Asymptotic pointwise contractive type in modular function spaces
© Golkarmanesh and Saeidi; licensee Springer 2013
Received: 9 September 2012
Accepted: 11 March 2013
Published: 17 April 2013
In this paper, we introduce asymptotic pointwise contractive type conditions in modular function spaces and present fixed point results for mappings under such conditions.
MSC:47H09, 47H10, 54H25.
where pointwise on M. Moreover, Kirk and Xu  proved that if C is a weakly compact convex subset of a Banach space E and an asymptotic pointwise contraction, then T has a unique fixed point , and for each , the sequence of Picard iterates converges in norm to v.
where pointwise on M.
Theorem 1.1 Let C be a nonempty weakly compact subset of a Banach space E, and let be a mapping of weak asymptotic pointwise contraction type. Then T has a unique fixed point and, for each , the sequence of Picard iterates converges in norm to v.
On the other hand, Khamsi and Kozlowski  studied the concept of asymptotic pointwise contractions in modular function spaces.
In this paper, motivated by Khamsi and Kozlowski [4, 5] and Saeidi , we study the notion of asymptotic pointwise contraction type in a modular function space. Moreover, we present fixed results which extend the earlier results in [3, 4].
Let Ω be a nonempty set, and let Σ be a nontrivial σ-algebra of subsets of Ω. Let be a δ-ring of subsets of Ω such that for any and . Let us assume that there exists an increasing sequence of sets such that . By ξ we denote the linear space of all simple functions with supports from . By we denote the space of all extended measurable function, i.e., all function such that there exists a sequence , and for all .
By we denote the characteristic function of the set A.
Definition 2.1 
ρ is monotone, i.e., for all implies , where ;
ρ is orthogonally subadditive, i.e., for any such that , ;
ρ has the Fatou property, i.e., for all implies , where ;
ρ is order continuous in ξ, i.e., and implies .
where each is actually an equivalence class of functions equal ρ-a.e. rather than an individual function. Where no confusion exists, we write ℳ instead of .
we say that ρ is a regular convex function semimodular if for every implies ρ-a.e.;
we say that ρ is a regular convex function modular if implies ρ-a.e.
The class of all nonzero regular convex function modulars on Ω is denoted by ℜ.
- (a)A modular function space is the vector space , or briefly , defined by
- (b)The following formula defines a norm in (frequently called Luxemburg norm):
In the following theorem, we recall some of the properties of modular function spaces that will be used later on in this paper.
has the Lebesgue property, i.e., , for , and ;
is the closure of ξ (in the sense of ).
We say that is ρ-convergent to f and write if and only if .
A sequence where is called ρ-Cauchy if as .
A set is called ρ-closed if for any sequence in C, the convergence implies that f belongs to C.
A set is called ρ-bounded if .
For a set , the mapping is called ρ-continuous if , then .
A set is called ρ-a.e. closed if for any sequence in C which ρ-a.e. converges to some f, then we must have .
A set is called ρ-a.e. compact if for any sequence in C, there exists a subsequence which ρ-a.e. converges to some .
- (h)Let and . The ρ-distance between f and C is defined as
Let us recall that ρ-convergence does not necessarily imply ρ-Cauchy condition. Also, does not imply in general , .
Definition 2.6 
We say that has the property if and only if every nonincreasing sequence of nonempty, ρ-bounded, ρ-closed, convex subsets of has nonempty intersection.
Definition 2.7 
Definition 2.8 
A function , where is nonempty and ρ-closed, is called ρ-lower semicontinuous if for any , the set is ρ-closed.
The following result plays an important role in the proof of the main results.
Lemma 2.9 
Let us recall the notion of ρ-type.
Definition 2.10 
Lemma 2.11 
Let be uniformly continuous. Let be nonempty, convex, ρ-closed and ρ-bounded. Then any ρ-type is ρ-lower semicontinuous in C.
3 Asymptotic pointwise contractive type conditions in modular function spaces
Definition 3.1 
If converges pointwise to , then T is called asymptotic pointwise ρ-contraction.
If for any , then T is called asymptotic pointwise nonexpansive.
If for any with , then T is called strongly asymptotic pointwise ρ-contraction.
Khamsi and Kozlowski proved the following results in modular function spaces.
Theorem 3.2 
Let be nonempty, ρ-closed and ρ-bounded. Let be an asymptotic pointwise ρ-contraction. Then T has at most one fixed point in C. Moreover, if is a fixed point of T, then the orbit is ρ-convergent to for any .
Theorem 3.3 
Let us assume that is uniformly continuous and has the property . Let be nonempty, convex, ρ-closed and ρ-bounded. Let be an asymptotic pointwise ρ-contraction. Then T has a unique fixed point . Moreover, the orbit is ρ-convergent to for any .
Below, we introduce the notion of asymptotic pointwise ρ-contraction type in modular function spaces.
where pointwise on M.
We will obtain fixed point results for these mappings in modular function spaces.
which implies that .
The following theorem is our main result.
Theorem 3.5 Let be uniformly continuous and have the property (R). Let be nonempty, convex, ρ-closed and ρ-bounded. Let be a mapping of weak asymptotic pointwise ρ-contraction type. Then T has a unique fixed point and, for each , the sequence of Picard iterates is ρ-convergent to v.
which forces as . Hence as . From this and the continuity of , for some , it follows that as . Since the ρ-limit of any ρ-convergent sequence is unique, we must have , namely, is a fixed point of . Now, repeating the above proof for instead of x, we deduce that is ρ-convergent to a member v of C; i.e., . But for all . Hence, and then .
Since the choice of is arbitrary and , we get .
Since and , we immediately get . □
Next, using the ρ-a.e. strong Opial property of the function modular, we prove a fixed point theorem which does not assume the uniform continuity of ρ.
Lemma 3.7 
Theorem 3.8 Let . Assume that has the ρ-a.e. strong Opial property. Let be a nonempty ρ-a.e. compact convex subset such that for some . Then any of weak asymptotic pointwise ρ-contraction type has a unique fixed point . Moreover, the orbit is ρ-convergent to for any .
The rest of the proof is like the one used for Theorem 3.5. □
The authors are grateful to the referees for their careful reading of the paper and several helpful comments.
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