In this paper, we give explicit formulas of the solutions of the two general systems of difference equations xn+1=f−1(ag(yn) +b f(xn−1) +cg(yn−2) +d f(xn−3

Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 22 (2021), No. 2, pp. 529–555 DOI: 10.18514/MMN.2021.3365

SOLUTIONS FORMULAS FOR SOME GENERAL SYSTEMS OF DIFFERENCE EQUATIONS

Y. AKROUR, M. KARA, N. TOUAFEK, AND Y. YAZLIK Received 31 May, 2020

Abstract. In this paper, we give explicit formulas of the solutions of the two general systems of difference equations

xn+1=f⁻¹(ag(y_n) +b f(x_n−1) +cg(y_n−2) +d f(x_n−3)), yn+1=g⁻¹(a f(xn) +bg(y_n−1) +c f(x_n−2) +dg(y_n−3)), and

xn+1=f⁻¹ a+ b

g(y_n)+ c

g(y_n)f(x_n−1)+ d

g(y_n)f(x_n−1)g(y_n−2)

! ,

y_n+1=g⁻¹ a+ b

f(x_n)+ c

f(x_n)g(y_n−1)+ d

f(x_n)g(y_n−1)f(x_n−2)

! ,

wheren∈N0, f,g:D−→Rare “1−1” continuous functions onD⊆R, the initial valuesx−i, y−i,i=0,1,2,3 are arbitrary real numbers inDand the parametersa,b,canddare arbitrary real numbers. Our results considerably extend some existing results in the literature.

2010Mathematics Subject Classification: 39A10, 40A05

Keywords: Systems of difference equations, form of solutions, stability of equilibrium points.

1. INTRODUCTION ANDPRELIMINARIES

Difference equations are used to describes real discrete models in various branches of modern sciences such as, biology, economy, control theory. This explain why a big number of papers is devoted to this subject, see for example ([1] - [20]). It is clear that if we want to understand our models, we need to know the behavior of the solutions of the equations of the models, and this fact will be possible if we can solve in closed form these equations. One can find in the literature a lot of works on difference equations where explicit formulas of the solutions are given, see for instance [1], [2], [5], [8], [9], [10], [7], [12], [11], [13], [16], [15], [14], [17], [18], [21], [20], [22].

Such type of difference equations and systems is called solvable difference equations.

In the present work, we continue our interest in solvable difference equations, more

© 2021 Miskolc University Press

precisely, we will solve the following two general systems of difference equations x_n+1=f⁻¹(ag(yn) +b f(xn−1) +cg(yn−2) +d f(xn−3)),

y_n+1=g⁻¹(a f(xn) +bg(yn−1) +c f(xn−2) +dg(yn−3)), and

x_n+1= f⁻¹

a+ b

g(yn)+ c

g(yn)f(xn−1)+ d

g(yn)f(xn−1)g(y_n−2)

, y_n+1=g⁻¹

a+ b

f(xn)+ c

f(xn)g(yn−1)+ d

f(xn)g(yn−1)f(x_n−2)

,

wheren∈N0, f,g:D−→Rare one to one (“1−1”) continuous functions onD⊆ R, the initial values x−i, y−i, i=0,1,2,3 are arbitrary real numbers in Dand the parametersa,b,canddare arbitrary real numbers.

In our study, we are inspired and motivated by the ideas, the equations and the systems of some recent published papers. The papers, [1], [2] and especially [15] are our main motivation in the present work. The obtained results considerably general- ize some existing results in the literature, see [1], [2], [3], [4], [5], [12], [11], [13], [16], [15], [14], [17], [18], [21].

Now, we will recall some known results that will be useful, in solving in closed form our systems. To this end, let consider the two following homogeneous fourth order linear difference equations

R_n+1=aRn+bRn−1+cRn−2+dRn−3,n∈N0, (1.1) S_n+1=−aS_n+bS_n−1−cS_n−2+dS_n−3,n∈N0, (1.2) where the initial valuesR₀,R₋₁,R₋₂,R₋₃,S₀,S₋₁,S₋₂andS₋₃ and the parameters a, b, candd,d̸=0 are real numbers. Noting that the formulas of the solutions of equations (1.1) and (1.2), see [2], are expressed in terms of the the sequence(Jn)^+∞_n=0 defined by the recurrent relation

J_n+4=aJ_n+3+bJ_n+2+cJ_n+1+dJ_n, n∈N0, (1.3) and the special initial values

J₀=0, J₁=0, J₂=1 andJ₃=a. (1.4) The closed form of the solutions of (Jn)^+∞_n=0 and many properties of them are well known in the literature see for example [19], [6].

The following result from [2], gave the general terms of (1.1) and (1.2) in terms of the sequence (Jn)^+∞_n=0. The obtained formulas will be very useful to obtain the formulas of the solutions of our systems.

Lemma 1. We have for all n∈N0,

Rn=dJ_n+1R−3+ (cJn+1+dJn)R−2+ (Jn+3−aJ_n+2)R−1+J_n+2R₀, (1.5) S_n= (−1)ⁿ⁺¹[dJ_n+1S₋₃−(cJ_n+1+dJ_n)S₋₂+ (J_n+3−aJ_n+2)S₋₁−J_n+2S₀]. (1.6)

Consider the following fourth order linear system of difference equations

u_n+1=av_n+bu_n−1+cv_n−2+du_n−3,v_n+1=au_n+bv_n−1+cu_n−2+dv_n−3,n∈N0, (1.7) where the initial valuesu_−i,v_−i,i=0,1,2,3 are nonzero real numbers.

In [2], the authors got the solutions of (1.7), in terms of the sequence(Jn)^+∞_n=0and the initial valuesu_−i,v_−i,i=0,1,2,3.

For the interest of the readers we show how we can solve system (1.7).

Lemma 2. The solutions of the system(1.7)are given by the following formulas u_2n−1=dJ_2nu₋₃+ (cJ_2n+dJ_2n−1)v₋₂+ (J2n+2−aJ_2n+1)u₋₁+J_2n+1v₀,n∈N,

(1.8) u_2n=dJ_2n+1v−3+ (cJ_2n+1+dJ_2n)u−2+ (J_2n+3−aJ_2n+2)v−1+J_2n+2u₀,n∈N0,

(1.9) v_2n−1=dJ_2nv₋₃+ (cJ_2n+dJ_2n−1)u₋₂+ (J2n+2−aJ_2n+1)v₋₁+J_2n+1u₀,n∈N,

(1.10) v_2n=dJ_2n+1u−3+ (cJ2n+1+dJ_2n)v−2+ (J2n+3−aJ_2n+2)u−1+J_2n+2v₀,n∈N0.

(1.11) Proof. From (1.7), we get

u_n+1+v_n+1=a(vn+un) +b(un−1+vn−1) +c(vn−2+un−2) +d(un−3+vn−3),n∈N0

and

u_n+1−v_n+1=a(vn−un) +b(un−1−vn−1) +c(vn−2−u_n−2) +d(un−3−v_n−3),n∈N0. Putting

Rn=un+vn, Sn=un−vn,n≥ −3, (1.12) we obtain two homogeneous linear difference equations of fourth order:

R_n+1=aR_n+bR_n−1+cR_n−2+dR_n−3,n∈N0, and

S_n+1=−aS_n+bS_n−1−cS_n−2+dS_n−3,n∈N0. (1.13) Using (1.12), we get

un=1

2(Rn+Sn),vn=1

2(Rn−Sn),n≥ −3.

From Lemma1, we obtain

u2n−1=dJ_2nu−3+ (cJ_2n+dJ2n−1)v−2+ (J_2n+2−aJ_2n+1)u−1+J_2n+1v₀,n∈N, u_2n=dJ_2n+1v₋₃+ (cJ2n+1+dJ_2n)u₋₂+ (J2n+3−aJ_2n+2)v₋₁+J_2n+2u₀,n∈N0, v_2n−1=dJ_2nv₋₃+ (cJ_2n+dJ_2n−1)u₋₂+ (J2n+2−aJ_2n+1)v₋₁+J_2n+1u₀,n∈N,

v_2n=dJ_2n+1u₋₃+ (cJ_2n+1+dJ_2n)v₋₂+ (J_2n+3−aJ_2n+2)u₋₁+J_2n+2v₀,n∈N0.

□

2. SOLUTIONS OF A FIRST GENERAL SYSTEM OF DIFFERENCE EQUATIONS

In this part, we will focus our interest on our first general system of difference equations, that is the system

(x_n+1= f⁻¹(ag(yn) +b f(xn−1) +cg(yn−2) +d f(xn−3)),

y_n+1=g⁻¹(a f(xn) +bg(yn−1) +c f(xn−2) +dg(yn−3)), (2.1) wheren∈N0, f,g:D−→Rare continuous functions, withD=Df =Dg, that is f andghave the same domain, and it is also assumed thatf,gare “1−1” onD⊆R, the initial valuesx_−i,y_−i,i=0,1,2,3 are arbitrary real numbers inDand the parameters a,b,canddare arbitrary real numbers.

2.1. Explicit formulas of solutions of system(2.1)with d̸=0 In the following result, we solve in closed form (2.1).

Definition 1. A solution{x_n,yn}_n≥−3of system (2.1) is said to be well-defined if for alln∈N0, we have

ag(yn) +b f(xn−1) +cg(yn−2) +d f(xn−3)∈D_f⁻¹, and

a f(xn) +bg(yn−1) +c f(xn−2) +dg(yn−3)∈D_g−1.

Theorem 1. Let{xn,yn}_n≥−3 be a well-defined solution of the system(2.1), then we have the following representation











x_2n−1 = f⁻¹(dJ_2nf(x−3) + (cJ_2n+dJ_2n−1)g(y−2) + (J2n+2−aJ_2n+1)f(x−1) +J_2n+1g(y₀)),n∈N,

x_2n = f⁻¹(dJ2n+1g(y−3) + (cJ2n+1+dJ_2n)f(x−2) + (J2n+3−aJ_2n+2)g(y−1) +J2n+2f(x₀)),n∈N0,

(2.2)











y_2n−1 =g⁻¹(dJ2ng(y−3) + (cJ2n+dJ_2n−1)f(x−2) + (J2n+2−aJ_2n+1)g(y−1) +J_2n+1f(x₀)),n∈N,

y_2n =g⁻¹(dJ_2n+1f(x−3) + (cJ_2n+1+dJ_2n)g(y−2) + (J_2n+3−aJ_2n+2)f(x−1) +J2n+2g(y0)),n∈N0.

(2.3) Proof. Since the functions f,gare “1−1”, then from (2.1) we get

(f(xn+1) =ag(yn) +b f(xn−1) +cg(yn−2) +d f(xn−3),

g(y_n+1) =a f(xn) +bg(yn−1) +c f(xn−2) +dg(yn−3), n∈N0. (2.4) By the change of variables

X_n= f(xn), Y_n=g(yn),n≥ −3, (2.5)

system (2.4) is transformed to the following one (

X_n+1=aYn+bXn−1+cYn−2+dXn−3,

Y_n+1=aXn+bY_n−1+cX_n−2+dY_n−3, n∈N0. (2.6) Clearly (2.6) is in the form of system (1.7), by Lemma (2), we obtain the following representation of solutions

(X_2n−1=dJ_2nX₋₃+ (cJ_2n+dJ_2n−1)Y−2+ (J2n+2−aJ_2n+1)X₋₁+J_2n+1Y₀,n∈N, X_2n=dJ_2n+1Y−3+ (cJ_2n+1+dJ_2n)X−2+ (J_2n+3−aJ_2n+2)Y−1+J_2n+2X₀,n∈N0,

(2.7) (

Y2n−1=dJ_2nY−3+ (cJ_2n+dJ2n−1)X−2+ (J_2n+2−aJ_2n+1)Y−1+J_2n+1X₀,n∈N, Y_2n=dJ_2n+1X−3+ (cJ2n+1+dJ_2n)Y−2+ (J2n+3−aJ_2n+2)X−1+J_2n+2Y₀,n∈N0.

(2.8) Now, by (2.5) we get that











x_2n−1 =f⁻¹[dJ2nf(x−3) + (cJ2n+dJ_2n−1)g(y−2) + (J2n+2−aJ_2n+1)f(x−1) +J_2n+1g(y₀)],n∈N,

x_2n =f⁻¹[dJ_2n+1g(y−3) + (cJ_2n+1+dJ_2n)f(x−2) + (J_2n+3−aJ_2n+2)g(y−1) +J2n+2f(x0)],n∈N0,

(2.9)











y2n−1 =g⁻¹[dJ_2ng(y−3) + (cJ_2n+dJ2n−1)f(x−2) + (J_2n+2−aJ_2n+1)g(y−1) +J2n+1f(x0)],n∈N,

y_2n =g⁻¹[dJ2n+1f(x−3) + (cJ2n+1+dJ_2n)g(y−2) + (J2n+3−aJ_2n+2)f(x−1) +J_2n+2g(y₀)],n∈N0.

(2.10)

□ Remark1. Moreover, ifg≡f andy−i=x−i,i=0,3 then, the system (2.1) will be the equation

x_n+1= f⁻¹(a f(xn) +b f(xn−1) +c f(xn−2) +d f(xn−3)) (2.11) and then the representation of the well-defined solutions are given by











x2n−1 = f⁻¹[dJ2nf(x−3) + (cJ2n+dJ2n−1)f(x−2) + (J2n+2−aJ_2n+1)f(x−1) +J2n+1f(x₀)],n∈N,

x_2n = f⁻¹[dJ_2n+1f(x−3) + (cJ_2n+1+dJ_2n)f(x−2) + (J_2n+3−aJ_2n+2)f(x−1) +J2n+2f(x0)],n∈N0.

(2.12)

In [15], Stevic studied the equation (2.11).

Now as applications of theorem1, we give the following examples.

Example1. Let

f(t) =t^2j+1,g(t) =t^2k+1, j,k∈N0. (2.13) Then,Df =Dg=R, clearly the functions f andgare “1−1” continuous functions onRand the system (2.1) becomes





 x_n+1=

h

ay^2k+1_n +bx²_n−1^j+1+cy^2k+1_n−2 +dx²_n−3^j+1 i_2j+1¹

, y_n+1=h

ax^2j+1n +by^2k+1_n−1 +cx²_n−2^j+1+dy^2k+1_n−3 i_2k+1¹

, n∈N0.

(2.14)

Then from (2.2) and (2.3), we obtain that general solution of the equation (2.14) is



















x2n−1 =h

dJ_2nx^2j+1₋₃ + (cJ_2n+dJ2n−1)y^2k+1₋₂ + (J_2n+2−aJ_2n+1)x^2j+1₋₁ +J_2n+1y^2k+1_{0 2j+1}¹

,n∈N, x_2n =

h

dJ_2n+1y^2k+1₋₃ + (cJ2n+1+dJ_2n)x²₋₂^j+1+ (J2n+3−aJ_2n+2)y^2k+1₋₁ +J_2n+2x²₀^j+1

i_2j+1¹

,n∈N0,

(2.15)



















y2n−1 =h

dJ_2ny^2k+1₋₃ + (cJ_2n+dJ2n−1)x²₋₂^j+1+ (J_2n+2−aJ_2n+1)y^2k+1₋₁ +J2n+1x²₀^j+1

i_2k+1¹

,n∈N, y_2n =h

dJ_2n+1x²₋₃^j+1+ (cJ_2n+1+dJ_2n)y^2k+1₋₂ + (J_2n+3−aJ_2n+2)x²₋₁^j+1 +J2n+2y^2k+1_{0 2k+1}¹

,n∈N0.

(2.16)

Example2. Let

f(t) = 1

t^2j+1,g(t) = 1

t^2k+1, j,k∈N0. (2.17) Then, D_f =D_g =R− {0}, clearly the functions f and g are “1−1” continuous functions onR− {0}and the system (2.1) becomes









 x_n+1=

a y^2k+1_n + ^b

x²_n−1^j+1+ ^c

y^2k+1_n−2 + ^d

x²_n−3^j+1

₂⁻¹_j+1 , y_n+1=

a x²n^j+1

+ ^b

y^2k+1_n−1 + ^c

x²_n−2^j+1+ ^d

y^2k+1_n−3

_2k+1⁻¹

, n∈N0,

(2.18)

or equivalently









 x_n+1=

y^2k+1_n x²_n−1^j+1y^2k+1_n−2x²_n−3^j+1

ax^2j+1_n−1y^2k+1_n−2x²_n−3^j+1+by^2k+1n y^2k+1_n−2x²_n−3^j+1+cy^2k+1n x²_n−1^j+1x²_n−3^j+1+dy^2k+1n x²_n−1^j+1y^2k+1_n−2

_2j+1¹ , y_n+1=

x²_n^j+1y^2k+1_n−1 x²_n−2^j+1y^2k+1_n−3

ay^2k+1_n−1x²_n−2^j+1y^2k+1_n−3 +bx²n^j+1x²_n−2^j+1y^2k+1_n−3+cx²n^j+1y^2k+1_n−1y^2k+1_n−3+dx²n^j+1y^2k+1_n−1x_n−2^2j+1

_2k+1¹

, n∈N0. (2.19) Then from (2.2) and (2.3), we obtain that general solution of system (2.19) is



















x2n−1 =h

dJ_2nx⁻⁽²₋₃ ^j+1)+ (cJ_2n+dJ2n−1)y^−(2k+1)₋₂ + (J_2n+2−aJ_2n+1)x^−(2j+1)₋₁ +J2n+1y^−(2k+1)₀

i−_2j+1¹

,n∈N, x_2n =h

dJ_2n+1y^−(2k+1)₋₃ + (cJ_2n+1+dJ_2n)x^−(2j+1)₋₂ + (J_2n+3−aJ_2n+2)y^−(2k+1)₋₁ +J2n+2x⁻⁽²₀ ^j+1)

i−_2j+1¹

,n∈N0,

(2.20)



















y2n−1 = h

dJ_2ny^−(2k+1)₋₃ + (cJ2n+dJ2n−1)x₋₂^−(2j+1)+ (J2n+2−aJ_2n+1)y^−(2k+1)₋₁ +J_2n+1x⁻⁽²₀ ^j+1)

i−_2k+1¹

,n∈N, y_2n =

h

dJ_2n+1x^−(2j+1)₋₃ + (cJ2n+1+dJ_2n)y^−(2k+1)₋₂ + (J2n+3−aJ_2n+2)x⁻⁽²₋₁ ^j+1) +J_2n+2y^−(2k+1)₀ i−_2k+1¹

,n∈N0.

(2.21) If j=k=0, then system (2.18) becomes

(x_n+1=_ax ^{ynxn−1yn−2xn−3}

n−1yn−2xn−3+bynyn−2xn−3+cynxn−1xn−3+dynxn−1yn−2, y_n+1=_ay ^{xnyn−1xn−2yn−3}

n−1xn−2yn−3+bxnxn−2yn−3+cxnyn−1yn−3+dxnyn−1xn−2, n∈N0. (2.22) The form of the well-defined solutions of (2.22), can be obtained by putting j=k=0, in the formulas of the solutions of (2.18). The solutions of the equation, see [15]

x_n+1= xnxn−1xn−2xn−3

axn−1xn−2xn−3+bxnxn−2xn−3+cxnxn−1xn−3+dxnxn−1xn−2

, n∈N0, (2.23) can be obtained from the solutions of (2.22) by takingy_−i=x_−i,i=0,1,2,3.

2.2. Particular cases of system(2.1) 2.2.1. The case d=0and c̸=0

In this case the system (2.1) takes the form:

(

x_n+1=f⁻¹(ag(yn) +b f(xn−1) +cg(yn−2)),

y_n+1=g⁻¹(a f(xn) +bg(yn−1) +c f(xn−2)), n∈N0. (2.24) Using the change of variables (2.5), withn≥ −2, we get the third order linear system X_n+1=aYn+bXn−1+cYn−2,Y_n+1=aXn+bYn−1+cXn−2,n≥ −2. (2.25) Consider the sequence

Jen

n≥0defined by

Je_n+3=aJe_n+2+bJe_n+1+cJen, n∈N0, (2.26) and the special initial values

Je₀=0,Je₁=1,Je₂=a.

The sequence Je_n

n≥0is obtained from the sequence(Jn)_n≥0defined by (1.3):

J_n+4=aJ_n+3+bJ_n+2+cJ_n+1+dJn,J₀=0, J₁=0, J₂=1 andJ₃=a,n∈N0. Ford=0, we obtain

J_n+4=aJ_n+3+bJ_n+2+cJ_n+1. Putting

Je_n=J_n+1,n∈N0,

we get the sequence (2.26). Noting that in this case, the corresponding sequences (Rn)n≥0,(Sn)n≥0will be

R_n+1=aR_n+bR_n−1+cR_n−2,S_n+1=−aS_n+bS_n−1−cS_n−2,n∈N0, with the initial valuesR₀,R−1, R−2,S₀,S−1,S−2. The formulas of the solutions of these equations are expressed using the sequence

Je_n

n≥0, see [2].

The formulas of the solutions of (2.25) and (2.24), can be obtaining from those of solutions of (1.7) and solutions of (2.1) by changingJ_nbyJe_n−1.

In summary we have the following result.

Corollary 1. Let{xn,yn}_n≥−2be a well-defined solution of system(2.24), then x_2n−1= f⁻¹

h

cJe_2n−1g(y−2) +

Je_2n+1−aJe_2n

f(x−1) +Je_2ng(y0) i

,n∈N, x_2n= f⁻¹

h

cJe_2nf(x−2) +

Je_2n+2−aJe_2n+1

g(y−1) +Je_2n+1f(x0) i

,n∈N0, y2n−1=g⁻¹

h

cJe2n−1f(x−2) +

Je_2n+1−aJe_2n

g(y−1) +Je_2nf(x₀)i

,n∈N,

y_2n=g⁻¹ h

cJe_2ng(y−2) +

Je_2n+2−aJe_2n+1

f(x−1) +Je_2n+1g(y₀) i

,n∈N0. Remark2. Ifg≡ f andy−i=x−i,i=0,1,2 then, system (2.24) becomes

x_n+1= f⁻¹[a f(xn) +b f(xn−1) +c f(xn−2)] (2.27) and by Corollary1, the every well defined solution is given by







x_2n−1=f⁻¹ h

cJe_2n−1f(x−2) +

Je_2n+1−aJe_2n

f(x−1) +Je_2nf(x₀) i

,n∈N, x_2n= f⁻¹

h

cJe_2nf(x−2) +

Je_2n+2−aJe_2n+1

f(x−1) +Je_2n+1f(x₀)i

,n∈N0, (2.28) which can written in a unified form as

xn=f⁻¹ h

cJenf(x−2) +

Je_n+2−aJe_n+1

f(x−1) +Je_n+1f(x₀)i

,n∈N0. (2.29) Noting again that this equation, was studied by Stevic in [15].

2.2.2. Case d=0,c̸=0and a=0 In this case we get the system

(

x_n+1= f⁻¹[b f(xn−1) +cg(yn−2)],

y_n+1=g⁻¹[bg(yn−1) +c f(xn−2)], n∈N0. (2.30) Here,

Je_n

n≥0will be the sequence defined by

Pn+3=bPn+1+cP^{n, n∈}N0, (2.31) and the special initial values

P0=0, P1=1 andP2=0, (2.32)

so, the solutions are expressed in terms of(P⁾n≥0and are given by x2n−1= f⁻¹[cP^{2n−1g(y−2) +}P2n+1f(x−1) +P2ng(y0)],n∈N,

x_2n= f⁻¹[cP2nf(x−2) +P2n+2g(y−1) +P2n+1f(x₀)],n∈N0, y_2n−1=g⁻¹[cP2n−1f(x−2) +P2n+1g(y−1) +P2nf(x₀)],n∈N,

y_2n=g⁻¹[cP2ng(y−2) +P2n+2f(x−1) +P2n+1g(y0)],n∈N0, for the system (2.30) and by

xn= f⁻¹[cP^{nf(x−2) +}Pn+2f(x−1) +Pn+1f(x₀)],n∈N0, for its one dimensional version, that is the equation

x_n+1= f⁻¹(b f(xn−1) +c f(xn−2)).

If b ̸=0, (P⁾n≥0 will be a generalized Padovan sequence and if b=c=1, then (Pⁿ⁾n≥0will be the famous Padovan sequence.

Noting that system (2.30) generalize for example the works of [5] and [21].

2.2.3. Case c=d=0and b̸=0 In this case, we get the system

(

x_n+1= f⁻¹[ag(yn) +b f(xn−1)],

y_n+1=g⁻¹[a f(xn) +bg(yn−1)], n∈N0. (2.33) By the same philosophy, we obtain the sequence

Fen

n≥0= Je_n+1

n≥0, defined by Fe_n+2=aFe_n+1+bFen,Fe₀=1,Fe₁=a,n∈N0,

and the solutions of (2.33) and its one dimensional version, are obtained from the solutions of (2.24) and (2.27), by writing Fen−1 instead of Jen. If a̸=0,

Fen

n≥0

is a generalized Fibonacci sequence and ifa=b=1, Fen

n≥0 will be the famous Fibonacci sequence.

System (2.33) and its one dimensional versions, generalize for example the works of [17], [18].

2.2.4. Case b=c=d=0and a̸=0 In this case, we get the system

x_n+1= f⁻¹(ag(yn))),y_n+1=g⁻¹(a f(xn)),n∈N0. (2.34) Using the fact that f,gare one to one functions, and the change of variables

X_n= f(xn),Y_n=g(yn),n≥0 the system (2.34), will be

X_n+1=aYn,Y_n+1=aXn,n∈N0. So,

X_2n=a²ⁿX₀,Y_2n=a²ⁿY₀,X_2n+1=a²ⁿ⁺¹Y₀,Y_2n+1=a²ⁿ⁺¹X₀,n∈N0. Hence

x_2n=f⁻¹(a²ⁿf(x0)),y_2n=g⁻¹(a²ⁿg(y0)),n∈N0

and

x_2n+1= f⁻¹(a²ⁿ⁺¹g(y0)),y_2n+1=g⁻¹(a²ⁿ⁺¹f(x0)),n∈N0.

3. SOLUTIONS OF A SECOND GENERAL SYSTEM OF DIFFERENCE EQUATIONS

In this part, we are interested in the following system of difference equations given by







x_n+1= f⁻¹

a+_g(y^b

n)+_g(y ^c

n)f(xn−1)+_g(y ^d

n)f(xn−1)g(yn−2)

, y_n+1=g⁻¹

a+_f(x^b

n)+ _f(x ^c

n)g(yn−1)+ _f(x ^d

n)g(yn−1)f(xn−2)

, (3.1)

wheren∈N0, f,g:D−→Rare continuous functions, withD=D_f =D_g, that is f andghave the same domain, in addition we assume that f,gare “1−1” onD⊆R, the initial valuesx_−i,y_−i,i=0,1,2, are arbitrary real numbers inDand the parameters a,b,canddare arbitrary real numbers.

Definition 2. A solution{x_n,yn}_n≥−2of system (3.1) is said to be well-defined if for alln∈N0, we have

a+ b

g(yn)+ c

g(yn)f(xn−1)+ d

g(yn)f(xn−1)g(yn−2) ∈D_f−1, and

a+ b

f(xn)+ c

f(xn)g(yn−1)+ d

f(xn)g(yn−1)f(xn−2) ∈D_g−1.

We solve in closed form (3.1) and we investigated particular cases of it. The philo- sophy is the same as in the previous section (2), so we will brief in presenting our formulas of the solutions.

3.1. Explicit formulas of solutions of system(3.1)with d̸=0

The following result is devoted to the formulas of well-defined solutions of (3.1).

Theorem 2. Let{xn,yn}_n≥−2be a well-defined solution of system(3.1). Then, for all n∈N0we have

x_2n+1=f⁻¹[(dJ_2n+2+ (cJ_2n+2+dJ_2n+1)g(y−2) + (J_2n+4−aJ_2n+3)f(x−1)g(y−2) +J_2n+3g(y₀)f(x−1)g(y−2))(dJ_2n+1+ (cJ_2n+1+dJ_2n)g(y−2)

+ (J_2n+3−aJ_2n+2)f(x−1)g(y−2) +J_2n+2g(y₀)f(x−1)g(y−2))⁻¹],

x2n+2=f⁻¹[(dJ_2n+3+ (cJ_2n+3+dJ2n+2)f(x₋₂) + (J_2n+5−aJ2n+4)g(y₋₁)f(x₋₂) +J_2n+4f(x₀)g(y−1)f(x−2))(dJ_2n+2+ (cJ_2n+2+dJ_2n+1)f(x−2)

+ (J_2n+4−aJ2n+3)g(y₋₁)f(x₋₂) +J2n+3f(x₀)g(y₋₁)f(x₋₂))⁻¹],

y_2n+1=g⁻¹[(dJ_2n+2+ (cJ_2n+2+dJ_2n+1)f(x−2) + (J_2n+4−aJ_2n+3)g(y−1)f(x−2) +J2n+3f(x₀)g(y₋₁)f(x₋₂))(dJ_2n+1+ (cJ_2n+1+dJ_2n)f(x₋₂)

+ (J_2n+3−aJ_2n+2)g(y−1)f(x−2) +J_2n+2f(x₀)g(y−1)f(x−2))⁻¹],

y_2n+2=g⁻¹[(dJ_2n+3+ (cJ_2n+3+dJ_2n+2)g(y−2) + (J_2n+5−aJ_2n+4)f(x−1)g(y−2) +J_2n+4g(y₀)f(x−1)g(y−2))(dJ_2n+2+ (cJ_2n+2+dJ_2n+1)g(y−2)

+ (J_2n+4−aJ_2n+3)f(x−1)g(y−2) +J_2n+3g(y₀)f(x−1)g(y−2))⁻¹].

Proof. Using the fact that the functions f,gare one to one and using the change of variables (2.5), withn≥ −2, the system (3.1) becomes







X_n+1=a+ b Yn

+ c YnXn−1

+ d

YnXn−1Yn−2

, Y_n+1=a+ b

Xn

+ c XnYn−1

+ d

XnYn−1Xn−2

, n∈N0,

(3.2)

or equivalently,







X_n+1=aYnX_n−1Y_n−2+bX_n−1Y_n−2+cY_n−2+d Y_nX_n−1Y_n−2 , Y_n+1= aX_nY_n−1X_n−2+bY_n−1X_n−2+cX_n−2+d

X_nY_n−1X_n−2 , n∈N0.

(3.3)

This system was solved in [2], and forn∈N0, the solutions of (3.3) takes the form X_2n+1=dJ_2n+2+ (cJ_2n+2+dJ_2n+1)Y−2+ (J_2n+4−aJ_2n+3)X−1Y−2+J_2n+3Y₀X−1Y−2

dJ_2n+1+ (cJ_2n+1+dJ_2n)Y−2+ (J_2n+3−aJ_2n+2)X−1Y−2+J_2n+2Y₀X−1Y−2

, (3.4) X_2n+2=dJ_2n+3+ (cJ_2n+3+dJ_2n+2)X−2+ (J_2n+5−aJ_2n+4)Y−1X−2+J_2n+4X₀Y−1X−2

dJ_2n+2+ (cJ2n+2+dJ_2n+1)X₋₂+ (J2n+4−aJ_2n+3)Y−1X₋₂+J_2n+3X₀Y₋₁X₋₂, (3.5) Y_2n+1=dJ_2n+2+ (cJ2n+2+dJ_2n+1)X−2+ (J2n+4−aJ_2n+3)Y−1X−2+J_2n+3X₀Y−1X−2

dJ_2n+1+ (cJ2n+1+dJ_2n)X₋₂+ (J2n+3−aJ_2n+2)Y−1X₋₂+J2n+2X₀Y₋₁X₋₂ , (3.6) Y_2n+2=dJ_2n+3+ (cJ2n+3+dJ_2n+2)Y−2+ (J2n+5−aJ_2n+4)X₋₁Y₋₂+J_2n+4Y₀X₋₁Y₋₂

dJ_2n+2+ (cJ_2n+2+dJ_2n+1)Y−2+ (J_2n+4−aJ_2n+3)X−1Y−2+J_2n+3Y₀X−1Y−2

, (3.7) where(Jn)_n∈

N0 is the sequence defined by (1.3).

Using (2.5), (3.4), (3.5), (3.6) and (3.7), we get that forn∈N0, every well-defined solution of system (3.1) has the following representation

x2n+1=f⁻¹[(dJ_2n+2+ (cJ_2n+2+dJ2n+1)g(y₋₂) + (J_2n+4−aJ2n+3)f(x₋₁)g(y₋₂) +J_2n+3g(y₀)f(x−1)g(y−2))(dJ_2n+1+ (cJ_2n+1+dJ_2n)g(y−2)

+ (J_2n+3−aJ2n+2)f(x₋₁)g(y₋₂) +J2n+2g(y₀)f(x₋₁)g(y₋₂))⁻¹],

x_2n+2=f⁻¹[(dJ_2n+3+ (cJ_2n+3+dJ_2n+2)f(x−2) + (J_2n+5−aJ_2n+4)g(y−1)f(x−2) +J2n+4f(x₀)g(y₋₁)f(x₋₂))(dJ_2n+2+ (cJ_2n+2+dJ2n+1)f(x₋₂)

+ (J_2n+4−aJ_2n+3)g(y−1)f(x−2) +J_2n+3f(x₀)g(y−1)f(x−2))⁻¹],

y_2n+1=g⁻¹[(dJ_2n+2+ (cJ_2n+2+dJ_2n+1)f(x−2) + (J_2n+4−aJ_2n+3)g(y−1)f(x−2) +J_2n+3f(x₀)g(y−1)f(x−2))(dJ_2n+1+ (cJ_2n+1+dJ_2n)f(x−2)

+ (J_2n+3−aJ_2n+2)g(y−1)f(x−2) +J_2n+2f(x₀)g(y−1)f(x−2))⁻¹],

y2n+2=g⁻¹[(dJ_2n+3+ (cJ_2n+3+dJ2n+2)g(y₋₂) + (J_2n+5−aJ2n+4)f(x₋₁)g(y₋₂) +J_2n+4g(y₀)f(x−1)g(y−2))(dJ_2n+2+ (cJ_2n+2+dJ_2n+1)g(y−2)

+ (J_2n+4−aJ2n+3)f(x₋₁)g(y₋₂) +J2n+3g(y₀)f(x₋₁)g(y₋₂))⁻¹].

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