THIRD-ORDER PRODUCT-TYPE SYSTEMS OF DIFFERENCE EQUATIONS SOLVABLE IN CLOSED FORM

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1 Electronic Journal of Differential Equations, Vol. 06 (06), No. 85, pp.. ISSN: URL: or THIRD-ORDER PRODUCT-TYPE SYSTEMS OF DIFFERENCE EQUATIONS SOLVABLE IN CLOSED FORM STEVO STEVIĆ Abstract. It is shown that a class of third order product-type systems of difference equations is solvable in closed form if initial values and multipliers are complex numbers, whereas the exponents are integers, by finding the formulas for the general solution in all possible cases. The main results complement some quite recent ones in the literature. The presented class of systems is the last one for whose investigation is not needed use of some associated polynomials of degree three or more, completing the investigation of such product-type systems.. Introduction Many recent publications are devoted to the study of nonlinear difference equations and systems of difference equations; see for example []-[3], [6]-[8], []-[35]. Papaschinopoulos and Schinas essentially initiated a serious study of some classes of concrete systems of difference equations in [3, 4, 5], which was later continued by several authors in numerous other papers; see for example [3,, 6, 7, 9,, 3, 5, 6, 8, 9, 30, 3, 3, 34, 35] and the related references therein. The study of solvability of difference equations and systems, which is a classical topic [4, 5, 9, 0], has re-attracted some recent interest; see for example []-[3], [8], [5]-[8], [30]-[35], where several methods have been used. One of them is transforming the original equation, which have been used and developed in several directions; see for example [, 3, 8, 7, 30, 3, 3, 33] and the related references therein. Having studied real-valued difference equations and systems whose right-hand sides are essentially obtained by acting with translations or some operators with maximum on product-type expressions [4, 9], we started studying some systems of difference equations in the complex domain (with complex initial values and/or parameters). One of the basic classes of difference equations and systems are product-type ones. The main obstacle in studying the equations and systems on the complex domain is the fact that many complex-valued functions are not single valued. Hence, we need to pose some conditions to prevent such a situation to obtain uniquely defined solutions. Also, the transformation method or its modifications [3, 30, 3, 3, 33] cannot be directly applied to product-type systems on the complex domain. 00 Mathematics Subject Classification. 39A0, 39A45. Key words and phrases. System of difference equations; product-type system; solvable in closed form. c 06 Texas State University. Submitted September 5, 06. Published October 6, 06.

2 S. STEVIĆ EJDE-06/85 In [8] we studied the following two-dimensional class of product-type systems of difference equations z n+ = wa n z b n, w n+ = zc n wn d, n N 0, where a, b, c, d Z, z, z 0, w, w 0 C \ {0}, and showed that it is solvable in closed form (a three dimensional extension of the system was investigated in [6]). A related product-type system was studied later in [34], whereas in paper [7] appeared some product-type equations during the study of a general difference equation. Soon after the publication of [6, 8, 34] we realized that some multipliers could be added in product-type systems so that the solvability of the systems is preserved. The first system of this type was studied in our paper [5]. Quite recently we have presented another such a system in [35]. Another thing that we have realized is that there is only a few product-type systems of difference equations which are solvable in closed form, which is connected to the inability of solving the polynomial equations of the degree five or more by radicals. This means that finding all the product-type systems of difference equations which are solvable in closed form is of some interest and importance. The purpose of this paper is to continue this research, by presenting another solvable product-type system of difference equations. More precisely, we will investigate the solvability of the system z n = αz a nw b n, w n = βw c nz d n 3, n N 0, (.) where a, b, c, d Z, α, β C \ {0}, z 3, z, z, w, w C \ {0}. Cases when α = 0 or β = 0 or if some of the initial values z 3, z, z, w, w is equal to zero are quite simple or produce not well-defined solutions, which is why they are excluded from our consideration. The presented class of systems is the last one for whose investigation is not needed use of some associated polynomials of degree three or more, completing the investigation of such product-type systems.. Main results In this section we prove our main results, which concern the solvability of system (.). Essentially there are six different results, but we incorporate them all in a theorem. Some of the formulas presented in the theorem hold on set N 0, some hold on set N, or even on the set N \ {} (such a situation appears, for example, if we have an expression of the form x n and x can be equal to zero and n = ). We will not specify which formula holds on which set and leave the minor observatory problem to the reader. What is interesting is that closed form formulas for solutions to the system of difference equations, although relatively complicated, are obtained in a more or less compact form, which is a rare case (one can note that it was not the case for the equation treated in [34]). Theorem.. Consider system of difference equation (.) where a, b, c, d Z, α, β C\{0} and z 3, z, z, w, w C\{0}. Then the following statements hold. (a) If ac = bd, a + c, then the general solution to system (.) is given by z n = α c a(a+c)n a c z n+ = α c a(a+c)n a c β b (a+c)n a c β b (a+c)n+ a c w b(a+c)n z a(a+c)n, (.) w bc(a+c)n z bd(a+c)n 3 z a(a+c)n, (.)

3 EJDE-06/85 SYSTEMS OF DIFFERENCE EQUATIONS 3 w n = α d (a+c)n a c β a c(a+c)n a c w n = α d (a+c)n a c w c (a+c) n β a c(a+c)n a c z cd(a+c)n 3 z d(a+c)n, (.3) w c(a+c)n z d(a+c)n. (.4) (b) If ac = bd, a + c =, then the general solution to system (.) is given by z n = α an+ β bn w b z a, (.5) z n+ = α an+ β b(n+) w bc z bd 3z a, (.6) w n = α d(n ) β ( a)n+ w c z cd 3z d, (.7) w n = α d(n ) β ( a)n+a w c z d. (.8) (c) If ac bd, (a + c) 4(ac bd) and bd (a )(c ), then the general solution to system (.) is given by where z n+ = α (t )(t c)t w bc t z n = α (t )(t c)t n+ (t )(t c)t n+ +(t t )( c) (t )(t )(t t ) β b (t )t n+ (t )t n+ +t t (t )(t )(t t ) w b t n+ t n+ t t z (t c)t n+ (t c)t n+ t t, n+ (t )(t c)t n+ +(t t )( c) (t )(t )(t t ) β b (t )tn+ (t )t n+ +t t (t )(t )(t t ) n+ t n+ t t z bd t n+ t n+ t t 3 w n = α d (t )t n (t )tn +t t (t )(t )(t t ) β (t )(t a)t z w c (t a)t n (t a)tn t t (t c)t n+ (t c)t n+ t t, n+ (t )(t a)t n+ +(t t )( a) (t )(t )(t t ) z d (t a)t n (t a)tn t t 3 z d t n tn t t, w n = α d (t )t n (t )tn +t t (t )(t )(t t ) β (t )(t a)tn (t )(t a)tn +(t t )( a) (t )(t )(t t ) (t a)t n (t a)tn t w t z d t n tn t t, (.9) (.0) (.) (.) t, = a + c ± (a + c) 4(ac bd). (.3) (d) If ac bd, (a+c) = 4(ac bd), bd (a )(c ), then the general solution to system (.) is given by c+t n ((n+)t (n(c+)+)t +c(n+)) ( t z n = α ) w b(n+)tn z (n(t c)+t c)tn, c+t n ((n+)t (n(c+)+)t +c(n+)) ( t z n+ = α ) w bc(n+)tn z bd(n+)tn 3 z (n(t c)+t c)tn, β b (n+)t n +ntn+ ( t ) β b (n+)t n+ +(n+)t n+ ( t ) (.4) (.5)

4 4 S. STEVIĆ EJDE-06/85 where w n = α d nt n +(n )t n ( t ) w c(n(t a)+t)tn w n = α d nt n +(n )t n ( t ) a+t n ((n+)t ((+a)n+)t +a(n+)) ( t β ) z d(n(t a)+t)tn 3 β w (n(t a)+t)tn z dntn, a+t n (nt ((+a)n+ a)t +an) ( t ) z dntn, (.6) (.7) t = a + c. (e) If ac bd, (a + c) 4(ac bd), bd = (a )(c ), a + c, then the general solution to system (.) is given by z n = α z n+ = α (t c)t n+ +((c )n+c)t +( c)n+ ( t ) β b t w b t n+ t z (t c)t n+ +c t, (t c)t n+ +((c )n+c)t +( c)n+ ( t ) w bc t n+ t z bd t n+ t 3 w n = α d t n nt +n ( t ) β w c (t a)t n +a t z β b t n+ (t c)t n+ +c t, (n+)t +n ( t ) n+ (n+)t +n+ ( t ) (t a)t n+ +((a )n+a)t +( a)n+ ( t ) z d (t a)t n +a t 3 z d t n t, w n = α d t n nt +n (t a)t n ( t ) +((a )n )t +( a)n+a ( t β ) (t a)t n +a t w z d t n t, (.8) (.9) (.0) (.) where t = a + c. (f) If ac bd, (a + c) = 4(ac bd), bd = (a )(c ), and a + c =, then the general solution to system (.) is given by z n = α (n+)(( c)n+) β b n(n+) z n+ = α (n+)(( c)n+) β b (n+)(n+) w n = α d (n )n β (n+)(( a)n+) w n = α d (n )n β n(( a)n++a) w b(n+) z ( c)n+ c, (.) w bc(n+) z bd(n+) 3 z ( c)n+ c w c(( a)n+) z d(( a)n+) 3 z dn w ( a)n+ z dn, (.3), (.4). (.5) Proof. Since α, β C \ {0} and z 3, z, z, w, w C \ {0}, using (.) and induction we easily get Hence, from (.) we have z n 0 for n 3, and w n 0 for n. w b n = z n αz a n, n N 0, (.6) w b n = β b w bc nz bd n 3, n N 0. (.7)

5 EJDE-06/85 SYSTEMS OF DIFFERENCE EQUATIONS 5 From (.6) and (.7) it follows that z n+ = α c β b z a+c n zbd ac n 3, n N. (.8) Let η := α c β b, u =, a = a + c, b = bd ac. (.9) From (.8) we have z (n+)+i = η u z a n+i zb (n )+i, (.30) for n N 0 and i = 0,. From (.30) it follows that for n N and i = 0,, where Assume that for a k it holds for n k and i = 0,, where z (n+)+i = η u (ηz a (n )+i zb (n)+i )a z b (n )+i = η u+a z aa+b (n )+i zba (n)+i = η u z a (n )+i zb (n)+i, (.3) u := u + a, a := a a + b, b := b a. (.3) z (n+)+i = η u k z a k (n k+)+i zb k (n k)+i, (.33) u k := u k + a k, a k := a a k + b k, b k := b a k. (.34) Using (.30) in (.33), it follows that for n k and i = 0,, where z (n+)+i = η u k z a k (n k+)+i zb k (n k)+i = η u k (ηz a (n k)+i zb (n k )+i )a k z b k (n k)+i = η u k+a k z aa k+b k (n k)+i zba k (n k )+i = η u k+ z a k+ (n k)+i zb k+ (n k )+i, (.35) u k+ := u k + a k, a k+ := a a k + b k, b k+ := b a k. (.36) Equalities (.3), (.3), (.35), (.36) along with induction show that (.33) and (.34) hold for all k, n N such that k n +. From (.33) we have z n+i = η un z an i z bn i, for n N and i = 0,, from which along with z 0 = αz a w b, it follows that z = αz a w b 0 = αz a (βw c z d 3) b = αβ b w bc z bd 3z a, z n = η un z an 0 zbn = (α c β b ) un (αz a w b ) an z bn = α ( c)un+an β bun w ban zaan+bn = α un+ cun β bun w ban zan+ can, z n+ = η un z an zbn = (α c β b ) un (αβ b wz bc 3z bd ) a an z bn = α ( c)un+an β bun+ban w bcan = α un+ cun β bun+ w bcan zbdan 3 zaan+bn zbdan 3 zan+ can, (.37) (.38)

6 6 S. STEVIĆ EJDE-06/85 for n N. From (.34) and since u =, we have a k = a a k + b a k, k 3, (.39) k u k = + a j, k N. (.40) Case ac = bd. Since b = bd ac = 0 equation (.39) is reduced to which implies j= a k = a a k = (a + c)a k, k 3, for k N (for k =, this is directly verified). Equalities (.40) and (.4) yield so that if a + c, whereas a k = a (a + c) k = (a + c) k, (.4) k u k = + (a + c) j, k N, u k = j= (a + c)k, k N, (.4) a c u k = k, k N, (.43) if a + c =. If a + c, then from (.37), (.38), (.4), (.4) and since c a(a + c)n u n+ cu n = a c a n+ ca n = a(a + c) n we obtain formulas (.) and (.), for n. If a + c =, then from (.37), (.38), (.4), (.43) and since u n+ cu n = ( c)n + = an + we obtain formulas (.5) and (.6), for n N. Case ac bd. Let t, be the roots of the characteristic polynomial P (t) = t (a + c)t + ac bd, (.44) associated with difference equation (.39). Note that they are given by the formulas in (.3). We have a n = c t n + c t n, n N, where c, c R, if (a + c) 4(ac bd), whereas u n = (d n + d )t n, n N, where d, d R, if (a + c) = 4(ac bd). Since a = t + t and a = (t + t ) t t = t + t t + t, it is easily obtained that a k = tk+ t k+, k N, (.45) t t

7 EJDE-06/85 SYSTEMS OF DIFFERENCE EQUATIONS 7 if (a + c) 4(ac bd), whereas if (a + c) = 4(ac bd). From (.40) and (.45) we have k u k = + j= a k = (k + )t k, k N, (.46) t j+ t j+ = (t )t k+ t t for k N, if (a + c) 4(ac bd) and bd (a )(c ). From (.40) and (.46) it follows that (t )t k+ + t t, (.47) (t )(t )(t t ) k u k = + (j + )t j = (k + )tk + kt k+ ( t ), (.48) j= for k N, if (a + c) = 4(ac bd) and bd (a )(c ). If (a + c) 4(ac bd), bd = (a )(c ) and a + c, then polynomial (.44) has exactly one zero equal to one, say t. From (.40) and (.45) we have k u k = + j= t j+ t = tk+ (k + )t + k (t ), (.49) for k N. Finally, if (a + c) = 4(ac bd), bd = (a )(c ) and a + c =, then both zeros of polynomial (.44) are equal to one. From (.40) and (.46) it follows that k u k = + (j + ) = j= k(k + ), (.50) for k N. If (a + c) 4(ac bd) and bd (a )(c ), then from (.37), (.38), (.45), (.47) and since u n+ cu n = (t )(t c)t n+ (t )(t c)t n+ + (t t )( c) (t )(t )(t t ) a n+ ca n = (t c)t n+ (t c)t n+, t t we obtain formulas (.9) and (.0). If (a + c) = 4(ac bd) and bd (a )(c ), then from (.37), (.38), (.46), (.48) and since u n+ cu n = c + tn ((n + )t ((c + )n + )t + c(n + )) ( t ) a n+ ca n = ((t c)n + t c)t n, we obtain formulas (.4) and (.5). If (a + c) 4(ac bd), bd = (a )(c ) and a + c, then from (.37), (.38), (.45) with t =, (.49) and since u n+ cu n = (t c)t n+ + ((c )n + c )t + ( c)n + ( t )

8 8 S. STEVIĆ EJDE-06/85 a n+ ca n = (t c)t n+ + c, t we obtain formulas (.8) and (.9), where t = ac bd = a + c. If (a + c) = 4(ac bd), bd = (a )(c ) and a + c =, then from (.37), (.38), (.46) with t =, (.50) and since we obtain formulas (.) and (.3). From (.) we also have (n + )(( c)n + ) u n+ cu n = a n+ ca n = ( c)n + c, z d n 3 = w n βw c n Thus, from (.5) and (.5) it follows that, n N 0, (.5) z d n = α d z ad nw bd n, n N 0. (.5) w n+3 = α d β a w a+c n+ wbd ac n, n N 0. (.53) Note that difference equations (.8) and (.53) have only different constant multipliers. Let ν := α d β a, û =, â = a + c, ˆb = bd ac. (.54) As above it is proved that for any k N it holds w (n+)+i = for n k and i =, 0, where νûk wâk (n k+)+i wˆb k (n k)+i, (.55) û k := û k + â k, â k := â â k + ˆb k, ˆbk = ˆb â k. (.56) Since initial conditions (.54) and system (.56) are the same as those in (.9) and (.34), it follows that for every k N. From (.55) we have â k = a k, ˆbk = b k, û k = u k, (.57) w n+i = ν un w an +i w bn i, for n and i =, 0, from which along with it follows that w n = ν un w an w bn 0 w 0 = βw c z d 3, w = βw c z d, w = βw c 0z d = β(βw c z d 3) c z d = β +c w c z cd 3z d, = (α d β a ) un (β +c w c z cd 3z d ) an (βw c z d 3) bn = α dun β ( a)un +(+c)an +bn w c a n +cb n = α dun β un+ aun w c(an aan ) z d(an aan ) 3 z dan, z cdan +dbn 3 z dan (.58)

9 EJDE-06/85 SYSTEMS OF DIFFERENCE EQUATIONS 9 w n = ν un w an w bn = (α d β a ) un (βw c z d ) an w bn = α dun β ( a)un +an w can +bn z dan (.59) = α dun β un aun w an aan z dan, for n. Case ac = bd. If a + c, then from (.4), (.4), (.58), (.59) and since a c(a + c)n u n au n = a c a n aa n = c(a + c) n we obtain formulas (.3) and (.4). If a + c =, then from (.4) with a + c =, (.43), (.58), (.59) and since we obtain formulas (.7) and (.8). u n au n = ( a)n + a Case ac bd. If ac bd, (a + c) 4(ac bd) and bd (a )(c ), then from (.45), (.47), (.58), (.59) and since u n au n = (t )(t a)t n (t )(t a)t n + (t t )( a) (t )(t )(t t ) a n aa n = (t a)t n (t a)t n t t we obtain formulas (.) and (.). If ac bd, (a + c) = 4(ac bd), bd (a )(c ), then from (.46), (.48), (.58), (.59) and since u n au n = a + tn (nt (( + a)n + a)t + an) ( t ) a n aa n = (n(t a) + t )t n we obtain formulas (.6) and (.7). If ac bd, (a + c) 4(ac bd), bd = (a )(c ) and a + c, then from (.45) with t =, (.49), (.58), (.59) and since u n au n = (t a)t n + ((a )n )t + ( a)n + a ( t ) a n aa n = (t a)t n + a t we obtain formulas (.0) and (.), where t = ac bd = a + c. If ac bd, (a + c) = 4(ac bd), bd = (a )(c ) and a + c =, then from (.46) with t =, (.50), (.58), (.59) and since we obtain formulas (.4) and (.5). n(( a)n + + a) u n au n = a n aa n = ( a)n +

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11 EJDE-06/85 SYSTEMS OF DIFFERENCE EQUATIONS [7] S. Stević; Solvable subclasses of a class of nonlinear second-order difference equations, Adv. Nonlinear Anal., 5 () (06), [8] S. Stević, M. A. Alghamdi, A. Alotaibi, E. M. Elsayed; Solvable product-type system of difference equations of second order, Electron. J. Differential Equations, Vol. 05, Article No. 69, (05), 0 pages. [9] S. Stević, M. A. Alghamdi, A. Alotaibi, N. Shahzad; Boundedness character of a max-type system of difference equations of second order, Electron. J. Qual. Theory Differ. Equ., Vol. 04, Atricle No. 45, (04), pages. [30] S. Stević, J. Diblik, B. Iričanin, Z. Šmarda; On a third-order system of difference equations with variable coefficients, Abstr. Appl. Anal. Vol. 0, Article ID 50853, (0), pages. [3] S. Stević, J. Diblik, B. Iričanin, Z. Šmarda; On some solvable difference equations and systems of difference equations, Abstr. Appl. Anal., Vol. 0, Article ID 5476, (0), pages. [3] S. Stević, J. Diblik, B. Iričanin, Z. Šmarda; On a solvable system of rational difference equations, J. Difference Equ. Appl. 0 (5-6) (04), [33] S. Stević, J. Diblik, B. Iričanin, Z. Šmarda; Solvability of nonlinear difference equations of fourth order, Electron. J. Differential Equations Vol. 04, Article No. 64, (04), 4 pages. [34] S. Stević, B. Iričanin, Z. Šmarda; On a product-type system of difference equations of second order solvable in closed form, J. Inequal. Appl., Vol. 05, Article No. 37, (05), 5 pages. [35] S. Stević, B. Iričanin, Z. Šmarda; Solvability of a close to symmetric system of difference equations, Electron. J. Differential Equations Vol. 06, Article No. 59, (06), 3 pages. Stevo Stević Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 000 Beograd, Serbia. Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 8003, Jeddah 589, Saudi Arabia address: sstevic@ptt.rs