Product-type system of difference equations of second-order solvable in closed form

Product-type system of difference equations of second-order solvable in closed form

Stevo Stevi´c

1Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia

2Operator Theory and Applications Research Group, Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 9 June 2015, appeared 12 September 2015 Communicated by Jeff R. L. Webb

Abstract. This paper presents solutions to the following product-type second-order system of difference equations

xn+1= ^y

na

z^b_n−1, yn+1= ^z

cn

x^d_n−1, zn+1= ^x

f n

y^g_n−1, n∈N0,

wherea,b,c,d,f,g∈_{Z, and}x−i,y−i,z−i ∈_C\ {0},i∈ {0, 1}, in closed form.

Keywords: solvable system of difference equations, second-order system, product-type system, complex initial values.

2010 Mathematics Subject Classification: 39A10, 39A20.

1 Introduction

Recently there has been some renewed interest in solving difference equations and systems of difference equations and their applications (see, e.g., [1–4,7,8,14,18,20,23–38,41–45]), especially after the publication of note [18] in which a method for solving a nonlinear difference equation of second-order was presented. Since the end of the 1990s there has been also some interest in concrete systems of difference equations (see, e.g., [8–13,15–17,23,24,26–28,30–40,42,43,45]).

In the line of our investigations [5,6,19,21,22] (see also the references therein) we studied the long-term behavior of several classes of difference equations related to the product-type ones.

Somewhat later we studied some systems which are extensions of these equations [39,40].

Long-term behavior of positive solutions to the following system x_n+1=max

c, y_n^p

z_n^p₋₁

, y_n+1 =max

c, z_n^p x_n^p₋₁

, z_n+1=max

c, x_n^p y_n^p₋₁

, (1.1)

BEmail: sstevic@ptt.rs

n ∈ _N0, with positive parameters cand p, was investigated in [40]. Note that system (1.1) is obtained by the action of the max-type operatormc(s) =max{c,s}on the right-hand side of the following product-type system of difference equations

x_n+1 = ^y

p n

z_n^p₋₁, y_n+1= ^z

p n

x_n^p₋₁, z_n+1 = ^x

p n

y_n^p₋₁, n∈ _N0. (1.2) An interesting feature of system (1.2) is that it can be solved in closed form in the case of positive initial values. Namely, a simple inductive argument shows that in this case

min{xn,yn,zn}>_{0, for everyn}≥ −1.

Hence, it is legitimate to take the logarithm of all the three equations in (1.2) and by using the change of variables

un=lnxn, vn=lnyn, wn =lnzn, n≥ −1, (1.3) the system is transformed into the following linear one

u_n+1= pvn−pw_n−1

v_n+₁= pwn−pu_n−1 (1.4)

w_n+1= pu_n−pv_n−1, n∈_N0.

Using the third equation of (1.4) in the first and second ones we obtain the following system of difference equations

u_n+1 = pvn−p²un−2+p²vn−3 (1.5) vn+₁ = (p²−p)un−1−p²vn−2, n∈_N0. (1.6) Using (1.6) in (1.5) we get

vn+3−(p³−3p²)vn+p³vn−3 =0, n∈_N0. (1.7) This is a linear difference equation which can be solved in closed form, from which along with (1.6) and the third equation in (1.4) closed form formulas for u_n,v_nandw_nare obtained, and consequently by using (1.3) we obtain formulas for xn, yn andzn. We leave the details to the reader as a simple exercise.

A natural question is whether system (1.2) can be solved in closed form if initial values x−i,y−i,z−i,i∈ {0, 1}, are complex numbers and for which values of parameter p.

Motivated by all above mentioned and by our recent paper [37], here we will study the solvability of the following system of difference equations

x_n+1 = ^y

an

z^b_n−1, y_n+1= ^z

cn

x^d_n−1, z_n+1 = ^x

nf

y_n^g₋₁, n∈ _{N0, (1.8)} wherea,b,c,d,f,g ∈Z, and when initial valuesx−i,y−i,z−i,i∈ {0, 1}, are complex numbers different from zero (it is easy to see that a solution to the system is well-defined if and only if all initial values are different from zero). We present a constructive method for solving the system. Conditiona,b,c,d, f,g∈_Zis naturally posed, in order not to deal with multi-valued sequences.

2 Main result

Here we present our main result in this paper.

Theorem 2.1. Consider system(1.8) with a,b,c,d,f,g ∈ _{Z. If x−i},y−i,z−i ∈ _C\ {0}, i∈ {0, 1}, then the system is solvable in closed form.

Proof. Let

a₁ =a, b₁ =b, c₁ =c, d₁= d, f₁= f, g₁ =g. (2.1) By using the equations in (1.8) we obtain

x_n+1= ^y

a1

n

z^b_n¹₋₁ = ^z

ca₁−b₁ n−1

x^da_n−¹₂ = ^z

a₂ n−1

x_n^b2₋₂, (2.2)

y_n+1= ^z

c₁ n

x^d_n¹₋₁ = ^x

f c1−d1

n−1

y^gc_n−¹₂ = ^x

c2

n−1

y^d_n²₋₂, (2.3)

z_n+1= ^x

f1

n

y^g_n¹₋₁ = ^y

a f₁−g₁ n−1

z^{b f}_n−¹₂ = ^y

f2

n−1

z^g_n²₋₂, (2.4)

where we definea₂,b₂,c₂,d₂, f₂andg₂as follows

a₂ := ca₁−b₁, b₂ := da₁, c₂:= f c₁−d₁, d₂ := gc₁, f₂:=a f₁−g₁, g₂ :=b f₁. By using (2.2), (2.3), (2.4) and the equations in (1.8), it follows that

x_n+1= ^z

a2

n−1

x^b_n²₋₂ = ^x

f a₂−b₂ n−2

y^ga_n−²₃ = ^x

a3

n−2

y^b_n³₋₃, (2.5)

y_n+1= ^x

c₂ n−1

y^d_n²₋₂ = ^y

ac2−d2

n−2

z^bc_n−²₃ = ^y

c₃ n−2

z^d_n³₋₃, (2.6)

z_n+1= ^y

f₂ n−1

z_n^g2₋₂ = ^z

c f2−g2

n−2

x^{d f}_n−²₃

= ^z

f3

n−2

x^g_n³₋₃, (2.7)

where we definea₃,b₃,c₃,d₃, f₃andg₃as follows

a₃ := f a₂−b₂, b₃:=ga₂, c₃:=ac₂−d₂, d₃ := bc₂, f₃:=c f₂−g₂, g₃:=d f₂. By using (2.5), (2.6), (2.7) and the equations in (1.8), we further get

x_n+1= ^x

a3

n−2

y^b_n³₋₃ = ^y

aa3−b3

n−3

z^ba_n−³₄ = ^y

a4

n−3

z^b_n⁴₋₄, (2.8)

y_n+1= ^y

c3

n−2

z^d_n³₋₃ = ^z

cc3−d3

n−3

x^dc_n−³₄ = ^z

c4

n−3

x^d_n⁴₋₄, (2.9)

z_n+1= ^z

f3

n−2

x^g_n³₋₃ = ^x

f f3−g3

n−3

y^{g f}_n−³₄

= ^x

f4

n−3

y^g_n⁴₋₄, (2.10)

where we definea₄,b₄,c₄,d₄, f₄ andg₄as follows

a₄:=aa₃−b₃, b₄:=ba₃, c₄:=cc₃−d₃, d₄:=dc₃, f₄:= f f₃−g₃, g₄ := g f₃. Let

xn+₁ = ^z

a3k−1

n−3k+2

x^b_n^3k₋⁻_3k¹₊₁, yn+₁= ^x

c3k−1

n−3k+2

y^d_n^3k₋⁻_3k¹₊₁, zn+₁= ^y

f3k−1

n−3k+2

z^g_n^3k₋⁻_3k¹₊₁, where

a_3k−1:=ca_3k−2−b_3k−2, b_3k−1 :=da_3k−2, c_3k−1:= f c_3k−2−d_3k−2, d_3k−1:=gc_3k−2, f_3k−1 := a f_3k−2−g_3k−2, g_3k−1:=b f_3k−2,

xn+₁= ^x

a_3k n−3k+1

y^b_n^3k_−3k , yn+₁ = ^y

c_3k n−3k+1

z^d_n^3k_−3k , zn+₁ = ^z

f3k

n−3k+1

x^g_n^3k_−3k , where

a_3k := f a_3k−1−b_3k−1, b_3k := ga_3k−1, c_3k := ac_3k−1−d_3k−1, d_3k :=bc_3k−1, f_3k := c f_3k−1−g_3k−1, g_3k :=d f_3k−1,

x_n+1 = ^y

a_3k+1

n−3k

z^b_n^3k₋⁺_3k¹₋₁, y_n+1 = ^z

c_3k+1

n−3k

x_n^d3k₋⁺_3k¹₋₁, z_n+1 = ^x

f_3k+1

n−3k

y_n^g3k₋⁺_3k¹₋₁ (2.11) where

a_3k+1:=aa_3k−b_3k, b_3k+1 :=ba_3k, c_3k+1:=cc_3k−d_3k, d_3k+1:=dc_3k, f_3k+1 := f f_3k−g_3k, g_3k+1:=g f_3k, for somek ∈_Nsuch thatn≥3k.

From the relations in (2.11) and by using the equations in (1.8) we obtain

xn+₁ = ^y

a3k+1

n−3k

z^b_n^3k₋⁺_3k¹₋₁ = ^z

ca3k+1−b3k+1

n−3k−1

x^da_n−^3k_3k⁺¹₋₂ = ^z

a3k+2

n−3k−1

x^b_n^3k₋⁺_3k²₋₂, (2.12) y_n+1 = ^z

c_3k+1

n−3k

x_n^d3k₋⁺_3k¹₋₁ = ^x

f c3k+1−d3k+1

n−3k−1

y^gc_n−^3k_3k⁺¹₋₂ = ^x

c_3k+2

n−3k−1

y^d_n^3k₋⁺_3k²₋₂, (2.13) z_n+1 = ^x

f_3k+1

n−3k

y^g_n^3k₋⁺_3k¹₋₁ = ^y

a f_3k+1−g_3k+1

n−3k−1

z^{b f}_n−^3k_3k⁺¹₋₂

= ^y

f_3k+2

n−3k−1

z^g_n^3k₋⁺_3k²₋₂, (2.14) where we definea_3k+₂,b_3k+₂,c_3k+₂,d_3k+₂, f_3k+₂ andg_3k+₂as follows

a_3k+2:=ca_3k+1−b_3k+1, b_3k+2 :=da_3k+1, c_3k+2:= f c_3k+1−d_3k+1, d_3k+2:=gc_3k+1, f_3k+2 := a f_3k+1−g_3k+1, g_3k+2:=b f_3k+1.

By using (2.12), (2.13), (2.14) and the equations in (1.8), it follows that x_n+1 =^z

a_3k+2

n−3k−1

x^b_n^3k₋⁺_3k²₋₂

= ^x

f a_3k+2−b_3k+2

n−3k−2

y^ga_n−^3k_3k⁺²₋₃ = ^x

a_3k+3

n−3k−2

y^b_n^3k₋⁺_3k³₋₃, (2.15) y_n+1 =^x

c_3k+2

n−3k−1

y^d_n^3k₋⁺_3k²₋₂

= ^y

ac_3k+2−d_3k+2

n−3k−2

z^bc_n−^3k_3k⁺²₋₃

= ^y

c_3k+3

n−3k−2

z^d_n^3k₋⁺_3k³₋₃, (2.16) z_n+1 =^y

f_3k+2

n−3k−1

z^g_n^3k₋⁺_3k²₋₂ = ^z

c f_3k+2−g_3k+2

n−3k−2

x^{d f}_n−^3k_3k⁺²₋₃

= ^z

f_3k+3

n−3k−2

x^g_n^3k₋⁺_3k³₋₃, (2.17) where we definea_3k+3,b_3k+3,c_3k+3,d_3k+3, f_3k+3andg_3k+3 as follows

a_3k+3:= f a_3k+2−b_3k+2, b_3k+3:= ga_3k+2, c_3k+3:=ac_3k+2−d_3k+2, d_3k+3:=bc_3k+2, f_3k+3:=c f_3k+2−g_3k+2, g_3k+3:=d f_3k+2.

By using (2.15), (2.16), (2.17) and the equations in (1.8) we further get x_n+1= ^x

a_3k+3

n−3k−2

y^b_n^3k₋⁺_3k³₋₃

= ^y

aa_3k+3−b_3k+3

n−3k−3

z^ba_n−^3k_3k⁺³₋₄

= ^y

a_3k+4

n−3k−3

z^b_n^3k₋⁺_3k⁴₋₄, (2.18) y_n+1= ^y

c3k+3

n−3k−2

z^d_n^3k₋⁺_3k³₋₃

= ^z

cc3k+3−d3k+3

n−3k−3

x^dc_n−^3k_3k⁺³₋₄

= ^z

c3k+4

n−3k−3

x_n^d3k₋⁺_3k⁴₋₄, (2.19) z_n+1= ^z

f3k+3

n−3k−2

x^g_n^3k₋⁺_3k³₋₃ = ^x

f f3k+3−g3k+3

n−3k−3

y^{g f}_n−^3k_3k⁺³₋₄

= ^x

f3k+4

n−3k−3

y^g_n^3k₋⁺_3k⁴₋₄, (2.20) where we definea_3k+4,b_3k+4,c_3k+4,d_3k+4, f_3k+4andg_3k+4 as follows

a_3k+₄:=aa_3k+₃−b_3k+₃, b_3k+₄:=ba_3k+₃, c_3k+₄:=cc_3k+₃−d_3k+₃, d_3k+4:=dc_3k+3, f_3k+4:= f f_3k+3−g_3k+3, g_3k+4:=g f_3k+3.

Hence, this inductive argument shows that sequences(an)_n∈N,(bn)_n∈N,(cn)_n∈N,(dn)_n∈N, (f_n)_n∈N, (g_n)_n∈N, satisfy the following recurrent relations

a_3k+2=ca_3k+1−b_3k+1, a_3k+3 = f a_3k+2−b_3k+2, a_3k+4= aa_3k+3−b_3k+3, (2.21) b_3k+2=da_3k+1, b_3k+3 =ga_3k+2, b_3k+4=ba_3k+3, (2.22) c_3k+2= f c_3k+1−d_3k+1, c_3k+3 =ac_3k+2−d_3k+2, c_3k+4=cc_3k+3−d_3k+3, (2.23) d_3k+2= gc_3k+1, d_3k+3 =bc_3k+2, d_3k+4=dc_3k+3, (2.24) f_3k+2= a f_3k+1−g_3k+1, f_3k+3 =c f_3k+2−g_3k+2, f_3k+4= f f_3k+3−g_3k+3, (2.25) g_3k+2=b f_3k+1, g_3k+3 =d f_3k+2, g_3k+4= g f_3k+3, (2.26) fork∈ _N0.

From (2.12)–(2.20) we easily obtain x_3n+1 = ^y

a3n+1

0

z^b₋³ⁿ₁⁺¹, x_3n+2 = ^z

a3n+2

0

x^b₋³ⁿ₁⁺², x_3n+3 = ^x

a3n+3

0

y^b₋³ⁿ₁⁺³, (2.27) y_3n+1 = ^z

c3n+1

0

x^d₋³ⁿ₁⁺¹, y_3n+2 = ^x

c3n+2

0

y^d₋³ⁿ₁⁺², y_3n+3 = ^y

c3n+3

0

z^d₋³ⁿ₁⁺³, (2.28) z_3n+1 = ^x

f3n+1

0

y₋^g3n₁⁺¹, z_3n+2 = ^y

f3n+2

0

z₋^g3n₁⁺², z_3n+3 = ^z

f3n+3

0

x₋^g3n₁⁺³, n ∈_N0. (2.29)

From (2.21) and (2.22) we have that

a_3k+2 =ca_3k+1−ba_3k, a_3k+3= f a_3k+2−da_3k+1, a_3k+4= aa_3k+3−ga_3k+2,

k∈ _N0, (here we regard thata₀ = _{1 if}b6= 0, due to the relationb_3k+4 =ba_3k+3 with k= −_1).

Using the first equation in the second and third ones it follows that

a_3k+3−(c f−d)a_3k+1+b f a_3k =0, k∈ _N0, (2.30) and

a_3k+4−aa_3k+3+cga_3k+1−bga_3k =0, k∈_N0. (2.31) Using (2.30) into (2.31) it follows that

a_3k+6+ (ad+b f+cg−ac f)a_3k+3+bdga_3k =0, k∈_N0. (2.32) Relation (2.32) means that the sequence(a_3k)_k∈N0 annihilate the linear operator

L(x_n) =x_n+2+ (ad+b f +cg−ac f)x_n+1+bdgx_n.

From this along with (2.30) it follows that sequence (a_3k+1)_k∈N0 annihilate the operator too.

Using these facts and the relationa_3k+2=ca_3k+1−ba_3k, it follows that that sequence(a_3k+2)_k∈N0 also annihilate the operator. This along with the relations in (2.22) implies that sequences (b_3k+i)_k∈N0, i = 0, 1, 2, annihilate the operator too. Since sequences (c_3k+2+i)_k∈N0 and (d_3k+2+i)_k∈N0, that is, (f_3k+1+i)_k∈N0 and(g_3k+1+i)_k∈N0,i= 0, 1, 2, satisfy the relations in (2.21) and (2.22) it follows that they also annihilate the operator.

Case b=0. In this case we have that sequences(a_3k+i)_k∈N0,(c_3k+i)_k∈N0,(f_3k+i)_k∈N0, satisfy the recurrent relation

x_n+1 = (ac f −ad−cg)x_n, n∈_N.

From this and since a₁ = a, c₁ = c, f₁ = f, a₂ = ac−b, c₂ = c f −d, f₂ = a f −g, a₃ = ac f −b f −ad, c₃ = ac f −ad−cg, f₃ = ac f −b f −cg, and by using the condition b = 0, it follows that

a_3k+1= a(ac f −ad−cg)^k, (2.33)

a_3k+₂= ac(ac f −ad−cg)^k, (2.34)

a_3k+3= a(c f−d)(ac f −ad−cg)^k, (2.35)

c_3k+1= c(ac f −ad−cg)^k, (2.36)

c_3k+2= (c f−d)(ac f −ad−cg)^k, (2.37)

c_3k+3= (ac f −ad−cg)^k+1, (2.38)

f_3k+1= f(ac f −ad−cg)^k, (2.39)

f_3k+₂= (a f −g)(ac f −ad−cg)^k, (2.40) f_3k+3= c(a f −g)(ac f −ad−cg)^k, k∈_N0. (2.41)

Using (2.33)–(2.41) into (2.22), (2.24) and (2.26), as well as the condition b = 0, it follows that

b_3k+1 =0, (2.42)

b_3k+2 =ad(ac f −ad−cg)^k, (2.43)

b_3k+3 =acg(ac f −ad−cg)^k, (2.44)

d_3k+1 =d(ac f −ad−cg)^k, (2.45)

d_3k+2 =_cg(_{ac f} −ad−cg)^k_{, (2.46)}

d_3k+3 =0, (2.47)

g_3k+₁ =cg(a f −g)(ac f −ad−cg)^k−1, (2.48)

g_3k+2 =0, (2.49)

g_3k+3 =d(a f −g)(ac f −ad−cg)^k, k∈_N0. (2.50) Employing (2.33)–(2.50) into (2.27), (2.28) and (2.29) we obtain that the well-defined solu- tions to system (1.8) in this case are given by the following formulas

x_3n+1 =y₀^{a(ac f−ad−cg)n}, (2.51)

x_3n+2 =z^ac₀^{(ac f−ad−cg)n}x⁻₋^ad₁ ^{(ac f−ad−cg)n}, (2.52) x_3n+3 =x^a₀^{(c f−d)(ac f−ad−cg)n}y⁻₋^acg₁ ^{(ac f−ad−cg)n}, (2.53) y_3n+1 =z^c₀^{(ac f−ad−cg)n}x⁻₋^d₁^{(ac f−ad−cg)n}, (2.54) y_3n+2 =x⁽₀^{c f−d)(ac f−ad−cg)n}y⁻₋₁^{cg(ac f−ad−cg)n}, (2.55)

y_3n+3 =y₀^{(ac f−ad−cg)n+1}, (2.56)

z_3n+1 =x₀^{f(ac f−ad−cg)n}y⁻₋^cg₁ ^{(a f−g)(ac f−ad−cg)n−1}, (2.57) z_3n+2 =y₀^{(a f−g)(ac f−ad−cg)n}, (2.58) z3n+3 =z^c₀^{(a f−g)(ac f−ad−cg)n}x⁻₋₁^{d(a f−g)(ac f−ad−cg)n}, n∈_N0. (2.59) Case d=0. In this case we have that sequences (a_3k+i)_k∈N0,(c_3k+i)_k∈N0,(f_3k+i)_k∈N0, satisfy the recurrent relation

x_n+1= (ac f −b f−cg)x_n, n∈_N.

From this and since a₁ = a, c₁ = c, f₁ = f, a₂ = ac−b, c₂ = c f −d, f₂ = a f −g, a₃ = ac f −b f−ad, c₃ = ac f −ad−cg, f₃ = ac f −b f −cg, and by using the condition d = _{0, it} follows that

a_3k+1= a(ac f −b f−cg)^k_{, (2.60)}

a_3k+₂= (ac−b)(ac f −b f −cg)^k, (2.61) a_3k+3= f(ac−b)(ac f −b f −cg)^k, (2.62)

c_3k+1= c(ac f −b f −cg)^k, (2.63)

c_3k+2= c f(ac f −b f −cg)^k, (2.64) c_3k+3= c(a f −g)(ac f −b f −cg)^k, (2.65)

f_3k+1= f(ac f −b f −cg)^k_{, (2.66)}