Nonlinear Analysis and Differential Equations, Vol. 5, 2017, no. 2, 89 - 98
HIKARI Ltd, www.m-hikari.com
https://doi.org/10.12988/nade.2017.711
New Fixed Point Theorems for Hybrid Kannan
Type Mappings and MT (λ)-Functions
Wei-Shih Du
Department of Mathematics
National Kaohsiung Normal University
Kaohsiung 82444, Taiwan
c 2017 Wei-Shih Du. This article is distributed under the Creative Commons
Copyright Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
Abstract
In this paper, we first establish a new fixed point theorem for hybrid
Kannan type mappings and MT (λ)-functions which generalizes and
improves Kannan’s fixed point theorem. As interesting applications of
our new result, many new fixed point theorems are given. Our these
new results in this paper are original and quite different from the well
known generalizations in the literature.
Mathematics Subject Classification: 47H10, 54H25
Keywords: MT -function (or R-function), eventually nonincreasing (resp.
nondecreasing) sequence, eventually strictly decreasing (resp. increasing) sequence, MT (λ)-function, Kannan’s fixed point theorem, hybrid Kannan type
mapping
1. Introduction and preliminaries
Let (X, d) be a metric space and T : X → X be a selfmapping. A point x
in X is a fixed point of T if T x = x. The set of fixed points of T is denoted
by F(T ). Throughout this paper, we denote by N and R, the sets of positive
integers and real numbers, respectively.
The celebrated Banach contraction principle [1] is one of the best known
fixed point theorem in metric fixed point theory. The Banach contraction
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Wei-Shih Du
principle plays an important role in nonlinear analysis and applied mathematical analysis and has been generalized in many various different directions; for
more detail, one can refer to [2-14] and references therein. In 1969, Kannan
[8] established his interesting fixed point theorem which is different from the
Banach contraction principle. Since then a number of generalizations of Kannan’s fixed point theorem have been investigated by several authors; see, e.g.,
[2, 6, 13] and references therein.
Theorem 1.1. (Kannan [8]) Let (X, d) be a complete metricspace
and
1
T : X → X be a self-mapping on X. Suppose that there exists γ ∈ 0, 2 such
that
d(T x, T y) ≤ γ(d(x, T x) + d(y, T y)) for all x, y ∈ X.
Then T admits a unique fixed point in X.
Let f be a real-valued function defined on R. For c ∈ R, we recall that
lim sup f (x) = inf
sup f (x).
ε>0 c<x<ε+c
x→c+
Definition 1.1. [2-6] A function ϕ : [0, ∞) → [0, 1) is said to be an MT f unction (or R-f unction) if lim sup ϕ(s) < 1 for all t ∈ [0, ∞).
s→t+
It is obvious that if ϕ : [0, ∞) → [0, 1) is a nondecreasing function or
a nonincreasing function, then ϕ is an MT -function. So the set of MT functions is a rich class.
Definition 1.2. [5] A real sequence {an }n∈N is called
(i) eventually strictly decreasing if there exists ` ∈ N such that an+1 < an
for all n ∈ N with n ≥ `;
(ii) eventually strictly increasing if there exists ` ∈ N such that an+1 > an
for all n ∈ N with n ≥ `;
(iii) eventually nonincreasing if there exists ` ∈ N such that an+1 ≤ an for all
n ∈ N with n ≥ `;
(iv) eventually nondecreasing if there exists ` ∈ N such that an+1 ≥ an for
all n ∈ N with n ≥ `.
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New fixed point theorems
Very recently, Du [5] presented some new characterizations of MT -functions
linked with eventually nonincreasing and eventually strictly decreasing sequences as follows. Although the following result was essentially proved in
[5], we give its proof here for the sake of completeness and the readers convenience.
Theorem 1.2 (see [5, Theorem 2.1]). Let ϕ : [0, ∞) → [0, 1) be a function. Then the following statements are equivalent.
(a) ϕ is an MT -function.
(1)
∈ [0, 1) and εt
(2)
∈ [0, 1) and εt
(3)
∈ [0, 1) and εt
(4)
∈ [0, 1) and εt
(b) For each t ∈ [0, ∞), there exist rt
(1)
(1)
ϕ(s) ≤ rt for all s ∈ (t, t + εt ).
(c) For each t ∈ [0, ∞), there exist rt
(2)
(2)
ϕ(s) ≤ rt for all s ∈ [t, t + εt ].
(d) For each t ∈ [0, ∞), there exist rt
(3)
(3)
ϕ(s) ≤ rt for all s ∈ (t, t + εt ].
(e) For each t ∈ [0, ∞), there exist rt
(4)
(4)
ϕ(s) ≤ rt for all s ∈ [t, t + εt ).
(1)
> 0 such that
(2)
> 0 such that
(3)
> 0 such that
(4)
> 0 such that
(f) For any nonincreasing sequence {xn }n∈N in [0, ∞), we have 0 ≤ sup ϕ(xn ) <
n∈N
1.
(g) ϕ is a function of contractive factor; that is, for any strictly decreasing
sequence {xn }n∈N in [0, ∞), we have 0 ≤ sup ϕ(xn ) < 1.
n∈N
(h) For any eventually nonincreasing sequence {xn }n∈N in [0, ∞), we have
0 ≤ sup ϕ(xn ) < 1.
n∈N
(i) For any eventually strictly decreasing sequence {xn }n∈N in [0, ∞), we
have 0 ≤ sup ϕ(xn ) < 1.
n∈N
Proof. The equivalence of statements (a)-(g) was indeed proved in [3, Theorem 2.1]. The implications ”(h) ⇒ (f)” and ”(i) ⇒ (g)” are obvious. Let us
prove ”(f) ⇒ (h)”. Suppose that (f) holds. Let {xn }n∈N be an eventually nonincreasing sequence in [0, ∞). Then there exists ` ∈ N such that xn+1 ≤ xn
for all n ∈ N with n ≥ `. Put yn = xn+`−1 for n ∈ N. So {yn }n∈N is a
nonincreasing sequence in [0, ∞). By (f), we obtain
0 ≤ sup ϕ(yn ) < 1.
n∈N
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Wei-Shih Du
Let γ := sup ϕ(yn ). Then
n∈N
0 ≤ ϕ(xn+`−1 ) = ϕ(yn ) ≤ γ < 1 for all n ∈ N.
Hence we get
0 ≤ ϕ(xn ) ≤ γ < 1 for all n ∈ N with n ≥ `.
Let
η := max{ϕ(x1 ), ϕ(x2 ), · · · , ϕ(x`−1 ), γ} < 1.
Then ϕ(xn ) ≤ η for all n ∈ N. Hence 0 ≤ sup ϕ(xn ) ≤ η < 1 and (h) holds.
n∈N
Similarly, we can varify ”(g) ⇒ (i)”. Therefore, from above, we prove that the
statements (a)-(i) are all logically equivalent. The proof is completed.
In [5], Du introduced the concept of MT (λ)-function which generalizes the
concept of P-function [11].
Definition 1.3 [5]. Let λ > 0. A function µ : [0, ∞) → [0, λ) is said to be
an MT (λ)-function if lim sup µ(s) < λ for all t ∈ [0, ∞).
s→t+
Remark 1.1.
(i) Obviously, an MT -function is an MT (1)-function and a P-function is
an MT 21 -function.
(ii) It is easy to see that µ is an MT (λ)-function if and only if λ−1 µ is an
MT -function.
The following characterizations of MT (λ)-functions is an immediate consequence of Theorem 1.2.
Theorem 1.3 (see [5, Theorem 2.4]). Let λ > 0 and let µ : [0, ∞) →
[0, λ) be a function. Then the following statements are equivalent.
(1) µ is an MT (λ)-function.
(2) λ−1 µ is an MT -function.
(1)
∈ [0, λ) and t
(2)
∈ [0, λ) and t
(3) For each t ∈ [0, ∞), there exist ξt
(1)
(1)
µ(s) ≤ ξt for all s ∈ (t, t + t ).
(4) For each t ∈ [0, ∞), there exist ξt
(2)
(2)
µ(s) ≤ ξt for all s ∈ [t, t + t ].
(1)
> 0 such that
(2)
> 0 such that
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New fixed point theorems
(3)
∈ [0, λ) and t
(4)
∈ [0, λ) and t
(5) For each t ∈ [0, ∞), there exist ξt
(3)
(3)
µ(s) ≤ ξt for all s ∈ (t, t + t ].
(6) For each t ∈ [0, ∞), there exist ξt
(4)
(4)
µ(s) ≤ ξt for all s ∈ [t, t + t ).
(3)
> 0 such that
(4)
> 0 such that
(7) For any nonincreasing sequence {xn }n∈N in [0, ∞), we have 0 ≤ sup µ(xn ) <
n∈N
λ.
(8) For any strictly decreasing sequence {xn }n∈N in [0, ∞), we have 0 ≤
sup µ(xn ) < λ.
n∈N
(9) For any eventually nonincreasing sequence {xn }n∈N in [0, ∞), we have
0 ≤ sup µ(xn ) < λ.
n∈N
(10) For any eventually strictly decreasing sequence {xn }n∈N in [0, ∞), we
have 0 ≤ sup µ(xn ) < λ.
n∈N
Remark 1.2. [11, Lemma 3.1] is a special case of Theorem 1.3 for λ = 12 .
In this work, we first establish a new fixed point theorem for hybrid Kannan
type mappings and MT (λ)-functions which generalizes and improves Kannan’s fixed point theorem. As interesting applications of our new result, many
new fixed point theorems are given. Consequently, our these new results in
this paper are original and quite different from the well known generalizations
in the literature.
2. Some new fixed point theorems
In this section, we first establish a new fixed point theorem for hybrid
Kannan type mappings and MT (λ)-functions which is one of the main results
of this paper and generalizes Kannan’s fixed point theorem.
Theorem 2.1. Let (X, d) be a complete metric space and T : X → X be a
selfmapping on X. Suppose that
(H) there exists an MT
1
2
-function η : [0, ∞) → 0, 21 such that
min{d(T x, T y), d(x, T x)} ≤ η(d(x, y))(d(x, T x) + d(y, T y))
for all x, y ∈ X with x 6= y.
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Wei-Shih Du
Then T admits a fixed point in X.
Proof. Given z ∈ X. If T z = z, then we are done. Otherwise, if T z 6= z,
we define a sequence {xn }n∈N by x1 = z and xn+1 = T xn for all n ∈ N.
Since
η(t) < 12 for all t ∈ [0, ∞), we can define a function ξ : [0, ∞) → 0, 21 by
1 1
ξ(t) =
+ η(t)
for all t ∈ [0, ∞).
2 2
Clearly, 0 ≤ η(t) < ξ(t) <
condition (H), we have
1
2
for all t ∈ [0, ∞). Since x2 = T x1 6= x1 , by
d(x3 , x2 ) = min{d(T x2 , T x1 ), d(x2 , T x2 )}
≤ η(d(x2 , x1 ))(d(x2 , T x2 ) + d(x1 , T x1 ))
< ξ(d(x2 , x1 ))(d(x3 , x2 ) + d(x2 , x1 )).
If T x2 = x2 then x2 ∈ F(T ) and the desired conclusion is proved. Assume
T x2 6= x2 . Then x3 6= x2 . By (H) again, we obtain
d(x4 , x3 ) = min{d(T x3 , T x2 ), d(x3 , T x3 )}
< ξ(d(x3 , x2 ))(d(x4 , x3 ) + d(x3 , x2 )).
So, by induction, if xn+1 6= xn then we can get
d(xn+2 , xn+1 ) < ξ(d(xn+1 , xn ))(d(xn+2 , xn+1 ) + d(xn+1 , xn )) for all n ∈ N.
(2.1)
Suppose there exists k ∈ N such that d(xk+1 , xk ) < d(xk+2 , xk+1 ). Thus the
inequality (2.1) implies
d(xk+2 , xk+1 ) < 2ξ(d(xk+1 , xk ))d(xk+2 , xk+1 ) < d(xk+2 , xk+1 ),
which leads a contradiction. Hence it must be
d(xn+2 , xn+1 ) ≤ d(xn+1 , xn )
for all n ∈ N.
(2.2)
By (2.2), we know that the sequence {d(xn+1 , xn )}n∈N is nonincreasing in
[0, ∞). Since η is an MT 21 -function, by applying Theorem 1.3, we have
0 ≤ sup η(d(xn+1 , xn )) < 21 and then deduces
n∈N
1 1
1
+ sup η(d(xn+1 , xn )) < .
0 < sup ξ(d(xn+1 , xn )) =
2 2 n∈N
2
n∈N
Let λ := sup ξ(d(xn+1 , xn )) and γ := 2λ. So γ ∈ (0, 1). For any n ∈ N, by
n∈N
(2.1) and (2.2), we get
d(xn+2 , xn+1 ) < ξ(d(xn+1 , xn ))(d(xn+2 , xn+1 ) + d(xn+1 , xn ))
≤ γd(xn+1 , xn )
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New fixed point theorems
and hence
d(xn+2 , xn+1 ) < γd(xn+1 , xn ) < · · · < γ n d(x2 , x1 ).
Let cn =
that
γ n−1
d(x2 , x1 ),
1−γ
(2.3)
n ∈ N. For m, n ∈ N with m > n, we get from (2.3)
d(xm , xn ) ≤
m−1
X
d(xj+1 , xj ) < cn .
(2.4)
j=n
Since 0 < γ < 1, lim cn = 0 and hence (2.4) implies
n→∞
lim sup{d(xm , xn ) : m > n} = 0.
n→∞
This proves that {xn }n∈N is a Cauchy sequence in X. By the completeness of
X, there exists v ∈ X such that xn → v as n → ∞. We will now finish the
proof by showing that v ∈ F(T ). Clearly, the condition (H) still holds for
x = y ∈ X. So, by (H), we have
1
(d(xn+1 , T v) + d(v, T v) − |d(xn+1 , T v) − d(v, T v)|)
2
= min{d(T xn , T v), d(v, T v)}
1
< (d(v, T v) + d(xn , xn+1 ))
for all n ∈ N.
2
Since xn → v as n → ∞, taking the limit as n tends to infinity on the last
inequality yields
1
d(v, T v) ≤ d(v, T v)
2
and hence deduces d(v, T v) = 0. So we obtain v ∈ F(T ). The proof is
completed.
The following new fixed point theorem is immediate from Theorem 2.1.
Corollary 2.1. Let (X, d) be a complete metric space
and T : X → X be a
1
selfmapping
on X. Suppose that there exists an MT 2 -function η : [0, ∞) →
1
0, 2 such that
d(T x, T y) ≤ η(d(x, y))(d(x, T x) + d(y, T y))
for all x, y ∈ X with x 6= y.
(2.5)
Then T admits a unique fixed point in X.
Proof. Applying Theorem 2.1, F(T ) 6= ∅. We claim that F(T ) is a singleton
set. Assume that there exist u, v ∈ F(T ) with u 6= v. By (2.5), we obtain
0 < d(u, v) = d(T u, T v) ≤ η(d(u, v))(d(u, T u) + d(v, T v)) = 0,
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Wei-Shih Du
a contradiction. Therefore F(T ) is a singleton set and T has a unique fixed
point in X.
Remark 2.1.
(i) Kannan’s fixed point theorem [8] is a special case of Theorem 2.1 and
Corollary 2.1;
(ii) If ” min{d(T x, T y), d(x, T x)}” is replaced with ” min{d(T x, T y), d(y, T y)}”
in condition (H) of Theorem 2.1, by symmetric, the conclusion of Theorem 2.1 still holds.
Corollary 2.2. Let (X, d) be a complete metric space
and T : X → X be a
1
on X. Suppose that there exists an MT 2 -function η : [0, ∞) →
selfmapping
0, 21 such that
d(x, T x) ≤ η(d(x, y))(d(x, T x) + d(y, T y))
for all x, y ∈ X with x 6= y.
Then T admits a fixed point in X.
Corollary 2.3. Let (X, d) be a complete metric space
and T : X → X be a
1
selfmapping on X. Suppose that there exists γ ∈ 0, 2 such that
d(x, T x) ≤ γ(d(x, T x) + d(y, T y))
for all x, y ∈ X with x 6= y.
Then T admits a fixed point in X.
We can establish the following new fixed point theorems by applying Theorem 2.1 immediately.
Theorem 2.2. Let (X, d) be a complete metric space and T : X → X be a
on X. Suppose that there exists an MT 21 -function η : [0, ∞) →
selfmapping
0, 21 such that
d(T x, T y)+d(x, T x) ≤ 2η(d(x, y))(d(x, T x)+d(y, T y))
for all x, y ∈ X with x 6= y.
(2.6)
Then T admits a fixed point in X.
Remark 2.2. The conclusion of Theorem 2.2 is still true if ”d(T x, T y) +
d(x, T x)” is replaced with ”d(T x, T y) + d(y, T y)” in (2.6).
Theorem 2.3. Let (X, d) be a complete metric space and T : X → X be a
on X. Suppose that there exists an MT 21 -function η : [0, ∞) →
selfmapping
1
0, 2 such that
p
d(T x, T y)d(x, T x) ≤ η(d(x, y))(d(x, T x)+d(y, T y)) for all x, y ∈ X with x 6= y.
(2.7)
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New fixed point theorems
Then T admits a fixed point in X.
p
d(T x, T y)d(x, T x)”
Remark 2.3. The
conclusion
of
Theorem
2.3
is
still
true
if
”
p
is replaced with ” d(T x, T y)d(y, T y)” in (2.7).
In fact, we can establish a wide generalization of Theorem 2.2 as follows.
Theorem 2.4. Let (X, d) be a complete metric spaceand T : X → X be a
self-mapping
on X. Suppose that there exists an MT 12 -function η: [0, ∞) →
1
0, 2 such that
αd(T x, T y) + βd(x, T x)
≤ η(d(x, y))(d(x, T x) + d(y, T y))
α+β
(2.8)
for all x, y ∈ X with x 6= y,
where α and β are nonnegative real numbers with α + β > 0. Then T admits
a fixed point in X.
Remark 2.4.
(i) We can also obtain Theorem 2.2 if we take α = β = 1 in Theorem 2.4;
y)+βd(x,T x)
(ii) The conclusion of Theorem 2.4 is still true if ” αd(T x,T α+β
” is rey)+βd(y,T y)
placed with ” αd(T x,Tα+β
” in (2.8).
Acknowledgements. This research was supported by grant no. MOST
105-2115-M-017-002 of the Ministry of Science and Technology of the Republic
of China.
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Received: January 20, 2017; Published: February 1, 2017
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