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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 (M,d) be a metric space. A mapping T:M\to M is called an asymptotic pointwise contraction if there exists a function \alpha :M\to [0,1) such that for each integer n\ge 1,
where {\alpha}_{n}\to \alpha pointwise on M. Moreover, Kirk and Xu [2] proved that if C is a weakly compact convex subset of a Banach space E and T:C\to C an asymptotic pointwise contraction, then T has a unique fixed point v\in C, and for each x\in C, the sequence of Picard iterates \{{T}^{n}x\} converges in norm to v.
Very recently, Saeidi [3] introduced the concept of (weak) asymptotic pointwise contraction type: Let (M,d) be a metric space. A mapping T:M\to M is said to be of asymptotic pointwise contraction type (resp. of weak asymptotic pointwise contraction type) if {T}^{N} is continuous for some integer N\ge 1 and there exists a function \alpha :M\to [0,1) such that for each x in M,
where {\alpha}_{n}\to \alpha 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 T:C\to C be a mapping of weak asymptotic pointwise contraction type. Then T has a unique fixed point v\in C and, for each x\in C, the sequence of Picard iterates \{{T}^{n}x\} 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 σalgebra of subsets of Ω. Let \mathcal{P} be a δring of subsets of Ω such that E\cap A\in \mathcal{P} for any E\in \mathcal{P} and A\in \mathrm{\Sigma}. Let us assume that there exists an increasing sequence of sets {K}_{n}\in \mathcal{P} such that \mathrm{\Omega}=\bigcup {K}_{n}. By ξ we denote the linear space of all simple functions with supports from \mathcal{P}. By {\mathcal{M}}_{\mathrm{\infty}} we denote the space of all extended measurable function, i.e., all function f:\mathrm{\Omega}\to [\mathrm{\infty},+\mathrm{\infty}] such that there exists a sequence \{{g}_{n}\}\in \xi, {g}_{n}\le f and {g}_{n}(\omega )\to f(\omega ) for all \omega \in \mathrm{\Omega}.
By {1}_{A} we denote the characteristic function of the set A.
Definition 2.1 [6]
Let \rho :{\mathcal{M}}_{\mathrm{\infty}}\to [0,\mathrm{\infty}] be a nontrivial, convex and even function. We say that ρ is a regular convex function pseudomodular if:

(a)
\rho (0)=0;

(b)
ρ is monotone, i.e., f(\omega )\le g(\omega ) for all \omega \in \mathrm{\Omega} implies \rho (f)\le \rho (g), where f,g\in {\mathcal{M}}_{\mathrm{\infty}};

(c)
ρ is orthogonally subadditive, i.e., \rho (f{1}_{A\cup B})\le \rho (f{1}_{A})+\rho (f{1}_{B}) for any A,B\in \mathrm{\Sigma} such that A\cap B\ne \mathrm{\varnothing}, f\in {\mathcal{M}}_{\mathrm{\infty}};

(d)
ρ has the Fatou property, i.e., {f}_{n}(\omega )\uparrow f(\omega ) for all \omega \in \mathrm{\Omega} implies \rho ({f}_{n})\uparrow \rho (f), where f\in {\mathcal{M}}_{\mathrm{\infty}};

(e)
ρ is order continuous in ξ, i.e., {g}_{n}\in \xi and {g}_{n}(\omega )\downarrow 0 implies \rho ({g}_{n})\downarrow 0.
Similarly as in the case of measure spaces, we say that a set A\in \mathrm{\Sigma} is ρnull if \rho (g{1}_{A})=0 for every g\in \xi. We say that a property holds ρalmost everywhere if the exceptional set is ρnull. As usual we identify any pair of measurable sets whose symmetric difference is ρnull as well as any pair of measurable functions differing only on a ρnull set. With this in mind, we define
where each f\in \mathcal{M}(\mathrm{\Omega},\mathrm{\Sigma},\mathcal{P},\rho ) is actually an equivalence class of functions equal ρa.e. rather than an individual function. Where no confusion exists, we write ℳ instead of \mathcal{M}(\mathrm{\Omega},\mathrm{\Sigma},\mathcal{P},\rho ).
Let ρ be a regular function pseudomodular;

(a)
we say that ρ is a regular convex function semimodular if \rho (\alpha f)=0 for every \alpha >0 implies f=0 ρa.e.;

(b)
we say that ρ is a regular convex function modular if \rho (f)=0 implies f=0 ρa.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 {L}_{\rho}(\mathrm{\Omega},\mathrm{\Sigma}), or briefly {L}_{\rho}, defined by
{L}_{\rho}=\{f\in \mathcal{M}:\rho (\lambda f)\to 0\text{as}\lambda \to 0\}. 
(b)
The following formula defines a norm in {L}_{\rho} (frequently called Luxemburg norm):
{\parallel f\parallel}_{\rho}=inf\{\alpha >0;\rho (f/\alpha )\le 1\}.
In the following theorem, we recall some of the properties of modular function spaces that will be used later on in this paper.
Let \rho \in \mathfrak{R}. Defining {L}_{\rho}^{0}=\{f\in {L}_{\rho};\rho (f,)\mathit{\text{is order continuous}}\} and {E}_{\rho}=\{f\in {L}_{\rho};\lambda f\in {L}_{\rho}^{0}\mathit{\text{for every}}\lambda >0\}, we have

(i)
{L}_{\rho}\supset {L}_{\rho}^{0}\supset {E}_{\rho};

(ii)
{E}_{\rho} has the Lebesgue property, i.e., \rho (\alpha f,{D}_{k})\to 0, for \alpha >0, f\in {E}_{\rho} and {D}_{k}\downarrow \mathrm{\varnothing};

(iii)
{E}_{\rho} is the closure of ξ (in the sense of {\parallel \parallel}_{\rho}).
Let \rho \in \mathfrak{R}.

(a)
We say that \{{f}_{n}\} is ρconvergent to f and write {f}_{n}\to f(\rho ) if and only if \rho ({f}_{n}f)\to 0.

(b)
A sequence \{{f}_{n}\} where {f}_{n}\in {L}_{\rho} is called ρCauchy if \rho ({f}_{n}{f}_{m})\to 0 as m,n\to \mathrm{\infty}.

(c)
A set C\subset {L}_{\rho} is called ρclosed if for any sequence \{{f}_{n}\} in C, the convergence {f}_{n}\to f(\rho ) implies that f belongs to C.

(d)
A set C\subset {L}_{\rho} is called ρbounded if sup\{\rho (fg);f\in C,g\in C\}<\mathrm{\infty}.

(e)
For a set C\subset {L}_{\rho}, the mapping T:C\to C is called ρcontinuous if {f}_{n}\to f(\rho ), then T({f}_{n})\to T(f)(\rho ).

(f)
A set C\subset {L}_{\rho} is called ρa.e. closed if for any sequence \{{f}_{n}\} in C which ρa.e. converges to some f, then we must have f\in C.

(g)
A set C\subset {L}_{\rho} is called ρa.e. compact if for any sequence \{{f}_{n}\} in C, there exists a subsequence \{{f}_{{n}_{k}}\} which ρa.e. converges to some f\in C.

(h)
Let f\in {L}_{\rho} and C\subset {L}_{\rho}. The ρdistance between f and C is defined as
{d}_{\rho}(f,C)=inf\{\rho (fg);g\in C\}.
Let us recall that ρconvergence does not necessarily imply ρCauchy condition. Also, {f}_{n}\to f does not imply in general \lambda {f}_{n}\to \lambda f, \lambda >1.
Definition 2.6 [4]
We say that {L}_{\rho} has the property (R) if and only if every nonincreasing sequence \{{C}_{n}\} of nonempty, ρbounded, ρclosed, convex subsets of {L}_{\rho} has nonempty intersection.
Definition 2.7 [4]
We say that the function modular ρ is uniformly continuous if for every \u03f5>0 and L>0, there exists \delta >0 such that
Definition 2.8 [4]
A function \lambda :C\to [0,\mathrm{\infty}], where C\subset {L}_{\rho} is nonempty and ρclosed, is called ρlower semicontinuous if for any \alpha >0, the set {C}_{\alpha}=\{f\in C;\lambda (f)\le \alpha \} is ρclosed.
It can be proved that ρlower 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 \rho \in \mathfrak{R} has the property (R). Let C\subset {L}_{\rho} be nonempty, convex, ρclosed and ρbounded. If \phi :C\to [0,\mathrm{\infty}) is a ρlower semicontinuous convex function, then there exists {x}_{0}\in C such that
Let us recall the notion of ρtype.
Definition 2.10 [4]
Let C\subset {L}_{\rho} be convex and ρbounded. A function \tau :C\to [0,\mathrm{\infty}) is called a (\rho )type (or shortly a type) if there exists a sequence \{{y}_{m}\} of elements of C such that for any z\in C, the following holds:
Lemma 2.11 [4]
Let \rho \in \mathfrak{R} be uniformly continuous. Let C\subset {L}_{\rho} be nonempty, convex, ρclosed and ρbounded. Then any ρtype \tau :C\to [0,\mathrm{\infty}) is ρlower semicontinuous in C.
3 Asymptotic pointwise contractive type conditions in modular function spaces
Definition 3.1 [4]
Let \rho \in \mathfrak{R} and C\subset {L}_{\rho} be nonempty and ρclosed. A mapping T:C\to C is called an asymptotic pointwise mapping if there exists a sequence of mappings {\alpha}_{n}:C\to [0,1] such that

(a)
If \{{\alpha}_{n}\} converges pointwise to \alpha :C\to [0,1), then T is called asymptotic pointwise ρcontraction.

(b)
If {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}(f)\le 1 for any f\in C, then T is called asymptotic pointwise nonexpansive.

(c)
If {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}(f)\le k for any f\in C with 0<k<1, then T is called strongly asymptotic pointwise ρcontraction.
Khamsi and Kozlowski proved the following results in modular function spaces.
Theorem 3.2 [4]
Let C\subset {L}_{\rho} be nonempty, ρclosed and ρbounded. Let T:C\to C be an asymptotic pointwise ρcontraction. Then T has at most one fixed point in C. Moreover, if {x}_{0} is a fixed point of T, then the orbit \{{T}^{n}x\} is ρconvergent to {x}_{0} for any x\in C.
Theorem 3.3 [4]
Let us assume that \rho \in \mathfrak{R} is uniformly continuous and has the property (R). Let C\subset {L}_{\rho} be nonempty, convex, ρclosed and ρbounded. Let T:C\to C be an asymptotic pointwise ρcontraction. Then T has a unique fixed point {x}_{0}\in C. Moreover, the orbit \{{T}^{n}x\} is ρconvergent to {x}_{0} for any x\in C.
Below, we introduce the notion of asymptotic pointwise ρcontraction type in modular function spaces.
Definition 3.4 Let C\subset {L}_{\rho} be nonempty, ρbounded and ρclosed. A mapping T:C\to C is said to be of asymptotic pointwise ρcontraction type (resp. of weak asymptotic pointwise ρcontraction type) if {T}^{N} is ρcontinuous for some integer N\ge 1 and there exists a function \alpha :C\to [0,1) such that, for each x in C,
where {\alpha}_{n}\to \alpha pointwise on M.
Taking
it can be easily seen from (3.1) (resp. (3.2)) that
for all x\in M, and
We will obtain fixed point results for these mappings in modular function spaces.
First, it is worth mentioning that the ρlimit of any ρconvergent sequence in {L}_{\rho} is unique. This fact follows from the following reasoning: Assume that \rho ({u}_{n}u)\to 0 and \rho ({u}_{n}v)\to 0. Then
which implies that u=v.
The following theorem is our main result.
Theorem 3.5 Let \rho \in \mathfrak{R} be uniformly continuous and have the property (R). Let C\subset {L}_{\rho} be nonempty, convex, ρclosed and ρbounded. Let T:C\to C be a mapping of weak asymptotic pointwise ρcontraction type. Then T has a unique fixed point v\in C and, for each x\in C, the sequence of Picard iterates \{{T}^{n}x\} is ρconvergent to v.
Proof Fix an x\in C and define a function τ by
By Lemma 2.11, τ is ρlower semicontinuous in C. By Lemma 2.9, then there exists {x}_{0}\in C such that
Let us prove that \tau ({x}_{0})=0. Indeed, for any n,m\ge 1, we have
which implies
Since T is of weak asymptotic pointwise ρcontraction type, by (3.4) we have {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{m}({x}_{0})\le 0. Thus, for a subsequence \{{r}_{{m}_{k}}({x}_{0})\} of \{{r}_{m}({x}_{0})\}, we have
Now, by (3.6) and (3.7), we obtain
which forces \tau ({x}_{0})=0 as \alpha ({x}_{0})<1. Hence \rho ({T}^{n}x{x}_{0})\to 0 as n\to \mathrm{\infty}. From this and the continuity of {T}^{N}, for some N\ge 1, it follows that \rho ({T}^{N+n}x{T}^{N}{x}_{0})\to 0 as n\to \mathrm{\infty}. Since the ρlimit of any ρconvergent sequence is unique, we must have {T}^{N}{x}_{0}={x}_{0}, namely, {x}_{0} is a fixed point of {T}^{N}. Now, repeating the above proof for {x}_{0} instead of x, we deduce that {T}^{n}{x}_{0} is ρconvergent to a member v of C; i.e., \rho ({T}^{n}{x}_{0}v)\to 0. But {T}^{kN}{x}_{0}={x}_{0} for all k\ge 1. Hence, v={x}_{0} and then {T}^{n}{x}_{0}\to {x}_{0}(\rho ).
We show that T{x}_{0}={x}_{0}; for this purpose, consider an arbitrary \u03f5>0. Then there exists a {k}_{0}>0 such that \rho ({T}^{n}{x}_{0}{x}_{0})<\u03f5 for all n>{k}_{0}. So, by choosing a natural number k>{k}_{0}/N, we obtain
Since the choice of \u03f5>0 is arbitrary and \rho \in \mathfrak{R}, we get T{x}_{0}={x}_{0}.
It is easy to verify that T can have only one fixed point. Indeed, if u,v\in C are fixed points of T, then by (3.5), we have
Taking lim inf in the above inequality, we obtain
Since \alpha (u)<1 and \rho \in \mathfrak{R}, we immediately get u=v. □
Next, using the ρa.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 {L}_{\rho} satisfies the ρa.e. strong Opial property (or shortly SOproperty) if for every \{{f}_{n}\}\in {L}_{\rho} which is ρa.e. convergent to zero such that there exists a \beta >1 for which
the following equality holds for any g\in {L}_{\rho}:
Lemma 3.7 [4]
Let \rho \in \mathfrak{R}. Assume that {L}_{\rho} has the ρa.e. strong Opial property. Let C\subset {E}_{\rho} be a nonempty, ρa.e. compact subset such that there exists \beta >1 such that {\delta}_{\rho}(\beta C)=sup\{\rho (\beta (xy));x,y\in C\}<\mathrm{\infty}. Let D\subset C be a nonempty ρa.e. closed subset. For any n\ge 1, let {\lambda}_{n}:D\to [0,\mathrm{\infty}) be such that for any y\in D, there exists a sequence \{{y}_{n}\}\subset C such that, for every n\ge 1, the following holds:
and \rho (x{y}_{n})\le {\lambda}_{n}(x) for every x\in D and n\ge 1. Let \lambda (x)={lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\lambda}_{n}(x) for any x\in D. Then there exists {x}_{0}\in D at which λ attains infimum, i.e.,
Theorem 3.8 Let \rho \in \mathfrak{R}. Assume that {L}_{\rho} has the ρa.e. strong Opial property. Let C\subset {E}_{\rho} be a nonempty ρa.e. compact convex subset such that {\delta}_{\rho}(\beta C)=sup\{\rho (\beta (xy));x,y\in C\}<\mathrm{\infty} for some \beta >1. Then any T:C\to C of weak asymptotic pointwise ρcontraction type has a unique fixed point {x}_{0}\in C. Moreover, the orbit \{{T}^{n}x\} is ρconvergent to {x}_{0} for any x\in C.
Proof Fix an x\in C and define a function τ by
By Lemma 3.7 applied with \lambda (u)=\tau (u), D=C, {\lambda}_{n}(u)=\rho ({T}^{n}xu), and with {y}_{n}={T}^{n}x chosen for all u\in C, there exists {x}_{0}\in C such that
The rest of the proof is like the one used for Theorem 3.5. □
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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/10.1186/168718122013101
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DOI: https://doi.org/10.1186/168718122013101
Keywords
 asymptotic pointwise ρcontraction type
 modular function space