1. Introduction
The introduction of chaos, fractal, and dynamical system could be found in many classical textbooks, such as Scheinerman [1] . A dynamical system has two parts, a state and a function. The second part of a dynamical system is a rule which tell us how the system changes over time. According to the time, we have the discrete and continuous system. The discrete dynamical system, in which we are interested, always does not have an analytical solution. Therefore, the behaviors of fixed points are very important. They play a vital role in the chaos, bifurcation, Julia sets problem in the dynamical system (see [2] [3] ). Those problems have been studied for last thirty years. Using the dynamics of functions near the real fixed points, the dynamics of functions in complex
plane were induced by the following researchers: The dynamics of families of entire functions
,
were studied by Prasad [4] , Kapoor and Prasad [2] , Sajid and Kapoor [5] , respectively. The dynamics of
is found in Devaney [6] . Recently, Sajid [7] [8] gave the results about the fixed points of one parameter family of function
. His work is motivated by the relationship of the function
with the well-known generating functions on base b by choosing
and
in the generalized Bernoulli generating function
The proofs in [7] and [8] are too
complicated. In this paper, we not only give a simple proof of the work of Sajid [7] , but also generalize his work.
2. Main Results
We will determine the fixed points of
where
(2-1)
i.e., we will solve the equation
Moreover, we also discuss the multiplicity and the behavior of the fixed points for two parameters b and n. For simplicity of notation, we denote
and
by
and
, respectively.
For
, the following results in Theorem 1 were in Sajid [7] , but we have a simpler proof.
Theorem 1. Let
where
is given in (2-1). Then
(1) The function
has a unique fixed point
for any ![]()
(2) The unique fixed point
is negative if
and positive if
Moreover, we have
is decreasing if
increasing if
as
is increasing and
(2-2)
(3) There exists
such that the fixed point
of the function
is (i) attracting, i.e.,
for
, (ii) rationally indifferent, i.e.,
at
, and (iii) repelling, i.e.,
for ![]()
Proof. Suppose that
and
. Statements (1) and (2) can be proved directly as follows. The expression
if and only if
because
Let
be the fixed
point of the function
for
Then the fixed point
(2-3)
is unique. Moreover, (2-3) easily implies statement (2).
Next, we proved statement (3). It is easy that
(2-4)
and the function
is contionus on
Therefore, substituting (2-3) into (2-4), we have
(2-5)
Hence,
(2-6)
(2-7)
(2-8)
Therefore, statement (3) are true by (2-6), (2-7), and (2-8).
The results about
depend on the parameters n and b. If n is odd, then the behavior of the
fixed points is similar to the case
If n is even, the behavior of the fixed points depends on the parameter b. We have the simple facts about the functions f and g. It is easy that
(2-9)
Hence, if the integer n is even, then
If the integer n is odd, then
(2-10)
Suppose that the fixed point of
exists. If n is odd and
, then the fixed point is negative. Otherwise, it is positive. Moreover,
(2-11)
and
is continuous in
Hence,
(2-12)
Lemma 2. Let
, where
, and
Then (1)
is concave downward in
if
is odd and
(2)
is concave upward in
otherwise.
Proof. Suppose that
and
.
![]()
![]()
Therefore, (2-9) implies that
and
have the opposite sign if and only if
is odd and
. Moreover,
since
Thus, it suffices to prove that
in
where ![]()
(2-4) implies that
We have
![]()
and
![]()
where
(2-13)
Let
Equation (2-13) is transformed to
(2-14)
Moreover, let
(2-14) is transformed to
(2-15)
In fact, the graph of
has 2 critical points, including
and
is the global minimum.
![]()
![]()
By the algorithm of bisection,
, and
It is easy to check that
and
Therefore,
is the minimum of H on
. This completes the proof of Lemma 2.
To study the behavior of the fixed points in Theorem 5, we need Lemma 3 and Lemma 4 as follows.
Lemma 3. Suppose that
(2-16)
Then (1) ![]()
and
(2) The function h is decreasing, and concave upward in
.
Proof. The statement (1) is easy. (2-16) implies
Specially, we have
. Therefore, the function h is decreasing in
. Moreover,
![]()
Let
Then
We have
for
, and
for
. Therefore,
, for
, and the function h is concave upward in
. The proof of Lemma 3 is completed.
Lemma 4. Suppose that
(2-17)
and
such that
(2-18)
and
(2-19)
Then there is a unique
, such that
(2-20)
Moreover, if
then
intersects
at exactly one point. If
then
does not intersect
If
then
intersects
at exactly two points.
Proof. Let
Then
(2-18) implies
The Intermediate Value Theorem of the continuous function implies
such that
i.e., Equation (2-20) holds
Suppose to the contrary that
is not unique.
There exist
such that
(2-21)
Then
by (2-17).
For
the Mean Value Theorem and (2-17) imply
such that
Therefore, the Equation (2-21) implies that
. It is impossible.
Suppose that
and
intersects
at
. Then we have
(2-20) and the uniqueness of
imply ![]()
Suppose to the contrary that
but
intersects
at the smallest
on
. Let
Then
and
. This implies ![]()
and there exists the minimum of
on
.
Let the minimum occurs at
. Then
and
. This contradicts to the unique- ness of
in (2-20).
Finally, suppose that
, we prove that
intersects
at exactly two points. Let
Then
and
. Intermediate Value Theorem and (2-19) imply that there exists
at which
intersects
.
Suppose to the contrary that
does not intersect
at exactly two points. By the previous proofs, the line
will intersect
at more than two points. Let
intersect
at the three consecutive points
, then
for
Without loss of generality, we may
suppose that
. Then
and
. It is impossible that
by (2-17). The proof of Lemma 4 is completed.
Theorem 5. Let
be even and
Then
(1) There exists a unique
such that
has a unique fixed point at
and
has two fixed points, say
,
and
for
, and no fixed points for ![]()
(2) Let n be fixed. If
, then
is decreasing, and
is increasing as
decreases.
(3) Let b be fixed. Then
is decreasing to 0 as n increases.
(4) Suppose
exist for fixed
. If
, then
is increasing, and
is decreasing as n increases. If
, then there exists a unique
, such
and
for any even n.
Moreover,
is decreasing if it is less than
, and
is increasing if it is less than
,
is increasing if it is greater than
, and
is decreasing if it is greater than
, as n increases.
(5) The fixed points
are attracting, rationally indifferent, and repelling, respectively.
Proof. Let
be even and ![]()
(1) We want to solve the equation
where
and
and
is given as in (2-1). Note that the equation
is equivalent to ![]()
Because of
and
we only consider ![]()
By
for
Lemma 4 implies that
the number of
intersections of
and
are exactly two for
, unique at
, and none for
. i.e.,
has two roots say
,
and
for
, one root, say
at
, and no root for
. Therefore,
(2-22)
(2) The statement (2) is easy by Lemma 2 and Part (1).
(3) In fact,
can be determined by solving the equation
(2-4) implies to solve
(2-23)
(2-1) and (2-4) imply to solve
(2-24)
Let
(2-24) is equivalent to
(2-25)
Lemma 3 and (2-25) imply that
is decreasing to 0 as n is increasing for any ![]()
(4) Suppose that
We have
Then
is increasing as even number increases by (2-9). Suppose that
Then
Let
be the solution of
. Therefore,
. Since
is increasing, and concave upward, Statement (4) holds.
(5) Let
be the fixed point of
Then
![]()
Let
Then
![]()
![]()
![]()
Lemma 3, (2-21) and
imply
i.e., the state- ment (5) holds.
Theorem 6. Let
be even,
Then
(1) The function
has a unique fixed point
which is increasing as
increases.
(2-27)
(2) Let
and
be fixed. If
then
is increasing as n increases. If
then there exists a unique number
, such
for any even n, and
is increasing if
, and decreasing if
as n increases.
(3) There exists
such that the fixed point
of the function
is (i) attracting, i.e.,
for
, (ii) rationally indifferent, i.e.,
at
, and (iii) repelling, i.e.,
for ![]()
Proof. The proof of Theorem 6 is similar to that of Theorem 5. We just mention some key points. The function f is positive, decreasing, and concave upward. Let
be the fixed point of
Then
(2-26)
Let
Then
for
and ![]()
Lemma 3 and (2-27) imply that there exists a unique
such that
, and
if
and
if ![]()
Theorem 7. Let n be odd. Then
(1)
has a unique fixed point
for any
The fixed point
is negative if
and positive if
Moreover, we have
(2-28)
(2) Let the parameter n be fixed. Then
is increasing if
and decreasing if
as
increases.
(3) Let the parameter
be fixed. If
then
is increasing as n increases. If
then there exists
a unique number
such
for any odd n, and
is increasing if
, and decreasing if
as n increases. If
, then
is increasing as n increases. If
, then there exists a unique
such that
for any odd n. Furthermore,
is decreasing if it is less than
, and increasing if it is greater than
as n increases.
(4) There exists
such that the fixed point
of the function
is (i) attracting, i.e.,
for
, (ii) rationally indifferent, i.e.,
at
, and (iii) repelling, i.e.,
for ![]()
Proof. The proof of Theorem 7 is similar to that of Theorem 5. We just also mention some key points. The function f is decreasing if
and f is positive, concave upward if
, and negative, concave
downward if
Let
be the fixed point of
Then
by (2-26). Let
Then
for
by statement (1). Therefore,
for
. Lemma 3 and (2-28) imply that there exists a unique
such that
, and
if and if
3. Discussion
The Sarkovskii’s theorem said that let the function
be continuous and it has points of prime period 3, then the function f has points of period k for all positive integer k. We also know that the dynamical behavior
of
is very complicated, see Scheinerman [1] . In our problem,
has points of prime period 3 if
,
, and
. We anticipate the problem we studied will have complicated dynamical behavior as that of
. On the other hand, the bifurcation of
will be interesting.