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NuriyeAtaseverand˙
IbrahimYalcinkaya
Hence,weobtainthatthereexist
n→∞
n→∞
lim
lim
x(3k+3)n1(3k+2)1I1j
x(3k+3)n1(3k+1)1I2j
...j
n→∞
lim
x(3k+3)n1I3k+3.
(b)(I1jI2j...jI3k+3jI1jI2j...jI3k+3j...)isasolutionwithperiod(3k+
3)ofEq.(1).
(c)InviewoftheEq.(1),weobtain
x(3k+3)n+m1
1+x(3k+3)n1(k+11m)x(3k+3)n1(2k+21m)
x(3k+3)n1(3k+31m)
form11j2j3j...j(k+1).
Takethelimitsonbothsidesoftheaboveequality
n→∞
lim
x(3k+3)n+m1lim
n→∞
1+x(3k+3)n1(k+11m)x(3k+3)n1(2k+21m)
x(3k+3)n1(3k+31m)
fromtheaboveequality,weobtain
Im·Im+(k+1)·Im+(2k+2)10.
(d)Ifthereexistsno∈Nsuchthat
xn1(2k+1)≥xn+1foralln>no
thenweget
I1≥I2k+3≥Ik+2≥I1≥I2k+3j
I2≥I2k+4≥Ik+3≥I2≥I2k+4j
.
.
.
Ik≥I3k+2≥I2k+1≥Ik≥I3k+2j
Ik+1≥I3k+3≥I2k+2≥Ik+1≥I3k+3.
Hence,
n→∞
lim
xn10.
(e)Subractingxn1(3k+2)fromboththeleftandright-handsidesofthe
Eq.(1)weobtain
xn+1−xn1(3k+2)1
1+xn1kxn1(2k+1)
1
(xn1k−xn1(4k+3))
