Note on difference equations with the right-hand side function nonincreasing in each variable

Stevi´ cetal.Journal of Inequalities and Applications (2022) 2022:25 https://doi.org/10.1186/s13660-022-02761-9 RESEARCH Open Access Note on difference equations with the right-hand side function nonincreasing in each variable Stevo Stevi´ c1,2*, Bratislav Iriˇ canin3,4, Witold Kosmala5and Zdenˇ ek Šmarda6 *Correspondence: [email protected] 1Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia 2Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China Full list of author information is available at the end of the article Abstract We present an example of a difference equation of arbitrary order, possessing the right-hand side function that is homogeneous to a certain degree and nonincreasing in each variable, which has a unique positive equilibrium, as well as solutions that do not converge to the equilibrium. The example shows that the main result in the paper: O. Moaaz, Dynamics of difference equation xn+1 =f(xn–l,xn–k)(Adv.Differ.Equ. 2018:447, 2018), is incorrect. MSC: 39A10; 39A22 Keywords: Difference equation; Unbounded solutions; Nondecreasing function; Homogeneous function 1 Introduction Let, as usual, Nbe the set of positive integers, Zthe set of integers, Rthe set of reals, and Cthe set of the complex numbers. By Nk,wherek∈Z,wedenotethesetofallj∈Zsuch that j≥k.Ifp,q∈Zand such that p≤q, then the notation j=p,qmeans that jtakes the values of all integers between pand q(including pand q). Difference equations and systems of difference equations have been studied analytically for more than three hundred years. The first important results were obtained by de Moivre [3,4] and D. Bernoulli [2]. For some classical results, see, for example, [5,7– 13,15].Allthesereferencesaretosomeextentdevotedtofindingclosed-formformulasfor solutions to linear or nonlinear difference equations and systems of difference equations. There has been some renewed interest in the solvability of difference equations and systems of difference equations, their invariants, and applications of obtained closed-form formulas for their solutions and/or invariants (see, for example, [1,16–28]andrelated references therein). It is a common fact that many solvable difference equations and systems are transformed by some suitable changes of variables to some solvable linear ones [1,19,22,23,25–28]. Ofthemanyclassesofdifferenceequationsandsystemsofdifferenceequationssolvable inclosedform,herewementionanimportantclass,whichisusedinthisnote.Inaddition, wealsomentionasimplemethodforgettingasequenceofdifferenceequationsorsystems ©The Author(s) 2022. 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Stevi´ cetal.Journal of Inequalities and Applications (2022) 2022:25 Page 2 of 7 of difference equations from a given one, which can sometimes be used to obtain some counterexamples in the theory of difference equations. 1.1 Product-type difference equations Somerecentinvestigationsonsolvabilityaredevotedtoproduct-typedifferenceequations andsystemsofdifferenceequations,ortosomedifferenceequationsandsystemsthatcan bereducedtothemusingsomesuitabletransformations(see,e.g.,[22–24]andtherelated references therein). General product-type difference equation is xn+k=anxα(k–1) n n+k–1xα(k–2) n n+k–2 ···xα(0) n n,n∈N0,(1) where the sequences (an)n∈N0,(α(j) n)n∈N0,j=0,k–1,aswellastheinitialvaluesxj,j= 0,k–1, are real or complex. In some cases, equation (1)canbesolvedbytakingthelogarithm,butinthecaseof complex initial values xj,j=0,k–1, coefficients and exponents, some other methods can beused(see,e.g.,[22–24] and the related reference therein). Since the equation is related to the general linear difference equation, it is of great importance. 1.2 Difference equations with interlacing indices There is a simple method enabling to construct a family of ‘cloned’ difference equations fromagivenone.Theequationsarecalledthedifferenceequationswithinterlacingindices [27]. The equations appear from time to time in the literature, and it seems that some authorsarenotawarethattheyareobtainedbycloningsomesimplerdifferenceequations (see the examples and analyses conducted in [25–28]). However, the equations can be useful in providing some counterexamples in the theory of difference equations (see, e.g., [6]). Systems of difference equations with interlacing indices can also be constructed by the cloning method. Now we briefly describe the method. The general form of the difference equation with interlacing indices is the following yn=h(yn–k,yn–2k,...,yn–lk), n∈N0,(2) where l∈N,andk≥2. If we define ksets of indices by Il j:=(s–l)k+j:s∈N0,j=0,k–1, we obviously have Il i∩Il j=∅,i=j, and k–1  j=0 Il j=N–lk. Stevi´ cetal.Journal of Inequalities and Applications (2022) 2022:25 Page 3 of 7 Let y(j) s:=y(s–l)k+j,s∈N0,j=0,k–1, then from (2), we have y(j) s=hy(j) s–1,y(j) s–2,...,y(j) s–l,s∈N0, for j=0,k–1. This means that the sequences (y(j) s)s∈N0,j=0,k–1areksolutions to the equation zn=h(zn–1,zn–2,...,zn–l), n∈N0,(3) with unrelated initial values. Usingthisprocedureinthereversedirection,fromequation(3),onecanobtainafamily of cloned equations in (2), i.e., a family of difference equations with interlacing indices. 1.3 Global convergence results and a claim One of the main problems in the theory of difference equations is the convergence of the solutions to the equations. There are many results on convergence in the literature, some of which can be found in the literature mentioned above. The following difference equation xn+1 =f(xn–k,xn–m), n∈N0,(4) where k,m∈N0, has been recently studied in [14]. Thefollowing claim isthemain result concerningthedifferenceequationsappearingin [14] (Theorem 3.3 therein). TheoremA Assume that f has non-positive partial derivatives and is homogeneous with degree s.Then equation (4)has a unique positive equilibrium x∗,and every solution to the equation converges to x∗. In this note, we show that the claim in Theorem Ais not true by giving an example of equation (4) such that the function fis homogeneous (with a degree to be chosen appropriately) and has non-positive partial derivatives, and equation (4) has a unique positive equilibrium x∗and solutions that do not converge to the equilibrium. 2 A counterexample to Theorem A Inthissection,wegiveacounterexampletoTheoremA.Toconstructthecounterexample, we are looking for a difference equation belonging to the above classes of equations; that is, we are looking for a product-type difference equation with interlacing indices that is solvable in the closed form. Example1 Consider the difference equation xn+k=1 xα n,n∈N0,(5) Stevi´ cetal.Journal of Inequalities and Applications (2022) 2022:25 Page 4 of 7 wherek∈N,andα> 1,whichisaspecialcaseofthegeneraldifferenceequationofhigher order xn+k=f(xn+k–1,...,xn), n∈N0,(6) where k∈N,andfis an arbitrary function. Here we are interested in positive solutions to equation (5). Hence, we assume that xj∈(0,+∞), j=0,k–1. For the case of equation (5), we have that f(t1,...,tk)= 1 tα k(7) is the right-hand side function, which generates their solutions along with the initial values. Since ∂f ∂tj(t1,...,tk)=0, j=1,k–1 and ∂f ∂tk(t1,...,tk)=– α tα+1 k<0, when tk> 0, we have that the function (7) has non-positive partial derivatives on the set (0,+∞)k, from which we choose our initial values. On the other hand, since f(λt1,...,λtk)= 1 (λtk)α=1 λα 1 tα k=λ–αf(t1,...,tk) holds for every λ> 0, we see that the function defined in (7) is homogeneous with degree –α. Further, if xn≡x∗,n∈N0 is an equilibrium solution to equation (5), then it must be x∗=1 (x∗)α, fromwhichitimmediatelyfollowsthatx∗=1,whichmeansthatequation(5)hasaunique positive equilibrium. Stevi´ cetal.Journal of Inequalities and Applications (2022) 2022:25 Page 5 of 7 Nownotethatequation(5) is a difference equation with interlacing indices, which is obtained by cloning the following product-type difference equation of first order yn+1 =1 yα n,n∈N0,(8) that is, equation (5)consistsofkcopies of equation (8) with initial values not connected to each other. Equation(8) obviously has the same equilibrium. Hence, to show that the claim of Theorem Ais not true, it is enough to show that equation (8) has solutions, which do not converge to the equilibrium. Now note that by iterating equation (8), we get y2n+2 =yα2 2n(9) and y2n+3 =yα2 2n+1, (10) for n∈N0. Byasimpleinductiveargumentfromrelations(9),(10),andequation(8)withn=1,one can easily obtain y2n=yα2n 0(11) and y2n+1 =yα2n 1=1 yα2n+1 0, (12) for n∈N0. Taking α>1,wehave lim n→+∞α2n=lim n→+∞α2n+1 =+∞. (13) Hence, if y0∈(0,1), then letting n→+∞in (11)and(12)andusing(13), it follows that lim n→+∞y2n=0 (14) and lim n→+∞y2n+1 =+∞. (15) Besides this, if y0> 1, then letting n→+∞in (11)and(12)andusing(13), we obtain lim n→+∞y2n=+∞(16) Stevi´ cetal.Journal of Inequalities and Applications (2022) 2022:25 Page 6 of 7 and lim n→+∞y2n+1 =0. (17) From the relations (14)–(17), we see that if any initial value of equation (8)belongsto the set (0,+∞)\{1}, the solution is unbounded. This fact implies that if any initial value of equation (5) belongs to the set (0,+∞)\{1}, the solution is unbounded. The above analysis shows that the claim of Theorem Ais not true. Remark 1 Note that the only bounded positive solution to equation (5) is the one which is generated by the initial values x0=x1=···=xk–1 =1. (18) The solution is, in fact, the equilibrium solution xn≡1, n∈N0,whichiseasilyproved using the initial values (18)inequation(5), together with a simple inductive argument. Remark2 Theaboveconsiderationalsoshowsthattheequilibriumsolutionistheunique positive solution to equation (5) that converges, which indicates to what extent the claim in Theorem Afails. Remark 3 Note also that the linearized equation associated with equation (8)aboutthe equilibriumpointis zn+1 =–αzn,n∈N0, (19) from which, along with the assumption α>1, it follows that the equilibrium is a repeller. Note that from (19), we have |zn|=αn|z0|,n∈N0, from which together with (13), we see that for each z0=0,thesolutionto(19) goes to infinity as n→+∞. Giventhatequation(8) is solvableand hasaclosed-formformulaforits solutions,from whichtheirlong-termbehavioriseasilydescribed,linearizationisnotrequired.However, the linearization argument also suggests in which direction counterexamples should be sought. Acknowledgements The work of Zdenˇ ek Šmarda was supported by the project FEKT-S-20-6225 of Brno University of Technology. Funding Brno University of Technology, project FEKT-S-20-6225. Availability of data and materials Not applicable. Declarations Competing interests The authors declare that they have no competing interests. Authors’ contributions SS investigated possibilities for finding a counterexample to Theorem A. BI, WK and ZŠ analyzed some original ideas in order to find as much as possible simpler counterexample to the theorem. All authors read and approved the final manuscript. Stevi´ cetal.Journal of Inequalities and Applications (2022) 2022:25 Page 7 of 7 Author details 1Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia. 2Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China. 3Faculty of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, 11000 Beograd, Serbia. 4Faculty of Mechanical and Civil Engineering in Kraljevo, University of Kragujevac, Kraljevo, Serbia. 5Department of Mathematical Sciences, Appalachian State University, Boone, NC 28608, USA. 6Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technicka 3058/10, CZ-616 00 Brno, Czech Republic. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 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