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Asymptotic pointwise contractive type in modular function spaces
Fixed Point Theory and Applications volume 2013, Article number: 101 (2013)
Abstract
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.
1 Introduction
The notion of asymptotic pointwise contraction was introduced by Kirk [1]: Let be a metric space. A mapping is called an asymptotic pointwise contraction if there exists a function such that for each integer ,
where pointwise on M. Moreover, Kirk and Xu [2] 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.
Very recently, Saeidi [3] introduced the concept of (weak) asymptotic pointwise contraction type: Let be a metric space. A mapping is said to be of asymptotic pointwise contraction type (resp. of weak asymptotic pointwise contraction type) if is continuous for some integer and there exists a function such that for each x in M,
where pointwise on M.
It is easy to see that an asymptotic pointwise contraction is of asymptotic pointwise contraction type, but the converse is not true [3]. The following result was proved in [3].
Theorem 1.1 [3]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 [4] studied the concept of asymptotic pointwise contractions in modular function spaces.
In this paper, motivated by Khamsi and Kozlowski [4, 5] and Saeidi [3], 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].
2 Preliminaries
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 [6]
Let be a nontrivial, convex and even function. We say that ρ is a regular convex function pseudomodular if:
-
(a)
;
-
(b)
ρ is monotone, i.e., for all implies , where ;
-
(c)
ρ is orthogonally subadditive, i.e., for any such that , ;
-
(d)
ρ has the Fatou property, i.e., for all implies , where ;
-
(e)
ρ is order continuous in ξ, i.e., and implies .
Similarly as in the case of measure spaces, we say that a set is ρ-null if for every . We say that a property holds ρ-almost everywhere if the exceptional set is ρ-null. As usual we identify any pair of measurable sets whose symmetric difference is ρ-null as well as any pair of measurable functions differing only on a ρ-null set. With this in mind, we define
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 .
Let ρ be a regular function pseudomodular;
-
(a)
we say that ρ is a regular convex function semimodular if for every implies ρ-a.e.;
-
(b)
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.
Let ρ be a convex function modular.
-
(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.
Let . Defining and , we have
-
(i)
;
-
(ii)
has the Lebesgue property, i.e., , for , and ;
-
(iii)
is the closure of ξ (in the sense of ).
Let .
-
(a)
We say that is ρ-convergent to f and write if and only if .
-
(b)
A sequence where is called ρ-Cauchy if as .
-
(c)
A set is called ρ-closed if for any sequence in C, the convergence implies that f belongs to C.
-
(d)
A set is called ρ-bounded if .
-
(e)
For a set , the mapping is called ρ-continuous if , then .
-
(f)
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 .
-
(g)
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 [4]
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 [4]
We say that the function modular ρ is uniformly continuous if for every and , there exists such that
Definition 2.8 [4]
A function , where is nonempty and ρ-closed, is called ρ-lower semicontinuous if for any , the set is ρ-closed.
It can be proved that ρ-lower semicontinuity is equivalent to the condition
The following result plays an important role in the proof of the main results.
Lemma 2.9 [4]
Assume that has the property . Let be nonempty, convex, ρ-closed and ρ-bounded. If is a ρ-lower semicontinuous convex function, then there exists such that
Let us recall the notion of ρ-type.
Definition 2.10 [4]
Let be convex and ρ-bounded. A function is called a -type (or shortly a type) if there exists a sequence of elements of C such that for any , the following holds:
Lemma 2.11 [4]
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 [4]
Let and be non-empty and ρ-closed. A mapping is called an asymptotic pointwise mapping if there exists a sequence of mappings such that
-
(a)
If converges pointwise to , then T is called asymptotic pointwise ρ-contraction.
-
(b)
If for any , then T is called asymptotic pointwise nonexpansive.
-
(c)
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 [4]
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 [4]
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.
Definition 3.4 Let be nonempty, ρ-bounded and ρ-closed. A mapping is said to be of asymptotic pointwise ρ-contraction type (resp. of weak asymptotic pointwise ρ-contraction type) if is ρ-continuous for some integer and there exists a function such that, for each x in C,
where pointwise on M.
Taking
it can be easily seen from (3.1) (resp. (3.2)) that
for all , and
We will obtain fixed point results for these mappings in modular function spaces.
First, it is worth mentioning that the ρ-limit of any ρ-convergent sequence in is unique. This fact follows from the following reasoning: Assume that and . Then
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.
Proof Fix an and define a function τ by
By Lemma 2.11, τ is ρ-lower semicontinuous in C. By Lemma 2.9, then there exists such that
Let us prove that . Indeed, for any , we have
which implies
Since T is of weak asymptotic pointwise ρ-contraction type, by (3.4) we have . Thus, for a subsequence of , we have
Now, by (3.6) and (3.7), we obtain
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 .
We show that ; for this purpose, consider an arbitrary . Then there exists a such that for all . So, by choosing a natural number , we obtain
Since the choice of is arbitrary and , we get .
It is easy to verify that T can have only one fixed point. Indeed, if are fixed points of T, then by (3.5), we have
Taking lim inf in the above inequality, we obtain
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.
We say that satisfies the ρ-a.e. strong Opial property (or shortly SO-property) if for every which is ρ-a.e. convergent to zero such that there exists a for which
the following equality holds for any :
Lemma 3.7 [4]
Let . Assume that has the ρ-a.e. strong Opial property. Let be a nonempty, ρ-a.e. compact subset such that there exists such that . Let be a nonempty ρ-a.e. closed subset. For any , let be such that for any , there exists a sequence such that, for every , the following holds:
and for every and . Let for any . Then there exists at which λ attains infimum, i.e.,
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 .
Proof Fix an and define a function τ by
By Lemma 3.7 applied with , , , and with chosen for all , there exists such that
The rest of the proof is like the one used for Theorem 3.5. □
References
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The authors are grateful to the referees for their careful reading of the paper and several helpful comments.
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Golkarmanesh, F., Saeidi, S. Asymptotic pointwise contractive type in modular function spaces. Fixed Point Theory Appl 2013, 101 (2013). https://doi.org/10.1186/1687-1812-2013-101
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DOI: https://doi.org/10.1186/1687-1812-2013-101