- Open Access
Fixed point of nonlinear contractions in modular spaces
© Chen and Wang; licensee Springer 2013
- Received: 31 August 2012
- Accepted: 30 July 2013
- Published: 22 August 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.
- nonlinear contractions
- fixed points
- 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.
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 ,
- (b)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 .
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 .
- (h)A subset is said to be ρ-bounded if
- (i)Define the ρ-distance between and as
- (j)Define the ρ-ball, , centered at with radius r as
Theorem 2.4 [, RNMP]
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.
Lemma 3.1 
Let , if and only if , where denotes the n-times repeated composition of ψ with itself.
for all . Then T has a unique fixed point.
Proof Let . At first, we show that the sequence converges to 0.
Consequently, is decreasing and bounded from below ().
Therefore, converges to a.
which is a contradiction, therefore, .
Obviously, and since , then by Well Ordering Principle, the minimum element of is denoted by , and clearly (3.9) holds.
Since as , therefore, and .
which implies that , so .
The proof is complete. □
The next corollary is immediate consequence of Theorem 3.2.
for all . Then T has a unique fixed point.
Next, we continue to generalize the above consequences.
The following lemmas will be helpful to prove the main result.
for all and .
for any .
for any .
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 .
so, we get , i.e., .
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 
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 .
This research was supported by the NSFC Grant No. 11071279.
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