Abstract
Let C be a ρbounded, ρclosed, convex subset of a modular function space ${L}_{\rho}$. We investigate the existence of common fixed points for asymptotic pointwise nonexpansive semigroups of nonlinear mappings ${T}_{t}:C\to C$, i.e. a family such that ${T}_{0}(f)=f$, ${T}_{s+t}(f)={T}_{s}\circ {T}_{t}(f)$ and
where ${lim\hspace{0.17em}sup}_{t\to \mathrm{\infty}}{\alpha}_{t}(f)\le 1$ for every $f\in C$. In particular, we prove that if ${L}_{\rho}$ is uniformly convex, then the common fixed point is nonempty ρclosed and convex.
MSC:47H09, 46B20, 47H10, 47E10.
1 Introduction
The purpose of this paper is to prove the existence of common fixed points for semigroups of nonlinear mappings acting in modular function spaces which 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, see the book by Kozlowski [1] for an extensive list of examples and special cases. Earlier studies of fixed point theory in modular function spaces can be found in [2–4], see also [5]. Recently, Khamsi and Kozlowski presented a series of fixed point results for pointwise contractions, asymptotic pointwise contractions, pointwise nonexpansive and asymptotic pointwise nonexpansive mappings acting in modular functions spaces [6, 7] (all these should be considered in the modular sense, not in the sense of the corresponding norms). These results are also new and of a big interest, even in a much simpler context of ‘plain’ modular contractions and nonexpansive mappings, i.e., without any pointwise and asymptotic complications.
In many cases, modular type conditions are much more natural as modular type assumptions can be more easily verified than their metric or norm counterparts. Furthermore, there are also important results that can be proved only using the apparatus of modular function spaces. Khamsi et al. demonstrated in [2] that a mapping T is normnonexpansive in a modular function space ${L}_{\rho}$ if and only if
They also gave an example of a mapping which is ρnonexpansive, but it is not normnonexpansive. 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.
Let us recall that a family ${\{{T}_{t}\}}_{t\ge 0}$ of mappings forms a semigroup if ${T}_{0}(x)=x$ and ${T}_{s+t}={T}_{s}\circ {T}_{t}$, see Definition 2.6 below for details. Such a situation is quite typical in mathematics and applications. For instance, in the theory of dynamical systems, the modular function space ${L}_{\rho}$ would define the state space and the mapping $(t,f)\to {T}_{t}(f)$ would represent the evolution function of a dynamical system. The question about the existence of common fixed points and about the structure of the set of common fixed points, can be interpreted as a question whether there exist points that are fixed during the state space transformation ${T}_{t}$ at any given point of time t, and if yes, what the structure of a set of such points may look like. In the setting of this paper, the state space may be an infinite dimensional vector space. Therefore, it is natural to apply these results not only to deterministic dynamical systems but also to stochastic dynamical systems.
The existence of common fixed points for families of contractions and nonexpansive mappings in the Banach spaces have been investigated since the early 1960s, see, e.g., Belluce and Kirk [8, 9], Browder [10], Bruck [11], DeMarr [12], Lim [13]. The asymptotic approach for finding common fixed points of semigroups of Lipschitzian (but not pointwise Lipschitzian) mappings has also been investigated for some time, see, e.g., Tan and Xu [14]. It is worthwhile mentioning the recent studies on the special case, when the parameter set for the semigroup is equal to $\{0,1,2,3,\dots \}$ and ${T}_{n}={T}^{n}$, the n th iterate of an asymptotic pointwise nonexpansive mapping, i.e., $T:C\to C$ such that there exists a sequence of functions ${\alpha}_{n}:C\to [0,\mathrm{\infty})$ with
and ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}(f)=1$ for any $f\in C$. Kirk and Xu [15] proved the existence of fixed points for asymptotic pointwise contractions and asymptotic pointwise nonexpansive mappings in the Banach spaces, while Hussain and Khamsi extended this result to metric spaces [16] and Khamsi and Kozlowski to modular function spaces [6, 7]. Kozlowski in [17] and [18] proved convergence to fixed points of some iterative algorithms, applied to asymptotic pointwise nonexpansive mappings in the Banach spaces, and the existence of common fixed points of semigroups of asymptotic pointwise nonexpansive semigroups in the Banach spaces [19]. Convergence of generalized Mann and Ishikawa algorithms to common points of such semigroups in Banach spaces was established in [20] and [21]. In the context of modular function spaces, convergence to fixed points of some iterative algorithms, applied to asymptotic pointwise nonexpansive mappings, was proven by Bin Dehaish and Kozlowski in [22].
In this paper, we extend the definition of asymptotic pointwise nonexpansive mappings to semigroups of mappings and prove some common fixed point results in the context of modular function spaces. Therefore, our results generalize the results of Kozlowski [23], who proved the existence of common fixed points for semigroups of nonexpansive mappings in modular functions spaces, to the pointwise asymptotic semigroups. However, methods used in the current paper are substantially different due to the asymptotic behavior of semigroups in question. It is worth noting that existence of semigroups of nonexpansive mappings in modular function spaces was discussed by Khamsi [24] in the context of MusielakOrlicz spaces and discussed applications to differential equations.
2 Preliminaries
Let us introduce basic notions related to modular function spaces and related notation, which will be used in this paper. For further details, we refer the reader to preliminary sections of the recent articles [6, 7, 22] or to the survey article [5], see also [1, 25, 26] for the standard framework of modular function spaces.
Let Ω be a nonempty set, and let Σ be a nontrivial σalgebra of subsets of Ω. Let 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 ℰ we denote the linear space of all simple functions with supports from . By ${\mathcal{M}}_{\mathrm{\infty}}$ we will denote the space of all extended measurable functions, i.e., all functions $f:\mathrm{\Omega}\to [\mathrm{\infty},\mathrm{\infty}]$ such that there exists a sequence $\{{g}_{n}\}\subset \mathcal{E}$, ${g}_{n}\le f$ 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 [1]
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:

(i)
$\rho (0)=0$;

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

(iii)
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 $A\cap B\ne \mathrm{\varnothing}$, $f\in \mathcal{M}$;

(iv)
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 $f\in {\mathcal{M}}_{\mathrm{\infty}}$;

(v)
ρ is order continuous in ℰ, i.e., ${g}_{n}\in \mathcal{E}$ 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 $g\in \mathcal{E}$. 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 $\mathcal{M}=\{f\in {\mathcal{M}}_{\mathrm{\infty}};f(\omega )<\mathrm{\infty}\phantom{\rule{0.25em}{0ex}}\rho \text{a.e.}\}$, where each element is actually an equivalence class of functions equal ρa.e. rather than an individual function.
Definition 2.2 [1]
We say that a regular function pseudomodular 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, defined on Ω will be denoted by ℜ.
Let ρ be a convex function modular. A modular function space is the vector space ${L}_{\rho}=\{f\in \mathcal{M};\rho (\lambda f)\to 0\text{as}\lambda \to 0\}$. In the vector space ${L}_{\rho}$, the following formula
defines a norm, frequently called Luxembourg norm.
The following notions will be used throughout the paper.
Definition 2.4 [2]
Let $\rho \in \mathrm{\Re}$.

(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 $n,m\to \mathrm{\infty}$.

(c)
We say that ${L}_{\rho}$ is ρcomplete if and only if any ρCauchy sequence in ${L}_{\rho}$ is ρconvergent.

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

(e)
A set $B\subset {L}_{\rho}$ is called ρbounded if $sup\{\rho (fg);f\in B,g\in B\}<\mathrm{\infty}$.
Since ρ fails in general the triangle identity, many of the known properties of limit may not extend to the ρconvergence. For example, the ρconvergence does not necessarily imply the ρCauchy condition. However, it is important to remember that the ρlimit is unique when it exists. The following proposition brings together few facts that will be often used in the proofs of our results.
Proposition 2.1 [1]
Let $\rho \in \mathrm{\Re}$.

(i)
${L}_{\rho}$ is ρcomplete.

(ii)
ρballs ${B}_{\rho}(f,r)=\{g\in {L}_{\rho};\rho (fg)\le r\}$ are ρclosed.

(iii)
If $\rho (\alpha {f}_{n})\to 0$ for an $\alpha >0$, then there exists a subsequence $\{{g}_{n}\}$ of $\{{f}_{n}\}$ such that ${g}_{n}\to 0$ ρa.e.

(iv)
$\rho (f)\le {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}\rho ({f}_{n})$, whenever ${f}_{n}\to f$ ρa.e. (Note: this property is equivalent to the Fatou property.)
Let us recall the definition of an asymptotic pointwise nonexpansive mapping acting in a modular function space.
Definition 2.5 [7]
Let $\rho \in \mathrm{\Re}$, and let $C\subset {L}_{\rho}$ be nonempty and ρclosed. A mapping $T:C\to C$ is called

(i)
a pointwise Lipschitzian mapping, if there exists $\alpha :C\to [0,\mathrm{\infty})$ such that
$$\rho (T(f)T(g))\le \alpha (f)\rho (fg)\phantom{\rule{1em}{0ex}}\text{for any}f,g\in C;$$ 
(ii)
an asymptotic pointwise nonexpansive, if there exists a sequence of mappings ${\alpha}_{n}:C\to [0,\mathrm{\infty})$ such that
$$\rho ({T}^{n}(f){T}^{n}(g))\le {\alpha}_{n}(f)\rho (fg)\phantom{\rule{1em}{0ex}}\text{for any}f,g\in C$$
and ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}(f)\le 1$ for any $f\in {L}_{\rho}$.
A point $f\in C$ is called a fixed point of T, whenever $T(f)=f$. The set of fixed points of T will be denoted by $F(T)$.
This definition is now extended to a oneparameter family of mappings.
Definition 2.6 A oneparameter family $\mathcal{F}=\{{T}_{t}:t\ge 0\}$ of mappings from C into itself is said to be a asymptotic pointwise nonexpansive semigroup on C if ℱ satisfies the following conditions:

(i)
${T}_{0}(f)=f$ for $f\in C$;

(ii)
${T}_{t+s}(f)={T}_{t}({T}_{s}(f))$ for $f\in C$ and $t,s\in [0,\mathrm{\infty})$;

(iii)
for each $t\ge 0$, ${T}_{t}$ is an asymptotic pointwise nonexpansive mapping, i.e., there exists a function ${\alpha}_{t}:C\to [0,\mathrm{\infty})$ such that
$$\rho ({T}_{t}(f){T}_{t}(g))\le {\alpha}_{t}(f)\rho (fg)\phantom{\rule{1em}{0ex}}\text{for all}f,g\in C$$(2.1)
such that ${lim\hspace{0.17em}sup}_{t\to \mathrm{\infty}}{\alpha}_{t}(f)\le 1$ for every $f\in C$;

(iv)
for each $f\in C$, the mapping $t\to {T}_{t}(f)$ is ρcontinuous.
For each $t\ge 0$, let $F({T}_{t})$ denote the set of its fixed points. Define then the set of all common fixed points set for mappings from ℱ as the following intersection
Note that without loss of generality, we may assume ${\alpha}_{t}(f)\ge 1$ for any $t\ge 0$ and $f\in C$, and ${lim\hspace{0.17em}sup}_{t\to \mathrm{\infty}}{\alpha}_{t}(f)={lim}_{t\to \mathrm{\infty}}{\alpha}_{t}(f)=1$.
3 Existence of common fixed points
The concept ρtype is a powerful technical tool, which is used in the proofs of many fixed point results. The definition of a ρtype is based on a given sequence. In this work, we generalize this definition to be adapted to oneparameter family of mappings.
Definition 3.1 Let $K\subset {L}_{\rho}$ be convex and ρbounded.

(1)
A function $\tau :K\to [0,\mathrm{\infty}]$ is called a ρtype (or shortly a type) if there exists a oneparameter family ${\{{h}_{t}\}}_{t\ge 0}$ of elements of K such that for any $f\in K$ there holds
$$\tau (f)=\underset{M>0}{inf}(\underset{t\ge M}{sup}\rho ({h}_{t}f)).$$ 
(2)
Let τ be a type. A sequence $\{{g}_{n}\}$ is called a minimizing sequence of τ if
$$\underset{n\to \mathrm{\infty}}{lim}\tau ({g}_{n})=inf\{\tau (f):f\in K\}.$$
Note that τ is convex, provided ρ is convex.
Let us recall the modular equivalents of uniform convexity introduced in [7].
Definition 3.2 Let $\rho \in \mathrm{\Re}$. We define the following uniform convexity (UC) type properties of the function modular ρ:

(i)
Let $r>0$, $\epsilon >0$. Define
$$D(r,\epsilon )=\{(f,g):f,g\in {L}_{\rho},\rho (f)\le r,\rho (g)\le r,\rho (fg)\ge \epsilon r\}.$$Let
$$\delta (r,\epsilon )=inf\{1\frac{1}{r}\rho \left(\frac{f+g}{2}\right):(f,g)\in D(r,\epsilon )\}\phantom{\rule{1em}{0ex}}\text{if}D(r,\epsilon )\ne \mathrm{\varnothing},$$and $\delta (r,\epsilon )=1$ if $D(r,\epsilon )=\mathrm{\varnothing}$. We say that ρ satisfies (UC) if for every $r>0$, $\epsilon >0$, $\delta (r,\epsilon )>0$. Note that for every $r>0$, $D(r,\epsilon )\ne \mathrm{\varnothing}$, for $\epsilon >0$ small enough.

(ii)
We say that ρ satisfies (UUC) if there exists $\eta (s,\epsilon )>0$, for every $s\ge 0$, and $\epsilon >0$ such that
$$\delta (r,\epsilon )>\eta (s,\epsilon )>0\phantom{\rule{1em}{0ex}}\text{for}rs.$$
The following technical lemma is very useful throughout this paper (see [7] for its proof).
Lemma 3.1 Let $\rho \in \mathrm{\Re}$ be (UUC). Let $R>0$. Assume that $\{{f}_{n}\}$ and $\{{g}_{n}\}$ are in ${L}_{\rho}$ such that
Then we must have ${lim}_{n\to \mathrm{\infty}}\rho ({f}_{n}{g}_{n})=0$.
The following property plays in the theory of modular function space a role similar to the reflexivity in the Banach spaces, see, e.g., [3].
Definition 3.3 We say that ${L}_{\rho}$ has property (R) if and only if every nonincreasing sequence $\{{C}_{n}\}$ of nonempty, ρbounded, ρclosed and convex subsets of ${L}_{\rho}$ has a nonempty intersection.
Similarly as in the Banach space case, the modular uniform convexity implies the property (R).
Theorem 3.1 [7]
Let $\rho \in \mathrm{\Re}$ be (UUC), then ${L}_{\rho}$ has a property (R).
The next lemma is the generalization of the minimizing sequence property for types defined by the sequences in Lemma 4.3 in [6] to the oneparameter semigroup case.
Lemma 3.2 Assume $\rho \in \mathrm{\Re}$ is (UUC). Let C be a nonempty, ρbounded, ρclosed and convex subset of ${L}_{\rho}$. Let τ be a type defined by a oneparameter family ${\{{h}_{t}\}}_{t\ge 0}$ in C.

(i)
If $\tau ({f}_{1})=\tau ({f}_{2})={inf}_{f\in C}\tau (f)$, then ${f}_{1}={f}_{2}$.

(ii)
Any minimizing sequence $\{{f}_{n}\}$ of τ is ρconvergent. Moreover, the ρlimit of $\{{f}_{n}\}$ is independent of the minimizing sequence.
Proof First, let us prove (i). Let ${f}_{1},{f}_{2}\in C$ such that $\tau ({f}_{1})=\tau ({f}_{2})={inf}_{f\in C}\tau (f)$. Let us consider two cases.
Case 1: ${inf}_{f\in C}\tau (f)=0$. Since
for any $t\ge 0$, we get
for any $M>0$. Since
for any $f\in C$, we get
which implies ${f}_{1}={f}_{2}$ as claimed.
Case 2: ${inf}_{f\in C}\tau (f)>0$. Assume to the contrary that ${f}_{1}\ne {f}_{2}$. Set
Let $\nu \in (0,R)$. Then $\rho ({f}_{1}{f}_{2})=2R\epsilon \ge (R+\nu )\epsilon $. Using the definition of τ, we deduce that there exists ${M}_{\nu}>0$ such that
Since ρ is (UUC), there exists $\eta (R,\epsilon )>0$ such that
for any $\nu \in (0,R)$. So for any $t\ge {M}_{\nu}$, we have
Hence
Since C is convex, we get
If we let $\nu \to 0$, we will get
which is impossible, since $R>0$ and $\eta (R,\epsilon )>0$. Therefore, we must have ${f}_{1}={f}_{2}$.
Next, we prove (ii). Denote $R={inf}_{g\in C}\tau (g)$. For any $n\ge 1$, let us set
where ${\overline{conv}}_{\rho}(A)$ is the intersection of all ρclosed convex subset of C, which contains $A\subset C$. Since C is itself ρclosed and convex, we get ${K}_{n}\subset C$ for any $n\ge 1$. Property (R) will then imply $\bigcap {K}_{n}\ne \mathrm{\varnothing}$. Let us fix then arbitrary $f\in \bigcap {K}_{n}$, $g\in C$ and $\epsilon >0$. By definition of $\tau (g)$, there exists ${M}_{\epsilon}>0$ such that ${sup}_{t\ge {M}_{\epsilon}}\rho (g{h}_{t})\le \tau (g)+\epsilon $. Let $n\ge {M}_{\epsilon}$. Then for any $t\ge n$, we have $\rho (g{h}_{t})\le \tau (g)+\epsilon $, i.e., ${h}_{t}\in {B}_{\rho}(g,\tau (g)+\epsilon )$. Since ${B}_{\rho}(g,\tau (g)+\epsilon )$ is ρclosed and convex, we get ${K}_{n}\subset {B}_{\rho}(g,\tau (g)+\epsilon )$. Hence $f\in {B}_{\rho}(g,\tau (g)+\epsilon )$, i.e.,
Since ε was taken arbitrarily greater than 0, we get $\rho (gf)\le \tau (g)$ for any $g\in C$. Let $\{{f}_{n}\}$ be a minimizing sequence for τ. If $R=0$, then, since $\{{f}_{n}\}$ is a minimizing sequence, we get ${lim}_{n\to \mathrm{\infty}}\tau ({f}_{n})=R=0$. Using (3.1), we can see that $\rho ({f}_{n}f)\le \tau ({f}_{n})$ for any $n\ge 1$. Hence $\{{f}_{n}\}$ is ρconvergent to f. Since selection of f was independent of $\{{f}_{n}\}$, it follows that any minimizing sequence is ρconvergent to f if $R=0$. We can assume, therefore, that $R>0$. For any $n\ge 1$, let us set
We claim that $\{{f}_{n}\}$ is ρCauchy. Assume to the contrary that this is not the case. Since the sequence $\{{d}_{n}\}$ is decreasing and $\{{f}_{n}\}$ is not ρCauchy, we get $d:={inf}_{n\ge 1}{d}_{n}>0$. Set $\epsilon =\frac{d}{4R}>0$. Let us fix arbitrary $\nu \in (0,R)$. Since ${lim}_{n\to \mathrm{\infty}}\tau ({f}_{n})=R$, there exists ${n}_{0}\ge 1$ such that for any $n\ge {n}_{0}$, we have
Let $n\ge {n}_{0}$. By (3.2), there exists ${i}_{n},{j}_{n}\ge 1$ such that
Using the definition of τ and (3.3), we deduce the existence of $M>0$ such that
and
Hence
for any $t\ge M$. Since ρ is (UUC), there exists ${\eta}_{1}(R,\epsilon )>0$ such that ${\delta}_{1}(R+\nu ,\epsilon )\ge {\eta}_{1}(R,\epsilon )$. Hence
for any $t\ge M$. Hence
Using the definition of R, we get
for any $\nu \in (0,R)$. If we let $\nu \to 0$, we get $R\le R(1{\eta}_{1}(R,\epsilon ))$. This contradiction implies that $\{{f}_{n}\}$ is ρCauchy. Since ${L}_{\rho}$ is ρcomplete, we deduce that $\{{f}_{n}\}$ is ρconvergent as claimed.
In order to finish the proof of (ii), let us show that the ρlimit of $\{{f}_{n}\}$ is independent of the minimizing sequence. Indeed, let $\{{g}_{n}\}$ be another minimizing sequence of τ. The previous proof will show that $\{{g}_{n}\}$ is also ρconvergent. In order to prove that the ρlimits of $\{{f}_{n}\}$ and $\{{g}_{n}\}$ are equal, let us show that ${lim}_{n\to \mathrm{\infty}}\rho ({f}_{n}{g}_{n})=0$. Assume not, i.e., ${lim}_{n\to \mathrm{\infty}}\rho ({f}_{n}{g}_{n})\ne 0$. Without loss of generality, we may assume that there exists $d>0$ such that $\rho ({f}_{n}{g}_{n})\ge d$ for any $n\ge 1$. Set $\epsilon =\frac{d}{2R}>0$. Let $\nu \in (0,R)$. Since ${lim}_{n\to \mathrm{\infty}}\tau ({f}_{n})={lim}_{n\to \mathrm{\infty}}\tau ({g}_{n})=R$, there exists ${n}_{0}\ge 1$ such that for any $n\ge 1$, we have $\tau ({f}_{n})\le R+\frac{\nu}{2}$ and $\tau ({g}_{n})\le R+\frac{\nu}{2}$. Fix $n\ge {n}_{0}$. Then
Using the definition of τ, we deduce the existence of $M>0$ such that
and
Hence
for any $t\ge M$. Since ρ is (UUC), there exists $\eta (R,\epsilon )>0$ such that $\delta (R+\nu ,\epsilon )\ge \eta (R,\epsilon )$ for any $\nu >0$. Hence
for any $t\ge M$. So
Using the definition of R, we get
for any $\nu \in (0,R)$. If we let $\nu \to 0$, we get $R\le R(1\eta (R,\epsilon ))$. This contradiction implies ${lim}_{n\to \mathrm{\infty}}\rho ({f}_{n}{g}_{n})=0$. The Fatou property will finally imply that
where f is the ρlimit of $\{{f}_{n}\}$ and g is the ρlimit of $\{{g}_{n}\}$. Hence $\rho (fg)=0$, i.e., $f=g$. □
Using Lemma 3.2, we are ready to prove our common fixed point result for asymptotic pointwise nonexpansive semigroups.
Theorem 3.2 Assume $\rho \in \mathrm{\Re}$ is (UUC). Let C be a ρclosed, ρbounded convex nonempty subset. Let $\mathcal{F}=\{{T}_{t}:t\ge 0\}$ be an asymptotic pointwise nonexpansive semigroup on C. Then ℱ has a common fixed point, and the set $F(\mathcal{F})$ of common fixed points is ρclosed and convex.
Proof Let us fix $f\in C$ and define the function
Since C is ρbounded, we have $\tau (g)\le {diam}_{\rho}(C)<+\mathrm{\infty}$ for any $g\in C$. Hence ${\tau}_{0}=inf\{\tau (g):g\in C\}$ exists and is finite. For any $n\ge 1$, there exists ${g}_{n}\in C$, such that
Therefore, ${lim}_{n\to \mathrm{\infty}}\tau ({z}_{n})={\tau}_{0}$, i.e., $\{{g}_{n}\}$ is a minimizing sequence for τ. By Lemma 3.2, there exists $g\in C$ such that $\{{g}_{n}\}$ ρconverges to g. Let us now prove that $g\in F(\mathcal{F})$. Note that
for $s,t>0$ and $h\in C$. Using the definition of τ, we get
for any $M>s$, which implies
Since ${lim}_{s\to \mathrm{\infty}}{\alpha}_{s}({g}_{1})=1$, there exists ${s}_{1}>0$ such that for any $s\ge {s}_{1}$, we have ${\alpha}_{s}({g}_{1})<1+1$. Repeating this argument, one will find ${s}_{2}>{s}_{1}+1$ such that for any $s\ge {s}_{2}$, we have ${\alpha}_{s}({g}_{2})<1+\frac{1}{2}$. By induction, we will construct a sequence $\{{s}_{n}\}$ of positive numbers such that ${s}_{n+1}<{s}_{n}+\frac{1}{n}$ and for any $s\ge {s}_{n}$, we have ${\alpha}_{s}({g}_{n})<1+\frac{1}{n}$. Let us fix $t\ge 0$. Then inequality (3.4) will imply
for any $n\ge 1$. In particular, we get that $\{{T}_{{s}_{n}+t}({g}_{n})\}$ is a minimizing sequence of τ. Therefore, Lemma 3.2 implies that $\{{T}_{{s}_{n}+t}({g}_{n})\}$ ρconverges to g for any $t\ge 0$. In particular, we have $\{{T}_{{s}_{n}}({g}_{n})\}$ ρconverges to g. Since
we get $\{{T}_{{s}_{n}+t}({g}_{n})\}$ ρconverges to ${T}_{t}(g)$. Finally, using
we get ${T}_{t}(g)=g$. Since t was arbitrarily positive, we get $g\in F(\mathcal{F})$, i.e., $F(\mathcal{F})$ is not empty. Next, let us prove that $F(\mathcal{F})$ is ρclosed. Let $\{{f}_{n}\}$ in $F(\mathcal{F})$ ρconvergent to f. Since
for any $n\ge 1$ and $s>0$, we get $\{{T}_{s}({f}_{n})\}$ ρconverges to ${T}_{s}(f)$. Since ${f}_{n}\in F(\mathcal{F})$, we get $\{{T}_{s}({f}_{n})\}=\{{f}_{n}\}$. In other words, $\{{f}_{n}\}$ ρconverges to ${T}_{s}(f)$ and f. The uniqueness of the ρlimit implies then that ${T}_{s}(f)=f$ for any $s\ge 0$, i.e., $f\in F(\mathcal{F})$. Therefore, $F(\mathcal{F})$ is ρclosed. Let us finish the proof of Theorem 3.2 by showing that $F(\mathcal{F})$ is convex. It is sufficient to show that
for any $f,g\in F(\mathcal{F})$. Without loss of generality, we will assume that $f\ne g$. Let $s>0$. We have
and
Since $\rho (fh)=\rho (gh)=\rho (\frac{fg}{2})$ and
we conclude that
Similarly, we have
and
Since
we conclude that
Therefore, we have
Lemma 3.1, applied to ${A}_{t}=f{T}_{s}(h)$ and ${B}_{t}={T}_{s}(h)g$, implies that $\rho ({A}_{t}{B}_{t})\to 0$. Hence
Clearly, we will get ${lim}_{s\to \mathrm{\infty}}\rho (h{T}_{s+t}(h))=0$, for any $t\ge 0$. Since
we get ${lim}_{s\to \mathrm{\infty}}\rho ({T}_{t}(h){T}_{s+t}(h))=0$. Finally, using the inequality
by letting $s\to \mathrm{\infty}$, we get ${T}_{t}(h)=h$ for any $t\ge 0$, i.e., $h\in F(\mathcal{F})$. □
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Acknowledgements
This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No. (247006D1433). The authors, therefore, acknowledge with thanks technical and financial support of DSR.
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Bin Dehaish, B.A., Khamsi, M.A. & Kozlowski, W.M. Common fixed points for pointwise Lipschitzian semigroups in modular function spaces. Fixed Point Theory Appl 2013, 214 (2013). https://doi.org/10.1186/168718122013214
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Keywords
 fixed point
 modular function space
 nonexpansive mapping
 Orlicz space
 pointwise Lipschitzian mapping
 pointwise nonexpansive mapping
 semigroup
 uniform convexity