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On common fixed points in modular vector spaces
Fixed Point Theory and Applications volume 2015, Article number: 229 (2015)
Abstract
In this work, we discuss the concept of Banach operator pairs in modular vector spaces. We prove the existence of common fixed points for these type of operators which satisfy a modular continuity in modular compact sets. On the basis of our result, we are able to give an analog of DeMarr’s common fixed point theorem for a family of symmetric Banach operator pairs in modular vector spaces.
Introduction
In recent years, there was an surge of interest in the study of electrorheological fluids, sometimes referred to as ‘smart fluids’ (for instance lithium polymetachrylate). For these fluids, modeling with sufficient accuracy using classical Lebesgue and Sobolev spaces, \(L^{p}\) and \(W^{1,p}\), where p is a fixed constant is not adequate, rather the exponent p should be able to vary [1–3]. One of the most interesting problem discussed in these spaces is the Dirichlet energy problem [4, 5]. One way to discuss this problem is to convert the energy functional, defined by a modular, to a problem which involves a convoluted and complicated norm (the Luxemburg norm). The modular approach is natural and has not been used.
The purpose of this paper is to discuss the existence of common fixed points of mappings defined on subsets of modular vector spaces, as introduced by Nakano [6] which are natural generalizations of many classical function spaces. The common fixed point problem of a pair of commuting mappings was investigated as early as the first fixed point results were proved [7, 8]. This problem becomes more challenging in view of the historically significant and negatively settled question that a pair of commuting continuous selfmappings defined on \([0, 1]\) may not have a common fixed point [9]. Over the years, many mathematicians have tried to find weaker forms of commutativity that imply the existence of a common fixed point for a pair of selfmappings. Weakly compatible mappings [10] and Banach operator pairs [11–17] were introduced and provided generalizing results in metric fixed point theory for singlevalued mappings. In this work, we discuss some of these results for multivalued in modular vector spaces.
A good reference for metric fixed point theory is the book [18]. For modular spaces, the interested reader may consult the books [19–21].
Preliminaries
Modular vector spaces have been studied in [6, 21].
Definition 2.1
[6]
Let X be a real vector space. A functional \(\rho: X \rightarrow[0,+\infty]\) is called a modular if

(1)
\(\rho(x) = 0\) if and only if \(x = 0\);

(2)
\(\rho(x) = \rho(x)\);

(3)
\(\rho(\alpha x + (1\alpha)y) \leq\rho(x) + \rho(y)\), for any \(x, y \in X\) and \(\alpha\in[0,1]\).
If we replace (3) by

(4)
\(\rho(\alpha x + (1\alpha)y) \leq\alpha\rho(x) + (1\alpha ) \rho(y)\), for any \(x, y \in X\) and \(\alpha\in[0,1]\),
then ρ is called a convex modular.
The concept of modular finds its roots in the work of Nakano [6] who expanded the earlier ideas of Orlicz and Birnbaum who tried to generalize the classical function spaces of the Lebesgue type \(L^{p}\). Orlicz and Birnbaum’s ideas consisted in considering spaces of functions with some growth properties different from the power type growth. In other words, they considered the space:
where \(\varphi:[0,\infty] \rightarrow[0,\infty]\) was assumed to be a convex function increasing to infinity, i.e., the function which to some extent behaves similar to power functions \(\varphi(t) = t^{p}\). The functional \(\rho: L^{\varphi} \rightarrow[0,+\infty]\) defined by
is a modular on \(L^{\varphi}\).
The classical book [20] contains many examples of modular function spaces. The interested reader may also consult the most recent book [19].
Definition 2.2
Let X be a vector space and ρ a convex modular defined on X. The set
is a vector subspace of X known as the associated modular vector space. On it we define the Luxemburg norm
for any \(x \in X_{\rho}\).
In the next definition, we give the basic definitions needed throughout this work.
Definition 2.3
Let X be a vector space and ρ a convex modular defined on X.

(1)
The sequence \(\{x_{n}\}_{n\in\mathbb{N}}\) in \(X_{\rho}\) is said to be ρconvergent to \(x\in X_{\rho}\) if and only if \(\rho (x_{n}x)\rightarrow0\), as \(n\rightarrow\infty\). x will be called the ρlimit of \(\{x_{n}\}\).

(2)
The sequence \(\{x_{n}\}_{n\in N}\) in \(X_{\rho}\) is said to be ρCauchy if \(\rho(x_{m}x_{n})\rightarrow0\), as \(m,n\rightarrow \infty\).

(3)
A subset M of \(X_{\rho}\) is said to be ρclosed if the ρlimit of a ρconvergent sequence of M always belong to M.

(4)
A subset M of \(X_{\rho}\) is said to be ρcomplete if any ρCauchy sequence in M is a ρconvergent sequence and its ρlimit is in M.

(5)
A subset M of \(X_{\rho}\) is said to be ρbounded if we have
$$\delta_{\rho}(M)= \sup \bigl\{ \rho(xy);x,y\in M \bigr\} < \infty. $$ 
(6)
A subset M of \(X_{\rho}\) is said to be sequentially ρcompact if for any \(\{x_{n}\}\) in M there exists a subsequence \(\{ x_{n_{k}}\}\) and \(x\in M\) such that \(\rho(x_{n_{k}}x)\rightarrow0\).

(7)
A nonempty subset K of \(X_{\rho}\) is said to be ρcompact if for any family \(\{A_{\alpha}; \alpha\in\Gamma\}\) of ρclosed subsets of \(X_{\rho}\) with \(K \cap A_{\alpha_{1}}\cap\cdots \cap A_{\alpha_{n}} \neq\emptyset\), for any \(\alpha_{1}, \ldots, \alpha_{n} \in\Gamma\), we have
$$K \cap \biggl(\bigcap_{\alpha\in\Gamma} A_{\alpha} \biggr) \neq \emptyset. $$ 
(8)
ρ is said to satisfy the Fatou property if and only if for any sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) in \(X_{\rho}\) which ρconverge, respectively, to x and y, we have
$$\rho(xy) \leq\liminf_{n\rightarrow\infty} \rho(x_{n}y_{n}). $$
In general if \(\lim_{n \rightarrow\infty} \rho(\lambda(x_{n}x)) = 0\), for some \(\lambda>0\), then we may not have \(\lim_{n \rightarrow \infty} \rho(\lambda (x_{n}x)) = 0\), for all \(\lambda>0\). Therefore, as it is done in modular function spaces, we will say that ρ satisfies \(\Delta_{2}\)condition if this is the case, i.e. \(\lim_{n \rightarrow\infty} \rho(\lambda (x_{n}x)) = 0\), for some \(\lambda>0\) implies \(\lim_{n \rightarrow\infty} \rho(\lambda(x_{n}x)) = 0\), for all \(\lambda>0\). In particular, we have
for any \(\{x_{n}\} \in X_{\rho}\) and \(x \in X_{\rho}\). In other words, we see that ρconvergence and \(\Vert \cdot \Vert _{\rho}\) convergence are equivalent if and only if the modular ρ satisfies the \(\Delta_{2}\)condition.
In the next section, we will prove Brouwer’s fixed point theorem in modular vector spaces and obtain a common fixed point result as was done in [22]. The following definition is therefore needed.
Definition 2.4
Let X be a vector space and ρ a convex modular defined on X. Let \(C \subset X_{\rho}\) be nonempty and ρclosed. Let \(T:C \rightarrow X_{\rho}\) be a map. We will say that T is ρcontinuous if \(\{T(x_{n})\}\) ρconverges to \(T(x)\) if \(\{x_{n}\}\) ρconverges to x. Moreover, we will say T is strongly ρcontinuous if T is ρcontinuous and
for any \(\{x_{n}\} \subset C\) such that \(\{x_{n}\}\) ρconverges to x and for any \(y \in C\). A point \(x \in C\) is called a fixed point of T if \(T(x) = x\). The set of fixed points of T is denoted by \(\operatorname {Fix}(T)\).
We now introduce the concept of Banach operator pairs [11, 17] in modular vector spaces.
Definition 2.5
Let X be a vector space and ρ a convex modular defined on X. Let \(C \subset X_{\rho}\) be nonempty. Let \(S, T: C \rightarrow C\) be two selfmappings. The ordered pair \((T,S)\) is said to be a Banach operator pair whenever the set \(\operatorname {Fix}(T)\) is Sinvariant, i.e.,
Note that if S and T commute, i.e., \(S\circ T = T \circ S\), then the ordered pairs \((S,T)\) and \((T,S)\) are Banach operator pairs. Therefore the concept of Banach operator pairs is seen as a weakening of the commutativity condition.
Common fixed points for Banach operators pairs
We prove the Brouwer fixed point theorem [23] in modular vector spaces via the theorem of KnasterKuratowskiMazurkiewicz (KKM) [24].
Let X be a vector space and ρ a convex modular defined on X. Let \(C \subset X_{\rho}\) be nonempty. The set of all subsets of C is denoted \(2^{C}\). The notation \(\operatorname {conv}(A)\) describes the convex hull of A, while \(\overline{\operatorname {conv}}_{\rho}(A)\) describes the smallest ρclosed convex subset of \(X_{\rho}\) which contains A. Recall that a family \(\{ A_{\alpha} \subset X_{0}; \alpha\in\Gamma\}\) is said to have the finite intersection property if the intersection of each finite subfamily is not empty. Throughout this section, we will assume that ρ is finite, i.e., \(\rho(x) < +\infty\), for any \(x \in X\), and ρ satisfies the Fatou property.
Definition 3.1
Let X be a vector space and ρ a convex modular defined on X. Let \(C \subset X_{\rho}\) be nonempty. A multivalued mapping \(G: C \rightarrow2^{X_{\rho}}\) is called a KnasterKuratowskiMazurkiewicz mapping (in short KKMmapping) if
for any \(f_{1},\ldots,f_{n} \in C\).
A subset \(A \subset X_{\rho}\) is called finitely ρclosed if for every \(x_{1},x_{2},\ldots,x_{n} \in X_{\rho}\), the set \(\overline {\operatorname {conv}}_{\rho}(\{x_{1},\ldots,x_{n}\}) \cap A\) is ρclosed.
We are now ready to prove the following result.
Theorem 3.1
Let X be a vector space and ρ a convex modular defined on X. Let \(C \subset X_{\rho}\) be nonempty, and \(G:C \rightarrow2^{X_{\rho}}\) be a KKMmapping such that for any \(x \in C\), \(G(x)\) is nonempty and finitely ρclosed. Then the family \(\{G(x); x \in C\}\) has the finite intersection property.
Proof
Assume not, i.e. there exist \(x_{1},\ldots,x_{n} \in C\) such that \(\bigcap_{1 \leq i \leq n} G(x_{i}) = \emptyset\). Set \(L = \overline{\operatorname {conv}}_{\rho}(\{x_{i}\})\) in \(X_{\rho}\). Our assumptions imply that \(L \cap G(x_{i})\) is ρclosed for every \(i=1,2,\ldots,n\). Since ρclosedness implies closedness for the Luxemburg norm \(\Vert \cdot \Vert _{\rho}\), \(L \cap G(x_{i})\) is closed for \(\Vert \cdot \Vert _{\rho}\), for any \(i \in\{1,\ldots,n\}\). Thus for every \(x \in L\), there exists \(i_{0}\) such that x does not belong to \(L \cap G(x_{i_{0}})\) since \(L \cap (\bigcap_{1 \leq i \leq n} G(x_{i}) ) = \emptyset\). Hence
because \(L \cap G(x_{i_{0}})\) is closed. We use the function
where \(x \in K = \operatorname {conv}\{x_{1},\ldots,x_{n}\}\), to define the map \(T: K\rightarrow K\) by
It is obvious that T is a continuous map. Using the compactness of the convex subset K of \((X_{\rho}, \Vert \cdot \Vert _{\rho})\), we can use Brouwer’s theorem to ensure the existence of a fixed point \(x_{0} \in K\) of T, i.e. \(T(x_{0}) = x_{0}\). Set
Clearly, we have
Hence \(x_{0} \notin\bigcup_{i \in I} G(x_{i})\) and \(x_{0} \in \operatorname {conv}(\{ x_{i}; i \in I\})\) contradict the assumption
□
As an immediate consequence of Theorem 3.1, we obtain the following result.
Theorem 3.2
Let X be a vector space and ρ a convex modular defined on X. Let \(C \subset X_{\rho}\) be nonempty, and \(G:C \rightarrow2^{X_{\rho}}\) be a KKMmapping such that for any \(x \in C\), \(G(x)\) is nonempty and ρclosed. Assume there exists \(f_{0} \in C\) such that \(G(f_{0})\) is ρcompact. Then we have \(\bigcap_{x\in C} G(x) \neq\emptyset\).
The following lemma is needed in proving the analog of Ky Fan fixed point result [24] in modular vector spaces.
Lemma 3.1
Let X be a vector space and ρ a convex modular defined on X. Let \(K \subset X_{\rho}\) be a nonempty convex, ρcompact subset. Let \(T:K \rightarrow X_{\rho}\) be strongly ρcontinuous. Then there exists \(x_{0} \in K\) such that
Proof
Let \(G: K \rightarrow2^{K} \subset2^{X_{\rho}}\) be defined by
Since T is strongly ρcontinuous and ρ is assumed to satisfy the Fatou property, we have
for any sequence \(\{x_{n}\}\) in \(G(y)\) which ρconverges to x. Hence \(G(y)\) is ρclosed for any \(y \in K\). Next, assume that G is not a KKMmapping. Then there exists \(\{y_{1},\ldots,y_{n}\} \subset K\) and \(x \in \operatorname {conv}(\{y_{i}\})\) such that \(x \notin\bigcup_{1 \leq i \leq n} G(y_{i})\). This clearly implies
Let \(\varepsilon> 0\) be such that \(\rho(y_{i}T(x)) \leq\rho (xT(x))\varepsilon \), for \(i=1,2,\ldots,n\). Using the convexity of ρ, we get
for any \(y \in \operatorname {conv}(\{y_{i}\})\). Since \(x \in \operatorname {conv}(\{y_{i}\})\), we get \(\rho (x,T(x)) \leq\rho(x,T(x))\varepsilon\). This is a contradiction which implies that G is a KKMmapping. Since K is ρcompact, we deduce that \(G(y)\) is also ρcompact for any \(y \in K\). Using Theorem 3.2, there exists \(x_{0}\) in \(\bigcap_{y \in K} G(y)\). Hence \(\rho(x_{0}T(x_{0})) \leq\rho(yT(x_{0}))\), for any \(y \in K\), which implies
□
Before we state the Ky Fan fixed point theorem [24] in modular vector spaces, we recall the definition of modular balls. Let \(x \in X_{\rho}\) and \(r \geq0\). The modular ball \(B_{\rho}(x,r)\) is defined by
Since ρ is convex and satisfies the Fatou property, the modular balls are convex and ρclosed.
Theorem 3.3
Let X be a vector space and ρ a convex modular defined on X. Let \(K \subset X_{\rho}\) be a nonempty convex, ρcompact subset. Let \(T:K \rightarrow X_{\rho}\) be strongly ρcontinuous such that for any \(x \in K\), with \(x \neq T(x)\), there exists \(\alpha\in(0,1)\) such that
Then T has a fixed point.
Proof
By Lemma 3.1, there exists \(x_{0} \in K\) such that
Assume that \(x_{0}\) is not a fixed point of T, i.e., \(x_{0} \neq T(x_{0})\). Then there exists \(\alpha\in(0,1)\) such that
Let \(y \in K_{0}\). Then \(\rho(yT(x_{0})) \leq(1\alpha) \rho (x_{0}T(x_{0}))\). This implies a contradiction to the property satisfied by \(x_{0}\). Therefore we must have \(T(x_{0}) = x_{0}\), i.e., \(x_{0}\) is a fixed point of T. □
Note that if \(T(K) \subset K\), then T satisfies the condition (3.1). The following theorem is an analog to Brouwer’s fixed point theorem in modular vector spaces.
Theorem 3.4
Let X be a vector space and ρ a convex modular defined on X. Let \(K \subset X_{\rho}\) be a nonempty convex, ρcompact subset. Let \(T:K \rightarrow K\) be strongly ρcontinuous. Then \(\operatorname {Fix}(T)\) is a nonempty ρcompact subset.
Proof
The existence of a fixed point of T is a direct consequence of Theorem 3.3. So \(\operatorname {Fix}(T)\) is nonempty. Next we show that \(\operatorname {Fix}(T)\) is ρcompact. Let us prove that \(\operatorname {Fix}(T)\) is ρclosed. Let \(\{x_{n}\}\) be a sequence in \(\operatorname {Fix}(T)\) such that \(\{ x_{n}\}\) ρconverges to x. Since T is ρcontinuous, \(\{ T(x_{n})\}\) ρconverges to \(T(x)\). Since \(T(x_{n}) = x_{n}\), we see that \(\{x_{n}\}\) ρconverges to x and \(T(x)\). Using the uniqueness of the ρlimit, we get \(T(x) = x\), i.e., \(x \in \operatorname {Fix}(T)\). Since K is ρcompact and \(\operatorname {Fix}(T)\) is ρclosed, we conclude that \(\operatorname {Fix}(T)\) is ρcompact. □
In order to prove our main result of this section, we need the following definition.
Definition 3.2
Let X be a vector space and ρ a convex modular defined on X. Let \(C \subset X_{\rho}\) be a nonempty subset. A mapping \(T: C \rightarrow C\) is said to be an Rmap if \(\operatorname {Fix}(T)\) is a ρcontinuous retraction of C, i.e., there exists a ρcontinuous mapping \(R: C \rightarrow \operatorname {Fix}(T)\) such that \(R\circ R = R\). Such a mapping is known as a retraction.
Next we discuss our first common fixed point result of this work.
Theorem 3.5
Let X be a vector space and ρ a convex modular defined on X. Let \(K \subset X_{\rho}\) be a nonempty convex, ρcompact subset. Let \(S,T: K \rightarrow K\) be two selfmappings. Assume that S and T are strongly ρcontinuous such that \((S,T)\) is a Banach operator pair. If we assume that T is an Rmap, then \(\operatorname {Fix}(T) \cap \operatorname {Fix}(S)\) is nonempty and ρcompact.
Proof
Theorem 3.4 implies that \(\operatorname {Fix}(T)\) is not empty and ρcompact. Since T is an Rmap, there exists a retraction \(R: K \rightarrow \operatorname {Fix}(T)\) which is ρcontinuous. Since \((S,T)\) is a Banach pair of operators, \(S(\operatorname {Fix}(T)) \subset \operatorname {Fix}(T)\). It is clear that \(S\circ R: K \rightarrow K\) is strongly ρcontinuous. To see this, let \(\{x_{n}\} \subset K\) be ρconvergent to x. Then \(\{R(x_{n})\}\) ρconverges to \(R(x)\) since R is ρcontinuous. Using the strong ρcontinuity of S, we have
for any \(y \in K\), i.e., \(S \circ R\) is strongly ρcontinuous. Theorem 3.4 implies that \(\operatorname {Fix}(S\circ R)\) is nonempty and ρcompact. Moreover, if \(x \in \operatorname {Fix}(S\circ R)\), then \(S \circ R(x) = S (R(x)) = x \in \operatorname {Fix}(T)\) because \(S\circ R(K) \subset \operatorname {Fix}(T)\). In particular, we have \(R(x) = x\). Hence \(S(x) = x\), i.e. \(x \in \operatorname {Fix}(T) \cap \operatorname {Fix}(S)\). It is easy to see that \(\operatorname {Fix}(T) \cap \operatorname {Fix}(S) = \operatorname {Fix}(S\circ R)\). Hence \(\operatorname {Fix}(T) \cap \operatorname {Fix}(S)\) is nonempty and ρcompact. □
Next we extend the conclusion of Theorem 3.5 to a family of Banach operator mappings. The following concept will be needed.
Definition 3.3
Let X be a vector space and ρ a convex modular defined on X. Let \(C \subset X_{\rho}\) be a nonempty subset. Let \(S,T: C \rightarrow C\) be two selfmappings. \((S, T )\) is called a symmetric Banach operator pair if \((S,T)\) and \((T, S )\) are Banach operator pairs, i.e.,
Recall that the set of common fixed points of a family of mappings \({\mathcal {F}}\) is the set \(\operatorname {Fix}(\mathcal {F}) = \bigcap_{T \in{ \mathcal {F}}} \operatorname {Fix}(T)\). The main result of this work states:
Theorem 3.6
Let X be a vector space and ρ a convex modular defined on X. Let \(K \subset X_{\rho}\) be nonempty convex and ρcompact. Let \({\mathcal {F}}\) be a family of selfmappings defined on K such that any map in \(\mathcal {F}\) is a strongly ρcontinuous Rmap. Assume that any two mappings in \(\mathcal {F}\) form a symmetric Banach operator pair. Then \(\operatorname {Fix}(\mathcal {F})\) is a nonempty ρcompact subset.
Proof
Theorem 3.5 implies that \(\operatorname {Fix}(T_{1}) \cap\cdots \cap \operatorname {Fix}(T_{n})\) is a nonempty ρcompact subset of K, for any \(T_{1}, T_{2}, \ldots, T_{n} \) in \(\mathcal {F}\). Therefore any finite family of the subsets \(\{\operatorname {Fix}(T); T \in{ \mathcal {F}}\}\) has a nonempty intersection. Since these sets are all ρclosed and K is ρcompact, we conclude that \(\operatorname {Fix}(\mathcal {F}) = \bigcap_{T \in{ \mathcal {F}}} \operatorname {Fix}(T)\) is not empty and is ρclosed. Therefore \(\operatorname {Fix}(\mathcal {F})\) is ρcompact. □
As commuting mappings are symmetric Banach operator pairs, we obtain an analog to DeMarr’s common fixed point theorem [8] in modular spaces as follows.
Corollary 3.1
Let X be a vector space and ρ a convex modular defined on X. Let \(K \subset X_{\rho}\) be a nonempty convex, ρcompact subset. Let \({\mathcal {F}}\) be a family of commuting selfmappings defined on K such that any map in \(\mathcal {F}\) is a strongly ρcontinuous Rmap. Then \(\operatorname {Fix}(\mathcal {F})\) is nonempty and ρcompact.
Remark 3.1
Since convexity plays a crucial role in the proof of Brouwer’s fixed point theorem, an analog of this profound theorem does not exist in metric spaces. In order to obtain an extension of this fundamental fixed point result, one needs to use some kind of convexity therein. This is the case for example of hyperconvex metric spaces [25]. So to obtain an extension of all the results obtained in this work for modular metric spaces [26–28], we will have to introduce something like hyperbolicity in modular metric spaces. This concept will be new and has not been introduced or investigated so far.
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Acknowledgements
This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. 372501D1435. The authors, therefore, gratefully acknowledge the DSR technical and financial support.
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MSC
 47H09
 46B20
 47H10
 47E10
Keywords
 Banach operator pair
 electrorheological fluids
 fixed point
 KKMmapping
 modular function space
 modular metric space
 retraction