) (q−1)2qt2.(5.23) If t=qiand s=qj, then one can verify that Green’s function is given by the following expression. If M=0, then G(t,s)=         1 2qN−1 2−qi+qj+1,if0≤j+1≤i≤N, 1 2qN−1 2−qj+1 +qi,if0≤i≤j≤N−2. (5.24) If M= 0, |M| = 1/(q−1)qkfor each k∈JN−2={0,1,...,N−2},1+ N−2 k=0(1 + M(q− 1)qk)= 0and1+N−2 k=0(1 −M(q−1)qk)= 0, then G(t,s)=                              1 2Mi−1 k=j+1 1−M(q−1)qk 1+N−2 k=01−M(q−1)qk−i−1 k=j+1 1+M(q−1)qk 1+N−2 k=01+M(q−1)qk, if 0 ≤j+1≤i≤N, 1 2Mk∈Ii,j1−M(q−1)qk 1+N−2 k=01−M(q−1)qk−k∈Ii,j1+M(q−1)qk 1+N−2 k=01+M(q−1)qk, if 0 ≤i≤j≤N−2, (5.25) with Ii,jdefined in (4.23). From the expression of Green’s function, we obtain the following upper bounds. If M=0, then (q−1) ·max 0≤i≤NN−2  j=0 qj Gqi,qj ≤maxK0,1,K0,2=:K0>0, (5.26) with K0,1 :=5qN−1−12 4, K0,2 :=1 2qN−1qN−1−1+1 2qN−1−12. (5.27) If |M|<1/(q−1)qN−2,M= 0, then (q−1) ·max 0≤i≤NN−2  j=0 qj Gqi,qj ≤maxKM,1,KM,2=:KM>0, (5.28) A. Cabada and D. R. Vivero 309 where KM,1 :=1 2|M|MN−2 k=01+M(q−1)qk−1 1+N−2 k=01+M(q−1)qk+1−N−2 k=01−M(q−1)qk 1+N−2 k=01−M(q−1)qk, KM,2 :=1 2|M|MN−1 k=01+M(q−1)qk−1−M(q−1)qN−1 1+N−2 k=01+M(q−1)qk +1−M(q−1)qN−1−N−1 k=01−M(q−1)qk 1+N−2 k=01−M(q−1)qk. (5.29) From the previous expressions, we have the following result. Corollary 5.4. Assume that M∈Ris such that |M|<1/(q−1)qN−2,αand βare a pair of coupled lower and upper solutions of the dynamic equation (¯ L2),and f:I×R→Rverifies that for every t∈I,f(t,·)∈C(R).If(3.16)(withM2instead of M)istrueforsomeK≥0 such that K<1/KM,withK0and KMdefined in (5.26)and(5.28), respectively, then the dynamic equation (¯ L2)has a unique solution in [α,β]. Acknowledgments The authors thank the referees of the paper for their interesting comments and suggestions. This research was partially supported by D. G. I. and F.E.D.E.R. project BFM2001- 3884-C02-01, and by Xunta de Galicia and F.E.D.E.R. project PGIDIT020XIC20703PN, Spain. References [1] R.P.Agarwal,M.Bohner,D.O’Regan,andA.Peterson,Dynamic equations on time scales: a survey, J. Comput. Appl. Math. 141 (2002), no. 1-2, 1–26. [2] R.P.Agarwal,M.Bohner,andA.Peterson,Inequalities on time scales: a survey, Math. Inequal. Appl. 4(2001), no. 4, 535–557. [3] R. P. Agarwal, A. Cabada, and V. Otero-Espinar, Existence and uniqueness results for nth order nonlinear difference equations in presence of lower and upper solutions, Arch. Inequal. Appl. 1(2003), no. 3-4, 421–431. [4] R. P. Agarwal, A. Cabada, V. Otero-Espinar, and S. Dontha, Existence and uniqueness of solutions for anti-periodic difference equations, to appear in Arch. Inequal. Appl. [5] M. Bohner and A. Peterson, Dynamic Equations on Time Scales. An Introduction with Applications,Birkh ¨ auser Boston, Massachusetts, 2001. [6] ,Advances in Dynamic Equations on Time Scales,Birkh ¨ auser Boston, Massachusetts, 2003. [7] H. Brezis, Analyse Fonctionnelle: Th´ eorie et Applications [Functional Analysis: Theory and Applications], Collection Math´ ematiques Appliqu´ ees pour la Maˆ ıtrise, Masson, Paris, 1983. [8] A. Cabada, Themethodofloweranduppersolutionsforperiodicandanti-periodicdifference equations, to appear in Elect. Trans. Num. Anal. [9] ,Extremal solutions and Green’s functions of higher order periodic boundary value problems in time scales, J. Math. Anal. Appl. 290 (2004), no. 1, 35–54. [10] A. Cabada and J. J. Nieto, Fixed points and approximate solutions for nonlinear operator equations. Fixed point theory with applications in nonlinear analysis,J.Comput.Appl.Math.113 (2000), no. 1-2, 17–25. 310 Higher-order antiperiodic dynamic equations [11] D. Franco, J. J. Nieto, and D. O’Regan, Anti-periodic boundary value problem for nonlinear first order ordinary differential equations, Math. Inequal. Appl. 6(2003), no. 3, 477–485. [12] S. Hilger, Ein Maßkettenkalk¨ ul mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. thesis, Universit¨ at W¨ urzburg, W¨ urzburg, 1988. [13] B. Kaymakc¸alan, V. Lakshmikantham, and S. Sivasundaram, Dynamical Systems on Measure Chains, Kluwer Academic Publishers, Dordrecht, 1996. [14] J. Mawhin, Analyse. Fondements, Techniques, ´ Evolution [Analysis. Foundations, Techniques, Evolution], second ed., Acc` es Sciences, De Boeck University, Brussels, 1997. Alberto Cabada: Departamento de An´ alise Matem´ atica, Facultade de Matem´ aticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Galicia, Spain E-mail address:[email protected] Dolores R. Vivero: Departamento de An´ alise Matem´ atica, Facultade de Matem´ aticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Galicia, Spain E-mail address:[email protected]