Remark on a Fixed-Point Theorem in the Lebesgue Spaces of Variable Integrability L^p(·)

Abstract

In a personal communication, Prof. Domínguez Benavides noted that a fixed-point theorem for modular nonexpansive mappings in $L^{p(·)}\left(\mathrm{\Omega }\right)$ obtained under the assumptions $p_{+}<\infty$ and the property $\left(R\right)$ satisfied by $\rho$ will force $p_{-}>1$. Therefore, the conclusion is well known. In this note, we establish said conclusion without the assumption $p_{+}<\infty$.

1. Introduction

In 1965, W. Kirk [1] established his celebrated fixed-point theorem for nonexpansive mappings. Specifically, he proved that any nonexpansive map:

$$T:C\to C$$

on a non-empty, closed, convex subset of a reflexive Banach space, which has the normal structure (see below), has a fixed point.

It is worthwhile to mention that in Kirk’s proof, the reflexivity of a Banach space X is used in the following equivalent form (established by Smulian): Every decreasing sequence of non-empty, bounded, closed, and convex subsets of X has a non-empty intersection.

In [2], a modular version of Kirk’s theorem was utilized in order to show a fixed-point property of variable-exponent Lebesgue spaces. Specifically, Theorem 5 in [2] reads as follows (we refer the reader to the body of the paper for the relevant terminology):

Theorem 1. 

Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be a bounded domain, and let $p\in \mathcal{P}\left(\mathrm{\Omega }\right)$; assume that $|{\mathrm{\Omega }}_1|=0$, $p_{+}<\infty$ and that ρ has property $\left(R\right)$. Let C be a non-empty, ρ-bounded, ρ-closed, and convex subset $L^{p(·)}\left(\mathrm{\Omega }\right)$. If a map $T:C\to C$ is ρ nonexpansive, then it has a fixed point.

It was rightly observed by Prof. Domínguez Benavídes that the assumption $p_{+}<\infty$ is equivalent to the ${\Delta }_2$ condition, which in turn, implies that the norm topology and the modular topology on $L^{p(·)}\left(\mathrm{\Omega }\right)$ coincide, from which it follows that the intersection property $\left(R\right)$, alluded to in the statement of Theorem 1 is just the intersection property $\left(R\right)$ for the norm. However, the latter implies the reflexivity of $L^{p(·)}\left(\mathrm{\Omega }\right)$, which is equivalent to $1<p_{-}\le p_{+}<\infty$. Under these conditions, the conclusion of Theorem 1 is already known [3].

In this note, we prove the conclusion of Theorem 1, without the condition $p_{+}<\infty$, to reveal the original modular nature of the result.

2. The Modular Geometry of the Variable-Exponent Lebesgue Spaces

In the interest of clarity, the definition of the variable-exponent Lebesgue spaces is recalled [4].

Definition 1. 

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a domain. As usual, $\mathcal{M}\left(\mathrm{\Omega }\right)$ will stand for the vector space of all real-valued Borel-measurable functions defined on $\mathrm{\Omega }$, and the Lebesgue measure of a subset $A\subset {\mathbb{R}}^n$ will be denoted by $\left|A\right|$. Let $\mathcal{P}\left(\mathrm{\Omega }\right)$ be the subset of $\mathcal{M}$ consisting of functions $p:\mathrm{\Omega }⟶[1,\infty ]$. For each such p, define the sets:

$${\mathrm{\Omega }}_1:=\{t\in \mathrm{\Omega }:p\left(t\right)=1\}and{\mathrm{\Omega }}_{\infty }:=\left\{x\in \mathrm{\Omega }:p\left(x\right)=\infty \right\}.$$

The function $\rho :\mathcal{M}\left(\mathrm{\Omega }\right)⟶[0,\infty ]$, defined by

$$\rho \left(u\right)=\underset{\mathrm{\Omega }\backslash {\mathrm{\Omega }}_{\infty }}{\int }{\left|u\left(x\right)\right|}^{p\left(x\right)}d\mu +\underset{x\in {\mathrm{\Omega }}_{\infty }}{sup}\left|u\left(x\right)\right|,$$

is a convex and continuous modular on $\mathcal{M}\left(\mathrm{\Omega }\right)$ in the sense of Nakano [2]. The associated modular vector space is denoted by $L^{p(·)}\left(\mathrm{\Omega }\right)$.

The first systematic treatment of the variable-exponent Lebesgue class is the work [4]; we refer the reader to [5] for a more recent survey on this topic. These spaces are by no means artificial constructions: their study has intensified due to their fairly recent applications to the hydrodynamics of electrorheological fluids and refined mathematical models used for image restoration, to name only two; see [5] and the references therein.

It has been recently observed that, under very mild conditions on the variable exponent $p(·)$, the modular ${\rho }_p$ is uniformly convex in every direction. More precisely, the following theorem holds:

Theorem 2 

([2,6]). Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be a bounded domain and let $p\in \mathcal{P}\left(\mathrm{\Omega }\right)$. Then the following properties are equivalent:

(a) 

$|{\mathrm{\Omega }}_1|=|{\mathrm{\Omega }}_{\infty }|=0$,

(b) 

The modular ρ is uniformly convex in every direction in the following sense: for any $z_1$ and $z_2$ in $L^{p(·)}\left(\mathrm{\Omega }\right)$ such that $z_1\ne z_2$ and $R>0$, there exists $\Delta (R,z_1,z_2)>0$ such that for any $u\in L^{p(·)}\left(\mathrm{\Omega }\right)$, we have

$$\rho \left(u-\frac{z_1+z_2}{2}\right)\le R(1-\Delta (z_1,z_2,R)),$$

provided $\rho (u-z_1)\le R$ and $\rho (u-z_2)\le R$.

As in the case of Banach spaces, the modular uniform convexity as stated in Theorem 2 implies the modular normal structure property [7]:

Proposition 1. 

Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be a bounded domain, and let $p\in \mathcal{P}\left(\mathrm{\Omega }\right)$. Assume $|{\mathrm{\Omega }}_1|=|{\mathrm{\Omega }}_{\infty }|=0$. Then, for any non-empty ρ-bounded, ρ-closed, and convex subset C of $L^{p(·)}\left(\mathrm{\Omega }\right)$ not reduced to one point, there exists $f\in C$ such that

$$\underset{g\in C}{sup}\rho (f-g)<{\delta }_{\rho }\left(C\right)=sup\{\rho (a-b),a\in C,b\in C\}.$$

This is known as the ρ-normal structure property.

For $p\in \mathcal{P}\left(\mathrm{\Omega }\right)$, set

$$p_{-}:=\underset{t\in \mathrm{\Omega }}{ess\; inf}p\left(t\right)\mathrm{and}{p}_{+}:=\underset{t\in \mathrm{\Omega }}{ess\; sup}p\left(t\right).$$

Clearly, in our setting, $1\le p_{-}\le p_{+}\le \infty .$ In particular, if $p_{-}>1$, a rather stronger form of modular uniform convexity holds for $L^{p(·)}\left(\mathrm{\Omega }\right).$

Theorem 3 

([8]). Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open and $p\in \mathcal{P}\left(\mathrm{\Omega }\right)$. If $|{\mathrm{\Omega }}_{\infty }|=0$ and $p_{-}>1$, then ρ satisfies the following modular uniform convexity property. Set

$$D(r,\epsilon )=\left\{(u,v)\in L^{p(·)}\left(\mathrm{\Omega }\right)\times L^{p(·)}\left(\mathrm{\Omega }\right):\rho \left(u\right)\le r,\rho \left(v\right)\le r,\rho \left(\frac{u-v}{2}\right)\ge \epsilon r\right\}$$

and

$$\delta (r,\epsilon )=inf\left\{1-\frac{1}{r}\rho \left(\frac{u+v}{2}\right):(u,v)\in D(r,\epsilon )\right\}.$$

(If $D(r,\epsilon )=\varnothing$, we define $\delta (r,\epsilon )=1$.) Then, for each $s\ge 0$, $\epsilon >0$, there exists $\eta (s,\epsilon )>0$ such that, for arbitrary $r>s>0$,

$$\delta (r,\epsilon )\ge \eta (s,\epsilon ).$$

For further reference, the following standard definition is recalled:

Definition 2. 

A family ${\left(C_i\right)}_{i\in I}$ of sets is said to have the finite intersection property if, for every finite subset $\{i_1,...i_k\}\subset I$, it holds that $\bigcap _{j=1}^k{C}_{i_j}\ne \varnothing .$

In this regard, the following theorem was proven in [8]:

Theorem 4. 

Assume that $p_{-}>1$. Then, ρ satisfies the strong-$\left(R\right)$ property, i.e., for any $C\subset L^{p(·)}\left(\mathrm{\Omega }\right)$ ρ-closed, ρ-bounded, and convex non-empty subset, then if ${\left(C_i\right)}_{i\in I}\subset 2^C$ is a family of ρ-closed convex subsets of C having the finite intersection property, it necessarily holds that ${\displaystyle \bigcap _{i\in I}}{C}_i\ne \varnothing$.

We will say that $\rho$ satisfies the property $\left(R\right)$ if the conclusion of Theorem 4 holds for countable families, i.e., for any ${\left(C_n\right)}_{n\in \mathbb{N}}$ decreasing sequence of $\rho$-closed, $\rho$-bounded, and convex non-empty subsets, it necessarily holds that ${\displaystyle \bigcap _{n\in \mathbb{N}}}{C}_n\ne \varnothing$.

3. Fixed-Point Theorems for $L^{p(·)}\left(\mathrm{\Omega }\right)$

In this section, the main fixed-point result of this work will be addressed. The following definition is a prerequisite:

Definition 3. 

Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be a bounded domain, and let $p\in \mathcal{P}\left(\mathrm{\Omega }\right)$. Let $\varnothing \ne C\subset L^{p(·)}\left(\mathrm{\Omega }\right)$ and $T:C\to C$ be a mapping. T is said to be ρ-nonexpansive if

$$\rho \left(T\right(x)-T(y\left)\right)\le \rho (x-y),foranyx,y\in C.$$

A point $x\in C$ that satisfies $T\left(x\right)=x$ is said to be a fixed point of T.

The field of the fixed-point theory of maps acting on modular function spaces is vast and deep; the interested reader is referred to [7] for a comprehensive treatment of the subject.

The next result is the modular version of Kirk’s celebrated fixed-point theorem [1]. The proof is constructive and was first used in the Banach-space setting by Kirk [9] and relaxes the compactness assumption in the above theorem. The main ingredient in Kirk’s constructive proof is a technical lemma due to Gillespie and Williams [10].

Theorem 5. 

Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be a bounded domain, and let $p\in \mathcal{P}\left(\mathrm{\Omega }\right)$; assume that $|{\mathrm{\Omega }}_1|=0$, $|{\mathrm{\Omega }}_{\infty }|=0$ and that ρ has property $\left(R\right)$. Let $\varnothing \ne C\subset L^{p(·)}\left(\mathrm{\Omega }\right)$ be ρ-bounded, ρ-closed, and convex. If a map $T:C\to C$ is ρ nonexpansive, then it has a fixed point.

Proof. 

Let $\mathcal{F}$ be the family of non-empty $\rho$-closed and convex subsets of C, which are T-invariant. The family $\mathcal{F}$ is not empty since $C\in \mathcal{F}$. Define $\tilde{\delta }:\mathcal{F}\to [0,+\infty )$ by

$$\tilde{\delta }\left(D\right)=inf\{{\delta }_{\rho }\left(B\right):B\in \mathcal{F}\mathrm{and}B\subset D\}.$$

Set $D_1=C$. By the definition of $\tilde{\delta }\left(D_1\right)$, there exists $D_2\in \mathcal{F}$ such that $D_2\subset D_1$ and ${\delta }_{\rho }\left(D_2\right)<\tilde{\delta }\left(D_1\right)+1$. Assume $D_1,D_2,\cdots ,D_n$, for $n\ge 1$, are constructed. Again by the definition of $\tilde{\delta }\left(D_n\right)$, there exists $D_{n+1}\in \mathcal{F}$ such that ${\delta }_{\rho }\left(D_{n+1}\right)<\tilde{\delta }\left(D_n\right)+\frac{1}{n}$ and $D_{n+1}\subset D_n$. The property $\left(R\right)$ implies $D_{\infty }={\displaystyle \bigcap _{n\ge 1}}{D}_n$ is not empty. Clearly, it holds that $D_{\infty }\in \mathcal{F}$. Assume that $D_{\infty }$ contains more than one point. Using Proposition 1, one derives the existence of $f_0\in D_{\infty }$ such that

$$r=\underset{g\in D_{\infty }}{sup}\rho (f_0-g)<{\delta }_{\rho }\left(D_{\infty }\right)=sup\{\rho (a-b),a\in D_{\infty },b\in D_{\infty }\}.$$

Hence, the set

$$D=\bigcap _{g\in D_{\infty }}{B}_{\rho }(g,r)\cap D_{\infty }$$

is a non-empty, $\rho$-closed and convex subset of $D_{\infty }$. Note that there is no reason for D to be T-invariant, i.e., $T\left(D\right)\subset D$. Consider the family ${\mathcal{F}}^{*}=\{M\in \mathcal{F}:D\subset M\}$. Obviously, ${\mathcal{F}}^{*}$ is not an empty since $C\in {\mathcal{F}}^{*}$. Set $L={\displaystyle \bigcap _{M\in {\mathcal{F}}^{*}}}M$. The set L is a non-empty, $\rho$-closed, and convex subset of C, which is T-invariant. Consider $B=D\cup T\left(L\right)$, and observe that ${\overline{conv}}_{\rho }\left(B\right)=L$ (where ${\overline{conv}}_{\rho }\left(B\right)$ is the intersection of all $\rho$-closed, convex subsets, which contain B). Indeed, since L contains D and is T-invariant, it is readily concluded that $B\subset L$. Since L is $\rho$-closed and convex, it follows that ${\overline{conv}}_{\rho }\left(B\right)\subset L$, whence

$$T\left({\overline{conv}}_{\rho }\left(B\right)\right)\subset T\left(L\right)\subset B\subset {\overline{conv}}_{\rho }\left(B\right).$$

Hence ${\overline{conv}}_{\rho }\left(B\right)\in {\mathcal{F}}^{*}$ and $L\subset {\overline{conv}}_{\rho }\left(B\right)$. This implies the desired equality $L={\overline{conv}}_{\rho }\left(B\right)$. Define $D^{*}={\displaystyle \bigcap _{g\in L}}{B}_{\rho }(g,r)\cap L$. Observe that $D^{*}$ is non-empty since it contains D (by the definition of D and $D_{\infty }\in {\mathcal{F}}^{*}$) and is a $\rho$-closed, convex subset of C. On the other hand, it is clear that ${\delta }_{\rho }\left(D^{*}\right)\le r$. Note that $D^{*}$ is T-invariant. Indeed, let $f\in D^{*}$. It is clear by the definition of $D^{*}$ that $L\subset B_{\rho }(f,r)$. Since T is $\rho$-nonexpansive, one has $T\left(L\right)\subset B_{\rho }(T\left(f\right),r)$. For any $g\in D$, it holds $L\subset B_{\rho }(g,r)$. However, $T\left(f\right)\in L$, so $T\left(f\right)\in B_{\rho }(g,r)$, which implies $g\in B_{\rho }(T\left(f\right),r)$. Hence, $D\subset B_{\rho }(T\left(f\right),r)$ holds. Since $B=D\cup T\left(L\right)$, it follows that $B\subset B_{\rho }(T\left(f\right),r)$. Therefore, one must have

$${\overline{conv}}_{\rho }\left(B\right)=L\subset B_{\rho }(T\left(f\right),r).$$

By the definition of $D^{*}$, it follows that $T\left(f\right)\in D^{*}$. In other words, $D^{*}$ is T-invariant. Since $L\subset D_{\infty }$ one has $D^{*}\subset D_{\infty }$. Therefore, the above construction yields $D^{*}\in \mathcal{F}$ and $D^{*}\subset D_{\infty }$ such that ${\delta }_{\rho }\left(D^{*}\right)\le r$. Since $D^{*}\subset D_n$, it is clear that

$${\delta }_{\rho }\left(D^{*}\right)\le {\delta }_{\rho }\left(D_{\infty }\right)\le {\delta }_{\rho }\left(D_{n+1}\right)\le \tilde{\delta }\left(D_n\right)+\frac{1}{n}\le {\delta }_{\rho }\left(D^{*}\right)+\frac{1}{n},$$

for any $n\ge 1$. Letting now $n\to \infty$, it is readily seen that ${\delta }_{\rho }\left(D^{*}\right)={\delta }_{\rho }\left(D_{\infty }\right)$, which implies ${\delta }_{\rho }\left(D_{\infty }\right)\le r$. This is in contradiction with the inequality $r<{\delta }_{\rho }\left(D_{\infty }\right)$. Hence, $D_{\infty }$ must consist of exactly one point, which is a fixed point of T since $D_{\infty }$ is T-invariant. □

Note that the conclusion of Theorem 5 requires only the validity of the intersection property $\left(R\right)$ for $\rho$ on C and not on the entire space $L^{p(·)}\left(\mathrm{\Omega }\right)$.

Remark 1. 

The proof of Theorem 5 can be significantly simplified in case ρ satisfies the so-called strong-$\left(R\right)$ property alluded to in Theorem 4.

Indeed, consider the family of non-empty, ρ-closed, and convex subsets of C, which are T-invariant $\mathcal{F}$, introduced in the previous proof. It is clear that $\mathcal{F}\ne \varnothing$, since $C\in \mathcal{F}$. The strong-$\left(R\right)$ property combined with Zorn’s lemma immediately yields the existence of a minimal element in $\mathcal{F}$. Let K be one such minimal element. It will be shown that K consists of exactly one point. First, notice that, since $T\left(K\right)\subset K$, it necessarily holds that $T\left({\overline{conv}}_{\rho }\left(T\left(K\right)\right)\right)\subset T\left(K\right)\subset {\overline{conv}}_{\rho }\left(T\left(K\right)\right)$. The minimality of K forces ${\overline{conv}}_{\rho }\left(T\left(K\right)\right)=K$. Fix $f_0\in K$; set $r=\underset{g\in K}{sup}\rho (f_0-g)$; define

$$K_r=\left\{f\in K;r_{\rho }\left(f\right)=\underset{g\in K}{sup}\rho (f-g)\le r\right\}.$$

Note that $K_r$ is a non-empty, ρ-closed, and convex subset of K ($f_0\in K_r$). $K_r$ is T-invariant. To see this, let $f\in K_r$, and observe that $K\subset B_{\rho }(f,r)$. Since T is ρ-nonexpansive, one must have $T\left(K\right)\subset B_{\rho }(T\left(f\right),r)$, which implies

$$K={\overline{conv}}_{\rho }\left(T\left(K\right)\right)\subset B_{\rho }(T\left(f\right),r).$$

Hence, $T\left(f\right)\in K_r$. Since $K_r$ is a T-invariant subset of K, it follows that $K=K_r$. This clearly implies $r=\underset{g\in K}{sup}\rho (f_0-g)={\delta }_{\rho }\left(K\right)$, which only holds for subsets that consist of exactly one point, on account of the ρ-normal structure property. In other words, T has a fixed point, as claimed.

Corollary 1. 

Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be a bounded domain, and let $p\in \mathcal{P}\left(\mathrm{\Omega }\right)$. Assume that $|{\mathrm{\Omega }}_1|=0$, $|{\mathrm{\Omega }}_{\infty }|=0$ and $p_{-}>1$. Let $C\subset L^{p(·)}\left(\mathrm{\Omega }\right)$ be a non-empty ρ-bounded, ρ-closed, and convex subset. If a map $T:C\to C$ is ρ-nonexpansive, then it has a fixed point.

Proof. 

If $p_{-}>1$, Theorem 4 asserts that $\rho$ satisfies the strong-$\left(R\right)$ property. The proof follows directly from the above remark. □