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Onelocal retract and common fixed point in modular function spaces
Fixed Point Theory and Applications volume 2012, Article number: 109 (2012)
Abstract
In this paper, we introduce and study the concept of onelocal retract in modular function spaces. In particular, we prove that any commutative family of ρnonexpansive mappings defined on a nonempty, ρclosed and ρbounded subset of a modular function space has a common fixed point provided its convexity structure of admissible subsets is compact and normal.
MSC: Primary 47H09; Secondary 46B20, 47H10.
Introduction
The purpose of this paper is to give an outline of a common fixed point theory for nonexpansive mappings defined on some subsets of modular function spaces. These spaces are natural generalizations of both function and sequence variants of many important, from applications perspective, spaces like Lebesgue, Orlicz, MusielakOrlicz, Lorentz, OrliczLorentz, CalderonLozanovskii spaces and many others. The current paper operates within the framework of convex function modulars. The importance for applications of nonexpansive mappings in modular function spaces consists in the richness of structure of modular function spaces, thatbesides being Banach spaces (or Fspaces in a more general settings)are equipped with modular equivalents of norm or metric notions, and also are equipped with almost everywhere convergence and convergence in submeasure. In many cases, particularly in applications to integral operators, approximation and fixed point results, modular type conditions are much more natural as modular type assumptions can be more easily verified than their metric or norm counterparts. There are also important results that can be proved only using the apparatus of modular function spaces. From this perspective, the fixed point theory in modular function spaces should be considered as complementary to the fixed point theory in normed spaces and in metric spaces.
The theory of contractions and nonexpansive mappings defined on convex subsets of Banach spaces has been well developed since the 1960s (see e.g. [1–6]), and generalized to other metric spaces (see e.g. [7–9]), and modular function spaces (see e.g. [10–12]).
In this paper, we invesigate the structure of the fixed point set of ρnonexpansive mappings. In particular, we introduce and investigate the concept of onelocal retracts in the framework of modular function spaces. Then we show a common fixed point in this setting.
Preliminaries
Let Ω be a nonempty set and ∑ be a nontrivial σalgebra of subsets of Ω. Let P be a δring of subsets of Ω, such that E\cap A\in \mathcal{P} for any E\in \mathcal{P} and A ∈ ∑. Let us assume that there exists an increasing sequence of sets {K}_{n}\in \mathcal{P} such that Ω = ∪K_{ n }. By ℰ we denote the linear space of all simple functions with supports from P. By ℳ_{∞} we will denote the space of all extended measurable functions, i.e. all functions f: Ω → [∞, ∞] such that there exists a sequence{g_{ n }} ⊂ ℰ, g_{ n } ≤  f  and g_{ n }(ω) → f(ω) for all ω ∈ Ω By 1_{ A } we denote the characteristic function of the set A.
Definition 2.1. Let ρ: ℳ_{∞} → [0, ∞] be a notrivial, convex and even function. We say that ρ is a regular convex function pseudomodular if:

(i)
ρ(0) = 0;

(ii)
ρ is monotone, i.e. f(ω) ≤ g(ω) for all ω ∈ Ω implies ρ(f) ≤ ρ(g), where f, g ∈ ℳ_{∞};

(iii)
ρ is orthogonally subadditive, i.e ρ(f 1_{ A∪B }) ≤ ρ(f 1_{ A })+ρ(f 1_{ B }) for any A, B ∈ ∑ such that A ∩ B ≠ ∅, f ∈ ℳ;

(iv)
ρ has the Fatou property, i.e. f_{ n }(ω)↑f(ω) for all ω ∈ Ω implies ρ(f_{ n }) ↑ρ(f), where f ∈ ℳ_{∞};

(v)
ρ is order continuous in ℰ, i.e. g_{ n } ∈ ℰ and g_{ n }(ω) ↓ 0 implies ρ(g_{ n }) ↓ 0.
Similarly as in the case of measure spaces, we we say that a set A ∈ ∑ is ρnull if ρ(g 1_{ A }) = 0 for every g ∈ ℰ. 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}\left(\Omega ,\mathrm{\Sigma},\mathcal{P},\rho \right) is actually an equivalence class of functions equal ρa.e. rather than an individual function. Where no confusion exists we will write ℳ instead of \mathcal{M}\left(\Omega ,\mathrm{\Sigma},\mathcal{P},\rho \right).
Definition 2.2. Let ρ be a regular function pseudomodular.

(1)
We say that ρ is a regular convex function semimodular if ρ(αf) = 0 for every α > 0 implies f = 0 ρ  a.e.;

(2)
We say that ρ is a regular convex function modular if ρ(f) = 0 implies f = 0 ρ  a.e.;
The class of all nonzero regular convex function modulars defined on Ω will be denoted by ℜ.
Let us denote ρ(f, E) = ρ (f 1_{ E }) for f ∈ ℳ, E ∈ ∑. It is easy to prove that ρ(f, E) is a function pseudomodular in the sense of Def. 2.1.1 in [13] (more precisely, it is a function pseudomodular with the Fatou property). Therefore, we can use all results of the standard theory of modular function spaces as per the framework defined by Kozlowski in [13–15], see also Musielak [16] for the basics of the general modular theory.
Definition 2.3. [13–15] Let ρ be a convex function modular.

(a)
A modular function space is the vector space L_{ ρ } (Ω, ∑), or briefly L_{ ρ }, defined by
{L}_{\rho}=\left\{f\in \mathcal{M};\rho \left(\lambda f\right)\to 0\phantom{\rule{2.77695pt}{0ex}}as\phantom{\rule{2.77695pt}{0ex}}\lambda \to 0\right\}. 
(b)
The following formula defines a norm in L_{ ρ } (frequently called Luxemburg norm):
{\u2225f\u2225}_{\rho}=\mathsf{\text{inf}}\left\{\alpha >0;\rho \left(f/\alpha \right)\le 1\right\}.
In the following theorem, we recall some of the properties of modular spaces that will be used later on in this paper.
Theorem 2.1. [13–15]Let ρ ∈ ℜ.

(1)
L_{ ρ },  f _{ ρ } is complete and the norm  · _{ ρ } is monotone w.r.t. the natural order in ℳ.

(2)
 f_{ n } _{ ρ } → 0 if and only if ρ(af_{ n }) → 0 for every α > 0.

(3)
If ρ (αf_{ n }) → 0 for an α > 0 then there exists a subsequence {g_{ n }} of {f_{ n }} such that g_{ n } → 0 ρ  a.e.

(4)
If {f_{ n }} converges uniformly to f on a set E\in \mathcal{P} then ρ (α (f_{ n }  f), E) → 0 for every α > 0.

(5)
Let f_{ n } → f ρ  a.e. There exists a nondecreasing sequence of sets {H}_{k}\in \mathcal{P} such that H_{ k } ↑ Ω and {fn} converges uniformly to f on every H_{ k } (Egoroff Theorem).

(6)
ρ(f) ≤ lim inf ρ(f_{ n }) whenever f_{ n } → f ρ  a.e. (Note: this property is equivalent to the Fatou Property).

(7)
Defining {L}_{\rho}^{0}=\{f\in {L}_{\rho};\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\rho \left(f,\cdot \right) is order continuous} and {E}_{\rho}=\left\{f\in {L}_{\rho};\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\lambda f\in {L}_{\rho}^{0}\phantom{\rule{0.3em}{0ex}}for\phantom{\rule{0.3em}{0ex}}every\phantom{\rule{0.3em}{0ex}}\lambda >0\right\} we have:

(a)
{L}_{\rho}\supset {L}_{\rho}^{0}\supset {E}_{\rho},

(b)
E_{ ρ } has the Lebesgue property, i.e. ρ (αf, D_{ k }) → 0 for α > 0, f ∈ E_{ ρ } and D_{ k } ↓ ∅.

(c)
E_{ ρ } is the closure of ℰ (in the sense of  · _{ ρ }).
The following definition plays an important role in the theory of modular function spaces.
Definition 2.4. Let ρ ∈ ℜ. We say that ρ has the Δ_{2}property if \underset{n}{\mathsf{\text{sup}}}\rho \left(2{f}_{n},{D}_{k}\right)\to 0 whenever D_{ k } ↓ ∅ and \underset{n}{\mathsf{\text{sup}}}\rho \left({f}_{n},{D}_{k}\right)\to 0.
Theorem 2.2. Let ρ ∈ ℜ. The following conditions are equivalent:

(a)
ρ has Δ_{2},

(b)
{L}_{\rho}^{0}is a linear subspace of L_{ ρ },

(c)
{L}_{\rho}={L}_{\rho}^{0}={E}_{\rho},

(d)
if ρ (f_{ n }) → 0 then ρ(2f_{ n }) → 0,

(e)
if ρ(αf_{ n }) → 0 for an α > 0 then  f_{ n }_{ ρ } → 0, i.e. the modular convergence is equivalent to the norm convergence.
The following definition is crucial throughout this paper.
Definition 2.5. Let ρ ∈ ℜ.

(a)
We say that {f_{ n }} is ρconvergent to f and write f_{ n } → 0 (ρ) if and only if ρ(f_{ n }  f) → 0.

(b)
A sequence {f_{ n }} where f_{ n } ∈ L_{ ρ } is called ρCauchy if ρ (f_{ n }  f_{ m }) → 0 as n, m → ∞.

(c)
A set B ⊂ L_{ ρ } is called ρclosed if for any sequence of f_{ n } ∈ B, the convergence f_{ n } → f (ρ) implies that f belongs to B.

(d)
A set B ⊂ L_{ ρ } is called ρbounded if sup{ρ (f  g); f ∈ B, g ∈ B} < ∞

(e)
Let f ∈ L_{ ρ } and C ⊂ L_{ ρ }. The ρdistance between f and C is defined as
{d}_{\rho}\left(f,C\right)=\mathsf{\text{inf}}\left\{\rho \left(fg\right);g\in C\right\}.
Let us note that ρconvergence does not necessarily imply ρCauchy condition. Also, f_{ n } → f does not imply in general λf_{ n } → λf, λ > 1. Using Theorem 2.1 it is not difficult to prove the following
Proposition 2.1. Let ρ ∈ ℜ.

(i)
L_{ ρ } is ρcomplete,

(ii)
ρballs B_{ ρ }(x, r) = {y ∈ L_{ ρ } ; ρ(x  y) ≤ r} are ρclosed.
The following property plays in the theory of modular function spaces a role similar to the reflexivity in Banach spaces (see e.g. [11]).
Definition 2.6. We say that L_{ ρ } has property (R) if and only if every nonincreasing sequence {C_{ n }} of nonempty, ρbounded, ρclosed, convex subsets of L_{ ρ } has nonempty intersection.
Throughout this paper we will need the following.
Definition 2.7. Let ρ ∈ ℜ and C ⊂ L_{ ρ } be nonempty.

(a)
By the ρdiameter of C, we will understand the number
{\delta}_{\rho}\left(C\right)=\mathsf{\text{sup}}\left\{\rho \left(fg\right);f,g\in C\right\}.
The subset C is said to be ρbounded whenever δ_{ ρ }(C) < ∞.

(b)
The quantity r_{ ρ }(f, C) = sup{ρ(f  g);g ∈ C} will be called the ρChebyshev radius of C with respect to f.

(c)
The ρChebyshev radius of C is defined by R_{ ρ }(C) = inf {r_{ ρ } (f, C); f ∈ C}.

(d)
The ρChebyshev center of C is defined as the set
{\mathcal{C}}_{\rho}\left(C\right)=\left\{f\in C;{r}_{\rho}\left(f,C\right)={R}_{\rho}\left(C\right)\right\}.
Note that R_{ ρ }(C) ≤ r_{ ρ } (f, C) ≤ δ_{ ρ }(C) for all f ∈ C and observe that there is no reason, in general, for {\mathcal{C}}_{\rho}\left(C\right) to be nonempty.
Let us finish this section with the modular definitions of ρnonexpansive mappings. The definitions are straightforward generalizations of their norm and metric equivalents.
Definition 2.8. Let ρ ∈ ℜ and C ⊂ L_{ ρ } be nonempty and ρclosed. A mapping T: C → C is called a ρnonexpansive mapping if
A point f ∈ C is called a fixed point of T whenever T(f) = f. The set of fixed point of T is denoted by Fix(T).
Penot compactness of admissible sets
The following definition is needed.
Definition 2.9. Let ρ ∈ ℜ and C ⊂ L_{ ρ } be nonempty and ρbounded. We say that A is an admissible subset of C if
where b_{ i } ∈ C, r_{ i } ≥ 0 and I is an arbitrary index set. By \mathcal{A}\left(C\right) we denote the family of all admissible subsets of C.
Note that if C is ρbounded, then C\in \mathcal{A}\left(C\right). In order to prove an analogue of Kirk's fixed point theorem [3], Penot [17] introduced the following definition.
Definition 2.10. Let ρ ∈ ℜ and C ⊂ L_{ ρ } be nonempty.

(1)
We will say that \mathcal{A}\left(C\right) is ρnormal if for any nonempty A\in \mathcal{A}\left(C\right), which has more than one point, we have R_{ ρ }(A) < δ_{ ρ }(A).

(2)
We will say that \mathcal{A}\left(C\right) is compact if for any family {\left\{{A}_{\alpha}\right\}}_{\alpha \in \Gamma}\subset \mathcal{A}\left(C\right) we have
\bigcap _{\alpha \in \Gamma}{A}_{\alpha}\ne \mathrm{\varnothing},
provided that \bigcap _{\alpha \in F}{A}_{\alpha}\ne \mathrm{\varnothing} for any finite subset F of Γ.
Clearly if \mathcal{A}\left({L}_{\rho}\right) is compact, then L_{ ρ } has property (R). In [18], the authors discussed the concept of uniform convexity in modular function spaces. In particular they proved that uniform convexity implies the property (R). Next, we show that uniform convexity implies compactness in the sense of Penot [17] of the family of convex sets. First, let us recall the definition of uniform convexity in modular function spaces. For more on this, the reader may consult [18].
Definition 2.11. Let ρ ∈ ℜ.

(i)
Let r > 0, ε > 0. Define
D\left(r,\epsilon \right)=\left\{\left(f,g\right);f,g\in {L}_{\rho},\rho \left(f\right)\le r,\rho \left(g\right)\le r,\rho \left(\frac{fg}{2}\right)\ge \epsilon r\right\}.
Let
and δ(r, ε) = 1 if D(r, ε) = ∅. We say that ρ satisfies (UC) if for every r > 0, ε > 0, δ(r, ε) > 0. Note, that for every r > 0, D(r, ε) ≠ ∅, for ε > 0 small enough.

(ii)
We say that ρ satisfies (UUC) if for every s ≥ 0, ε > 0 there exists
\eta \left(s,\epsilon \right)>0
depending on s and ε such that

(iii)
We say that ρ is Strictly Convex, (SC), if for every f, g ∈ L_{ ρ } such that ρ(f) = ρ(g) and
\rho \left(\frac{f+g}{2}\right)=\frac{\rho \left(f\right)+\rho \left(g\right)}{2}
there holds f = g.
Note that in [11], the authors proved that in Orlicz spaces over a finite, atomless measure space, both conditions (UC) and (UUC) are equivalent. Typical examples of Orlicz functions that do not satisfy the Δ_{2} condition but are uniformly convex are: φ_{1}(t) = e^{t}t1 and {\phi}_{2}\left(t\right)={e}^{{t}^{2}}1. In these cases, the associated modular is (UUC).
It is shown in [18], that if ρ ∈ ℜ is (UUC), then for any nonempty, convex, and ρclosed C ⊂ L_{ ρ } , and any f ∈ L_{ ρ } such that d = d_{ ρ } (f, C) < ∞, there exists a unique best ρapproximant of f in C, i.e. a unique g_{0} ∈ C such that
Moreover it is also shown in [18] that if ρ ∈ ℜ is (UUC), then for any nonincreasing sequence {C_{ n }} of nonempty, convex, and ρclosed subsets of L_{ ρ }, we have ∩_{n ≥ 1}C_{ n } ≠ ∅, provided there exists f ∈ L_{ ρ } such that \underset{n\ge 1}{\mathsf{\text{sup}}}{d}_{\rho}\left(f,{C}_{n}\right)\infty. The authors in [18] did not show that such conclusion is still valid for any decreasing family. A property useful to get the compactness of the admissible subsets.
Theorem 2.3. Let ρ ∈ ℜ. Assume ρ ∈ ℜ is (UUC). Let {C_{ α }}_{α∈Γ}be a decreasing family of nonempty, convex, ρclosed subsets of L_{ ρ }, where (Γ,≺) is upward directed. Assume that there exists f ∈ L_{ ρ } such that \underset{\alpha \in \Gamma}{\mathsf{\text{sup}}}{d}_{\rho}\left(f,{C}_{\alpha}\right)\infty. Then, ∩_{α∈Γ}C_{ α }≠ ∅.
Proof. Set d=\underset{\alpha \in \Gamma}{\mathsf{\text{sup}}}{d}_{\rho}\left(f,{C}_{\alpha}\right). Without loss of generality, we may assume d > 0. For Any n ≥ 1, there exists a_{ n } ∈ Γ such that
Since (Γ,≺) is upward directed, we may assume α_{ n } ≺ α_{n+1}. In particular we have {C}_{{\alpha}_{n+1}}\subset {C}_{{\alpha}_{n}} for any n ≥ 1. Since ρ is (UUC), we get {C}_{0}={\cap}_{n\ge 1}{C}_{{\alpha}_{n}}\ne \mathrm{\varnothing}. Clearly C_{0} is ρclosed and
Again using the property (UUC) satisfied by ρ, there exists g_{0} ∈ C_{0} unique such that d_{ ρ }(f, C_{0}) = ρ (f  g_{0}). Let us prove that g_{0} ∈ C_{ α } for any α ∈ Γ. Fix α ∈ Γ. If for some n ≥ 1 we have α ≺ α_{ n }, then obviously we have {g}_{0}\in {C}_{{\alpha}_{n}}\subset {C}_{\alpha}.
Therefore let us assume that α⊀ α_{ n }, for any n ≥ 1. Since Γ is upward directed, there exists β_{ n } ∈ Γ such that α_{ n }≺ β_{ n } and α ≺ β_{ n } for any n ≥ 1. We can also assume that β_{ n } ≺ β_{n+1}for any n ≥ 1. Again we have {C}_{1}={\cap}_{n\ge 1}{C}_{{\beta}_{n}}\ne \mathrm{\varnothing}. Since {C}_{{\beta}_{n}}\subset {C}_{{\alpha}_{n}}, for any n ≥ 1, we get C_{1} ⊂ C_{0}. Moreover we have
Hence, d_{ ρ }(f, C_{1}) = d which implies the existence of a unique point g_{1} ∈ C_{1} such that d_{ ρ }(f, C_{1}) = ρ(f  g_{1}) = d. Since ρ is uniformly convex, it must be (SC). Hence, g_{0} = g_{1}. In particular, we have {g}_{0}\in {C}_{{\beta}_{n}}, for any n ≥ 1. Since α ≺ β_{ n }, we get {C}_{{\beta}_{n}}\subset {C}_{\alpha}, for any n ≥ 1, which implies g_{0} ∈ C_{ α }. Since α was taking arbitrary in Γ, we get g_{0} ∈ ∩_{α∈Γ} C_{α}, which implies ∩_{α∈Γ} C_{α} ≠ ∅. □
Since ρ is convex, ρclosed balls are convex. Theorem 2.3 implies the following.
Corollary 2.1. Let ρ ∈ ℜ and C ⊂ L_{ ρ } be nonempty, convex, ρclosed, and ρbounded. Assume ρ is (UUC). Then \mathcal{A}\left(C\right) is compact.
Remark 2.1. Note that under the above assumptions, \mathcal{A}\left(C\right) is ρnormal. Indeed let A\in \mathcal{A}\left(C\right) nonempty and not reduced to one point. Let f, g ∈ A such that f ≠ g. Then \rho \left(\frac{fg}{2}\right)>0. Since ρ is (UUC), there exists η > 0 such that for any h ∈ A, we have
Hence, {r}_{\rho}\left(\frac{f+g}{2},A\right)\le \left(1\eta \right){\delta}_{\rho}\left(A\right), which implies R_{ ρ }(A) < δ_{ ρ }(A).
Finally, we state Penot's formulation of Kirk's fixed point theorem in modular function spaces. For the sake of completeness we will give its proof.
Theorem 2.4. Let ρ ∈ ℜ and C ⊂ L_{ ρ } be nonempty, ρclosed, and ρbounded. Assume that \mathcal{A}\left(C\right) is compact and ρnormal. Then any ρnonexpansive T: C → C has a fixed point.
Proof. Since C is ρbounded, then we have C\in \mathcal{A}\left(C\right). Since \mathcal{A}\left(C\right) is compact, the family \mathcal{F}=\left\{A\in \mathcal{A}\left(C\right);T\left(A\right)\subset A\right\} has a minimal element K. Set
Note that T(K) ⊂ K_{0}. This implies that K_{0} is nonempty and belongs to \mathcal{A}\left(C\right). Moreover since K_{0} ⊂ K, we get T(K_{0}) ⊂ T(K) ⊂ K_{0}. Hence K_{0} ∈ ℱ. The minimality of K implies that K = K_{0}. Next let f ∈ K. By definition of the ρChebyshev radius r_{ ρ }(f, K), we have K ⊂ B_{ ρ }(f, r_{ ρ }(f, K)). Since T is ρ nonexpansive, we have T(K) ⊂ B_{ ρ }(T(f), r_{ ρ }(f, K)). The definition of K_{0} implies K_{0} ⊂ B_{ ρ }(T(f), r_{ ρ }(f, K)). Since K = K_{0}, we get K ⊂ B_{ ρ } (T (f), r_{ ρ }(f, K)), which implies r_{ ρ }(T(f), K) ≤ r_{ ρ }(f, K). Fix f ∈ K and set r = r_{ ρ } (f, K). We have
Clearly, we have T(K_{1}) ⊂ K_{1} and {K}_{1}\in \mathcal{A}\left(C\right). Since K is minimal, we get K = K_{1} which implies that the ρChebyshev radius r_{ ρ }(f, K) is constant. In particular, we have r_{ ρ }(f, K) = δ_{ ρ } (K), for any f ∈ K. Since \mathcal{A}\left(C\right) is ρnormal, we conclude that K does not have more than one point. Therefore, K = {f} which forces T (f) = f. □
In the next section, we investigate the structure of the fixed point set of ρnonexpansive mappings.
Onelocal retract subsets in modular function spaces
Let ρ ∈ ℜ and C ⊂ L_{ ρ } be nonempty. A nonempty subset D of C is said to be a onelocal retract of C if for every family {B_{ i }; i ∈ I} of ρballs centered in D such that C ∩ (∩_{i∈I}B_{ i }) ≠ ∅, it is the case that D ∩ (∩_{i∈I}B_{ i }) ≠ ∅. It is immediate that each ρnonexpansive retract of L_{ ρ } is a onelocal retract (but not conversely). Recall that D ⊂ C is a ρnonexpansive retract of C if there exists a ρnonexpansive map R: C → D such that R(f) = f, for every f ∈ D.
The following result will shed some light on the interest generated around this concept.
Theorem 2.5. Let ρ ∈ ℜ and C ⊂ L_{ ρ } be nonempty, ρclosed, and ρbounded. Assume that \mathcal{A}\left(C\right) is compact and ρnormal. Then for any ρnonexpansive mapping T: C → C, the fixed point set Fix(T) is a nonempty onelocal retract of C.
Proof. Theorem 2.4 shows that Fix(T) is nonempty. Let us complete the proof by showing it is a onelocal retract of C. Let {B_{ ρ }(f_{ i }, r_{ i })}_{i∈I}be any family of ρclosed balls such that f_{ i } ∈ Fix(T), for any i ∈ I, and
Let us prove that Fix (T) ∩ (∩_{i∈I}B_{ ρ }(f_{ i }, r_{ i })) ≠ ∅. Since {fi}_{i∈I}⊂ Fix(T), and T is ρnonexpansive, then T(C_{0}) ⊂ C_{0}. Clearly, {C}_{0}\in \mathcal{A}\left(C\right) and is nonempty. Then we have \mathcal{A}\left({C}_{0}\right)\subset \mathcal{A}\left(C\right). Therefore, \mathcal{A}\left({C}_{0}\right) is compact and ρnormal. Theorem 2.4 will imply that T has a fixed point in C_{0} which will imply
□
This result gives some information to the structure of the fixed point set. To the best of our knowledge this is the first attempt done in modular function spaces. Next we discuss some properties of onelocal retract subsets.
Theorem 2.6. Let ρ ∈ ℜ and C ⊂ L_{ ρ } be nonempty. Let D be a nonempty subset of C. The following are equivalent.

(i)
D is a onelocal retract of C.

(ii)
D is a ρnonexpansive retract of D ∪ {f}, for every f ∈ C.
Proof. let us prove (i) ⇒ (ii). Let f ∈ C. We may assume that f ∉ D. In order to construct a ρnonexpansive retract R: D ∪ {f} → D, we only need to find R(f) ∈ D such that
Since f ∈ ∩_{g∈D}B_{ ρ }(g, ρ(fg)) and f ∈ C, then
Since D is a onelocal retract of C, we get
Any point in D_{0} will work as R(f).
Next, we prove that (ii) ⇒ (i). In order to prove that D is a onelocal retract of C, let {B_{ ρ }(f_{ i }, r_{ i })}_{i∈I}be any family of ρclosed balls such that f_{ i } ∈ D, for any i ∈ I, and
Let us prove that D ∩ (∩_{i∈I}B_{ ρ }(f_{ i }, r_{ i })) = ∅. Let f ∈ C_{0}. If f ∈ D, we have nothing to prove. Assume otherwise that f ∉ D. Property (ii) implies the existence of a ρnonexpansive retract R: D ∪ {f} → C. It is easy to check that R(f) ∈ D ∩ (∩_{i∈I}B_{ ρ }(f_{ i }, r_{ i })) = ∅, which completes the proof of our theorem. □
The following technical lemma will be useful for the next results.
Lemma 2.1. Let ρ ∈ ℜ and C ⊂ L_{ ρ } be nonempty, and ρbounded. Let D be a nonempty onelocal retract of C. Set c{o}_{C}\left(D\right)=C\cap (\cap \{A;A\in \mathcal{A}\left(C\right) and D ⊂ A}). Then

(i)
r_{ ρ }(f, D) = r_{ ρ }(f, co_{ C }(D)), for any f ∈ C;

(ii)
R_{ ρ }(co_{ C }(D)) = R_{ ρ }(D);

(iii)
δ_{ ρ }(co_{ C }(D)) = δ_{ ρ }(D).
Proof. Let us first prove (i). Fix f ∈ C. Since D ⊂ co_{ C }(D), we get r_{ ρ }(f, D) ≤ r_{ ρ }(f, co_{ C }(D)). Set r = r_{ ρ }(f, D). We have D\subset {B}_{\rho}\left(f,r\right)\in \mathcal{A}\left(C\right). The definition of co_{ C }(D) implies co_{ C }(D) ⊂ B_{ ρ }(f, r). Hence r_{ ρ }(f, co_{ C }(D)) ≤ r = r_{ ρ }(f, D), which implies r_{ ρ }(f, D) = r_{ ρ }(f, co_{ C }(D)).
Next, we prove (ii). Let f ∈ D. We have f ∈ co_{ C }(D). Using (i), we get r_{ ρ }(f, D) = r_{ ρ }(f, co_{ C }(D)) ≥ R_{ ρ }(co_{ C }(D)). Hence, R_{ ρ }(D) ≥ R_{ ρ }(co_{ C }(D)). Next, let f ∈ co_{ C }(D) and set r = r_{ ρ }(f, co_{ C }(D)). We have D ⊂ co_{ C }(D) ⊂ B_{ ρ }(f, r). Hence, f ∈ ∩_{g∈D}B_{ ρ }(g, r). Hence, C ∩ (∩_{g∈ D}B_{ ρ }(g, r)) = ∅. Since D is a onelocal retract of C, we get D_{0} = D ∩ (∩_{g∈ D}B_{ ρ }(g, r)) = ∅. Let g ∈ D_{0}. Then it is easy to see that r_{ ρ }(g, D) ≤ r. Hence, R_{ ρ }(D) ≤ r. Since f was arbitrary taken in co_{ C }(D), we get R_{ ρ }(D) ≤ R_{ ρ }(co_{ C }(D)), which implies R_{ ρ }(D) = R_{ ρ }(co_{ C }(D)).
Finally, let us prove (iii). Since D ⊂ co_{ C }(D), we get δ_{ ρ }(D) ≤ δ_{ ρ }(co_{ C }(D)). Next set d = δ_{ ρ }(D). Then, for any f ∈ D, we have D ⊂ B_{ ρ }(f, d). Hence co_{ C }(D) ⊂ B_{ ρ }(f, d). This implies f\in {\cap}_{g\in c{o}_{C}\left(D\right))}{B}_{\rho}\left(g,d\right). Sice f was taken arbitrary in D, we get D\subset {\cap}_{g\in c{o}_{C}\left(D\right))}{B}_{\rho}\left(g,d\right). The definition of co_{ C }(D) implies c{o}_{C}\left(D\right)\subset {\cap}_{g\in c{o}_{C}\left(D\right))}{B}_{\rho}\left(g,d\right). So for any f, g ∈ co_{ C }(D), we have ρ(f  g) ≤ d. Hence δ_{ ρ }(co_{ C }(D)) ≤ d = δ_{ ρ }(D), which implies δ_{ ρ } (D) = δ_{ ρ } (co_{ C }(D)). □
As an application of this lemma we get the following result.
Theorem 2.7. Let ρ ∈ ℜ and C ⊂ L_{ ρ } be nonempty, ρclosed, and ρbounded. Assume that \mathcal{A}\left(C\right) is compact and ρnormal. If D is a nonempty onelocal retract of C, then \mathcal{A}\left(D\right) is compact and ρnormal.
Proof. Using the definition of onelocal retract, it is easy to see that \mathcal{A}\left(D\right) is compact. Let us show that \mathcal{A}\left(D\right) is ρnormal. Let {A}_{0}\in \mathcal{A}\left(D\right) nonempty and not reduced to one point. Set c{o}_{C}\left({A}_{0}\right)=C\cap \left(\cap \left\{A;A\in \mathcal{A}\left(C\right)\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}{A}_{0}\subset A\right\}\right). Then from the Lemma 2.1, we get
Since c{o}_{C}\left({A}_{0}\right)\in \mathcal{A}\left(C\right), then we must have R_{ ρ }(co_{ C }(A_{0})) < δ_{ ρ }(co_{ C }(A_{0})) because \mathcal{A}\left(C\right) is ρnormal. Therefore, we have R_{ ρ }(A_{0}) < δ_{ ρ }(A_{0}), which completes the proof of our claim. □
The next result is amazing and has found many applications in metric spaces. Most of the ideas in its proof go back to Baillon's work [8].
Theorem 2.8. Let ρ ∈ ℜ and C ⊂ L_{ ρ } be nonempty, ρclosed, and ρbounded. Assume that \mathcal{A}\left(C\right) is compact and ρnormal. Let (C_{ β })_{β∈Γ}be a decreasing family of onelocal retracts of C, where (Γ, ≺) is totally ordered. Then ∩_{β∈ Γ}C_{ β } is not empty and is a onelocal retract of C.
Proof. First, let us prove that ∩_{β∈ Γ}C_{ β } is not empty. Consider the family
ℱ is not empty since \prod _{\beta \in \Gamma}{C}_{\beta}\in \mathcal{F}. ℱ will be ordered by inclusion, i.e., \prod _{\beta \in \Gamma}{A}_{\beta}\subset \prod _{\beta \in \Gamma}{B}_{\beta} if and only if A_{ β } ⊂ B_{ β } for any β ∈ Γ. From Theorem 2.7, we know that \mathcal{A}\left({C}_{\beta}\right) is compact, for every β ∈ Γ. Therefore, ℱ satisfies the hypothesis of Zorn's lemma. Hence for every D ∈ℱ, there exists a minimal element A ∈ℱ such that A ⊂ D. We claim that if A=\prod _{\beta \in \Gamma}{A}_{\beta} is minimal, then there exists β_{0} ∈ Γ such that δ(A_{ β }) = 0 for every β ≻ β_{0}. Assume not, i.e., δ(A_{ β }) > 0 for every β ∈ Γ. Fix β ∈ Γ. For every K ⊂ C, set
Consider {A}^{\prime}=\prod _{\alpha \in \Gamma}{A}_{\alpha}^{\prime} where
The family \left({A}_{\alpha \ge \beta}^{\prime}\right) is decreasing since A ∈ℱ. Let α ≤ γ ≤ β. Then {A}_{\gamma}^{\prime}\subset {A}_{\alpha}^{\prime} since A_{ γ } ⊂ A_{ α } and A_{ β } = co_{ β } (A_{ β }) ∩ A_{ β }. Hence the family \left({A}_{\alpha}^{\prime}\right) is decreasing. On the other hand if α ≺ β, then c{o}_{\beta}\left({A}_{\beta}\right)\cap {A}_{\alpha}\in \mathcal{A}\left({C}_{\alpha}\right)since C_{ β } ⊂ C_{ α }. Hence {A}_{\alpha}^{\prime}\in \mathcal{A}\left({C}_{\alpha}\right). Therefore, we have A' ∈ ℱ. Since A is minimal, then A = A'. Hence
Let f ∈ C_{ β } and a ≺ β. Since A_{ β } ⊂ A_{ α }, then r_{ ρ }(f, A_{ β }) ≤ r_{ ρ }(f, A_{ α }). Because c{o}_{\beta}\left({A}_{\beta}\right)={\cap}_{g\in {C}_{\beta}}{B}_{\rho}\left(g,{r}_{\rho}\left(g,{A}_{\beta}\right)\right), then we have co_{ β }(A_{ β }) ⊂ B_{ ρ }(g, r_{ ρ }(g, A_{ β })) which implies r_{ ρ }(g, A_{ β }) ≤ r_{ ρ }(g, A_{ α }). Since A_{ α } ⊂ co_{ β }(A_{ β }), then
Therefore, we have r_{ ρ }(g, A_{ α }) ≤ r_{ ρ }(g, A_{ β }) for every g ∈ C_{ β }. Using the definition of the ρChebyshev radius R_{ ρ }, we get
Let f ∈ A_{ α } and set s = r_{ ρ }(f, A_{ α }). Then f ∈ co_{ β }(A_{ β }) since A_{ α } ⊂ co_{ β }(A_{ β }). Hence, f\in \left({\cap}_{g\in {A}_{\beta}}{B}_{\rho}\left(g,s\right)\right)\cap c{o}_{\beta}\left({A}_{\beta}\right). Since C_{ β } is a onelocal retract of C, then
Since A_{ β } = C_{ β } ∩ co_{ β }(A_{ β }), then we have
Let h ∈ S_{ β }, then h\in \bigcap _{g\in {A}_{\beta}}{B}_{\rho}\left(g,s\right). Hence, r_{ ρ }(h, A_{ β }) ≤ s which implies R_{ ρ }(A_{ β }) ≤ s = r_{ ρ }(f, A_{ α }), for every f ∈ A_{ α }. Hence R_{ ρ }(A_{ β }) ≤ R_{ ρ }(A_{ α }). Therefore we have
Since δ_{ ρ }(A_{ β }) > 0 for every β ∈ Γ. Set {A}_{\beta}^{\prime \prime} to be the ρChebyshev center of A_{ β }, i.e., {A}_{\beta}^{\prime \prime}={\mathcal{C}}_{\rho}\left({A}_{\beta}\right), for every β ∈ Γ. Since R_{ ρ }(A_{ β }) = R_{ ρ }(A_{ α }), for every α, β ∈ Γ, then the family \left({A}_{\beta}^{\prime \prime}\right) is decreasing. Indeed, let α≺ β and f\in {A}_{\beta}^{\prime \prime}. Then we have r_{ ρ }(f, A_{ β }) = R_{ ρ }(A_{ β }). Since we proved that r_{ ρ }(g, A_{ β }) = r_{ ρ }(g, A_{ α }), for every g ∈ C_{ β }, then
which implies that f\in {A}_{\alpha}^{\prime \prime}. Therefore, we have {A}^{\prime \prime}=\prod _{\beta \in \Gamma}{A}_{\beta}^{\prime \prime}\in \mathcal{F}. Since A'' ⊂ A and A is minimal, we get A = A^{''}. Therefore, we have {\mathcal{C}}_{\rho}\left({A}_{\beta}\right)={A}_{\beta} for every β ∈ Γ. This contradicts the fact that \mathcal{A}\left({C}_{\beta}\right) is normal for every β ∈ Γ. Hence there exists β_{0} ∈ Γ such that
The proof of our claim is therefore complete. Then we have A_{ β } = {f }, for every β ≻ β_{0}. This clearly implies that f ∈ ∩_{β∈Γ}C_{ β } ≠ ∅. In order to complete the proof, we need to show that S = ∩_{β∈Γ}C_{ β } is a onelocal retract of C. Let (B_{ i })_{i∈I}be a family of ρballs centered in S such that ∩_{i∈I}B_{ i } ≠ ∅. Set D_{ β } = (∩_{i∈I}B_{ i }) ∩ C_{ β }, for any β ∈ Γ. Since C_{ β } is a onelocal retract of C, and the family (B_{ i }) is centered in C_{ β }, then D_{ β } is not empty and {D}_{\beta}\in \mathcal{A}\left({C}_{\beta}\right). Therefore, D=\prod _{\beta \in \Gamma}{D}_{\beta}\in \mathcal{F}. Let A=\prod _{\beta \in \Gamma}{A}_{\beta}\subset D be a minimal element of ℱ. The above proof shows that
The proof of Theorem 2.8 is therefore complete. □
The next theorem will be useful to prove the main result of the next section.
Theorem 2.9. Let ρ ∈ ℜ and C ⊂ L_{ ρ } be nonempty, ρclosed, and ρbounded. Assume that \mathcal{A}\left(C\right) is compact and ρnormal. Let (C_{ β })_{β∈Γ}be a family of onelocal retracts of C such that for any finite subset I of Γ, ∩_{β ∈ Γ}C_{ β } is not empty and is a onelocal retract of C. Then ∩_{β ∈ Γ}C_{ β } is not empty and is a onelocal retract of C.
Proof. Consider the family ℱ of subsets I ⊂ Γ such that for any finite subset J ⊂ Γ (empty or not), we have ∩_{α∈I∪J}C_{ α } is a nonempty onelocal retract of C. Note that ℱ is not empty since any finite subset of Γ is in ℱ. Using Theorem 2.8, we can show that ℱ satisfies the hypothesis of Zorn's lemma. Hence ℱ has a maximal element I ⊂ Γ. Assume I ≠ Γ. Let α ∈ Γ \ I. Obviously we have I ∪ {α}∈ ℱ. This is a clear contradiction with the maximality of I. Therefore we have I = Γ ∈ ℱ, i.e., ∩_{β∈Γ}C_{ β } is not empty and is a onelocal retract of C.
□
Common fixed point result
In the previous section, we showed that under suitable conditions, any ρnonexpansive mapping has a fixed point. In this section we will discuss the existence of a fixed point common to a family of a commutative ρnonexpansive mappings. First we will need to discuss the case of finite families.
Theorem 2.10. Let ρ ∈ ℜ and C ⊂ L_{ ρ } be nonempty, ρclosed, and ρbounded. Assume that \mathcal{A}\left(C\right) is compact and ρnormal. Then for any finite family ℱ = {T_{1}, T_{2},...T_{ n }} of commutative ρnonexpansive mappings defined on C has a common fixed point, i.e., Fix (T_{1}) ∩ ··· ∩ Fix(T_{n}) ≠ ∅. Moreover, the set of common fixed point set, denoted Fix(ℱ) = Fix(T_{1})) ∩ ··· ∩ Fix(T_{ n }), is a one local retract of C.
Proof. Let us first prove Theorem 2.10 for two mappings T_{1} and T_{2}. Using Theorem 2.5, we know that Fix(T_{1}) is a nonempty onelocal retract of C. Since T_{1} and T_{2} are commutative, then T_{2}(Fix(T_{1})) ⊂ Fix(T_{1}). Theorems 2.4 and 2.7 show that the restriction of T_{2} to Fix(T_{1}) has a fixed point. Again Theorem 2.5 will imply that the common fixed point set Fix(T_{1}) ∩ Fix(T_{2}) is a nonempty onelocal retract of C. Using the same argument will show that the conclusion of Theorem 2.10 is valid for any finite number of mappings. □
Next we prove the main result of this section.
Theorem 2.11. Let ρ ∈ ℜ and C ⊂ L_{ ρ } be nonempty, ρclosed, and ρbounded. Assume that \mathcal{A}\left(C\right) is compact and ρnormal. Then for any family ℱ = {T_{i}; i∈I}of commutative ρnonexpansive mappings defined on C has a common fixed point, i.e., ∩_{ i∈I } Fix(T_{i}) ≠ ∅. Moreover the set of common fixed point set, denoted Fix(ℱ) = ∩_{ i∈I } Fix(T_{i}), is a onelocal retract of C.
Proof. Let Γ = {β; β is a nonempty finite subset of I}. Theorem 2.10 implies that for every β ∈ Γ, the set F_{ β } of common fixed point set of the mappings T_{ i }, i ∈ β, is a nonempty onelocal retract of C. Clearly the family (F_{ β })_{β∈Γ}is decreasing and satisfies the assumptions of Theorem 2.9. Therefore, we have ∩_{β∈Γ}F_{ β } is nonempty and is a onelocal retract of C. The proof of Theorem 2.11 is complete.
□
Using Corollary 2.1 and Remark 2.1 we get the following result.
Corollary 2.2. Let ρ ∈ ℜ and C ⊂ L_{ ρ } be nonempty, convex, ρclosed, andρbounded. Assume ρ is (UUC). Then for any family ℱ = {T_{i}; i∈I} of commutative ρnonexpansive mappings defined on C has a common fixed point, i.e., ∩_{ i∈I } Fix(T_{i}) ≠ ∅. Moreover the set of common fixed point set, denoted Fix(ℱ) = ∩_{ i∈I } Fix(T_{i}), is a onelocal retract of C.
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The authors gratefully acknowledge the financial support provided by the University of Tabuk through the project of international cooperation with the University of Texas at El Paso.
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AlMezel, S.A., AlRoqi, A. & Khamsi, M.A. Onelocal retract and common fixed point in modular function spaces. Fixed Point Theory Appl 2012, 109 (2012). https://doi.org/10.1186/168718122012109
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DOI: https://doi.org/10.1186/168718122012109
Keywords
 convexity structure
 fixed point
 modular function space
 nonexpansive mappings
 normal structure
 retract