# Exact Solutions of Three-Dimensional Max-Type System of Difference Equations

Authors: Abdul Khaliq; Stephen Sadiq; Abdul Qadeer Khan; Tarek F. Ibrahim
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this paper, we deal with the form and the periodicity of the solutions of the max-type system of difference equationsun+1=maxAn/vn−1,un−1,vn+1=maxBn/wn−1,wn+1=maxCn/un−1,wn−1 where the initial conditions u−1,u0∈0,∞,v−1,v0∈0,∞,w−1,w0∈0,∞andAnn∈N0,Bnn∈N0,Cnn∈N0 are positive two-periodic sequences.


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## Body

## 1. Introduction

Difference equations and systems which do not stem from the differential ones have attracted some attention in the last few decades (see, e.g., [1–26]). Some of the systems that are of interest are symmetric or those retrieved from symmetric by alteration of their parameters. Recently, there has been enormous interest in various types of nonlinear difference equations and the references cited therein. One of the reasons for this is a necessity for some techniques which can be used in inspecting equations arising in mathematical models illustrating real life situations in biology, control theory, economics, physics, sociology, and so on (see, e.g., [21, 23, 24]). Among these equations, the so-called max-type difference equations also attracted some attention.The study of the system of max-type difference equations attracted recently a considerable attention; see, for example, [1–16], and the references listed therein. This type of difference equations stemming from, for example, certain models are useful in automatic control theory (see [27]). In the beginning, on the study of these equations, experts have been focused on the investigation of the behavior of some particular cases.In [3], Berenhaut et al. explained the boundedness nature of positive solutions of the following max-type difference equation system:(1)xn+1=maxAnxn,Bnxn−1,where AnandBn are periodic parameters. In 2020, Balibrea et al. [7] obtained in an elegant way the general solution of the following max-type system of difference equations:(2)xn=maxA,yn−txn−s,yn=maxB,xn−tyn−s,n∈N0,where An,Bn∈0,+∞,t,s∈1,2,… with gcds,t=1. The initial conditions x−d,y−d,x−d+1,y−d+1,…,x−1,y−1∈0,+∞ and d=maxt,sIn [8], Cunningham et al. evaluated the solution of the following max-type difference equation with period 2:(3)xn=maxA,yn−1xn−2,yn=maxB,xn−1yn−2,n∈N0,where An,Bn∈0,+∞ are periodic sequences with period 2.In 2015, Grove et al. [13] obtained the solution of the following max-type difference equation system:(4)xn+1=max1xn,min1,Aynyn+1=max1yn,min1,Axn,n∈N0,where N0=N∪0. The parameter A is positive real number.In [15], Su et al. obtained the solution of the following max-type difference equation system with a period 3 parameter:(5)xn+1=max1xn,Anxn−1.In [26], C.M. Kent and M.A. Radin explained the boundedness nature of positive solutions of the following difference equation:(6)xn+1=maxAnxn,Bnxn−1,where AnandBn are periodic parameters.Motivated by the above study, our purpose in this paper is to evaluate the eventual periodicity of the following max-type3D-system of difference equations:(7)un+1=maxAnvn−1,un−1,(8)vn+1=maxBnwn−1,vn−1,(9)wn+1=maxCnun−1,wn−1,where n∈N0,N0=N∪0,Ann∈N0,Bnn∈N0, and Cnn∈N0 are positive periodic sequences and initial conditions u0,u−1,v0,v−1,w0,w−1∈0,+∞.Theorem 1.
Suppose thatun,vn,wn is a solution of system (1)–(3) such that A0/v−1>u−1, B0/w−1>v−1, and C0/u−1>w−1. Then, the following statements hold: (1)
IfA1/v0>u0, B1/w0≥v0, and w0≥C1/u0, then(10)u2n+1=A0v−1,v2n+1,=B0w−1,w2n+1,=C0u−1,u2n,=u0,v2n,=v0,w2n,=w0,n∈N0.2IfA1/v0≥u0, B1/w0≥v0, and w0≥C1/u0, then(11)u2n+1=A0v−1,v2n+1,=B0w−1,w2n+1,=C0u−1,u2n=u0,v2n,=v0,w2n+2,=c1u0.3IfA1/v0≥u0, B1/w0≥v0, and w0≥C1/u0, then(12)u2n+1,=A0v−1,v2n+1,=B0w−1,w2n+1,=C0u−1,u2n+2,=A1w0B1,v2n+2,=B1w0,w2n,=w0.Proof.
(1)
From the following conditions:(13)A0v−1≥u−1,B0w−1≥v−1,C0u−1≥w−1,we have(14)u1=maxA0v−1,u−1,=A0v−1,v1=maxB0w−1,v−1,=B0w−1,w1=maxC0u−1,w−1,=C0u−1,u3=maxA0v1,u1,=maxA0w−1B0,A0v−1,=A0v−1,v3=maxB0w1,v1,=maxB0u−1C0,B0w−1,=B0w−1,w3=maxC0u1,w1,=maxC0v−1A0,C0u−1,=C0u−1.By induction, we obtained formula as follows:(15)u2n+1=A0v−1,v2n+1,=B0w−1,w2n+1,=C0u−1,u2n,=u0,v2n,=v0,w2n,=w0,n∈N0,which are formulas of odd terms in (4)–(6). Hence, it remains only to prove the formulas for even terms in (4)–(6).Similarly, we can find for even terms.(16)u2=maxA1v0,u0,=u0,v2=maxB1w0,v0,=v0,w2=maxC1u0,w0,=w0,u4=maxA1v2,u2,=maxA1v0,u0,=u0,v4=maxB1w2,v2,=maxB1w0,v0,=v0,w4=maxC1u2,w2,=maxC1u0,w0,=w0.By induction,(17)u2n=u0,v2n,=v0,w2n,=w0,n∈ℕo.(2)
Becauseu0≥A1/v0, v0≥B1/w0, and w0≤C1/u0, we have(18)1w0≥u0C1.Then,(19)v0≥B1w0≥B1u0C1,u2=maxA1v0,u0,=u0,v2=maxB1w0,v0,=v0,w2=maxC1u0,w0,=C1u0.(20)u4=maxA1v2,u2,=maxA1v0,u0,=u0,v4=maxB1w2,v2,=maxB1C1u0,v0,=v0,w4=maxC1u2,w2,=maxC1u0,C1u0,=C1u0,u6=maxA1v0,u0,=maxA1v0,u0,=u0,v6=maxB1w4,v4,=maxB1u0C1,v0,=v0,w6=maxC1u4,w4,=maxC1u0,C1u0=C1u0.By induction, we obtain formulas for even terms as given in (11).(3)
Becauseu0≥A1/v0, v0≤B1/w0, and w0≥C1/u0, then(21)1v0≥w0B1,u0≥A1×w0B1,u0≥A1w0B1,u2=maxA1v0,u0,=u0,v2=maxB1w0,v0,=B1w0,w2=maxC1u0,w0,=w0.
So, we have(22)u4=maxA1v2,u2,=maxA1B1w0,u0,=u0,v4=maxB1w2,v2,=maxB1w0,B1v0,=B1w0,w4=maxC1u2,w2,=maxC1u0,w0,=w0,u6=maxA1v4,u4,=maxA1w0B1,u0,=u0,v6=maxB1w4,v4,=maxB1w0,B1w0,=B1w0,w6=maxC1u4,w4,=maxC1u0,w0,=w0.
By induction, we obtain formulas as given in (26):(23)u2n+2=u0,v2n+2,=B1w0,w2n,=w0.
The proof is completed.Theorem 2.
Suppose thatun,vn,wn is a solution of systems (1)–(3) such that A0/v−1≤u−1, B0/w−1≤v−1, and C0/u−1≤w−1. Then, the following statements hold:(1)
IfA1/v0≤u0, B1/w0≤v0, and w0≥C1/u0, then(24)u2n+1=u−1,v2n+1,=v−1,w2n+1,=w−1,u2n,=u0,v2n,=v0,w2n,=w0.2IfA1/v0≥u0, B1/w0≥v0, and w0≤C1/u0,(25)u2n+1=u−1,v2n+1,=v−1,w2n+1,=w−1,u2n,=u0,v2n,=v0,w2n+2,=C1u0.3IfA1/v0≤u0,B1/w0≤v0, and w0≥C1/u0,(26)un+1=u−1,v2n+1,=v−1,w2n+1,=w−1,u2n+2,=A1v0,v2n,=v0,w2n,=w0.4IfA1/v0≥u0,B1/w0≤v0, and w0≥C1/u0,(27)u2n+1=u−1,v2n+1,=v−1,w2n+1,=w−1,u2n,=u0,v2n+2,=B1w0,w2n,=w0.Proof.
(1)
From the conditionsA0,/v−1≤u−1, B0/w−1≤v−1, and C0/u−1,≤w−1, we have(28)u1=maxA0v−1,u−1,=u−1,v1=maxB0w−1,v−1,=v−1,w1=maxC0u−1,w−1,=w−1,u3=maxA0v1,u1,=maxA0v−1,u−1,=u−1,v3=maxB0w1,v1,=maxB0w−1,v−1,=v−1,w3=maxC0u1,w1,=maxC0u−1,w−1,=w−1.By induction, we obtained formula as follows:(29)u2n+1=u−1,v2n+1,=v−1,w2n+1,=w−1.Similarly, we can find the proof for even terms.(30)u2=maxA1v0,u0=u0,v2=maxB1w0,v0=v0,w2=maxC1u0,w0=w0,u4=maxA1v2,u2=maxA1v0,u0=u0,v4=maxB1w2,v2=maxB1w0,v0=v0,w4=maxC1u2,w2=maxC1u0,w0=w0.By induction,(31)u2n=u0,v2n,=v0,w2n,=w0,as given in (24).(2)
Becauseu0≥A1/v0, v0≥B1/w0, and w0≤C1/u0, then we have(32)1w0≥u0C1,v0≥B1×u0C1,u2=maxA1v0,u0,=u0,v2=maxB1w0,v0=v0,w2=maxC1u0,w0=C1u0,u4=maxA1v2,u2=maxA1v0,u0=u0,v4=maxB1w2,v2=maxB1u0C1,v0=v0,w4=maxC1u2,w2=maxC1u0,C1u0=C1u0,u6=maxA1v4,u4=maxA1u0,u0=u0,v6=maxB1w4,v4=maxB1u0C1,v0=v0,w6=maxC1u4,w4=maxC1u0,C1u0=C1u0.By induction, we obtain the formulas as stated in (25).(3)
Becauseu0≤A1/v0, v0≥B1/w0, and w0≥C1/u0, then(33)1u0≥v0A1,w0≥C1×1u0,w0≥C1v0A1,u2=maxA1v0,u0,=A1v0,v2=maxB1w0,v0,=v0,w2=maxC1u0,w0,=w0,u4=maxA1v2,u2,=maxA1v0,A1v0,=A1v0,v4=maxB1w2,v2,=maxB1w0,v0,=v0,w4=maxC1u2,w2,=maxC1v0A1,w0,=w0,u6=maxA1v4,u4,=maxA1v0,A1v0,=A1v0,v6=maxB1w4,v4,=maxB1w0,v0,=v0,w6=maxC1u4,w4,=maxC1v0A1,w0,=w0.By induction, we obtain formula as in (26).(4)
Becauseu0≥A1/v0, v0≤B1/w0, and w0≥C1/u0, then(34)1v0≥w0B1,u0≥A1w0B1,u2=maxA1v0,u0,=u0,v2=maxB1w0,v0,=B1w0,v2=maxB1w0,v0,=B1w0,u4=maxA1v2,u2,=maxA1w0B1,u0,=u0,v4=maxB1w2,v2,=maxB1w0,B1w0,=B1w0,w4=maxC1u2,w2,=maxC1u0,w0m=w0,u6=maxA1v4,u4,=maxA1w0B1,u0,=u0,v6=maxB1w4,v4,=maxB1w0,B1w0,=B1w0,w6=maxC1u4,w4,=maxC1u0,w0,=w0.
Then, by induction, we get formulas as given in (27).Theorem 3.
Suppose thatun,vn,wn is a solution of systems (1)–(3) such that(35)A0v−1≤u−1,B0w−1≥v−1,C0u−1≥w−1,then we have(36)1v−1≥w−1B0,u−1≥A0w−1B0.
Then, the following system holds for odd solutions:(37)u2n+1=u−1,v2n+1,=B0w−1,w2n+1,=C0u−1.Theorem 4.
Suppose thatun,vn,wn is a solution of systems (1)–(3) such that(38)A0v−1≥u−1,B0w−1≤v−1,C0u−1≥w−1,then the following statement holds:(39)u2n+1=A0v−1,v2n+1,=v−1,w1,=C0u−1,w2n+3,=C0A0v−1.Theorem 5.
Suppose thatun,vn,wn is a solution of systems (1)–(3) such that(40)A0v−1≥u−1,B0w−1≤v−1,C0u−1≤w−1,then the following statement holds:(41)u2n+1=A0v−1,v2n+1,=v−1,w2n+1,=w−1.Theorem 6.
Suppose thatun,vn,wn is a solution of systems (1)–(3) such that(42)A0v−1≤u−1,B0w−1≥v−1,C0u−1≤w−1,then the following statement holds:(43)u2n+1=u−1,v2n+1,=B0w−1,w2n+1,=w−1.


## 2. Conclusion

We investigate the closed form solutions of an important type of difference equation. The eventual periodicity of the following max-type3D-system of difference equations is(44)un+1=maxAnvn−1,un−1vn+1,=maxBnwn−1,vn−1wn+1,=maxCnun−1,wn−1,where n∈No,No=N∪0,Ann∈No,Bnn∈No, and Cnn∈No are positive periodic sequences and initial conditions u0,u−1,v0,v−1,w0,w−1∈0,+∞.



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