# A no-go theorem for superpositions of causal orders

Fabio Costa<EMAIL_ADDRESS>Centre for Engineered Quantum Systems, School
of Mathematics and Physics, The University of Queensland, St Lucia, QLD 4072,
Australia

###### Abstract

The causal order of events need not be fixed: whether a bus arrives before or
after another at a certain stop can depend on other variables—like traffic.
Coherent quantum control of causal order is possible too and is a useful
resource for several tasks. However, quantum control implies that a
controlling system carries the which-order information—if the control is
traced out, the order of events remains in a probabilistic mixture. Can the
order of two events be in a pure superposition, uncorrelated with any other
system? Here we show that this is not possible for a broad class of processes:
a pure superposition of any pair of Markovian, unitary processes with equal
local dimensions and different causal orders is not a valid process, namely it
results in non-normalised probabilities when probed with certain operations.
The result imposes constraints on novel resources for quantum information
processing and on possible processes in a theory of quantum gravity.

##  1 Introduction

Quantum superpositions can be viewed as generalisations of classical
probabilities: if classically we can be uncertain between two alternatives,
assigning to each a probability, quantumly we should be able to consider a
superposition of the two, replacing probabilities with complex amplitudes.
Feynman’s sum-over-histories approach famously leverages this intuition [1].
Despite this view, one typically considers superpositions of _states_ ,
whereas classical probabilities can be assigned to _any_ logical statement. To
what extent is it possible to generalise the superposition principle, beyond
its original range of applicability? Providing a general, principled answer to
this question appears problematic because, unlike classical probabilities,
quantum superpositions have no clear interpretation in terms of subjective
lack of knowledge.

Of particular interest is the case of causal relations. A main motivation
comes from combining the principles of quantum theory and general relativity.
In a regime where spacetime itself becomes subject to the laws of quantum
mechanics, we expect that causal structure, too, will become nonclassical [2,
3, 4]. A second, more practical, motivation is that quantum causal structures
can also be realised experimentally (on a classical spacetime background) [5,
6, 7, 8, 9, 10, 11, 12] possibly leading to an advantage in solving
computational and information-theoretic tasks [13, 14, 15, 16, 17, 18, 19, 20,
21].

Figure 1: The quantum switch. The state of a control system determines the
order of two operations on a target system.

Despite much recent research, the scenarios considered so far do not support a
direct interpretation as superpositions of causal orders. The most discussed
example is the so-called “quantum switch” [13], Fig. 1, where a control system
determines the order in which a set of operations are performed on a target.
Preparing the control in a superposition produces a nonclassical causal
structure for the order of operations. Although sometimes colloquially
described as a “superposition of causal orders”, the quantum switch is in fact
more appropriately interpreted as “entanglement” between causal relations and
the control system. Indeed, if the control is discarded, the quantum switch is
indistinguishable from a probabilistic mixture of causal orders. Only by
measuring the control can one verify the nonclassicality of the order of
operations.

Figure 2: Pure superposition of causal orders. Is it possible to superpose the
causal order of two events, without the aid of a control system?

Is a ‘pure’ superposition of causal orders possible (Fig. 2), not relying on
any additional control system? Perhaps, this would provide a novel resource,
enabling tasks not achievable by the quantum switch. It might also suggest
types of processes arising in a theory of quantum gravity.

Here, we propose a definition of superpositions of causal orders and present a
negative result, ruling out the most intuitive candidates. We show that for
unitary, Markovian processes where all systems have equal dimension,
superpositions of two different causal orders are not possible, in the sense
that they do not constitute valid processes. In particular, the result applies
to simple sequences of operations on a target system—as in the switch—showing
that an additional system is necessary for performing such operations in an
indefinite causal order. It remains an open question whether a superposition
is possible under weaker conditions.

##  2 Superpositions of unitaries

Before considering superpositions of different causal orders, it is useful to
look at a simpler case: superpositions of unitary transformations connecting
an event in the past—a state preparation—to one in the future—a measurement.
Given two unitaries $U_{j}$, $j=1,2$, it is natural to define their
superposition as a linear combination $\alpha U_{1}+\beta U_{2}$ for some
complex numbers $\alpha$, $\beta$, interpreted as probability amplitudes.
However, such a ‘superposition’ is not necessarily a unitary operator. For
example, no linear combination of $U_{1}=\mathds{1}$ and
$U_{2}=\frac{\sigma_{x}+\mathrm{i}\sigma_{y}}{\sqrt{2}}$ (where $\sigma_{x}$,
$\sigma_{y}$, $\sigma_{z}$ denote the Pauli matrices) is unitary, for _any_
complex amplitudes $\alpha\neq 0$, $\beta\neq 0$.

One could interpret arbitrary linear combinations of unitaries in the the
following way [22]: Introduce the ‘controlled unitary’
$U_{0}\otimes\left[0\right]+U_{1}\otimes\left[1\right]$, which acts on an
additional ‘control’ system and where we use the shorthand
$\left[\psi\right]=\left|\psi\middle\rangle\middle\langle\psi\right|$.
Starting from a state of the form
$\left|\psi\right\rangle\left(\alpha\left|0\right\rangle+\beta\left|1\right\rangle\right)$,
and postselecting the control on $\left(\left\langle 0\right|+\left\langle
1\right|\right)/\sqrt{2}$, the target system is left into $\left(\alpha
U_{0}+\beta U_{1}\right)\left|\psi\right\rangle/\sqrt{2}$, which one can
interpret as resulting from a superposition of unitaries. The problem with
this interpretation is that the postselection step requires obtaining one out
of a set of possible measurement outcomes, which in general only succeeds with
some probability that depends on the initial state. As such a probability can
be very small, or even vanish, realising a process through postselection can
nullify the associated advantages. Furthermore, postselection can induce
apparent signalling, making postselected processes unsuitable for studies of
causal relations 111For example, one can place bets on a roulette and consider
only the winning spins, effectively predicting the roulette’s outcome in each
of the postselected spins. This is unlikely to be a remunerative strategy in
any casino.. For these reasons, here we are only interested in processes that
can be realised _without_ postselection.

We see that, for processes, superposition is not a universal possibility, as
it is for states. However, there are unitaries for which superpositions _are_
possible. For example, the _Hadamard gate_
$H:=\frac{\sigma_{x}+\sigma_{z}}{\sqrt{2}}$ is a rightful superposition of
$\sigma_{x}$ and $\sigma_{z}$, not requiring any postselection. $H$ is a
ubiquitous resource in quantum information and computation, precisely for its
ability to convert basis states into superpositions. Therefore, even if we do
not expect to be able to superpose arbitrary processes with different causal
orders, we can still ask whether _any_ such a superposition is possible.

##  3 General quantum processes

As we will see, superpositions of causal orders become meaningful for
processes connecting more than two events. These can be described conveniently
in the so-called “process matrix formalism” [23] (see also other closely
related frameworks [24, 25, 26, 27, 28]). A simple example of a multi-event
process is one where a sequence of unitaries $U_{1},U_{2},\dots$ connects a
set of events at times $t_{1},t_{2},\dots$ At each time $t_{j}$, after unitary
$U_{j-1}$ and before unitary $U_{j}$, it is in principle possible to perform
arbitrary measurements or transformations on the system. The particular
operation taking place at a given time is what we call an ‘event’, while the
set of unitaries connecting the events is the ‘process’. More generally, we
consider a scenario where the labels that identify possible events are not
necessarily associated with time instants (for example, events might be
delocalised in time [29, 30] or take place on a nonclassical background [4]).
A single ‘event location’ with label $A$ (also referred to as “region”,
“party”, or “laboratory”) is therefore identified with the set of possible
operations from an input to an output Hilbert space, $\mathcal{H}^{A}_{I}$ and
$\mathcal{H}^{A}_{O}$, respectively.

The most general operation [31] is described by a Completely Positive (CP)
trace non-increasing map $\mathcal{M}^{A}:A_{I}\rightarrow A_{O}$, where
$A_{I,O}\equiv\mathcal{L}(\mathcal{H}^{A}_{I,O})$ are the spaces of linear
operators over the respective Hilbert spaces. A deterministic operation, i.e.,
one that happens with probability one, is CP and trace preserving (CPTP):
$\operatorname{tr}\circ\mathcal{M}^{A}=\operatorname{tr}$, where
$\operatorname{tr}$ and $\circ$ denote operator trace and function
composition, respectively. A collection of CP maps
$\\{\mathcal{M}^{A}_{a}\\}_{a}$, where $a$ labels the measurement outcome, is
called an _instrument_ if $\sum_{a}\mathcal{M}^{A}_{a}$ is CPTP, which implies
that there is unit probability that at least one of the outcomes will occur.
It is convenient to represent CP maps as operators, using a version of the
Choi-Jamiołkowksi isomorphism [32, 33], $\mathcal{M}^{A}\mapsto M^{A}\in
A_{I}\otimes A_{O}$:

$M^{A}:=\left[\sum_{jk}\left|j\middle\rangle\middle\langle
k\right|\otimes\mathcal{M}^{A}\left(\left|j\middle\rangle\middle\langle
k\right|\right)\right]^{T},$ (1)

where $\\{\left|j\right\rangle\\}_{j}$ is a basis of $\mathcal{H}^{A}_{I}$ and
T denotes transposition.

The local validity of quantum mechanics implies that the joint probability for
outcomes $a_{1},\dots,a_{n}$ in regions $A^{1},\dots,A^{n}$ is given by [23,
34]

$P(a_{1},\dots,a_{n})=\operatorname{tr}\left[W\cdot\left(M_{a_{1}}^{A^{1}}\otimes\dots\otimes
M_{a_{n}}^{A_{n}}\right)\right],$ (2)

where $W\in A^{1}\otimes\dots\otimes A^{n}$ is called the _process matrix_ (or
simply _process_) and we use the shorthand $A^{j}\equiv A^{j}_{I}\otimes
A^{j}_{O}$. $W$ encodes all information relevant to the possible events, such
as initial state, transformations, and causal relations. Positivity of
probabilities (and the possibility to extend the local operations to
additional systems) implies that the process matrix is positive semidefinite,
$W\geq 0$, while normalisation requires

$\operatorname{tr}\left[W\cdot\left(\bar{M}^{A^{1}}\otimes\dots\otimes\bar{M}^{A^{n}}\right)\right]=1$
(3)

for all CPTP maps $\bar{M}^{A^{1}},\dots,\bar{M}^{A^{n}}$. Crucially, Eq. (3)
imposes on $W$ more constraints than the normalisation condition for states,
$\operatorname{tr}\rho=1$. Therefore, whereas all positive semidefinite
operators represent states (up to normalisation) the same is not true for
process matrices. As we will see, it is this constraint that obstructs general
superpositions of processes.

The key property of the formalism is that it encodes signalling correlations:
a choice of instrument in a region can change the marginal probability for
measurement outcomes in another region. We say that a process is _causally
ordered_ if there is a permutation $\sigma$ of $n$ elements such that
$A^{\sigma(j)}$ cannot signal to any group of parties
$\left\\{A^{\sigma(k)}\right\\}_{k}$ with $\sigma(k)<\sigma(j)$. Conversely,
we say that $W$ is _incompatible_ with (the causal order induced by) $\sigma$
if it possible to signal from some $A^{\sigma(j)}$ to a group of parties
$\left\\{A^{\sigma(k)}\right\\}_{k}$, all with $\sigma(k)<\sigma(j)$. We say
that two processes $W_{1}$, $W_{2}$ are _differently ordered_ if they are
causally ordered relative to permutations $\sigma_{1}$, $\sigma_{2}$, but
incompatible with $\sigma_{2}$, $\sigma_{1}$, respectively. We are interested
in superpositions of differently ordered processes.

A related concept is that of _causally nonseparable process_ , which
formalises the notion of indefinite causal order [23, 35, 36]. However, the
existence of locally classical, causally nonseparable processes [37, 38, 39,
40] makes this definition unsuitable to characterise _nonclassical_ causal
order.

##  4 Superpositions of pure processes

Process matrices are a generalisation of density matrices and, as such, cannot
be directly superposed. In analogy to states, we can define superpositions for
‘pure’ processes, namely, for rank-1 process matrices, with
$W=\left[\omega\right]$ for some _process vector_
$\left|\omega\right\rangle\in\mathcal{H}^{A^{1}}_{I}\otimes\mathcal{H}^{A^{1}}_{O}\otimes\dots$
Given two pure process $\left[\omega_{1}\right]$, $\left[\omega_{2}\right]$,
we define their superposition through a linear combination
$\left|\omega\right\rangle=\alpha\left|\omega_{1}\right\rangle+\beta\left|\omega_{2}\right\rangle$,
with $\alpha,\beta\in\mathbb{C}$, _if $\left[\omega\right]$ is a valid process
matrix_. That is to say, if and only if

$\displaystyle\left\langle\omega\right|$
$\displaystyle\left(\bar{M}^{A^{1}}\otimes\dots\otimes\bar{M}^{A^{n}}\right)\left|\omega\right\rangle=1$
(4) $\displaystyle\forall$ $\displaystyle\textrm{ CPTP
}\bar{M}^{A^{1}},\dots,\bar{M}^{A^{n}}.$

If all parties have only input space (or, equivalently, if all output spaces
are trivial, namely they have dimension 1), the process matrix reduces to a
state, and superposition of processes reduces to superposition of states.
Another special case is a process that connects a region’s output, $A_{O}$, to
another region’s input, $B_{I}$, of equal dimension. In this case, a pure
process represents a unitary transformation (see, e.g., Appendix A of Ref.
[41]), for which we use the notation

${|{U}\rangle\\!\rangle}^{A_{O}B_{I}}:=\sum_{j}\left|j\right\rangle^{A_{O}}\otimes\left(U\left|j\right\rangle\right)^{B_{I}}.$
(5)

Because of the linearity of the representation (5), a ‘superposition of
unitary processes’ reduces to the ‘superposition of unitaries’ discussed
earlier:
$\alpha{|{U_{1}}\rangle\\!\rangle}+\beta{|{U_{2}}\rangle\\!\rangle}={|{\alpha
U_{1}+\beta U_{2}}\rangle\\!\rangle}$.

More generally, a pure process defines an _isometry_ from all the output to
all the input spaces: if all parties prepare a pure state, they all receive a
pure state, related to the prepared ones through an isometry. In the
particular case where the total input and output dimensions are equal, the
process defines a unitary transformation from outputs to inputs [42]. We will
use the term _unitary process_ for such cases 222Ref. [42] used the term “pure
process” for what we call here unitary process..

##  5 Superpositions of causal orders

We are now in a position to discuss superpositions of differently ordered
processes. Let us first analyse the quantum switch, to see what type of
superposition it represents.

There are two different versions relevant to the present discussion. The first
is a tripartite process, where $A^{1}\equiv A$ and $A^{2}\equiv B$ act on a
target system, while a third party, $A^{3}\equiv F$ (for “future”), receives
the target system resulting from $A$ and $B$’s operations, as well as a
control system. In this version, the initial state of the target is set to
some fixed state $\left|\psi\right\rangle$, while the control is fixed to
$\left|+\right\rangle=\frac{1}{\sqrt{2}}\left(\left|0\right\rangle+\left|1\right\rangle\right)$.
The resulting process vector is [41]

$\left|\omega^{(3)}_{\textrm{switch}}\right\rangle=\frac{1}{\sqrt{2}}\left(\left|0\right\rangle^{F_{\textrm{c}}}\left|\psi\right\rangle^{A_{I}}{|{\mathds{1}}\rangle\\!\rangle}^{A_{O}B_{I}}{|{\mathds{1}}\rangle\\!\rangle}^{B_{O}F_{\textrm{t}}}\right.\\\
\left.+\left|1\right\rangle^{F_{\textrm{c}}}\left|\psi\right\rangle^{B_{I}}{|{\mathds{1}}\rangle\\!\rangle}^{B_{O}A_{I}}{|{\mathds{1}}\rangle\\!\rangle}^{A_{O}F_{\textrm{t}}}\right),$
(6)

where $F_{\textrm{c}}$ and $F_{\textrm{t}}$ respectively denote the control
and target subsystems of $F$. Here $F$ has only input space, while its output
is trivial, as no influence from $F$ to $A$ and $B$ is possible.

Expression (6) manifestly represents a superposition of two differently
ordered processes, with the target system going first to $A$ and then $B$ or
vice versa. However, seen as an isometry, the process maps the two-qubit space
($A_{O}B_{O}$) to the larger four-qubit space
($A_{I}B_{I}F_{\textrm{c}}F_{\textrm{t}}$). This means that the isometry does
not describe a ‘pure transformation’, but also the specification of a state.
The natural interpretation is that process (6) displays entanglement between
the control system $F_{\textrm{c}}$ and the causal order of $A$ and $B$;
therefore, it does not represent a superposition of causal orders alone.
Indeed, if we trace out the control system, the resulting reduced process is
causally separable (i.e., a classical mixture of causally ordered ones) [41,
35].

The second version of the switch features an extra laboratory, $A^{0}\equiv P$
(for ‘past’), where both the control and target systems can be prepared
(corresponding to subsystems $P_{\textrm{c}}$, $P_{\textrm{t}}$,
respectively). As $P$ does not receive any state, its input space can be taken
to be trivial. The resulting unitary process is

$\left|\omega^{(4)}_{\textrm{switch}}\right\rangle=\left(\left|0\right\rangle^{P_{\textrm{c}}}\left|0\right\rangle^{F_{\textrm{c}}}{|{\mathds{1}}\rangle\\!\rangle}^{P_{\textrm{t}}A_{I}}{|{\mathds{1}}\rangle\\!\rangle}^{A_{O}B_{I}}{|{\mathds{1}}\rangle\\!\rangle}^{B_{O}F_{\textrm{t}}}\right.\\\
\left.+\left|1\right\rangle^{P_{\textrm{c}}}\left|1\right\rangle^{F_{\textrm{c}}}{|{\mathds{1}}\rangle\\!\rangle}^{P_{\textrm{t}}B_{I}}{|{\mathds{1}}\rangle\\!\rangle}^{B_{O}A_{I}}{|{\mathds{1}}\rangle\\!\rangle}^{A_{O}F_{\textrm{t}}}\right).$
(7)

This process is the sum of two vectors, each responsible for a different
causal order. However, the two vectors do not define individually valid
processes, as they are not normalised for arbitrary CPTP maps. Therefore,
process (7) does not represent a superposition of processes with different
causal orders, but rather describes _quantum control_ of the order of $A$ and
$B$.

The above observations lead us to the following:

######  Definition 1.

A process vector $\left|\omega\right\rangle$ for parties
$P,A^{1},\dots,A^{n},F$ (where $P$ and $F$ have trivial input and output
space, respectively) is a _pure superposition of two causal orders_ if

  * •

$\left|\omega\right\rangle$ represents a unitary process;

  * •

there are two nonvanishing complex numbers $\alpha$, $\beta$ and two
differently ordered, unitary process vectors $\left|\omega_{1}\right\rangle$,
$\left|\omega_{2}\right\rangle$ for the same set of parties as
$\left|\omega\right\rangle$ such that

$\left|\omega\right\rangle=\alpha\left|\omega_{1}\right\rangle+\beta\left|\omega_{2}\right\rangle.$
(8)

The presence of a past ($P$) and future ($F$) party in the definition follows
from the fact that, in a causally ordered process, there must be an initial
party, which cannot receive signals from any other, and a final party, which
cannot signal to the rest. If the process is unitary, such parties must have
trivial input and output space, respectively. As such parties can only be
respectively first and last in any causally ordered process, the simplest
candidate superposition of orders has to be four-partite, as in the unitary
switch (7).

Although we do not know if a superposition of causal orders is possible in
general, we can rule it out for the broad class of _Markovian_ processes.
These describe evolution from each region to the next, without any memory
carried over through an external environment across different steps. A
unitary, Markovian process is simply a sequence of unitaries, as in the
example discussed in Sec. 3.

A Markovian process matrix is the tensor product of processes describing the
individual time evolutions from each region to the next [43, 44, 45].
Therefore, a unitary, Markovian process matrix compatible with the causal
order $\sigma$ has the form

$\left|\omega\right\rangle=\bigotimes_{j=0}^{n}{|{U_{j}}\rangle\\!\rangle}^{A^{\sigma(j)}_{O}A^{\sigma(j+1)}_{I}},$
(9)

where $U_{j}$ are unitary matrices and we extend the permutation to
$\sigma(0)=0$, $\sigma(n+1)=n+1$.

Note that, for such a process, each party can always signal to any future one
(conditioned on the intermediate parties performing appropriate operations).
Therefore, a process of this form compatible with $\sigma$ is necessarily
incompatible with any $\sigma^{\prime}\neq\sigma$.

Our core result is that, for the simplest four-partite case, a linear
combination of two unitary, Markovian processes with different orders cannot
satisfy the process vector normalisation constraints, Eq. (4). In particular,
a CPTP map for the output-only space $P$ is an arbitrary density matrix
$\rho^{P}\geq 0$, with $\operatorname{tr}\rho^{P}=1$ (describing a state
preparation), while the only CPTP map for the input-only space $F$ is
$\mathds{1}^{F}$ (describing an arbitrary POVM measurement whose outcome is
ignored). For $X=A,B$, a CPTP map is represented by a matrix
$\xi^{X_{I}X_{O}}\geq 0$ such that
$\operatorname{tr}_{X_{O}}\xi^{X_{I}X_{O}}=\mathds{1}^{X_{I}}$. The technical
statement of our result, proved in the Appendix333While completing the current
manuscript, the author was made aware of related results in Ref. [46]. In
particular, Lemma 2 can be deduced from Corollary 5 in Ref. [46], although the
consequence as a no go theorem for superpositions of orders is not discussed
there. The argument and the proof in the present manuscript were originally
presented at [47]., is then as follows:

######  Lemma 2.

For every set of unitaries $\left\\{U_{j},V_{j}\right\\}_{j=1,2,3}$, and
complex numbers $\alpha,\beta\neq 0$, it is possible to find matrices $\rho$,
$\xi$, $\eta\geq 0$, with $\operatorname{tr}\rho=1$,
$\operatorname{tr}_{A_{O}}\xi^{A_{I}A_{O}}=\mathds{1}^{A_{I}}$, and
$\operatorname{tr}_{B_{O}}\eta^{B_{I}B_{O}}=\mathds{1}^{B_{I}}$, such that

$\left\langle\omega\right|\left(\rho^{P}\otimes\xi^{A_{I}A_{O}}\otimes\eta^{B_{I}B_{O}}\otimes\mathds{1}^{F}\right)\left|\omega\right\rangle\neq
1,$ (10)

where $\left|\omega\right\rangle$ is defined by Eq. (8) and

$\displaystyle\left|\omega_{1}\right\rangle$
$\displaystyle={|{U_{1}}\rangle\\!\rangle}^{PA_{I}}{|{U_{2}}\rangle\\!\rangle}^{A_{O}B_{I}}{|{U_{3}}\rangle\\!\rangle}^{B_{O}F},$
(11) $\displaystyle\left|\omega_{2}\right\rangle$
$\displaystyle={|{V_{1}}\rangle\\!\rangle}^{PB_{I}}{|{V_{2}}\rangle\\!\rangle}^{B_{O}A_{I}}{|{V_{3}}\rangle\\!\rangle}^{A_{O}F}.$
(12)

This result rules out the simplest, most intuitive superpositions of causal
orders. In particular, it proves that the control system is necessary for the
quantum switch to be a valid process. Note that, because each step is unitary,
all input and output spaces must have equal dimension 444The unitaries in Eq.
(11) imply $d^{P}=d^{A_{I}}$, $d^{A_{O}}=d^{B_{I}}$, and $d^{B_{O}}=d^{F}$,
while from Eq. (12) we get $d^{P}=d^{B_{I}}$, $d^{B_{O}}=d^{A_{I}}$, and
$d^{A_{O}}=d^{F}$. Taken together, the two sets of equations imply that all
dimensions are the same.. This is not guaranteed for arbitrarily many parties.
As it turns out, we need the additional assumption of equal dimensions to
prove our general result:

######  Theorem 3.

Consider a set of parties $P,A^{1},\dots,A^{n},F$, where $P$ has trivial
input, $F$ has trivial output, and all nontrivial input and output dimensions
are equal. For every pair of differently ordered, unitary, Markovian process
vectors $\left|\omega_{1}\right\rangle$, $\left|\omega_{2}\right\rangle$, and
complex numbers $\alpha,\beta\neq 0$, the linear combination
$\left|\omega\right\rangle=\alpha\left|\omega_{1}\right\rangle+\beta\left|\omega_{2}\right\rangle$
is not a valid process vector.

###### Proof.

We can prove the statement by reducing the general case to the $n=2$ one.
Indeed, let $\sigma_{1}\neq\sigma_{2}$ be the permutations defining the causal
orders of $\left|\omega_{1}\right\rangle$ and $\left|\omega_{2}\right\rangle$.
As the two permutations are different, we can find $j\neq k$ such that
$\sigma_{1}(j)<\sigma_{1}(k)$ and $\sigma_{2}(k)<\sigma_{2}(j)$. Inserting
identity unitaries in all regions except $j$, $k$, we obtain the bipartite
reduced processes [41]
$\left|\tilde{\omega}_{\sharp}\right\rangle:=\left(\bigotimes_{i\neq
j,k}{\langle\\!\langle{\mathds{1}}|}^{A^{i}_{I}A^{i}_{O}}\right)\left|\omega_{\sharp}\right\rangle$,
with
$\left|\omega_{\sharp}\right\rangle=\left|\omega\right\rangle,\left|\omega_{1}\right\rangle,\left|\omega_{2}\right\rangle$.
Now, we have
$\left|\tilde{\omega}\right\rangle=\alpha\left|\tilde{\omega}_{1}\right\rangle+\beta\left|\tilde{\omega}_{2}\right\rangle$,
where $\left|\tilde{\omega}_{1}\right\rangle$ and
$\left|\tilde{\omega}_{2}\right\rangle$ are unitary, Markovian, and
differently ordered. Lemma 2 then implies that
$\left|\tilde{\omega}\right\rangle$ cannot be a valid process, which in turn
implies that $\left|\omega\right\rangle$ cannot be a valid process either. ∎

##  6 Conclusions

We have seen that superpositions of processes with different causal orders are
strongly constrained, if we require that the order of events is not correlated
with any additional system. In particular, it is not possible to superpose
sequences of equal-dimension unitaries connecting events in different
orders—arguably the most intuitive candidate superposition of causal orders.
Relaxing the assumptions in the no-go theorem leads to some open
possibilities: it is currently unclear whether it is possible to superpose
unitary, differently ordered processes that do not satisfy Markovianity or
where the local input and output spaces do not have equal dimensions. Another
interesting direction is to consider superpositions of more than two
processes. Furthermore, it is an intriguing possibility to develop a notion of
‘coherence’ of causal orders that applies to nonpure processes, possibly
generalising corresponding resource theories for states [48, 49].

The author thanks Mateus Araújo, Cyril Branciard, Časlav Brukner, Matthew
Palmer, and Magdalena Zych for useful discussions. This work was partially
supported through an Australian Research Council Discovery Early Career
Researcher Award (DE170100712). The author acknowledges the traditional owners
of the land on which the University of Queensland is situated, the Turrbal and
Jagera people.

## References

  * [1] R. P. Feynman, “Space-Time Approach to Non-Relativistic Quantum Mechanics,” Rev. Mod. Phys. 20, 367–387 (1948).
  * [2] J. Butterfield and C. Isham, “Spacetime and the philosophical challenge of quantum gravity,” in Physics meets philosophy at the Planck scale, C. Callender and N. Huggett, eds., p. 33.  Cambridge University Press, 2001.  arXiv:gr-qc/9903072.
  * [3] L. Hardy, “Towards quantum gravity: a framework for probabilistic theories with non-fixed causal structure,” J. Phys. A: Math. Gen. 40, 3081–3099 (2007).
  * [4] M. Zych, F. Costa, I. Pikovski, and Č. Brukner, “Bell’s theorem for temporal order,” Nat. Commun. 10, 3772 (2019).
  * [5] L. M. Procopio, A. Moqanaki, M. Araújo, F. Costa, I. A. Calafell, E. G. Dowd, D. R. Hamel, L. A. Rozema, Č. Brukner, and P. Walther, “Experimental superposition of orders of quantum gates,” Nat. Commun. 6, 7913 (2015).
  * [6] G. Rubino, L. A. Rozema, A. Feix, M. Araújo, J. M. Zeuner, L. M. Procopio, Č. Brukner, and P. Walther, “Experimental verification of an indefinite causal order,” Sci. Adv. 3, e1602589 (2017).
  * [7] G. Rubino, L. A. Rozema, F. Massa, M. Araújo, M. Zych, Č. Brukner, and P. Walther, “Experimental Entanglement of Temporal Orders,” arXiv:1712.06884 [quant-ph].
  * [8] K. Goswami, C. Giarmatzi, M. Kewming, F. Costa, C. Branciard, J. Romero, and A. G. White, “Indefinite Causal Order in a Quantum Switch,” Phys. Rev. Lett. 121, 090503 (2018).
  * [9] K. Goswami, Y. Cao, G. A. Paz-Silva, J. Romero, and A. G. White, “Increasing communication capacity via superposition of order,” Phys. Rev. Research 2, 033292 (2020).
  * [10] Y. Guo, X.-M. Hu, Z.-B. Hou, H. Cao, J.-M. Cui, B.-H. Liu, Y.-F. Huang, C.-F. Li, G.-C. Guo, and G. Chiribella, “Experimental Transmission of Quantum Information Using a Superposition of Causal Orders,” Phys. Rev. Lett. 124, 030502 (2020).
  * [11] K. Wei, N. Tischler, S.-R. Zhao, Y.-H. Li, J. M. Arrazola, Y. Liu, W. Zhang, H. Li, L. You, Z. Wang, Y.-A. Chen, B. C. Sanders, Q. Zhang, G. J. Pryde, F. Xu, and J.-W. Pan, “Experimental Quantum Switching for Exponentially Superior Quantum Communication Complexity,” Phys. Rev. Lett. 122, 120504 (2019).
  * [12] M. M. Taddei, J. Cariñe, D. Martínez, T. García, N. Guerrero, A. A. Abbott, M. Araújo, C. Branciard, E. S. Gómez, S. P. Walborn, L. Aolita, and G. Lima, “Computational Advantage from the Quantum Superposition of Multiple Temporal Orders of Photonic Gates,” PRX Quantum 2, 010320 (2021).
  * [13] G. Chiribella, G. M. D’Ariano, P. Perinotti, and B. Valiron, “Quantum computations without definite causal structure,” Phys. Rev. A 88, 022318 (2013).
  * [14] G. Chiribella, “Perfect discrimination of no-signalling channels via quantum superposition of causal structures,” Phys. Rev. A 86, 040301(R) (2012).
  * [15] T. Colnaghi, G. M. D’Ariano, S. Facchini, and P. Perinotti, “Quantum computation with programmable connections between gates,” Phys. Lett. A 376, 2940–2943 (2012).
  * [16] M. Araújo, F. Costa, and Č. Brukner, “Computational Advantage from Quantum-Controlled Ordering of Gates,” Phys. Rev. Lett. 113, 250402 (2014).
  * [17] A. Feix, M. Araújo, and Č. Brukner, “Quantum superposition of the order of parties as a communication resource,” Phys. Rev. A 92, 052326 (2015).
  * [18] P. A. Guérin, A. Feix, M. Araújo, and Č. Brukner, “Exponential Communication Complexity Advantage from Quantum Superposition of the Direction of Communication,” Phys. Rev. Lett. 117, 100502 (2016).
  * [19] D. Ebler, S. Salek, and G. Chiribella, “Enhanced Communication with the Assistance of Indefinite Causal Order,” Phys. Rev. Lett. 120, 120502 (2018).
  * [20] S. Salek, D. Ebler, and G. Chiribella, “Quantum communication in a superposition of causal orders,” arXiv:1809.06655v2 [quant-ph].
  * [21] M. K. Gupta and U. Sen, “Transmitting quantum information by superposing causal order of mutually unbiased measurements,” arXiv:1909.13125v1 [quant-ph].
  * [22] Y. Aharonov, J. Anandan, S. Popescu, and L. Vaidman, “Superpositions of time evolutions of a quantum system and a quantum time-translation machine,” Phys. Rev. Lett. 64, 2965 (1990).
  * [23] O. Oreshkov, F. Costa, and Č. Brukner, “Quantum correlations with no causal order,” Nat. Commun. 3, 1092 (2012).
  * [24] R. Oeckl, “A “general boundary” formulation for quantum mechanics and quantum gravity,” Phys. Lett. B 575, 318–324 (2003).
  * [25] Y. Aharonov, S. Popescu, J. Tollaksen, and L. Vaidman, “Multiple-time states and multiple-time measurements in quantum mechanics,” Phys. Rev. A 79, 052110 (2009).
  * [26] J. Cotler, C.-M. Jian, X.-L. Qi, and F. Wilczek, “Superdensity operators for spacetime quantum mechanics,” J. High Energ. Phys. 2018, 93 (2018).
  * [27] R. Silva, Y. Guryanova, A. J. Short, P. Skrzypczyk, N. Brunner, and S. Popescu, “Connecting processes with indefinite causal order and multi-time quantum states,” New J. Phys. 19, 103022 (2017).
  * [28] J. Barrett, R. Lorenz, and O. Oreshkov, “Quantum Causal Models,” arXiv:1906.10726v1 [quant-ph].
  * [29] P. A. Guérin and Č. Brukner, “Observer-dependent locality of quantum events,” New J. Phys. 20, 103031 (2018).
  * [30] O. Oreshkov, “Time-delocalized quantum subsystems and operations: on the existence of processes with indefinite causal structure in quantum mechanics,” Quantum 3, 206 (2019).
  * [31] T. Heinosaari and M. Ziman, The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement.  Cambridge University Press, 2011.
  * [32] M.-D. Choi, “Completely positive linear maps on complex matrices,” Linear Algebra Appl. 10, 285–290 (1975).
  * [33] A. Jamiołkowski, “Linear transformations which preserve trace and positive semidefiniteness of operators,” Rep. Math. Phys 3, 275–278 (1972).
  * [34] S. Shrapnel, F. Costa, and G. Milburn, “Updating the Born rule,” New J. Phys. 20, 053010 (2018).
  * [35] O. Oreshkov and C. Giarmatzi, “Causal and causally separable processes,” New J. of Phys. 18, 093020 (2016).
  * [36] J. Wechs, A. A. Abbott, and C. Branciard, “On the definition and characterisation of multipartite causal (non)separability,” New J. of Phys. 21, 013027 (2019).
  * [37] Ä. Baumeler, A. Feix, and S. Wolf, “Maximal incompatibility of locally classical behavior and global causal order in multi-party scenarios,” Phys. Rev. A 90, 042106 (2014).
  * [38] Ä. Baumeler and S. Wolf, “The space of logically consistent classical processes without causal order,” New J. of Phys. 18, 013036 (2016).
  * [39] Ä. Baumeler, F. Costa, T. C. Ralph, S. Wolf, and M. Zych, “Reversible time travel with freedom of choice,” Class. Quantum Grav. 36, 224002 (2019).
  * [40] G. Tobar and F. Costa, “Reversible dynamics with closed time-like curves and freedom of choice,” Classical and Quantum Gravity 37, 205011 (2020).
  * [41] M. Araújo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi, and Č. Brukner, “Witnessing causal nonseparability,” New J. Phys. 17, 102001 (2015).
  * [42] M. Araújo, A. Feix, M. Navascués, and Č. Brukner, “A purification postulate for quantum mechanics with indefinite causal order,” Quantum 1, 10 (2017).
  * [43] F. Costa and S. Shrapnel, “Quantum causal modelling,” New J. of Phys. 18, 063032 (2016).
  * [44] F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, “Operational Markov Condition for Quantum Processes,” Phys. Rev. Lett. 120, 040405 (2018).
  * [45] C. Giarmatzi and F. Costa, “Witnessing quantum memory in non-Markovian processes,” Quantum 5, 440 (2021).
  * [46] W. Yokojima, M. T. Quintino, A. Soeda, and M. Murao, “Consequences of preserving reversibility in quantum superchannels,” Quantum 5, 441 (2021).
  * [47] F. Costa, “A no-go theorem for superpositions of causal order,” Space-time and Information, Manitoulin Island, Ontario, Canada, 2017.
  * [48] T. Theurer, N. Killoran, D. Egloff, and M. B. Plenio, “Resource Theory of Superposition,” Phys. Rev. Lett. 119, 230401 (2017).
  * [49] F. Bischof, H. Kampermann, and D. Bruß, “Resource Theory of Coherence Based on Positive-Operator-Valued Measures,” Phys. Rev. Lett. 123, 110402 (2019).

##  Appendix A Proof of Lemma 2

Here we provide the proof of Lemma 2, which we reformulate for convenience:

######  Lemma.

For every set of unitaries $\left\\{U_{j},V_{j}\right\\}_{j=1,2,3}$, and
complex numbers $\alpha,\beta\neq 0$, it is possible to find

$\displaystyle\chi:=$
$\displaystyle\rho^{P}\otimes\xi^{A_{I}A_{O}}\otimes\eta^{B_{I}B_{O}}\otimes\mathds{1}^{F},\quad\textrm{with}$
(13) $\displaystyle\operatorname{tr}\rho^{P}=$ $\displaystyle
1,\;\operatorname{tr}_{A_{O}}\xi^{A_{I}A_{O}}=\mathds{1}^{A_{I}},\;\operatorname{tr}_{B_{O}}\eta^{B_{I}B_{O}}=\mathds{1}^{B_{I}},$

such that

$\langle\chi\rangle:=\left\langle\omega\right|\chi\left|\omega\right\rangle\neq
1,$ (14)

where
$\left|\omega\right\rangle=\alpha\left|\omega_{1}\right\rangle+\beta\left|\omega_{2}\right\rangle$
and

$\displaystyle\left|\omega_{1}\right\rangle$
$\displaystyle={|{U_{1}}\rangle\\!\rangle}^{PA_{I}}{|{U_{2}}\rangle\\!\rangle}^{A_{O}B_{I}}{|{U_{3}}\rangle\\!\rangle}^{B_{O}F},$
(15) $\displaystyle\left|\omega_{2}\right\rangle$
$\displaystyle={|{V_{1}}\rangle\\!\rangle}^{PB_{I}}{|{V_{2}}\rangle\\!\rangle}^{B_{O}A_{I}}{|{V_{3}}\rangle\\!\rangle}^{A_{O}F}.$
(16)

As $\left|\omega_{1}\right\rangle$ and $\left|\omega_{2}\right\rangle$ are
normalised process vectors, we have
$\left\langle\omega_{1}\right|\chi\left|\omega_{1}\right\rangle=\left\langle\omega_{2}\right|\chi\left|\omega_{2}\right\rangle=1$
for all $\chi$ of the form (13), so that

$\langle\chi\rangle=\left|\alpha\right|^{2}+\left|\beta\right|^{2}+2\mathrm{Re}\left(\alpha^{*}\beta\left\langle\omega_{1}\right|\chi\left|\omega_{2}\right\rangle\right).$
(17)

For $\left|\omega\right\rangle$ to be a normalised process, we need
$\langle\chi\rangle=1$ for all $\chi$ of the form (13), which requires
$\mathrm{Re}\left(\alpha^{*}\beta\left\langle\omega_{1}\right|\chi\left|\omega_{2}\right\rangle\right)$
to be constant. Writing
$\alpha^{*}\beta=\mathrm{e}^{\mathrm{i}\phi}\left|\alpha^{*}\beta\right|$, we
obtain that the cross term
$\mathrm{Re}\left(\mathrm{e}^{\mathrm{i}\phi}\left\langle\omega_{1}\right|\chi\left|\omega_{2}\right\rangle\right)$
should be constant for all $\chi$.

In order to simplify the cross term, we restrict the CPTP maps to a pure state
preparation $\left|\psi\right\rangle$ for $P$ and unitaries $R$, $S$ for $A$
and $B$, respectively, so that

$\chi=\left[\psi^{*}\right]^{P}\otimes[[R^{*}]]^{A_{I}A_{O}}\otimes[[S^{*}]]^{B_{I}B_{O}}\otimes\mathds{1}^{F},$
(18)

with the notation
$[[A]]\equiv{{|{A}\rangle\\!\rangle}\\!{\langle\\!\langle{A}|}}$ and where the
complex conjugations in $\left|\psi^{*}\right\rangle$, $R^{*}$, $S^{*}$,
result from the transposition in our definition of the Choi representation for
local operations, Eq. (1). After some linear algebra, we arrive at

$\left\langle\omega_{1}\right|\chi\left|\omega_{2}\right\rangle=\left\langle\psi\right|U_{1}^{{\dagger}}R^{{\dagger}}U_{2}^{{\dagger}}S^{{\dagger}}U_{3}^{{\dagger}}V_{3}RV_{2}SV_{1}\left|\psi\right\rangle.$
(19)

The crucial step of the proof is to simplify this expression through an
appropriate choice of local operations. We can do this using the substitutions

$\displaystyle S\mapsto S^{\prime}$
$\displaystyle=U_{3}^{{\dagger}}V_{3}U_{2}^{{\dagger}}V_{1}SV_{1}^{{\dagger}},$
(20) $\displaystyle R\mapsto R^{\prime}$
$\displaystyle=U_{2}^{{\dagger}}V_{1}RV_{1}^{{\dagger}}U_{2}V_{3}^{{\dagger}}U_{3}V_{2}^{{\dagger}},$
(21)

under which the cross term takes the form

$\displaystyle\mathrm{e}^{\mathrm{i}\phi}$
$\displaystyle\left\langle\omega_{1}\right|\chi\left|\omega_{2}\right\rangle=\left\langle\psi\right|VT\left|\psi\right\rangle,$
(22) $\displaystyle V$
$\displaystyle=\mathrm{e}^{\mathrm{i}\phi}U_{1}^{{\dagger}}V_{2}U_{3}^{{\dagger}}V_{3}U_{2}^{{\dagger}}V_{1},$
(23) $\displaystyle T$ $\displaystyle=R^{{\dagger}}S^{{\dagger}}RS.$ (24)

Now the normalisation of $\left|\omega\right\rangle$ reduces to the condition
that, given $V$as in Eq. (23),
$\mathrm{Re}\left(\left\langle\psi\right|VT\left|\psi\right\rangle\right)$ has
to be constant for all unitaries $T$ of the form (24) and all
$\left|\psi\right\rangle$. This implies that $VT+T^{{\dagger}}V^{{\dagger}}$
should be proportional to the identity or, equivalently,

$TVT+V^{{\dagger}}=\lambda T,$ (25)

where $\lambda$ is a number that depends on $\alpha$ and $\beta$:

$\lambda=\frac{1-\left|\alpha\right|^{2}-\left|\beta\right|^{2}}{2\left|\alpha^{*}\beta\right|}.$
(26)

To show that no unitary $V$ can satisfy condition (25) for all $R$, $S$, it is
sufficient to take block-diagonal unitaries $R$, $S$ of the form
$\sigma\oplus\mathds{1}$, where $\sigma$ acts on a two-dimensional subspace
spanned by eigenstates of $V$ (so that the restriction of $V$ to that subspace
is also unitary). In other words, it is sufficient to show that condition (25)
cannot be satisfied by any single-qubit unitary $V$. We will omit the
$\oplus\mathds{1}$ for notational convenience.

Let us consider first the case where $\lambda=0$; that is, a superposition of
processes with normalised amplitudes
$\left|\alpha\right|^{2}+\left|\beta\right|^{2}=1$. Condition (25) reduces to

$TVT+V^{{\dagger}}=0,$ (27)

which has to hold for all $T$ of the form (24). We have to find a set of
unitaries $T$ of that form such that a single unitary $V$ cannot satisfy Eq.
(27) for all of them. First, choose $R=S=\mathds{1}$ to set $T=\mathds{1}$, so
that Eq. (27) gives

$V+V^{{\dagger}}=0.$ (28)

Substituting this into condition (27), we now have

$TVT=V.$ (29)

We can see that this is not possible by choosing $R_{j}$, $S_{j}$ such that
$\mathrm{i}\sigma_{j}=T_{j}=R_{j}^{{\dagger}}S_{j}^{{\dagger}}R_{j}S_{j}$,
with $j=x,y,z$ labelling the three Pauli matrices. (For example, for $j=x$, we
can choose $R_{x}=\sigma_{y}$,
$S_{x}=\frac{\mathds{1}+\mathrm{i}\sigma_{x}}{\sqrt{2}}$555The general
construction is $R_{j}=\sigma_{k}$ (with $k\neq j$) and
$S_{j}=\frac{\mathds{1}+\mathrm{i}\sigma_{j}}{\sqrt{2}}$. This gives
$R_{j}S_{j}=\frac{\sigma_{k}-\sum_{l}\epsilon_{kjl}\sigma_{l}}{\sqrt{2}}$ and
$R^{{\dagger}}_{j}S^{{\dagger}}_{j}=\frac{\sigma_{k}+\sum_{l}\epsilon_{kjl}\sigma_{l}}{\sqrt{2}}$.
As $(\sigma_{k}\sigma_{j})^{2}=-\mathds{1}$ for $k\neq j$, we get
$R_{j}^{{\dagger}}S_{j}^{{\dagger}}R_{j}S_{j}=\frac{\mathrm{i}}{2}\left(\sigma_{j}-\sigma_{k}\sigma_{j}\sigma_{k}\right)=\mathrm{i}\sigma_{j}$..)
Now, $V$ has to satisfy

$\sigma_{j}V\sigma_{j}=-V,\qquad j=x,y,z.$ (30)

This can only hold for $V=0$, which is not a unitary.

Let us now go back to the $\lambda\neq 0$ case. As we have seen above, we can
choose $R$ and $S$ such that $T=\mathrm{i}\sigma_{x}$. But we can also set
$T=-\mathrm{i}\sigma_{x}$ by choosing $R=\sigma_{y}$,
$S=\frac{\mathds{1}-\mathrm{i}\sigma_{x}}{\sqrt{2}}$. Thus, we get the two
equations

$\displaystyle-\sigma_{x}$ $\displaystyle
V\sigma_{x}+V^{{\dagger}}=i\lambda\sigma_{x},$ (31) $\displaystyle-\sigma_{x}$
$\displaystyle V\sigma_{x}+V^{{\dagger}}=-i\lambda\sigma_{x}.$ (32)

These can be satisfied simultaneously only for $\lambda=0$, which brings us
back to the previous case, concluding the proof. ∎


# Non-potential systems with relativistic operators and maximal monotone
boundary conditions

Petru<EMAIL_ADDRESS>- corresponding author and Călin
<EMAIL_ADDRESS>
aInstitute for Advanced Environmental Research, bDepartment of Mathematics
West University of Timişoara,
Blvd. V. Pârvan, no. 4, 300223 - Timişoara, Romania

###### Abstract

We are concerned with solvability of a non-potential system involving two
relativistic operators, subject to boundary conditions expressed in terms of
maximal monotone operators. The approach makes use of a fixed point
formulation and relies on a priori estimates and convergent to zero matrices.

Mathematics Subject Classification: 34B15, 34L30, 47H05, 47N20

Keywords and phrases: relativistic operator systems; maximal monotone
operator; fixed point; a priori estimates; convergent to zero matrix

##  1 Introduction

We deal with solvability of a system of type

$\left\\{\begin{array}[]{ll}-\left[\phi_{n}(u^{\prime})\right]^{\prime}=f_{1}(t,u,v)\\\
-\left[\phi_{m}(v^{\prime})\right]^{\prime}=f_{2}(t,u,v)\end{array}\quad\mbox{
in }[0,T],\right.$ (1.1)

associated with the multivalued boundary conditions

$\left\\{\begin{array}[]{ll}\left(\phi_{n}\left(u^{\prime}\right)(0),-\phi_{n}\left(u^{\prime}\right)(T)\right)\in\gamma(u(0),u(T))\\\
\left(\phi_{m}\left(v^{\prime}\right)(0),-\phi_{m}\left(v^{\prime}\right)(T)\right)\in\eta(v(0),v(T))\end{array},\right.$
(1.2)

where, for $q\in\\{n,m\\}$, the homeomorphism
$\phi_{q}:\mathbb{B}_{1}(q)\to\mathbb{R}^{q}$ is given by

$\displaystyle\phi_{q}(y)=\frac{y}{\sqrt{1-|y|^{2}}}\quad(y\in\mathbb{B}_{1}(q));$

here and hereafter, $\mathbb{B}_{r}(q)$ denotes the open ball of radius $r$ in
$\mathbb{R}^{q}$ endowed with the Euclidean norm $|\cdot|$ corresponding to
the usual inner product $\langle\cdot|\cdot\rangle$. Also, on the product
space $\mathbb{R}^{q}\times\mathbb{R}^{q}$ we consider the usual inner product
$\langle\\!\langle\cdot|\cdot\rangle\\!\rangle$. The operators
$\gamma:\mathbb{R}^{n}\times\mathbb{R}^{n}\to
2^{\mathbb{R}^{n}\times\mathbb{R}^{n}}$ and
$\eta:\mathbb{R}^{m}\times\mathbb{R}^{m}\to
2^{\mathbb{R}^{m}\times\mathbb{R}^{m}}$ are maximal monotone, with
$0_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\in\gamma(0_{\mathbb{R}^{n}\times\mathbb{R}^{n}})$,
$0_{\mathbb{R}^{m}\times\mathbb{R}^{m}}\in\eta(0_{\mathbb{R}^{m}\times\mathbb{R}^{m}})$
and the mappings
$f_{1}:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{m}\to\mathbb{R}^{n}$,
$f_{2}:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{m}\to\mathbb{R}^{m}$ are
continuous. The notations $D(\gamma)$, $D(\eta)$ will stand for the domain of
$\gamma$ and $\eta$, respectively (recall, this means
$D(\gamma)=\\{z\in\mathbb{R}^{n}\times\mathbb{R}^{n}\,:\,\gamma(z)\neq\emptyset\\}$;
similar definition for $D(\eta)$) and we denote
$C_{q}^{1}:=C^{1}(\left[0,T\right];\mathbb{R}^{q})$ ($q\in\\{n,m\\}$).

By a solution of problem (1.1)-(1.2) we mean a couple of functions $(u,v)\in
C_{n}^{1}\times C_{m}^{1}$ with $|u^{\prime}(t)|<1$, $|v^{\prime}(t)|<1$ for
all $t\in[0,T]$, such that $\phi_{n}(u^{\prime})\in C_{n}^{1}$,
$\phi_{m}(v^{\prime})\in C_{m}^{1}$, $(u(0),u(T))\in D\left(\gamma\right)$,
$(v(0),v(T))\in D\left(\eta\right)$ and which satisfies (1.1) and (1.2).

In recent years a special attention was paid to various qualitative aspects
for boundary value problems involving the relativistic operator:
$u\mapsto\left[\phi_{q}(u^{\prime})\right]^{\prime}$, which is known that
occurs in the dynamics of special relativity. Mainly, the results obtained for
this operator use both topological techniques as well as variational methods.
Thus, among others and far from being exhausted, related to existence and
multiplicity of solutions of nonlinear systems or scalar equations subject to
classical boundary conditions, such as Dirichlet, Neumann or periodic, under
various assumptions on the perturbing nonlinearity, we refer the reader to
e.g. [1] \- [6], [10], [13], [14] and the references therein. This work is
mainly motivated by paper [8], which is concerned with a system having the
form

$-\left[\phi_{n}(u^{\prime})\right]^{\prime}=\nabla_{u}F(t,u)\quad\mbox{in
}[0,T],$ (1.3)

subject to the boundary condition

$\left(\phi_{n}\left(u^{\prime}\right)(0),-\phi_{n}\left(u^{\prime}\right)(T)\right)\in\partial
j(u(0),u(T)),$ (1.4)

under the assumptions that
$j:\mathbb{R}^{n}\times\mathbb{R}^{n}\to(-\infty,+\infty]$ is a proper lower
semicontinuous convex function with
$j(0_{\mathbb{R}^{n}\times\mathbb{R}^{n}})=0$ and
$0_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\in\partial
j(0_{\mathbb{R}^{n}\times\mathbb{R}^{n}})$; here, $\partial j$ stands for the
subdifferential of $j$ in the sense of convex analysis [20]. The mapping
$F=F(t,u):[0,T]\times\mathbb{R}^{n}\to\mathbb{R}$ is supposed to be
continuous, with $\nabla_{u}F$ continuous on the set
$[0,T]\times\mathbb{R}^{n}$ and $F(\,\cdot\,,0_{\mathbb{R}^{n}})=0.$ Notice
that by the lower semicontinuity assumption on the proper convex function $j$,
the multivalued operator $\partial
j:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow
2^{\mathbb{R}^{n}\times\mathbb{R}^{n}}$ is maximal monotone [19]. It is known
(see e.g. [7] \- [9], [15] and the references therein) that different choices
of $j$ yield various boundary conditions, including the classical ones
(Dirichlet, Neumann, periodic, antiperiodic, etc.) as well as other ones of
special interest.

Observe that problem (1.3)-(1.4) is entirely of potential type, both through
the system and through the boundary condition. As shown in [8], this allows a
variational formulation in the frame of critical point theory for convex,
lower semicontinuous perturbations of $C^{1}$-functionals. Then, taking the
advantage of this key feature, therein one obtains the existence of minimum
energy as well as saddle-point solutions of the problem. It is worth to notice
that, as seen by concrete examples of applications, the generality of
(1.3)-(1.4) is quite broad. However, it is clear that firstly the potentiality
of the system (1.3) and secondly of the boundary condition (1.4) restrict the
applicability of the results from [8]. In this view, it appears as being
natural to consider a non-potential system

$-\left[\phi_{n}(u^{\prime})\right]^{\prime}=f(t,u)\quad\mbox{in }[0,T],$
(1.5)

associated with the non-potential boundary condition

$\left(\phi_{n}\left(u^{\prime}\right)(0),-\phi_{n}\left(u^{\prime}\right)(T)\right)\in\gamma(u(0),u(T)),$
(1.6)

where the maximal monotone operator $\gamma$ is as above and
$f:[0,T]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ is continuous. For the sake of
clarity of the fact that (1.6) is really more general than (1.4), we give here
a simple example of a boundary condition which is expressed by a maximal
monotone operator that is not of subdifferential type. In this respect, let
$A$ be a real ($2n\times 2n$)-matrix which is positive semi-definite (i.e.
$\langle\\!\langle Az\,|\,z\rangle\\!\rangle\geq 0$, for all
$z\in\mathbb{R}^{n}\times\mathbb{R}^{n}$) and

$\gamma_{A}(z)=A\,z\qquad(z\in\mathbb{R}^{n}\times\mathbb{R}^{n}).$ (1.7)

Then $\gamma_{A}$ is maximal monotone [17, Example 1.5 (b)]. If $A$ is not
symmetric, then $\gamma_{A}$ will not be cyclically monotone (see p. 240 in
[20]) and thus, by a classical result on the potentiality of maximal monotone
operators [19], there is no lower semicontinuous proper convex function
$j:\mathbb{R}^{n}\times\mathbb{R}^{n}\to(-\infty,+\infty]$ such that
$\gamma_{A}=\partial j$. Summarizing, if $A$ is positive semi-definite, but it
is not symmetric, the boundary condition

$\left(\begin{array}[]{clrc}\phi_{n}\left(u^{\prime}\right)(0)\\\\[6.0pt]
-\phi_{n}\left(u^{\prime}\right)(T)\end{array}\right)=A\
\left(\begin{array}[]{clrc}u(0)\\\\[6.0pt] u(T)\end{array}\right)$ (1.8)

is of type (1.6), with $\gamma$ which is not the subdifferential of a proper
lower semicontinuous convex function $j$.

Coming back to system (1.1)-(1.2), let us note that it contains as a
particular case problem (1.5)-(1.6). Indeed, given be $f$ and $\gamma$, this
can be seen by taking $m=n$, $\eta=\gamma$, $f_{1}(t,u,v)=f(t,u)$ and
$f_{2}(t,u,v)=f(t,v)$. Therefore, a qualitative result about (1.1)-(1.2) can
be easily transposed to problem (1.5)-(1.6) (see Corollaries 3.1 and 3.2). On
the other hand, our main result (Theorem 3.1) is of type of [11, Theorem 2.1]
which is concerned with classical vector $p$ and $q$-Laplacian operators. In
this view, the brought novelties consist in the fact that the differential
operators in (1.1) are singular and the boundary conditions (1.2) are of non-
potential type. As we will see, both technically as well as structurally these
bring specific approaches and features, even if broadly speaking, it is about
a similar strategy to the one in [11] \- this makes use of a priori estimates
combined with convergent to zero matrices. We mention that unlike the refered
classical case, for system (1.1) subject to a whole class of boundary
conditions of type (1.2), we obtain an ”universal” existence result, meaning
that no additional assumptions must be imposed on the nonlinearities $f_{1}$,
$f_{2}$ to ensure the existence of the solution (see Theorem 3.1 $(i)$).
Another notable aspect is that various choices of operators $\gamma$ and
$\eta$ in (1.2) allow the coupling of boundary conditions of different types.

The rest of the paper is organized as follows. In Section 2 we give the fixed
point formulation of problem (1.1)-(1.2). Then, in Section 3 we present the
main result concerning solvability of (1.1)-(1.2) and derive existence results
for system (1.5)-(1.6).

##  2 A fixed point approach

Let $q\in\\{n,m\\}$. By $\|\cdot\|_{\infty}$ we denote the sup-norm on
$C_{q}:=C([0,T];\mathbb{R}^{q})$ and the usual norm on
$L^{r}:=L^{r}([0,T];\mathbb{R}^{q})$ will be denoted by $\|\cdot\|_{L^{r}}$
($1\leq r\leq\infty$). The Sobolev space
$H^{1}_{q}:=W^{1,2}([0,T];\mathbb{R}^{q})$ will be considered with the norm

$\|w\|_{H^{1}_{q}}=\left(\|w^{\prime}\|_{L^{2}}^{2}+\|w\|_{L^{2}}^{2}\right)^{1/2}.$

The product spaces $\mathcal{C}:=C_{n}\times C_{m}$ and
$\mathcal{H}:=H^{1}_{n}\times H^{1}_{m}$ will be considered with the usual
product topology; we choose to see $\mathcal{H}$ as being endowed with the
norm

$\|(u,v)\|_{\mathcal{H}}=\|u\|_{H^{1}_{n}}+\|v\|_{H^{1}_{m}}\quad((u,v)\in\mathcal{H}).$

According to [8, Theorem 2.8], we have that for any $h\in C_{n}$, the system

$-\left[\phi_{n}(u^{\prime})\right]^{\prime}+u=h(t)\quad\mbox{in }[0,T],$
(2.1)

has an unique solution satisfying the boundary condition (1.6). This enables
us to define the solution operator $S_{\gamma}:C_{n}\to H^{1}_{n}$ by
$S_{\gamma}(h):=\mbox{the unique solution of
\eqref{eqpb2}-\eqref{bcsyspnew1}}\,(h\in C_{n}).$ In a similar way we
introduce $S_{\eta}:C_{m}\to H^{1}_{m}$ by $S_{\eta}(l):=\mbox{the unique
solution of}$ problem

$-\left[\phi_{m}(v^{\prime})\right]^{\prime}+v=l(t)\ \ \mbox{in
}[0,T],\quad\left(\phi_{m}\left(v^{\prime}\right)(0),-\phi_{m}\left(v^{\prime}\right)(T)\right)\in\eta(v(0),v(T))\qquad(l\in
C_{m}).$

###### Proposition 2.1

The operators $S_{\gamma}$ and $S_{\eta}$ are continuous.

Proof. We refer to $S_{\gamma}$; similar argument for $S_{\eta}$. Let
$\\{h_{k}\\}\subset C_{n}$ be a sequence such that $h_{k}\to h\in C_{n}$ as
$k\to\infty$, and set $u:=S_{\gamma}(h)$, $u_{k}:=S_{\gamma}(h_{k})$
($k\in\mathbb{N}$). As $\gamma$ is (maximal) monotone, by virtue of (1.6) one
has

$\displaystyle 0$ $\displaystyle\leq$
$\displaystyle\langle\\!\langle\left(\phi_{n}\left(u_{k}^{\prime}\right)(0),-\phi_{n}\left(u_{k}^{\prime}\right)(T)\right)-\left(\phi_{n}\left(u^{\prime}\right)(0),-\phi_{n}\left(u^{\prime}\right)(T)\right)|\left(u_{k}(0),u_{k}(T)\right)-\left(u(0),u(T)\right)\rangle\\!\rangle$
(2.2) $\displaystyle=$
$\displaystyle\langle\\!\langle\left(\phi_{n}\left(u_{k}^{\prime}\right)(0)-\phi_{n}\left(u^{\prime}\right)(0),-\phi_{n}\left(u_{k}^{\prime}\right)(T)+\phi_{n}\left(u^{\prime}\right)(T)\right)|\left(u_{k}(0)-u(0),u_{k}(T)-u(T)\right)\rangle\\!\rangle$
$\displaystyle=$
$\displaystyle\langle\phi_{n}\left(u_{k}^{\prime}\right)(0)-\phi_{n}\left(u^{\prime}\right)(0)|u_{k}(0)-u(0)\rangle-\langle\phi_{n}\left(u_{k}^{\prime}\right)(T)-\phi_{n}\left(u^{\prime}\right)(T)|u_{k}(T)-u(T)\rangle.$

On the other hand, multiplying the equality

$-\left[\phi_{n}(u_{k}^{\prime})-\phi_{n}(u^{\prime})\right]^{\prime}+u_{k}-u=h_{k}(t)-h(t)$

by $u_{k}-u$, integrating over $[0,T]$ and using the integration by parts
formula, we derive

$\langle\phi_{n}\left(u_{k}^{\prime}\right)(0)-\phi_{n}\left(u^{\prime}\right)(0)|u_{k}(0)-u(0)\rangle-\langle\phi_{n}\left(u_{k}^{\prime}\right)(T)-\phi_{n}\left(u^{\prime}\right)(T)|u_{k}(T)-u(T)\rangle+$
$+\int_{0}^{T}\langle\phi_{n}\left(u_{k}^{\prime}\right)-\phi_{n}\left(u^{\prime}\right)|u_{k}^{\prime}-u^{\prime}\rangle+\|u_{k}-u\|_{L^{2}}^{2}=\int_{0}^{T}\langle
h_{k}(t)-h(t)|u_{k}-u\rangle$

and from (2.2), one obtains

$\int_{0}^{T}\langle\phi_{n}\left(u_{k}^{\prime}\right)-\phi_{n}\left(u^{\prime}\right)|u_{k}^{\prime}-u^{\prime}\rangle+\|u_{k}-u\|_{L^{2}}^{2}\leq\sqrt{T}\|h_{k}-h\|_{\infty}\|u_{k}-u\|_{L^{2}}.$

Then, since $\phi_{n}$ satisfies [10, Lemma 2.1]:

$\langle\phi_{n}(x)-\phi_{n}(y)|x-y\rangle\geq|x-y|^{2}\quad(x,y\in\mathbb{B}_{1}(n)),$

we get that $\\{\|u_{k}-u\|_{L^{2}}\\}$ is bounded and

$\|u_{k}-u\|_{H^{1}_{n}}^{2}\leq\sqrt{T}\|h_{k}-h\|_{\infty}\|u_{k}-u\|_{L^{2}},$

which, as $h_{k}\to h$ in $C_{n}$, yields that $u_{k}\to u$ in $H^{1}_{n}$ and
the continuity of $S_{\gamma}$ is proved.

Next, we introduce the mappings
$g_{1}:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{m}\to\mathbb{R}^{n}$,
$g_{2}:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{m}\to\mathbb{R}^{m}$ by

$g_{1}(t,x,y)=f_{1}(t,x,y)+x,\ \quad
g_{2}(t,x,y)=f_{2}(t,x,y)+y\quad\,(t\in[0,T],\,x\in\mathbb{R}^{n},\
y\in\mathbb{R}^{m})$

and define the Nemytskii operators $N_{g_{1}}:\mathcal{C}\to C_{n}$ and
$N_{g_{2}}:\mathcal{C}\to C_{m}$ by

$N_{g_{1}}(u,v)(t)=g_{1}(t,u(t),v(t)),\ \quad
N_{g_{2}}(u,v)(t)=g_{2}(t,u(t),v(t))\quad\,(t\in[0,T],\ (u,v)\in\mathcal{C}).$

It is well known that $N_{g_{1}}$, $N_{g_{2}}$ are continuous and bounded
(i.e., takes bounded sets from $\mathcal{C}$ into bounded sets in $C_{q}$
($q\in\\{n,m\\}$)). Then, let $Q_{\gamma}:\mathcal{H}\to H^{1}_{n}$ and
$Q_{\eta}:\mathcal{H}\to H^{1}_{m}$ be given by

$Q_{\gamma}=S_{\gamma}\circ N_{g_{1}}\circ i_{d},\quad Q_{\eta}=S_{\eta}\circ
N_{g_{2}}\circ i_{d},$ (2.3)

where $i_{d}:\mathcal{H}\to\mathcal{C}$ is the identity map.

###### Proposition 2.2

$(i)$ The operator $Q=(Q_{\gamma},Q_{\eta}):\mathcal{H}\to\mathcal{H}$ is
compact.

$(ii)$ A couple of functions $(u,v)\in\mathcal{H}$ is a solution of problem
(1.1)-(1.2) iff it is a fixed point of $Q$.

Proof. Statement $(i)$ follows from Proposition 2.1, the continuity of
$N_{g_{1}}$, $N_{g_{2}}$ and the compactness of $i_{d}$, while $(ii)$ is
straightforward.

Before concluding this section, we note for reader convenience some simple
characterizations of the convergent to zero matrices, which will be needed in
the sequel. Recall, a square matrix $M$ is said to be convergent to zero if
$M^{k}\to 0$ as $k\to\infty$. The following result is proved in [18, Lemma 2].

###### Lemma 2.1

Let $M$ be a square matrix of non-negative numbers. The following statements
are equivalent:

  1. $($i$)$

$M$ is a convergent to zero matrix;

  2. $($ii$)$

$I-M$ is non-singular and

$(I-M)^{-1}=I+M+M^{2}+\ldots;$

  3. $($iii$)$

$|\lambda|<1$ for every $\lambda\in\mathbb{C}$ with $\det(M-\lambda I)=0$;

  4. $($iv$)$

$I-M$ is non-singular and $(I-M)^{-1}$ has non-negative elements.

##  3 Main result

We denote

$\mathcal{K}(q):=\left\\{w\in
W^{1,\infty}([0,T];\mathbb{R}^{q})\,:\,\|w^{\prime}\|_{{}_{L^{\infty}}}\leq
1\right\\}\quad(q\in\\{n,m\\})$

and define the first eigenvalue-like constants:

$\lambda_{1}(\gamma):=\inf\left\\{\frac{\|u^{\prime}\|_{L^{2}}^{2}}{\|u\|_{L^{2}}^{2}}\,:\,u\in\mathcal{K}(n)\setminus\\{0_{\mathbb{R}^{n}}\\},\,(u(0),u(T))\in
D(\gamma)\right\\},$ (3.1)
$\lambda_{1}(\eta):=\inf\left\\{\frac{\|v^{\prime}\|_{L^{2}}^{2}}{\|v\|_{L^{2}}^{2}}\,:\,v\in\mathcal{K}(m)\setminus\\{0_{\mathbb{R}^{m}}\\},\,(v(0),v(T))\in
D(\eta)\right\\}.$ (3.2)

The following main result provides sufficient conditions which ensures the
solvability of the boundary value problem (1.1)-(1.2).

###### Theorem 3.1

Assume that one of the following four conditions is satisfied:

$(i)$ $\lambda_{1}(\gamma)>0$, $\lambda_{1}(\eta)>0;$

$(ii)$ $\lambda_{1}(\eta)>0$ and there are constants
$a\in[0,1),b,\delta\in\mathbb{R}_{+}$ such that

$\langle
f_{1}(t,x,y)\,|\,x\rangle\leq(a-1)|x|^{2}+b|y|^{2}+\delta\quad(t\in[0,T],\
x\in\mathbb{R}^{n},\ y\in\mathbb{R}^{m});$ (3.3)

$(iii)$ $\lambda_{1}(\gamma)>0$ and there are constants
$c,\delta\in\mathbb{R}_{+},d\in[0,1)$ such that

$\langle f_{2}(t,x,y)\,|\,y\rangle\leq
c|x|^{2}+(d-1)|y|^{2}+\delta\quad(t\in[0,T],\ x\in\mathbb{R}^{n},\
y\in\mathbb{R}^{m});$ (3.4)

$(iv)$ there are constants $a,b,c,d,\delta\in\mathbb{R}_{+}$ such that the
matrix

$M:=\left[\begin{array}[]{clrc}a&b\\\ c&d\end{array}\right]$ (3.5)

is convergent to zero and

$\begin{array}[]{ll}\langle
f_{1}(t,x,y)\,|\,x\rangle\leq(a-1)|x|^{2}+b|y|^{2}+\delta\\\ \langle
f_{2}(t,x,y)\,|\,y\rangle\leq
c|x|^{2}+(d-1)|y|^{2}+\delta\end{array}\quad(t\in[0,T],\ x\in\mathbb{R}^{n},\
y\in\mathbb{R}^{m}).$ (3.6)

Then problem (1.1)-(1.2) has at least one solution.

Proof. On account of Proposition 2.2 $(ii)$, it suffices to show that under
any of the assumptions $(i)-(iv)$, the operator $Q$ has a fixed point. In this
view, according to Schaefer’s theorem (see e.g. [16, Corollary 4.4.12]), it
suffices to prove that the set

$\\{(u,v)\in\mathcal{H}:\ \exists\ \lambda\in(0,1]\mbox{ such that
}(u,v)=\lambda Q(u,v)\\}$

is bounded in $(\mathcal{H},\|\cdot\|_{\mathcal{H}})$.

Let $(u,v)\in\mathcal{H}$ be such that $Q(u,v)=\lambda^{-1}(u,v)$, with some
$\lambda\in(0,1]$. From (2.3) and definition of operators $S_{\gamma}$ and
$S_{\eta}$, this means that $|\lambda^{-1}u^{\prime}(t)|<1$,
$|\lambda^{-1}v^{\prime}(t)|<1$ for all $t\in[0,T]$,
$(\lambda^{-1}u(0),\lambda^{-1}u(T))\in D\left(\gamma\right)$,
$(\lambda^{-1}v(0),\lambda^{-1}v(T))\in D\left(\eta\right)$,

$-\left[\phi_{n}\left(\lambda^{-1}u^{\prime}\right)\right]^{\prime}+\lambda^{-1}u=g_{1}(t,u,v)\quad\,(t\in[0,T]),$
(3.7)
$-\left[\phi_{m}\left(\lambda^{-1}v^{\prime}\right)\right]^{\prime}+\lambda^{-1}v=g_{2}(t,u,v)\quad\,(t\in[0,T])$
(3.8)

and

$\left(\phi_{n}\left(\lambda^{-1}u^{\prime}\right)(0),-\phi_{n}\left(\lambda^{-1}u^{\prime}\right)(T)\right)\in\gamma(\lambda^{-1}u(0),\lambda^{-1}u(T)),$
(3.9)
$\left(\phi_{m}\left(\lambda^{-1}v^{\prime}\right)(0),-\phi_{m}\left(\lambda^{-1}v^{\prime}\right)(T)\right)\in\eta(\lambda^{-1}v(0),\lambda^{-1}v(T)).$
(3.10)

$(i)$ From (3.1) and (3.2), we have that

$\|u\|_{L^{2}}^{2}\leq\frac{\|u^{\prime}\|_{L^{2}}^{2}}{\lambda_{1}(\gamma)}<\frac{T}{\lambda_{1}(\gamma)}\
\mbox{ and }\
\|v\|_{L^{2}}^{2}\leq\frac{\|v^{\prime}\|_{L^{2}}^{2}}{\lambda_{1}(\eta)}<\frac{T}{\lambda_{1}(\eta)},$

which gives
$\|(u,v)\|_{\mathcal{H}}<\left(T+T/\lambda_{1}(\gamma)\right)^{1/2}+\left(T+T/\lambda_{1}(\eta)\right)^{1/2}$.

$(ii)$ Multiplying (3.7) by $\lambda^{-1}u$, then integrating over $[0,T]$ and
using integration by parts formula, we infer

$\left\langle\phi_{n}\left(\lambda^{-1}u^{\prime}\right)(0)|\lambda^{-1}u(0)\right\rangle-\left\langle\phi_{n}\left(\lambda^{-1}u^{\prime}\right)(T)|\lambda^{-1}u(T)\right\rangle+$
$+\int_{0}^{T}\left\langle\phi_{n}\left(\lambda^{-1}u^{\prime}\right)|\lambda^{-1}u^{\prime}\right\rangle+\lambda^{-2}\int_{0}^{T}|u|^{2}=\lambda^{-1}\int_{0}^{T}\langle
g_{1}(t,u,v)|u\rangle.$ (3.11)

Using that
$0_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\in\gamma(0_{\mathbb{R}^{n}\times\mathbb{R}^{n}})$,
the monotonicity of $\gamma$ and (3.9), one has

$\left\langle\phi_{n}\left(\lambda^{-1}u^{\prime}\right)(0)|\lambda^{-1}u(0)\right\rangle-\left\langle\phi_{n}\left(\lambda^{-1}u^{\prime}\right)(T)|\lambda^{-1}u(T)\right\rangle\geq
0,$

which, on account of $\langle\phi_{n}(y)|y\rangle\geq|y|^{2}$
($y\in\mathbb{B}_{1}(n)$) and (3.11), yields

$\|u\|_{H^{1}_{n}}^{2}\leq\lambda\int_{0}^{T}\langle
g_{1}(t,u,v)|u\rangle=\lambda\int_{0}^{T}\langle
f_{1}(t,u,v)|u\rangle+\lambda\|u\|_{L^{2}}^{2}.$ (3.12)

Then, from (3.2), since $\lambda_{1}(\eta)>0$, one has
$\|v\|_{L^{2}}^{2}<T/\lambda_{1}(\eta)$ and by virtue of (3.3) and (3.12), we
get

$\|u\|_{H^{1}_{n}}^{2}\leq a\|u\|_{L^{2}}^{2}+b\|v\|_{L^{2}}^{2}+\delta
T<a\|u\|_{H^{1}_{n}}^{2}+k_{1},$

with $k_{1}=k_{1}(\eta):=(b/\lambda_{1}(\eta)+\delta)T$. Hence,
$\|(u,v)\|_{\mathcal{H}}<\left(k_{1}/(1-a)\right)^{1/2}+\left(T+T/\lambda_{1}(\eta)\right)^{1/2}$.

$(iii)$ Similarly to $(ii)$, using (3.8), (3.10) and the monotonicity of
$\eta$, we obtain

$\|v\|_{H^{1}_{m}}^{2}\leq\lambda\int_{0}^{T}\langle
g_{2}(t,u,v)|v\rangle\leq\lambda\int_{0}^{T}\langle
f_{2}(t,u,v)|v\rangle+\lambda\|v\|_{L^{2}}^{2},$ (3.13)

which, together with (3.1) and (3.4), imply
$\|(u,v)\|_{\mathcal{H}}<\left(T+T/\lambda_{1}(\gamma)\right)^{1/2}+\left(k_{2}/(1-d)\right)^{1/2},$
where $k_{2}=k_{2}(\gamma):=(c/\lambda_{1}(\gamma)+\delta)T$.

$(iv)$ From (3.6), (3.12) and (3.13), we derive

$\|u\|_{H^{1}_{n}}^{2}\leq
a\|u\|_{H^{1}_{n}}^{2}+b\|v\|_{H^{1}_{m}}^{2}+\delta T\ \mbox{ and }\
\|v\|_{H^{1}_{m}}^{2}\leq c\|u\|_{H^{1}_{n}}^{2}+d\|v\|_{H^{1}_{m}}^{2}+\delta
T,$

which can be written in the following vector form:

$(I-M)\left[\begin{array}[]{c}\|u\|_{H^{1}_{n}}^{2}\\\\[6.0pt]
\|v\|_{H^{1}_{m}}^{2}\end{array}\right]\leq\left[\begin{array}[]{c}\delta
T\\\\[6.0pt] \delta T\end{array}\right].$ (3.14)

Then, since matrix $M$ is convergent to zero, Lemma 2.1 guarantees that $I-M$
is invertible and the elements of $(I-M)^{-1}$ are all non-negative.
Therefore, we may multiply (3.14) by $(I-M)^{-1}$ without changing the sense
of the inequality and so, we get

$\left[\begin{array}[]{c}\|u\|_{H^{1}_{n}}^{2}\\\\[6.0pt]
\|v\|_{H^{1}_{m}}^{2}\end{array}\right]\leq(I-M)^{-1}\left[\begin{array}[]{c}\delta
T\\\\[6.0pt] \delta T\end{array}\right],$

which shows that $\|(u,v)\|_{\mathcal{H}}$ is bounded by a constant depending
on $a,b,c,d,\delta$ and independent of $\lambda\in(0,1]$.

###### Remark 3.1

Denote by $cone\,D(\gamma)$ the conical hull of the set $D(\gamma)$, that is

$cone\,D(\gamma):=\\{\alpha z\,:\,\alpha\geq 0,\,z\in D(\gamma)\\}.$

A sufficient condition to have $\lambda_{1}(\gamma)>0$ is

$\overline{cone\,D(\gamma)}\cap\\{(\xi,\xi)\,:\,\xi\in\mathbb{R}^{n}\\}=\\{0_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\\}.$
(3.15)

To see this, we argue as in the proof of Corollary 4.2 from [8]. Thus, setting
${\mathcal{M}}(n):=\\{u\in
H^{1}_{n}\,:\,(u(0),u(T))\in\overline{cone\,D(\gamma)}\\}$ and

$\underline{\lambda}_{1}(\gamma):=\inf\left\\{\frac{\|u^{\prime}\|_{L^{2}}^{2}}{\|u\|_{L^{2}}^{2}}\,:\,u\in{\mathcal{M}}(n)\setminus\\{0_{\mathbb{R}^{n}}\\}\right\\},$

from [12, Theorem 3.1], we know that $\underline{\lambda}_{1}(\gamma)>0$ and
the conclusion follows from
$\lambda_{1}(\gamma)\geq\underline{\lambda}_{1}(\gamma).$

Similarly to (3.15), condition

$\overline{cone\,D(\eta)}\cap\\{(\xi,\xi)\,:\,\xi\in\mathbb{R}^{m}\\}=\\{0_{\mathbb{R}^{m}\times\mathbb{R}^{m}}\\}$

is a sufficient one to have that $\lambda_{1}(\eta)>0$.

###### Example 3.1

Let $a,b\in\mathbb{R}$, $h\in C_{n}$, $l\in C_{m}$ and consider the system

$\left\\{\begin{array}[]{ll}-\displaystyle\left[\phi_{n}(u^{\prime})\right]^{\prime}=(a-1)u+\displaystyle\frac{b}{1+|u|^{2}}\left[|v|^{2}u+h(t)\right]\\\
-\displaystyle\left[\phi_{m}(v^{\prime})\right]^{\prime}=\displaystyle\frac{b}{1+|v|^{2}}\left[|u|^{2}v+l(t)\right]+(a-1)v\end{array}\quad\mbox{
in }[0,T].\right.$ (3.16)

Then (3.16) associated with the Dirichlet and antiperiodic boundary conditions

$\left\\{\begin{array}[]{ll}u(0)=0_{\mathbb{R}^{n}}=u(T)\\\
v(0)+v(T)=0_{\mathbb{R}^{m}}=v^{\prime}(0)+v^{\prime}(T)\end{array}\right.$

has at least one solution for any constants $a,b\in\mathbb{R}.$ This easily
follows from Theorem 3.1 $(i)$ and Remark 3.1, with

$\gamma(x,y)=\left\\{\begin{array}[]{ll}\mathbb{R}^{n}\times\mathbb{R}^{n}&\mbox{
if }x=0_{\mathbb{R}^{n}}=y\\\ \emptyset&\mbox{ otherwise,
}\end{array}\right.\qquad\eta(x,y)=\left\\{\begin{array}[]{ll}\\{(\xi,\xi)\,:\,\xi\in\mathbb{R}^{m}\\}&\mbox{
if }x+y=0_{\mathbb{R}^{m}}\\\ \emptyset&\mbox{ otherwise. }\end{array}\right.$

On the other hand, if we take $\gamma(x,y)=(y,-x)$
($(x,y)\in\mathbb{R}^{n}\times\mathbb{R}^{n}$), which is a maximal monotone
operator [17, Example 2.23 (a)] of type $\gamma_{A}$ (see (1.7)) with

$A=\left[\begin{array}[]{clrc}\mathbb{O}_{n}&\mathbb{I}_{n}\\\
-\mathbb{I}_{n}&\mathbb{O}_{n}\end{array}\right],$

and

$\eta(x,y)=\left\\{\begin{array}[]{ll}\\{(\xi,-\xi)\,:\,\xi\in\mathbb{R}^{m}\\}&\mbox{
if }x=y\\\ \emptyset&\mbox{ otherwise, }\end{array}\right.$

then the boundary conditions (1.2) become (see (1.8)):

$\left\\{\begin{array}[]{ll}\phi_{n}(u^{\prime}(0))=u(T),\
\phi_{n}(u^{\prime}(T))=u(0)\\\
v(0)-v(T)=0_{\mathbb{R}^{m}}=v^{\prime}(0)-v^{\prime}(T).\end{array}\right.$
(3.17)

In this case $\lambda_{1}(\gamma)=0=\lambda_{1}(\eta)$ and by Theorem 3.1
$(iv)$ we get that problem (3.16)-(3.17) has at least one solution provided
that $a,b\in\mathbb{R}_{+}$ and $a+b<1$.

###### Corollary 3.1

Problem (1.5)-(1.6) has at least one solution if one of the following two
conditions is satisfied:

$(i)$ $\lambda_{1}(\gamma)>0;$

$(ii)$ there are constants $a\in[0,1)$, $\delta\in\mathbb{R}_{+}$ such that

$\langle f(t,x)\,|\,x\rangle\leq(a-1)|x|^{2}+\delta,\quad\mbox{ for all
}t\in[0,T],\,x\in\mathbb{R}^{n}.$ (3.18)

Proof. Theorem 3.1 applies with $m=n$, $\eta=\gamma$, $f_{1}(t,u,v)=f(t,u)$
and $f_{2}(t,u,v)=f(t,v)$. If $(u,v)$ solves (1.1)-(1.2) with these choices of
the data, then each of the functions $u$ or $v$ is a solution of (1.5)-(1.6).
Clearly, (3.18) implies that (3.6) hold true with $b=c=0$ and $d=a$. The
matrix $M$ in (3.5) becomes

$M=\left[\begin{array}[]{clrc}a&0\\\ 0&a\end{array}\right]$

and it is convergent to zero because $a\in[0,1).$

###### Corollary 3.2

If

$\limsup_{|x|\to\infty}\frac{\langle
f(t,x)|x\rangle}{|x|^{2}}<0,\quad\mbox{uniformly with }t\in[0,T],$ (3.19)

then condition $(ii)$ in Corollary 3.1 is fulfilled, and hence problem
(1.5)-(1.6) has at least one solution.

Proof. From (3.19) we can find constants $\sigma\in(0,1]$ and $\rho>0$ such
that

$\langle f(t,x)|x\rangle\leq-\sigma|x|^{2},\quad\mbox{for all }\
t\in[0,T]\mbox{ and }x\in\mathbb{R}^{n}\mbox{ with }|x|>\rho.$

Putting $k=k(\rho):=\max\\{|\langle
f(t,x)|x\rangle|\,:\,(t,x)\in[0,T]\times\mathbb{B}_{\rho}(n)\\},$ one has

$\langle f(t,x)|x\rangle\leq-\sigma|x|^{2}+\sigma\rho^{2}+k,\quad\mbox{for all
}\ t\in[0,T]\mbox{ and }x\in\mathbb{R}^{n}.$

Then, (3.18) holds true with $a=1-\sigma$ and $\delta=\sigma\rho^{2}+k$.

## References

  * [1] A. Ambrosetti and D. Arcoya, On the relativistic pendulum-type equation, Differential Integral Equations 33 (2020), 91–112.
  * [2] D. Arcoya, C. Bereanu and P.J. Torres, Critical point theory for the Lorentz force equation, Arch. Rational Mech. Anal. 232 (2019), 1685–1724.
  * [3] A. Boscaggin, W. Dambrosio and G. Feltrin, Periodic solutions to a perturbed relativistic Kepler problem, SIAM J. Math. Anal. 53 (2021), 5813–5834.
  * [4] A. Boscaggin, W. Dambrosio and D. Papini, Infinitely many periodic solutions to a Lorentz force equation with singular electromagnetic potential, J. Differential Equations 383 (2024), 190–213.
  * [5] H. Brézis and J. Mawhin, Periodic solutions of the forced relativistic pendulum, Differential Integral Equations 23 (2010), 801–810.
  * [6] H. Brézis and J. Mawhin, Periodic solutions of Lagrangian systems of relativistic oscillators, Commun. Appl. Anal. 15 (2011), 235–250.
  * [7] L. Gasinski and N.S. Papageorgiou, Nonlinear second-order multivalued boundary value problems, Proc. Indian Acad. Sci. (Math. Sci.) 113 (2003), 293–319.
  * [8] P. Jebelean, Potential systems with singular $\phi$-Laplacian, http://arxiv.org/abs/2406.09090.
  * [9] P. Jebelean, Variational methods for ordinary $p$-Laplacian systems with potential boundary conditions, Adv. Differential Equations 14 (2008), 273–322.
  * [10] P. Jebelean, J. Mawhin and C. Şerban, Multiple periodic solutions for perturbed relativistic pendulum systems, Proc. Amer. Math. Soc. 143 (2015), 3029–3039.
  * [11] P. Jebelean and R. Precup, Solvability of $p,q$-Laplacian systems with potential boundary conditions, Appl. Anal. 89 (2010), 221–228.
  * [12] P. Jebelean and R. Precup, Poincaré inequality in reflexive cones, Appl. Math. Lett. 24 (2011), 359–363.
  * [13] P. Jebelean and R. Precup, Symmetric positive solutions to a singular $\phi$-Laplace equation, J. Lond. Math. Soc. 99 (2019), 495–515.
  * [14] P. Jebelean and C. Şerban, Relativistic type systems with parametric odd nonlinearities, Commun. Pure Appl. Anal. 22 (2023), 736–750.
  * [15] D.A. Kandilakis and N.S. Papageorgiou, Existence theorems for nonlinear boundary value problems for second order differential inclusions, J. Differential Equations 132 (1996), 107–125.
  * [16] N.G. Lloyd, Degree Theory, Cambridge University Press, Cambridge, 1978.
  * [17] R.R. Phelps, Lectures on maximal monotone operators, Extracta Math. 12 (1997), 193–230.
  * [18] R. Precup, The role of matrices that are convergent to zero in the study of semilinear operator systems, Math. Comput. Modelling 49 (2009), 703–708.
  * [19] R.T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 209–216.
  * [20] R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1972.