Treść książki
Przejdź do opcji czytnikaPrzejdź do nawigacjiPrzejdź do informacjiPrzejdź do stopki
8
ElsayedM.Elsayed
x5n+3=
x5n+1(1+x5n+2)
x5n+2
=
(k+kh+h)h(1+
k(1+h)
(k(1+h)+h)
1
)
=
h(k+kh+h+1)
k(1+h)
=
h(k+1)(h+1)
k(1+h)
=
h(k+1)
k
.
Thus,theproofiscompleted.
.
Theorem2.Supposethat{xn}∞
nl11beasolutionofequation(3).Then
allsolutionsofequation(3)areperiodicwithperiodfive.
Proof.FromEq(3),weseethat
xn+1=
xn11(1+xn)
xn
j
xn+2=
xn(1+xn+1)
xn+1
=
xnxn11(1+xn)(1+
xn
xn11(1+xn)
xn
)
xn+3=
=
(xn11(1+xn)+xn)
xn+1(1+xn+2)
xn+2
1
.
xn11(1+xn)
=
(xn11+xnxn11+xn)xn(1+
(xn11+xnxn11+xn)
1
)
=
xn(xn11+xnxn11+xn+1)
xn11(1+xn)
=
xn(1+xn11)
xn11
.
xn+4=
xn+2(1+xn+3)
xn+3
=
xn(1+xn11)(1+
xn11(xn11+xnxn11+xn)
xn(1+xn11)
xn11
)
=
xn11(xn11+xnxn11+xn)
(xn(1+xn11)+xn11)
=xn11.
xn+5=
xn+3(1+xn+4)
xn+4
=
xn11(1+xn11)xn
xn11(1+xn11)
=xn.
Thiscompletestheproof.
.
Theorem3.Eq(3)havethreeequilibriumpointswhichare0,
√511
2
,
1√511
2
.
Proof.FortheequilibriumpointsofEq(3),wecanwrite
x=
x(x+1)
x
.
