Marini, Mauro

Matucci, Serena

Rehak, P.

English

Florence Research

Equadiff 10, August 27–31, 2001Prague, Czech RepublicEquadiff 10 CD ROMPapers, pp. 289–296Nonoscillatory Solutions for Nonlinear DiscreteSystemsMauro Marini1, Serena Matucci2 and Pavel Řehák31 Department of Electronics and Telecommunications, Faculty of Engineering,University of Florence, via S. Marta, I-50139 Florence, ItalyEmail: marini@ing.unifi.it2 Department of Electronics and Telecommunications, Faculty of Engineering,University of Florence, via S. Marta, I-50139 Florence, ItalyEmail: matucci@det.unifi.it3 Mathematical Institute, Academy of Sciences of Czech Republic,Žižkova 22, CZ-616 62 Brno, Czech RepublicEmail: rehak@math.muni.czAbstract. We investigate some asymptotic properties of the nonlinearforced difference system∆(rkΦα(∆xk))− σϕkf(yk+1) = σϕˆk,∆(qkΦβ(∆yk))− ψkg(xk+1) = ψˆk.In particular we give necessary and sufficient conditions for existence of theso-called regularly decaying solutions and thereby we complete the resultspresented in [10].MSC 2000. 39A10, 39A11.Keywords. (singular) nonlinear difference system, asymptotic behavior,positive decreasing solutions.1 IntroductionIn [10] the authors investigated certain discrete asymptotic boundary value prob-lems on the discrete interval [m,∞) := {m,m + 1, ..},m ∈ Z, associated to the Supported by the Grants No. 201/01/P041 and No. 201/01/0079 of the Czech GrantAgency and by C.N.R. of ItalyThis is the preliminary version of the paper.290 M. Marini, S. Matucci and P. Řeháknonlinear forced difference system∆(rkΦα(∆xk))− σϕkf(yk+1) = σϕˆk,∆(qkΦβ(∆yk))− ψkg(xk+1) = ψˆk. (1)In particular, necessary and sufficient conditions for the existence of the so-calledstrongly decaying solutions of (1) were presented. The principal aim of this con-tribution is to complete those results examining also the existence of the so-calledregularly decaying solutions. For the definitions of these concepts see Definition 1.In system (1) we assume that {ϕk}, {ψk}, {rk}, {qk} are real positive sequences de-fined for any k ≥ m, the forcing terms {ϕ̂k}, {ψ̂k} are real nonnegative sequencesdefined for any k ≥ m, Φp(u) = |u|p−1sgnu with p > 1 is the one-dimensionalp-laplacian operator, f, g : R+ → R+ are monotone continuous functions andσ ∈ {−1, 1}.There are several motivations for the investigation of (1):• System (1) arises in the discretization process of differential systems withp-laplacian operator. For results concerning the role of system (1) in various appli-cations, we refer the reader to [6], [9], [12] and to the references contained therein.• The study of system (1) is motivated also by certain results obtained forscalar second order difference equations of the form∆(rkΦα(∆xk)) = σϕkf(xk+1),which attracted a considerable attention in recent years, see, e.g., [2], [3], [4], [7], [8],[11], [15]. Other interesting contributions can be found also in the monograph [1].• System (1) can be rewritten in the form of a fourth-order nonlinear equationof the type∆2(rk∆2uk) + pkf(uk+2) = 0.For results from this point of view see for instance [13], [14].Here we are interested in the existence of positive nonoscillatory solutions,which are asymptotically decreasing towards zero. A nonoscillatory solution of (1)is a vector sequence (x, y) = ({xk}, {yk}) satisfying (1) for k ≥ m, and such thatboth components {xk}, {yk} are eventually of fixed sign.Definition 1. A solution (x, y) of (1) is said to be– decaying, if x, y are eventually positive decreasing andlimk→∞xk = 0 = limk→∞yk;– regularly decaying, if it is decaying andlimk→∞rkΦα(∆xk) = −Ax, limk→∞qkΦβ(∆yk) = −By,where Ax, By are positive constants;Nonoscillatory Solutions for Nonlinear Discrete Systems 291– strongly decaying, if it is decaying and limk→∞ qkΦβ(∆yk) = 0,limk→∞rkΦα(∆xk) ={−Cx for σ = −1,0 for σ = 1,where Cx is a positive constant.Note that the constant Cx cannot be zero when σ = −1, since in this case thequasidifference rkΦα(∆xk) is eventually negative decreasing.As already mentioned at the beginning of this section, the main purpose of thispaper is to complete the results presented in [10], in which the existence of stronglydecaying solutions is considered. To this end we will give here necessary and suf-ficient conditions for existence of regularly decaying solutions (see Section 2), inview of their crucial role in a variety of physical applications, as already claimed.We will also restate the main results proved in [10] in order to have a comparisonwith our new results (see Section 3). Note that both the regular case (i.e. with thenonlinearities f, g bounded in a right neighborhood of zero) and the singular case(i.e. with f or g unbounded in a right neighborhood of zero) will be considered.Some remarks and comments are given throughout this contribution.Finally, nontrivial prototypes of nonlinearities f, g are the one-dimensionalLaplacians Φγ , Φδ, respectively, with γ, δ = 1 (i.e., with possible singularities atzero). Then (1) leads to the Emden-Fowler type system∆(rkΦα(∆xk)) = σϕkΦγ(yk+1)∆(qkΦβ(∆yk)) = ψkΦδ(xk+1)(2)as a special case.Some of our main results remain valid for more general systems of the form∆(rkα(∆xk)) = σF (k, yk+1, xk+1)∆(qkβ(∆yk)) = G(k, xk+1, yk+1),(3)where the functions α and β are monotone continuous increasing with α(0) = 0 =β(0), and F,G are positive continuous on {m,m+1, . . .}× (0, ε]× (0, ε] with someε > 0 and bounded with respect to the third variable.2 Regularly Decaying SolutionsDenote with α∗ the conjugate number of α i.e. 1/α+ 1/α∗ = 1, and analogouslyfor β. The following necessary and sufficient condition holds:The o re m 2 . System (1) has at least one regularly decaying solution if and onlyif∞∑k=mΦα∗(1rk)<∞,∞∑k=mΦβ∗(1qk)<∞, (4)∞∑k=mϕˆk <∞,∞∑k=mψˆk <∞, (5)292 M. Marini, S. Matucci and P. Řehákand there exist A > 0, B > 0 such that∞∑k=mϕk f ∞∑j=k+1Φβ∗(Aqj) <∞, (6)∞∑k=mψk g ∞∑j=k+1Φα∗(Brj) <∞. (7)In addition, if σ = 1, then this solution is decreasing for any k ≥ m.Proof. The “if part”. Choose an integer T ≥ m such that∞∑k=Tϕk f ∞∑j=k+1Φβ∗(1qj) < 1/2, ∞∑k=Tψk g ∞∑j=k+1Φα∗(1rj) < 1/2.Denote with ^∞T the Banach space of all bounded sequences defined for k ≥ T ,endowed with the topology of the supremum norm, and consider the set Ω ⊂^∞T × ^∞T given byΩ =(u, v) = ({uk}, {vk}) ∈ ^∞T × ^∞T :∞∑j=k+1Φα∗(M1rj)≤≤ uk ≤∞∑j=k+1Φα∗(M2rj),∞∑j=k+1Φβ∗(N1qj)≤ vk ≤∞∑j=k+1Φβ∗(N2qj) ,where M1, N1,M2, N2 are suitable positive constants which will be determinedlater. Consider the operator T : Ω → ^∞T × ^∞T defined byT (u, v) = (T1(v), T2(u)) = ({(T1(v))k}, {(T2(u))k}),where(T1(v))k =∞∑j=kΦα∗Mτ(σ)rj+σrj∞∑i=j[ϕif(vi+1) + ϕˆi] ,(T2(u))k =∞∑j=kΦβ∗N1qj+1qj∞∑i=j[ψig(ui+1) + ψˆi]with τ(σ) = 1 for σ = 1 and τ(σ) = 2 for σ = −1. In order to show that T hasa fixed point in Ω, for the Schauder-Tychonoff fixed point theorem it is sufficientto verify that: (i) Ω is a nonempty, closed and convex subset of ^∞T × ^∞T , (ii)T (Ω) ⊆ Ω, (iii) T (Ω) is relatively compact, (iv) T is continuous in Ω.Nonoscillatory Solutions for Nonlinear Discrete Systems 293The validity of (i) is obvious, and furthermore it is not difficult to show thatT maps Ω into itself, i.e. (ii) holds. To show it, if f, g are nondecreasing it issufficient to choose M1 = N1 = 1/2 and M2 = N2 = 1. If f, g are nonincreasingwe choose M1 = N1 = 1,M2 = N2 = 3/2, and finally, when f is nondecreasingand g is nonincreasing, or vice versa, it is sufficient to choose M1 = N2 = 1,M2 =3/2, N1 = 1/2, or N1 = M2 = 1,M1 = 1/2, N2 = 3/2, respectively.(iii) To show that T (Ω) is relatively compact it is sufficient to prove that T (Ω)is uniformly Cauchy in the topology of ^∞T × ^∞T by [4, Theorem 3.3], i.e. for anyε > 0 there exists N ≥ T such that for any k, l ≥ N it holds|(T1(v))k − (T1(v))l| < ε and |(T2(u))k − (T2(u))l| < εfor (u, v) ∈ Ω. The details are left to the reader.(iv) The continuity of T in Ω can be proved using a similar argument to that inthe proof of [10, Theorem 1], namely using the discrete analogue of the Lebesguedominated convergence theorem, since the series occuring in the definition of theoperator T are totally convergent.Thus the Schauder fixed point theorem can be applied and the operator T hasa fixed point (x, y) ∈ Ω. It is easy to see that (x, y) is a regularly decaying solutionof (1) for k ≥ T .To show that this solution can be extended to the left in a decreasing mannerfor σ = 1 we use the fact that system (1) is actually a recurrence relation. Weproceed in the same way as it is done in the proof of [10, Theorem2].The “only if part”. Let (x, y) be a regularly decaying solution of (1). Thenthere exist positive constants M1,M2, N1, N2 and T ≥ m such that∑∞j=k Φα∗(M1rj)≤ xk ≤∑∞j=k Φα∗(M2rj),∑∞j=k Φβ∗(N1qj)≤ yk ≤∑∞j=k Φβ∗(N2qj) (8)for k ≥ T . Assume f, g nondecreasing; the remaining cases can be treated similarly.Summing twice both equations in (1) from k to ∞ we getxk = σ∑∞j=k Φα∗(1rj∑∞i=j(ϕif(yi+1) + ϕˆi)),yk =∑∞j=k Φβ∗(1qj∑∞i=j(ψig(xi+1) + ψˆi)).(9)Now (8) and (9) imply the conditions (4), (5), (6) and (7), that leads to theassertion. @unionsqRemark 3. Note that the necessary and sufficient conditions in Theorem 2 are thesame for both cases σ = ±1 in spite of the fact that different sign condition causesa different dynamical behavior as regards other types of solutions.Remark 4. Using similar arguments to those in the above proof, we are able toprove the “if part” for more general system (3), where the first condition in (4)294 M. Marini, S. Matucci and P. Řehákis replaced by∑∞k=m α−1(1/rj) < ∞, α−1 being the inverse of the function α.Instead of (6) we suppose the existence of certain “upper” functions ϕ¯, F¯ suchthat F (k, u, v) ≤ ϕ¯kF¯ (u) on {m,m+ 1, . . . } × (0, ε]× (0, ε], where ϕ¯, F¯ satisfy∞∑k=mϕ¯k F¯( ∞∑j=k+1β−1(1qj))<∞,β−1 being the inverse of β. The latter condition in (4) and (7) would be rearrangedin a similar way. Obviously, a necessary condition in the above sense cannot bestated in this case.Remark 5. Closer examination of the proof enables us to obtain an asymptoticestimate for regularly decaying solutions. Indeed, it is not difficult to see that xkis asymptotic to∑∞j=k Φα∗(1/rj), while yk is asymptotic to∑∞j=k Φβ∗(1/qj).Remark 6. Finally note that, besides the regular case, the statement of Theo-rem 2 includes the singular one as well, i.e. when f, g are unbounded in a rightneigborhood of zero. It should be emphasized that we do not require any additionalconditions to treat this case.3 Strongly Decaying SolutionsFor sake of completeness in this section we recall some results that were provedin [10] in order to have “complementary” statements. We start with a necessaryand sufficient criterion guaranteeing the existence of strongly decaying solutionsof system (1) with σ = −1.The o re m 7 ([ 10], Theorem 1). System (1) with σ = −1 has strongly decayingsolutions if and only if condition (5) and∞∑k=mΦα∗(1rk)<∞hold, and there exists a constant A > 0 such that∞∑k=mϕkf( ∞∑j=k+1ωj(A))<∞, (10)whereωk(A) = Φβ∗{1qk∞∑j=k[ψj g( ∞∑i=j+1Φα∗(Arj))+ ψˆj]}<∞.Remark 8. In the contrast to the existence of regularly decaying solutions, thefirst condition in (4) is not necessary in Theorem 7. On the other hand, if thefirst condition in (4) holds, then condition (10) can be relaxed to simpler (butonly sufficient) conditions (6), (7). Thus we have analogous sufficient conditionsguaranteeing the existence of both regular and strongly decaying solutions of (1)with σ = −1. See Remark 13 for additional information concerning the case σ = 1.Nonoscillatory Solutions for Nonlinear Discrete Systems 295Remark 9. The proof of Theorem 7 gives an asymptotic estimate for stronglydecaying solutions. However, in the contrast to the case in Section 2, we havean asymptotic estimate only for the first component. In fact, xk is asymptotic to∑∞j=k Φα∗(1/rj).In the case of system (1) with σ = 1, the following existence result holds.The o re m 1 0 ( [ 10], Theorem 2). Let f, g be nondecreasing. Suppose thatat least one of the forcing terms {ϕˆk}, {ψˆk} is eventually nontrivial. (11)If∞∑k=mΦα∗(1rk∞∑j=k(ϕj + ϕˆj))<∞,∞∑k=mΦβ∗(1qk∞∑j=k(ψj + ψˆj))<∞, (12)then system (1) with σ = 1 has at least one strongly decaying solution that isdecreasing for any k ≥ m.Remark 11. Observe that condition (4) is not necessary. On the other hand, The-orem 10 requires the convergence of the series∑∞j=m ϕj and∑∞j=m ψj .When at least one of the series∑∞j=m ϕj and∑∞j=m ψj diverges, we can statethe following criterion.The o re m 1 2 ( [ 10], Theorem 3). Let f, g be nondecreasing. If (11), (4), (5),(6), (7) hold, then system (1) with σ = 1 has at least one strongly decaying solutionthat is decreasing for any k ≥ m.Remark 13. If f, g are nondecreasing and (11) holds, then the conditions of The-orem 2 are sufficient for the existence of both regularly and strongly decayingsolution of (1) with σ = 1 and with σ = −1 as well.Remark 14. In the contrast to Theorems 2 and 7, the presence of forcing termsplays an important role in Theorems 10 and 12 for proving the existence of stronglydecaying solutions. On the other hand, as it is shown in [10], there exist discretenonforced systems having strongly decaying solutions. The problem of finding suf-ficient conditions for nonforced discrete systems to have singular solutions remainsopen; in the continuous case this problem can be solved using the concept of sin-gular solution (see [12]) that however has no discrete counterpart.References1. R.P. Agarwal, Difference Equations and Inequalities, 2nd edition, Pure Appl. Math.228, Marcel Dekker, New York, 2000.2. M. Cecchi, Z. Došlá and M. Marini, Positive decreasing solutions of quasilinear dif-ference equations, Comp. Math. Appl 42 (2001), 1401–1410.296 M. Marini, S. Matucci and P. Řehák3. M. Cecchi, Z. Došlá and M. Marini, Unbounded solutions of quasilinear differenceequations, to appear in Comp. Math. Appl.4. S. S. Cheng and W.T. Patula, An existence theorem for a nonlinear difference equa-tion, Nonlinear Analysis 20 (1993), 193–203.5. P. Clément, J. Fleckinger, E. Mitidieri and F. de Thélin, Existence of positive solu-tions for a nonvariational quasilinear elliptic system, J. Diff. Eq. 166 (2000), 455–477.6. J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries, Volume I:Elliptic equations, Research Notes in Math. 106, Pitman Publ. Ltd., London, 1985.7. J. R. Graef and E. Thandapani, Oscillation of two-dimensional difference systems,Comp. Ath. Appl. 36 (1999), 157–165.8. B. Liu and S. S. Cheng, Positive solutions of second order nonlinear difference equa-tions, J. Math. Anal. Appl. 204 (1996), 482–493.9. M. Marini, S. Matucci and P. Řehák, On decay of coupled nonlinear systems, toappear in Adv. Math. Sci. Appl.10. M. Marini, S. Matucci and P. Řehák, Strongly decaying solutions of nonlinear forceddiscrete systems, to appear in Proc. of ICDEA 2001, Augsburg.11. R. Medina, Nonoscillatory solutions for the one-dimensional p-laplacian, J. Comp.Applied Math. 98 (1998), 27–33.12. T. Tanigawa, Positive decreasing solutions of systems of second order quasilineardifferential equations, Funkc. Ekvac. 43 (2000), 361–380.13. E. Thandapani and I.M. Arockiasamy, Some oscillation and non-oscillation theoremsfor fourth order difference equations, Zeitricht für Analysis und ihre Anwend. 19(2000), 863–872.14. J. Yan and B. Liu, Oscillatory and asymptotic behaviour of fourth order nonlineardifference equations, Acta Math. Sinica 13 (1997), 105–115.15. P. J. Y. Wong and R.P. Agarwal, Oscillation and monotone solutions of second orderquasilinear difference equations, Funk. Ekv. 39 (1996), 491–517.

NONOSCILLATORY SOLUTIONS FOR NONLINEAR DISCRETE SYSTEMS

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NONOSCILLATORY SOLUTIONS FOR NONLINEAR DISCRETE SYSTEMS

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