The 22nd Annual Meeting in Mathematics (AMM 2017)
Department of Mathematics, Faculty of Science
Chiang Mai University, Chaing Mai, Thailand
Common fixed point theorems for generalized α-ϕ
Geraghty contraction mappings with two metrics in
complete metric spaces
K. Sukma†‡ , C. Thangthong, and P. Charoensawan
Department of Mathematics, Faculty of Science
Chiang Mai University, Chiang Mai 50200, Thailand
Abstract
The purpose of this paper is to present some existence and uniqueness results for common
fixed point theorems for generalized α-ϕ Geraghty contraction mappings with two metrics.
In addition, Our results generalize and extend the results presented in [6–8].
Keywords: Fixed point, Coincidence point, Geraghty, Admissible.
2010 MSC: 47H04, 47H10.
1
Introduction and Preliminaries
Let F be the family of all functions β : [0, ∞) → [0, 1) which satisfies the condition
lim β(tn ) = 1 implies lim tn = 0.
n→∞
n→∞
In 1973, Geraghty [2] proved the following theorem, which is a generalization of Banach’s
contraction principle [1].
Theorem 1.1. [2] Let (X, d) be a complete metric space and T : X → X be a mapping. If T
satisfies the following inequality:
d(T x, T y) ≤ β(d(x, y))d(x, y)
for any x, y ∈ X, where β ∈ F. Then T has a unique fixed point.
In 2012, Samet et al. [7] introduced a concept of α-admissible as follows.
Definition 1.2. [7] Let T : X → X be a mapping and α : X × X → R be a function. Then T
is said to be α- admissible mapping, if for all x, y ∈ X, we have
α(x, y) ≥ 1 implies α(T x, T y) ≥ 1.
∗
This research was financially supported by...
Corresponding author.
‡
Speaker.
E-mail address:
[email protected] (K. Sukma),
[email protected] ( P. Charoensawan ).
†
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[email protected] (C. Thangthong),
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Later, Karapinar et al. [5] defined the following.
Definition 1.3. [5] An α-admissible mapping T is said to be triangular α-admissible mapping,
if for all x, y, z ∈ X, we have
α(x, z) ≥ 1 and α(z, y) ≥ 1 imply α(x, y) ≥ 1.
Recently, Karapinar [4] investigated the existence of fixed point for the mapping satisfying
a generalization of α-ψ Geraghty contraction as follows.
Let Φ denote the class of the functions ϕ : [0, ∞) → [0, ∞) which satisfy the following
condition:
(1) ϕ is nondecreasing;
(2) ϕ is continuous;
(3) ϕ(t) = 0 ⇔ t = 0.
Theorem 1.4. [4] Let (X, d) be a metric space, and let α : X × X → R be a function, and let
T : X → X be a mapping. Suppose that the following conditions are satisfied:
(1) there exists β ∈ F such that
α(x, y)ϕ(d(T x, T y)) ≤ β(ϕ(M (x, y)))ϕ(M (x, y)), for all x, y ∈ X
where M (x, y) = max{d(x, y), d(x, T x), d(y, T y)} and ϕ ∈ Φ;
(2) T is triangular α-admissible;
(3) there exists x1 ∈ X such that α(x1 , T x1 ) ≥ 1;
(4) T is continuous.
Then T has a fixed point x∗ ∈ X and {T n x1 } converges to x∗ .
In 2015, P. Shahi et al. [8] gave the concept of α-admissible with respect to g and studied
coincidence and common fixed points for generalized α-ψ contractive type mappings.
Definition 1.5. [8] Let f, g : X → X and α : X × X → [0, ∞). We say that f is α-admissible
with respect to g if, for all x, y ∈ X, we have
α(gx, gy) ≥ 1 implies α(f x, f y) ≥ 1
In 1986, G. Jungck [3] defined the following.
Definition 1.6. [3] Let (X, d) be a metric space and f, g : X → X be two mappings. The
mappings g and f are said to be d-compatible, if
lim d(gf xn , f gxn ) = 0
n→∞
whenever {xn } is sequences in X such that lim f xn = lim gxn .
n→∞
n→∞
Definition 1.7. [3] Let (X, dX ) and (Y, dY ) be two metric spaces and f : X → Y and
g : X → X be two mappings. A mapping f is said to be g-uniformly continuous on X, if for
any real number ϵ > 0, there exists δ > 0 such that dY (f x, f y) < ϵ whenever x, y ∈ X and
dX (gx, gy) < δ. If g is the identity mapping, then f is said to be uniformly continuous on X.
Very recently, J. Martinez-Moreno et al. [6] use the concept of d-compatible and g-uniformly
continuous to show the common fixed point theorem for Geraghty’s type contraction mappings
using the monotone property with two metrics as follows.
Let f, g : X → X be two mappings. A mapping f is said to be g-non-decreasing (resp.,
g-non-increasing), if for all x, y ∈ X, gx ≼ gy implies f x ≼ f y (resp., f y ≼ f x). If g is the
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identity mapping, then f is said to be non-decreasing (resp., non-increasing). Let d, d be two
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metrics on X. By d < d (resp., d ≤ d ), we mean d(x, y) < d (x, y) (resp., d(x, y) ≤ d (x, y))
for all x, y ∈ X. For more details and examples of two metrics, see e.g. [3].
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Theorem 1.8. [6] Let (X, d , ≼) be a complete partially ordered metric space, d be another
metric on X and f, g : X → X be two mappings such that f has the g-monotone property.
Suppose that the following conditions hold:
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(1) g : (X, d ) → (X, d ) is continuous and g(X) is d -closed;
(2) f (X) ⊆ g(X);
(3) there exists x0 ∈ X such that gx0 ≼ f x0 ;
(4) there exists β ∈ F such that
d(f x, f y) ≤ β(d(gx, gy))d(gx, gy)
for all x, y ∈ X with gx ≼ gy or gx ≽ gy;
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(5) if d d , assume that f : (X, d) → (X, d ) is g-uniformly contunuous;
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(6) if d ̸= d , assume that f : (X, d ) → (X, d ) is continuous and f and g are d -compatible;
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(7) if d = d , assume that (a) f is continuous and f and g are compatible or (b) for any
non-decreasing sequence {xn } in X, if xn → x as n → ∞, then xn ≼ x for all n ≥ 1.
Then there exists u ∈ X such that gu = f u, i.e., g and f have a coincidence point.
Theorem 1.9. [6] In addition to the hypotheses of Theorem 1.8, assume that
(8) for any x, u ∈ X, there exists y ∈ X such that f y is comparable to both f x and f u.
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If f and g are d -compatible, then g and f have a common fixed point.
Motivated by these works, we will study some new common fixed point theorems for generalized α-ϕ Geraghty contraction mappings with two metrics, which generalize and extend the
results in [6–8].
2
Main Results
We recollect the following auxiliary result which will be used efficiently in the proof of main
results.
Lemma 2.1. Let f : X → X be a triangular α- admissible w.r.t g and f (X) ⊆ g(X). Assume
that there exists x1 ∈ X such that α(gx1 , f x1 ) ≥ 1. Define a sequence {gxn } by gxn+1 = f xn .
Then, we have α(gxn , gxm ) ≥ 1 for all m, n ∈ N with n < m
We introduce the following definition.
Definition 2.2. Let (X, d) be a complete metric space and f, g : X → X be given mappings.
The pair (f, g) is called generalized α-ϕ Geraghty contraction type map w.r.t d if there exists
two functions α : X × X → R and ϕ ∈ Φ such that for all x, y ∈ X,
α(gx, gy)ϕ(d(f x, f y)) ≤ β(M (gx, gy))ϕ(M (gx, gy)),
(2.1)
}
{
d(gx, f y) + d(gy, f x)
and β ∈ F.
where M (gx, gy) = max d(gx, gy), d(gx, f x), d(gy, f y),
2
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Theorem 2.3. Let (X, d ) be a complete metric space, d be another metric on X, α : X ×X → R
be a function, and let f, g : X → X be given mappings such that f (X) ⊆ g(X). Suppose that
the following conditions are satisfied:
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(1) g : (X, d ) → (X, d ) is continuous and g(X) is closed;
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(2) f is triangular α-admissible w.r.t g;
(3) there exists x1 ∈ X such that α(gx1 , f x1 ) ≥ 1;
(4) (f, g) is generalized α-ϕ Geraghty contraction type map w.r.t d;
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(5) if d d , assume that f : (X, d) → (X, d ) is g-uniformly continuous;
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(6) if d ̸= d , assume that f : (X, d ) → (X, d ) is continuous and f and g are d -compatible;
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(7) if d = d , assume that (a) f is continuous and f and g are compatible or (b) for any {gxn }
in X such that α(gxn , gxn+1 ) ≥ 1 for all n and gxn → gz ∈ g(X) as n → ∞, then there
exists a subsequence {gxnk } of {gxn } such that α(gxnk , gz) ≥ 1 for all k.
Then f and g have a coincidence point.
Proof. By condition (3), let x1 ∈ X be such that α(gx1 , f x1 ) ≥ 1. Since f (X) ⊆ g(X), we
can choose a point x2 ∈ X such that f x1 = gx2 . Continuing this process having chosen
x2 , x3 , . . . , xn , we choose xn+1 ∈ X such that
f xn = gxn+1 , n = 1, 2, 3, . . .
(2.2)
Since f is α- admissible w.r.t g, we get
α(gx1 , gx2 ) = α(gx1 , f x1 ) ≥ 1 ⇒ α(gx2 , gx3 ) = α(f x1 , f x2 ) ≥ 1
Using mathematical induction, we obtain that
α(gxn , gxn+1 ) ≥ 1, ∀n = 1, 2, 3, . . .
(2.3)
If f xn0 +1 = f xn0 for some n0 ∈ N, then by (3.1)
f xn0 +1 = gxn0 +1 ,
that is, f and g have a coincidence point at x = xn0 +1 , and so we have finished.
Suppose that f xn+1 ̸= f xn , ∀n ∈ N. Then d(gxn+1 , gxn+2 ) = d(f xn , f xn+1 ) > 0 for all n.
From the inequality (2.1), we have
ϕ(d(gxn+1 , gxn+2 )) = ϕ(d(f xn , f xn+1 ))
≤ α(gxn , gxn+1 )ϕ(d(f xn , f xn+1 ))
≤ β(M (gxn , gxn+1 ))ϕ(M (gxn , gxn+1 ))
(2.4)
< ϕ(M (gxn , gxn+1 ))
On the other hand, we get
{
M (gxn , gxn+1 ) = max d(gxn , gxn+1 ), d(gxn , f xn ), d(gxn+1 , f xn+1 ),
d(gxn , f xn+1 ) + d(gxn+1 , f xn )
2
{
}
d(gxn , gxn+2 )
= max d(gxn , gxn+1 ), d(gxn+1 , gxn+2 ),
2
{
}
≤ max d(gxn , gxn+1 ), d(gxn+1 , gxn+2 )
}
If for some n ≥ 1, M (gxn , gxn+1 ) = d(gxn+1 , gxn+2 ). Then by (2.4), we obtain that
ϕ(d(gxn+1 , gxn+2 )) < ϕ(d(gxn+1 , gxn+2 ))
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a contradiction. So, for all n ≥ 1, we have
M (gxn , gxn+1 ) = d(gxn , gxn+1 )
(2.5)
Notice that in view of (2.4), we get for all n ≥ 1 that ϕ(d(gxn+1 , gxn+2 )) < ϕ(d(gxn , gxn+1 )).
From (a) of property ϕ, we have
d(gxn+1 , gxn+2 ) ≤ d(gxn , gxn+1 ), ∀n ∈ N.
Hence, we deduce that the sequence {d(gxn , gxn+1 )} is nonnegative and non-increasing.
Consequently, there exists r ≥ 0 such that lim d(gxn , gxn+1 ) = r. We claim that r = 0.
n→∞
Suppose, on the contrary, that r > 0. Then, due to (2.4), we have
ϕ(d(gxn+1 , gxn+2 ))
ϕ(d(gxn+1 , gxn+2 ))
=
≤ β(M (gxn , gxn+1 )) < 1.
ϕ(d(gxn , gxn+1 ))
ϕ(M (gxn , gxn+1 ))
In what follows that lim β(M (gxn , gxn+1 )) = 1. Owing to the fact that β ∈ F, we get
n→∞
lim M (gxn , gxn+1 ) = 0 imply that lim d(gxn , gxn+1 ) = 0 a contradiction. So, we conclude
n→∞
n→∞
that
lim d(gxn , gxn+1 ) = 0
(2.6)
n→∞
We assert that {gxn } is a Cauchy sequence. Suppose, on the contrary, that {gxn } is not a
Cauchy sequence. Thus, there exists ϵ > 0 such that, for all k ∈ N, there exists n(k) > m(k) ≥ k
with the smallest number satisfying the condition below
d(gxnk , gxmk ) ≥ ϵ and d(gxnk −1 , gxmk ) < ϵ
.
Then, we have
ϵ ≤ d(gxmk , gxnk )
≤ d(gxmk , gxnk −1 ) + d(gxnk −1 , gxnk )
< ϵ + d(gxnk −1 , gxnk )
Letting k → ∞ in the above inequality. By (2.6) , we obtain that
lim d(gxmk , gxnk ) = ϵ > 0
(2.7)
n→∞
By Lemma 2.1, α(gxm(k) , gxn(k) ) ≥ 1, ∀k. Thus, we have
ϕ(d(gxm(k)+1 , gxn(k)+1 )) = ϕ(d(f xm(k) , f xn(k) ))
≤ α(gxm(k) , gxn(k) )ϕ(d(f xm(k) , f xn(k) ))
≤ β(M (gxm(k) , gxn(k) ))ϕ(M (gxm(k) , gxn(k) )),
where
{
M (gxm(k) , gxn(k) ) = max d(gxm(k) , gxn(k) ), d(gxm(k) , f xm(k) ), d(gxn(k) , f xn(k) ),
d(gxm(k) , f xn(k) ) + d(gxn(k) , f xm(k) )
2
{
}
= max d(gxm(k) , gxn(k) ), d(gxm(k) , gxm(k)+1 ), d(gxn(k) , gxn(k)+1 ),
d(gxm(k) , gxn(k)+1 ) + d(gxn(k) , gxm(k)+1 )
2
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}
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Keeping (2.6),(2.7) in mind and letting k → ∞, we derive that
lim M (gxm(k) , gxn(k) ) = ϵ > 0
k→∞
.
Hence, we conclude that
ϕ(d(gxm(k)+1 , gxn(k)+1 ))
≤ β(M (gxm(k) , gxn(k) )) < 1.
ϕ(M (gxm(k) , gxn(k) ))
(2.8)
Keeping (2.6),(2.7) in mind and letting k → ∞, we derive that
lim M (gxm(k) , gxn(k) ) = ϵ > 0
k→∞
.
By inequality (2.8), we get
lim β(M (gxm(k) , gxn(k) )) = 1
k→∞
and hence lim M (gxm(k) , gxn(k) ) = 0 a contradiction. Hence, we conclude that {gxn } is Cauchy
k→∞
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sequence in (X, d). Also, we claim that {gxn } is a Cauchy sequence with respect to d .
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If d ≥ d , it is trivial. Thus, suppose d d . Let ε > 0 there exists δ > 0 such that
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d (f x, f y) < ε
(2.9)
whenever x, y ∈ X and d(gx, gy) < δ. Since {gxn } is a Cauchy sequence with respect to d, then
there exists N0 ∈ N with
d(gxn , gxm ) < δ
(2.10)
whenever n, m ≥ N0 . Now, (2.9) and (2.10) imply that
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d (gxn+1 , gxm+1 ) = d (f xn , f xm ) < ε
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whenever n, m ≥ N0 and so {gxn } is a Cauchy sequence with respect to d .
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Since g(X) is a d -closed subset of the complete metric space (X, d ), there exists u = gx ∈
g(X) such that
lim gxn = lim f xn = u
n→∞
n→∞
Finally, we prove that u is a coincidence point of f and g. We consider two case:
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Case I: d ̸= d .
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By the d -compatible of f and g, we have
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lim d (gf xn , f gxn ) = 0
(2.11)
n→∞
Using the triangular inequality, we have
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d (gu, f u) ≤ d (gu, gf xn ) + d (gf xn , f gxn ) + d (f gxn , f u)
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Letting n → ∞, from (2.11) and the continuous of f and g, it follows that d (gu, f u) = 0 imply
that gu = f u. So u is a coincidence point of f and g.
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Case II: d = d .
In order to avoid the repetition, we can only consider (b) of the condition (7). Since {gxn }
is a Cauchy sequence in (X, d) and g(X) is closed in X, then there exists gz ∈ g(X) such that
lim gxn = gz = lim f xn
n→∞
n→∞
(2.12)
Now, we show that z is a coincidence point of f and g. On contrary, assume that f z ̸= gz.
Then d(f z, gz) > 0. Since assumption (b), we have
α(gxn(k) , gz) ≥ 1, ∀k.
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By using the triangular inequality, we derive
d(gz, f z) ≤ d(gz, f xn(k) ) + d(f xn(k) , f z)
imply that
d(gz, f z) − d(gz, f xn(k) ) ≤ d(f xn(k) , f z)
Since (a) of property ϕ, we obtain that
ϕ(d(gz, f z) − d(gz, f xn(k) )) ≤ ϕ(d(f xn(k) , f z))
≤ α(gxn(k) , gz)ϕ(d(f xn(k) , f z))
≤ β(M (gxn(k) , gz))ϕ(M (gxn(k) , gz))
(2.13)
where
{
M (gxn(k) , gz) = max d(gxn(k) , gz), d(gxn(k) , f xn(k) ), d(gz, f z),
d(gxn(k) , f z) + d(gz, f xn(k) )
2
}
Since inequality (2.12), we obtain that
lim M (gxn(k) , gz) = d(gz, f z) > 0
k→∞
Letting k → ∞ in inequality (2.13), we obtain that lim β(M (gxn(k) , gz)) = 1 and so
k→∞
lim M (gxn(k) , gz) = 0 a contradiction. Therefore f z = gz. Consequently, we conclude that f
k→∞
and g have a coincidence point.
For the uniqueness of a common fixed point for generalized α-ϕ Geraghty contraction mapping, we will consider the following condition.
(K) For any x, y ∈ X, there exists v ∈ X such that α(gx, f v) ≥ 1, α(gy, f v) ≥ 1 and
α(gv, f v) ≥ 1
Theorem 2.4. Adding condition (K) to the hypotheses of Theorem 2.3, then f and g have a
unique common fixed point.
Proof. Due to Theorem 2.3 implies that there exists a coincidence point x ∈ X, that is, gx = f x.
Suppose that there exists another coincidence point y ∈ Xand hence gy = f y.
Now, we prove that gx = gy. From the condition (K), it follows that there exists v ∈ X
such that α(f x, f v) ≥ 1, α(f y, f v) ≥ 1 and α(gv, f v) ≥ 1
Put v0 = v and, analogously to the proof of Theorem 2.3, choose a sequence {vn } in X
satisfying
f vn = gvn+1
for all n ∈ N.
Following the proof of Theorem 2.3, we have lim gvn = v ∗ such that v ∗ is a coincidence
n→∞
point of f and g.
Since 1 ≤ α(gx, f v) = α(gx, gv1 ). Then by f is α-admissible w.r.t g, we have α(gx, gv2 ) =
α(f x, f v1 ) ≥ 1.
Using mathematical induction, we have
α(gx, gvn ) ≥ 1
for all n.
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We show that gx = v ∗ . Suppose, on the contrary, that gx ̸= v ∗ . Then d(gx, v ∗ ) > 0. Thus
we can apply the contraction (2.1) to obtain
ϕ(d(gx, gvn+1 )) = ϕ(d(f x, f vn ))
≤ α(gx, gvn )ϕ(d(f x, f vn ))
≤ β(M (gx, gvn ))ϕ(M (gx, gvn ))
< ϕ(M (gx, gvn ))
(2.14)
{
where
d(gx, f vn ) + d(gvn , f x)
M (gx, gvn ) = max d(gx, gvn ), d(gx, f x), d(gvn , f vn ),
2
N.
}
for all n ∈
So, we get lim M (gx, gvn ) = d(gx, v ∗ ) > 0. Since inequality (2.14), we have lim β(M (gx, gvn )) =
n→∞
n→∞
1 imply lim M (gx, gvn ) = 0 a contradiction. Hence, we conclude that gx = v ∗ .
n→∞
Similarly, gy = v ∗ . Consequently, we get
gx = gy
(2.15)
Next, let p = gx such that x is a coincidence point of f and g. Since f (X) ⊆ g(X), let
x0 = x, we choose the sequence {xn } satisfying gxn = f xn−1 for each n ∈ N.
Hence, we have gp = ggx = gf x. Let xn = x for all n, i.e., gxn = f x. Then gxn = f x =
f xn−1 for all n ∈ N and so lim f xn = lim gxn = f x = gx
n→∞
n→∞
Since f and g are compatible, we have
lim d(gf xn , f gxn ) = 0
n→∞
that is, gf x = f gx. Therefore, we get gp = ggx = gf x = f gx = f p. This implies that p is
another coincidence point of the mappings f and g.
It follows that f p = gp = gx = p and hence p is a common fixed point of f and g.
Next, we prove that f and g have a unique common fixed point. Let p and q have a common
fixed point. Then by definition of common fixed point, we get p and q have a coincidence point
of the mappings f and g. From (2.15), we have p = gp = gq = q. Therefore we conclude that
f and g have a unique common fixed point. This completes the proof.
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Corollary 2.5. [6] Let (X, d , ≼) be a complete partially ordered metric space, d be another
metric on X and f, g : X → X be two mappings such that f has the g-monotone property.
Suppose that the following conditions hold:
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(1) g : (X, d ) → (X, d ) is continuous and g(X) is d -closed;
(2) f (X) ⊆ g(X);
(3) there exists x0 ∈ X such that gx0 ≼ f x0 ;
(4) there exists β ∈ F such that
d(f x, f y) ≤ β(d(gx, gy))d(gx, gy)
for all x, y ∈ X with gx ≼ gy or gx ≽ gy;
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(5) if d d , assume that f : (X, d) → (X, d ) is g-uniformly continuous;
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(6) if d ̸= d , assume that f : (X, d ) → (X, d ) is continuous and f and g are d -compatible;
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(7) if d = d , assume that (a) f is continuous and f and g are compatible or (b) for any
non-decreasing sequence {xn } in X, if xn → x as n → ∞, then xn ≼ x for all n ≥ 1.
Then there exists u ∈ X such that gu = f u, i.e., g and f have a coincidence point.
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Proof. Define a function α : X × X → [0, ∞) by
{
1, if x ≼ y or y ≼ x,
α(x, y) =
0, otherwise.
If we take ϕ(t) = t and M (x, y) = d(x, y) in Theorem 2.3, then we get the result.
3
Some Consequences
We start this section with following definition.
Definition 3.1. Let (X, d) be a complete metric space and f, g : X → X be given mappings.
The pair (f, g) is called a α-ϕ Geraghty contraction type map w.r.t d if there exists two functions
α : X × X → R and ϕ ∈ Φ such that for all x, y ∈ X,
α(gx, gy)ϕ(d(f x, f y)) ≤ β(d(gx, gy))ϕ(M (gx, gy)),
{
}
d(gx, f y) + d(gy, f x)
where M (gx, gy) = max d(gx, gy), d(gx, f x), d(gy, f y),
and β ∈ F.
2
The proof of the following theorem is similar to that of Theorem 2.3
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Theorem 3.2. Let (X, d ) be a complete metric space, d be another metric on X, α : X ×X → R
be a function, and let f, g : X → X be given mappings such that f (X) ⊆ g(X). Suppose that
the following conditions are satisfied:
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(1) g : (X, d ) → (X, d ) is continuous and g(X) is closed;
(2) f is triangular α-admissible w.r.t g;
(3) there exists x1 ∈ X such that α(gx1 , f x1 ) ≥ 1;
(4) (f, g) is α-ϕ Geraghty contraction type map w.r.t d;
′
′
(5) if d d , assume that f : (X, d) → (X, d ) is g-uniformly continuous.
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′
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If f : (X, d ) → (X, d ) is continuous and f and g are d -compatible. Then f and g have a
coincidence point.
Proof. By condition (3), let x1 ∈ X be such that α(gx1 , f x1 ) ≥ 1. Since f (X) ⊆ g(X), we
can choose a point x2 ∈ X such that f x1 = gx2 . Continuing this process having chosen
x2 , x3 , . . . , xn , we choose xn+1 ∈ X such that
f xn = gxn+1 , n = 1, 2, 3, . . .
(3.1)
Since f is α- admissible w.r.t g, we get
α(gx1 , gx2 ) = α(gx1 , f x1 ) ≥ 1 ⇒ α(gx2 , gx3 ) = α(f x1 , f x2 ) ≥ 1
Using mathematical induction, we obtain that
α(gxn , gxn+1 ) ≥ 1, ∀n = 1, 2, 3, . . .
(3.2)
If f xn0 +1 = f xn0 for some n0 ∈ N, then by (3.1)
f xn0 +1 = gxn0 +1 ,
that is f and g have a coincidence point at x = xn0 +1 , and so we have finished.
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Suppose that f xn+1 ̸= f xn , ∀n ∈ N. Then d(gxn+1 , gxn+2 ) = d(f xn , f xn+1 ) > 0 for all n.
From condition (2.1), we have
ϕ(d(gxn+1 , gxn+2 )) = ϕ(d(f xn , f xn+1 ))
≤ α(gxn , gxn+1 )ϕ(d(f xn , f xn+1 ))
≤ β(d(gxn , gxn+1 ))ϕ(M (gxn , gxn+1 ))
< ϕ(M (gxn , gxn+1 ))
(3.3)
On the other hand, we get
{
M (gxn , gxn+1 ) = max d(gxn , gxn+1 ), d(gxn , f xn ), d(gxn+1 , f xn+1 ),
d(gxn , f xn+1 ) + d(gxn+1 , f xn )
2
{
}
d(gxn , gxn+2 )
= max d(gxn , gxn+1 ), d(gxn+1 , gxn+2 ),
2
{
}
≤ max d(gxn , gxn+1 ), d(gxn+1 , gxn+2 )
}
If for some n ≥ 1, M (gxn , gxn+1 ) = d(gxn+1 , gxn+2 ). Then by (3.3), we obtain that
ϕ(d(gxn+1 , gxn+2 )) < ϕ(d(gxn+1 , gxn+2 ))
a contradiction. So, for all n ≥ 1, we have
M (gxn , gxn+1 ) = d(gxn , gxn+1 )
(3.4)
Notice that in view of (3.3), we get for all n ≥ 1 that ϕ(d(gxn+1 , gxn+2 )) < ϕ(d(gxn , gxn+1 )).
From property ϕ, we have
d(gxn+1 , gxn+2 ) ≤ d(gxn , gxn+1 ), ∀n ∈ N.
Hence, we deduce that the sequence {d(gxn , gxn+1 )} is nonnegative and non-increasing.
Consequently, there exists r ≥ 0 such that lim d(gxn , gxn+1 ) = r. We claim that r = 0.
n→∞
Suppose, on the contrary, that r > 0. Then, due to (3.3), we have
ϕ(d(gxn+1 , gxn+2 ))
ϕ(d(gxn+1 , gxn+2 ))
=
≤ β(d(gxn , gxn+1 )) < 1.
ϕ(d(gxn , gxn+1 ))
ϕ(M (gxn , gxn+1 ))
In what follows that lim β(d(gxn , gxn+1 )) = 1. Owing to the fact that β ∈ F, we get
n→∞
lim d(gxn , gxn+1 ) = 0
n→∞
a contradiction. So, we conclude that
lim d(gxn , gxn+1 ) = 0
n→∞
(3.5)
We assert that {gxn } is a Cauchy sequence. Suppose, on the contrary, that {gxn } is not a
Cauchy sequence. Thus, there exists ϵ > 0 such that, for all k ∈ N, there exists n(k) > m(k) ≥ k
with the smallest number satisfying the condition below
d(gxnk , gxmk ) ≥ ϵ and d(gxnk −1 , gxmk ) < ϵ
.
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Then, we have
ϵ ≤ d(gxmk , gxnk )
≤ d(gxmk , gxnk −1 ) + d(gxnk −1 , gxnk )
< ϵ + d(gxnk −1 , gxnk )
Letting k → ∞ in the above inequality. By (3.5) , we obtain that
lim d(gxmk , gxnk ) = ϵ > 0
(3.6)
n→∞
By Lemma 2.1, α(gxm(k) , gxn(k) ) ≥ 1, ∀k. Thus, we have
ϕ(d(gxm(k)+1 , gxn(k)+1 )) = ϕ(d(f xm(k) , f xn(k) ))
≤ α(gxm(k) , gxn(k) )ϕ(d(f xm(k) , f xn(k) ))
≤ β(d(gxm(k) , gxn(k) ))ϕ(M (gxm(k) , gxn(k) )),
where
{
M (gxm(k) , gxn(k) ) = max d(gxm(k) , gxn(k) ), d(gxm(k) , f xm(k) ), d(gxn(k) , f xn(k) ),
d(gxm(k) , f xn(k) ) + d(gxn(k) , f xm(k) )
2
{
}
= max d(gxm(k) , gxn(k) ), d(gxm(k) , gxm(k)+1 ), d(gxn(k) , gxn(k)+1 ),
d(gxm(k) , gxn(k)+1 ) + d(gxn(k) , gxm(k)+1 )
2
}
Hence, we conclude that
ϕ(d(gxm(k)+1 , gxn(k)+1 ))
≤ β(d(gxm(k) , gxn(k) )) < 1.
ϕ(M (gxm(k) , gxn(k) ))
Keeping (3.5),(3.6) in mind and letting k → ∞ in the above inequality, we derive that
lim β(d(gxm(k) , gxn(k) )) = 1, and so lim d(gxm(k) , gxn(k) ) = 0. Hence, ϵ = 0, which is a
k→∞
k→∞
contradiction. Thus, we conclude that {gxn } is Cauchy sequence in (X, d). Also, we claim that
′
{gxn } is a Cauchy sequence with respect to d .
′
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If d ≥ d , it is trivial. Thus, suppose d d . Let ε > 0 there exists δ > 0 such that
′
d (f x, f y) < ε
(3.7)
whenever x, y ∈ X and d(gx, gy) < δ. Since {gxn } is a Cauchy sequence with respect to d, then
there exists N0 ∈ N with
d(gxn , gxm ) < δ
(3.8)
whenever n, m ≥ N0 . Now, (3.7) and (3.8) imply that
′
′
d (gxn+1 , gxm+1 ) = d (f xn , f xm ) < ε
′
whenever n, m ≥ N0 and so {gxn } is a Cauchy sequence with respect to d .
′
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Since g(X) is a d -closed subset of the complete metric space (X, d ), there exists u = gx ∈
g(X) such that
lim gxn = lim f xn = u
n→∞
n→∞
′
Finally, we prove that u is a coincidence point of f and g. By the d -compatible of f and g, we
have
′
lim d (gf xn , f gxn ) = 0
(3.9)
n→∞
Proceedings of AMM 2017
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Using the triangular inequality, we have
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′
′
′
d (gu, f u) ≤ d (gu, gf xn ) + d (gf xn , f gxn ) + d (f gxn , f u)
′
Letting n → ∞, from (3.9) and the continuous of f and g, it follows that d (gu, f u) = 0 imply
that gu = f u. So u is a coincidence point of f and g.
We give an example to illustrate Theorem 3.2
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Example 3.3. Let X = [0, ∞) ⊆ R, and let metrics d, d : X × X → [0, ∞) be defined by
′
d(x, y) = |x − y|, d (x, y) = k|x − y|
1
with k ∈ (1, ∞) for all x, y ∈ X. Let β(t) = 1+t
, ∀t > 0 and β(0) = 0. Then, β ∈ F. Let
t
ϕ(t) = 2 , ∀t ≥ 0. Then, ϕ ∈ Φ. Define the mappings f : X → X and g : X → X by
f (x) =
x
, ∀x ∈ X
8
g(x) =
x
, ∀x ∈ X
2
and
.
Now, we define the mapping α : X × X → [0, ∞) by
{
α(x, y) =
1, if x, y ∈ [0, 1],
0, otherwise.
Proof. Clearly, f (X) ⊆ g(X).
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(1) Clearly, g : (X, d ) → (X, d ) is continuous and g(X) is closed.
(2) We show that f is triangular α- admissible w.r.t g. In so doing, let x, y ∈ X such that
α(gx, gy) ≥ 1. This imply that gx, gy ∈ [0, 1] and by the definition of g, we get x, y ∈ [0, 2]. So,
by definition of f , we have f (x) = x8 ∈ [0, 1] and f (y) = y8 ∈ [0, 1]. Therefore, α(f x, f y) ≥ 1.
Thus, f is α- admissible w.r.t g.
If x, y, z ∈ X such that α(x, z) ≥ 1 and α(z, y) ≥ 1. Then by the definition of α, we have
x, y, z ∈ [0, 1] and hence α(x, y) ≥ 1.
So, we conclude that f is triangular α- admissible w.r.t g.
(3) There exists x1 ∈ X such that α(gx1 , f x1 ) ≥ 1. In fact, for x1 = 1, we have α(g1, f 1) =
α( 21 , 18 ) = 1.
(4) The pair (f, g) is a α-ϕ Geraghty contraction pair of mapping w.r.t d. In fact, if
0 ≤ x, y ≤ 2, then α(gx, gy) = α( x2 , y2 ) = 1.
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Consider
(
)(
)
1
M (gx, gy)
β(d(gx, gy))ϕ(M (gx, gy)) − α(gx, gy)ϕ(d(f x, f y)) =
1 + d(gx, gy)
2
(
)
d(f x, f y)
−
2
(
)(
)
1
d(gx, gy)
≥
1 + d(gx, gy)
2
(
)
d(f x, f y)
−
2
)(
(
)
| x2 − y2 |
1
=
1 + | x2 − y2 |
2
)
(
| x8 − y8 |
−
2
|x − y|
|x − y|
−
2(2 + |x − y|)
16
|x − y|(6 − |x − y|)
=
16(2 + |x − y|)
|x − y|(6 − 4)
≥
16(2 + |x − y|)
≥ 0
=
Therefore, we derive that α(gx, gy)ϕ(d(f x, f y)) ≤ β(d(gx, gy))ϕ(M (gx, gy)) for all x, y ∈
[0, 2].
If x > 2 or y > 2, then α(gx, gy) = 0. Obviously, we have
α(gx, gy)ϕ(d(f x, f y)) ≤ β(d(gx, gy))ϕ(M (gx, gy))
Thus, (f, g) is a α-ϕ Geraghty contraction map.
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(5) Since d d , we show that a mapping f : (X, d) → (X, d ) is g-uniformly continuous. Let
ϵ > 0 be given and choose δ := 4ϵ
k with k ∈ (1, ∞). Assume that x, y ∈ X with d(gx, gy) < δ.
Then we have
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d (f x, f y) = k|f x − f y|
x y = k − 8 8
k x y =
− 4 2 2
k
=
d(gx, gy)
4
k
<
δ
4
= ϵ
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This imply that f : (X, d) → (X, d ) is g-uniformly continuous.
′
′
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(6) We prove that f : (X, d ) → (X, d ) is continuous and f and g are d -compatible. It is
′
easy to see that f : (X, d) → (X, d ) is g-uniformly continuous.
′
So we will only show that f and g are d -compatible. Let {xn } is a sequence in X such that
lim gxn = lim f xn = a
n→∞
Proceedings of AMM 2017
n→∞
ANA-05-13
Then we obtain
a
2
=
a
8
imply that a = 0. Now, we have
′
d (gf xn , f gxn ) = k|gf xn − f gxn |
(x )
( x )
n
n = k g
−f
8
x 8 x n
n
= k
−
16
16
= 0
′
for all n ∈ N. So, we get d (gf xn , f gxn ) → 0 as n → ∞.
Consequently, all assumptions of Theorem 3.2 are satisfied, and hence f and g have a
coincidence point at 0.
Acknowledgment. This research was supported by Chiang Mai University.
References
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[3] G. Jungck, Compatible mappings and common fixed points,Int. J. Math. Math. Sci.
9(1986), 771-779.
[4] E. Karapinar, A discussion on ”α-ψ Geraghty contraction type mappings”, Filomat, 28:4
(2014), 761-766.
[5] E. Karapinar, α-ψ Geraghty contraction type mappings and some related fixed point results, Filomat, 28:1 (2014), 37-48.
[6] J. Martinez-Moreno, W. Sintunavarat and Y. Je Cho, Common fixed point theorem for
Geraghty’s type contraction mappings using the monotone property with two metric, Fixed
Point Theory and Applications, 2015, 2015:174.
[7] B. Samet, C. Vetro and P. Vetro, Fixed point theorems for α-ψ-contractive mappings,
Nonlinear Anal., 4(2012), 2154-2165.
[8] P. Shahi, J. kaur and S. S. Bhatia, Coincidence and Common Fixed Point Results for
Generalized α-ψ Contractive Type Mappings with Applications, Bulletin of the Belgian
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