# Liquid-liquid displacement in slippery liquid-infused membranes (SLIMs)

Hanieh Bazyar  Department of science and technology, Soft matter, Fluidics and
Interfaces (SFI), University of Twente, Enschede, The Netherlands  Wetsus,
European centre of excellence for sustainable water technology, Leeuwarden,
The Netherlands  Pengyu Lv  Department of Science and Technology, Physics of
Fluids (POF), Max Planck - University of Twente Center for Complex Fluid
Dynamics, University of Twente, Enschede, The Netherlands  Jeffery A. Wood
Department of science and technology, Soft matter, Fluidics and Interfaces
(SFI), University of Twente, Enschede, The Netherlands  Slawomir Porada
Department of science and technology, Soft matter, Fluidics and Interfaces
(SFI), University of Twente, Enschede, The Netherlands  Wetsus, European
centre of excellence for sustainable water technology, Leeuwarden, The
Netherlands  Detlef Lohse  Department of Science and Technology, Physics of
Fluids (POF), Max Planck - University of Twente Center for Complex Fluid
Dynamics, University of Twente, Enschede, The Netherlands  Rob G. H.
Lammertink<EMAIL_ADDRESS>Department of science and technology,
Soft matter, Fluidics and Interfaces (SFI), University of Twente, Enschede,
The Netherlands

###### Abstract

Liquid-infused membranes inspired by slippery liquid-infused porous surfaces
(SLIPS) have been recently introduced to membrane technology. The gating
mechanism of these membranes is expected to give rise to anti-fouling
properties and multi-phase transport capabilities. However, the long-term
retention of the infusion liquid has not yet been explored. To address this
issue, we investigate the retention of the infusion liquid in slippery liquid-
infused membranes (SLIMs) via liquid-liquid displacement porometry (LLDP)
experiments combined with microscopic observations of the displacement
mechanism. Our results reveal that pores will be opened corresponding to the
capillary pressure, leading to preferential flow pathways for water transport.
The LLDP results further suggest the presence of liquid-lined pores in SLIM.
This hypothesis is analyzed theoretically using an interfacial pore flow
model. We find that the displacement patterns correspond to capillary
fingering in immiscible displacement in porous media. The related physics
regarding two-phase flow in porous media is used to confirm the permeation
mechanism appearing in SLIMs. In order to experimentally observe liquid-liquid
displacement, a microfluidic chip mimicking a porous medium is designed and a
highly ramified structure with trapped infusion liquid is observed. The
remaining infusion liquid is retained as pools, bridges and thin films around
pillar structures in the chip, which further confirms liquid-lining. Fractal
dimension analysis, along with evaluation of the fluid (non-wetting phase)
saturation, further confirms that the fractal patterns correspond to capillary
fingering, which is consistent with an invasion percolation with trapping
(IPT) model.

Bio-inspired interfacial materials with non-wetting properties have broad
technological implications for areas ranging from biomedical devices and fuel
transport to architecture Quéré (2008). Lotus-leaf inspired superhydrophobic
surfaces are well-known, owing to properties such as drag reduction, anti-
icing, anti-frosting and self-cleaning Darmanina and Guittard (2014); Lv _et
al._ (2014); Hemeda and Tafreshi (2016); Elsharkawy _et al._ (2016); Cao _et
al._ (2016); Solomon _et al._ (2014); Subramanyam _et al._ (2016). However,
superhydrophobic surfaces are prone to failure due to elevated pressures and
temperatures and by dissolution of the trapped air into the surrounding fluid
Hemeda and Tafreshi (2016); Okada and Shiratori (2014). This is particularly
significant for low surface tension liquids Nguyen _et al._ (2010). Recently,
a novel class of functional surfaces known as slippery liquid-infused porous
surfaces (SLIPS) has been introduced Wong _et al._ (2011). Inspired by
Nepenthes pitcher plants Bohn and Federle (2004), the nano/microstructured
substrate is used to lock-in an intermediary liquid. This liquid, stabilized
by capillary forces, forms a smooth, low hysteresis lubrication layer which is
responsible for the non-wetting properties Kim _et al._ (2012); Hou _et al._
(2015); Epstein _et al._ (2012). In comparison with superhydrophobic surfaces,
SLIPS can potentially improve anti-icing or suppress frost accretion Kim _et
al._ (2012); Elsharkawy _et al._ (2016); Stone (2012), operate at high
pressures and temperatures Wong _et al._ (2011); Okada and Shiratori (2014),
and possibly reduce drag Solomon _et al._ (2014); Fu _et al._ (2017). Self-
healing by capillary wicking, repelling a variety of liquids, and anti-
biofouling are other potentially advantageous properties of SLIPS Epstein _et
al._ (2012); Wong _et al._ (2011). Recently, it has been shown that the
capillary-stabilized liquid in a membrane pore can form a reconfigurable gate
which can selectively let fluids pass through. This so-called gating mechanism
gives SLIPS the capability of multiphase transport without clogging Hou _et
al._ (2015). The liquid-lined gating mechanism was previously explored via
gas-liquid porometry where the remaining liquid film thickness on the pore
wall was estimated experimentally and theoretically Bazyar _et al._ (2016).

In previous studies, the fabrication methods of liquid infused surfaces were
complex, time-consuming and the substrate selectivity to match solid and
liquid chemistry was limited. Some methods required multi-step processing,
high temperatures and drying Stone (2012); Anand _et al._ (2012). Using well-
matched membrane-liquid combinations with low surface energies for water, e.g.
membranes prepared from fluorinated polymers, avoids the extra step of solid
surface energy reduction. Examples of these membranes are polyvinylidene
fluoride (PVDF) and polytetrafluoroethylene (PTFE). Okada and Shiratori
(2014); Hou _et al._ (2015); Bazyar _et al._ (2016).

Introduction of SLIPS to membranes provides potentially anti-fouling
properties, as well as pressure responsive pores for selective fluid transport
Hou _et al._ (2015); Hou (2016). These properties are attractive for
separation applications. The capability of these membranes in efficient gas-
liquid sorting for three phase air-water-oil mixtures Hou _et al._ (2015), for
example, is relevant for oily waste water treatment. Before the potential of
slippery liquid infused membranes (SLIM) can be realized, a thorough
understanding of the retention of the infusion liquid under dynamic conditions
of transport of the immiscible fluid is required.

Two-phase flow or immiscible displacement in any porous media, such as porous
membranes, is typically governed by viscous and capillary forces,
characterized by two dimensionless numbers, the viscosity ratio

$M=\frac{\mu_{i}}{\mu_{d}},\ $ (1)

and the capillary number

$Ca=\frac{\mu_{i}v}{\gamma}.\ $ (2)

Here $\mu_{i}$ and $\mu_{d}$ are the viscosities of invading (water) and
defending (oil) fluids respectively, $v$ is the superficial velocity and
$\gamma$ is the interfacial tension between the two immiscible fluids. The
domain of validity of different basic mechanisms, i.e. capillary fingering,
viscous fingering, or stable displacement, can be mapped on to a phase diagram
Lenormand _et al._ (1988). The displacement can also be classified according
to drainage or imbibition, where the defending (displaced) or invading
(displacing) fluid preferentially wets the solid surface, respectively
Holtzman and Segre (2015); Lenormand _et al._ (1988). Microfluidic techniques
have been used to investigate the displacement mechanisms. These techniques
offer the opportunity to fabricate micromodels resembling a porous medium with
regular as well as irregular pore shapes, anisotropy and pore sizes. Pore-
scale multi-phase displacement phenomena have been experimentally observed in
pore network patterns fabricated in materials such as silicon Chomsurin and
Werth (2003); Zhang _et al._ (2011), glass Ferer _et al._ (2004), PDMS
Berejnovz _et al._ (2010); Cottin _et al._ (2011); Berejnov _et al._ (2008);
Cottin _et al._ (2010), polyester and thiolene-based resin Lenormand (1990);
Lenormand _et al._ (1988); Odier _et al._ (2017).

Liquid-gated membranes are expected to possess anti-fouling characteristics
and multi-phase transport capabilities Hou _et al._ (2015). However, knowledge
regarding the retention of the infusion liquid within these membranes is still
limited. To address this issue, we report on retention of the infusion liquid
in SLIM and microscopic observation of the displacement mechanism. Liquid-
liquid displacement porometry (LLDP) is done in a flux-controlled mode by
pushing pure water through PVDF membranes infused with perfluoropolyether oil
(Krytox 101). The results are further analyzed theoretically using an
interfacial pore flow model. Finally, a microfluidic chip resembling the
porous medium has been used to microscopically investigate the displacement
mechanism under identical capillary number and viscosity ratio. The observed
flow pattern is characterized using fractal dimension analysis and fluid (non-
wetting phase) saturation, methods known from porous media analysis. Here, the
permeation through SLIMs is related to two-phase flow in porous media to
confirm the observed displacement mechanisms that are crucial for their
promising membrane applications.

##  I Results and discussion

###  I.1 Membrane experimental results

Liquid-infused membranes are made by infusing different types of liquids
(fluorinated or hydrocarbon) into dry PVDF membranes (see section Materials
and Methods). The physical properties of the used liquids are shown in Table 1
(see SI for more information on the measurement methods).

Table 1: Physical properties of different infusion liquids Infusion liquids | Chemical structure |  Surface tension [$mN/m$] |  Absolute viscosity at 20∘C [$mPa$ $s$] |  Kinematic viscosity at 20∘C [$mm^{2}/s$] |  Density at 24∘C [$g/cm^{-3}$]
---|---|---|---|---|---
Fluorinert FC-43 | Perfluorocarbon (PFC) |  16.3$\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}$0.05 |  5.6$\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}$0.08 |  3.0 |  1.88$\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}$0.01
Galpore | Perfluoropolyether (PFPE) |  15.5$\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}$0.05 |  8.6$\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}$0.08 |  4.8 |  1.83$\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}$0.01
Krytox GPL oil 101 | Perfluoropolyether (PFPE) |  16.3$\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}$0.13 |  25$\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}$0.1 |  13.5 |  1.85$\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}$0.01
Silicone oil AR20 | Polyphenylmethylsiloxane |  21.8$\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}$0.03 |  19.6$\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}$0.1 |  19.4 |  1.00$\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}$0.002

All liquid-infused membranes display hydrophobic behavior with a static
contact angle of about 120∘ and contact angle hysteresis $<5^{\circ}$ using
standard contact angle Goniometer (the water contact angle values on liquid
infused PVDF membranes are shown in Table S1). These are the key design
parameters, since a high contact angle and a low contact angle hysteresis are
desirable for high droplet mobility Semprebon _et al._ (2017). Liquid-liquid
displacement experiments are done successively in a flux controlled mode by
pushing pure water (displacing fluid) through the liquid infused membrane. The
schematic illustration of the set-up is shown in Figure 1.

Figure 1: Schematic illustration of the membrane experiment and SEM image of
the PVDF membrane (Scale bar is 10 $\mu$m).

The result of LLDP on infused PVDF membrane with perfluoropolyether oil
(Krytox 101) is shown in Figure 2. The results of other liquid infused
membranes are shown in Figure S4. The LLDP experiment is performed in five
different cycles. Each cycle is carried out in a flux-increasing mode, which
is done twice starting from zero up to a certain flux value, i.e. the cycle
consists of two runs (1${}^{\text{st}}$ run and 2${}^{\text{nd}}$ run). In
each run, the flux is increased step-wise and pressure is measured
simultaneously. At each step, flux is kept constant for 100 s and the pressure
is reported for the last 40 s of each step.

In each cycle, a certain number of pores have been opened in the
1${}^{\text{st}}$ run corresponding to the Laplace pressure

$\Delta P=\frac{2\gamma\lvert\cos\theta_{E}\rvert}{r},$ (3)

leading to preferential flow pathways.

Here $\Delta P$ is the transmembrane pressure, $r$ is the pore radius,
$\gamma$ is the interfacial tension between displacing and displaced fluid,
and $\theta_{E}$ is the advancing contact angle of the displacing fluid on a
surface of smooth and dense PVDF (total non-wetting ($\theta_{E}$=180∘) with
displacing fluid is considered for liquid-infused membranes).

The opened pores remain open during the 2${}^{\text{nd}}$ run, confirmed by
the linear relation between flux and pressure Mietton-Peuchot _et al._ (1997).
An initial critical pressure of approximately $0.57\times 10^{5}$ Pa is
required to open the biggest pore sizes. Based on the pore size distribution
of the membrane obtained from capillary flow porometer (see Figures S2), the
largest pore radius is
1.77$\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}$0.11
$\mu$m. According to Young-Laplace equation (equation 3) and by considering
total non-wetting, the corresponding pressure to open the largest pores is
$0.61\times 10^{5}$ Pa (see Table S3 for surface and interfacial tension
values), which is in good agreement with the LLDP experiments.

Figure 2: Successive liquid-liquid displacement porometry (LLDP) experiment in
a flux controlled mode on Krytox 101 infused PVDF membrane. LLDP is done in
five different cycles (I-V) and each cycle consists of a 1${}^{\text{st}}$
(filled symbols) and a 2${}^{\text{nd}}$ run (open symbols). In each run flux
is increased step-wise from zero up to a certain maximum value and pressure is
measured correspondingly. A typical sequence of the measurement is shown for
cycle V.

In order to check for the presence of the infusion liquid in the membrane
after the experiment, the obtained results are compared with the results of
the pre-wetted membrane (see Figure 2). The permeability of pre-wetted as well
as liquid-infused membranes is calculated using Darcy’s law Darcy (1856)

$Q=\frac{\kappa A}{\mu}\ \frac{dp}{dx}.\ $ (4)

Here and in the following, $Q$ is the volumetric flow of the permeating fluid
(in this case water), $\kappa$ is permeability (L2), $A$ is the total area of
the membrane, $\mu$ is the viscosity of the displacing fluid, $\frac{dp}{dx}$
is the pressure gradient across the membrane thickness, $r$ is the mean pore
radius, and $\phi$ is porosity.

A simple model is then used to relate permeability to porosity $\phi$ (see SI
for derivation), namely

$\kappa=\frac{\phi r^{2}}{24}.\ $ (5)

The calculated porosity is a measure of opened and active pores for water
transport in the membrane. The results of the calculated permeability and the
estimated fraction of the active pores are shown in Table 2.

Table 2: Comparison of permeability and fraction of active pores for five
different cycles of liquid infused membrane and for the pre-wetted case.

Membrane |  Permeability ($\kappa$) (Darcy∗) |  Estimated fraction of active pores
---|---|---
SLIM-cycle 1 |  3.12$\times$10-3 |  0.09
SLIM-cycle 2 |  5.23$\times$10-3 |  0.15
SLIM-cycle 3 |  9.19$\times$10-3 |  0.26
SLIM-cycle 4 |  1.51$\times$10-2 |  0.43
SLIM-cycle 5 |  2.01$\times$10-2 |  0.57
Pre-wet |  3.54$\times$10-2 |  1

  * *

$1$ $Darcy=0.987\times 10^{-12}$ $m^{2}$

This fraction is the ratio between the calculated porosity for the liquid
infused membrane in each cycle and that of the pre-wetted membrane. The
permeability of the liquid-infused membrane increases in each cycle, revealing
that new pores are opened for water transport. The lower permeability values
of SLIM in comparison with pre-wetted membranes suggests incomplete removal of
the infusion liquid with around 43% of retained infusion liquid in the
membrane. Direct observations using the microfluidic chip shows that 27% of
the infused liquid remained in the chip at the corresponding flux value
($1.8\times 10^{-2}$ m3/m2 s) (see Table 3).

The average pore radius of the membrane based on its pore size distribution is
1.65$\mathbin{\mathchoice{\ooalign{\raisebox{0.1pt}{$\displaystyle+$}\cr\smash{\raisebox{-0.6pt}{$\displaystyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\textstyle+$}\cr\smash{\raisebox{-0.6pt}{$\textstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptstyle-$}}\cr}}{\ooalign{\raisebox{0.1pt}{$\scriptscriptstyle+$}\cr\smash{\raisebox{-0.6pt}{$\scriptscriptstyle-$}}\cr}}}$0.11
$\mu$m (Figure S3). Based on the Young-Laplace equation (equation 3) and by
assuming total non-wetting, all the pores should be opened at pressures beyond
$0.61\times 10^{5}$ Pa. As the used membranes are tortuous porous media with
interconnected pores and areas, infused liquid can be trapped. The trapped
liquid will not be removed by further flux or pressure increase and thus leads
to a lower permeability compared to the pre-wet one. In order to observe the
re-infusion of the opened pores, long-term experiments are performed. In these
experiments the pause time between each cycle was set to 12 hr. However, re-
infusion is not observed at the high flux values in which the experiments have
been already conducted. This is evident from the linear flux-pressure
relation, that would be different when re-infused.

The experimental results are analyzed theoretically using an interfacial pore
flow model to relate the flux ($J$) to pressure ($\Delta P$), namely Antón _et
al._ (2014)

$J(\Delta P)=\frac{N\pi\Delta P}{8\sqrt{2\pi}\ln(S)\mu
l}\int_{r_{\text{min}}}^{r_{\text{max}}}r^{3}\exp\left(-\frac{1}{2}\left(\frac{\ln(r/R)}{\ln(S)}\right)^{2}\right)\;\mathrm{d}r.\
$ (6)

Figure 3: Fitting of experimental results of LLDP to the interfacial pore flow
model (equation 6 of reference Antón _et al._ (2014)). Symbols and lines show
the experimental and fitting results respectively. Fitting parameter are
$N=10^{7}$ pore/m2 (total number of pores per unit area), $R=1.65$ $\mu$m
(geometric mean radius) and $S=1.2$ (geometric standard deviation)).

Here $N$ is the total number of the pores per unit area, $R$ is the geometric
mean radius, and $S$ is the geometric standard deviation. $J$ is the
volumetric flux of displacing fluid per membrane area through the membrane,
$l$ is the membrane thickness. $r_{\text{{min}}}$ is the equilibrium radius of
curvature, i.e. Kelvin radius, which can be calculated using Young-Laplace
equation (equation 3). $r_{\text{{max}}}$ is taken at least one order of
magnitude larger than the largest pore size corresponding to the lowest
measurable pressure by the set-up. A log-normal pore size distribution is
considered for this model Antón _et al._ (2014); Morison (2008) and was found
to describe the measurements well (see SI for more details on fitting). The
pore size distribution of the PVDF membrane is shown in Figure S3.

Figure 3 shows good agreement between experimental values and fitted curves
using the interfacial pore flow model (equation 6). It accurately describes
the flux behavior in both liquid-infused and dry membranes. The result for a
silicone oil (SO AR20)-infused membrane is also shown. The difference in
pressure at a given flux arises purely from different Laplace pressures due to
the corresponding interfacial tension values (Table S2). Since the liquid-
infused membrane is preferentially wetted by the infusion liquid (this is one
of the criteria for fabrication of SLIM Wong _et al._ (2011)), total non-
wetting can be assumed for the invading liquid ($\theta_{\text{E}}$=180∘) (see
Figure 4). For validation, the advancing contact angle of water on a dense
membrane which was immersed in the oil was measured.
$\theta_{\text{{Adv}}}$=175∘ and $\theta_{\text{{Adv}}}$=160∘ were obtained
respectively for K101 and SO AR20 (Table S1). In a dry membrane due to the
absence of the wetting layer and hydrophobic character of the pore wall,
partial wetting occurs. In this case, the advancing contact angle of water on
smooth and dense membrane should be considered ($\theta_{\text{{Adv}}}$=122∘)
(Table S1). Thus, the numerator of equation 3 for a dry pore ($\gamma\lvert
cos\theta_{\text{{E}}}\rvert$=38 mN/m) is between that of K101 and that of SO
AR20 infused pores ($\gamma_{\text{{K101-water}}}=54$ mN/m and
$\gamma_{\text{{SO-water}}}=18$ mN/m Kobayashi _et al._ (2012)). As it is
schematically shown in Figure 4(b), this further suggests the presence of
remaining liquid film on the pore wall (liquid-lined pores) in SLIM.

(a)

(b)

Figure 4: Schematic illustration of water pushing through (a) dry membrane
pore and (b) liquid-infused pore.

###  I.2 Microfluidic experimental results

In order to observe the displacement mechanism and the resulting flow paths, a
microfluidic chip has been used as a mimic of the porous medium (Figure 5).
The chip is fabricated in silicon using standard photolithography and reactive
ion etching (see SI for detailed fabrication procedure). The pore network
contains a uniform distribution of square pillars with 20 $\mu$m center to
center spacing, 8 $\mu$m diameter and 50 $\mu$m height giving a porosity of
0.84.

(a)

(b)

Figure 5: Scanning electron microscopy (SEM) images of (a) the inlet of the
microfluidic chip, (b) pillars.

To mimic the experimental conditions of the membrane experiments, the chip is
further hydrophobized with perfluorinated silane and filled with SO AR20
labeled with 8.7 mM fluorescent dye (Perylene). Water (labeled with 17.7
$\mu$M fluorescent dye (Rhodamine 6G)) is pushed through the chip from the
right side using a syringe pump (Harvard apparatus PHD 2000 infuse/withdraw).
The addition of the aforementioned dyes neither changed the surface tension of
water nor SO AR20 considerably (see Table S3). Laser scanning confocal
microscope (LSCM) is used for observation of liquid-liquid displacement. The
schematic of the experimental set-up is shown in Figure 6.

Figure 6: Schematic illustration of the microfluidic experiment.

The viscosity ratio (equation 1) is kept the same as in membrane experiments
by using similar fluid pairs. The experiment is done at different flow rates,
corresponding to the same capillary number (equation 2) as the membrane
experiments (Table 3).

A typical result of the microfluidic displacement is shown in Figure 7. Figure
7(a) displays the last time frame of the corresponding movie (see SI for the
movie). In this figure, yellow and blue colors correspond to the infused
liquid (SO AR20) and water, respectively. A highly ramified pattern with
trapped infusion liquid is observed (Figure 7(b)). Based on the used capillary
number ($Ca\simeq 10^{-4}$ see Table 3) and viscosity ratio ($M=0.05$), the
displacement pattern corresponds to the capillary fingering invasion regime in
drainage. Fingers in capillary fingering show spreading across considerable
part of the chip and growth is seen in all directions. When capillary
fingering occurs some wetting fluid becomes entrapped due to the complex
displacement pattern Zhang _et al._ (2011). The residual wetting fluid can be
observed as pools, bridges and thin films around the pillars (Figure 7(b)).
Observation of thin films around pillars (insert of Figure 7(b)) further
confirms liquid-lining after displacing with water (Figures 4(b) and S6). The
presence of this liquid layer is crucial for any potential anti-fouling
properties of liquid-infused surfaces due to minimum contact of foulants with
solid material of the surface Hou _et al._ (2015).

(a)

(b)

Figure 7: Experiment using laser scanning confocal microscope (LSCM). (a) Image after water transport through the liquid infused chip at $Q$=0.2 $\mu$l/s ($Ca$=1.23$\times$10-4) (yellow color is the oil phase and blue color is the water phase). (b) Image at 20x magnification showing different configurations of the residual wetting fluid. Table 3: Irreducible water saturation values (S${}_{\text{nwr}}$) for the microfluidic experiment done at different flow rates. Microfluidic Q ($\mu$l/s) |  Membrane flux (m3/m2 s) |  $Ca$ |  S${}_{\text{nwr}}$
---|---|---|---
0.2 |  0.0013 |  1.23$\times$10-4 |  0.56
0.4 |  0.0026 |  2.46$\times$10-4 |  0.62
0.8 |  0.0053 |  4.91$\times$10-4 |  0.63
1.6 |  0.011 |  9.28$\times$10-4 |  0.69
2.9 |  0.018 |  1.72$\times$10-3 |  0.73
Figure 8: Total Saturation of the non-wetting phase (S${}_{\text{nw}}$) and
fractal dimension ($D$) as a function of time for the corresponding
displacement movie. During phase (I) formation of the fingers takes place.
During phase (II) the saturation profile is evolving. Phase (III) corresponds
to the breakthrough point and phase (IV) corresponds to the steady-state
regime where the saturation profile is preserved.

For quantitative description of the observed patterns and better understanding
of the displacement mechanism, the patterns are analyzed in terms of total
saturation S${}_{\text{nw}}$ of the non-wetting phase (water in this case),
the local saturation (and their evolution over time) and the fractal dimension
$D$. The total saturation S${}_{\text{nw}}$ is the ratio of the total area of
the non-wetting phase to the total available area of the chip Berejnovz _et
al._ (2010); Zhang _et al._ (2011). For measuring the local saturation, each
image is divided vertically into 33 slices and S${}_{\text{nw}}$ is calculated
for each slice. The MATLAB bio-format toolbox is used for image processing
Olsson and Hartley (2016).

The fractal dimension $D$ is given by

$D=\frac{\log B}{\log a}.$ (7)

Here, $B$ is the number of boxes that cover the pattern and $a$ is the
magnification factor which is the inverse of the box size. A box counting
algorithm is used to calculate $D$ (Hausdorff (Box-Counting) fractal dimension
code) (See SI for more details). $D$ for this kind of analysis is a non-
integer value which ranges between 1 and 2. $D$=1 for a straight line and
$D$=2 for a fractal pattern which completely fills up a 2D plane Feder (1988).

Figure 8 shows the total saturation S${}_{\text{nw}}$ and the fractal
dimension $D$ as a function of time. There are four distinct phases in the
total saturation plot. In phase I, a sharp increase is observed corresponding
to the formation of the fingers. Phase II corresponds to the evolution of the
formed fingers before the breakthrough point. During phase III the
breakthrough point has been reached. At this point a decrease in specific
interfacial length for the boundary between two phases is observed corresponds
to the removal of the larger mobile oil blobs (Figure S5(b)). In phase IV,
S${}_{\text{nw}}$ reaches a constant water saturation value of 0.6 after the
breakthrough point where the oil blobs are completely immobile. The
corresponding saturation value for the displacing liquid is the saturation at
which the displaced liquid goes from being mobile to being immobile, i.e.
irreducible saturation (S${}_{\text{nwr}}$).

The irreducible saturation values (S${}_{\text{nwr}}$) of other experiments
done at higher flow rates are shown in Table 3. The results show that at the
highest flow rate which corresponds to the highest flux value of the membrane
experiment ($1.8\times 10^{-2}$ m3/m2 s), 27% of the infused liquid will
remain in the chip. This is lower than the amount of remaining infused liquid
in the membrane experiments (Table 2). The reason can be attributed to the
presence of interconnected pores and irregular shaped pores in the membrane
where more infused liquid can be trapped.

The fractal dimension $D$ reaches its steady state value ($D_{s}$) of 1.8 at
the breakthrough point. This confirms that the displacement mechanism falls
within the flow regime of invasion percolation with trapping (IPT), a
statistical model which is used to describe capillary fingering. It is
established that IPT patterns can be identified with fractal dimension between
1.8 and 1.83 Lenormand _et al._ (1988); Lenormand and Zarcone (1989).

In order to better quantify the formation of the fingers in phase I, local
saturation of the non-wetting phase is plotted as a function of time and
location (see Figure S8). Local saturation provides a detailed description of
the dynamics of liquid-liquid displacement during the invasion. The initial
formation of the fingers corresponds to the movement of a shock front toward
the outlet (Figure 9). In order to model the displacement behaviour and
movement of the shock front, the Buckley-Leverett model is used Buckley and
Leverett (1942),

$\frac{\phi}{u}\frac{\partial S_{\text{nw}}}{\partial
t}+\frac{df_{\text{nw}}}{dS_{\text{nw}}}\frac{\partial S_{\text{nw}}}{\partial
x}=0.$ (8)

Here $\phi$ is the porosity, and $u$ is the total fluid velocity in the
direction of the flow. $f_{\text{nw}}$ is fractional flow of the non-wetting
phase, which is defined as the ratio of the non-wetting phase velocity to the
total velocity, i.e., $f_{\text{nw}}=u_{\text{nw}}/u$. This model is the best
known analytical approach for investigation and modelling of two-phase flow in
porous media Pinder and Gray (2008); Sahimi (2011); Marle (1981) (see SI for
derivation and solution details).

The modelling result is shown in Figure 9. The model can predict the
experimental results for the movement of the shock front. The differences may
originate from the simplifications in the model as well as experimental
accuracy. The displacement observations using LSCM is done at 4x magnification
and each time frame of the corresponding movie is divided to 33 slices to
obtain the local saturation plot. This is the smallest achievable slice size
in order to avoid noticeable experimental noise.

Figure 9: Buckley-Leverett experimental results (symbols) and modelling
results (solid line) of the shock front movement for four different times,
i.e., t${}_{\text{1}}$ = 1 s (black), t${}_{\text{2}}$ = 2 s (blue),
t${}_{\text{3}}$ = 3 s (red), and t${}_{\text{4}}$ = 4 s (green).

##  II Conclusions

We have reported on the retention of the infusion liquid in slippery liquid
infused membranes (SLIM) during water permeation and microscopic observation
of the displacement mechanism. The membrane experiments have been done via
liquid-liquid displacement porometry (LLDP) by pushing pure water through SLIM
in a flux-controlled mode. The pressure-flux results confirmed the presence of
remaining infusion liquid (43%) after displacing with water. Comparison of
different liquid-infused membranes with a dry one further indicates the
presence of liquid-lined pores. Infused pores were displaced according to the
corresponding capillary pressure and remained open once flow was stopped. This
suggested the preferential flow path ways for water transport through the
membranes which corresponds to the capillary fingering displacement regime.
The related physics regarding two-phase flow in porous media was used to
confirm the observed displacement mechanism in SLIMs. In order to better
understand the physical displacement process, a microfluidic chip was used for
direct visualization using laser scanning confocal microscope (LSCM). A highly
ramified structure with trapped infusion liquid was observed. Residual wetting
structures were observed in the form of pools, bridges, and thin films around
pillars. The presence of thin films further confirmed liquid-lining after
displacing with water. Fractal breakthrough patterns were analyzed in terms of
total and local saturation (S${}_{\text{nw}}$) and fractal dimension ($D$).
The saturation profiles indicated the dynamic distribution of the infusion
liquid during the displacement process. Local saturation provides details on
the formation of the fingers which corresponds to the movement of a shock
front toward the outlet. Buckley-Leverett model was used to predict the
experimental results of the shock front movement. The observed patterns along
with the fractal analysis confirmed that the experiment falls within the flow
regime of capillary fingering which can be described by an invasion
percolation with trapping (IPT) model. The saturation of the water phase
reached the steady-state value of 0.73 at the same capillary number as
membrane experiments corresponds to the highest flux value. This showed that
27% of the oil still remains in the chip. This study showed the retention of
liquid-lining under cyclic pressure-flux testing during immiscible
displacement process. The presence of the liquid-lined pores after
displacement with water is crucial for anti-fouling characteristics of SLIM,
which makes it a potential candidate for separation processes.

##  III Materials and Methods

###  III.1 Materials

PVDF (Solef 6020/1001) was received from Solvay Solexis, France.
1-methyl-2-pyrrolidinone (NMP) (99% extra pure) was purchased from Acros
Organics, The Netherlands. Ethanol (99.8%) was supplied from Atlas and Assink
chemical company, The Netherlands. Krytox GPL oil 101 was purchased from MAVOM
chemical industry, The Netherlands. Silicone oil AR20 was purchased from
Sigma-Aldrich, The Netherlands. Trichloro(1H,1H,2H,2H-perfluorooctyl)silane
(FOTS, 97%) was purchased from Sigma-Aldrich, The Netherlands. Perylene
(sublimed grade, 99.5%) and Rhodamine 6G (dye content 99%) as fluorescent dyes
for oil and water phases respectively were purchased from Sigma-Aldrich, The
Netherlands. n-Hexane and 2-propanol (analysis grade) were purchased from
Merck milipore, The Netherlands. Sulfuric acid (H2SO4 95-98%) was purchased
from Merck milipore, The Netherlands. Hydrogen peroxide (H2O2) solution 30%
(w/w) in H2O was supplied from Sigma-Aldrich, The Netherlands.

###  III.2 Membrane fabrication

The polymer solution was prepared by mixing 15 wt.% PVDF in NMP with a
mechanical stirrer overnight at 80∘C. The solution was cast on a glass plate
using a casting knife at an initial thickness of 500 $\mu m$. The cast
membrane was immediately submerged in water/NMP (30:70 vol.%) as the
coagulation bath for 60 min. To remove the remaining NMP from the membranes,
they were kept in ethanol subsequently for another 60 min. The films were then
taped to a piece of paper to prevent curling and left to dry in fume hood (60
min) before placing them in a 30∘C vacuum oven overnight.

###  III.3 Fabrication of dense PVDF

Dense PVDF was made by casting the polymer dope solution on a glass plate
using the same procedure as described for membrane fabrication. The cast
polymer solution was placed in a box and dried with a flow of nitrogen for two
days.

###  III.4 Fabrication of SLIM and pre-wet membrane

An overcoat layer (15.5 $\mu$l cm-2) of the low surface tension liquid (Krytox
101) was added to the membranes using a micropipette. The liquid spontaneously
infiltrated the pores via capillary wicking. The samples were further placed
vertical (2-3 h) for gravity-induced removal of the excess liquid.

The pre-wet membrane was prepared using the same procedure by adding ethanol
to the membrane. To replace ethanol with water in membrane pores, the sample
was placed in a beaker of water (200 $m$l) and left overnight.

###  III.5 Membrane characterization

The membrane was characterized using contact angle Goniometer (Dataphysics
OCA20), capillary flow porometer (Porolux-1000), and scanning electron
microscope (SEM) (JEOL 5600 LV). See SI for details on the techniques.

###  III.6 Laser scanning confocal microscopy (LSCM) experiments

The liquid-liquid displacement in the microfluidic chip is observed using an
inverted laser scanning confocal microscope (LSCM) (A1 system, Nikon
Corporation, Japan) with a 4x dry objective (CFI Plan Fluor 4x/0.13, numerical
aperture (NA) = 0.13, working distance (WD) = 17.2 mm). The scanning area is
chosen the same as the area of the pillar structure in the microfluidic chip,
i.e. $5\times 2.5$ mm2.

###  III.7 Hydrophobization of the microfluidic chip

In order to render the microfluidic chip hydrophobic, it was hydrophobized
using deposition of trichloro(1H,1H,2H,2H-perfluorooctyl)silane (FOTS) via
vapor-induced method. To achieve this, a glass bottle with gas inlet and
outlet was filled with 12 $\mu$L of FOTS. The inlet was connected to nitrogen
gas and the outlet was connected to the chip using a chip holder. The whole
set-up was then placed in oven at temperature of 100∘C. The outflow from the
chip was directed to a beaker containing water to neutralize and absorb the
permeate gas. The acidity of water was checked to further ensure the presence
of FOTS vapor in the chip via formation of hydrochloric acid (HCl) in water.
Before hydrophobization, the chip was cleaned with piranha solution
(H2SO4:H2O2 (3:1) vol.) and rinsed with pure water. Since the chip should be
completely dry before hydrophobization, water was replaced with 2-propanol and
then n-hexane. The chip was dried in oven (temperature of 100∘C) overnight
while having a flow of nitrogen gas through the channels.

##  IV ACKNOWLEDGMENTS

We thank Gabriela Berenice Diaz Cortes from Delft University of Technology,
The Netherlands for invaluable advice on Buckley-Leverett analysis using
MATLAB reservoir simulation tool box and Yanshen Li, guest PhD at Physics of
Fluids (POF) group at University of Twente for help on laser scanning confocal
microscope. We also thank Jan van Nieuwkasteele and BIOS lab-on-a-chip group
at Univerity of Twente for the design and fabrication of the microfluidic
chip. This work was performed in the cooperation framework of Wetsus, European
Centre of Excellence for Sustainable Water Technology (www.wetsus.nl). Wetsus
is co-funded by the Dutch Ministry of Economic Affairs and Ministry of
Infrastructure and Environment, the European Union Regional Development Fund,
the Province of Fryslân, the Northern Netherlands Provinces. This work is part
of a project that has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No. 665874. This work was also supported by the Netherlands Center
for Multiscale Catalytic Energy Conversion (MCEC), an NWO Gravitation program
funded by the Ministry of Education, Culture and Science of the government of
the Netherlands. This work is part of the Vici project STW 016.160.312 which
is financed by the Netherlands Organisation for Scientific Research (NWO).

## References

  * Quéré (2008) D. Quéré, _Annu. Rev. Mater. Res._ , 2008, 38, 71–99.
  * Darmanina and Guittard (2014) T. Darmanina and F. Guittard, _J. Mater. Chem. A_ , 2014, 2, 16319–16359.
  * Lv _et al._ (2014) J. Lv, Y. Song, L. Jiang and J. Wang, _ACS Nano_ , 2014, 8, 3152–3169.
  * Hemeda and Tafreshi (2016) A. A. Hemeda and H. V. Tafreshi, _Langmuir_ , 2016, 32, 2955–2962.
  * Elsharkawy _et al._ (2016) M. Elsharkawy, D. Tortorella, S. Kapatral and C. M. Megaridis, _Langmuir_ , 2016, 32, 4278–4288.
  * Cao _et al._ (2016) M. Cao, D. Guo, C. Yu, K. Li, M. Liu and L. Jiang, _ACS Appl. Mater. Interfaces_ , 2016, 8, 3615–3623.
  * Solomon _et al._ (2014) B. R. Solomon, K. S. Khalil and K. K. Varanasi, _Langmuir_ , 2014, 30, 10970–10976.
  * Subramanyam _et al._ (2016) S. B. Subramanyam, V. Kondrashov, J. Rühe and K. K. Varanasi, _ACS Appl. Mater. Interfaces_ , 2016, 8, 12583–12587.
  * Okada and Shiratori (2014) I. Okada and S. Shiratori, _ACS Appl. Mater. Interfaces_ , 2014, 6, 1502–1508.
  * Nguyen _et al._ (2010) T. P. N. Nguyen, P. Brunet, Y. Coffinier and R. Boukherroub, _Langmuir_ , 2010, 26, 18369–18373.
  * Wong _et al._ (2011) T.-S. Wong, S. H. Kang, S. K. Tang, E. J. Smythe, B. D. Hatton, A. Grinthal and J. Aizenberg, _Nature_ , 2011, 477, 443–447.
  * Bohn and Federle (2004) H. F. Bohn and W. Federle, _Proc. Natl. Acad. Sci. U. S. A._ , 2004, 101, 14138–14143.
  * Kim _et al._ (2012) P. Kim, T.-S. Wong, J. Alvarenga, M. J. Kreder, W. E. Adorno-Martinez and J. Aizenberg, _ACS Nano_ , 2012, 6, 6569–6577.
  * Hou _et al._ (2015) X. Hou, Y. Hu, A. Grinthal, M. Khan and J. Aizenberg, _Nature_ , 2015, 519, 70–73.
  * Epstein _et al._ (2012) A. K. Epstein, T.-S. Wong, R. A. Belisle, E. M. Boggs and J. Aizenberg, _Proc. Natl. Acad. Sci. U. S. A._ , 2012, 109, 13182–13187.
  * Stone (2012) H. A. Stone, _ACS Nano_ , 2012, 6, 6536–6540.
  * Fu _et al._ (2017) M. K. Fu, I. Arenas, S. Leonardi and M. Hultmark, _J. Fluid Mech._ , 2017, 824, 688–700.
  * Bazyar _et al._ (2016) H. Bazyar, S. Javadpour and R. G. H. Lammertink, _Adv. Mater. Interfaces_ , 2016, 3, 1600025.
  * Anand _et al._ (2012) S. Anand, A. T. Paxson, R. Dhiman, J. D. Smith and K. K. Varanasi, _ACS Nano_ , 2012, 6, 10122–10129.
  * Hou (2016) X. Hou, _Adv. Mater._ , 2016, 28, 7049–7064.
  * Lenormand _et al._ (1988) R. Lenormand, E. Touboul and C. Zarcone, _J. Fluid Mech._ , 1988, 189, 165–187.
  * Holtzman and Segre (2015) R. Holtzman and E. Segre, _Phys. Rev. Lett._ , 2015, 115, 164501\.
  * Chomsurin and Werth (2003) C. Chomsurin and C. J. Werth, _Water Resour. Res._ , 2003, 39, 1265\.
  * Zhang _et al._ (2011) C. Zhang, M. Oostrom, T. W. Wietsma, J. W. Grate and M. G. Warner, _Energ. Fuel_ , 2011, 25, 3493–3505.
  * Ferer _et al._ (2004) M. Ferer, C. Ji, G. S. Bromhal, J. Cook, G. Ahmadi and D. H. Smith, _Phys. Rev. E_ , 2004, 70, 016303.
  * Berejnovz _et al._ (2010) V. Berejnovz, A. Bazylaka, D. Sinton and N. Djilali, _J. Electrochem. Soc._ , 2010, 157, B760–B767.
  * Cottin _et al._ (2011) C. Cottin, H. Bodiguel and A. Colin, _Phys. Rev. E_ , 2011, 84, 026311\.
  * Berejnov _et al._ (2008) V. Berejnov, N. Djilali and D. Sinton, _Lab Chip_ , 2008, 8, 689–693.
  * Cottin _et al._ (2010) C. Cottin, H. Bodiguel and A. Colin, _Phys. Rev. E_ , 2010, 82, 046315\.
  * Lenormand (1990) R. Lenormand, _J. Phys.: Condens. Matter_ , 1990, 2, SA79–SA88.
  * Odier _et al._ (2017) C. Odier, B. Levaché, E. Santanach-Carreras and D. Bartolo, _Phys. Rev. Lett._ , 2017, 119, 208005.
  * Semprebon _et al._ (2017) C. Semprebon, G. McHale and H. Kusumaatmaja, _Soft Matter_ , 2017, 13, 101–110.
  * Mietton-Peuchot _et al._ (1997) M. Mietton-Peuchot, C. Condat and T. Courtois, _J. Membr. Sci._ , 1997, 133, 73–82.
  * Darcy (1856) H. Darcy, _Les Fontaines Publiques de la Ville de Dijon_ , Victor Dalmont, Paris, Paris, 1856.
  * Antón _et al._ (2014) E. Antón, J. I. Calvo, J. R. Álvarez, A. Hernández and S. Luque, _J. Membr. Sci._ , 2014, 470, 219–228.
  * Morison (2008) K. R. Morison, _J. Membr. Sci._ , 2008, 325, 301–310.
  * Kobayashi _et al._ (2012) M. Kobayashi, Y. Terayama, H. Yamaguchi, M. Terada, D. Murakami, K. Ishihara and A. Takahara, _Langmuir_ , 2012, 28, 7212–7222.
  * Olsson and Hartley (2016) T. S. G. Olsson and M. Hartley, _PeerJ_ , 2016, 4, e2674.
  * Feder (1988) J. Feder, _Fractals_ , Plenum press, New York, New York, 1988, pp. 6–27.
  * Lenormand and Zarcone (1989) R. Lenormand and C. Zarcone, _Transport Porous Media_ , 1989, 4, 599–612.
  * Buckley and Leverett (1942) S. E. Buckley and M. C. Leverett, _Trans. AIME_ , 1942, 146, 107–116.
  * Pinder and Gray (2008) G. F. Pinder and W. G. Gray, _Essentials of Multiphase Flow and Transport in Porous Media_ , John Wiley and Sons, Inc., Hoboken, New Jersey, 2008, pp. 83–164.
  * Sahimi (2011) M. Sahimi, _Flow and Transport in Porous Media and Fractured Rock_ , Wiley-VCH Verlag GmbH and Co. KGaA, Weinheim, Germany, 2011, pp. 519–573.
  * Marle (1981) C. Marle, _Multiphase Flow in Porous Media_ , Gulf Publishing Company, Houston, Texas, 1981, pp. 35–92.


# Multiple fixed point theorems for contractive and Meir-Keeler type mappings
defined on partially ordered spaces with a distance

Mitrofan M. Choban1 and Vasile Berinde2,3

###### Abstract.

We introduce and study a general concept of multiple fixed point for mappings
defined on partially ordered distance spaces in the presence of a contraction
type condition and appropriate monotonicity properties. This notion and the
obtained results complement the corresponding ones from [Choban, M., Berinde,
V., A general concept of multiple fixed point for mappings defined on spaces
with a distance (submitted)] and also simplifies some concepts of multiple
fixed point considered by various authors in the last decade or so.

##  1\. Introduction

In a previous paper [32], the authors have introduced and studied a general
concept of multidimensional fixed point for mappings defined on a distance
space and satisfying a certain contraction condition.

Several interesting new results that generalise, extend and unify
corresponding related results from literature for the case of non ordered
distance spaces were obtained. However, the great majority of the
multidimensional fixed point theorems existing in literature were established
in the setting of a partially ordered metric space or of a partially ordered
generalised metric space. Therefore, the main aim of this paper is to study
the concept of multidimensional fixed point introduced in [32] for the case of
mappings defined on partially ordered distance space, thus extending and
complementing most of the results established in [32].

We start by presenting a brief survey on the notion of multidimensional fixed
point, which naturally emerged from the rich literature produced in the last
four decades devoted to coupled fixed points. The concept itself of coupled
fixed point has been first introduced and studied by V. I. Opoitsev, in a
series of papers he published in the period 1975-1986, see [61]-[65]. Opoitsev
has been inspired by some concrete problems arising in the dynamics of
collective behaviour in mathematical economics and considered the coupled
fixed point problem for mixed monotone nonlinear operators which also satisfy
a nonexpansive type condition.

In 1987, Guo and Lakshmikantham [41], apparently not being aware of Opoitsev’s
previous results [61]-[65], have studied coupled fixed points in connection
with coupled quasi-solutions of an initial value problem for ordinary
differential equations. Amongst the subsequent developments we quote the
following works: [39]; [30], containing coupled fixed point results of
$\frac{1}{2}$-$\alpha$-condensing and mixed monotone operators, where $\alpha$
denotes the Kuratowski’s measure of non compactness, thus extending some
previous results from [41] and [79]; [29], which discusses some existence
results and iterative approximation of coupled fixed points for mixed monotone
condensing set-valued operators; [28] where the authors obtained coupled fixed
point results of $\frac{1}{2}$-$\alpha$-contractive and generalized condensing
mixed monotone operators.

More recently, Gnana Bhaskar and Lakshmikantham in [37] established coupled
fixed point results for mixed monotone operators in partially ordered metric
spaces in the presence of a Bancah contraction type condition. Essentialy, the
results by Bhaskar and Lakshmikantham in [37] combined, in the context of
bivariable mixed monotone mappings, the main fixed point results previously
obtained by Nieto and Rodriguez-Lopez [58] and [59], for the case of one
variable increasing and decreasing nonlinear operator, respectively. The last
two papers are, in turn, a continuation of the hybrid fixed point theorem
established in the seminal paper of Ran and Reurings [66], which has the merit
to combine a metrical fixed point theorem (the contraction mapping principle)
and an order theoretic fixed point result (Tarski’s fixed point theorem).

Various applications of the theoretical results in coupled fixed point theory
were also considered, for the case of: a) Uryson integral equations [63]; b) a
system of Volterra integral equations [30], [28]; c) a class of functional
equations arising in dynamic programming [29]; d) initial value problems for
first order differential equations with discontinuous right hand side [41]; e)
(two point) periodic boundary value problems [17], [37], [33], [82]; f)
integral equations and systems of integral equations [3], [6], [9], [24],
[38], [42], [78], [80], [85]; g) nonlinear elliptic problems and delayed
hematopoesis models [84]; h) nonlinear Hammerstein integral equations [76]; i)
nonlinear matrix and nonlinear quadratic equations [4], [24]; j) initial value
problems for ODE [8], [75] etc.

For a very recent account on the developments of coupled fixed point theory,
we also refer to [22].

On the other hand, in 2010, Samet and Vetro [74] apart of some coupled fixed
point results they have established, considered a concept of fixed point of
m-order as a natural extension of the notion of coupled fixed point. Then, in
2011, mainly inspired by [37], Berinde and Borcut [18] introduced the concept
of triple fixed point and proved existence and existence and uniqueness triple
fixed point theorems for three-variable mixed monotone mappings, while, in
2012, Karapinar and Berinde [47], have studied quadruple fixed points of
nonlinear contractions in partially ordered metric spaces.

After these starting papers, a substantial number of articles were dedicated
to the study of triple fixed points, quadruple fixed points, as well as to
multiple fixed points (also called fixed point of m-order, or ”a
multidimensional fixed point”, or ”an $m$-tuplet fixed point”, or ”an
$m$-tuple fixed point”), see [1], [2], [7], [48], [49], [50], [53], [60],
[67]-[72], [81], [83], [86], which form a very selective list contributions.

Starting from this background, the main aim of the present paper is to study
the concept of multidimensional fixed point introduced in [32] but for
mappings defined on partially ordered distance space, in the presence of a
contraction type condition and appropriate monotonicity properties, thus
extending and complementing the results established in [32].

This approach is based on the idea to reduce the study of multidimensional
fixed points and coincidence points to the study of usual one-dimensional
fixed points for an associate operator. Note that, the first author who
reduced the problem of finding a coupled fixed point of mixed monotone
operators to the problem of finding a fixed point of an increasing one
variable operator was Opoitsev, see for example [63].

##  2\. Preliminaries

By a space we understand a topological $T_{0}$-space. We use the terminology
from [36, 40, 73, 31].

Let $X$ be a non-empty set and $d:X\times X\rightarrow\mathbb{R}$ be a mapping
such that:

($i_{m}$) $d(x,y)\geq 0$, for all $x,y\in X$;

($ii_{m}$) $d(x,y)+d(y,x)=0$ if and only if $x=y$.

Then $d$ is called a distance on $X$, while $(X,d)$ is called a distance
space.

Let $d$ be a distance on $X$ and $B(x,d,r)$ = $\\{y\in X:d(x,y)<r\\}$ be the
ball with the center $x$ and radius $r>0$. The set $U\subset X$ is called
$d$-open if for any $x\in U$ there exists $r>0$ such that $B(x,d,r)\subset U$.
The family $\mathcal{T}(d)$ of all $d$-open subsets is the topology on $X$
generated by $d$. A distance space is a sequential space, i.e., a space for
which a set $B\subseteq X$ is closed if and only if together with any sequence
it contains all its limits [36].

Let $(X,d)$ be a distance space, $\\{x_{n}\\}_{n\in\mathbb{N}}$ be a sequence
in $X$ and $x\in X$. We say that the sequence $\\{x_{n}\\}_{n\in\mathbb{N}}$
is:

1) convergent to $x$ if and only if $\lim_{n\rightarrow\infty}d(x,x_{n})=0$.
We denote this by $x_{n}\rightarrow x$ or $x=\lim_{n\rightarrow\infty}x_{n}$
(really, we may denote $x\in\lim_{n\rightarrow\infty}x_{n}$);

2) convergent if it converges to some point $x$ in $X$;

3) Cauchy or fundamental if $\lim_{n,m\rightarrow\infty}d(x_{n},x_{m})=0$.

A distance space $(X,d)$ is called complete if every Cauchy sequence in $X$
converges to some point $x$ in $X$.

Let $X$ be a non-empty set and $d$ be a distance on $X$. Then:

  * •

$(X,d)$ is called a symmetric space and $d$ is called a symmetric on $X$ if

($iii_{m}$) $d(x,y)$ = $d(y,x)$, for all $x,y\in X$;

  * •

$(X,d)$ is called a quasimetric space and $d$ is called a quasimetric on $X$
if

($iv_{m}$) $d(x,z)\leq d(x,y)+d(y,z)$, for all $x,y,z\in X$;

  * •

$(X,d)$ is called a metric space and $d$ is called a metric if $d$ is a
symmetric and a quasimetric, simultaneously.

Let $X$ be a non-empty set and $d(x,y)$ be a distance on $X$ with the
following property:

($N$) for each point $x\in X$ and any $\varepsilon>0$ there exists
$\delta=\delta(x,\varepsilon)>0$ such that from $d(x,y)\leq\delta$ and
$d(y,z)\leq\delta$ it follows $d(x,z)\leq\varepsilon$.

Then $(X,d)$ is called an N-distance space and $d$ is called an N-distance on
$X$. If $d$ is a symmetric, then we say that $d$ is an $N$-symmetric.

Spaces with $N$-distances were studied by V. Niemyzki [56] and by S. I. Nedev
[55]. If $d$ satisfies the condition

($F$) for any $\varepsilon>0$ there exists $\delta=\delta(\varepsilon)>0$ such
that from $d(x,y)\leq\delta$ and $d(y,z)\leq\delta$ it follows
$d(x,z)\leq\varepsilon$,

then $d$ is called an F-distance or a Fréchet distance and $(X,d)$ is called
an F-distance space. Any $F$-distance $d$ is an $N$-distance, too. If $d$ is a
symmetric and an $F$-distance on a space $X$, then we say that $d$ is an
$F$-symmetric.

######  Remark 2.1.

If $(X,d)$ is an $F$-symmetric space, then any convergent sequence is a Cauchy
sequence. For $N$-symmetric spaces and for quasimetric spaces this assertion
is not more true.

If $s>0$ and $d(x,y)\leq s[d(x,z)+d(z,y)]$ for all points $x,y,z\in X$, then
we say that $d$ is an $s$-distance. Any $s$-distance is an $F$-distance.

A distance space $(X,d)$ is called an H-distance space if, for any two
distinct points $x,y\in X$, there exists $\delta=\delta(x,y)>0$ such that
$B(x,d,\delta)\cap B(y,d,\delta)=\varnothing$.

######  Remark 2.2.

A distance space $(X,d)$ is an $H$-distance space if and only if any
convergent sequence in $X$ has a unique limit point.

##  3\. Ordering on Cartesian product of distance spaces

Let $(X,d)$ be a distance space, $m\in\mathbb{N}$ = $\\{1,2,...\\}$. On
$X^{m}$ consider the distances

$d^{m}((x_{1},...,x_{m}),(y_{1},...,y_{m}))=sup\\{d(x_{i},y_{i}):i\leq m\\}$

and

$\bar{d}^{m}((x_{1},...,x_{m}),(y_{1},...,y_{m}))=\sum_{i=1}^{m}d(x_{i},y_{i}).$

Obviously, $(X^{m},d^{m})$ and $(X^{m},\bar{d}^{m})$ are distance spaces, too.

######  Proposition 3.1.

([32]) Let $(X,d)$ be a distance space. Then:

1\. If $d$ is a symmetric, then $(X^{m},d^{m})$ and $(X^{m},\bar{d}^{m})$ are
symmetric spaces, too.

2\. If $d$ is a quasimetric, then $(X^{m},d^{m})$ and $(X^{m},\bar{d^{m}})$
are quasimetric spaces, too.

3\. If $d$ is a metric, then $(X^{m},d^{m})$ and $(X^{m},\bar{d}^{m})$ are
metric spaces, too.

4\. If $d$ is an $F$-distance space, then $(X^{m},d^{m})$ and
$(X^{m},\bar{d}^{m})$ are $F$-distance spaces, too.

5\. If $d$ is an $N$-distance space, then $(X^{m},d^{m})$ and
$(X^{m},\bar{d}^{m})$ are $N$-distance spaces, too.

6\. If $d$ is an $H$-distance space, then $(X^{m},d^{m})$ and
$(X^{m},\bar{d}^{m})$ are $H$-distance spaces, too.

7\. If $(X,d)$ is a $C$-distance space, then $(X^{m},d^{m})$ and
$(X^{m},\bar{d}^{m})$ are $C$-distance spaces, too.

8\. If $(X,d)$ is a complete distance space, then $(X^{m},d^{m})$ and
$(X^{m},\bar{d}^{m})$ are complete distance spaces, too.

9\. If $d$ is an $s$-distance space, then $(X^{m},d^{m})$ and
$(X^{m},\bar{d}^{m})$ are $s$-distance spaces, too.

10\. The spaces $(X^{m},d^{m})$ and $(X^{m},\bar{d}^{m})$ share the same
convergent sequences and the same Cauchy sequences. Moreover, the distances
$d^{m}$ and $\bar{d^{m}}$ are uniformly equivalent, i.e., for each
$\varepsilon>0$, there exists $\delta$ = $\delta(\varepsilon)>0$ such that:

\- from $d^{m}(x,y)\leq\delta$ it follows $\bar{d^{m}}(x,y)\leq\varepsilon$;

\- from $\bar{d^{m}}(x,y)\leq\delta$ it follows $d^{m}(x,y)\leq\varepsilon$.

Let $\preceq$ be a (partial) order on a distance space $(X,d)$. A sequence
$\\{x_{n}\\}_{n\in\mathbb{N}}$ is called:

  * •

non-decreasing if $x_{n}\preceq x_{n+1}$ for each $n\in\mathbb{N}$;

  * •

non-increasing if $x_{n}\succeq x_{n+1}$ for each $n\in\mathbb{N}$;

  * •

monotone if it is either non-decreasing or non-increasing.

If $(X,d,\preceq)$ is an ordered distance space and $g:X\rightarrow X$ is a
mapping, then $(X,d,\preceq)$ is said to have the sequential $g$-monotone
property [37, 68] if it verifies:

i) if $\\{x_{n}\\}_{n\in\mathbb{N}}$ is a non-decreasing sequence and
$\lim\limits_{n\rightarrow\infty}d(x,x_{n})$ = $0$, then $g(x_{n})\preceq
g(x)$ for all $n\in\mathbb{N}$;

ii) if $\\{x_{n}\\}_{n\in\mathbb{N}}$ is a non-increasing sequence and
$\lim\limits_{n\rightarrow\infty}d(x,x_{n})$ = $0$, then $g(x_{n})\preceq
g(x)$ for all $n\in\mathbb{N}$.

An ordered distance space $(X,d,\preceq)$ is called monotonically complete if
any monotone Cauchy sequence in $X$ converges to some point in $X$.

Fix $m\in\mathbb{N}$ and a subset $L\subseteq\\{1,2,...,m\\}$. Like in [67,
68, 69], we introduce on $X$ the ordering $\preceq_{L}$:
$(x_{1},...,x_{m})\preceq_{L}(y_{1},...,y_{m})$ iff $x_{i}\preceq y_{i}$ for
$i\in L$ and $y_{j}\preceq x_{j}$ for $j\notin L$.

By construction, $(X^{m},d^{m},\preceq_{L})$ and
$(X^{m},\bar{d^{m}},\preceq_{L})$ are ordered distance spaces.

If $L\subseteq\\{1,2,...,m\\}$ and $M$ = $\\{1,2,...,m\\}\setminus L$, then
$x\preceq_{L}y$ if and only if $y\preceq_{M}x$ for $x,y\in X^{m}$. Hence
$\preceq_{M}$ is the dual (inverse) order of the order $\preceq_{L}$.

######  Proposition 3.2.

Let $(X,d,\preceq)$ be a monotonically complete distance space. Then
$(X^{m},d^{m},\preceq_{L})$ and $(X^{m},\bar{d^{m}},\preceq_{L})$ are ordered
monotonically complete distance spaces, too.

###### Proof.

It is obvious. ∎

##  4\. Multiple fixed point principles for monotone type operators

Fix $m\in\mathbb{N}$. Denote by $\lambda$ = $(\lambda_{1},...,\lambda_{m})$ a
collection of mappings $\\{\lambda_{i}:\\{1,2,...,m\\}$
$\longrightarrow\\{1,2,...,m\\}:1\leq i\leq m\\}$.

Let $(X,d)$ be a distance space and $F:X^{m}\longrightarrow X$ be an operator.
The operator $F$ and the mappings $\lambda$ generate the operator $\lambda
F:X^{m}\longrightarrow X^{m}$, where

$\lambda F(x_{1},....,x_{m})=(y_{1},...,y_{m})\textnormal{ and
}y_{i}=F(x_{\lambda_{i}(1)},...,x_{\lambda_{i}(m)}),$

for each point $(x_{1},...,x_{m})\in X^{m}$ and any index
$i\in\\{1,2,...,m\\}$.

A point $a$ = $(a_{1},...,a_{m})\in X^{m}$ is called a $\lambda$-multiple
fixed point of the operator $F$ if $a$ = $\lambda F(a)$, i.e., $a_{i}$ =
$F(a_{\lambda_{i}(1)},...,a_{\lambda_{i}(m)})$ for any $i\in\\{1,2,...,m\\}$.

Let $(X,d,\preceq)$ be a (partially) ordered distance space, $m\in\mathbb{N}$,
$L\subseteq\\{1,2,...,m\\}$ and $F:X^{m}\longrightarrow X$ be an operator. In
this context, we consider the following sets of assumptions that include a
symmetric type contractive condition, similar to the symmetric contraction
introduced and used by the second author in [14].

Conditions $\Omega_{1}$:
1\. $(X,\preceq)$ is a lattice;

2\. If $x,y,z\in X$ and $x\preceq y\preceq z$, then $d(x,y)+d(y,x)\leq
d(x,z)+d(z,x)$;

3\. If $x,y\in X_{m}$, $x\not=y$ and $x\preceq_{L}y$, then $\lambda
F(x)\preceq_{L}\lambda F(y)$ and $d^{m}(\lambda F(x),\lambda F(y))$ \+
$d^{m}(\lambda F(y),\lambda F(x))$ $<d^{m}(x,y)$ \+ $d^{m}(y,x)$.

Conditions $\Omega_{2}$:
1\. $(X,\preceq)$ is a lattice;

2\. If $x,y,z\in X$ and $x\preceq y\preceq z$, then $d(x,y)+d(y,x)\leq
d(x,z)+d(z,x)$;

3\. If $x,y\in X_{m}$, $x\not=y$ and $x\preceq_{L}y$, then $\lambda
F(y)\preceq_{L}\lambda F(x)$ and $d^{m}(\lambda F(x),\lambda F(y))$ \+
$d^{m}(\lambda F(y),\lambda F(x))$ $<d^{m}(x,y)$ \+ $d^{m}(y,x)$.

Conditions $\Omega_{3}$:
1\. $(X,\preceq)$ is a lattice;

2\. If $x,y,z\in X$ and $x\preceq y\preceq z$, then $d(x,y)+d(y,x)\leq
d(x,z)+d(z,x)$;

3\. For any $i\in\\{1,2,\dots,m\\}$ the mapping $\lambda_{i}$ is a surjection
or, more generally, $|\cup\\{\lambda_{i}^{-1}(j):1\leq j\leq m\\}|=m$, for
each $i\in\\{1,2,\dots,m\\}$;

4\. If $x,y\in X_{m}$, $x\not=y$ and $x\preceq_{L}y$, then $\lambda
F(x)\preceq_{L}\lambda F(y)$ and $\bar{d}^{m}(\lambda F(x),\lambda F(y))$ \+
$\bar{d}^{m}(\lambda F(y),\lambda F(x))$ $<\bar{d}^{m}(x,y)$ \+
$\bar{d}^{m}(y,x)$.

Conditions $\Omega_{4}$:
1\. $(X,\preceq)$ is a lattice;

2\. If $x,y,z\in X$ and $x\preceq y\preceq z$, then $d(x,y)+d(y,x)\leq
d(x,z)+d(z,x)$;

3\. For any $i\in\\{1,2,\dots,m\\}$ the mapping $\lambda_{i}$ is a surjection
or, more generally, $|\cup\\{\lambda_{i}^{-1}(j):1\leq j\leq m\\}|=m$, for
each $i\in\\{1,2,\dots,m\\}$;

4\. If $x,y\in X_{m}$, $x\not=y$ and $x\preceq_{L}y$, then $\lambda
F(y)\preceq_{L}\lambda F(x)$ and $\bar{d}^{m}(\lambda F(x),\lambda F(y))$ \+
$\bar{d}^{m}(\lambda F(y),\lambda F(x))$ $<\bar{d}^{m}(x,y)$ \+
$\bar{d}^{m}(y,x)$.

Now we can state concisely the following general and comprehensive
multidimensional fixed point result.

######  Theorem 4.1.

Let $a\in X^{m}$ be a multidimensional fixed point of the operator of
$F:X^{m}\longrightarrow X$. If any of the Conditions $\Omega_{i}$,
$i\in\\{1,2,3,4\\}$, is satisfied, then the operator $F$ has a unique
multidimensional fixed point.

###### Proof.

Obviously, $(X^{m},\preceq_{L})$ is a lattice, too. Let $\rho$ = $d^{m}$, for
$i\in\\{1,2\\}$, and $\rho$ = $\bar{d}^{m}$, for $i\in\\{3,4\\}$. Then for
$x,y,z\in X^{m}$ and $x\preceq_{L}y\preceq_{L}z$ we have
$\rho(x,y)+\rho(y,x)\leq\rho(x,z)+\rho(z,x)$. Assume that $b\in X^{m}$ is a
multidimensional fixed point of the operator $F$ with $b\not=a$.

Case 1. The points $a$ and $b$ are comparable.

Assume that $a\preceq_{L}b$. Then $\rho(a,b)$ \+ $\rho(b,a)$ = $\rho(\lambda
F(a),\lambda F(b))$ \+ $\rho(\lambda F(b),\lambda F(a))$ $<\rho(a,b)$ \+
$\rho(b,a)$, a contradiction.

Case 2. The points $a$ and $b$ are not comparable.

Fix $c=\max\\{a,b\\}\in X_{n}$. We put $d$ = $\lambda F(\lambda F(c))$. By
construction, $a\preceq_{L}d$ and $b\preceq_{L}d$. Hence, $c\preceq_{L}d$ and
$\rho(a,c)\leq\rho(a,d)$, $\rho(b,c)\leq\rho(b,d)$. Therefore
$\rho(a,c)+\rho(c,a)$ $\leq\rho(a,d)+\rho(d(a)$. By virtue of the conditions
$\Omega_{i}$, we then have $\rho(a,d)+\rho(d,a)$ $<\rho(a,c)+\rho(c,a)$, a
contradiction. ∎

Now, according to [52], consider the following two classes of Meir-Keeler type
assumptions.

Conditions $MK_{1}$:
1\. For any two points $x,y\in X$ there exists an upper bound and a lower
bound;

2 (Meir-Keeler monotone contraction condition). There exists a function
$\delta:(0,+\infty)\longrightarrow(0,+\infty)$ such that from $r>0$, $x,y\in
X$, $d(x,y)<r+\delta(r)$ and $x\preceq y$ it follows that $d(x,y)<r$;

3\. If $x,y\in X_{m}$, $x\preceq_{L}y$, then $\lambda F(x)\preceq_{L}\lambda
F(y)$.

Conditions $MK_{2}$:
1\. For any two points $x,y\in X$ there exist an upper bound and a lower
bound;

2 (Meir-Keeler monotone contraction condition). There exists a function
$\delta:(0,+\infty)\longrightarrow(0,+\infty)$ such that from $r>0$, $x,y\in
X$, $d(x,y)<r+\delta(r)$ and $x\preceq y$ it follows that $d(x,y)<r$;

3\. If $x,y\in X_{m}$, $x\preceq_{L}y$, then $\lambda F(y)\preceq_{L}\lambda
F(x)$.

######  Theorem 4.2.

Let $(X,d)$ be an $H$-distance space and let $a\in X^{m}$ be a
multidimensional fixed point of the operator of $F:X^{m}\rightarrow X$. Then
in any of the Conditions $MK_{i}$, $i\in\\{1,2\\}$, the operator $F$ has a
unique multidimensional fixed point.

###### Proof.

Obviously, in $(X^{m},\preceq_{L})$, for any two points $x,y\in X^{m}$, there
exist an upper bound and a lower bound. Let $\rho$ = $d^{m}$. In this case for
any two points $x,y\in X^{m}$, from the condition $\rho(x,y)<r+\delta(r)$ and
$x\preceq y$, it follows that $\rho(x,y)<r$.

Assume that $b\in X^{m}$ is a multidimensional fixed point of the operator of
$F$ and that $b\not=a$.

Case 1. The points $a$ and $b$ are comparable.

Assume that $a\preceq Lb$ and $\rho(a,b)=r>0$. Since $\rho(a,b)<r+\delta(r)$,
we have $r$ = $\rho(a,b)$ = $\rho(\lambda F(a),\lambda F(b))<r$, a
contradiction.

Case 2. The points $a$ and $b$ are not comparable. We put $r$ =
$\inf\\{max\\{\rho(a,c),\rho(b,c)\\}:c\in
X^{m},a\preceq_{L}c,b\preceq_{L}c\\}$. We claim that $r$ = 0. Assume that
$r>0$. Then $\delta(r)>0$ and there exists $c$ such that
$\max\\{\rho(a,c),\rho(b,c)\\}<r+\delta(r)$, $a\preceq_{L}c$, $b\preceq_{L}c$.
We put $e$ = $\lambda F(\lambda F(c))$. Then $a\preceq_{L}e$ and
$b\preceq_{L}e$. Since $\lambda F(\lambda F(a))$ = $a$ and $\lambda F(\lambda
F(b))$ = $b$, we have $\max\\{\rho(a,e),\rho(b,e)\\}<r$, a contradiction. Thus
$r$ = $0$. For each $n\in\mathbb{N}$ there exists a point $c_{n}\in X^{n}$
such that $a\preceq_{L}c_{n}$, $b\preceq_{L}c_{n}$ and
$max\\{\rho(a,c_{n}),\rho(b,c_{n})\\}<2^{-n}$. We can construct a sequence
$\\{c_{n}:n\in\mathbb{N}\\}$ for which $a$ = $\lim_{n\rightarrow\infty}c_{n}$
and $b$ = $\lim_{n\rightarrow\infty}c_{n}$, a contradiction. ∎

In the particular case $m=2$, the following theorems were proved in [21].
Their proofs in the general case are similar and we omit them.

######  Theorem 4.3.

Let $(X,d,\preceq)$ be an ordered metric space, $m\in\mathbb{N}$,
$F:X^{m}\rightarrow X$ be an operator. Suppose that:

a) there exists a function $\delta:(0,+\infty)\longrightarrow(0,+\infty)$ such
that from $r>0$, $x,y\in X^{m}$, $d^{m}(x,y)<r+\delta(r)$ and $x\preceq y$ it
follows that $d^{m}(\lambda F(x),\lambda F(y))<r$;

b) for any two points $x,y\in X$ there exists an upper bound and a lower
bound.

Suppose also that one of the following sets of conditions is satisfied:

1\. $(X,d,\preceq)$ is monotonically complete; from $x,y\in X^{m}$ and
$x\preceq_{L}y$ it follows that $\lambda F(x)\preceq_{L}\lambda F(y)$; there
exists $a\in X^{m}$ such that $a\preceq_{L}\lambda F(a)$.

2\. $(X,d,\preceq)$ is complete; from $x,y\in X^{m}$ and $x\preceq_{L}y$ it
follows that $\lambda F(y)\preceq L\lambda F(x)$; there exists $a\in X^{m}$
such that $a\preceq_{L}\lambda F(a)$ or $F(a)\preceq_{L}a$.

Then there exists a unique multidimensional fixed point of the operator of
$F$.

######  Theorem 4.4.

Let $(X,d,\preceq)$ be an ordered metric space, $m\in\mathbb{N}$,
$F:X^{m}\rightarrow X$ be an operator. Suppose that:

a) there exists a function $\delta:(0,+\infty)\longrightarrow(0,+\infty)$ such
that from $r>0$, $x,y\in X^{m}$, $\bar{d}^{m}(x,y)<r+\delta(r)$ and $x\preceq
y$ it follows that $\bar{d}^{m}(\lambda F(x),\lambda F(y))<r$;

b) for any two points $x,y\in X$, there exist an upper bound and a lower
bound; for any $i\in\\{1,2,\dots,m\\}$, the mapping $\lambda_{i}$ is a
surjection or, more generally, $|\cup\\{\lambda_{i}^{-1}(j):1\leq j\leq
m\\}|=m$, for each $i\in\\{1,2,\dots,m\\}$.

Suppose also that one of the following sets of conditions is satisfied:

1\. $(X,d,\preceq)$ is monotonically complete; from $x,y\in X^{m}$ and
$x\preceq_{L}y$ it follows that $\lambda F(x)\preceq_{L}\lambda F(y)$; there
exists $a\in X^{m}$ such that $a\preceq_{L}\lambda F(a)$.

2\. $(X,d,\preceq)$ is complete; from $x,y\in X^{m}$ and $x\preceq_{L}y$ it
follows that $\lambda F(y)\preceq L\lambda F(x)$; there exists $a\in X^{m}$
such that $a\preceq_{L}\lambda F(a)$ or $F(a)\preceq_{L}a$.

Then there exists a unique multidimensional fixed point of the operator of
$F$.

##  5\. Some particular cases and a generic application of multiple fixed
points

If we take concrete values of $m\in\mathbb{N}$ and consider various particular
functions $\lambda$ =
$\\{\lambda_{i}:\\{1,...,m\\}\rightarrow\\{1,...,m\\}:1\leq i\leq m\\}$ then,
most of the concepts of coupled, triple, quadruple,…, multiple fixed point
theory existing in literature are obtained as particular cases of the concept
of multiple fixed point considered in [32] and the present paper.

For example, if $m=2$, $\lambda_{1}(1)=1$, $\lambda_{1}(2)=2$;
$\lambda_{2}(1)=2$, $\lambda_{2}(2)=1$, we obtain the incept of coupled fixed
point studied in [37] and in various subsequent papers. If $m=3$,
$\lambda_{1}(1)=1$, $\lambda_{1}(2)=2$, $\lambda_{1}(3)=3$;
$\lambda_{2}(1)=2$, $\lambda_{2}(2)=1$, $\lambda_{2}(3)=2$;
$\lambda_{3}(1)=3$, $\lambda_{3}(2)=2$, $\lambda_{3}(3)=1$, then the concept
of multiple fixed point studied in the present paper reduces to that of triple
fixed point, first introduced in [18] and intensively studied in many other
research works emerging from it.

We note that, as pointed out in [77], the notion of tripled fixed point due to
Berinde and Borcut [18] is different from the one defined by Samet and Vetro
[74] for $N=3$, since in the case of ordered metric spaces in order to keep
the mixed monotone property working, it is necessary to take
$\lambda_{2}(3)=2$ and not $\lambda_{2}(3)=3$.

It is also important to mention here that some cases of multidimensional
coincidence points results (that extend multiple fixed point theorems) are not
compatible with the mixed monotone property (see [7]).

For other concepts of multiple fixed points considered in literature the
condition ” $\lambda_{i}$ is a surjection, for each $i\leq m$” is no more
valid, see for example [18] and the research papers emerging from it, while
the second condition, $|\cup\\{\lambda_{i}^{-1}(j):1\leq j\leq m\\}|=m$, for
each $i\leq m$, is satisfied.

Finally, we point out the fact that our approach in [32] and in this paper is
based on the idea to obtain general multiple fixed point theorems by reducing
this problem to a unidimensional fixed point problem and by simultaneously
working in a more general and very reliable setting, i.e., that of distance
space.

Many other related and relevant results could be obtained in the same way, by
reducing the multidimensional fixed point problem to many other independent
unidimensional fixed point principles, like the ones established in [5], [11],
[12], [13], [15], [16], [19], [21], [23] etc.

We end the paper by indicating an interesting generic application of multiple
fixed points in game theory.

Fix an orderable distance space $(X,d,\preceq)$ and a positive integer number
$m\geq 2$. We put $\mathbb{N}$ = $\\{1,2,...\\}$ and $\mathbb{N}_{m}$ =
$\\{1,2,...,m\\}$. For $L\subseteq\mathbb{N}$, we introduce on $X$ the
ordering $\preceq_{L}$: $(x_{1},...,x_{m})\preceq_{L}(y_{1},...,y_{m})$ iff
$x_{i}\preceq y_{i}$ for $i\in L$ and $y_{j}\preceq x_{j}$ for $j\notin L$.

Denote by $\Lambda$ = $(\lambda_{1},...,\lambda_{m})$ a collection of mappings
$\\{\lambda_{i}:\mathbb{N}_{m}$ $\longrightarrow\mathbb{N}_{m}:i\leq m\\}$.

Let $F:X^{m}\longrightarrow X$ be an operator. The operator $F$ and the
mappings $\Lambda$ generate the operator $\Lambda F:X^{m}\longrightarrow
X^{m}$, where $\lambda F(x_{1},....,x_{m})$ = $(y_{1},...,y_{m})$ and $y_{i}$
= $F(x_{\lambda_{i}(1)},...,x_{\lambda_{i}(m)})$ for each point
$(x_{1},...,x_{m})\in X^{m}$ and any index $i\leq m$.

Assume now that $\mathbb{N}_{m}$ is the set of players and
$i\in\mathbb{N}_{m}$ is the symbol of the $i$th player. In this case we say
that:

\- $X$ is the space of the positions (decisions) of the players;

\- $d(x,y)$ is the measure of the non-convenience of the position $x$
relatively to the position $y$;

\- ordering $\leq$ is the relation of domination of positions;

\- a point $x$ = $(x_{1},x_{2},...,x_{m})\in X^{m}$ is a selection of
positions, where $x_{i}$ is the position of the player $i$;

\- the operator $F$ is the operator of correction of the positions.

Every selection of positions $x$ = $(x_{1},x_{2},...,x_{m})\in X^{m}$
determines the selection of positions $y$ = $(y_{1},y_{2},...,y_{m})$ =
$\Lambda F(x)\in X^{m}$.

For any player $i$ the number $d(x_{i},y_{i})$ is the measure of the non-
convenience of the position $x_{i}$ relatively to the position $y_{i}$ for the
$i^{th}$ player.

One considers that the selection of the positions $x$ =
$(x_{1},x_{2},...,x_{m})\in X^{m}$ is optimal if $d(x_{i},y_{i})$ is minimal
for each $i$.

In particular, if $\Lambda F(x)$ = $x$, then the selection of positions $x$ is
optimal.

One can find distinct concrete examples of the above general model in [54],
[51], [61], [62].

## Acknowledgements

This second author acknowledges the support provided by the Deanship of
Scientific Research at King Fahd University of Petroleum and Minerals for
funding this work through the projects IN151014 and IN141047.

## References

  * [1] Agarwal, R., Karapinar, E., Roldán-López-de-Hierro, A.-F., Some remarks on ‘Multidimensional fixed point theorems for isotone mappings in partially ordered metric spaces’. Fixed Point Theory Appl. 2014, 2014:245, 13 pp.
  * [2] Agarwal, R., Karapinar, E., Roldán-López-de-Hierro, A.-F., Fixed point theorems in quasi-metric spaces and applications to multidimensional fixed point theorems on $G$-metric spaces. J. Nonlinear Convex Anal. 16 (2015), no. 9, 1787–1816.
  * [3] Aghajani, A., Abbas, M. and Kallehbasti, E. P., Coupled fixed point theorems in partially ordered metric spaces and application, Math. Commun., 17 (2002), No. 2, 497–509
  * [4] Aghajani, A. and Arab, R., Fixed points of $(\psi,\varphi,\theta)$-contractivemappings in partially ordered $b$-metric spaces and application to quadratic integral equations, Fixed Point Theory Appl. 2013, 2013:245 doi:10.1186/1687-1812-2013-245
  * [5] Alghamdi, M. A., Berinde, V., Shahzad, N., Fixed Points of Multivalued Nonself Almost Contractions, J. Appl. Math. 2013, 2013: 621614.
  * [6] Alghamdi, M. A., Hussain, N. and Salimi, P., Fixed point and coupled fixed point theorems on $b$-metric-like spaces, J. Ineq. Appl. 2013, 2013:402 doi:10.1186/1029-242X-2013-402
  * [7] Al-Mezel, S. A., Alsulami, H. H., Karapinar, E. and Lopez-de-Hierro, A.-F. R., Discussion on ”Multidimensional Coincidence Points” via recent publications, Abstr. Appl. Anal. 2014, Art. ID 287492, 13 pp.
  * [8] Amini-Harandi, A., Coupled and tripled fixed point theory in partially ordered metric spaces with application to initial value problem, Math. Comput. Model., 57 (2013), Nos 9–10, 2343–2348.
  * [9] Aydi, H., Samet, B. and Vetro, C. Coupled fixed point results in cone metric spaces for $\tilde{w}$-compatible mappings, Fixed Point Theory Appl. 2011, 2011:27 doi:10.1186/1687-1812-2011-27
  * [10] Bakhtin, I. A., The contraction mapping principle in almost metric spaces (in Russian), Funct. Anal., Ulianovskii Gosud. Pedag. Inst. 30 (1989), 26–37.
  * [11] Berinde, V., A common fixed point theorem for compatible quasi contractive self mappings in metric spaces. Appl. Math. Comput. 213 (2009), no. 2, 348–354.
  * [12] Berinde, V., Approximating common fixed points of noncommuting discontinuous weakly contractive mappings in metric spaces. Carpathian J. Math. 25 (2009), no. 1, 13–22.
  * [13] Berinde, V., Common fixed points of noncommuting discontinuous weakly contractive mappings in cone metric spaces. Taiwanese J. Math. 14 (2010), no. 5, 1763–1776.
  * [14] Berinde, V., Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces, Nonlinear Anal. 74 (2011) 7347–7355.
  * [15] Berinde, V., Stability of Picard iteration for contractive mappings satisfying an implicit relation. Carpathian J. Math. 27 (2011), no. 1, 13–23.
  * [16] Berinde, V., Coupled coincidence point theorems for mixed monotone nonlinear operators. Comput. Math. Appl. 64 (2012), no. 6, 1770–1777.
  * [17] Berinde, V., Coupled fixed point theorems for $\Phi$-contractive mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal. 75 (2012), no. 6, 3218–3228.
  * [18] Berinde, V., Borcut, M., Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. Nonlinear Anal. 74 (2011), no. 15, 4889–4897.
  * [19] Berinde, V. and Choban, M. M., Remarks on some completeness conditions involved in several common fixed point theorems, Creat. Math. Inform. 19 (2010), no. 1, 1–10.
  * [20] Berinde, V. and Choban, M. M., Generalized distances and their associate metrics. Impact on fixed point theory, Creat. Math. Inform. 22 (2013), no. 1, 23–32.
  * [21] Berinde, V., Păcurar, M., Coupled fixed point theorems for generalized symmetric Meir-Keeler contractions in ordered metric spaces. Fixed Point Theory Appl. 2012, 2012:115, 11 pp.
  * [22] Berinde, V., Păcurar, M., Coupled and triple fixed points theorems for mixed monotone almost contractive mappings in partially ordered metric spaces (submitted)
  * [23] Berinde, V., Păcurar, M., A constructive approach to coupled fixed point theorems in metric spaces, Carpathian J. Math. 31 (2015), No. 3, 269–275
  * [24] Berzig, M, Samet, B. An extension of coupled fixed points concept in higher dimension and applications, Comput. Math. Appl. 63 (2012), 1319–1334.
  * [25] Borcut, M., Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces, Appl. Math. Comput. 218 (2012), 7339-7346.
  * [26] Borcut, M., Puncte triple fixe pentru operatori definiţi pe spaţii metrice parţial ordonate, Risoprint, Cluj-Napoca, 2016.
  * [27] Borcut, M., Berinde, V., Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces, Appl. Math. Comput. 218 (2012), 5929–5936.
  * [28] Chang, S.-S., Cho, Y. J., Huang, N. J., Coupled fixed point theorems with applications. J. Korean Math. Soc. 33 (1996), no. 3, 575–585.
  * [29] Chang, S.-S., Ma, Y. H., Coupled fixed points for mixed monotone condensing operators and an existence theorem of the solutions for a class of functional equations arising in dynamic programming. J. Math. Anal. Appl. 160 (1991), no. 2, 468–479.
  * [30] Chen, Y. Z., Existence theorems of coupled fixed points. J. Math. Anal. Appl. 154 (1991), no. 1, 142–150.
  * [31] Choban, M., Fixed points of mappings defined on spaces with distance, Carpathian J. Math., 32 (2016), no. 2, 173–188.
  * [32] Choban, M., Berinde, V., A general concept of multiple fixed point for mappings defined on spaces with a distance (submitted)
  * [33] Ćirić, L., Damjanović, B. , Jleli, M. and B. Samet, Coupled fixed point theorems for generalized Mizoguchi-Takahashi contractions with applications, Fixed Point Theory Appl. 2012, 2012:51 doi:10.1186/1687-1812-2012-51
  * [34] Czerwik, S., Fixed Points Theorems and Special Solutions of Functional Equations, Katowice, 1980.
  * [35] Dalal, S., Khan, L.A., Masmali, I., Radenovic, S., Some remarks on multidimensional fixed point theorems in partially ordered metric spaces. J. Adv. Math. 7 (2014), no. 1, 1084–1094.
  * [36] Engelking, R., General topology. Translated from the Polish by the author. Second edition. Sigma Series in Pure Mathematics, 6. Heldermann Verlag, Berlin, 1989.
  * [37] Gnana Bhaskar, T., Lakshmikantham, V., Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 65 (2006), no. 7, 1379–1393.
  * [38] Gu, F. and Yin, Y., A new common coupled fixed point theorem in generalized metric space and applications to integral equations, Fixed Point Theory Appl. 2013, 2013:266 doi:10.1186/1687-1812-2013-266
  * [39] Guo, D. J., Fixed points of mixed monotone operators with applications. Appl. Anal. 31 (1988), no. 3, 215–224.
  * [40] Granas, A. and Dugundji, J., Fixed point theory, Springer, Berlin, 2003.
  * [41] Guo, D. J., Lakshmikantham, V., Coupled fixed points of nonlinear operators with applications. Nonlinear Anal. 11 (1987), no. 5, 623–632.
  * [42] Hussain, N., Salimi, P. and Al-Mezel, S., Coupled fixed point results on quasi-Banach spaces with application to a system of integral equations, Fixed Point Theory Appl. 2013, 2013:261 doi:10.1186/1687-1812-2013-261
  * [43] Imdad, M., Soliman, A. H., Choudhury, B. S. and Das, P., On n- tupled coincidence and common fixed points results in metric spaces. J. Oper. 2013, Article ID 532867, 9 pages.
  * [44] Imdad, M., Sharma, A., Rao, K.P.R., n-tupled coincidence and common fixed point results for weakly contractive mappings in complete metric spaces. Bull. Math. Anal. Appl. 5 (2013), no. 4, 19–39.
  * [45] Imdad, M., Alam, A., Soliman, A.H., Remarks on a recent gen- eral even-tupled coincidence theorem. J. Adv. Math. 9 (2014), no. 1, 1787–1805.
  * [46] Karapinar, E., Quartet fixed points theorems for nonlinear contractions in partially ordered metric space, arXiv:1106.5472v1 [math.GN] 27 Jun 2011, 10 p.
  * [47] Karapinar E. and Berinde, V., Quadruple fixed points theorems for nonlinear contractions in partially ordered spaces, Banach J. Math. Anal. 6 (2012), no. 1, 74–89.
  * [48] Karapinar, E., Roldán, A., A note on ‘n-tuplet fixed point theorems for contractive type mappings in partially ordered metric spaces’. J. Inequal. Appl. 2013, 2013:567, 7 pp.
  * [49] Karapinar, E., Roldán, A., Martínez-Moreno, J., Roldán, C., Meir-Keeler type multidimensional fixed point theorems in partially ordered metric spaces. Abstr. Appl. Anal. 2013, Art. ID 406026, 9 pp.
  * [50] Lee, H., Kim, S., Multivariate coupled fixed point theorems on ordered partial metric spaces. J. Korean Math. Soc. 51 (2014), no. 6, 1189–1207.
  * [51] Malishevsky, A. V., Models of many goal-seeking elements combined functioning. I, Automation and Remote Control 33, 1972, no. 11, 1828–1845 (Transl. from Russian: Avtomat. i Telemeh. 1972, no. 11, 92–110).
  * [52] Meir, A., Keeler, E., A theorem on contraction mappings. J. Math. Anal. Appl. 28 (1969), 326–329.
  * [53] Mutlu, A. Gürdal, U., An infinite dimensional fixed point theorem on function spaces of ordered metric spaces. Kuwait J. Sci. 42 (2015), no. 3, 36–49.
  * [54] Nikaido, H., Convex Structures and Economic Theory, Mathematics in Science and Engineering. 51. New York, Academic Press, 1968.
  * [55] Nedev, S. I., o-metrizable spaces, Trudy Moskov. Mat.Ob-va 24 (1971), 201–236 (English translation: Trans. Moscow Math. Soc. 24 (1974), 213–247).
  * [56] Niemytzki, V., On the third axiom of metric spaces, Trans Amer. Math. Soc. 29 (1927), 507–513.
  * [57] Niemytzki, V., Über die Axiome des metrischen Raumes, Math. Ann. 104 (1931), 666–671.
  * [58] Nieto, J. J., Rodriguez-Lopez, R., Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005), no. 3, 223–239 (2006).
  * [59] Nieto, J. J., Rodriguez-Lopez, R., Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta. Math. Sin., (Engl. Ser.) 23(2007), no. 12, 2205–2212.
  * [60] Olaoluwa, H., Olaleru, J., Multipled fixed point theorems in cone metric spaces. Fixed Point Theory Appl. 2014, 2014:43, 15 pp.
  * [61] Opoitsev, V. I., Heterogeneous and combined-concave operators, Sib. Mat. Zh. 16 (1975), nr. 4, 597–605 (Transl. from Russian: Sib. Matem. Jurn. 16 (1975), no. 4, 781–792).
  * [62] Opoitsev, V. I., Dynamics of collective behavior. III. Heterogenic systems, translated from Avtomat. i Telemeh. 1975, no. 1, 124–138 Automat. Remote Control 36 (1975), no. 1, 111–124.
  * [63] Opoitsev, V. I., Generalization of the theory of monotone and concave operators, Tr. Mosk. Mat. Obs., 36 (1978), 237–273.
  * [64] Opoitsev, V. I.; Khurodze, T. A., Nelineinye operatory v prostranstvakh s konusom. [Nonlinear operators in spaces with a cone, Tbilis. Gos. Univ., Tbilisi, 1984. 271 pp.
  * [65] Opoitsev, V. I., Nelineinaya sistemostatika. [Nonlinear systemostatics] Ekonomiko-Matematicheskaya Biblioteka [Library of Mathematical Economics], 31. Nauka, Moscow, 1986.
  * [66] Ran, A. C. M., Reurings, M. C. B., A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004), no. 5, 1435–1443.
  * [67] Roldán, A., Martínez-Moreno, J., Roldán, C., Multidimensional fixed point theorems in partially ordered complete metricspaces, J. Math. Anal. Appl. 396 (2012), 536–545.
  * [68] Roldán, A., Martínez-Moreno, J., Roldán C. and Karapinar, E., Multidimensional Fixed-Point Theorems in Partially Ordered Complete Partial Metric Spaces under $(\psi,\varphi)$-Contractivity Conditions, Abstr. Appl. Anal. 2013, Art. ID 634371, 12 pp.
  * [69] Roldán, A., Martínez-Moreno, J., Roldán, C. and Karapinar, E., Meir-Keeler type multidimensional fixed point theorems inpartially ordered metric spaces, Abstr. Appl. Anal. 2013, Art. ID 406026, 9 pp.
  * [70] Roldán, A., Martínez-Moreno, J., Roldán, C., Karapinar, E. Some remarks on multidimensional fixed point theorems. Fixed Point Theory 15 (2014), no. 2, 545 558.
  * [71] Roldán, A., Martínez-Moreno, J., Roldán, C., Cho, Y. J., Multidimensional fixed point theorems under $(\psi,\phi)$-contractive conditions in partially ordered complete metric spaces. J. Comput. Appl. Math. 273 (2015), 76–87.
  * [72] Rus, M.-D., The fixed point problem for systems of coordinate-wise uniformly monotone operators and applications. Mediterr. J. Math. 11 (2014), no. 1, 109 122.
  * [73] Rus, I. A., Petruşel, A. and Petruşel, G., Fixed Point Theory, Cluj University Press, Cluj-Napoca, 2008.
  * [74] Samet, B., Vetro, C., Coupled fixed point, f-invariant set and fixed point of N-order, Ann. Funct. Anal. 1 (2010), 46–56.
  * [75] Samet, B., Vetro, C. and Vetro, P., Fixed point theorems for alpha-psi-contractive type mappings, Nonlinear Anal. 75 (2012), No. 4, 2154–2165.
  * [76] Sang, Y., A class of $\varphi$-concave operators and applications, Fixed Point Theory Appl. 2013, 2013:274 doi:10.1186/1687-1812-2013-274
  * [77] Sharma, A., Imdad, M., Alam, A., Shorter proofs of some recent even-tupled coincidence theorems for weak contractions in ordered metric spaces. Math. Sci. (Springer) 8 (2014), no. 4, 131–138.
  * [78] Shatanawi, W., Samet, B. and Abbas, M., Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces, Math. Comput. Model. 55 (2012), Nos 3-4, 680–687
  * [79] Shendge, G. R., Dasare, V. N., Existence of maximal and minimal quasifixed points of mixed monotone operators by iterative technique. in Methods of functional analysis in approximation theory (Bombay, 1985), pp. 401–410, Internat. Schriftenreihe Numer. Math., 76, Birkhäuser, Basel, 1986.
  * [80] Sintunavarat, W., Kumam, P. and Cho, Y. J., Coupled fixed point theorems for nonlinear contractions without mixed monotone property, Fixed Point Theory Appl. 2012, 2012:170 doi:10.1186/1687-1812-2012-170
  * [81] Soleimani R. G., Shukla, S. and Rahimi, H., Some relations between $n$-tuple fixed point and fixed point results. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 109 (2015), no. 2, 471–481.
  * [82] Urs, C., Coupled fixed point theorems and applications to periodic boundary value problems, Miskolc Math. Notes, 14 (2013), no. 1, 323–333
  * [83] Wang, S., Multidimensional fixed point theorems for isotone mappings in partially ordered metric spaces. Fixed Point Theory Appl. 2014, 2014:137, 13 pp.
  * [84] Wu, J. and Liu, Y., Fixed point theorems for monotone operators and applications to nonlinear elliptic problems, Fixed Point Theory Appl. 2013, 2013:134 doi:10.1186/1687-1812-2013-134
  * [85] Xiao, J.-Z., Zhu, X.-H. and Shen, Z.-M., Common coupled fixed point results for hybrid nonlinear contractions in metric spaces, Fixed Point Theory  14 (2013), No. 1, 235–249
  * [86] Zhu, L., Zhu, C.-X., Chen, C.-F., Stojanović, Ž., Multidimensional fixed points for generalized $\psi$-quasi-contractions in quasi-metric-like spaces. J. Inequal. Appl. 2014, 2014:27, 15 pp.

a Department of Physics, Mathematics and Information Technologies
Tiraspol State University
Gh. Iablocikin 5., MD2069 Chişinău, Republic of Moldova

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b Department of Mathematics and Computer Science

North University Center at Baia Mare

Technical University of Cluj-Napoca

Victoriei 76, 430072 Baia Mare ROMANIA

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c Department of Mathematics and Statistics

King Fahd University of Petroleum and Minerals

Dhahran, Saudi Arabia

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