Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 907951, 5 pages
doi:10.1155/2012/907951
Review Article
Quasi-Contractive Mappings in Modular
Metric Spaces
Yeol J. E. Cho,1 Reza Saadati,2 and Ghadir Sadeghi3
1
Department of Mathematics Education and RINS, Gyeongsang National University,
jinju 660-701, Republic of Korea
2
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
3
Department of Mathematics and Computer Sciences, Sabzevar Tarbiat Moallem University, Sabzevar, Iran
Correspondence should be addressed to Reza Saadati, rsaadati@eml.cc
Received 6 October 2011; Accepted 30 November 2011
Academic Editor: Rudong Chen
Copyright q 2012 Yeol J. E. Cho et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We prove the existence of fixed point and uniqueness of quasi-contractive mappings in modular
metric spaces which was introduced by Ćirić
1. Introduction and Preliminaries
In this paper, we prove the existence and uniqueness of fixed points of quasi-contractive
mappings in modular metric spaces which develop the theory of metric spaces generated
by modulars. Throughout the paper X is a nonempty set and λ > 0. The notion of a metric
modular was introduced by Chistyakov 1 as follows.
Definition 1.1. A function ω : 0, ∞ × X × X → 0, ∞ is said to be a metric modular on X or,
simply, a modular if no ambiguity arises if it satisfies three axioms:
i for any x, y ∈ X, ωλ x, y 0 for all λ > 0 if and only if x y;
ii ωλ x, y ωλ y, x for all λ > 0, and x, y ∈ X;
iii ωλμ x, y ≤ ωλ x, z ωμ z, y for all λ, μ > 0 and x, y ∈ X.
Definition 1.2. Let X, ω be a metric modular space.
2
Journal of Applied Mathematics
1 A sequence {xn } in Xω is said to be ω-convergent to a point x ∈ X if, for all λ > 0,
ωλ xn , x −→ 0
1.1
as n → ∞.
2 A subset C of Xω is said to be ω-closed if the ω-limit of a ω-convergent sequence of
C always belongs to C.
3 A subset C of Xω is said to be ω-complete if every ω-Cauchy sequence in C is ωconvergent and its ω-limit is in C.
Definition 1.3. The metric modular ω is said to have the Fatou property if
ωλ x, y ≤ lim inf ω xn , y
n→∞
1.2
for all y ∈ Xω and λ ∈ 0, ∞, where {xn } ω-converges to x.
2. Main Results
Definition 2.1. Let X, ω be a metric modular space, and let C be a nonempty subset of Xω .
The self-mapping T : C → C is said to be quasi-contraction if there exists 0 < k < 1 such that
ωλ T x, T y ≤ k max ωλ x, y , ωλ x, T x, ωλ y, T y , ωλ x, T y , ωλ T x, y
2.1
for any x, y ∈ X and λ ∈ 0, ∞.
Let T : C → C be a mapping, and let C be a nonempty subset of Xω . For any x ∈ C,
define the orbit
Ox x, T x, T 2 x, . . .
2.2
δω x diamOx sup{ωλ T n x, T m x : n, m ∈ N }.
2.3
and its ω-diameter by
Lemma 2.2. Let X, ω be a metric modular space, and let C be a nonempty subset of Xω . Let T :
C → C be a quasi-contractive mapping, and let x ∈ C be such that δω x < ∞. Then, for any n ≥ 1,
one has
δω T x ≤ kn δω x,
2.4
where k is the constant associated with the mapping of T . Moreover, one has
ωλ T n x, T nm x ≤ kn δω x
2.5
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3
for any n, m ≥ 1 and λ ∈ 0, ∞.
Proof. For each n, m ≥ 1, we have
ωλ T n x, T m y ≤ k max ωλ T n−1 x, T m−1 y , ωλ T n−1 x, T n x ,
ωλ T m−1 y , T m y , ωλ T n−1 x, T m y , ωλ T n x, T m−1 y
2.6
for any x, y ∈ C and λ ∈ 0, ∞. This obviously implies that
δω T n x ≤ kδω T n−1 x
2.7
for any n ≥ 1. Hence, for any n ≥ 1, we have
δω T n x ≤ kn δω x.
2.8
Moreover, for any n, m ≥ 1, we have
ωλ T n x, T nm x ≤ δω T n x ≤ kn δω x.
2.9
This completes the proof.
The next lemma is helpful to prove the main result in this paper.
Lemma 2.3. Let X, ω be a modular metric space, and let C be a ω-complete nonempty subset of
Xω . Let T : C → C be quasi-contractive mapping, and let x ∈ C be such that δω x < ∞. Then
{T n x} ω-converges to a point ν ∈ C. Moreover, one has
ωλ T n X − ν ≤ kn δω x
2.10
for all n ≥ 1 and λ ∈ 0, ∞.
Proof. From Lemma 2.2, we know that {T n x} is a ω-Cauchy sequence in C. Since C is ωcomplete, then there exists ν ∈ C such that {T n x} ω-converges to ν. Since
ωλ T n x, T nm x ≤ kn δω x
2.11
for any n, m ≥ 1 and ω satisfies the Fatou property, and letting m → ∞, we have
ωλ T n x, ν ≤ lim inf ωλ T n x, T nm x ≤ kn δω x.
m→∞
This completes the proof.
2.12
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Journal of Applied Mathematics
Next, we prove that ν is, in fact, a fixed point of T and it is unique provided some extra
assumptions.
Theorem 2.4. Let T, C, and x be as in Lemma 2.3. Suppose that ωλ ν, T ν < ∞ and ωλ x, T x <
∞ for all λ ∈ 0, ∞. Then the ω-limit of {T n x} is a fixed point of T , that is, T ν ν. Moreover, if
ν∗ is any fixed point of T in C such that ωλ ν, ν∗ < ∞ for all λ ∈ 0, ∞, then one has ν ν∗ .
Proof. We have
ωλ T x, T ν ≤ k max{ωλ x, ν, ωλ x, T x, ωλ ν, T ν, ωλ x, T ν, ωλ T x, ν}.
2.13
From Lemma 2.3, it follows that
ωλ T x, T ν ≤ k max{δω x, ωλ ν, T ν, ωλ x, T ν}.
2.14
Suppose that, for each n ≥ 1,
ωλ Tn x, T ν ≤ max{kn δω x, kωλ ν, T ν, kn ωλ x, T ν}.
2.15
Then we have
ωλ T n1 x, T ν
≤ k max ωλ T n x, ν, ωλ T n x, T n1 x , ωλ ν, T ν, ωλ T n x, T ν, ωλ T n1 x, ν .
2.16
Hence we have
ωλ T n1 x, T ν ≤ k max{kn δω x, kωλ ν, T ν, ωλ T n x, T ν}.
2.17
Using our previous assumption, we get
ωλ T n1 x, T ν ≤ max kn1 δω x, kωλ ν, T ν, kn1 ωλ x, T ν .
2.18
Thus, by induction, we have
ωλ T n x, T ν ≤ max{kn δω x, kωλ ν, T ν, kn ωλ x, T ν}
2.19
for any n ≥ 1 and λ ∈ 0, ∞. Therefore, we have
lim sup ωλ T n x, T x ≤ ων, T ν
n→∞
2.20
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5
for all λ ∈ 0, ∞. Using the Fatou property for the metric modular ω, we get
ωλ ν, T νlim inf ωλ T n x, T ν ≤ kων, T ν
n→∞
2.21
for all λ ∈ 0, ∞. Since k < 1, we get ωλ ν, T ν 0 for all λ ∈ 0, ∞, and so T ν ν.
Let ν∗ be another fixed point of T such that ωλ ν, ν∗ < ∞ for all λ ∈ 0, ∞. Then we
have
ωλ ν, ν∗ ωλ T ν, T ν∗ ≤ kωλ ν, ν∗ ,
2.22
ωλ ν, ν∗ 0
2.23
which implies that
for all λ ∈ 0, ∞. Hence ν ν∗ . This complete the proof.
Acknowledgments
The authors are thankful to the anonymous referees and the area editor Professor Rudong
Chen for their critical remarks which helped greatly to improve the presentation of this
paper. The first author was supported by the Basic Science Research Program through the
National Research Foundation of Korea NRF funded by the Ministry of Education, Science
and Technology Grant no. 2011-0021821.
References
1 V. V. Chistyakov, “Modular metric spaces I: basic concepts,” Nonlinear Analysis, vol. 74, pp. 1–14, 2010.
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