Fixed Points of Two-Parameter Family of Function 1 n x x b λ −

We establish sufficient conditions of the multiplicity of real fixed points of two-parameter family ( ) ( ) ( )       =           1 ; , if 0, 0; , : , 0, 0, 1 1 ln n x n x f x b n x f b n n b b b b λ λ λ λ = ≠ = ∈ > > ≠ −   . Moreover, the behaviors of these fixed points are studied.


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

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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.

Main Results
We will determine the fixed points of ; , ; , , and ; , ; , 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 λ > (2) The unique fixed point x λ is negative if 0 1, b < < and positive if (3) There exists * 0 λ > such that the fixed point x λ of the function ( ) = , and (iii) repelling, i.e., ( ) 1, 1) and ( 2) can be proved directly as follows.The expression ( ) λ > Then the fixed point ( ) is unique.Moreover, (2-3) easily implies statement (2).Next, we proved statement (3).It is easy that and the function g′ is contionus on .
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 1. n = 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 ( ) ( ) Hence, if the integer n is even, then Suppose that the fixed point of ( )  , where ( ) Thus, it suffices to prove that ( ) ( ) where ( ) We have ( ) By the algorithm of bisection,  .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 ( ) The function h is decreasing, and concave upward in ( ) Proof.The statement (1) is easy.(2-16) implies ( ) ( ) Therefore, the function h is decreasing in ( ) Then there is a unique Intermediate Value Theorem of the continuous function implies  holds.Suppose to the contrary that * x is not unique.There exist * * 1 2 ,  the Mean Value Theorem and (2-17) imply ( ) and y mx = intersects ( ) x .Then we have ( ) (2-20) and the uniqueness of * x imply * 0 .x x = Suppose to the contrary that ( ) and there exists the minimum of 1 Let the minimum occurs at * 1 . This contradicts to the unique- Finally, suppose that ( ) , we prove that y mx = intersects ( ) Suppose to the contrary that y mx = does not intersect ( ) y h x = at exactly two points.By the previous proofs, the line y mx = will intersect ( ) Without loss of generality, we may suppose that ( ) ( ) (2-17).The proof of Lemma 4 is completed.Theorem 5. Let n be even and 0  x λ is decreasing if it is less than 0 x , and * , , x , and * , , u n x λ is decreasing if it is greater than 0 x , as n increases.
Proof.Let n be even and 0 (1) We want to solve the equation is given as in (2-1).Note that the equation ( ) (2) The statement (2) is easy by Lemma 2 and Part (1).
x u b = (2-24) is equivalent to ( ) Lemma 3 and (2-25) imply that * * ,n x λ is decreasing to 0 as n is increasing for any ( ) We have ( ) x be the solution of ( ) f x is increasing, and concave upward, Statement (4) holds.
(5) Let x λ be the fixed of ).
(3) There exists * 0 λ > such that the fixed point x λ of the function ( ) = , and (iii) repelling, i.e., ( ) 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 * x λ be the fixed point of ( ).
Lemma 3 and (2-27) imply that there exists a unique * λ such that ( ) λ λ > Theorem 7. Let n be odd.Then (1) (3) Let the parameter λ be fixed.x such that ( ) 0 1 f x = − for any odd n.Furthermore, * x λ is decreasing if it is less than 0 x , and in- creasing if it is greater than 0 x as n increases.

Discussion
The Sarkovskii's theorem said that let the function : f →   be continuous and it has points of prime period by Prasad [4],Kapoor and Prasad [2], Sajid and Kapoor [5], respec- tively.The dynamics of e z λ is found in Devaney[6].Recently, Sajid [7] [8]  gave the results about the fixed points of one parameter family of function 1 the following results in Theorem 1 were in Sajid [7], but we have a simpler proof.fixed point x λ for any 0.
increasing as λ decreases.
Let λ and b be fixed.If , b e ≥ then * x λ is increasing as n increases.If 1, for any even n, and * The proof of Theorem 7 is similar to that of Theorem 5. We just also mention some key points.The function f