Volume 2010, Article ID 317107,10pages 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 Beygmohammadi
1and Abdolrahman Razani
21Department of Mathematics, Islamic Azad University-Kermanshah Branch, Iran
2Department of Mathematics, Faculty of Science, Imam Khomeini International University, Iran
Correspondence should be addressed to Maryam Beygmohammadi, [email protected]
Received 16 February 2010; Revised 22 July 2010; Accepted 28 October 2010 Academic Editor: Oscar Blasco
Copyrightq2010 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 Khamsi2008and Ciric1974.
1. Introduction
In1, Branciari established that a functionfdefined on a complete metric space satisfying a contraction condition of the form
dfx,fy
0
ϕtdt≤c dx,y
0
ϕtdt 1.1
has a unique attractive fixed point whereϕ:R → Ris a Lebesgue-integrable mapping and c∈0,1.
In2, Rhoades extended this result to a quasicontraction functionf. The purpose of this paper is to extend these theorems in modular space.
First, we introduce the notion of modular space.
Definition 1.1. LetX be an arbitrary vector space overK RorC. A functionalρ : X → 0,∞is called modular if
1ρx 0 if and only ifx0;
2ραx ρxforα∈Kwith|α|1, for allx, y∈X;
3ραxβy≤ρx ρyifα, β≥0, αβ1, for allx, y∈X.
If2.14in Definition1.1is replaced by ρ
αxβy
≤αsρx βsρ y
1.2 forα, β≥0, αsβs 1 with ans∈0,1, then the modularρis called ans-convex modular;
and ifs1,ρis called a convex modular.
Definition 1.2. A modularρdefines a corresponding modular space, that is, the spaceXρis given by
Xρ
x∈X |ρλx−→0 asλ−→0
. 1.3
Definition 1.3. LetXρbe a modular space.
1A sequence{xn}ninXρis said to be
aρ-convergent toxifρxn−x → 0 asn → ∞, bρ-Cauchy ifρxn−xm → 0 asn, m → ∞.
2Xρisρ-complete if anyρ-Cauchy sequence isρ-convergent.
3A subsetB⊂ Xρis said to beρ-closed if for any sequence{xn}n ⊂Bwithxn → x thenx∈B.Bρdenotes the closure ofBin the sense ofρ.
4A subsetB⊂Xρis calledρ-bounded if δρB sup
x,y∈Bρ x−y
<∞, 1.4
whereδρBis called theρ-diameter ofB.
5We say thatρhas Fatou property if ρ
x−y
≤lim infρ xn−yn
1.5
whenever
xn
−→ρ x, yn
−→ρ y. 1.6
6ρis said to satisfy theΔ2-condition if:ρ2xn → 0 asn → ∞wheneverρxn → 0 asn → ∞.
Remark 1.4. Note that sinceρdoes not satisfy a priori the triangle inequality, we cannot expect that if{xn}and{yn}areρ-convergent, respectively, toxandythen{xnyn}isρ-convergent toxy, neither that aρ-convergent sequence isρ-Cauchy.
2. Main Result
Theorem 2.1. LetXρbe a complete modular space, whereρsatisfies theΔ2-condition. Assume that ψ:R → 0,∞is an increasing and upper semicontinuous function satisfying
ψt< t, ∀t >0. 2.1
Letϕ : 0,∞ → 0,∞be a nonnegative Lebesgue-integrable mapping which is summable on each compact subset of0,∞and such that forε > 0,ε
0ϕtdt > 0 and letf : Xρ → Xρbe a mapping such that there arec, l∈Rwherel < c,
ρcfx−fy
0
ϕtdt≤ψ
ρlx−y
0
ϕtdt , 2.2
for eachx, y∈Xρ. Thenfhas a unique fixed point inXρ.
Proof. First, we show that forx ∈Xρ, the sequence{ρcfnx−fn−1x}converges to 0. For n∈N, we have
ρcfnx−fn−1x 0
ϕtdt≤ψ
ρlfn−1x−fn−2x
0
ϕtdt
<
ρlfn−1x−fn−2x 0
ϕtdt
<
ρcfn−1x−fn−2x
0
ϕtdt.
2.3
Consequently, {ρcfnx−fn−1x
0 ϕtdt} is decreasing and bounded from below. Therefore {ρcfnx−fn−1x
0 ϕtdt}converges to a nonnegative pointa.
Now, ifa /0,
a lim
n→ ∞
ρcfnx−fn−1x
0
ϕtdt
≤ lim
n→ ∞ψ
ρlfn−1x−fn−2x
0
ϕtdt
≤ lim
n→ ∞ψ
ρcfn−1x−fn−2x
0
ϕtdt ,
2.4
then
a≤ψa, 2.5
which is a contradiction, soa0 and ρcfnx−fn1x
0
ϕtdt−→0 asn−→∞. 2.6
This concludesρcfnx−fn1x → 0. Suppose that
nlim→ ∞supρ c
fnx−fn1x
ε >0 2.7
then there exist aνε∈Nand a sequencefnνxν≥νε such that
ρ c
fnνx−fnν1x
−→ε >0, ν−→ ∞, ρ
c
fnνx−fnν1x
≥ ε
2, ∀ν≥νε,
2.8
then we get the following contradiction:
0 lim
ν→ ∞
ρcfnνx−fnν1x 0
ϕtdt≥ ε/2
0
ϕtdt >0. 2.9
Now, we prove for eachx∈Xρthe sequence{fnx}n∈Nis aρ-Cauchy sequence.
Assume that there is anε > 0 such that for each ν ∈ Nthere exist mν, nν ∈ Nthat mν> nν> ν,
ρ l
fmνx−fnνx
≥ε. 2.10
Then we choose the sequencemνν∈N and nνν∈N such that for each ν ∈ N,mν is minimal in the sense that
ρ l
fmνx−fnνx
≥ε. 2.11
But
ρ l
fhx−fnνx
< ε, 2.12
for eachh∈ {nν1, . . . , mν−1}.
Now, letα∈Rbe such thatl/c1/α1, then we have ε
0
ϕtdt≤
ρlfmνx−fnνx 0
ϕtdt
≤
ρcfmνx−fnν1x 0
ϕtdt
ραlfnν1x−fnνx 0
ϕtdt
≤ψ
ρlfmν−1x−fnνx
0
ϕtdt
ραlfnν1x−fnνx 0
ϕtdt
≤
ρlfmν−1x−fnνx 0
ϕtdt
ραlfnν1x−fnνx 0
ϕtdt
≤ ε
0
ϕtdt
ραlfnν1x−fnνx 0
ϕtdt.
2.13
Thus, asν → ∞, byΔ2-condition,ραlfnν1x−fnνx
0 ϕtdt → 0. Therefore
ρlfmνx−fnνx 0
ϕtdt−→ε, ν−→ ∞. 2.14
Now, ρlfmνx−fnνx
0
ϕtdt≤
ρcfmν1x−fnν1x 0
ϕtdt
ρ2αlfmνx−fmν1x 0
ϕtdt
ρ2αlfnν1x−fnνx 0
ϕtdt
≤ψ
ρlfmνx−fnνx 0
ϕtdt
ρ2αlfmνx−fmν1x 0
ϕtdt
ρ2αlfnν1x−fnνx
0
ϕtdt.
2.15
Ifν → ∞we get
ε
0
ϕtdt≤ψ ε
0
ϕtdt
, 2.16
which is a contradiction for ε > 0. Therefore {lfnx} is a ρ-Cauchy sequence and by Δ2- condition{fnx}isρ-Cauchy. By the fact thatXρisρ-complete, there is az ∈ Xρsuch that ρfnz−z → 0 asn → ∞. Furthermore,zis the fixed point forf. In fact
ρc 2
z−fz
≤ρ c
z−fnz ρ
c
fnz−fz
−→0, n−→ ∞ 2.17
thenρc/2z−fz 0 andfzz.
Now, assume that we have more than one fixed point forf. Letzandube two distinct fixed points, then
ρcz−u
0
ϕtdt
ρcfz−fu
0
ϕtdt≤ψ
ρlz−u
0
ϕtdt
<
ρlz−u
0
ϕtdt≤
ρcz−u
0
ϕtdt,
2.18
which is a contradiction. Sozuand the proof is complete.
Corollary 2.2see1. LetXρbe a complete modular space, whereρsatisfies theΔ2-condition. Let f:Xρ → Xρbe a mapping such that there exists anλ∈0,1andc, l∈Rwherel < cand for each x, y∈Xρ,
ρcfx−fy
0
ϕtdt≤λ
ρlx−y
0
ϕtdt , 2.19
thenfhas a unique fixed point.
Corollary 2.3see3. LetXρ be a complete modular space, where ρ satisfies theΔ2-condition.
Assume thatψ:R → 0,∞is an increasing and upper semicontinuous function satisfying
ψt< t, ∀t >0. 2.20
LetBbe aρ-closed subset ofXρandT : B → Bbe a mapping such that there existc, l∈ Rwith c > l,
ρ c
Tx−Ty
≤ψ ρ
l
x−y
2.21
for allx, y∈B. ThenThas a fixed point.
In the next theorem we use the following notation:
m x, y
max
ρ x−y
, ρx−Tx, ρ
y−Ty ,ρ
1/2
x−Ty ρ
1/2
y−Tx 2
. 2.22
Theorem 2.4. LetXρ, ρbe aρ-complete modular space thatρ satisfies the Δ2-condition and let T :Xρ → Xρbe a mapping such that for eachx, y∈Xρ,
ρTx−Ty
0
φtdt≤ψ
mx,y
0
φtdt , 2.23
whereφ:R → Randψ:R → 0,∞are as in Theorem2.1. ThenThas a unique fixed point.
Proof. Letx∈Xρ, we will show that{Tnx}is a Cauchy sequence. First, we prove that{ρTnx−
Tn−1x}converges to 0. From2.23, ρTnx−Tn−1x
0
φtdt≤ψ
mTn−1x,Tn−2x
0
φtdt . 2.24
By the definition ofmx, y,
m
Tn−1x, Tn−2x max
ρ
Tnx−Tn−1x , ρ
Tn−1−Tn−2x ,ρ
1/2
Tnx−Tn−2x 2
, ρ
1/2
Tnx−Tn−2x
2 ≤ ρ
Tnx−Tn−1x ρ
Tn−1−Tn−2x 2
≤max ρ
Tnx−Tn−1x , ρ
Tn−1−Tn−2x .
2.25
Hence, m
Tn−1x, Tn−2x
max ρ
Tnx−Tn−1x , ρ
Tn−1−Tn−2x
2.26
and therefore,
ρTnx−Tn−1x 0
φtdt≤ψ
mTn−1x,Tn−2x
0
φtdt
≤
mTn−1x,Tn−2x
0
φtdt
max{ρTnx−Tn−1x,ρTn−1−Tn−2x}
0
φtdt
max
ρTnx−Tn−1x
0
φtdt,
ρTn−1−Tn−2x
0
φtdt
ρTn−1−Tn−2x 0
φtdt.
2.27
This means that{ρTnx−Tn−1x}is decreasing and since it is bounded from below, it is a convergent sequence. Similarly to Theorem2.1, it is easy to show that
ρ
Tnx−Tn−1x
−→0. 2.28
Now, we show that{Tnx}is Cauchy. If not, then there exist anε >0 and subsequences {mp}and{np}such thatmp< np< mp1with
ρ
Tmpx−Tnpx
≥ε, ρ 2
Tmpx−Tnp−1x
< ε. 2.29
From2.22,
m
Tmp−1x, Tnp−1x max
ρ
Tmp−1x−Tnp−1x , ρ
Tmpx−Tmp−1x , ρ
Tnpx−Tnp−1x , ρ
1/2
Tmpx−Tnp−1x ρ
1/2
Tnpx−Tmp−1x 2
. 2.30 By using2.28, we get
limp
ρTmpx−Tmp−1x 0
φtdtlim
p
ρTnpx−Tnp−1x 0
φtdt0. 2.31
On the other hand,
ρ
Tmp−1x−Tnp−1x
≤ρ 2
Tmp−1x−Tmpx ρ
2
Tmpx−Tnp−1x
≤ρ 2
Tmp−1x−Tmpx ε,
2.32
thus by theΔ2-condition,
limp
ρTmp−1x−Tnp−1x 0
φtdt≤ ε
0
φtdt. 2.33
For the last term inmTmp−1x, Tnp−1xby the fact thatρcxis an increasing function ofcwe have
vm, n: ρ 1/2
Tmpx−Tnp−1x ρ
1/2
Tnpx−Tmp−1x 2
≤ ρ
Tmpx−Tmp−1x ρ
2
Tnpx−Tnp−1x 2
ρ 2
Tmpx−Tnp−1x ρ
1/2
Tmpx−Tnp−1x 2
≤ερ
Tmpx−Tmp−1x ρ
2
Tnpx−Tnp−1x
2 .
2.34
Hence, from2.28we get
limp
vm,n
0
φtdt≤ ε
0
φtdt. 2.35
Therefore from2.31,2.33, and2.35it can be concluded that ε
0
φtdt≤
ρTmpx−Tnpx 0
φtdt≤ψ
mTmp−1x,Tnp−1x
0
φtdt
<
mTmp−1x,Tnp−1x 0
φtdt≤ ε
0
φtdt
2.36
which is a contradiction, whenpis large enough. Therefore,{Tnx}is Cauchy and sinceXρis ρ-complete there is anz∈XρthatTnx → z. Now, we should prove thatzis the fixed point forT. In fact,
ρ1/2Tz−z
0
φtdt≤
ρTz−Tnz
0
φtdt
ρTnz−z
0
φtdt
≤ψ
mz,Tn−1z
0
φtdt
ρTnz−z
0
φtdt−→0 asn−→ ∞,
2.37
by the definition ofm. It follows thatTzz.
Letw∈Xρbe another fixed point ofT. Then, ρw−z
0
φtdt
ρTw−Tz
0
φtdt≤ψ
mw,z
0
φtdt
<
mw,z
0
φtdt ρw−z
0
φtdt.
2.38
That is because mw, z max
ρz−w, ρz−z, ρw−w,ρ1/2z−w ρ1/2w−z 2
ρw−z,
2.39
thuszw.
Corollary 2.5see2. LetX, dbe complete metric space,k∈0,1,f :X → Xa mapping such that, forx, y∈X,
dfx,fy
0
φtdt≤k mx,y
0
φtdt, 2.40
whereφ : R → R is a Lebesgue-integrable mapping which is summable, nonnegative, and such that
ε
0
φtdt >0 ∀ε >0, 2.41
and where
m x, y
max
d x, y
, d x, fx
, d y, fy
,d x, fy
d y, fy 2
. 2.42
Thenfhas a unique fixed point.
Corollary 2.6see4. LetX, ρbe a modular space such thatρsatisfies the Fatou property. Let Cbe aρ-complete nonempty subset ofXρandT :C → Cbe quasicontraction. Letx∈Csuch that δρx<∞. Then{Tnx}ρ-converges toω∈C. Hereδρx sup{ρTnx−Tmx;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 withoutΔ2-condition,” Fixed Point Theory and Applications, vol. 2008, Article ID 916187, 6 pages, 2008.
