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We establish sufficient conditions of the multiplicity of real fixed points of two-parameter family . Moreover, the behaviors of these fixed points are studied.

The introduction of chaos, fractal, and dynamical system could be found in many classical textbooks, such as Scheinerman [

plane were induced by the following researchers: The dynamics of families of entire functions

complicated. In this paper, we not only give a simple proof of the work of Sajid [

We will determine the fixed points of

i.e., we will solve the equation

For

Theorem 1. Let

(1) The function

(2) The unique fixed point

(3) There exists

Proof. Suppose that

point of the function

is unique. Moreover, (2-3) easily implies statement (2).

Next, we proved statement (3). It is easy that

and the function

Hence,

Therefore, statement (3) are true by (2-6), (2-7), and (2-8).

The results about

fixed points is similar to the case

Hence, if the integer n is even, then

Suppose that the fixed point of

and

Lemma 2. Let

Proof. Suppose that

Therefore, (2-9) implies that

(2-4) implies that

and

where

Let

Moreover, let

In fact, the graph of

By the algorithm of bisection,

To study the behavior of the fixed points in Theorem 5, we need Lemma 3 and Lemma 4 as follows.

Lemma 3. Suppose that

Then (1)

Proof. The statement (1) is easy. (2-16) implies

Let

Lemma 4. Suppose that

and

Then there is a unique

Moreover, if

Proof. Let

Suppose to the contrary that

There exist

Then

For

Suppose that

Suppose to the contrary that

and there exists the minimum of

Let the minimum occurs at

Finally, suppose that

Suppose to the contrary that

suppose that

Theorem 5. Let

(1) There exists a unique

(2) Let n be fixed. If

(3) Let b be fixed. Then

(4) Suppose

Moreover,

(5) The fixed points

Proof. Let

(1) We want to solve the equation

Because of

By

intersections of

(2) The statement (2) is easy by Lemma 2 and Part (1).

(3) In fact,

(2-1) and (2-4) imply to solve

Let

Lemma 3 and (2-25) imply that

(4) Suppose that

(5) Let

Let

Lemma 3, (2-21) and

Theorem 6. Let

(1) The function

(2) Let

(3) There exists

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

Let

Lemma 3 and (2-27) imply that there exists a unique

Theorem 7. Let n be odd. Then

(1)

(2) Let the parameter n be fixed. Then

(3) Let the parameter

a unique number

(4) There exists

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

downward if

The Sarkovskii’s theorem said that let the function

of