Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2010, Article ID 317107, 10 pages
doi:10.1155/2010/317107
Research Article
Two Fixed-Point Theorems for Mappings Satisfying
a General Contractive Condition of Integral Type in
the Modular Space
Maryam Beygmohammadi1 and Abdolrahman Razani2
1
2
Department of Mathematics, Islamic Azad University-Kermanshah Branch, Iran
Department of Mathematics, Faculty of Science, Imam Khomeini International University, Iran
Correspondence should be addressed to Maryam Beygmohammadi,
maryambmohamadi@mathdep.iust.ac.ir
Received 16 February 2010; Revised 22 July 2010; Accepted 28 October 2010
Academic Editor: Oscar Blasco
Copyright q 2010 M. Beygmohammadi and A. Razani. 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.
First we prove existence of a fixed point for mappings defined on a complete modular space
satisfying a general contractive inequality of integral type. Then we generalize fixed-point theorem
for a quasicontraction mapping given by Khamsi 2008 and Ciric 1974 .
1. Introduction
In 1, Branciari established that a function f defined on a complete metric space satisfying a
contraction condition of the form
dfx,fy
0
ϕtdt ≤ c
dx,y
ϕtdt
1.1
0
has a unique attractive fixed point where ϕ : R → R is a Lebesgue-integrable mapping and
c ∈ 0, 1.
In 2, Rhoades extended this result to a quasicontraction function f. The purpose of
this paper is to extend these theorems in modular space.
First, we introduce the notion of modular space.
Definition 1.1. Let X be an arbitrary vector space over K R or C. A functional ρ : X →
0, ∞ is called modular if
2
International Journal of Mathematics and Mathematical Sciences
1 ρx 0 if and only if x 0;
2 ραx ρx for α ∈ K with |α| 1, for all x, y ∈ X;
3 ραx βy ≤ ρx ρy if α, β ≥ 0, α β 1, for all x, y ∈ X.
If 2.14 in Definition 1.1 is replaced by
ρ αx βy ≤ αs ρx βs ρ y
1.2
for α, β ≥ 0, αs βs 1 with an s ∈ 0, 1, then the modular ρ is called an s-convex modular;
and if s 1, ρ is called a convex modular.
Definition 1.2. A modular ρ defines a corresponding modular space, that is, the space Xρ is
given by
Xρ x ∈ X | ρλx −→ 0 as λ −→ 0 .
1.3
Definition 1.3. Let Xρ be a modular space.
1 A sequence {xn }n in Xρ is said to be
a ρ-convergent to x if ρxn − x → 0 as n → ∞,
b ρ-Cauchy if ρxn − xm → 0 as n, m → ∞.
2 Xρ is ρ-complete if any ρ-Cauchy sequence is ρ-convergent.
3 A subset B ⊂ Xρ is said to be ρ-closed if for any sequence {xn }n ⊂ B with xn → x
ρ
then x ∈ B. B denotes the closure of B in the sense of ρ.
4 A subset B ⊂ Xρ is called ρ-bounded if
δρ B sup ρ x − y < ∞,
x,y∈B
1.4
where δρ B is called the ρ-diameter of B.
5 We say that ρ has Fatou property if
ρ x − y ≤ lim inf ρ xn − yn
1.5
whenever
ρ
xn →
− x,
ρ
yn →
− y.
1.6
6 ρ is said to satisfy the Δ2 -condition if: ρ2xn → 0 as n → ∞ whenever ρxn → 0
as n → ∞.
Remark 1.4. Note that since ρ does not satisfy a priori the triangle inequality, we cannot expect
that if {xn } and {yn } are ρ-convergent, respectively, to x and y then {xn yn } is ρ-convergent
to x y, neither that a ρ-convergent sequence is ρ-Cauchy.
International Journal of Mathematics and Mathematical Sciences
3
2. Main Result
Theorem 2.1. Let Xρ be a complete modular space, where ρ satisfies the Δ2 -condition. Assume that
ψ : R → 0, ∞ is an increasing and upper semicontinuous function satisfying
ψt < t,
∀t > 0.
2.1
Let ϕ : 0, ∞ → 0, ∞ be a nonnegative Lebesgue-integrable
mapping which is summable on
ε
each compact subset of 0, ∞ and such that for ε > 0, 0 ϕtdt > 0 and let f : Xρ → Xρ be a
mapping such that there are c, l ∈ R where l < c,
ρcfx−fy
ϕtdt ≤ ψ
ρlx−y
2.2
ϕtdt ,
0
0
for each x, y ∈ Xρ . Then f has a unique fixed point in Xρ .
Proof. First, we show that for x ∈ Xρ , the sequence {ρcf n x − f n−1 x} converges to 0. For
n ∈ N, we have
ρcf n x−f n−1 x
ϕtdt ≤ ψ
ρlf n−1 x−f n−2 x
0
ϕtdt
0
ρlf n−1 x−f n−2 x
<
ϕtdt
2.3
0
ρcf n−1 x−f n−2 x
<
ϕtdt.
0
ρcf n x−f n−1 x
ϕtdt} is decreasing and bounded from below. Therefore
Consequently, { 0
ρcf n x−f n−1 x
ϕtdt} converges to a nonnegative point a.
{0
Now, if a /
0,
a lim
ρcf n x−f n−1 x
n→∞
≤ lim ψ
n→∞
≤ lim ψ
n→∞
ϕtdt
0
ρlf n−1 x−f n−2 x
2.4
ϕtdt
0
ρcf n−1 x−f n−2 x
ϕtdt ,
0
then
a ≤ ψa,
2.5
4
International Journal of Mathematics and Mathematical Sciences
which is a contradiction, so a 0 and
ρcf n x−f n1 x
ϕtdt −→ 0
as n −→ ∞.
2.6
0
This concludes ρcf n x − f n1 x → 0. Suppose that
lim sup ρ c f n x − f n1 x ε > 0
n→∞
2.7
then there exist a νε ∈ N and a sequence f nν xν≥νε such that
ρ c f nν x − f nν 1 x −→ ε > 0,
ε
ρ c f nν x − f nν 1 x ≥ ,
2
ν −→ ∞,
∀ν ≥ νε ,
2.8
then we get the following contradiction:
0 lim
ρcf nν x−f nν 1 x
ν→∞
0
ϕtdt ≥
ε/2
ϕtdt > 0.
2.9
0
Now, we prove for each x ∈ Xρ the sequence {f n x}n∈N is a ρ-Cauchy sequence.
Assume that there is an ε > 0 such that for each ν ∈ N there exist mν , nν ∈ N that
mν > nν > ν,
ρ l f mν x − f nν x ≥ ε.
2.10
Then we choose the sequence mν ν∈N and nν ν∈N such that for each ν ∈ N, mν is
minimal in the sense that
ρ l f mν x − f nν x ≥ ε.
2.11
ρ l f h x − f nν x < ε,
2.12
But
for each h ∈ {nν 1, . . . , mν − 1}.
International Journal of Mathematics and Mathematical Sciences
5
Now, let α ∈ R be such that l/c 1/α 1, then we have
ε
ϕtdt ≤
0
ρlf mν x−f nν x
ϕtdt
0
≤
ρcf mν x−f nν 1 x
ραlf nν 1 x−f nν x
ϕtdt ϕtdt
0
≤ψ
0
ρlf mν −1 x−f nν x
ϕtdt
ραlf nν 1 x−f nν x
ϕtdt
0
≤
0
ρlf mν −1 x−f nν x
0
≤
ε
ϕtdt 2.13
ραlf nν 1 x−f nν x
ϕtdt
0
ϕtdt 0
ραlf nν 1 x−f nν x
ϕtdt.
0
Thus, as ν → ∞, by Δ2 -condition,
ρlf mν x−f nν x
ραlf nν 1 x−f nν x
0
ϕtdt −→ ε ,
ϕtdt → 0. Therefore
2.14
ν −→ ∞.
0
Now,
ρlf mν x−f nν x
0
ϕtdt ≤
ρcf mν 1 x−f nν 1 x
ϕtdt ρ2αlf mν x−f mν 1 x
0
ρ2αlf nν 1 x−f nν x
ϕtdt
0
≤ψ
ρlf mν x−f nν x
ϕtdt
0
ϕtdt
0
2.15
ρ2αlf mν x−f mν 1 x
ϕtdt
0
ρ2αlf nν 1 x−f nν x
ϕtdt.
0
If ν → ∞ we get
ε
ϕtdt ≤ ψ
0
ε
ϕtdt ,
2.16
0
which is a contradiction for ε > 0. Therefore {lf n x} is a ρ-Cauchy sequence and by Δ2 condition {f n x} is ρ-Cauchy. By the fact that Xρ is ρ-complete, there is a z ∈ Xρ such that
ρf n z − z → 0 as n → ∞. Furthermore, z is the fixed point for f. In fact
ρ
c
2
z − fz ≤ ρ c z − f n z ρ c f n z − fz −→ 0,
then ρc/2z − fz 0 and fz z.
n −→ ∞
2.17
6
International Journal of Mathematics and Mathematical Sciences
Now, assume that we have more than one fixed point for f. Let z and u be two distinct
fixed points, then
ρcz−u
ϕtdt 0
ρcfz−fu
ϕtdt ≤ ψ
ρlz−u
0
ϕtdt
0
ρlz−u
<
ϕtdt ≤
0
2.18
ρcz−u
ϕtdt,
0
which is a contradiction. So z u and the proof is complete.
Corollary 2.2 see 1. Let Xρ be a complete modular space, where ρ satisfies the Δ2 -condition. Let
f : Xρ → Xρ be a mapping such that there exists an λ ∈ 0, 1 and c, l ∈ R where l < c and for each
x, y ∈ Xρ ,
ρcfx−fy
ϕtdt ≤ λ
0
ρlx−y
ϕtdt ,
2.19
0
then f has a unique fixed point.
Corollary 2.3 see 3. Let Xρ be a complete modular space, where ρ satisfies the Δ2 -condition.
Assume that ψ : R → 0, ∞ is an increasing and upper semicontinuous function satisfying
ψt < t,
∀t > 0.
2.20
Let B be a ρ-closed subset of Xρ and T : B → B be a mapping such that there exist c, l ∈ R with
c > l,
ρ c Tx − Ty ≤ ψ ρ l x − y
2.21
for all x, y ∈ B. Then T has a fixed point.
In the next theorem we use the following notation:
ρ 1/2 x − T y ρ 1/2 y − T x
m x, y max ρ x − y , ρx − T x, ρ y − T y ,
.
2
2.22
International Journal of Mathematics and Mathematical Sciences
7
Theorem 2.4. Let Xρ , ρ be a ρ-complete modular space that ρ satisfies the Δ2 -condition and let
T : Xρ → Xρ be a mapping such that for each x, y ∈ Xρ ,
ρT x−T y
φtdt ≤ ψ
mx,y
2.23
φtdt ,
0
0
where φ : R → R and ψ : R → 0, ∞ are as in Theorem 2.1. Then T has a unique fixed point.
Proof. Let x ∈ Xρ , we will show that {T n x} is a Cauchy sequence. First, we prove that {ρT n x−
T n−1 x} converges to 0. From 2.23,
ρT n x−T n−1 x
φtdt ≤ ψ
mT n−1 x,T n−2 x
2.24
φtdt .
0
0
By the definition of mx, y,
ρ1/2T n x − T n−2 x
n
n−1
n−1
n−2
,
m T x, T x max ρ T x − T x , ρ T
−T x ,
2
ρ 1/2 T n x − T n−2 x
ρ T n x − T n−1 x ρ T n−1 − T n−2 x
≤
2
2
≤ max ρ T n x − T n−1 x , ρ T n−1 − T n−2 x .
n−1
n−2
2.25
Hence,
m T n−1 x, T n−2 x max ρ T n x − T n−1 x , ρ T n−1 − T n−2 x
2.26
and therefore,
ρT n x−T n−1 x
φtdt ≤ ψ
mT n−1 x,Tn−2 x
0
φtdt
0
≤
mT n−1 x,T n−2 x
φtdt
0
max{ρT n x−T n−1 x,ρT n−1 −T n−2 x}
max
ρT n x−T n−1 x
φtdt,
φtdt
0
ρT n−1 −T n−2 x
φtdt.
0
ρT n−1 −T n−2 x
0
2.27
φtdt
0
8
International Journal of Mathematics and Mathematical Sciences
This means that {ρT n x − T n−1 x} is decreasing and since it is bounded from below, it is a
convergent sequence. Similarly to Theorem 2.1, it is easy to show that
ρ T n x − T n−1 x −→ 0.
2.28
Now, we show that {T n x} is Cauchy. If not, then there exist an ε > 0 and subsequences
{mp} and {np} such that mp < np < mp 1 with
ρ T mp x − T np x ≥ ε,
ρ 2 T mp x − T np−1 x < ε.
2.29
From 2.22,
mp−1
np−1
m T
x, T
x max ρ T mp−1 x − T np−1 x ,
ρ T mp x − T mp−1 x , ρ T np x − T np−1 x ,
ρ 1/2 T mp x − T np−1 x ρ 1/2 T np x − T mp−1 x
.
2
2.30
By using 2.28, we get
ρT mp x−T mp−1 x
lim
p
φtdt lim
p
0
ρT np x−T np−1 x
φtdt 0.
2.31
0
On the other hand,
ρ T mp−1 x − T np−1 x ≤ ρ 2 T mp−1 x − T mp x ρ 2 T mp x − T np−1 x
≤ ρ 2 T mp−1 x − T mp x ε,
2.32
thus by the Δ2 -condition,
ρT mp−1 x−T np−1 x
lim
p
0
φtdt ≤
ε
φtdt.
0
2.33
International Journal of Mathematics and Mathematical Sciences
9
For the last term in mT mp−1 x, T np−1 x by the fact that ρcx is an increasing function
of c we have
ρ 1/2 T mp x − T np−1 x ρ 1/2 T np x − T mp−1 x
2
mp
ρ T
x − T mp−1 x ρ 2 T np x − T np−1 x
≤
2
mp
ρ 2 T
x − T np−1 x ρ 1/2 T mp x − T np−1 x
2
mp
ρ T
x − T mp−1 x ρ 2 T np x − T np−1 x
.
≤ε
2
vm, n : 2.34
Hence, from 2.28 we get
vm,n
lim
p
ε
φtdt ≤
2.35
φtdt.
0
0
Therefore from 2.31, 2.33, and 2.35 it can be concluded that
ε
ρT mp x−T np x
φtdt ≤
0
φtdt ≤ ψ
mT mp−1 x,T np−1 x
φtdt
0
0
mT mp−1 x,T np−1 x
<
φtdt ≤
2.36
ε
φtdt
0
0
which is a contradiction, when p is large enough. Therefore, {T n x} is Cauchy and since Xρ is
ρ-complete there is an z ∈ Xρ that T n x → z. Now, we should prove that z is the fixed point
for T . In fact,
ρ1/2T z−z
φtdt ≤
0
ρT z−T n z
φtdt 0
≤ψ
ρT n z−z
φtdt
0
mz,T n−1 z
φtdt
0
ρT n z−z
2.37
φtdt −→ 0
as n −→ ∞,
0
by the definition of m. It follows that T z z.
Let w ∈ Xρ be another fixed point of T . Then,
ρw−z
φtdt 0
ρT w−T z
φtdt ≤ ψ
mw,z
0
mw,z
<
0
φtdt
0
φtdt ρw−z
φtdt.
0
2.38
10
International Journal of Mathematics and Mathematical Sciences
That is because
ρ1/2z − w ρ1/2w − z
mw, z max ρz − w, ρz − z, ρw − w,
2
2.39
ρw − z,
thus z w.
Corollary 2.5 see 2. Let X, d be complete metric space, k ∈ 0, 1, f : X → X a mapping such
that, for x, y ∈ X,
dfx,fy
0
φtdt ≤ k
mx,y
φtdt,
2.40
0
where φ : R → R is a Lebesgue-integrable mapping which is summable, nonnegative, and such
that
ε
φtdt > 0 ∀ε > 0,
2.41
0
and where
d x, fy d y, fy
.
m x, y max d x, y , d x, fx , d y, fy ,
2
2.42
Then f has a unique fixed point.
Corollary 2.6 see 4. Let X, ρ be a modular space such that ρ satisfies the Fatou property. Let
C be a ρ-complete nonempty subset of Xρ and T : C → C be quasicontraction. Let x ∈ C such that
δρ x < ∞. Then {T n x} ρ-converges to ω ∈ C. Here δρ x sup{ρT n x − T m x; n, m ∈ N}.
Acknowledgments
The authors would like to thank the anonymous referees for helpful comments to improve
this paper. The first author thanks the Islamic Azad University-Kermanshah branch for
supporting this research.
References
1 A. Branciari, “A fixed point theorem for mappings satisfying a general contractive condition of integral
type,” International Journal of Mathematics and Mathematical Sciences, vol. 29, no. 9, pp. 531–536, 2002.
2 B. E. Rhoades, “Two fixed-point theorems for mappings satisfying a general contractive condition of
integral type,” International Journal of Mathematics and Mathematical Sciences, no. 63, pp. 4007–4013, 2003.
3 A. Razani, E. Nabizadeh, M. B. Mohamadi, and S. H. Pour, “Fixed points of nonlinear and asymptotic
contractions in the modular space,” Abstract and Applied Analysis, vol. 2007, Article ID 40575, 10 pages,
2007.
4 M. A. Khamsi, “Quasicontraction mappings in modular spaces without Δ2 -condition,” Fixed Point
Theory and Applications, vol. 2008, Article ID 916187, 6 pages, 2008.