Int. Journal of Math. Analysis, Vol. 8, 2014, no. 18, 891 - 894
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ijma.2014.4383
On Fixed Points of One Parameter Family
of Function
x
bx −1
II
Mohammad Sajid
College of Engineering, Qassim University
Buraidah, Al-Qassim, Saudi Arabia
c 2014 Mohammad Sajid. This is an open access article distributed under
Copyright the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
Abstract
In this paper, the real fixed points of one parameter family T =
fλ (x) = λ bxx−1 and fλ (0) = lnλb : λ < 0, x ∈ R, b > 0, b = 1 are investigated. Further, the nature of these fixed points of fλ ∈ T are shown
for b > 0 except b = 1.
Mathematics Subject Classification: 37C25
Keywords: Fixed points
1
Introduction
The fixed points play very important role in the dynamics of functions. Many
researchers are induced the dynamics of functions in complex plane using real
dynamics near to real fixed points [2, 3, 4, 5, 7]. The theory of fixed points of
functions and its applications can be seen in [1].
A point x is said to be a fixed point of function f (x) if f (x) = x. A fixed
point x0 is called an attracting, neutral (indifferent) or repelling if |f (x0 )| < 1,
|f (x0 )| = 1 or |f (x0 )| > 1 respectively.
The real fixed points of the family of functions λ bxx−
892
Mohammad Sajid
T =
x
λ
and fλ (0) =
: λ < 0, x ∈ R, b > 0, b = 1 .
fλ (x) = λ x
b −1
ln b
This paper is organized as follows: In Theorem 2.1, the real fixed points of
fλ ∈ T are determined. The nature of these fixed points of fλ ∈ T are shown
in Theorem 2.3.
2
Fixed Points of fλ ∈ T and their Nature
In the following theorem, it is found that the function fλ ∈ T has a unique
real fixed point:
Theorem 2.1 Let fλ ∈ T . Then, the function fλ (x) has a unique real fixed
point xλ for −1 < λ < 0 and no real fixed points for λ < −1. The fixed point
xλ is positive if 0 < b < 1 and negative if b > 1.
Proof. For fixed points of fλ (x), we have fλ (xλ ) = xλ . The solution of this
equation is xλ = ln(1+λ)
. This solution is real if −1 < λ < 0 and no real
ln b
solution if λ < −1. Moreover, xλ > 0 if 0 < b < 1 and xλ < 0 if b > 1.
Therefore, the function fλ (x) has a unique real fixed point. This real fixed
point of fλ (x) is positive for 0 < b < 1 and negative for b > 1. The proof is
completed.
x
Let fλ (x) = λf (x), where f (x) = bx −1 . Further, consider the function
ψ(x) = xf (x) + f (x). It is easily found that the function ψ(x) is continuous
at x = 0 and ψ(0) = ln1b . In [6], the following lemma is proved which is used
in the sequel:
Lemma 2.2 (a) For 0 < b < 1,
⎧
⎨ < 0 for x∗1 < x < 0
x
x
[(2 − x ln b)b − 2] = 0 for x = x∗1
ψ(x) = x
2
⎩
(b − 1)
> 0 for x < x∗1
where x∗1 is the unique negative real root of the equation (2 − x ln b)bx − 2 =
0.
(b) For b > 1
⎧
⎨ > 0 for 0 < x < x∗2
x
x
[(2 − x ln b)b − 2] = 0 for x = x∗2
ψ(x) = x
⎩
(b − 1)2
< 0 for x > x∗2
where x∗2 is the unique positive real root of the equation (2−x ln b)bx −2 = 0.
On fixed points of one parameter family of function
893
Let us denote
λ∗ = −
x∗
∗
= −(bx − 1)
∗
f (x )
(1)
where x∗ is the unique real root of the equation (2 − x ln b)bx − 2 = 0. The
root x∗ is negative if 0 < b < 1 and positive if b > 1.
The nature of fixed points of fλ ∈ T are shown in the following theorem:
Theorem 2.3 Let fλ ∈ T . Then, the fixed point xλ of the function fλ (x)
is (i) an attracting for λ∗ < λ < 0 (ii) rationally indifferent for λ = λ∗ (iii)
repelling for −1 < λ < λ∗ .
x
Proof. The function − f (x)
is increasing on R+ for 0 < b < 1 and is decreasing
d
x
− f (x)
is positive for x ∈ R+ , 0 < b < 1 and negative
on R− for b > 1 since dx
for x ∈ R− , b > 1. It is also seen that f (x) < 0 for 0 < b < 1 and f (x) > 0 for
b > 1.
For 0 < b < 1, the following cases are proved :
x
(i) For λ∗ < λ < 0, since the function − f (x)
is increasing on R+ and λ =
∗
xλ
xλ
x∗
xλ
− f (x
, we have − f (x
< bx . Hence, xλ > x∗ .
∗ ) < − f (x ) . It gives that b
λ)
λ
λ)
By Lemma 2.2(a), ψ(xλ ) < 0. Since fλ (xλ ) = − ψ(x
+ 1, it shows that
f (xλ )
λ)
< 0. Since fλ (x) is positive on R+ , it follows that
fλ (xλ ) − 1 = − ψ(x
f (xλ )
0 < fλ (xλ ) < 1. Thus, the fixed point xλ of fλ (x) is an attracting for
λ∗ < λ < 0 .
(ii) For λ = λ∗ , it is easily seen that xλ = x∗ . By Lemma 2.2(a), it concludes
λ)
that fλ (xλ ) − 1 = − ψ(x
= 0 which implying fλ ∗ (xλ ) = 1. Therefore, it
f (xλ )
gives that the fixed point xλ of fλ (x) is rationally indifferent for λ = λ∗ .
(iii) For −1 < λ < λ∗ , by using analogous arguments as (i), it shows that
xλ < x∗ . By Lemma 2.2(a), we get ψ(xλ ) > 0. It deduce that fλ (xλ ) −
λ)
> 0 and consequently fλ (xλ ) > 1. Thus, xλ is repelling fixed
1 = − ψ(x
f (xλ )
point of f (xλ ) for −1 < λ < λ∗ .
For b > 1, the proof of all cases (i), (ii) and (iii) are easily deduced similar as
above by using Lemma 2.2(b).
From Theorem 2.3, it is realized that when parameter λ crosses certain
parameter value λ∗ , then the nature of the fixed point is changed.
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2000.
894
Mohammad Sajid
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Received: March 19, 2014