# UserLibri: A Dataset for ASR Personalization Using Only Text

###### Abstract

Personalization of speech models on mobile devices (on-device personalization)
is an active area of research, but more often than not, mobile devices have
more text-only data than paired audio-text data. We explore training a
personalized language model on text-only data, used during inference to
improve speech recognition performance for that user. We experiment on a user-
clustered LibriSpeech corpus, supplemented with personalized text-only data
for each user from Project Gutenberg. We release this User-Specific
LibriSpeech (UserLibri) dataset to aid future personalization research.
LibriSpeech audio-transcript pairs are grouped into 55 users from the test-
clean dataset and 52 users from test-other. We are able to lower the average
word error rate per user across both sets in streaming and nonstreaming
models, including an improvement of 2.5 for the harder set of test-other users
when streaming.

Index Terms: speech recognition, personalization, language modeling

##  1 Introduction

End-to-end (E2E) automatic speech recognition (ASR) systems can now run
inference entirely on-device, thereby having important implications for
latency, reliability and privacy [1, 2]. The models themselves, which are
typically trained server-side on large amounts of audio data, can perform
quite well during inference on-device if the target domain is a reasonable
match to the training data. However, they may perform poorly on rare words, or
in applications where there is not a wealth of similar labeled training data
available server-side. New research techniques are being developed to address
this, such as by leveraging an external language model (LM) [3].

On-device learning techniques such as Federated Learning (FL) and
personalization (p13n111In this paper, we use “p13n” to mean both
“personalization” and “personalized”, as in “p13n LM”.) can train models on
the actual user data on-device without ever sending the raw user data to a
central server. While a few active users of speech recognition services on
mobile devices may have significant amounts of audio data on their device
($\approx 40\%$ of the U.S. population uses a digital voice assistant at least
once a month according to Statista [4]), users tend to interact more with
their mobile devices using virtual keyboards rather than speech recognition.
Further, typical supervised training of ASR models requires ground-truth text
labels for each utterance, which may not be available on mobile devices to use
for p13n. However, by utilizing more of the on-device text data to learn user-
preferred words or phrases, we can enable the on-device ASR models to possibly
recognize them correctly the first time without access to any audio training
examples containing that information.

In this work, we first generate a simulation dataset of ~100 users, based on
the existing LibriSpeech dataset. We then explore the use of shallow fusion
with personalized LMs to improve average word error rate (WER) per user on
this data.

##  2 Our Contributions

We present the UserLibri
dataset222https://www.kaggle.com/datasets/google/userlibri, a re-formatting of
LibriSpeech containing paired audio-transcripts and extra text-only data, that
can be useful in a variety of experiments. We hope that this dataset will help
futher personalization research as to our knowledge there is no existing open-
source dataset containing both user-specific audio and additional text-only
personalized data. We report our initial experiments leveraging this dataset,
which aim to improve WER of an ASR system via shallow fusion with personalized
(p13n11footnotemark: 1) LMs. Our top findings are:

  * •

p13n LM fusion can greatly improve WER, especially for certain users, and is
better than non-p13n LM fusion,

  * •

streaming ASR models, which perform worse than non-streaming models, see
greater absolute WER improvements with p13n LM fusion than non-streaming,

  * •

larger LMs perform better than smaller ones,

  * •

the number of LM training examples per user affects fusion performance, but
even fine tuning a p13n LM on only 500 examples per user can beat the
baseline,

  * •

p13n speech models can be combined with p13n LM fusion for even better
results.

Background and related work motivating our research is in Section 3. Dataset
creation details are in Section 4, followed by our models and experiments in
Section 5. We also include some specific wins and losses in Section 5.5.

##  3 Background

Machine learning models frequently have to handle the question of bias, where
model performance on certain classes of data or certain users may be much
worse than others, even despite excellent metrics on large test datasets [5,
6]. Previous work has improved ASR performance for underrepresented classes of
users via ASR p13n, by fine tuning a speaker-independent base model on a
single speaker’s audio examples [7, 8].

One specific type of data that can be difficult for natural language models is
less common proper names or other tail words, which may not occur frequently
in the training data and occur disproportionately depending on the user [9].
Previous work has used shallow fusion [10] to leverage larger text corpora and
combined it with biasing techniques that can target improvements for contexts
frequently followed by proper nouns, such as “call” and “message” [11], or
even without such context constraints [12]. Recent work has specifically
explored personalizing end-to-end ASR models on-device by incorporating a
biased LM on-the-fly, or by model fine-tuning on synthesized speech [13].
These approaches only require the target proper nouns in text form, possibly
in a user-provided list. By contrast, we aim to improve ASR performance via
personalization of an RNN LM requiring no active input from the user at all,
and no real or synthesized user-specific audio data.

There are several ways of using LMs to improve ASR. One can interpolate the
scores with the ASR model score, use them during the beam search, or use the
LM as a rescoring model after the beam search. In this work, we choose to
always interpolate the language model scores prior to the beam search, which
is known as shallow fusion [10].

Since one of the advantages of shallow fusion is the ability to leverage a
much larger text dataset to improve an ASR model, most literature explores
fusing with large LMs trained on the order of a million examples [14].
However, in the on-device LM p13n setting, both training data and model size
are much more limited. Streaming models are preferred in the on-device setting
due to their low latency at inference, despite their generally worse
performance compared to nonstreaming models, which can analyze the entire
input sequence before deciding the output. Recent work showed that shallow
fusion of an external RNN LM can be successful with streaming models, again
training on hundreds of millions or even billions of examples [15].

Related work in personalizing LMs has often focused on the dilemma of
mismatched domains and sparseness of personalized text data, and incorporates
adaptation or transfer learning techniques during p13n [16]. While our
simulation datasets all draw from matching text domains, the work in this
paper introduces a new challenge of evaluating p13n LMs not on text-only
tasks, where optimizing LM perplexity may suffice, but on ASR. To our
knowledge, there is limited literature discussing this topic in the realm of
p13n, although there are many works on optimizing or adapting LMs for speech
tasks [17, 18, 19].

##  4 User Specific LibriSpeech Dataset

LibriSpeech [20] is a widely-used dataset containing 970 hours of paired
audio-transcript data, which are recordings of various speakers reading aloud
from Project Gutenberg e-books [21]. The Project Gutenberg (PG) raw book text
data which served as the source for these recordings can also be downloaded
from the LibriSpeech resources at https://www.openslr.org/12. The raw metadata
files are also available which map audio examples to the corresponding PG
source book texts.

The audio recordings are based on only a subset of text from each book, and
the rest of the book text will contain similar vocabulary, character names,
and writing style as the recordings. Therefore, we can create a dataset for a
”user” containing the original paired audio-transcripts from a book as well as
additional text-only data from the remainder of the book, which will match the
user’s domain. This multi-user dataset can be used for a variety of
personalization applications beyond shallow fusion with personalized LMs, such
as data augmentation studies using Text-to-Speech to create more audio
training examples for a user [22]. While the LibriSpeech audio data is stored
by speaker ID and chapter ID, we combine audio examples from chapters of the
same book, read by the same speaker, into a single “user” dataset. This
results in more data per user, with an average of 52 audio examples per user
(see more in Table 1).

Table 1: UserLibri dataset utterance & p13n LM text metadata Metadata | Test-Other | Test-Clean
---|---|---
$\\#$ Users | $52$ | $55$
Audio-Transcript Utts |  |
Avg. $\\#$ per User | $56.5$ | $47.1$
Median $\\#$ per User | $52$ | $46$
$\\#$ Users with $\geq$10 | $50$ | $52$
Max $\\#$ for $1$ User | $144$ | $108$
LM Train Text Sentences |  |
Total $\\#$ | $444,520$ | $377,049$
Avg. $\\#$ per User | $8,548$ | $6,855$
Median $\\#$ per User | $5,299$ | $3,750$
$\\#$ Users with $\geq$3k | $43$ | $38$
Max $\\#$ for $1$ User | $38,498$ | $54,306$

To process the raw PG book text files into LM train sets of sentences, the
text file encodings are standardized and boilerplate is removed. Then, the
book text is broken into sentences, ignoring newlines mid-sentence that exist
for readability in the raw data. For each sentence, we remove word-leading or
trailing punctuation and sentence-leading or trailing whitespace, and
uppercase the ascii characters. We then discard any sentences containing a
sequence of $80\%$ of the sentence tokens where the same sequence can be found
in the test audio examples which came from the same book.

In the UserLibri dataset, we only consider the users found in the LibriSpeech
test-clean (considered easier for ASR) and test-other (considered harder/more
noisy for ASR) datasets, as this allows us to use the LibriSpeech train and
dev sets to train the ASR model that we use for our fusion experiments without
training on any of the test speakers’ data. We note that although
LibriSpeech’s train, dev, and test audio sets have no overlap in speakers,
there are a few speakers in each set who read chapters from the same book, so
the ASR model does see some text snippets from these books, read by different
speakers. See Appendix A for further details on the dataset generation
process.

While we focus on the English LibriSpeech here, the Multilingual LibriSpeech
could be similarly processed in the future.

##  5 Experiments

We use the UserLibri dataset to explore whether shallow fusion with p13n LMs
can improve per-user WER, as well as the effects of fusing with streaming ASR
models vs. nonstreaming, and using different LM sizes trained on various
amounts of data. We then combine p13n LM fusion with a p13n ASR model.

###  5.1 Models

We train 86M parameter Conformer Hybrid Autoregressive Transducer (HAT) models
[23] on the 960 hours of audio train data in LibriSpeech. The audio data is
preprocessed similarly to [24], by extracting 80-channel filterbanks features
computed from a 25ms window with a stride of 10ms. We use SpecAugment [25]
with mask parameter $(F=27)$, and ten time masks with maximum time-mask ratio
($p_{S}$ = 0.05), where the maximum-size of the time mask is set to $p_{S}$
times the length of the utterance. Our models consist of 12 encoder layers
with dimension 512, 4 attention heads, a convolution kernel size of 32, and a
HAT decoder like in [23] with a single RNN layer of dimension 640. Each label
is embedded with 128 dimensions, and inputs are tokenized with a 1k Word-Piece
Model (WPM) trained on the LibriSpeech text-only data. The models are trained
with Adam [26] and use GroupNorm [27].

We train two Conformer HAT models, one with streaming where the right context
is 0 (no lookahead) which uses causal convolution and local self attention,
and one nonstreaming with multi-headed attention. To make sure that the
encoder is streaming, we remove all the sub-architecture components which were
benefiting from right context. Namely, we remove the convolution sub-sampling
layer and also force the stacking layers to only stack within the left
context.

For both models, we always use time-synchronous beam search during decoding,
resulting in slightly different baseline WERs than similar studies like [24],
which uses label-synchronous beam search and a right context of 1 frame.

We experiment with three RNN LM sizes with LSTM cells, all using the same 1024
WPM as the Conformer. The 3M parameter model uses 192 embeddings and 1 RNN
layer of size 670. The 10M and 25M models use 384 embeddings and an RNN of
size 1340, with 1 and 2 layers respectively. We train general non-p13n LMs on
the LibriSpeech LM text data (40M examples)333See Appendix B for alternative
general LM results. using Adam with a learning rate of 1e-4, and batch size
4096. We then fine tune all weights for 1k steps on the user’s p13n LM data,
with batch size 32.

###  5.2 Room for Improvement with Personalization

We ran hyperparameter sweeps over the ASR models above to select the best
top-K and beam width for the beam search, and the best smoothing temperature
for the baseline models without shallow fusion (baseline-1 or BL1).
Additionally, we swept over the HAT decoding hyperparameters from [23] \- the
internal LM weight, the external LM weight, and internal LM smoothing
temperature - for the same models fused with a 10M general LM, trained on the
entire LibriSpeech LM text corpus (BL2).

Table 2: Baseline WERs, for test-clean (CL) and test-other (OT), without p13n LM fusion. Average WERs per user (with 95$\%$ CIs) are higher than the WER on the full set of users. Model | Set | Full Set | Average WER
---|---|---|---
|  | WER | per User
Streaming |  |  |
No Fusion (BL1) | CL | $5.8$ |  $6.0$ $[5.3,~{}~{}~{}6.7]$
| OT | $10.4$ |  $11.2$ $[9.6,~{}13.0]$
Gen. 10M LM (BL2) | CL | $5.3$ | $\mathbf{5.4~{}[4.8,~{}~{}6.0]}$
| OT | $9.0$ | $\mathbf{9.7~{}[8.3,11.1]}$
Nonstreaming |  |  |
No Fusion (BL1) | CL | $2.4$ |  $2.5$ $[2.1,~{}~{}2.8]$
| OT | $5.8$ |  $6.8$ $[5.6,~{}~{}8.1]$
Gen. 10M LM (BL2) | CL | $2.1$ | $\mathbf{2.1~{}[1.8,~{}2.5]}$
| OT | $5.0$ | $\mathbf{5.8~{}[4.8,~{}6.9]}$
Figure 1: Histogram of per-user baseline WERs (no fusion). Baseline WER for
the full test-clean is shown as a solid vertical line, and for test-other as a
dotted vertical line. Many users have much higher baseline WERs, and p13n may
help.

With the best hyperparameters for the streaming and the nonstreaming models,
both with and without shallow fusion using this non-p13n LM, we get the WERs
reported in Table 2. 95$\%$ confidence intervals (CIs) for averages throughout
the paper are calculated using the bootstrap technique [28], sampling the per-
user WERs with replacement 10k times.

Our models achieve similar results on the full LibriSpeech test-other and
test-clean sets compared to other studies such as [24] (note the differences
as described in Section 5.1). However, average per-user WERs are worse than
WERs of the full set, especially in the test-other users, and even in the
nonstreaming model despite its generally lower WERs. We hope to improve
performance for the high-WER users (Figure 1).

Fusion with the general LM (BL2) improves performance, but still shows higher
per-user WER averages than on the full sets. We will now describe how further
per-user improvements can be made using p13n LMs.

###  5.3 Results

Table 3: Average WERs and 95$\%$ CIs from fusing with general (Gen) and p13n LMs of different sizes, for streaming (Str.) and nonstreaming (Nonstr.). BL1 is the baseline with no fusion. Model | Test-Clean | Test-Other | All Users
---|---|---|---
Str. |  |  |
BL1 |  $6.0$ $[5.3,~{}6.7]$ |  $11.2$ $[9.6,~{}13.0]$ |  $8.5$ $[7.5,~{}9.6]$
3M Gen |  $5.8$ $[5.1,~{}6.4]$ |  $10.5$ $[9.0,~{}12.0]$ |  $8.1$ $[7.2,~{}9.0]$
3M p13n |  $5.6$ $[4.9,~{}6.2]$ |  $10.0$ $[8.4,~{}11.6]$ |  $7.7$ $[6.8,~{}8.7]$
10M Gen | $5.4~{}[4.8,~{}6.0]$ |  $9.7$ $[8.3,~{}11.1]$ |  $7.5$ $[6.7,~{}8.4]$
10M p13n |  $5.4$ $[4.8,~{}6.1]$ | $9.4~{}[8.0,~{}10.9]$ | $7.4~{}[6.5,~{}8.3]$
25M Gen | $\mathbf{5.2~{}[4.7,5.9]}$ |  $9.1$ $[7.8,~{}10.4]$ |  $7.1$ $[6.3,~{}8.0]$
25M p13n | $\mathbf{5.2~{}[4.5,5.9]}$ | $\mathbf{8.7~{}[7.4,10.2]}$ | $\mathbf{6.9~{}[6.1,7.8]}$
Nonstr. |  |  |
BL1 |  $2.5$ $[2.1,~{}2.8]$ |  $6.8$ $[5.6,~{}8.1]$ |  $4.5$ $[3.8,~{}5.4]$
3M Gen |  $2.3$ $[2.1,~{}2.7]$ |  $6.4$ $[5.3,~{}7.6]$ |  $4.3$ $[3.6,~{}5.1]$
3M p13n |  $2.0$ $[1.7,~{}2.4]$ |  $5.8$ $[4.8,~{}7.0]$ |  $3.9$ $[3.2,~{}4.5]$
10M Gen |  $2.1$ $[1.8,~{}2.5]$ |  $5.8$ $[4.8,~{}6.9]$ |  $3.9$ $[3.3,~{}4.6]$
10M p13n | $\mathbf{1.9~{}[1.6,2.3]}$ |  $5.3$ $[4.3,~{}6.4]$ |  $3.6$ $[3.0,~{}4.2]$
25M Gen |  $2.0$ $[1.7,~{}2.3]$ |  $5.5$ $[4.6,~{}6.6]$ |  $3.7$ $[3.2,~{}4.4]$
25M p13n | $\mathbf{1.9~{}[1.6,2.3]}$ | $\mathbf{4.6~{}[3.8,5.6]}$ | $\mathbf{3.2~{}[2.7,3.8]}$

We sweep external LM weights of [0.15, 0.22, 0.36, 0.45, 0.55] for p13n LM
shallow fusion with both the streaming and nonstreaming ASR models and notice
that values above 0.22 quickly perform worse than the baseline, and for most
users, 0.15 performs better than 0.22. We fuse using LM weight 0.15 for all
users and leave the question of p13n acceptance criteria, or if/how to fuse
per user, for future work. Results can be found in Table 3, with p13n LMs
outperforming the general LMs in almost every category. While the streaming
model shows larger effects, even the nonstreaming model benefits from p13n LM
fusion. Fusing with the 3M model already surpasses the baselines, with greater
gains as model size increases. Per-User WER histograms are included in
Appendix C.

We also experiment with fine tuning the p13n LMs on varying amounts of user LM
examples: train sets of 200, 500, 1k, and 3k examples per user (see Appendix
D), which we compared to fine tuning on the full amount of LM data we have
available for that user, only for the 81 users that have at least 3k examples
total. In Figure 2, we see that training with 3k examples performs almost as
well as using all available data, and we can beat the baselines as long as we
personalize on at least 500 examples for streaming models and 1k for
nonstreaming.

Figure 2: Average WER per user (and 95$\%$ CI) for p13n 10M LM fusion on
limited user LM train sets, for Test-Clean, Test-Other and all users combined,
for the 81 users with at least 3k LM train examples. Horizontal dotted lines
show the baseline WERs (no fusion) for each group.

###  5.4 Interaction with Speech Personalization

As mentioned in Section 3, speech p13n by fine tuning full or partial models
on-device can greatly improve the recall of proper nouns such as named
entities [13]. We briefly try combining speech p13n and p13n LM fusion444We
don’t sweep any hyperparameters for these speech experiments..

We fine tune the joint layer of the streaming ASR model for each user via
5-fold cross validation, for the 102 users with at least 10 audio-transcript
pairs, and average the per-fold WERs. Speech p13n alone improves upon the
baseline, with non-p13n speech + p13n LM fusion performing a little better.
Combining the two approaches gets the best results (Table 4).

Table 4: Average WERs from personalizing the streaming ASR model on each user’s test utterances via 5-fold cross validation, with and without p13n LM fusion. No p13n means no speech p13n or p13n LM fusion (BL1). Only the 102 users with at least 10 audio-transcript pairs are included. Reports $95\%$ CIs. Model | Test-Clean | Test-Other | All Users
---|---|---|---
No p13n |  $6.0$ $[5.3,~{}6.6]$ |  $11.3$ $[9.6,~{}13.1]$ |  $8.6$ $[7.6,~{}9.7]$
+10M LM |  $5.6$ $[5.0,~{}6.3]$ |  $9.3$ $[7.9,~{}10.8]$ |  $7.4$ $[6.6,~{}8.3]$
p13n ASR |  $5.7$ $[5.1,~{}6.3]$ |  $10.4$ $[9.0,~{}12.1]$ |  $8.0$ $[7.1,~{}9.0]$
+10M LM | $\mathbf{5.4~{}[4.8,6.0]}$ | $\mathbf{8.7~{}[7.5,10.0]}$ | $\mathbf{7.0~{}[6.2,7.8]}$

###  5.5 Wins and Losses

If we examine specific cases where the p13n LM is able to correctly predict
examples that the baseline or general LM predicted incorrectly (Table 5), we
see that proper nouns seen more frequently in the p13n LM training data can be
a source of the win, but there are also some losses due to over-predicting
words seen in the p13n LM data. Combining speech and LM p13n may allow wins on
tail words while avoiding losses on small common words (Table 6). See Appendix
E for more.

Table 5: Example predictions. p13n LM fusion can win (W) on names, but may lose (L) if a test word rarely appears in the rest of the p13n LM data, but a similar word appears many times. W/L | Baseline | General | P13n | Count in
---|---|---|---|---
|  | LM | LM | p13n data
W | king | king | king | sharkan: 0
| sharkan | sharkan | sharrkan | sharrkan: 336
W | mardock | murdock | murdoch | m[a/u]rdock: 0
| blinked | blinked | blinked | murdoch: 92
W | thanks | thanks | thanks | is he: 72
| is he | is he | izzy | izzy: 155
L | on the | on the | on the | navel: 0
| navel | navel | naval | naval: 2
L |  tied to |  tied to |  tide to | tied: 0
| a woman | a woman | a woman | tide: 4
Table 6: Predictions on one utterance for speech p13n combined with LM p13n, which may allow similar tail word wins from Table 5 while avoiding losses on smaller common words. Experiment | Prediction
---|---
Baseline |  ten years farewell vintage is none
General LM |  ten years farewell vintage is done
p13n LM only |  panniers farewell vintage is none
Speech p13n only |  panniers farewell vintage is none
Speech & LM p13n |  panniers farewell vintage is done

##  6 Conclusions

We share our new personalization simulation dataset, UserLibri, based on
LibriSpeech and useful for a variety of personalization research. We
demonstrate a use case where we are able to improve average per-user WER in
both streaming and nonstreaming models, bringing down streaming WER on the
hardest group of users by 1.2 by performing shallow fusion with personalized
LMs of only 3M parameters, or an improvement of 2.5 with 25M LMs. We show that
the technique can still work with limited p13n LM examples and can be combined
with speech p13n for even better results.

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UserLibri Appendix

Interspeech 2022 Submission, Breiner et. al.

##  Appendix A Processing the Raw Book Data

While Project Gutenberg (PG) is a highly useful and well organized data
resource, there are some inconsistencies that come along with such a large
crowd-sourced dataset that we had to handle during the UserLibri dataset
generation. We were only processing the 99 books that were included as
transcript sources for the LibriSpeech test-clean and test-other data as
mentioned in Section 4, so we only know of the fixes we had to make for that
small subset of the PG corpus.

The PG book texts can be downloaded from the LibriSpeech resources site at
https://www.openslr.org/12. Most of the book text files are named with their
book ID number as referenced in the BOOKS.txt metadata file that is provided,
but some book texts are not named that way, although they do appear in a
directory correctly named with the book ID. To ease processing, we first
locally renamed all of the book texts to match the directory name.

We then had to standardize the file encodings. Most of the PG book text files
are encoded in ascii format, with some encoded in UTF-8 which appear in a
separate directory. However, some of the files are encoded in Windows-1252
encoding and were not readable by our pipelines until we first converted to
UTF-8. Namely, we noticed this problem with the following book IDs: 19019,
3436, 3440, 3441.

Within each book text, there were also various boilerplate conventions giving
the metadata about PG, the book, the licensing information or other non-book-
content information, which we did not want to include in our LM train
examples. We found the majority to use variations on simple lines such as
“***START OF THIS PROJECT GUTENBERG EBOOK” and “*** END OF THIS PROJECT
GUTENBERG EBOOK” to demarcate the beginning and end of the actual book
content, with small differences such as “THE” instead of “THIS” or varying
numbers of asterisks. A minority of the books used similar boilerplate
although in lowercase, or with a different demarcation for the end of the
book, while a couple had no boilerplate at all. We were able to leverage
regular expressions to remove the boilerplate for our 99 target books at
scale.

We note in Section 4 that while the speakers are properly unique across
LibriSpeech train, dev and test set splits and only appear in one of these
splits, the book text sources are not kept separate in this way. One book
often serves as the source of transcripts for more than one speaker, and the
different speakers may appear in different data splits. Specifically, 87 books
are read by different speakers to create audio data in both the train and test
sets, 66 books in both dev and test, and 98 books in both train and dev. This
should not have a large effect on our experiments, but it does mean that a
character name or other tail words seen in a user’s p13n LM data (pulled from
LibriSpeech test) may still occur in the LibriSpeech audio and LM examples
used to train the ASR model and the General LM. However, these long tail words
would appear rarely in the ASR and General LM train data, and
disproportionately frequently in the p13n LM data and the test utterances,
which is why we are still able to see gains with p13n LM fusion.

##  Appendix B Additional General LM Fusion Experiments

In our main paper, we only report two baselines for fusion without p13n LMs,
BL1 with no fusion and BL2 where we fuse with a 10M general LM trained on all
LibriSpeech LM data (Table 2). We explored a third baseline as well, where we
trained a general 10M LM on only the LM train data from our 107 users that
were generated in Section 4, combined into a single set (650k examples). While
there are a few books that serve as the p13n data for more than one user and
thus appear multiple times in the full UserLibri set, these examples are only
included once when training this smaller LM. Training an LM on this set
parallels the type of LM we might be able to train if we have no central in-
domain data and use a technique such as FL to train only on the text found on-
device. We call this BL3 and compare results in Table 7.

Table 7: Baseline WERs, for test-clean (CL) and test-other (OT), without p13n LM fusion. Average WERs per user, with 95$\%$ confidence intervals (CIs), are higher in almost every case than the WER on the full set. Model | Set | Full Set | Average WER
---|---|---|---
|  | WER | per User
Streaming |  |  |
No Fusion (BL1) | CL | $5.8$ |  $6.0$ $[5.3,~{}~{}~{}6.7]$
| OT | $10.4$ |  $11.2$ $[9.6,~{}13.0]$
General 10M LM (BL2) | CL | $5.3$ | $\mathbf{5.4~{}[4.8,~{}~{}6.0]}$
| OT | $9.0$ | $\mathbf{9.7~{}[8.3,11.1]}$
User-Only 10M LM (BL3) | CL | $5.5$ |  $5.7$ $[5.0,~{}~{}~{}6.3]$
| OT | $9.4$ |  $10.2$ $[8.7,~{}11.9]$
Nonstreaming |  |  |
No Fusion (BL1) | CL | $2.4$ |  $2.5$ $[2.1,~{}~{}2.8]$
| OT | $5.8$ |  $6.8$ $[5.6,~{}~{}8.1]$
General 10M LM (BL2) | CL | $2.1$ | $\mathbf{2.1~{}[1.8,~{}2.5]}$
| OT | $5.0$ | $\mathbf{5.8~{}[4.8,~{}6.9]}$
User-Only 10M LM (BL3) | CL | $2.4$ |  $2.4$ $[2.0,~{}~{}2.8]$
| OT | $4.9$ | $\mathbf{5.8~{}[4.8,~{}7.0]}$

Fusion with the general LM (BL2) gives best improvements overall, but even
fusing with the LM which was only trained on the far smaller set of user p13n
text data (BL3) improves upon the baselines, especially when streaming.
However, all baselines still show higher per-user WER averages than on the
full set of combined users. For the remainder of our experiments, we chose to
focus on personalizing the general LM which trained on the full dataset (BL2)
as its baseline performance is best, and leave personalizing a model trained
only with the union of user-specific text examples to future work.

##  Appendix C Per-User WER Histograms

To compare to Figure 1 showing the baseline per-user WER distributions, some
additional histograms showing the per-user WERs for various experiments are
shown in Figures 3, 4, and 5. The WER on the full sets of combined users are
shown as vertical lines, solid for WER on test-clean and dotted for WER on
test-other. We see the mass of each distribution, and particularly the high
WER outliers on the right end of the x-axis, shift left from General LM fusion
to p13n LM fusion, but also as we increase the model size from 3M to 10M to
25M parameters.

Figure 3: Histograms of per-user WERs with 3M General and p13n LM fusion.

Figure 4: Histograms of per-user WERs with 10M General and p13n LM fusion.

Figure 5: Histograms of per-user WERs with 25M General and p13n LM fusion.

##  Appendix D Limited Training Examples Experiments

In order to conduct the limited training examples experiments, we created new,
separate LM datasets containing only 200, 500, 1k or 3k examples for each
user. Each of these train sets is a strict subset of the next size set, so for
example the 500 example set is created by taking the same 200 example set and
adding 300 more randomly selected examples from the full p13n LM training data
set for that user. This ensures that the experiments are more directly
comparable and depend less on having included lucky examples in certain of the
sets but not others.

##  Appendix E Additional Wins and Losses

In Table 8 we show a larger set of examples building on Table 5, where p13n
LMs are able to improve upon the baseline and General LM experiments, for
example with long tail proper names, but may overfit to smaller common words
in the p13n LM dataset and also cause some losses. In some examples, such as
“horse sense”, there are roughly equal occurrences in the p13n LM data of both
the ground truth word and the predicted word, which may be why the p13n LM did
not help there. However, for proper names the p13n still often is able to beat
the baseline and general LM.

In Table 9 we show a few more examples not included in the main paper to
compare p13n LM fusion with and without speech p13n. The wins that the speech
p13n and combined speech p13n & LM p13n see are mostly not on proper nouns,
but often allow recovery from predictions that were correct with General LM
fusion, incorrect with p13n LM fusion, but are then correct with the
combination speech p13n and p13n LM fusion. Combining these approaches may
allow the p13n LM fusion to show wins on proper nouns elsewhere in the user
test data but not over-predict incorrect common words from the p13n LM data.

Table 8: Predictions per approach. p13n LM fusion is able to get wins (W) on proper names, but may lose (L) if a word only appears once in a test example and a similar word appears many times in the p13n LM data. W/L | Baseline | General | P13n | Count in
---|---|---|---|---
|  | LM | LM | p13n data
W | king | king | king | sharkan: 0
| sharkan | sharkan | sharrkan | sharrkan: 336
W | lord of | lord of | lord of | bagdad: 0
| bagdad | bagdad | baghdad | baghdad: 71
W | mister | mister | mister | bell: 3
| bell | bell | beale | beale: 340
W | mardock | murdock | murdoch | m[a/u]rdock: 0
| blinked | blinked | blinked | murdoch: 92
W | thanks | thanks | thanks | is he: 72
| is he | is he | izzy | izzy: 155
W |  a peck of |  a peck of |  a bag | peck: 0
| fields | fields | eels | bag: 9
L | on the | on the | on the | navel: 0
| navel | navel | naval | naval: 2
L |  tied to |  tied to |  tide to | tied: 0
| a woman | a woman | a woman | tide: 4
L | nearer | nearer | nearer | virtuous: 15
| virtuous | virtuous | virtues | virtues: 24
L | poor | horse | poor | poor: 7
| sense | sense | cents | horse: 9
Table 9: Additional predictions using speech p13n with and without LM p13n, which may allow similar tail word wins from Table 8 while avoiding losses on smaller common words. Experiment | Prediction
---|---
Baseline |  ten years farewell vintage is none
General LM |  ten years farewell vintage is done
p13n LM only |  panniers farewell vintage is none
Speech p13n only |  panniers farewell vintage is none
Speech & LM p13n |  panniers farewell vintage is done
Baseline |  but this our banishment utterly
General LM |  but this our banishment utterly
p13n LM only |  but this our vanity utterly
Speech p13n only |  but this shall banish it utterly
Speech & LM p13n |  but this shall banish it utterly
Baseline |  wars that have been raged
General LM |  wars that have been waged
p13n LM only |  wars that have been raged
Speech p13n only |  wars that have been waged
Speech & LM p13n |  wars that have been waged
Baseline |  he does never made man moral
General LM |  it has never made man moral
p13n LM only |  it is never made man moral
Speech p13n only |  it has never made man moral
Speech & LM p13n |  it has never made man moral
Baseline |  her legion has been cried
General LM |  religion has been tried
p13n LM only |  religion has been cried
Speech p13n only |  her legion has been tried
Speech & LM p13n |  religion has been tried


# Orbital Lipschitzian mappings and semigroup actions on metric spaces

Rafael Espínola, Maria Japón, Daniel Souza  Dedicated to the memory of
Professors K. Goebel and W. A. Kirk for all the moments, teachings and love
they gave us.  R. Espínola Departamento de Análisis Matemático, Universidad de
Sevilla, Spain<EMAIL_ADDRESS>M. Japón Departamento de Análisis Matemático,
Universidad de Sevilla, Spain<EMAIL_ADDRESS>D. Souza Departamento de Análisis
Matemático, Universidad de Sevilla, Spain<EMAIL_ADDRESS>
###### Abstract.

In this paper we study some results on common fixed points of families of
mappings on metric spaces by imposing orbit Lipschitzian conditions on them.
These orbit Lipschitzian conditions are weaker than asking the mappings to be
Lipschitzian in the traditional way. We provide new results under the two
classic approaches in the theory of fixed points for uniformly Lipschitzian
mappings: the one under the normal structure property of the space (which can
be regarded as the Cassini-Maluta’s approach) and the one after the Lifschitz
characteristic of the metric space (Lifschitz’s approach). Although we focus
on the case of semigroup of mappings, our results are new even when a mapping
is considered by itself.

###### Key words and phrases:

Fixed points, actions of semigroups, metric spaces, uniform Lipschitzian
mappings, Lifschitz constant, uniform normal structure, orbit-nonexpansive
mappings, orbit Lipschitzian actions.

###### 2010 Mathematics Subject Classification:

47H09, 47H10, 47H20, 54E40

The authors were partially supported by Ministerio de Ciencia e Innovación,
Grant PGC2018-098474-B-C21, and Junta de Andalucía Grants P20-00637,
US-1380969 and FQM-127.

##  1\. Introduction

Given a metric space $(X,d)$, a mapping $T:X\to X$ is said to be uniformly
$k$-Lipschitzian if there exists some $k>0$ such that

$d(T^{n}x,T^{n}y)\leq kd(x,y)$

for all $x,y\in X$ and for all $n\in\mathbb{N}$. Uniformly Lipschitzian
mappings were introduced by K. Goebel and W. A. Kirk in 1973 in their seminal
work [13] (see also Chapter 16 in [12]). As a result, they proposed a
generalization of nonexpansive mappings (which is the case when $k=1$ in the
above definition) and for which fixed point results were still possible under
adequate geometrical conditions on the space. Since then, many authors have
worked on this class of mappings providing a very fruitful line of research
for decades (see, for instance, [3, 4, 7, 14, 15, 16, 19, 20, 23, 24, 25] and
references therein).

Inspired by the notion of orbit-nonexpansivity [1, 10, 22], in this paper we
introduce some orbit uniformly Lipschitzian conditions which we use to revisit
the two main approaches for obtaining fixed points for uniformly Lipschitzian
mappings: Lifschitz’s and Casini-Maluta’s approaches. In our setting, the
Lipschitz condition is not evaluated for every pair of points $x$ and $y$ in
$X$ in terms of their distance, but we consider orbital assumptions instead
(see Section 3 for definitions), which result in weaker requirements for the
mappings. The operators here considered are very far from being continuous as
it is shown with several examples along the work. Additionally, we will
consider semigroups acting over a metric space with the purpose of finding a
common fixed point for all the operators involved. Lifschitz and Casini-Maluta
Theorems, and many others in the literature, will be now obtained as
particular cases of the results.

Our work is organized as follows: in the Preliminaries section, we provide
some details on the history of fixed points for uniformly Lipschitzian
mappings as well as we introduce the notation and definitions that will be
used along the paper. Additionally, Lifschitz and Casini-Maluta approaches
will be fully stated. In Section $3$ we introduce actions by semigroups over a
metric space considering the case of a single mapping as a particular example.
We define orbit Lipschitzian actions as the natural extension of orbit
nonexpansivity for uniformly Lipschitzian mappings. Also, a strong version of
this notion is introduced that will be of very much importance for us.
Nevertheless, we show that both notions coincide in a wide class of
semigroups. In our two main sections, Sections $4$ and $5$, we focus on
extending Lifschitz and Casini-Maluta Theorems, respectively, in metric spaces
to provide common fixed points for semigroups of mappings satisfying our orbit
Lipschitzian conditions. We also display examples that illustrate the
applicability of our Lipschitz condition on the orbits as well as the
optimality of our results.

##  2\. Preliminaries

We start this section by recalling some well-known definitions and fixed point
theorems for uniformly Lipschitzian mappings. Throughout all the article, by
$X$ and $B(x,r)$ we will denote an arbitrary metric space and the closed ball
centered at $x\in X$ with radius $r\in(0,+\infty)$, respectively.

The starting point for this work are uniformly Lipschitzian mappings, which
have already been introduced in the previous section. In the seminal paper
[13], K. Goebel and W. A. Kirk proved a fixed point theorem for uniformly
$k$-Lipschitzian mappings in the framework of uniformly convex Banach spaces
for which $k(1-\delta(1/k))<1$, where $\delta(\cdot)$ denotes the modulus of
uniform convexity of $X$. For the particular case of a Hilbert space, Goebel-
Kirk result provides fixed point for uniformly $k$-Lipschitzian mappings with
$k<5^{1/2}/2$ (see [12, Chapter 16]). A sharper estimation for $k$, still
providing the existence of fixed points for uniformly $k$-Lipschitzian
mappings in Hilbert spaces, was obtained by Lifschitz in [19]. By using the
geometric characteristic $\kappa(X)$ introduced in the same work, Lifschitz
proved as a consequence of its main result that every uniformly
$k$-Lipschitzian self-mapping defined on closed convex bounded subset of a
Hilbert space has a fixed point whenever $k<2^{1/2}$. In general, the best
constant $k$ in Hilbert spaces for which the existence of fixed points for
uniformly $k$-Lipschitzian mappings can be assured is unknown, although this
constant has to be less that $\pi/2$ by Baillon’s example [12, Example 16.1]).

Lifschitz Theorem was proved in the broader context of general metric spaces
by using the notion of regular balls that we recall next.

######  Definition 2.1.

Balls in a metric space $X$ are said to be $c$-regular, for $c\geq 1$, if the
following holds: For any $k<c$ there are numbers $\mu,\alpha\in(0,1)$ such
that for any $x,y\in X$ and $r>0$ with $d(x,y)\geq(1-\mu)r$, there exists a
$z\in X$ such that

$B(x,(1+\mu)r)\cap B(y,k(1+\mu)r)\subset B(z,\alpha r).$

The Lifschitz characteristic of $X$ is finally defined as

$\kappa(X)=\sup\\{c\geq 1:\mbox{ the balls in $X$ are $c$-regular}\\}.$

######  Remark 2.2.

It happens that $\kappa(X)\in[1,2]$ for any metric space $X$. In particular,
extreme values are reached as $\kappa(X)=1$ for
$X=({\mathbb{R}}^{2},\|\cdot\|_{\rm max})$ and $\kappa(X)=2$ for $X$ the real
line with the usual metric. For a Hilbert space $X$, $\kappa(X)=2^{1/2}$. In
the framework of Banach spaces, it holds that $\kappa(X)>1$ if and only if
$\epsilon_{0}(X)$, the characteristic of convexity of $X$, is smaller than $1$
(see, for instance, [8, Theorem 5]). For more on Lifschitz constant, check [4,
8] or [12, Chapter 16]. In particular, if $X$ is a complete CAT$(0)$ space,
then $\kappa(X)\geq\sqrt{2}$ with $\kappa(X)=2$ if $X$ is an $\mathbb{R}$-tree
(see [4, Theorem 5]). Remember that $\mathbb{R}$-trees are a particular
instance of CAT$(0)$ spaces.

The following result is the main one in [19] and it is the one regarded as
Lifschitz Theorem (see also [12, Theorem 16.2]).

######  Theorem 2.3.

Let $X$ be a bounded complete metric space and $T:X\to X$ a uniformly
$k$-Lipschitzian mapping. If $k<\kappa(X)$, then $T$ has a fixed point.

For a nonempty subset $C$ of a metric space $X$, we recall the following
notation:

$\displaystyle D(x,C)=$ $\displaystyle\sup\\{d(x,y)\;:\;y\in C\\},\;\;x\in X;$
$\displaystyle r(C)=$ $\displaystyle\inf\\{D(x,C)\;:\;x\in C\\};$
$\displaystyle\delta(C)=$ $\displaystyle\sup\\{d(x,y)\;:\;x,y\in C\\};$

Note that $D(x,C)$ is also denoted by $r_{x}(C)$ in some other references. We
prefer $D(x,C)$ since it makes definitions and proofs simpler (see [9]).

If $X$ is a Banach space, the normal structure coefficient $N(X)$ is defined
as the supremum of the quotients $r(C)/\delta(C)$ when $C$ is a closed convex
bounded subset of $X$ with positive diameter. It is said that $X$ has uniform
normal structure when $N(X)<1$. It is well known, for instance, that $N(X)<1$
implies that the Banach space is reflexive. Now we can state Casini-Maluta
Theorem which was originally proved in [3].

######  Theorem 2.4.

Let $X$ be a Banach space with uniform normal structure. Then, for every
closed convex bounded subset $C$ of $X$, any $T:C\to C$ which is uniform
$k$-Lipschitzian with $k<{N}(X)^{-1/2}$ has a fixed point.

In their paper, E. Casini and E. Maluta showed some examples of Banach spaces
where their approach by using the coefficient $N(X)$ provides better
estimations for obtaining fixed points than the Lifschitz’s characteristic
$\kappa(X)$. Therefore, Lifschitz Theorem and Casini-Maluta Theorem are not
related in general, both providing a different spectrum when it comes to their
possible applications.

Casini-Maluta Theorem was later extended to metric spaces by T. C. Lim and H.
K. Xu in [20]. In order to do that, the notion of uniform normal structure was
redefined replacing the family of closed convex bounded subsets by the family
of admissible subsets of a metric space $X$. Recall that a subset $A$ of $X$
is said to be admissible if $A$ is a nonempty intersection of closed balls of
$X$. We denote by $\mathcal{A}(X)$ the family of all admissible subsets of
$X$. Admissible sets are a natural substitute for convex sets in the lack of
linear structure of the underlying space. In fact, every admissible set in a
Banach space is additionally closed convex and bounded. Also, for a nonempty
subset $C$ of a metric space $X$, we denote

$\displaystyle{\rm cov}(C)=$ $\displaystyle\cap\\{B\;:\;B\text{ is a closed
ball and $C\subseteq B$}\\},$

where ${\rm cov}(C)$ is known as the admissible cover or envelope of $C$ in
$X$. For a general metric space $X$, the (metric) normal structure coefficient
$\tilde{N}(X)$ is defined as follows:

$\tilde{N}(X):=\sup\left\\{\frac{r(A)}{\delta(A)}\right\\},$

where the supremum is taken over all admissible subsets $A$ of $X$ with
positive diameter. The metric space $X$ is said to have (metric) uniform
normal structure when $\tilde{N}(X)<1$.

In a similar way as reflexivity was implied by uniform normal structure in
Banach spaces, uniform normal structure implies that the family of admissible
sets $\mathcal{A}(X)$ is compact [17, Theorem 5.4]. This means that the
intersection of any collection of elements of $\mathcal{A}(X)$ is nonempty
provided that it satisfies the finite intersection property.

T. C. Lim and H. K. Xu would still require an extra assumption for their
counterpart of Casini-Maluta Theorem: the so-called property (P) for
sequences.

######  Definition 2.5.

A metric space $X$ is said to have property (P) for sequences if given any two
bounded sequences $\\{x_{n}\\}$ and $\\{z_{n}\\}$ in $X$, such that
$z_{n}\in{\rm cov}(\\{x_{m}:m\geq n\\})$, there is $z\in\bigcap_{n}{\rm
cov}(\\{z_{m}\;:\;m\geq n\\})$ such that

$\limsup_{n}d(z,x_{n})\leq\limsup_{m}\limsup_{n}d(z_{m},x_{n}).$

Then, Lim-Xu Theorem reads as follows.

######  Theorem 2.6.

Let $X$ be a bounded metric space with uniform normal structure and property
(P). Let $T:X\to X$ be a uniformly $k$-Lipschitzian mapping. If
$k<\tilde{N}(X)^{-1/2}$, then $T$ has a fixed point.

######  Remark 2.7.

Property (P) can be understood as a replacement in metric spaces of the weakly
lower semicontinuity for the type functions in Banach spaces. In fact,
property (P) basically gives a formal condition for Theorem 2.4 to hold when
convex weakly compact domains are no longer available (see Section $5$ where
property (P) is also required). It is a general fact that convex weakly
compact sets have property (P).

##  3\. Orbit Lipschitzian actions

With the purpose of obtaining a common fixed point for a family of mappings
acting on a same metric space, we are going to consider actions defined by
semigroups. Therefore, we recall some semigroup notions:

A semigroup $\mathcal{S}$ is a set with an inner product, $\cdot$, that is
associative. Additionally, if the semigroup contains an element $1\in S$ such
that $s\cdot 1=1\cdot s=s$ for all $s\in\mathcal{S}$ and for every
$s\in\mathcal{S}$ there is $s^{-1}\in\mathcal{S}$ with $s\cdot
s^{-1}=s^{-1}\cdot s=1$, then $\mathcal{S}$ is called a group. Trivially,
every group is a semigroup. Following standard semigroup notation (see for
instance the monographs [2, 21]), we denote by $\mathcal{S}^{1}$ the semigroup
$\mathcal{S}$ with an identity adjoined, that is
$\mathcal{S}^{1}:=\mathcal{S}\cup\\{1\\}$ with $s\cdot 1=1\cdot s=s$ for all
$s\in\mathcal{S}$. If $\mathcal{S}$ already has an identity, then
$\mathcal{S}^{1}=\mathcal{S}$.

Every family of mappings generates a semigroup under the composition operation
and a common fixed point for all the operators of the family is also a common
fixed point for all the elements of the corresponding semigroup. Because of
that, it is said that a semigroup $\mathcal{S}$ generates an action over a
space $X$ when there exists a mapping from ${\mathcal{S}}\times X$ to $X$ such
that $(s,x)\to s(x)\in X$, verifying $s\cdot t(x)=s(t(x))$ for all
$s,t\in\mathcal{S}$ and $x\in X$. As it is standard, we will denote $s(x)$
simply by $sx$ and by $st$ the product $s\cdot t$ for every
$s,t\in\mathcal{S}$ and $x\in X$.

In what follows, we will denote by $\mathcal{S}$ a given semigroup and we will
denote by $(\mathcal{S},X)$ an action generated by $\mathcal{S}$ over a metric
space $(X,d)$. The orbit of $x\in X$ under the action $(\mathcal{S},X)$ is
defined by

$o(x)=\\{x\\}\cup\\{sx:s\in\mathcal{S}\\}=\\{sx:s\in\mathcal{S}^{1}\\}.$

######  Definition 3.1.

An action $({\mathcal{S}},X)$ is said to be $k$-Lipschitzian for $k>0$ if

$d(sx,sy)\leq kd(x,y)$

for all $x,y\in X$ and $s\in{\mathcal{S}}$.

######  Remark 3.2.

Taking into account that a single mapping $T:X\to X$ generates the semigroup
$\mathcal{S}=\\{T^{n},n\geq 1\\}$ and the semigroup action $\mathcal{S}\times
X\to X$, $(T^{n},x)\to T^{n}x\in X$, the definitions of $T$ being uniformly
$k$-Lipschitzian and that $\mathcal{S}:=\\{T^{n}:n\geq 1\\}$ being a
$k$-Lipschitzian action coincide.

When a single mapping $T:X\to X$ is considered, the orbit of $x\in X$ under
$T$ is just the orbit of $x$ under the action $\\{T^{n}:n\geq 1\\}$.

The following definition was introduced by A. Nicolae in [22] and widely
studied by the authors in [10] for metric spaces under metric normal structure
assumptions:

######  Definition 3.3.

A mapping $T:X\to X$ is said to be orbit-nonexpansive if

$d(Tx,Ty)\leq D(x,o(y))$

for all $x,y\in X$.

The following concept naturally extends the notion of uniformly Lipschitzian
mappings and encompasses orbit-nonexpansive operators as a particular case:

######  Definition 3.4.

A mapping $T:X\to X$ is said to be orbit uniformly $k$-Lipschitzian  if there
is some $k\in(0,+\infty)$ such that

$d(T^{n}x,T^{n}y)\leq kD(x,o(y))$

for all $x,y\in X$ and $n\in\mathbb{N}$.

It is easy to prove that every orbit-nonexpansive mapping is orbit uniformly
$1$-Lipschitzian. We next show an example of a noncontinuous orbit uniformly
$k$-Lipschitzian mapping. The proof follows the same arguments as those of
[10, Example 2.2].

######  Example 3.5.

Let $0<a<1$, $a\in\mathbb{Q}$ and $S_{a}:[-1,1]\to[-1,1]$ given by

$S_{a}(x)=\left\\{\begin{array}[]{ll}\ \ {ax}&\mbox{if $x$ is irrational}\\\
{-ax}&\mbox{ if $x$ is rational}\\\ \end{array}\right.$

Then $S_{a}$ is orbit uniform $k$-Lipschitzian for $k=3a$.

Being orbit uniformly $k$-Lipschitzian can also be considered on the action of
a semigroup as the next definition states.

######  Definition 3.6.

An action $({\mathcal{S}},X)$ is said to be orbit $k$-Lipschitzian if

$d(sx,sy)\leq kD(x,o(y))$

for all $x,y\in X$ and $s\in\mathcal{S}$.

The following assertions are now straightforward:

######  Lemma 3.7.

  1. (1)

Let $T\colon X\to X$ be uniformly $k$-Lipschitzian. Then $T$ is orbit
uniformly $k$-Lipschitzian.

  2. (2)

Let $({\mathcal{S}},X)$ be a $k$-Lipschitzian action. Then $(\mathcal{S},X)$
is orbit $k$-Lipschitzian.

We introduce next one more orbit Lipschitzian condition on a family of
mappings that will play a main role in our exposition.

######  Definition 3.8.

An action $(\mathcal{S},X)$ is said to be strong-orbit $k$-Lipschitzian  if

(3.1) $D(sx,o(sy))\leq kD(x,o(y)).$

for all $s\in\mathcal{S}$ and $x,y\in X$.

It is clear that a strong-orbit $k$-Lipschitzian action is an orbit
$k$-Lipschitzian action too since $d(sx,sy)\leq D(sx,o(sy))$ for all
$s\in\mathcal{S}$ and $x,y\in X$. We will next check that, for a wide range of
semigroups, both notions are in fact equivalent. The next result explores the
case of a single mapping.

######  Lemma 3.9.

Assume that $T\colon X\to X$ is a uniformly $k$-Lipschitzian mapping or, more
generally, an orbit uniformly $k$-Lipschitzian mapping. Then the semigroup
${\mathcal{S}}=\\{T^{n}\colon n\in{\mathbb{N}}\\}$ satisfies the strong-orbit
$k$-Lipschitzian condition.

###### Proof.

Fix $n\in\mathbb{N}$. Then for all $m\in\mathbb{N}\cup\\{0\\}:$

$d(T^{n}x,T^{m+n}y)\leq kd(x,T^{m}y)\leq kD(x,o(y))$

which concludes that for all $m\in\mathbb{N}$:

$D(T^{n}x,o(T^{n}y))\leq kD(x,o(y))$

for all $(x,y)\in X\times X$. ∎

Both notions will also be equivalent of some classes of semigroups of
mappings.

######  Lemma 3.10.

Assume that $\mathcal{S}$ is a semigroup with the property that for all
$s\in\mathcal{S}$ it must be the case that

$\mathcal{S}s\subset s\mathcal{S}.$

Then, if $\mathcal{S}$ is an orbit $k$-Lipschitzian action, it is a strong-
orbit $k$-Lipschitzian action too. In particular, this holds whenever
$\mathcal{S}$ is a group or a commutative semigroup.

###### Proof.

Let $s\in\mathcal{S}$ and $x,y\in X$. Take $p\in\mathcal{S}$. Since
$ps\in\mathcal{S}s\subset s\mathcal{S}$, there exists $q\in\mathcal{S}$ such
that $ps=sq$. Then

$d(sx,psy)=d(sx,qsy)\leq kD(x,o(sy))\leq kD(x,o(y)).$

Taking supremum with respect to $p\in\mathcal{S}^{1}$, we finally obtain that

$D(x,o(sy))\leq kD(x,o(y)).$

∎

######  Remark 3.11.

The condition ${\mathcal{S}}s\subset s{\mathcal{S}}$ for all
$s\in{\mathcal{S}}$ is verified for larger families of semigroups others than
commutative ones and groups as, for instance, the left (right) quasi
commutative semigroups (a semigroup is called left (right) quasi commutative
if for every $a,b\in\mathcal{S}$, there exists a positive integer $r$ such
that $ab=b^{r}a$ ($ab=ba^{r}$)). Additionally, if $\mathcal{S}$ is a normal
semigroup of a group ($t^{-1}st\in\mathcal{S}$ for all $s\in\mathcal{S}$) the
above condition also holds. $\mathcal{H}$-commutative semigroups also fall
within the the scope of Lemma 3.10 (see [21, Chapter 5]).

We close this section with a last remark about the boundedness of orbits.

######  Remark 3.12.

In case that a semigroup action $\mathcal{S}$ provides unbounded orbits for
every $x\in X$, it is clear that there is no fixed point and that the action
provided by $\mathcal{S}$ trivially satisfies all orbit conditions above
considered. Nevertheless, in the next two sections we will show common fixed
point results as long as adequate conditions are given and there exists an
orbit which is bounded.

##  4\. Strong orbital Lipschitzian actions and the Lifschitz characteristic
$\kappa(X)$

In this section we study the existence of common fixed points for strong-orbit
$k$-Lipschitzian actions in terms of the Lifschitz characteristic $\kappa(X)$
of the metric space. In fact, we will consider a weaker Lipschitzian condition
on the orbits of the action than this one. This is given by the next result
which is the main one in this section.

######  Theorem 4.1.

Let $X$ be a complete metric space and $\mathcal{S}$ a semigroup. Assume that
$({\mathcal{S}},X)$ is an action verifying the following condition: There
exists $k<\kappa(X)$ such that:

(4.1) $\begin{array}[]{c}\mbox{ For $(x,y)\in X\times X$ with $D(x,o(y))\leq
D(x,o(x))$ and $\forall s\in\mathcal{S}$:}\\\ \\\
\inf_{t\in\mathcal{S}^{1}}D(sx,o(tsy))\leq kD(x,o(y)).\end{array}$

Then either all the orbits are unbounded or there exists some $x\in X$ such
that $sx=x$ for all $s\in\mathcal{S}$.

###### Proof.

Take some $k_{0}\in(k,\kappa(X))$.

Assume that there is $z\in X$ for which the orbit is bounded. For $x\in X$,
define

$r(x):=\inf\\{D(x,o(y)):y\in X\\}$

which, from our assumption, gives a nonnegative finite number.

Firstly, let us check that $x\in X$ is a common fixed point for $\mathcal{S}$
whenever $r(x)=0$. Fix $s\in\mathcal{S}$.

Let $\epsilon>0$ be arbitrary. We want to show that $d(x,s(x))\leq\epsilon$
for any $s\in\mathcal{S}$. Take $y\in X$ such that $\displaystyle
D(x,o(y))\leq{\epsilon\over 1+k_{0}}$. If $D(x,o(x))<D(x,o(y))$, since
$D(x,o(y))<\epsilon$, we are done. Otherwise, we can consider condition (4.1)
and so there is some $t_{0}\in\mathcal{S}$ such that

$D(sx,o(t_{0}sy))\leq k_{0}D(x,o(y)).$

Therefore

$\begin{array}[]{lll}d(x,sx)&\leq&d(x,t_{0}sy)+d(sx,t_{0}sy))\leq
D\left(x,o(y)\right)+D\left(sx,o(t_{0}sy)\right)\\\
&\leq&D\left(x,o(y)\right)+k_{0}D\left(x,o(y)\right)\\\
&\leq&(1+k_{0})D\left(x,o(y)\right)\leq\epsilon,\end{array}$

as we wanted to prove.

We can assume now, with no loss of generality, that there exists $x\in X$ such
that $r(x)\in(0,+\infty)$. We will show that there exists $p\in X$ with
$r(p)=0$ and so our proof will be complete. Set $r:=r(x)>0$ and consider
$\mu,\alpha\in(0,1)$ as in Definition 2.1. In particular, $r\leq D(x,o(x))$,
which implies that there exists some $s_{0}\in\mathcal{S}$ with

$(1-\mu)r<d(x,s_{0}x).$

Take $z\in X$ given by Definition 2.1 such that

(4.2) $B(x,(1+\mu)r)\cap B(s_{0}x,k_{0}(1+\mu)r)\subset B(z,\alpha r).$

Note that $z$ depends on the point $x$. We next prove that $r(z)\leq\alpha r$.
We split the proof in two cases:

Case 1: $D(x,o(x))<r(1+\mu)$.

From condition (4.1), take $t_{0}\in\mathcal{S}$ verifying

$D(s_{0}x,o(t_{0}s_{0}x))\leq k_{0}D(x,o(x))<k_{0}r(1+\mu).$

In particular, $o(t_{0}s_{0}x)\subset o(x)\subset B(x,r(1+\mu))$.
Additionally, the above inequalities imply $o(t_{0}s_{0}x)\subset
B(s_{0}x,k_{0}(1+\mu)r)$. From $(\ref{1})$ we deduce that
$o(t_{0}s_{0}x)\subset B(z,\alpha r)$ and $r(z)\leq\alpha r$. Additionally,

$d(z,x)\leq d(z,t_{0}s_{0}x)+d(t_{0}s_{0}x,x)\leq\alpha r+r(1+\mu)=Ar$

where $A:=\alpha+1+\mu$.

We next prove that $D(z,o(z))<(1+k_{0})\alpha r$:

Again we consider two subcases:

Case 1.1: Assume $D(z,o(z))\leq D(z,o(t_{0}s_{0}x))$, which directly implies
that

$D(z,o(z))<\alpha r<(1+k_{0})\alpha r.$

Case 1.2: Assume otherwise that $D(z,o(t_{0}s_{0}x))\leq D(z,o(z))$. Then fix
any $t\in\mathcal{S}$. According to condition (4.1) applied to the points $z$
and $t_{0}s_{0}x$, there is some $t_{1}\in\mathcal{S}$ with

$D(tz,o(t_{1}tt_{0}s_{0}x))\leq k_{0}D(z,o(t_{0}s_{0}x)).$

Therefore,

$\begin{array}[]{lll}d(z,tz)&\leq&d(z,t_{1}tt_{0}s_{0}x)+d(tz,t_{1}tt_{0}s_{0}x)\\\
&\leq&D(z,o(t_{0}s_{0}x))+D(tz,o(t_{1}tt_{0}s_{0}x))\\\
&\leq&D(z,o(t_{0}s_{0}x))+k_{0}D(z,o(t_{0}s_{0}x))\\\ &\leq&(1+k_{0})\alpha
r.\\\ \end{array}$

Taking supremum when $t\in\mathcal{S}$, we finally deduce

(4.3) $D(z,o(z))\leq(1+k_{0})\alpha r.$

Case 2: Assume now that Case 1 fails so $r(1+\mu)\leq D(x,o(x)).$ Now we can
select some $y\in X$ verifying

$D(x,o(y))<r(1+\mu)\leq D(x,o(x)).$

According to condition (4.1), we can choose $t_{1}\in\mathcal{S}$ such that

$D(s_{0}x,o(t_{1}s_{0}y))\leq k_{0}D(x,o(y))\leq k_{0}(1+\mu)r.$

Note that $o(t_{1}s_{0}y)\subset o(y)\subset B(x,(1+\mu)r)$ and

$o(t_{1}s_{0}y)\subset B(s_{0}x,k_{0}(1+\mu)r).$

Together with (4.2), we deduce that $o(t_{1}s_{0}y)\subset B(z,\alpha r)$ and

$r(z)\leq\alpha r.$

Besides,

$d(x,z)\leq d(x,t_{1}s_{0}y)+d(t_{1}s_{0}y,z)\leq Ar.$

Likewise, replacing $x$ by $y$ in a previous argument, we can deduce that
$D(z,o(z))\leq(1+k)\alpha r.$

In any case, we have found $z\in X$ such that $r(z)\leq\alpha r$, $d(x,z)\leq
Ar$ and $D(z,o(z))\leq(1+k)\alpha r.$

From this moment on, we will proceed following some standard arguments in
order to find a Cauchy sequence whose limit $p$ verifies $r(p)=0$ and
therefore is necessarily a common fixed point for the action. Let us call
$x_{0}:=x$ and take $z=z(x)$, the point $z$ depending on $x$ as above. Then,
we define $x_{1}:=z$. If $r(x_{1})=0$ we make $x_{1}=p$ and we are done,
otherwise we can reason as above and choose $x_{2}=z(x_{1})$. Following this
inductive argument, we construct a sequence $(x_{n})\subset X$ verifying the
following properties:

  * i)

$r(x_{n})\leq\alpha r(x_{n-1})$

  * ii)

$d(x_{n},x_{n-1})\leq Ar(x_{n-1})$.

  * iii)

$D(x_{n},o(x_{n}))\leq(1+k)\alpha r(x_{n-1})$

for all $n\in\mathbb{N}$.

Properties i) and ii) imply that $(x_{n})$ is a Cauchy sequence in $X$ and, by
completeness, there is $p\in X$ such that $\lim_{n}d(x_{n},p)=0$. Note that

$D(p,o(x_{n}))\leq d(p,x_{n})+D(x_{n},o(x_{n})).$

Therefore $D(p,o(x_{n}))$ tends to $0$ as $n$ goes to infinity. This finally
shows that $r(p)=0$ and so $p$ is a common fixed point for the action
$\mathcal{S}$ on $X$, which completes the proof. ∎

Since it is direct to check that being a strong-orbit $k$-Lipschitzian action
implies that the action satisfies condition (4.1), the next corollary is
immediate.

######  Corollary 4.2.

Let $X$ be a complete metric space and $\mathcal{S}$ a semigroup. Assume that
$({\mathcal{S}},X)$ is a strong-orbit $k$-Lipschitzian action with
$k<\kappa(X)$. Then either all the orbits are unbounded or there exists some
$x\in X$ such that $sx=x$ for all $s\in\mathcal{S}$.

The next example compares some of the Lipschitzian conditions studied in this
work.

######  Example 4.3.

For $X=[0,1]$ define the mapping $T(x)=x^{2}$ if $x\in[0,1)$ and $T(1)=0$. We
prove that the semigroup $\\{T^{n}:n\in\mathbb{N}\\}$ satisfies condition
(4.1) for $k=1$ while it fails to be a Lipschitzian and orbit uniformly
$k$-Lipschitzian action for any $k<2$.

###### Proof.

Notice that for all $x\in[0,1]$, $0\in\overline{o(x)}$. This implies that
$D(x,o(x))=x$ and $D(x,o(y))=\max\\{x,y-x\\}$.

Let $y\in[0,1)$ with $D(x,o(y))\leq D(x,o(x))$. This implies that
$D(x,o(y))=\max\\{x,y-x\\}\leq D(x,o(x))=x$ and $D(x,o(y))=x$. Fix
$n\in\mathbb{N}$,

$\begin{array}[]{lll}\inf_{m\in\mathbb{N}}D(T^{n}x,o(T^{m}T^{n}y))&=&\inf_{m\geq
n}D(T^{n}x,o(T^{m}y))\\\ &=&\inf_{m\geq n}\max\\{x^{2n},y^{2m}-x^{2n}\\}\\\
&\leq&x^{2n}\leq x=D(x,o(y)).\\\ \end{array}$

In case that $y=1$, $T^{m}y=0$ for all $m\geq 2$ and the previous argument
also follows. Therefore condition (4.1) is satisfied for $k=1$.

It is straightforward to show that $T$ is not uniform Lipschitzian for any $k$
since $T$ is not continuous. Additionally $\\{T^{n}:n\in\mathbb{N}\\}$ is not
orbit $k$-Lipschitz for any $k\in(0,2)$ either.

Indeed, consider $0<y<1$ and $x=y/2$. In this case $D(x,o(y))=y-x=d(x,y)=y/2$.
Additionally

${d(T^{n}(y/2),T^{n}y)\over d(y/2,y)}={y^{2n}-\left({y\over
2}\right)^{2n}\over{y\over 2}}=2y^{2n-1}\left(1-{1\over 2^{2n}}\right).$

Considering for instance $y=e^{-1/n^{2}}$, it is not difficult to check that

$\sup\left\\{{d(T^{n}(y/2),T^{n}y)\over d(y/2,y)}:n\in\
\mathbb{N},y\in[0,1)\right\\}=2,$

from where it follows that the semigroup generated by $T$ is not an orbit
uniformly $k$-Lipschitzian action for any $k<2$. ∎

In a case of a single mapping, we deduce the following:

######  Corollary 4.4.

Let $X$ be a complete metric space and $T:X\to X$ a mapping such that there
exists some $k<\kappa(X)$ for which the following holds:

(4.4) $\begin{array}[]{c}\mbox{ For $(x,y)\in X\times X$ with $D(x,o(y))\leq
D(x,o(x))$:}\\\ \\\ \lim_{m}D(T^{n}x,o(T^{m}y))\leq kD(x,o(y))\end{array}$

for all $n\in\mathbb{N}$. Then either all the orbits are unbounded or there
exists some $x\in X$ such that $Tx=x$.

Since $\kappa(X)\geq 1$ for any metric space $X$, the above corollary implies
the existence of a fixed point in a complete metric space as long as condition
(4.4) holds for some $k<1$. The following is an extension of Lifschitz Theorem
(Theorem 2.3 above):

######  Corollary 4.5.

Let $X$ be a complete metric space and $T\colon X\to X$ such that there exists
$k<\kappa(X)$ with

$d(T^{n}x,T^{n}y)\leq kD(x,o(y))$

for all $n\in\mathbb{N}$ and for all $(x,y)\in X\times X$ with $D(x,o(y))\leq
D(x,o(x))$. If some orbit is bounded, then $T$ has a fixed point.

######  Remark 4.6.

Note that Theorem 4.1 is optimal since the following example shows that the
Lipschitz constant cannot be improved. Indeed, consider the semigroup
generated by the mapping defined on $[0,1]$ given by

$T(x)=\begin{cases}1,&\text{if $x\in[0,1)$,}\\\ 0,&\text{if
$x=1$.}\end{cases}$

To check the orbit Lipschitzian character of $T$, the worst case happens for
$x=1/2$ and $y=1$. In this case, we have that $o(y)=\\{1,0\\}$, and so
$D(x,o(y))=1/2$. From where, $d(Tx,Ty)=2D(x,o(y))$. As a consequence, the
semigroup generated by $\\{T\\}$ enjoys the orbit $2$-Lipschitzian condition
(and so condition (4.1) in Theorem 4.1 too), while it fails to have a fixed
point. Remember that $\kappa([0,1])=2$.

######  Example 4.7.

Let $X=B_{\ell_{p}}$ be the closed unit ball of $\ell_{p}$ endowed with the
usual metric provided by the norm $\|\cdot\|_{p}$ for $1\leq p<+\infty$. Let
us denote by $\\{e_{n}\\}$ the standard basic vectors and define
$T:B_{\ell_{p}}\to B_{\ell_{p}}$ as $T(x)=e_{1}$ if $x\neq e_{n}$ for all
$n\in\mathbb{N}$, and $T(e_{n})=e_{n+1}$. It is clear that $T$ lacks of any
fixed point and so does the semigroup generated by
$\\{T^{n}:n\in\mathbb{N}\\}$. We next prove that the action satisfies the
strong-orbit $2^{1/p}$-Lipschitzian condition.

Indeed, for every $x\in B_{\ell_{p}}$ we have that
$\limsup_{n}\|x-e_{n}\|=(\|x\|_{p}+1)^{1/p}$. Moreover, if $x\neq x_{n}$ for
all $n\in\mathbb{N}$, $o(x)=\\{x,e_{1},e_{2},\cdots\\}$ and
$o(Tx)=\\{e_{1},e_{2},\cdots\\}$. In particular, setting $x:=0$, $D(x,o(x))=1$
and $D(x,o(y))=1$ for all $y\in B_{\ell_{p}}$. Additionally, for every
$n\in\mathbb{N}$, $y\neq 0$

$D(T0,o(T^{n}y))=D(e_{1},o(T^{n}y))=2^{1\over p}=2^{1\over p}D(0,o(y))$

which shows that the semigroup satisfies the strong-orbit $2^{1/p}$-Lipschitz
condition and its extension considered in Theorem 4.1 given by condition
(4.1). The above shows in particular that $\kappa(\ell_{p})\geq 2^{1/p}$ for
$1\leq p<+\infty$ and from the example included in Remark 5.8 (see Section 5),
$\kappa(\ell_{\infty})=1$.

##  5\. Strong-orbit Lipschitzian actions and uniform normal structure

The main goal of this section is to obtain a common fixed point result for
strong-orbit Lipschitzian actions in metric spaces satisfying the (metric)
uniform normal structure (see Section $2$ for definitions) and, therefore, to
achieve extensions of Casini-Maluta Theorem [3], Lim-Xu Theorem [20] and some
others published in the literature.

Given a semigroup $\mathcal{S}$, a preorder can be defined as follows:

$s\leq t\text{ if }t\in\mathcal{S}^{1}s=\\{s\\}\cup\mathcal{S}s.$

Notice that $\leq$ is a reflexive and transitive relation, but it need not be
antisymmetric. In fact, when $\mathcal{S}$ is a group, $t\in\mathcal{S}s$ for
every $t,s\in\mathcal{S}$.

Along this section we assume that the preorder is total, that is, for every
$s,t\in\mathcal{S}$, either $s\geq t$ or $t\geq s$. In particular, this
implies that $(\mathcal{S},\leq)$ is a directed set. Particular examples are
groups, $(\mathbb{N},+)$, $((1,+\infty),+)$, $((1,+\infty),\cdot)$.

Property (P) has already been recalled in Section $2$ for sequences (see
Definition 2.5), we will need it for nets in this section.

######  Definition 5.1.

A metric space $X$ is said to have property (P) for nets if given two bounded
nets $\\{x_{s}\\}_{s}$ and $\\{z_{s}\\}_{s}$ in $X$, such that $z_{s}\in{\rm
cov}(\\{x_{j}:j\geq s\\})$, there is $z\in\bigcap_{s}{\rm
cov}(\\{z_{j}\;:\;j\geq s\\})$ such that

$\limsup_{s}d(z,x_{s})\leq\limsup_{t}\limsup_{s}d(z_{t},x_{s}).$

Recall that a metric space is said to be proper if its closed balls are
compact. Then the following lemma is easy to prove.

######  Lemma 5.2.

Every proper metric space has property (P) for nets.

###### Proof.

Let $\\{x_{s}\\}$, $\\{z_{s}\\}$ two bounded nets as in Definition 5.1. We can
assume that they lie in a compact closed ball. Let $z$ be a cluster point of
the net $\\{z_{s}\\}$ for the metric topology. Then $z\in\bigcap_{s}{\rm
cov}(\\{z_{j}\;:\;j\geq s\\})$ and there exists a subnet
$\\{z_{s_{\alpha}}\\}$ such that $\lim_{\alpha}d(z,z_{s_{\alpha}})=0$. Thus,

$\begin{array}[]{lll}\limsup_{s}d(z,x_{s})&\leq&\lim_{\alpha}d(z,z_{t_{\alpha}})+\liminf_{\alpha}\limsup_{s}d(z_{t_{\alpha}},x_{s})\\\
&\leq&\limsup_{t}\limsup_{s}d(z_{t},x_{s}).\\\ \end{array}$

∎

Although we are strongly inspired by the arguments used in [3] and [20], we
will proceed introducing all the details for the sake of a better
understanding.

######  Remark 5.3.

Note that if $A$ is a subset of $X$ and $z\in{\rm cov}(A)$ then $d(y,z)\leq
D(y,A)$ for all $y\in X$. This, in particular, implies that if $(A_{s})_{s}$
is a net of subsets of $X$ such that $z\in\bigcap_{s}{\rm cov}(A_{s})$, then,
for all $y\in X$, it must be the case that

$d(z,y)\leq\inf_{s}D(y,A_{s}).$

Additionally, $\delta(A)=\delta({\rm cov}(A))$ for all $A\subset X$.

######  Theorem 5.4.

Let $X$ be a complete metric space with (metric) uniform normal structure and
property (P) for nets. Let $(\mathcal{S},\leq)$ be a totally preordered
semigroup such that $(\mathcal{S},X)$ is a strong-orbit $k$-Lipschitzian
action. If $k<\tilde{N}(X)^{-1/2}$ and some orbit is bounded, there is $z\in
X$ such that $sz=z$ for all $s\in\mathcal{S}$.

###### Proof.

We will start the proof by making some preliminary observations. Let $x\in X$:

  * i)

If $s,t\in\mathcal{S}$, $s\leq t$ then $o(tx)\subset o(sx)$ and
$o(sx)=\\{ux:u\geq s\\}$.

  * ii)

$\\{sx:s\in\mathcal{S}\\}$ is a net in $X$ and $\\{o(sx):s\in\mathcal{S}\\}$
forms a nonincreasing net of subsets of $X$. Additionally for all $y\in X$:

$\limsup_{s}d(y,sx)=\lim_{s}D(y,o(sx))=\inf_{s}D(y,o(sx)).$

We choose a constant $c$ such that $1>c>\tilde{N}(X)$ and $k<c^{-1/2}$. Fix
$x_{0}\in X$ for which the orbit is bounded and consider the net
$\\{sx_{0}:s\in\mathcal{S}\\}$. From the definition of the uniform normal
structure coefficient $\tilde{N}(X)$, for $s\in\mathcal{S}$, there exists

$z_{s}\in{\rm cov}(o(sx_{0}))\subset B(z_{s},c\delta(o(sx_{0}))).$

Applying the condition (P) to the nets $\\{sx_{0}:s\in\mathcal{S}\\}$ and
$\\{z_{s}:s\in\mathcal{S}\\}$, there exists $z_{0}\in\bigcap_{s}{\rm
cov}(o(sx_{0}))$ such that:

$\limsup_{s}d(z_{0},sx_{0})\leq\limsup_{t}\limsup_{s}d(z_{t},sx_{0}).$

Writing the above inequality in terms of the orbits:

$\begin{array}[]{lll}\lim_{s}D(z_{0},o(sx_{0}))&\leq&\limsup_{t}\lim_{s}D(z_{t},o(sx_{0}))=\limsup_{t}\inf_{s}D(z_{t},o(sx_{0}))\\\
&\leq&\limsup_{t}D(z_{t},o(tx_{0}))\leq c\limsup_{t}\delta(o(tx_{0}))\\\
&\leq&c\delta(o(x_{0})).\\\ \end{array}$

Thus, if we call $x_{1}:=z_{0}$ we have just prove that

$x_{1}\in\bigcap_{s}{\rm cov}(o(sx_{0}))$

and, by Remark 5.3,

$\lim_{s}D(x_{1},o(sx_{0}))\leq c\delta(o(x_{0})).$

Repeating the process, we can find a sequence $(x_{j})\subset X$ such that:

(5.1) $x_{j+1}\in\bigcap_{s}{\rm cov}(o(sx_{j}))$

and, again by Remark 5.3,

(5.2) $\lim_{s}D(x_{j+1},o(sx_{j}))\leq c\delta(o(x_{j})).$

We next find an upper estimation for $\delta(o(x_{j}))$:

Let $s,t\in\mathcal{S}$ with $t\neq s$. Since the preorder is total, we can
assume that $s\geq t$ which implies that $s=at$ for some $a\in\mathcal{S}$.
Then

$d(sx_{j},tx_{j})=d(atx_{j},tx_{j})\leq D(tx_{j},o(tx_{j}))\leq
kD(x_{j},o(x_{j}))$

The above implies that

(5.3) $\delta(o(x_{j}))\leq kD(x_{j},o(x_{j})).$

Additionally, using (5.1) and Remark 5.3, for every $s\in\mathcal{S}$:

$\begin{array}[]{lll}d(x_{j},sx_{j})&\leq&\inf_{t\in\mathcal{S}}D(sx_{j},o(tx_{j-1}))\leq\inf_{t\in\mathcal{S}}D(sx_{j},o(stx_{j-1}))\\\
&\leq&\inf_{t\in\mathcal{S}}kD(x_{j},o(tx_{j-1})).\\\ \end{array}$

Taking supremum in $s\in\mathcal{S}$, we deduce that

(5.4) $D(x_{j},o(x_{j}))\leq
k\inf_{t\in\mathcal{S}}D(x_{j},o(tx_{j-1}))=k\lim_{t}D(x_{j},o(tx_{j-1})).$

For every $j\geq 0$, we define $D_{j}:=\lim_{s}D(x_{j+1},o(sx_{j}))$.
Considering (5.2), (5.3) and (5.4), we obtain that:

(5.5) $D_{j}\leq ck^{2}D_{j-1},$

for every $j\in\mathbb{N}$. Set $r:=ck^{2}<1$. In particular $\lim_{j}D_{j}=0$
and $\lim_{j}D(x_{j},o(x_{j}))=0$ by (5.4).

We next check that $(x_{j})$ is a Cauchy sequence by using (5.1), Remark 5.3,
(5.4) and (5.5):

$\begin{array}[]{lll}d(x_{j+1},x_{j})&\leq&\inf_{s\in\mathcal{S}}D(x_{j},o(sx_{j}))\leq
D(x_{j},o(x_{j}))\\\ &\leq&kD_{j-1}\leq krD_{j-2}\leq
kr^{j-1}D_{0}.\end{array}$

Set $w=\lim_{j}x_{j}$, which exists by the completeness of the metric space.
We will prove that $w$ is a common fixed point for $\mathcal{S}$. Firstly, let
us check that $\lim_{j}D(w,o(x_{j}))=0$. Indeed, let $t\in\mathcal{S}$:

$d(w,tx_{j})\leq d(w,x_{j})+d(x_{j},tx_{j})\leq d(w,x_{j})+D(x_{j},o(x_{j})).$

The above implies that

$D(w,o(x_{j}))\leq d(w,x_{j})+D(x_{j},o(x_{j}))\to_{j}0.$

Finally, given $s\in\mathcal{S}$:

$\begin{array}[]{lll}d(w,sw)&\leq&d(w,x_{j})+d(x_{j},sx_{j})+d(sx_{j},sw)\\\
&\leq&d(w,x_{j})+D(x_{j},o(x_{j}))+D(sw,o(sx_{j}))\\\
&\leq&d(w,x_{j})+D(x_{j},o(x_{j}))+kD(w,o(x_{j}))\to_{j}0.\\\ \end{array}$

Therefore the proof is concluded.

∎

Notice that when $\mathcal{S}$ is countable, it is enough to consider property
(P) for sequences. When $\mathcal{S}$ is additionally a commutative semigroup,
we have the following corollary:

######  Corollary 5.5.

Let $X$ be a metric space with uniform normal structure and property (P) for
nets. Let $\mathcal{S}$ be a commutative semigroup, or more generally, such
that $\mathcal{S}s\subset s\mathcal{S}$ for all $s\in\mathcal{S}$. Assume that
$(\mathcal{S},\leq)$ is total. Then every $(\mathcal{S},X)$ action which is
orbit $k$-Lipschitzian with $k<\tilde{N}(X)^{-1/2}$ has a common fixed point.

For a single mapping $T$, we have the next result.

######  Corollary 5.6.

Let $X$ be a metric space with uniform normal structure and property (P) for
sequences. Let $T:X\to X$ be orbit uniform $k$-Lipschitzian with
$k<\tilde{N}(X)^{-1/2}$. Then $T$ has a fixed point.

###### Proof.

It is enough to apply Theorem 5.4 to the action generated by $T$ and its
iterates. ∎

######  Remark 5.7.

Given $\mathcal{S}$ a semigroup, if we fix some $t\in\mathcal{S}$ and consider

$\mathcal{S}_{t}:=S^{1}t=\\{t\\}\cup\mathcal{S}t=\\{s\in\mathcal{S}:s\geq
t\\},$

It turns out that $\mathcal{S}_{t}$ is also a semigroup. Assume that
$\mathcal{S}$ is an action over a metric space $X$ and all the mappings in
$\mathcal{S}_{t}$ share a common fixed point $w\in X$. Then $w$ is a common
fixed point for all $s\in\mathcal{S}$. Indeed $st(w)=w=t(w)$ so $s(w)=w$ for
all $s\in\mathcal{S}$. This implies in particular that in order to obtain a
fixed point for a semigroup action, it is enough to verify the required
hypotheses for some $\mathcal{S}_{t}$ with $t\in\mathcal{S}$. In particular,
if $\mathcal{S}$ is generated by a single mapping $T$, the existence of a
common fixed point for $\mathcal{S}_{n_{0}}=\\{T^{n}:n\geq n_{0}\\}$ for some
$n_{0}\in\mathbb{N}$, implies the existence of a fixed point for $T$.

######  Remark 5.8.

We would like to point out the fact that only one bounded orbit is needed for
our results in contrast to the fact that bounded metric spaces are usually
required in the literature (see for instance in [3, 7, 20]). This is somehow
surprising if we consider the following example, given by S. Prus (see for
instance [18, p. 412]: Let $T:\ell_{\infty}\to\ell_{\infty}$ be defined by

$T(x)=(1+\limsup_{n}x_{n},x_{1},x_{2},x_{3},\cdots,....).$

It is very easy to check that $T$ is an isometry fixed point free mapping,
where all the orbits are bounded. Additionally, $\ell_{\infty}$ is a
hyperconvex metric space, which implies that
$\displaystyle\tilde{N}(\ell_{\infty})=\frac{1}{2}$ (see [11] for details).
Therefore, if $\ell_{\infty}$ verified property $(P)$ for sequences, $T$ would
have to have a fixed point. In conclusion, $\ell_{\infty}$ does not verify
property $(P)$ for sequences.

In case that $\mathbb{R}^{n}$ is endowed with the supremum norm and
$d(x,y):=\|x-y\|_{\infty}$ for $x,y\in\mathbb{R}^{n}$, we know from
hyperconvexity that $\tilde{N}(\mathbb{R}^{n})=1/2$ and that property (P)
holds since $\mathbb{R}^{n}$ is a proper metric space (note that in this case
$\kappa(\mathbb{R}^{n})=1$ when $n>1$). Thus, Theorem 5.4 can be applied and
we can deduce the existence of a common fixed point for a strong-orbit
$k$-Lipschitzian action or a fixed point for an orbit uniformly
$k$-Lipschitzian mapping, when $k<\sqrt{2}$. Besides, Theorem 5.4 can also be
applied to the Isbell’s hyperconvex hull of every compact metric space (see
for instance [11, Section 8]), since Isbell’s hyperconvex hull of a compact
metric space is hyperconvex and compact and therefore property (P) holds.

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