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Fixed point of nonlinear contractions in modular spaces
Journal of Inequalities and Applications volume 2013, Article number: 399 (2013)
Razani et al. studied the fixed points of nonlinear and asymptotic contractions in the modular space in 2007. In this paper, we generalize the kind of nonlinear contraction that is the result of Razani et al. (Abstr. Appl. Anal. 2007:40575, 2007) and prove the existence and uniqueness of fixed points for the generalized nonlinear contractions in modular spaces.
The notion of modular spaces, as a generalization of metric spaces, was introduced by Nakano  in 1950 in connection with the theory of order spaces and redefined and generalized by Musielak and Orlicz  in 1959. These spaces were developed following the successful theory of Orlicz spaces, which replaces the particular integral form of the nonlinear functional, which controls the growth of members of the space, by an abstractly given functional with some good properties. The monographic exposition of the theory of Orlicz spaces may be found in the book of Krasnosel’skii and Rutickii . For a current review of the theory of Musielak-Orlicz spaces and modular spaces, the reader is referred to the books of Musielak  and Kozlowski .
Fixed point theorems in modular spaces, generalizing the classical Banach fixed point theorem in metric spaces, have been studied extensively. In 2007, Razani et al.  studied some fixed points of nonlinear and asymptotic contractions in the modular spaces. In addition, quasi-contraction mappings in modular spaces without -condition were considered by Khamsi  in 2008. In 2011, Kuaket and Kumam  proved the existence of fixed points of asymptotic pointwise contractions in modular spaces. Recently, we proved the fixed points of asymptotic pointwise nonexpansive mappings in modular spaces .
In this paper, we introduce the notions of nonlinear contractions in modular spaces and establish their fixed points theorems in modular spaces.
Definition 2.1 Let X be an arbitrary vector space over K (=R or C).
A functional is called modular if
if and only if ;
for with , for all ;
if , for all .
If (iii) is replaced by
(iii′) , for , , for all ,
then the modular ρ is called convex modular.
A modular ρ defines a corresponding modular space, i.e., the space given by
Remark 2.2 Note that ρ is an increasing function. Suppose . Then, property (iii) with shows that .
Definition 2.3 Let be a modular space.
A sequence is said to be ρ-convergent to , and write if as .
A sequence is called ρ-Cauchy whenever as .
is called ρ-complete if any ρ-Cauchy sequence is ρ-convergent.
A subset is called ρ-closed if for any sequence ρ-convergent to , we have .
A ρ-closed subset is called ρ-compact if any sequence has a ρ-convergent subsequence.
ρ is said to satisfy the -condition if whenever as .
We say that ρ has the Fâtou property if , whenever and as .
A subset is said to be ρ-bounded if
where is called the ρ-diameter of B.
Define the ρ-distance between and as
Define the ρ-ball, , centered at with radius r as
Theorem 2.4 [, RNMP]
Let be a ρ-complete modular space, where ρ satisfies the -condition. Assume that is an increasing and upper semi-continuous function satisfying
Let B be a ρ-closed subset of , and let be a mapping such that there exist with ,
for all . Then T has a fixed point.
The Banach contraction mapping principle shows the existence and uniqueness of a fixed point in a complete metric space; this has been generalized by many mathematicians such as Arandelovic , Edelstein , Ciric , Rakotch , Reich , Kirk , and so forth. In addition, Boyd and Wong  studied mappings, which are nonlinear contractions in the metric space. It is necessary to mention that the applications of contraction, generalized contraction principle for self-mappings, and the applications of nonlinear contractions are well known. In the next section, we will prove the existence fixed points theorems for nonlinear contractions in modular space.
3 Fixed points of nonlinear contractions
In the sequel, we assume that is an increasing and upper semi-continuous function satisfying
Lemma 3.1 
Let , if and only if , where denotes the n-times repeated composition of ψ with itself.
Theorem 3.2 Let be a ρ-complete modular space, where ρ satisfies the -condition. Let ψ be as in the previous definition, let C be a ρ-closed subset of , and let be a mapping such that there exist with ,
for all . Then T has a unique fixed point.
Proof Let . At first, we show that the sequence converges to 0.
For , we have
By (3.3) and (3.4), therefore, we have
Consequently, is decreasing and bounded from below ().
Therefore, converges to a.
which is a contradiction, therefore, .
Now, we show that is a ρ-Cauchy sequence for . Suppose that is not a ρ-Cauchy sequence. Then, there are an and sequences of integers , , with , and such that
we can assume that
Let be the smallest number exceeding , for which (3.8) holds, and
Obviously, and since , then by Well Ordering Principle, the minimum element of is denoted by , and clearly (3.9) holds.
Now, we assume that and , then we have
If and by -condition, . Hence, . Now,
Thus, as , we obtain , which is a contradiction for . So, is a ρ-Cauchy sequence, and by -condition, is a ρ-Cauchy sequence, and is ρ-complete, there is a such that as . Now, it is enough to show that ω is a fixed point of T. Indeed,
Since as , therefore, and .
Next, we prove that T has a unique fixed point. Letting be another fixed point of T, we have
which implies that , so .
The proof is complete. □
The next corollary is immediate consequence of Theorem 3.2.
Corollary 3.3 Let be a ρ-complete modular space, where ρ satisfies the -condition. Let C be a ρ-closed subset of , and let be a mapping such that there exist with and ,
for all . Then T has a unique fixed point.
Next, we continue to generalize the above consequences.
Firstly, for any , define the orbit
and its ρ-diameter by
The following lemmas will be helpful to prove the main result.
Lemma 3.4 Let , C, ψ be as in the definitions above, let be a mapping such that there exist with ,
for all . Let such that . Then for any , one has
Moreover, one has
for all and .
Proof Letting , we have
for all . This obviously implies the following
for any .
Hence, for any , we have
Moreover, for any and , we have
Lemma 3.5 Let , T, C, ψ, x be as in the previous lemma, and let ρ satisfy the Fâtou property. Then ρ-converges to . Moreover, one has
for any .
Proof From the previous lemma, it is easy to know that is ρ-Cauchy. Since C is ρ-closed, then there exists such that . Since
for any , , and ρ satisfies the Fâtou property, we let to get
Theorem 3.6 Let , ρ, T, C, ψ be as in Lemma 3.5. Assume that is the ρ-limit of , and , . Then ω is a fixed point of T, i.e., . Moreover, if is another fixed point of T in C such that , then we have .
Proof We have
From (3.15) and the previous results, we get
Assume that for , we have
Using our previous assumption, we get
So, by induction, we have
for any . Therefore, we have
Using the Fâtou property, satisfied by ρ, we get
so, we get , i.e., .
Let be another fixed point of T such that . Then we get
which implies , i.e., .
We complete the proof of our theorem. □
The following corollary is an immediate consequence of Theorem 3.6.
Corollary 3.7 
Let be a ρ-complete modular space, where ρ satisfies the Fâtou property. Let C be a ρ-closed subset of , and let be a mapping such that there exist with and ,
for all . Assume that is the ρ-limit of , and , . Then ω is a fixed point of T, i.e., . Moreover, if is another fixed point of T in C such that , then we have .
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This research was supported by the NSFC Grant No. 11071279.
The authors declare that they have no competing interests.
RC was introduce and design the question, proof of theorem by XW, RC finally to revise, finalized, submit it to journal.