General solution to a higher-order linear difference equation and existence of bounded solutions

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1 Stević Advanes in Differene Equations :377 DOI /s R E S E A R C H Open Aess General solution to a higher-order linear differene equation and existene of bounded solutions Stevo Stević * * Correspondene: sstevi@pttrs Mathematial Institute of the Serbian Aademy of Sienes, Knez Mihailova 36/III, Beograd, 11000, Serbia Abstrat We present a losed-form formula for the general solution to the differene equation x n+ q n x n f n, n N 0, where N,q n n N0,f n n N0 C,intheaseq n q, n N 0, q C \{0} Using the formula, we show the existene of a unique bounded solution to the equation when q >1andsup n N0 f n < by finding a solution in losed form By using the formula for the bounded solution we introdue an operator that, together with the ontration mapping priniple, helps in showing the existene of a unique bounded solution to the equation in the ase where the sequene q n n N0 is real and nononstant, whih shows that, in this ase, there is an elegant method of proving the result in a unified way We also obtain some interesting formulas MSC: Primary 39A14; seondary 05A10 Keywords: linear differene equation; general solution; existene of bounded solutions; ontration mapping priniple 1 Introdution Differene equations are an area of onsiderable interest Some lassial results an be found, for example, in [1 4] There has been some renewed interest in solvable differene equations [5 9] andsystems[6, 10 13] and some losely related topis suh as finding their invariants or some appliations [14 18]; see also numerous referenes therein In many of these papers on solvability, the equations and systems are nonlinear and are transformed into some solvable linear ones by using suitable hanges of variables A frequent situation is that a differene equation is transformed into a linear first-order one [5 9], whih is solvable [4] ontains a nie presentation of some methods for solving theequation;seealso[1], as well as [19] where the ase of onstant oeffiients is onsidered Moreover, an analysis shows that many systems are also essentially redued to the equation see, for example, [6, 10] and the referenes therein In our reent papers on produt-type differene equations and systems, we frequently use the orresponding produt-type first-order equation, whih, in some ases, is also solvable see, for example, [11 13] and the referenes therein The Authors 2017 This artile is distributed under the terms of the Creative Commons Attribution 40 International Liense whih permits unrestrited use, distribution, and reprodution in any medium, provided you give appropriate redit to the original authors and the soure, provide a lin to the Creative Commons liense, and indiate if hanges were made

2 Stević Advanes in Differene Equations :377 Page 2 of 12 One of the simplest inhomogenous higher-order linear equations is the following relative of the linear first-order differene equation: x n+ q n x n f n, n N 0, 1 where q n n N0 and f n n N0 are real or omplex sequenes If q n q 0andf n 0,n N 0,thenwehave x n+ qx n 0, n N 0 2 Sine the assoiated harateristi polynomial to equation 2is P λλ q, its general solution an be written in the following form [1 4]: x n j qε j 1 n n q j1 j ε j 1 n, n N0, 3 j1 where j, j 1,, are arbitrary onstants, q is one of the th roots of q,and ε e 2πi the notation will be used from now on Formula 3 shows that every solution to equation 2 onverges to zero when q < 1, every solution to the equation is bounded when q 1, and all nontrivial solutions to the equation are unbounded when q >1 In [20] we presented some of our old results related to equation 1 forthease 2, whih had been presented at some tals and/or onferenes during the last deade Some of them seem follore, but there are some nie ideas behind them Let us briefly desribe the results of [20] Namely, we have studied, among other problems, the existene of bounded solutions to equation 1 when 2 in two different ways Using a routine method, it was shown that when q n q C \{0}, the equation has the general solution x n n 1 q n f q n d q n f 2 q +2, n N 0, 4 where 0, d 0 C, and q is one of two possible square roots of q Employing4, it was shown that the equation in the ase q n q, n N 0, has a unique bounded solution when q > 1 by finding its losed-form formula The formula motivated us to introdue an operator that, together with the ontration mapping priniple [21], helped us in showing the existene of a unique bounded solution to the equation under some onditions on q n n N0 A more general linear seond-order differene equation was later studied in a similar way in [22] A natural problem is to try to generalize the results by using the proedure for arbitrary, whih is tehnially not so easy Reently in [23], we have managed to solve the problem for the ase 3, whih suggested us that there was a related solution in the

3 Stević Advanes in Differene Equations :377 Page 3 of 12 general ase However, the problem for the ase of an arbitrary wasleftopenbeause of many tehnial diffiulties This paper is devoted to solving the problem Namely, following the ideas in [20], we first present a losed-form formula for the general solution to equation 1 intheaseq n q, n N 0, q C \{0} Usingtheformula,weshowthe existene of a unique bounded solution to the equation when q >1andsup n N0 f n < by finding the solution in losed form Then, using the formula for the bounded solution, we introdue an operator that, together with the ontration mapping priniple, helps in showing the existene of a unique bounded solution to the equation when the sequene q n n N0 is real, nononstant and satisfies some additional onditions whih will be speified later We also obtain some interesting formulas Some other appliations of fixed-point theorems in investigation of differene equations an be found in [1, 24, 25] see also the related referenes therein, where a variant of the Shauder fixed-point theorem [24, 25] is frequently applied The majority of suh papers onstrut suitable operators by using some summations, whih an be regarded as some ind of solvability methods As usual, by l N 0 we denote the Banah spae of bounded sequenes u u n n N0 with the supremum norm u sup u n 5 By V t 1, t 2,,t we denote the Vandermonde determinant of th order: t 1 t 2 t 3 t V t 1, t 2,,t : t1 2 t2 2 t3 2 t 2 t1 1 t2 1 t3 1 t 1 It is well nown that V t 1, t 2,,t 1 l<j t j t l 6 2 Main results Our first result shows that there is a losed-form formula for general solutions to equation 1 whenq n n N0 is a onstant sequene From the theoretial point of view, we now that for eah inhomogeneous linear differene equation with onstant oeffiients, suh a formula exists On the other hand, we now that the polynomial equations of order greater than or equal to five need not be solved by radials, whih implies that there are linear differene equations with onstant oeffiients for whih we annot find a losedform formula for their general solutions The result shows that there is a lass of linear equations of arbitrary order for whih it is possible to find suh a losed-form formula Moreover, the result gives only one formula that inludes all the solutions to the equation Lemma 1 Consider the differene equation x n+ qx n f n, n N 0, 7

4 Stević Advanes in Differene Equations :377 Page 4 of 12 where N, q C \{0}, and f n n N0 is a given sequene of omplex numbers Then, the general solution to the equation is x n qε s n s + 1 n 1 ε sj f j q j+, n N 0, 8 where s, s 0, 1,are arbitrary numbers, and q is one of the th roots of q Proof Based on 3, we try to find the general solution to equation 7intheform x n s s n, n N0, 9 where s n n N0, s 0, 1, are some undetermined sequenes To do this, we pose the following onditions: x n+1 x n+2 s n+1 qε s n+1 s n+1 qε s n+2 s s n+1, s s n+2, 10 x n+ 1 s n+1 qε s n+ 1 for n N 0, from whih it follows that s s n+ 1 s n+1 s s n+1 0, s n+1 s s n+2 0, 11 s n+1 s s n+ 1 0 for n N 0 From 7, 9, and the last equality in 10withn n +1,weeasilyget s n+1 s s n+ fn 12 for n N 0

5 Stević Advanes in Differene Equations :377 Page 5 of 12 Sine q 0,foreahfixedn N 0,equations11and12 are equivalent to the following -dimensional linear system: s n+1 s n ε s n+1 0, s n+1 s n ε s n+2 0, 13 s n+1 s n ε s n+ 1 0, s n+1 s n ε s n+ f n q n+ in unnown variables s n+1 s n, s 0, 1 The determinant of the system is 1 ε n+1 ε 2n+1 ε 1n+1 1 ε n+2 ε 2n+2 ε 1n+2 n 1 ε n+3 ε 2n+3 ε 1n+3 1 ε n+ ε 2n+ ε 1n+ ε n ε ε 2 ε 1 1 ε 2 ε 4 ε ε 1 ε 2 1 ε 1 1 ε n V 1, ε,,ε 1 14 From 13and14, by some alulation and using properties of determinants, we get s n+1 s n ε n+11 2 V 1, ε,,ε 1 1 ε n+1 ε s 1n+1 0 ε s+1n+1 ε 1n+1 1 ε n+2 ε s 1n+2 0 ε s+1n+2 ε 1n+2 1 ε n+3 ε s 1n+3 0 ε s+1n+3 ε 1n+3 1 ε n+ 1 ε s 1n+ 1 0 ε s+1n+ 1 ε 1n+ 1 1 ε n+ ε s 1n+ f n ε s+1n+ ε 1n+ q n+

6 Stević Advanes in Differene Equations :377 Page 6 of 12 ε sn+1 V 1, ε,,ε ε ε s 1 0 ε s+1 ε 1 1 ε 2 ε s ε s+1 2 ε ε 2 ε s ε s+1 2 ε ε 1 ε s 1 1 f n ε s+1 1 ε 1 1 q n+ 1 +s+1 ε sn+1 f n q n+ V 1, ε,,ε ε ε s 1 ε s+1 ε 1 1 ε 2 ε s 1 2 ε s+1 2 ε ε 2 ε s 1 2 ε s+1 2 ε s+1 ε sn+1 V 1 1, ε,,ε s 1, ε s+1,,ε 1 f n 15 q n+ V 1, ε,,ε 1 By using formula 6 it follows that V 1 1, ε,,ε s 1, ε s+1,,ε 1 V 1, ε,,ε 1 Now note that 1 ε s 1 ε s ε s 1 ε s+1 ε s ε 1 ε s 1 1 s ε s 1 ε s ε s 1 ε s ε s+1 ε s ε 1 16 z 1z 1 z ε s 1 z ε s z ε s+1 z ε 1, 17 from whih it follows that ε s 1 ε s ε s 1 ε s ε s+1 ε s ε 1 lim z ε s z 1 z ε s 1 z ε s+1 z ε 1 z 1 lim z ε s z ε s εs 1 18 From 15, 16, and 18, sine ε 1, it follows that s n+1 s n ε sn f n q n+ 19 for n N 0 and s 0, 1

7 Stević Advanes in Differene Equations :377 Page 7 of 12 Summing up 19from0ton 1,weobtain s n s + 1 n 1 ε sj f j q j+ 20 for n N 0 and s 0, 1,where s : s 0, s 0, 1 Employing 20 in9, we get formula 8, as desired Remar 1 It is interesting that the determinant V 1, ε,,ε 1 anbealulatedin losed form see, for example, [26,p46],[27, p 61] Namely, we have the following formula: V 1, ε,,ε 1 2 e πi From 16, 18, and 21we obtain V s 1 : V 1 1, ε,,ε s 1, ε s+1,,ε s 2 2 e πi s for s 0, 1 Our next result gives an appliation of Lemma 1 in the investigation of the existene of a bounded solution to equation 7 when q >1andf n n N0 is a bounded sequene of omplex numbers Theorem 1 Assume that q >1and f : f n n N0 C is a given bounded sequene Then, there is a unique bounded solution to equation 7 Proof Employing 8, itfollowsthat x m+l qε s m+l s + 1 q m+l 1 m+l 1 s ε sl + 1 ε sj f j q j+ m+l 1 ε sl j f j q j+ 22 for all m N 0 and l 0, 1 Sine q >1andf is bounded, we have m+l 1 ε sl j f j q j+ f q j+ f q 1 q 1 < 23 for eah l 0, 1

8 Stević Advanes in Differene Equations :377 Page 8 of 12 From 22, 23, and the assumption q > 1, we see that, for a bounded solution x n n N0 to 7, there must be s ε sl 1 ε sl j f j q j+ : S l 24 for l 0, 1 Equalities 24 area-dimensional linear system in variables s, s 0, 1,whosedeterminant is V 1, ε,,ε 1 :V By solving the system we have S ε ε s 2 S 1 ε s ε 1 s ε 2 ε s 2 2 S 2 ε s 2 ε 1 2 V 1 ε 2 ε s 2 2 S 2 ε s 2 ε ε 1 ε s 2 1 S 1 ε s 1 ε 1 1 l1 1l+s S l 1 W ls V 1, ε,,ε 1 25 for s 1,,whereW ls, l, s 1,,are 1-dimensional minors of the determinant in 25 orresponding to the element on the position l, s They an be obtained by the oeffiients of the following polynomial of 1thorder, whih is defined by the Vandermonde determinant: ε ε s 2 x ε s ε 1 1 ε 2 ε s 2 2 x 2 ε s 2 ε 1 2 P 1 x: 1 ε 2 ε s 2 2 x 2 ε s 2 ε ε 1 ε s 2 1 x 1 ε s 1 ε s x 1 W s + 1 +s 1 x 2 W 1 s s+1 W 1s 1 s x 1 x ε s 2 x ε s x ε 1 V 1 1, ε,,ε s 2, ε s,,ε 1, 26 where the seond equality is obtained by expanding the determinant along the sth olumn, whereas the third one follows from 6 First, note that from 26 it follows that W s V 1 1, ε,,ε s 2, ε s,,ε 1 27 Now note the following equality: ε sj 0, s m,, s m, 28 for m Z

9 Stević Advanes in Differene Equations :377 Page 9 of 12 From 28and26wehave W 1 s V 1 1, ε,,ε s 2, ε s,,ε 1,j s 1 V 1 1, ε,,ε s 2, ε s,,ε 1 ε j ε s 1 ε j ε s 1 V 1 1, ε,,ε s 2, ε s,,ε 1 29 Now note that from 17 and the Viète formula, it follows that ε j 1 ε j2 ε j t j 1 <j 2 < <j t 1 for t 1, 1 From 26, 30witht 2, and the alulation in 29wehave W 2 s V 1 1, ε,,ε s 2, ε s,,ε 1 V 1 1, ε,,ε s 2, ε s,,ε 1 1 j 1 <j 2 1,j 1,j 2 s 1 0 j 1 <j 2 1 ε j 1 ε j 2 ε j 1 ε j 2 ε s 1,j s 1 ε 2s 1 V 1 1, ε,,ε s 2, ε s,,ε 1 31 ε j Assume that, for an m {2,, 2},wehaveprovedthat W ms 1 m ε ms 1 V 1 1, ε,,ε s 2, ε s,,ε 1 32 and ε j1 ε j m 1 m ε ms j 1 < <j m 1,j 1,,j m s 1 Then, from 26, 30witht m +1,and33wehave W m 1 s V 1 1, ε,,ε s 2, ε s,,ε 1 V 1 1, ε,,ε s 2, ε s,,ε 1 ε j1 ε j m+1 ε s 1 0 j 1 < <j m j 1 < <j m+1 1,j 1,,j m+1 s 1 0 ĵ 1 < < j m 1,ĵ 1,, j m s 1 ε j1 ε j m+1 j εĵ1 ε m 1 m+1 ε m+1s 1 V 1 1, ε,,ε s 2, ε s,,ε 1 34 From 29, 34, and the method of indution we see that 32holds

10 Stević Advanes in Differene Equations :377 Page 10 of 12 Employing 18, 24, and 32in25, we get s f j q j+ 1 1 ε ls ε tj l l0 t0 ε sj f j q j+ 35 for s 0, 1, where we have also used that 1 1 ε ls l0 ε tj l t0 t0 ε t sl, ε tj 1 l0 and, then applied 28intheasest s and t s note also that t s < Using 35in8, we get x n 1 qε s n ε sj f j 36 q j+ jn for n N 0 Using 28, by a diret alulation we verify that 36 presents a solution to equation 7 Also, we have that x n f q 1 q 1 <, n N 0, showing the boundedness of the solution The uniqueness of the bounded solution follows fromtheuniquehoieofonstants s, s 0, 1,in35 Remar 2 Note that by using 28in24 it follows that S l m0 for l 0, 1 f l+m q l+m+ Now, motivated by 36 and some operator theory tehnique, we prove a result on the unique existene of bounded solutions to equation 1 Theorem 2 Consider equation 1 where 1<a q n b, n N 0, 37 or b q n a < 1, n N 0, 38 for some positive numbers a and b, and f n n N0 is a bounded sequene of omplex numbers Then the equation has a unique bounded solution

11 Stević Advanes in Differene Equations :377 Page 11 of 12 Proof We may assume that 37 holds The reasoning in the ase 38 is similar Choose a number q suh that q max { a,b +1/2 }, b 39 and write 1 as follows: x n+ qx n q n qx n + f n, n N 0 40 Now we introdue the following operator: Au q n 1 jn 1 εsn j q j qu j + f j q j+ 41 Assume that u l N 0 Then 41, together with some simple estimates, implies Au sup n 1 1 q εsn j q j qu j + f j q j+ 1 sup jn jn b + q u + f q 1 q 1 q + q u + f q j+ n < Hene, Al N 0 l N 0 Assume that u, v l N 0 Then, using 28and41, we have Au Av sup n 1 q jn sup n q sup Bythehoieofq it follows that ˆq : max{q a, b q} q 1 so 42anbewrittenas 1 εsn j q j qu j v j q j+ q n+j qu n+j v n+j q n+j+ q n+j q u n+j v n+j q j+1 max{q a, b q} u v 42 q 1 0, 1, Au Av ˆq u v 43 for u, v l N 0, whih means that A : l N 0 l N 0 isaontration

12 Stević Advanes in Differene Equations :377 Page 12 of 12 The Banah fixed point theorem says that the operator has a unique fixed point, say x x n l N 0, that is, Ax x,orequivalently x n q n 1 jn 1 εsn j q j qx j + f j q j+, n N 0 44 It is not diffiult to verify that 44 is a bounded solution to 1forn N 0 Competing interests The author delares that he has no ompeting interests Authors ontributions The author has ontributed solely to the writing of this paper He read and approved the manusript Publisher s Note Springer Nature remains neutral with regard to jurisditional laims in published maps and institutional affiliations Reeived: 6 Otober 2017 Aepted: 25 November 2017 Referenes 1 Agarwal, RP: Differene Equations and Inequalities: Theory, Methods, and Appliations, 2nd edn Deer, New Yor Jordan, C: Calulus of Finite Differenes Chelsea Pub Co, New Yor Levy, H, Lessman, F: Finite Differene Equations Dover, New Yor Mitrinović, DS, Kečić, JD: Methods for Calulating Finite Sums Naučna Knjiga, Beograd 1984 in Serbian 5 Papashinopoulos, G, Stefanidou, G: Asymptoti behavior of the solutions of a lass of rational differene equations Int J Differene Equ 52, Stević,S, Dibli,J, Iričanin, B, Šmarda, Z: On some solvable differene equations and systems of differene equations Abstr Appl Anal 2012,Artile ID Stević, S, Dibli, J, Iričanin, B, Šmarda, Z: On the differene equation x n+1 x n x n /x n +1 a + bx n x n Abstr Appl Anal 2012, Artile ID Stević, S, Dibli, J, Iričanin, B, Šmarda, Z: On the differene equation x n a n x n /b n + n x n 1 x n Abstr Appl Anal 2012, Artile ID Stević,S, Dibli,J, Iričanin, B, Šmarda, Z: Solvability of nonlinear differene equations of fourth order Eletron J Differ Equ 2014, Artile ID Berg, L, Stević, S: On some systems of differene equations Appl Math Comput 218, Stević,S, Iričanin, B, Šmarda, Z: On a produt-type system of differene equations of seond order solvable in losed formjinequalappl2015, Artile ID Stević,S, Iričanin, B, Šmarda, Z: Solvability of a lose to symmetri system of differene equations Eletron J Differ Equ 2016, Artile ID Stević,S, Iričanin, B, Šmarda, Z: Two-dimensional produt-type system of differene equations solvable in losed form Adv Differ Equ 2016, Artile ID Berezansy, L, Braverman, E: On impulsive Beverton-Holt differene equations and their appliations J Differ Equ Appl 109, Iričanin, B, Stević, S: Eventually onstant solutions of a rational differene equation Appl Math Comput 215, Papashinopoulos, G, Shinas, CJ: Invariants for systems of two nonlinear differene equations Differ Equ Dyn Syst 7, Papashinopoulos, G, Shinas, CJ: Invariants and osillation for systems of two nonlinear differene equations Nonlinear Anal, Theory Methods Appl 46, Papashinopoulos, G, Shinas, CJ, Stefanidou, G: On a -order system of Lyness-type differene equations Adv Differ Equ 2007, Artile ID Krehmar, VA: A Problem Boo in Algebra Mir, Mosow Stević, S: Existene of a unique bounded solution to a linear seond order differene equation and the linear first order differene equation Adv Differ Equ 2017, ArtileID Banah, S: Sur les opérations dans les ensembles abstraits et leur appliation aux équations intégrales Fundam Math 3, Stević, S: Bounded solutions to nonhomogeneous linear seond-order differene equations Symmetry 9, Artile ID Stević,S, Iričanin, B, Šmarda, Z: Note on bounded solutions to a lass of nonhomogenous linear differene equations Eletron J Differ Equ 2017, Artile ID Dibli, J, Shmeidel, E: On the existene of solutions of linear Volterra differene equations asymptotially equivalent to a given sequene Appl Math Comput 218, Drozdowiz, A, Popenda, J: Asymptoti behavior of the solutions of the seond order differene equation Pro Am Math So 991, Mitrinović, DS: Matries and Determinants Naučna Knjiga, Beograd 1989 in Serbian 27 Prosuryaov, IV: Problems in Linear Algebra Naua, Mosow 1984 in Russian