# Bell’s theorem in time without inequalities

Md. Manirul Ali  0000-0002-5076-7619<EMAIL_ADDRESS>Centre for Quantum
Science and Technology, Chennai Institute of Technology, Chennai 600069, India

###### Abstract

Bell’s theorem revealed that a local hidden-variable model cannot completely
reproduce the quantum mechanical predictions. Bell’s inequality provides an
upper bound under the locality and reality assumptions that can be violated by
correlated measurement statistics of quantum mechanics. Greenberger, Horne,
and Zeilinger (GHZ) gave a more compelling proof of Bell’s theorem without
inequalities by considering perfect correlations rather than statistical
correlations. This work presents a temporal analog of the GHZ argument that
establishes Bell’s theorem in time without inequalities.

###### pacs:

03.65.Ta,03.65.Ud,03.67.-a

Einstein, Podolski, and Rosen (EPR) in 1935 remarked [1] that the description
of nature depicted by quantum mechanics is incomplete, whereas a more complete
realistic theory is possible where physical observables can have predetermined
definite values even in the absence of measurement. In realistic hidden-
variables models [2], the very occurrence of a definite outcome in an
individual measurement of an observable is predetermined by some hidden-
variables. Bell’s theorem showed [3] that a local hidden-variables model
cannot completely reproduce some spatially-separated measurement statistics or
statistical correlations of a pair of two-level quantum systems. Motivated by
Bell’s theorem, a temporal analogue of Bell inequalities was provided by
Leggett and Garg [4] in terms of time-separated measurement statistics or
temporal statistical correlation functions. Leggett-Garg inequality was
constructed under two classical assumptions (a1) macrorealism: macroscopic
systems are always in a definite state with well-defined pre-existing value
and (a2) noninvasive measurability: that pre-existing value can be measured
without disturbing the system. These classical realism and noninvasive
measurability assumptions together allow us to consider a single joint
probability distribution for all time-separated measurement observables in an
experiment, from which the marginal probability distributions can also be
obtained [4, 5, 6, 7]. Hence, Bell-inequalities and Leggett-Garg inequalities
have a common origin, they are related by their implicit assumption of joint
realism that there exists a classical joint probability distribution
associated to the joint measurements that determines all measurement
statistics [4, 5, 6, 7]. More precisely, the common assumption is that there
exists a classical probability distribution that can reproduce all spatially-
separated or time-separated measurement statistics in quantum mechanics. Bell-
inequalities and the Leggett-Garg inequalities both were shown to be violated
by certain quantum mechanical statistical predictions [8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18].

A more powerful demonstration of Bell’s theorem was given by Greenberger,
Horne, and Zeilinger [19, 20, 21] using three or more correlated spin-1/2
quantum particles. Unlike Bell’s original theorem, GHZ analysis examines
perfect correlations rather than statistical correlations or imperfect
correlations, and it is independent of any inequality bound. Hence GHZ-
nonlocality tests Bell’s local realism hypothesis without inequalities. To
test Bell inequalities or Leggett-Garg inequalities one needs to repeat the
measurements many times with the access of many copies of the state, to
estimate the spatial or temporal correlation functions using the probabilities
of measurement outcomes, whereas the GHZ experiment needs only be repeated
once in principle. In this work, we are providing a temporal analogue of GHZ
argument which is surprisingly unexplored in the literature. GHZ nonlocality
argument tests Bell’s local realism without inequalities, in the same spirit
our result in this work tests Leggett-Garg’s classical realism and noninvasive
measurability hypothesis without any statistical correlation functions or
inequalities. We provide here a non-probabilistic argument. Leggett-Garg
inequalities and Bell-inequalities have identical structures and are
intimately related. Bell-inequalities provide upper bounds on correlations
between measurements on spatially separated systems, whereas Leggett-Garg
inequalities provide upper bounds on correlations of measurements at different
times. If one perceives Leggett-Garg inequality as the temporal version of
Bell-inequality [13, 22, 23, 24, 25], then our result or proof can be regarded
as a stronger test of Bell’s theorem in time without a statistical inequality.

The GHZ experiment [26, 27, 28] works as follows. Consider three spin-1/2
particles prepared in an entangled state

$\displaystyle|\Psi_{GHZ}\rangle=\frac{1}{\sqrt{2}}\left(|\uparrow\rangle_{1}|\uparrow\rangle_{2}|\uparrow\rangle_{3}+|\downarrow\rangle_{1}|\downarrow\rangle_{2}|\downarrow\rangle_{3}\right),$
(1)

where $|\uparrow\rangle$ and $|\downarrow\rangle$ represent the eigenstates of
spin measurement operator $\sigma_{z}$ along z-axis with eigenvalues $+1$ and
$-1$ respectively. Let us now consider four mutually commuting Hermitian
operators in the three-qubit Hilbert space as

$\displaystyle\sigma_{x}^{1}\leavevmode\nobreak\
\sigma_{x}^{2}\leavevmode\nobreak\ \sigma_{x}^{3},$ (2)
$\displaystyle\sigma_{x}^{1}\leavevmode\nobreak\
\sigma_{y}^{2}\leavevmode\nobreak\ \sigma_{y}^{3},$ (3)
$\displaystyle\sigma_{y}^{1}\leavevmode\nobreak\
\sigma_{y}^{2}\leavevmode\nobreak\ \sigma_{x}^{3},$ (4)
$\displaystyle\sigma_{y}^{1}\leavevmode\nobreak\
\sigma_{x}^{2}\leavevmode\nobreak\ \sigma_{y}^{3}.$ (5)

These four operators commute with each other for which they can have
simultaneous eigenstates and eigenvalues. The state $|\Psi_{GHZ}\rangle$ is an
eigenstate of the operators $\sigma_{x}^{1}\sigma_{y}^{2}\sigma_{y}^{3}$,
$\sigma_{y}^{1}\sigma_{y}^{2}\sigma_{x}^{3}$, and
$\sigma_{y}^{1}\sigma_{x}^{2}\sigma_{y}^{3}$ with a common eigenvalue $-1$.
One can also verify that $|\Psi_{GHZ}\rangle$ is an eigenstate of
$\sigma_{x}^{1}\sigma_{x}^{2}\sigma_{x}^{3}$ with eigenvalue $+1$. According
to quantum mechanics, the eigenvalue equations for the four mutually commuting
Hermitian operators (2-5) are given by

$\displaystyle\sigma_{x}^{1}\leavevmode\nobreak\
\sigma_{x}^{2}\leavevmode\nobreak\ \sigma_{x}^{3}\leavevmode\nobreak\
|\Psi_{GHZ}\rangle$ $\displaystyle=$ $\displaystyle(+1)\leavevmode\nobreak\
|\Psi_{GHZ}\rangle,$ (6) $\displaystyle\sigma_{x}^{1}\leavevmode\nobreak\
\sigma_{y}^{2}\leavevmode\nobreak\ \sigma_{y}^{3}\leavevmode\nobreak\
|\Psi_{GHZ}\rangle$ $\displaystyle=$ $\displaystyle(-1)\leavevmode\nobreak\
|\Psi_{GHZ}\rangle,$ (7) $\displaystyle\sigma_{y}^{1}\leavevmode\nobreak\
\sigma_{y}^{2}\leavevmode\nobreak\ \sigma_{x}^{3}\leavevmode\nobreak\
|\Psi_{GHZ}\rangle$ $\displaystyle=$ $\displaystyle(-1)\leavevmode\nobreak\
|\Psi_{GHZ}\rangle,$ (8) $\displaystyle\sigma_{y}^{1}\leavevmode\nobreak\
\sigma_{x}^{2}\leavevmode\nobreak\ \sigma_{y}^{3}\leavevmode\nobreak\
|\Psi_{GHZ}\rangle$ $\displaystyle=$ $\displaystyle(-1)\leavevmode\nobreak\
|\Psi_{GHZ}\rangle.$ (9)

GHZ argument was that these quantum results are incompatible with local
hidden-variables model. Suppose the outcomes given by the eigenvalues of the
local spin measurements $\sigma_{x}^{1}$, $\sigma_{x}^{2}$, $\sigma_{x}^{3}$,
$\sigma_{y}^{1}$, $\sigma_{y}^{2}$, $\sigma_{y}^{3}$ are fixed by some local
hidden-variables model as $v_{r}^{i}=\pm 1$, $r\in{x,y,z}$ and $i\in{1,2,3}$.
Those predetermined fixed values must satisfy the relations

$\displaystyle v_{x}^{1}\leavevmode\nobreak\ v_{x}^{2}\leavevmode\nobreak\
v_{x}^{3}=+1,$ (10) $\displaystyle v_{x}^{1}\leavevmode\nobreak\
v_{y}^{2}\leavevmode\nobreak\ v_{y}^{3}=-1,$ (11) $\displaystyle
v_{y}^{1}\leavevmode\nobreak\ v_{y}^{2}\leavevmode\nobreak\ v_{x}^{3}=-1,$
(12) $\displaystyle v_{y}^{1}\leavevmode\nobreak\
v_{x}^{2}\leavevmode\nobreak\ v_{y}^{3}=-1,$ (13)

to reproduce the quantum mechanical results given by the equations (6-9).

Now, the product of measurement outcomes for the four observables
$\sigma_{x}^{1}\sigma_{x}^{2}\sigma_{x}^{3}$,
$\sigma_{x}^{1}\sigma_{y}^{2}\sigma_{y}^{3}$,
$\sigma_{y}^{1}\sigma_{y}^{2}\sigma_{x}^{3}$, and
$\sigma_{y}^{1}\sigma_{x}^{2}\sigma_{y}^{3}$ should be $-1$, according to
quantum mechanical results (6-9). Whereas according to the left hand sides of
Eqs. (10-13), the product of the predetermined measurement values comes out to
be
$(v_{x}^{1})^{2}(v_{y}^{1})^{2}(v_{x}^{2})^{2}(v_{y}^{2})^{2}(v_{x}^{3})^{2}(v_{y}^{3})^{2}=+1$.
This happens because the values $v_{r}^{i}$ are fixed under a local hidden-
variables scheme and each local spin measurement along a particular axis
appears twice in the overall set of four product measurements. This is in
direct contradiction with the results of quantum mechanics. GHZ experiment
demonstrates the nonlocal nature of quantum mechanics that violates our
classical intuition of local realism. Hence the results of quantum
measurements on three spin-1/2 particles prepared in an entangled state
$|\Psi_{GHZ}\rangle$ cannot be explained if each of the particles are assigned
predetermined values for their local spin components before the measurement.
GHZ argument provides the sharpest possible contradiction between quantum
mechanics and EPR reality criterion [21].

Next, we present a temporal version of GHZ situation where the measurements
are now time-separated. Let us consider a single spin-1/2 system evolving
under a suitable Hamiltonian $H$. Analogous to the three-particle version of
GHZ argument, here we consider single-particle spin measurements at three
successive times $t_{1}$, $t_{2}$, and $t_{3}$ with $t_{3}>t_{2}>t_{1}$. We
consider the following four joint measurement operators in time as

$\displaystyle\sigma_{x}(t_{3})\leavevmode\nobreak\
\sigma_{x}(t_{2})\leavevmode\nobreak\ \sigma_{x}(t_{1}),$ (14)
$\displaystyle\sigma_{x}(t_{3})\leavevmode\nobreak\
\sigma_{y}(t_{2})\leavevmode\nobreak\ \sigma_{y}(t_{1}),$ (15)
$\displaystyle\sigma_{y}(t_{3})\leavevmode\nobreak\
\sigma_{y}(t_{2})\leavevmode\nobreak\ \sigma_{x}(t_{1}),$ (16)
$\displaystyle\sigma_{y}(t_{3})\leavevmode\nobreak\
\sigma_{x}(t_{2})\leavevmode\nobreak\ \sigma_{y}(t_{1}).$ (17)

Quantum mechanics prescribes that the measurement results of an observable are
given by the eigenvalues of the Hermitian operator associated to that
observable. Consider an attempt to assign values (pre-existing values that can
be measured without disturbing the system) to the six individual Hermitian
operators $\sigma_{x}(t_{1})$, $\sigma_{x}(t_{2})$, $\sigma_{x}(t_{3})$,
$\sigma_{y}(t_{1})$, $\sigma_{y}(t_{2})$, $\sigma_{y}(t_{3})$ as the
corresponding eigenvalues of the operators $m_{x}^{1}$, $m_{x}^{2}$,
$m_{x}^{3}$, $m_{y}^{1}$, $m_{y}^{2}$, $m_{y}^{3}$ respectively. This attempt
of value-assignment will fail as is shown below.

We consider a quantum spin-1/2 system evolving under the Hamiltonian
$H=\frac{\hbar}{2}\omega\sigma_{z}$. The time evolution dynamics of the Pauli
spin operators are given by the Heisenberg equations of motion

$\displaystyle\frac{d}{dt}\sigma_{x}(t)=\frac{1}{i\hbar}\left[\sigma_{x}(t),H\right],$
(18)
$\displaystyle\frac{d}{dt}\sigma_{y}(t)=\frac{1}{i\hbar}\left[\sigma_{y}(t),H\right],$
(19)

where the Heisenberg picture operators are

$\displaystyle\sigma_{x}(t)=e^{iHt/\hbar}\leavevmode\nobreak\
\sigma_{x}\leavevmode\nobreak\ e^{-iHt/\hbar},$ (20)
$\displaystyle\sigma_{y}(t)=e^{iHt/\hbar}\leavevmode\nobreak\
\sigma_{y}\leavevmode\nobreak\ e^{-iHt/\hbar}.$ (21)

The solutions of the Heisenberg equations of motion are given by

$\displaystyle\sigma_{x}(t)=\sigma_{x}\cos(\omega t)-\sigma_{y}\sin(\omega
t),$ (22) $\displaystyle\sigma_{y}(t)=\sigma_{y}\cos(\omega
t)+\sigma_{x}\sin(\omega t),$ (23)

where

$\displaystyle\sigma_{x}=\left(\begin{array}[]{cc}0&1\\\ 1&0\\\
\end{array}\right),\sigma_{y}=\left(\begin{array}[]{cc}0&-i\\\ i&0\\\
\end{array}\right),\sigma_{z}=\left(\begin{array}[]{cc}1&0\\\ 0&-1\\\
\end{array}\right).$ (30)

We consider the time-dependent dynamical observables $\sigma_{x}(t)$,
$\sigma_{y}(t)$ at three different times $t_{1}=0$,
$t_{2}=\frac{\pi}{2\omega}$, and $t_{3}=\frac{3\pi}{2\omega}$. Then according
to the time evolution equations (22) and (23), the four operators given by
Eqs. (14-17) become Hermitian operators for the above choice of times $t_{1}$,
$t_{2}$, and $t_{3}$. Moreover, under that situation all the four operators
(14-17) also commute with each other. Because they all commute, the four
operators can have simultaneous eigenstates and eigenvalues. In this
situation, one can verify that the state
$|\Phi\rangle=(1/\sqrt{2})(|\uparrow\rangle+|\downarrow\rangle)$ is an
eigenstate of the three operators
$\sigma_{x}(t_{3})\sigma_{x}(t_{2})\sigma_{x}(t_{1})$,
$\sigma_{x}(t_{3})\sigma_{y}(t_{2})\sigma_{y}(t_{1})$, and
$\sigma_{y}(t_{3})\sigma_{y}(t_{2})\sigma_{x}(t_{1})$ with a common eigenvalue
$-1$. Similarly, the state $|\Phi\rangle$ is also an eigenstate of the fourth
operator $\sigma_{y}(t_{3})\sigma_{x}(t_{2})\sigma_{y}(t_{1})$ with eigenvalue
$+1$. Then we can write down the eigenvalue equations for the four mutually
commuting Hermitian operators given by Eqs. (14-17) as follows

$\displaystyle\sigma_{x}(t_{3})\leavevmode\nobreak\
\sigma_{x}(t_{2})\leavevmode\nobreak\ \sigma_{x}(t_{1})\leavevmode\nobreak\
|\Phi\rangle=(-1)\leavevmode\nobreak\ |\Phi\rangle,$ (31)
$\displaystyle\sigma_{x}(t_{3})\leavevmode\nobreak\
\sigma_{y}(t_{2})\leavevmode\nobreak\ \sigma_{y}(t_{1})\leavevmode\nobreak\
|\Phi\rangle=(-1)\leavevmode\nobreak\ |\Phi\rangle,$ (32)
$\displaystyle\sigma_{y}(t_{3})\leavevmode\nobreak\
\sigma_{y}(t_{2})\leavevmode\nobreak\ \sigma_{x}(t_{1})\leavevmode\nobreak\
|\Phi\rangle=(-1)\leavevmode\nobreak\ |\Phi\rangle,$ (33)
$\displaystyle\sigma_{y}(t_{3})\leavevmode\nobreak\
\sigma_{x}(t_{2})\leavevmode\nobreak\ \sigma_{y}(t_{1})\leavevmode\nobreak\
|\Phi\rangle=(+1)\leavevmode\nobreak\ |\Phi\rangle.$ (34)

We now impose the idea of classical realism$-$the results of measuring
observables must have already been specified prior to measurement. We assume
that the measurement values of the individual measurement observables
$\sigma_{x}(t_{1})$, $\sigma_{y}(t_{1})$, $\sigma_{x}(t_{2})$,
$\sigma_{y}(t_{2})$, $\sigma_{x}(t_{3})$, and $\sigma_{y}(t_{3})$ are
predetermined, and measurements merely reveal those predetermined values.
Whether we perform any measurement or not, the x-component of spin
$\sigma_{x}$ has predetermined values $m_{x}^{1}$, $m_{x}^{2}$, and
$m_{x}^{3}$ at times $t_{1}$, $t_{2}$, and $t_{3}$ respectively. Similarly, we
assign the predetermined measurement values of the operators
$\sigma_{y}(t_{1})$, $\sigma_{y}(t_{2})$, $\sigma_{y}(t_{3})$ as $m_{y}^{1}$,
$m_{y}^{2}$, and $m_{y}^{3}$ respectively. Any such predetermined value is
assumed to be independent of the sequence at which the sets of time-separated
spin measurements are done on this single-qubit system. This means measurement
at one time does not influence the subsequent dynamics of the system and the
measurement outcomes at a later time. This is Leggett-Garg’s noninvasive
measurability assumption which states that the predetermined values of the
observables can be measured without disturbing the system. Now for our
suitably chosen three different times $t_{1}$, $t_{2}$ and $t_{3}$ and for a
given initial state $|\Phi\rangle$, the four mutually commuting Hermitian
operators $\sigma_{x}(t_{3})\sigma_{x}(t_{2})\sigma_{x}(t_{1})$,
$\sigma_{x}(t_{3})\sigma_{y}(t_{2})\sigma_{y}(t_{1})$,
$\sigma_{y}(t_{3})\sigma_{y}(t_{2})\sigma_{x}(t_{1})$, and
$\sigma_{y}(t_{3})\sigma_{x}(t_{2})\sigma_{y}(t_{1})$ have a set of
simultaneous eigenvalues $-1$, $-1$, $-1$ and $+1$ respectively. Hence
according to the requirement of quantum mechanics, consistent with the
eigenvalue equations Eqs. (31-34) we then have the following relations to
satisfy

$\displaystyle m_{x}^{3}\leavevmode\nobreak\ m_{x}^{2}\leavevmode\nobreak\
m_{x}^{1}=-1,$ (35) $\displaystyle m_{x}^{3}\leavevmode\nobreak\
m_{y}^{2}\leavevmode\nobreak\ m_{y}^{1}=-1,$ (36) $\displaystyle
m_{y}^{3}\leavevmode\nobreak\ m_{y}^{2}\leavevmode\nobreak\ m_{x}^{1}=-1,$
(37) $\displaystyle m_{y}^{3}\leavevmode\nobreak\
m_{x}^{2}\leavevmode\nobreak\ m_{y}^{1}=+1.$ (38)

In the above relations (35-38) the value of any particular quantity
$m_{x}^{1}$, $m_{x}^{2}$, $m_{x}^{3}$, $m_{y}^{1}$, $m_{y}^{2}$, or
$m_{y}^{3}$ is the same irrespective of the equation in which it appears. For
instance, the value of $m_{x}^{2}$ is the same in Eqs. (35) and (38), which is
fixed according to the assumption of classical realism. However multiplying
the left hand sides of equations (35-38) yields
$(m_{x}^{1})^{2}(m_{x}^{2})^{2}(m_{x}^{3})^{2}(m_{y}^{1})^{2}(m_{y}^{2})^{2}(m_{y}^{3})^{2}=+1$,
which is in contradiction with the product of right hand sides. The
contradiction is sharp and extreme in the sense that for some particular
measurement, $+1$ result is expected whereas quantum mechanics results $-1$
instead. This result establishes a GHZ version of Bell’s theorem in time
without inequalities. The temporal GHZ version of the Bell’s theorem,
discovered in this work can be experimentally verified with the present
quantum technology [28, 29, 30, 31, 32]. Experimental realization of
Greenberger-Horne-Zeilinger correlations is demonstrated in Ref. [33] using
nuclear magnetic resonance (NMR), where the perfect correlations given by

$\displaystyle\langle\Psi_{GHZ}|\sigma_{x}^{1}\leavevmode\nobreak\
\sigma_{x}^{2}\leavevmode\nobreak\ \sigma_{x}^{3}\leavevmode\nobreak\
|\Psi_{GHZ}\rangle$ $\displaystyle=$ $\displaystyle+1,$ (39)
$\displaystyle\langle\Psi_{GHZ}|\sigma_{x}^{1}\leavevmode\nobreak\
\sigma_{y}^{2}\leavevmode\nobreak\ \sigma_{y}^{3}\leavevmode\nobreak\
|\Psi_{GHZ}\rangle$ $\displaystyle=$ $\displaystyle-1,$ (40)
$\displaystyle\langle\Psi_{GHZ}|\sigma_{y}^{1}\leavevmode\nobreak\
\sigma_{y}^{2}\leavevmode\nobreak\ \sigma_{x}^{3}\leavevmode\nobreak\
|\Psi_{GHZ}\rangle$ $\displaystyle=$ $\displaystyle-1,$ (41)
$\displaystyle\langle\Psi_{GHZ}|\sigma_{y}^{1}\leavevmode\nobreak\
\sigma_{x}^{2}\leavevmode\nobreak\ \sigma_{y}^{3}\leavevmode\nobreak\
|\Psi_{GHZ}\rangle$ $\displaystyle=$ $\displaystyle-1$ (42)

were experimentally verified with respect to the GHZ state 111The GHZ state
considered in Ref. [33] is
$|\psi\rangle=(1/\sqrt{2})\left(|\uparrow\rangle_{1}|\uparrow\rangle_{2}|\uparrow\rangle_{3}-|\downarrow\rangle_{1}|\downarrow\rangle_{2}|\downarrow\rangle_{3}\right)$,
for which the GHZ correlations $\langle\psi|\sigma_{x}^{1}\leavevmode\nobreak\
\sigma_{x}^{2}\leavevmode\nobreak\ \sigma_{x}^{3}|\psi\rangle=-1$,
$\langle\psi|\sigma_{x}^{1}\leavevmode\nobreak\
\sigma_{y}^{2}\leavevmode\nobreak\ \sigma_{y}^{3}|\psi\rangle=+1$,
$\langle\psi|\sigma_{y}^{1}\leavevmode\nobreak\
\sigma_{y}^{2}\leavevmode\nobreak\ \sigma_{x}^{3}|\psi\rangle=+1$, and
$\langle\psi|\sigma_{y}^{1}\leavevmode\nobreak\
\sigma_{x}^{2}\leavevmode\nobreak\ \sigma_{y}^{3}|\psi\rangle=+1$, confirming
quantum prediction of GHZ correlations. The perfect GHZ correlations (39-42)
are obtained from equations (6-9). In the temporal situation, with the same
spirit one can experimentally verify the three-time temporal correlation
functions with respect to the state $|\Phi\rangle$ as follows

$\displaystyle\langle\Phi|\sigma_{x}(t_{3})\leavevmode\nobreak\
\sigma_{x}(t_{2})\leavevmode\nobreak\ \sigma_{x}(t_{1})\leavevmode\nobreak\
|\Phi\rangle$ $\displaystyle=$ $\displaystyle-1,$ (43)
$\displaystyle\langle\Phi|\sigma_{x}(t_{3})\leavevmode\nobreak\
\sigma_{y}(t_{2})\leavevmode\nobreak\ \sigma_{y}(t_{1})\leavevmode\nobreak\
|\Phi\rangle$ $\displaystyle=$ $\displaystyle-1,$ (44)
$\displaystyle\langle\Phi|\sigma_{y}(t_{3})\leavevmode\nobreak\
\sigma_{y}(t_{2})\leavevmode\nobreak\ \sigma_{x}(t_{1})\leavevmode\nobreak\
|\Phi\rangle$ $\displaystyle=$ $\displaystyle-1,$ (45)
$\displaystyle\langle\Phi|\sigma_{y}(t_{3})\leavevmode\nobreak\
\sigma_{x}(t_{2})\leavevmode\nobreak\ \sigma_{y}(t_{1})\leavevmode\nobreak\
|\Phi\rangle$ $\displaystyle=$ $\displaystyle+1,$ (46)

where the perfect temporal correlations (43-46) are obtained from the
eigenvalue equations (31-34). NMR technique to measure such temporal
correlation functions are already available [35, 36]. NMR is one of the most
mature technologies for implementing isolated quantum spin-1/2 dynamics in a
magnetic field [37, 38]. The NMR realization of GHZ correlations [33] provides
an added motivation for possible experimental verification of our temporal GHZ
result.

## References

  * Einstein  _et al._ [1935] A. Einstein, B. Podolsky, and N. Rosen, Can quantum-mechanical description of physical reality be considered complete?, Physical Review 47, 777 (1935).
  * Bell [1966] J. S. Bell, On the problem of hidden variables in quantum mechanics, Reviews of Modern physics 38, 447 (1966).
  * Bell [1964] J. S. Bell, On the einstein podolsky rosen paradox, Physics 1, 195 (1964).
  * Leggett and Garg [1985] A. J. Leggett and A. Garg, Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks?, Physical Review Letters 54, 857 (1985).
  * Das  _et al._ [2014] S. Das, S. Aravinda, R. Srikanth, and D. Home, Unification of bell, leggett-garg and kochen-specker inequalities: hybrid spatio-temporal inequalities, Europhysics Letters 104, 60006 (2014).
  * Markiewicz  _et al._ [2014] M. Markiewicz, P. Kurzyński, J. Thompson, S.-Y. Lee, A. Soeda, T. Paterek, and D. Kaszlikowski, Unified approach to contextuality, nonlocality, and temporal correlations, Physical Review A 89, 042109 (2014).
  * Emary  _et al._ [2014] C. Emary, N. Lambert, and F. Nori, Leggett–garg inequalities, Reports on Progress in Physics 77, 016001 (2014).
  * Freedman and Clauser [1972] S. J. Freedman and J. F. Clauser, Experimental test of local hidden-variable theories, Physical Review Letters 28, 938 (1972).
  * Aspect  _et al._ [1982] A. Aspect, J. Dalibard, and G. Roger, Experimental test of bell’s inequalities using time-varying analyzers, Physical Review Letters 49, 1804 (1982).
  * Hensen  _et al._ [2015] B. Hensen, H. Bernien, A. E. Dréau, A. Reiserer, N. Kalb, M. S. Blok, J. Ruitenberg, R. F. Vermeulen, R. N. Schouten, C. Abellán, _et al._ , Loophole-free bell inequality violation using electron spins separated by 1.3 kilometres, Nature 526, 682 (2015).
  * Giustina  _et al._ [2015] M. Giustina, M. A. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, _et al._ , Significant-loophole-free test of bell’s theorem with entangled photons, Physical Review Letters 115, 250401 (2015).
  * Shalm  _et al._ [2015] L. K. Shalm, E. Meyer-Scott, B. G. Christensen, P. Bierhorst, M. A. Wayne, M. J. Stevens, T. Gerrits, S. Glancy, D. R. Hamel, M. S. Allman, _et al._ , Strong loophole-free test of local realism, Physical Review Letters 115, 250402 (2015).
  * Palacios-Laloy  _et al._ [2010] A. Palacios-Laloy, F. Mallet, F. Nguyen, P. Bertet, D. Vion, D. Esteve, and A. N. Korotkov, Experimental violation of a bell’s inequality in time with weak measurement, Nature Physics 6, 442 (2010).
  * Xu  _et al._ [2011] J.-S. Xu, C.-F. Li, X.-B. Zou, and G.-C. Guo, Experimental violation of the leggett-garg inequality under decoherence, Scientific Reports 1, 101 (2011).
  * Goggin  _et al._ [2011] M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde, Violation of the leggett–garg inequality with weak measurements of photons, Proceedings of the National Academy of Sciences 108, 1256 (2011).
  * Knee  _et al._ [2012] G. C. Knee, S. Simmons, E. M. Gauger, J. J. Morton, H. Riemann, N. V. Abrosimov, P. Becker, H.-J. Pohl, K. M. Itoh, M. L. Thewalt, _et al._ , Violation of a leggett–garg inequality with ideal non-invasive measurements, Nature Communications 3, 606 (2012).
  * Knee  _et al._ [2016] G. C. Knee, K. Kakuyanagi, M.-C. Yeh, Y. Matsuzaki, H. Toida, H. Yamaguchi, S. Saito, A. J. Leggett, and W. J. Munro, A strict experimental test of macroscopic realism in a superconducting flux qubit, Nature Communications 7, 13253 (2016).
  * Zhan  _et al._ [2023] T. Zhan, C. Wu, M. Zhang, Q. Qin, X. Yang, H. Hu, W. Su, J. Zhang, T. Chen, Y. Xie, _et al._ , Experimental violation of the leggett-garg inequality in a three-level trapped-ion system, Physical Review A 107, 012424 (2023).
  * Greenberger  _et al._ [1989] D. M. Greenberger, M. A. Horne, and A. Zeilinger, Going beyond bell’s theorem, in  _Bell’s theorem, quantum theory and conceptions of the universe_ (Springer, 1989) pp. 69–72.
  * Greenberger  _et al._ [1990] D. M. Greenberger, M. A. Horne, A. Shimony, and A. Zeilinger, Bell’s theorem without inequalities, American Journal of Physics 58, 1131 (1990).
  * Mermin [1990] N. D. Mermin, Quantum mysteries revisited, American Journal of Physics 58, 731 (1990).
  * Paz and Mahler [1993] J. P. Paz and G. Mahler, Proposed test for temporal bell inequalities, Physical Review Letters 71, 3235 (1993).
  * Ruskov  _et al._ [2006] R. Ruskov, A. N. Korotkov, and A. Mizel, Signatures of quantum behavior in single-qubit weak measurements, Physical Review Letters 96, 200404 (2006).
  * Emary  _et al._ [2012] C. Emary, N. Lambert, and F. Nori, Leggett-garg inequality in electron interferometers, Physical Review B 86, 235447 (2012).
  * Souza  _et al._ [2011] A. Souza, I. Oliveira, and R. Sarthour, A scattering quantum circuit for measuring bell’s time inequality: a nuclear magnetic resonance demonstration using maximally mixed states, New Journal of Physics 13, 053023 (2011).
  * Pan  _et al._ [2000] J.-W. Pan, D. Bouwmeester, M. Daniell, H. Weinfurter, and A. Zeilinger, Experimental test of quantum nonlocality in three-photon greenberger–horne–zeilinger entanglement, Nature 403, 515 (2000).
  * Erhard  _et al._ [2018] M. Erhard, M. Malik, M. Krenn, and A. Zeilinger, Experimental greenberger–horne–zeilinger entanglement beyond qubits, Nature Photonics 12, 759 (2018).
  * Liu  _et al._ [2021] Z.-H. Liu, J. Zhou, H.-X. Meng, M. Yang, Q. Li, Y. Meng, H.-Y. Su, J.-L. Chen, K. Sun, J.-S. Xu, _et al._ , Experimental test of the greenberger–horne–zeilinger-type paradoxes in and beyond graph states, npj Quantum Information 7, 66 (2021).
  * Clarke and Wilhelm [2008] J. Clarke and F. K. Wilhelm, Superconducting quantum bits, Nature 453, 1031 (2008).
  * Häffner  _et al._ [2008] H. Häffner, C. F. Roos, and R. Blatt, Quantum computing with trapped ions, Physics Reports 469, 155 (2008).
  * Kirchmair  _et al._ [2009] G. Kirchmair, F. Zähringer, R. Gerritsma, M. Kleinmann, O. Gühne, A. Cabello, R. Blatt, and C. F. Roos, State-independent experimental test of quantum contextuality, Nature 460, 494 (2009).
  * Lapkiewicz  _et al._ [2011] R. Lapkiewicz, P. Li, C. Schaeff, N. K. Langford, S. Ramelow, M. Wieśniak, and A. Zeilinger, Experimental non-classicality of an indivisible quantum system, Nature 474, 490 (2011).
  * Nelson  _et al._ [2000] R. J. Nelson, D. G. Cory, and S. Lloyd, Experimental demonstration of greenberger-horne-zeilinger correlations using nuclear magnetic resonance, Physical Review A 61, 022106 (2000).
  * Note [1] The GHZ state considered in Ref. [33] is $|\psi\rangle=(1/\sqrt{2})\left(|\uparrow\rangle_{1}|\uparrow\rangle_{2}|\uparrow\rangle_{3}-|\downarrow\rangle_{1}|\downarrow\rangle_{2}|\downarrow\rangle_{3}\right)$, for which the GHZ correlations $\langle\psi|\sigma_{x}^{1}\leavevmode\nobreak\ \sigma_{x}^{2}\leavevmode\nobreak\ \sigma_{x}^{3}|\psi\rangle=-1$, $\langle\psi|\sigma_{x}^{1}\leavevmode\nobreak\ \sigma_{y}^{2}\leavevmode\nobreak\ \sigma_{y}^{3}|\psi\rangle=+1$, $\langle\psi|\sigma_{y}^{1}\leavevmode\nobreak\ \sigma_{y}^{2}\leavevmode\nobreak\ \sigma_{x}^{3}|\psi\rangle=+1$, and $\langle\psi|\sigma_{y}^{1}\leavevmode\nobreak\ \sigma_{x}^{2}\leavevmode\nobreak\ \sigma_{y}^{3}|\psi\rangle=+1$.
  * Katiyar  _et al._ [2013] H. Katiyar, A. Shukla, K. R. K. Rao, and T. Mahesh, Violation of entropic leggett-garg inequality in nuclear spins, Physical Review A 87, 052102 (2013).
  * Singh  _et al._ [2022] D. Singh, K. Dorai, _et al._ , Experimental demonstration of the violation of the temporal peres-mermin inequality using contextual temporal correlations and noninvasive measurements, Physical Review A 105, 022216 (2022).
  * Ernst  _et al._ [1990] R. R. Ernst, G. Bodenhausen, and A. Wokaun,  _Principles of nuclear magnetic resonance in one and two dimensions_ (Oxford University Press, 1990).
  * Oliveira  _et al._ [2011] I. Oliveira, R. Sarthour Jr, T. Bonagamba, E. Azevedo, and J. C. Freitas,  _NMR quantum information processing_ (Elsevier, 2011).


_Electronic Journal of Differential Equations_ , Vol. 2020 (2020), No. 111,
pp. 1–14.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

# Convergence of solutions of fractional differential equations to power-type
functions

Mohammed Dahan Kassim, Nasser Eddine Tatar  Mohammed Dahan Kassim
Department of Basic Engineering Sciences, College of Engineering, Imam
Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam 31441, Saudi Arabia
<EMAIL_ADDRESS>Nasser Eddine Tatar
King Fahd University of Petroleum and Minerals, Department of Mathematics and
Statistics, Dhahran, 31261, Saudi Arabia<EMAIL_ADDRESS>
###### Abstract.

In this article we study the asymptotic behavior of solutions of some
fractional differential equations. We prove convergence to power type
functions under some assumptions on the nonlinearities. Our results extend and
generalize some existing well-known results on solutions of ordinary
differential equations. Appropriate estimations and lemmas such as a
fractional version of L’Hopital’s rule are used.

###### Key words and phrases:

Asymptotic behavior; boundedness; fractional differential equation; Caputo
fractional derivative; Riemann-Liouville fractional derivative

###### 2010 Mathematics Subject Classification:

34E10, 26A33, 34A08

©2020 Texas State University.

Submitted September 15, 2020. Published November 4, 2020.

##  1\. Introduction

We consider the initial value problems

$\begin{gathered}(^{C}\mathfrak{D}_{0}^{\alpha}x)^{\prime}(\tau)=f(\tau,x(\tau),^{C}\mathfrak{D}_{0}^{\beta}x(\tau)),\quad
0<\beta<\alpha<1,\;\tau>0\\\
{}^{C}\mathfrak{D}_{0}^{\alpha}x(\tau)\big{|}_{\tau=0}=b_{2},\quad=x(\tau)|_{\tau=0}=b_{1},\quad
b_{1},\quad b_{2}\in\mathbb{R},\end{gathered}$ (1.1)

and

$\begin{gathered}{}^{C}\mathfrak{D}_{0}^{\alpha}x(\tau)=f(\tau,x(\tau),{}^{C}\mathfrak{D}_{0}^{\beta}x(\tau)),\quad
0\leq\beta<\alpha<1,\;\tau>0\\\ x(\tau)|_{\tau=0}=b,\end{gathered}$ (1.2)

where ${}^{C}\mathfrak{D}_{0}^{\alpha}$ is the Caputo fractional derivative.
The definition of the Caputo fractional derivative is given in the next
section. We prove that the solutions of (1.1) approach power type functions
and the solutions of (1.2) are bounded. To this end, the fractional
differential problems (1.1) and (1.2) are first transformed into equivalent
integral equations in appropriate underlying spaces. Various appropriate
estimates, comparison theorems and lemmas are used. Moreover, we prove a
Caputo fractional version of L’Hopital’s rule. Our arguments here are quite
different from those used so far in the literature.

The behavior of solutions of various classes of ODEs (ordinary differential
equations) has been discussed in fairly a large number of papers in the
literature. For example the equation

$x^{\prime\prime}(\tau)+f(\tau,x(\tau))=0,$ (1.3)

has been studied in [17, 18, 7, 30, 32, 9, 31] and other papers. The authors
proved that, under various conditions, all solutions of (1.3) are asymptotic
to $c\tau+b$ as $\tau\to\infty$, $c,b\in\mathbb{R}$. For the equation

$x^{\prime\prime}(\tau)+f(\tau,x(\tau),x^{\prime}(\tau))=0,$ (1.4)

see, for instance [8, 10, 19, 22, 23, 25, 27, 28]. It is proved that all
solutions of (1.4) can be expressed asymptotically as $c\tau+b$ as
$\tau\to\infty$, $c,b\in\mathbb{R}$.

Medveď and Pekárková [23], studied the one-dimensional p-Laplacian equation

$(|x^{\prime}|^{p-1}x^{\prime})^{\prime}=f(\tau,x,x^{\prime}),\quad p>1.$
(1.5)

They demonstrated that any solution of (1.5) behaves asymptotically as
$b+c\tau$ as $\tau\to\infty$ for some real numbers $b$, $c$.

In [22], the initial value problem

$\begin{gathered}(\Phi_{p}(x^{\prime})\Psi(\tau))^{\prime}+f(\tau,x,x^{\prime})=0,\quad
1<p<2,\\\ x(\tau_{0})=x_{0},\quad x^{\prime}(\tau_{0})=x_{1},\quad\tau_{0}\geq
1,\end{gathered}$ (1.6)

was studied, where $\Phi_{p}(u)=|u|^{p-2}u$ and $\Psi(\tau)$ is a continuous
positive function. Sufficient conditions under which all solutions of (1.6)
obey the asymptotic expansion $x(\tau)=b+c\tau$ are established.

In contrast, the fractional case of equations (1.3) and (1.4) have been
studied by comparatively a only few researchers; see, for instance [1, 2, 3,
4, 5, 6, 11, 12, 13, 14, 15, 20, 21, 24]. In 2009, Băleanu and Mustafa [1]
studied the nonlinear fractional differential equation

${}^{C}\mathfrak{D}_{0}^{\alpha}x(\tau)=f(\tau,x(\tau)),\quad
0<\alpha<1,\quad\tau>0.$ (1.7)

They showed that the solutions of (1.7) are asymptotic to $o(\tau^{c\alpha})$
as $\tau\to\infty$, for some $c$.

In 2012, Medveď [20] studied the problem

$\begin{gathered}{}^{C}\mathfrak{D}_{a}^{\alpha+1}x(\tau)=f(\tau,x(\tau)),\quad
0<\alpha<1,\quad\tau\geq a>1\\\ x(a)=c_{1},\quad
x^{\prime}(a)=c_{2}.\end{gathered}$ (1.8)

He demonstrated that any solution of (1.8) has the asymptotic property
$x(\tau)=b+c\tau$ as $\tau\to\infty$, for some $c,b\in\mathbb{R}$.

Also, in 2013, Medveď [21] discussed the equation

${}^{C}\mathfrak{D}_{a}^{\alpha+1}x(\tau)=f(\tau,x(\tau),x^{\prime}(\tau)),\quad
0<\alpha<1,\quad\tau\geq a>1$ (1.9)

and proved that every solution of (1.9) can be expressed asymptotically as
$b+c\tau$ as $\tau\to\infty$, for some $c,b\in\mathbb{R}$.

Brestovanská and Medveď [6] studied the problem

$\begin{gathered}x^{\prime\prime}(\tau)+f(\tau,x(\tau),x^{\prime}(\tau))+\sum_{i=1}^{m}r_{i}(\tau)\int_{0}^{\tau}(\tau-s)^{\alpha_{i}-1}f_{i}(s,x(s),x^{\prime}(s))ds=0\\\
x(1)=b_{1},\quad x^{\prime}(1)=b_{2},\quad
0<\alpha_{i}<1,\;i=1,2,\dots,m.\end{gathered}$ (1.10)

They showed that any solution enjoys the asymptotic expansion
$x(\tau)=b+c\tau$ as $\tau\to\infty$, for some $c,b\in\mathbb{R}$.

In 2015, Medveď and Pospíšil [24] considered the equation

${}^{C}\mathfrak{D}_{a}^{\alpha}x(\tau)=f(\tau,x(\tau),{}^{C}\mathfrak{D}_{0}^{\beta}x(\tau)),\quad\tau>a.$
(1.11)

They proved that any solution $x(\tau)$ with $0<\beta<\alpha<1$, has the
asymptotic property $x(\tau)=c\tau^{\beta}+o(\tau^{\beta})$ as
$\tau\to\infty$, for some $c\in\mathbb{R}$. Also they proved that any solution
$x(\tau)$ of (1.11), $0<\beta<1<\alpha<2$, has the asymptotic property
$x(\tau)=c\tau+o(\tau)$ as $\tau\to\infty$, for some $c\in\mathbb{R}$.
Moreover, they proved that there exists a constant $c\in\mathbb{R}$ such that
any global solution $x(\tau)$ of the initial value problem

$\begin{gathered}{}^{C}\mathfrak{D}_{a}^{\alpha}x(\tau)=f\big{(}\tau,x(\tau),x^{\prime}(\tau),\dots,x^{(n-1)}(\tau),{}^{C}\mathfrak{D}_{0}^{\beta_{1}}x(\tau),\dots,{}^{C}\mathfrak{D}_{0}^{\beta_{m}}x(\tau)\big{)},\\\
\tau>a.x^{(n-1)}(a)=c_{i},\quad i=0,1,\dots,n-1,\end{gathered}$ (1.12)

has the asymptotic property $x(\tau)=c\tau^{k}+o(\tau^{k})$ as
$\tau\to\infty$, where $k\in\max\\{n-1,\beta_{m}\\}$,
$0<\beta_{1}<\dots.<\beta_{m}<\alpha<n$, and $n$, $m\in\mathbb{N}$.

In Sections 2 and 3, we prepare some material which will be needed later in
our proofs. Sections 4, 5 and 6 are devoted to the main results on the
asymptotic behavior results and boundedness of solutions for non-fractional
and fractional source terms, respectively.

##  2\. Preliminaries

In this section, we introduce some basic definitions, notation, properties and
lemmas to be used in our results. We refer the reader to
citeKilbas,Podl-1999,Samko for more details.

######  Definition 2.1 ([16]).

We introduce the space

$C_{\eta}[a,b]=\\{h:(a,b]\to\mathbb{R}:\quad h(\tau)(\tau-a)^{\eta}\in
C[a,b]\\},\quad 1>\eta\geq 0.$

######  Definition 2.2 ([16]).

The left-sided Riemann-Liouville fractional integral of order $\alpha>0$ is
defined by

$\mathfrak{I}_{a}^{\alpha}f(\tau):=\frac{1}{\Gamma(\alpha)}\int_{a}^{\tau}\frac{f(s)}{(\tau-s)^{1-\alpha}}ds,\quad\tau>a,$

provided that the right hand side exists.

######  Definition 2.3 ([16]).

The left-sided Riemann-Liouville fractional derivative of order $\alpha\geq
0$, $n-1\leq\alpha<n$, $n=-[-\alpha]$, is defined by

$\displaystyle\mathfrak{D}_{a}^{\alpha}f(\tau)$
$\displaystyle=D^{n}\mathfrak{I}_{a}^{n-\alpha}f(\tau)=(\frac{d}{d\tau})^{n}\mathfrak{I}_{a}^{n-\alpha}f(\tau)$
$\displaystyle=\frac{1}{\Gamma(n-\alpha)}\big{(}\frac{d}{d\tau}\big{)}^{n}\int_{a}^{\tau}\frac{f(s)}{(\tau-s)^{\alpha-n+1}}ds,\quad\tau>a.$

In particular, when $\alpha=n$ we have $\mathfrak{D}_{a}^{\alpha}f=D^{n}f$,
and when $\alpha=0$, $\mathfrak{D}_{a}^{0}f=f$.

######  Definition 2.4 ([16]).

The left-sided Caputo fractional derivative of order $\alpha\geq 0$,
$n-1\leq\alpha<n$, $n=-[-\alpha]$, is defined by

${}^{C}\mathfrak{D}_{a}^{\alpha}f(\tau)=\mathfrak{I}_{a}^{n-\alpha}f^{(n)}(\tau),\;\tau>a.$

The fractional integral and fractional derivative of power functions have the
same effect as the integer-order integral and derivative. Namely,

######  Lemma 2.5 ([16]).

If $\beta>0$ and $\alpha\geq 0$, then

$\displaystyle\mathfrak{I}_{a}^{\alpha}(\tau-a)^{\beta-1}=\frac{\Gamma(\beta)}{\Gamma(\beta+\alpha)}(\tau-a)^{\alpha+\beta+1},\quad\alpha>0,\;\tau>a,$
${}^{C}\mathfrak{D}_{a}^{\alpha}(\tau-a)^{\beta-1}=\frac{\Gamma(\beta)}{\Gamma(\beta-\alpha)}(\tau-a)^{\beta-\alpha-1},\quad\alpha\geq
0,\;\tau>a.$

If $\beta=1$, then $(^{C}\mathfrak{D}_{a}^{\alpha}1)(\tau)=0$, $\tau>a$.

The Riemann-Liouville fractional integral (Definition 2.2) satisfies the
following semigroup property.

######  Lemma 2.6.

[16]] Let $0\leq\eta<1$, $\alpha>0$ and $\beta>0$. If $h\in C_{\eta}[a,b]$,
then

$\mathfrak{I}_{a}^{\beta}\mathfrak{I}_{a}^{\alpha}h(\tau)=\mathfrak{I}_{a}^{\beta+\alpha}h(\tau),\quad\tau>a.$

The following result provides another composition of the fractional
integration operator $\mathfrak{I}_{a}^{\alpha}$ with the fractional
differentiation operator $\mathfrak{D}_{a}^{\alpha}$.

######  Lemma 2.7 ([16]).

Let $\alpha>0$, $0\leq\eta<1$, $n=-[-\alpha]$. If $h\in C_{\eta}[a,b]$ and
$\mathfrak{I}_{a}^{n-\alpha}h\in C_{\eta}^{n}[a,b]$, then

$\mathfrak{I}_{a}^{\alpha}\mathfrak{D}_{a}^{\alpha}h(\tau)=h(\tau)-\sum_{i=1}^{n}\frac{(\mathfrak{D}^{n-i}\mathfrak{I}_{a}^{n-\alpha}h)(a)}{\Gamma(\alpha-i+1)}(\tau-a)^{\alpha-i},\quad\tau>a.$

######  Lemma 2.8 ([16]).

Let $\alpha>0$, $n=-[-\alpha]$. If $h\in C^{n}[a,b]$ or $h\in AC^{n}[a,b]$,
then

$\mathfrak{I}_{a}^{\alpha}{}^{C}\mathfrak{D}_{a}^{\alpha}h(\tau)=h(\tau)-\sum_{k=0}^{n-1}\frac{h^{(k)}(a)}{k!}(\tau-a)^{k},\quad\tau>a.$

######  Lemma 2.9.

Let $0<\beta\leq\alpha<1$. If $h\in AC[a,b]$, then

${}^{C}\mathfrak{D}_{0}^{\beta}h=\mathfrak{I}_{0}^{\alpha-\beta}{}^{C}\mathfrak{D}_{0}^{\alpha}h.$

###### Proof.

From Definition 2.4 and Lemma 2.6, we have

${}^{C}\mathfrak{D}_{0}^{\beta}h=\mathfrak{I}_{0}^{1-\beta}h^{\prime}=\mathfrak{I}_{0}^{\alpha-\beta}\mathfrak{I}_{0}^{1-\alpha}h^{\prime}=\mathfrak{I}_{0}^{\alpha-\beta}{}^{C}\mathfrak{D}_{0}^{\alpha}h.$

∎

######  Lemma 2.10 ([12]).

Let $f\in L_{1}(0,\infty)$. Then

$\lim_{\tau\to\infty}\frac{1}{\tau^{\alpha}}\mathfrak{I}_{0}^{\alpha+1}f(\tau)=\frac{1}{\Gamma(\alpha+1)}\int_{0}^{\infty}f(s)ds=\frac{1}{\Gamma(\alpha+1)}\mathfrak{I}_{0}^{1}f(\infty),\
\alpha>0.$

######  Lemma 2.11.

Let $0<\alpha<1$ and $0\leq\eta<1$. Assume that $x\in AC[0,\infty)$ and
$\mathfrak{I}_{0}^{1-\alpha}x^{\prime}\in C_{\eta}^{1}[0,\infty)$. Then

$\lim_{\tau\to\infty}\frac{x(\tau)}{\tau^{\alpha}}=\lim_{\tau\to\infty}\frac{{}^{C}\mathfrak{D}_{0}^{\alpha}x(\tau)}{\Gamma(1+\alpha)}.$
(2.1)

###### Proof.

Since $x\in AC[0,\infty)\subset C_{\eta}[0,\infty)$ and
$\mathfrak{I}_{0}^{1-\alpha}x^{\prime}\in C_{\eta}^{1}[0,\infty)$, we can use
Lemma 2.7 to obtain

$\mathfrak{I}_{0}^{\alpha}\mathfrak{D}_{0}^{\alpha}x^{\prime}(\tau)=x^{\prime}(\tau)-\frac{\mathfrak{I}_{0}^{1-\alpha}x^{\prime}(0)}{\Gamma(\alpha)}\tau^{\alpha-1},\quad\tau>0.$
(2.2)

Applying $\mathfrak{I}_{0}^{1}$ to both sides of (2.2), using Lemma 2.5 (with
$\beta=\alpha$) and Lemma 2.6, we obtain

$x(\tau)=x(0)+\frac{\mathfrak{I}_{0}^{1-\alpha}x^{\prime}(0)}{\Gamma(\alpha+1)}\tau^{\alpha}+\mathfrak{I}_{0}^{1+\alpha}\mathfrak{D}_{0}^{\alpha}x^{\prime}(\tau),\quad\tau>0.$
(2.3)

Dividing both sides of (2.3) by $\tau^{\alpha}$, we obtain

$\frac{x(\tau)}{\tau^{\alpha}}=\frac{x(0)}{\tau^{\alpha}}+\frac{\mathfrak{I}_{0}^{1-\alpha}x^{\prime}(0)}{\Gamma(\alpha+1)}+\frac{1}{\tau^{\alpha}}\mathfrak{I}_{0}^{1+\alpha}\mathfrak{D}_{0}^{\alpha}x^{\prime}(\tau),\quad\tau>0.$

Next, we take the limit as $\tau\to\infty$, we arrive at

$\lim_{\tau\to\infty}\frac{x(\tau)}{\tau^{\alpha}}=\frac{\mathfrak{I}_{0}^{1-\alpha}x^{\prime}(0)}{\Gamma(1+\alpha)}+\frac{1}{\Gamma(1+\alpha)}\lim_{\tau\to\infty}\mathfrak{I}_{0}^{1}\mathfrak{D}_{0}^{\alpha}x^{\prime}(\tau),$
(2.4)

where we used Lemma 2.10. Moreover, we conclude that

$\mathfrak{I}_{0}^{1}\mathfrak{D}_{0}^{\alpha}x^{\prime}(\tau)=\mathfrak{I}_{0}^{1}D\mathfrak{I}_{0}^{1-\alpha}x^{\prime}(\tau)=\mathfrak{I}_{0}^{1-\alpha}x^{\prime}(\tau)-\mathfrak{I}_{0}^{1-\alpha}x^{\prime}(0)={}^{C}\mathfrak{D}_{0}^{\alpha}x(\tau)-{}^{C}\mathfrak{D}_{0}^{\alpha}x(0),\quad\tau>0,$
(2.5)

and (2.1) follows directly from (2.4) and (2.5). ∎

##  3\. Some useful inequalities

First, we define the following special classes of functions

$\displaystyle H_{k}=\\{h\in L_{1}(0,\infty):h\text{ is positive and
}s^{k}h\in L_{1}(1,\infty),\;k>-1\\},$ (3.1) $\displaystyle\begin{aligned}
M=\big{\\{}&F:(0,\infty)\times\mathbb{R}_{+}\to\mathbb{R}_{+}\text{ where
}0\leq F(\tau,s)-F(\tau,r)\leq H(\tau)(s-r),\\\ &\text{for some continuous
function $H$ on }\mathbb{R}_{+},\;s\geq r\geq 0\text{ and
}\tau>0\big{\\}}.\end{aligned}$ (3.2) $\displaystyle\begin{aligned}
\Phi=\big{\\{}&\varphi\in C(0,\infty):\varphi\text{ is nondecreasing and
positive on }(0,\infty),\\\
&\frac{1}{v}\varphi(w)\leq\varphi(\frac{w}{v}),\;w>0,\;v\geq
1\big{\\}}.\end{aligned}$ (3.3)

The proofs of the following lemmas are based on an application of the Bihari
inequality which is a generalization of the Gronwall inequality.

######  Lemma 3.1 ([12]).

Let $g(\tau)$ and $z(\tau)$ be nonnegative continuous functions defined for
$\tau\geq 0$, $\varphi\in\Phi$ and $c_{i}\in\mathbb{R}$, $i=1,2,3$. Then

$z(\tau)\leq
c_{1}+c_{2}\tau^{\gamma}+c_{3}\tau^{\gamma}\int_{0}^{\tau}g(s)\varphi(z(s))ds,\text{\
\ }\tau,\quad\gamma\geq 0,$ (3.4)

implies

$z(\tau)\leq\begin{cases}E^{-1}(E(|c_{1}|+|c_{2}|)+|c_{3}|\int_{0}^{\tau}g(s)ds),&0\leq\tau<1\\\\[4.0pt]
\tau^{\gamma}E^{-1}(E(A)+|c_{3}|\int_{1}^{\tau}s^{\gamma}g(s)ds),&\tau\geq
1,\end{cases}$ (3.5)

where

$\displaystyle A=|c_{1}|+|c_{2}|+|c_{3}|\varphi(E^{-1}(C))\int_{0}^{1}g(s)ds,$
$\displaystyle C=E(|c_{1}|+|c_{2}|)+|c_{3}|\int_{0}^{1}g(s)ds<\infty,$

and $E^{-1}$ is the inverse function of

$E(\mathfrak{\xi})=\int_{\mathfrak{\xi}_{0}}^{\mathfrak{\xi}}\frac{ds}{\varphi(s)}.$

######  Lemma 3.2 ([12]).

Let $z(\tau)$ satisfy

$z(\tau)\leq
c_{1}\tau^{\gamma}+c_{2}\tau^{\gamma}\int_{0}^{\tau}[F_{1}(s,z(s)+c_{3})+F_{2}(s,z(s)+c_{4})+h(s)]ds,\quad\tau\geq
0,$ (3.6)

where $h:C[\mathbb{R}_{+},\mathbb{R}_{+}]$, $F_{j}\in M$, $j=1,2$ and
$\gamma$, $c_{i}>0$, $i=1,2,3,4$. Then

$z(\tau)\leq\tau^{\gamma}f(\tau),\quad\tau>0,$ (3.7)

where

$\displaystyle f(\tau)$
$\displaystyle=\Big{(}c_{1}+c_{2}\int_{0}^{\tau}[F_{1}(s,c_{3})+F_{2}(s,c_{4})+h(s)]ds\Big{)}$
(3.8)
$\displaystyle\quad\times\exp\Big{(}c_{2}\int_{0}^{\tau}s^{\gamma}[N_{1}(s)+N_{2}(s)]ds\Big{)},\quad\tau>0,$

with $N_{1}$ and $N_{2}$ are as in the definition of $M$ corresponding to
$F_{1}$ and $F_{2}$, respectively.

######  Remark 3.3.

If $p$, $q>1$ and $\frac{1}{p}+\frac{1}{q}=1$, then for $\alpha>0$,
$p(\alpha-1)+1>0\Longleftrightarrow q\alpha>1$.

######  Lemma 3.4 ([13]).

If $\upsilon,\lambda+1>1/r$, for some $r>1$, and $g$ is a nonnegative
continuous function defined on $[0,\infty)$, then

$\int_{0}^{\tau}(\tau-s)^{\upsilon-1}s^{\lambda}g(s)ds\leq
C\tau^{\upsilon+\lambda-1/r}\Big{(}\int_{0}^{\tau}g^{r}(s)ds\Big{)}^{1/r},\quad\tau>0,$
(3.9)

where

$C=K_{p(\upsilon-1)+1,p\lambda}=\frac{\Gamma(p\lambda+1)\Gamma(p(\upsilon-1)+1)}{\Gamma(p\lambda+p(\upsilon-1)+2)},\quad\frac{1}{p}+\frac{1}{r}=1.$

######  Lemma 3.5 ([13]).

Let $h$ and $z$ be nonnegative continuous functions defined on $[0,\infty)$.
Let $\varphi_{i}(z)>0$ on $(0,\infty)$, $i=1,2$, and $\varphi_{i}(z)$ are
continuous nondecreasing functions defined on $[0,\infty)$. If

$z(\tau)\leq
K_{1}+K_{2}(\int_{0}^{\tau}h^{q}(s)\varphi_{1}^{q}(z(s))\varphi_{2}^{q}(z(s))ds)^{1/q},\quad
q>1,\;\tau>0,$ (3.10)

where $K_{i}\in\mathbb{R}_{+}$, $i=1,2$, then

$z(\tau)\leq\Big{[}E^{-1}\Big{(}E(2^{q-1}K_{1})+2^{q-1}K_{2}\int_{0}^{\tau}h^{q}(s)ds\Big{)}\Big{]}^{1/q},\quad\tau>0,$

where $E^{-1}$ is the inverse of

$E(\mathfrak{\xi})=\int_{\mathfrak{\xi}_{0}}^{\mathfrak{\xi}}\frac{ds}{\varphi_{1}^{q}(s^{1/q})\varphi_{2}^{q}(s^{1/q})},\quad\mathfrak{\xi}>\mathfrak{\xi}_{0}>0.$

##  4\. Source without fractional derivatives

We consider the asymptotic behavior of solutions of the equation

$(^{C}\mathfrak{D}_{0}^{\alpha}x)^{\prime}(\tau)=f(\tau,x(\tau)),\quad
0<\alpha<1\text{, }\tau\geq 0,$ (4.1)

subject to

$x(0)=b_{1},\quad{}^{C}\mathfrak{D}_{0}^{\alpha}x(\tau)|_{\tau=0}=b_{2},$
(4.2)

in the space

$C_{1-\alpha}^{\alpha,1}[0,\infty)=\\{x\in
AC[0,\infty),(^{C}\mathfrak{D}_{0}^{\alpha}x)^{\prime}\in
C_{1-\alpha}[0,\infty)\\}.$ (4.3)

We assume the following conditions:

  * (C1)

$f(\tau,x)\in C[[0,\infty)\times\mathbb{R},\mathbb{R}]$ is such that
$f(\cdot,x(\cdot))\in C_{1-\alpha}[0,\infty)$ for any $x\in AC[0,\infty)$.

  * (C2)

There are continuous functions $P,\varphi:[0,\infty)\to[0,\infty)$ such that

$|f(\tau,x(\tau))|\leq\varphi(|x(\tau)|)P(\tau),\quad\tau\geq 0,$ (4.4)

where

$\int_{1}^{\infty}s^{\alpha}P(s)ds<\infty,$ (4.5)

and $\varphi\in\Phi$.

######  Theorem 4.1.

Suppose $f$ satisfies (C1), (C2) and $x\in AC[0,\infty)$ is a solution of
(4.1)-(4.2). Then

$\lim_{\tau\to\infty}\frac{x(\tau)}{\tau^{\alpha}}=a\in\mathbb{R},\quad\text{as
}\tau\to\infty.$

###### Proof.

Integrating both sides of (4.1), we find

${}^{C}\mathfrak{D}_{0}^{\alpha}x(\tau)=b_{2}+\int_{0}^{\tau}f(s,x(s))ds=b_{2}+\mathfrak{I}_{0}^{1}f(\tau,x(\tau)).$
(4.6)

Applying $\mathfrak{I}_{0}^{\alpha}$ to both sides of (4.6), and using Lemmas
2.8 and 2.5, we arrive at

$x(\tau)=b_{1}+\frac{b_{2}}{\Gamma(\alpha+1)}\tau^{\alpha}+\frac{1}{\Gamma(\alpha+1)}\int_{0}^{\tau}(\tau-s)^{\alpha}f(s,x(s))ds,\quad\tau\geq
0.$ (4.7)

By using (4.4) we obtain

$|x(\tau)|\leq|b_{1}|+\frac{|b_{2}|}{\Gamma(\alpha+1)}\tau^{\alpha}+\frac{1}{\Gamma(\alpha+1)}\tau^{\alpha}\int_{0}^{\tau}P(s)\varphi(|x(s)|)ds,\quad\tau\geq
0.$ (4.8)

Applying Lemma 3.1 to (4.8) we obtain

$|x(\tau)|\leq\begin{cases}E^{-1}\big{(}E(|b_{1}|+\frac{|b_{2}|}{\Gamma(\alpha+1)})+\frac{1}{\Gamma(\alpha+1)}\int_{0}^{\tau}P(s)ds\big{)},&0\leq\tau<1\\\
\tau^{\alpha}E^{-1}\big{(}E(A)+\frac{1}{\Gamma(\alpha+1)}\int_{1}^{\tau}s^{\alpha}P(s)ds\big{)},&\tau\geq
1,\end{cases}$

where

$\displaystyle
A=|b_{1}|+\frac{|b_{2}|}{\Gamma(\alpha+1)}+\frac{1}{\Gamma(\alpha+1)}\varphi(E^{-1}(K))\int_{0}^{1}P(s)ds,$
$\displaystyle
K=E(|b_{1}|+\frac{|b_{2}|}{\Gamma(\alpha+1)})+\frac{1}{\Gamma(\alpha+1)}\int_{0}^{1}P(s)ds<\infty.$

From (4.5) and the continuity of $P$ on $\mathbb{R}_{+}$, we see that

$|x(\tau)|\leq\begin{cases}C_{1},&0\leq\tau<1,\\\ \tau^{\alpha}C_{2},&\tau\geq
1,\end{cases}$ (4.9)

with

$\displaystyle
C_{1}=E^{-1}\Big{(}E\big{(}|b_{1}|+\frac{|b_{2}|}{\Gamma(\alpha+1)}\big{)}+\frac{1}{\Gamma(\alpha+1)}\int_{0}^{1}P(s)ds\Big{)}<\infty,$
$\displaystyle
C_{2}=E^{-1}\Big{(}E(A)+\frac{1}{\Gamma(\alpha+1)}\int_{1}^{\infty}s^{\alpha}P(s)ds\Big{)}<\infty.$

Next, it is clear that

$\displaystyle\int_{0}^{\tau}|f(s,x(s))|ds$
$\displaystyle\leq\int_{0}^{\tau}P(s)\varphi(|x(s)|)ds$ (4.10)
$\displaystyle\leq\int_{0}^{1}P(s)\varphi(|x(s)|)ds+\int_{1}^{\tau}P(s)\varphi(|x(s)|)ds$
$\displaystyle\leq\int_{0}^{1}P(s)\varphi(|x(s)|)ds+\int_{1}^{\tau}s^{\alpha}P(s)\varphi\big{(}\frac{|x(s)|}{s^{\alpha}}\big{)}ds,\quad\tau>0.$

By (4.9) and (4.10), we have

$\lim_{\tau\to\infty}\int_{0}^{\tau}f(s,x(s))ds<\infty.$ (4.11)

On the other hand, integrating (4.1) yields

${}^{C}\mathfrak{D}_{0}^{\alpha}x(\tau)=b_{2}+\int_{0}^{\tau}f(s,x(s))ds,\quad\tau>0.$
(4.12)

From (4.11) and (4.12), we conclude

$\lim_{\tau\to\infty}{}^{C}\mathfrak{D}_{0}^{\alpha}x(\tau)=c,\quad
c\in\mathbb{R}.$

Further, by Lemma 2.11, we can write

$\lim_{\tau\to\infty}\frac{x(\tau)}{\tau^{\alpha}}=\lim_{\tau\to\infty}\frac{{}^{C}\mathfrak{D}_{0}^{\alpha}x(\tau)}{\Gamma(\alpha+1)}=a,$

for some real number $a$. ∎

######  Example 4.2.

All solutions of

$(^{C}\mathfrak{D}_{0}^{\alpha}x)^{\prime}(\tau)=e^{-\tau}x^{r}(\tau),\quad
0<\alpha,r\leq 1,\;\tau>0.$ (4.13)

satisfy $\lim_{\tau\to\infty}\frac{x(\tau)}{\tau^{\alpha}}=a$, as
$\tau\to\infty$, for some real number $a$.

To prove this claim, let $\varphi(\tau)=\tau^{r}$ and $P(\tau)=e^{-\tau}$.
Then

$\int_{1}^{\infty}s^{\alpha}P(s)ds\leq\int_{0}^{\infty}s^{\alpha}e^{-s}ds=\Gamma(\alpha+1)<\infty.$

Obviously $\varphi$ is a nondecreasing and positive function with

$u\varphi(v)=uv^{r}\leq(vu)^{r}=\varphi(vu),\quad u\geq 1,\;v>0,$

and

$\int_{0}^{\infty}\frac{ds}{\varphi(s)}=\int_{0}^{\infty}\frac{ds}{s^{r}}=\infty.$

Then $\varphi\in\Phi$. All the conditions of Theorem 4.1 are satisfied,
therefore every solution $x$ of (4.13) has satisfy
$\lim_{\tau\to\infty}\frac{x(\tau)}{\tau^{\alpha}}=a$, , $a\in\mathbb{R}$, as
$\tau\to\infty$.

##  5\. Equations with fractional source terms

We study problem (1.1) in the space $C_{1-\alpha}^{\alpha,1}[0,\infty)$
defined in (4.3) with the following assumptions:

  * (C3)

$f(\tau,v,w):[0,\infty)\times\mathbb{R}^{2}\to\mathbb{R}$ is so that
$f(\cdot,v(\cdot),w(\cdot))\in C_{1-\alpha}[0,\infty)$ for every $v,w\in
AC[0,\infty)$.

  * (C4)

$|f(\tau,u(\tau),v(\tau))|\leq
F_{1}(\tau,|u(\tau)|)+F_{2}(\tau,\tau^{\beta}|v(\tau)|),\quad\tau\geq 0,$
(5.1)

where $F_{i}\in M$, $i=1,2$.

######  Lemma 5.1.

Suppose that $f$ satisfies (C3), (4) and $x\in AC[0,\infty)$ is a solution of
(1.1). Then

$\max\big{\\{}|x(\tau)|,\tau^{\beta}|^{C}\mathfrak{D}_{0}^{\beta}x(\tau)|\big{\\}}\leq|b_{1}|+z(\tau),\quad\tau>0,$
(5.2)

where

$\begin{gathered}z(\tau)=C_{2}\tau^{\alpha}+C_{3}\tau^{\alpha}\int_{0}^{\tau}[F_{1}(s,|x(s)|)+F_{2}(s,\tau^{\beta}|^{C}\mathfrak{D}_{0}^{\beta}x(s)|)]ds,\quad\tau>0,\\\
C_{3}=\max\big{\\{}\frac{1}{\Gamma(\alpha+1)},\frac{1}{\Gamma(\alpha-\beta+1)}\big{\\}},\quad
C_{2}=|b_{2}|C_{3}.\end{gathered}$ (5.3)

###### Proof.

Applying $\mathfrak{I}_{0}^{1}$ to (1.1), we obtain

$\displaystyle{}^{C}\mathfrak{D}_{0}^{\alpha}x(\tau)$
$\displaystyle=b_{2}+\mathfrak{I}_{0}^{1}f\big{(}\tau,x(\tau),{}^{C}\mathfrak{D}_{0}^{\beta}x(\tau)\big{)}$
(5.4)
$\displaystyle=b_{2}+\int_{0}^{\tau}f\big{(}s,x(s),{}^{C}\mathfrak{D}_{0}^{\beta}x(s)\big{)}ds,\quad\tau>0.$

Next, we apply $\mathfrak{I}_{0}^{\alpha}$ to both sides of (5.4), using
Lemmas 2.6, 2.8 and 2.5, we find

$\displaystyle x(\tau)$
$\displaystyle=b_{1}+\frac{b_{2}}{\Gamma(\alpha+1)}\tau^{\alpha}+\mathfrak{I}_{0}^{1+\alpha}f\big{(}\tau,x(\tau),{}^{C}\mathfrak{D}_{0}^{\beta}x(\tau)\big{)}$
(5.5)
$\displaystyle=b_{1}+\frac{b_{2}}{\Gamma(\alpha+1)}\tau^{\alpha}+\frac{1}{\Gamma(\alpha+1)}\int_{0}^{\tau}(\tau-s)^{\alpha}f(s,x(s),{}^{C}\mathfrak{D}_{0}^{\beta}x(s))ds,$

for $\tau>0$. Thus, from (5.5) and (5.1) we have

$\displaystyle|x(\tau)|$
$\displaystyle\leq|b_{1}|+\frac{|b_{2}|}{\Gamma(\alpha+1)}\tau^{\alpha}+\frac{\tau^{\alpha}}{\Gamma(\alpha+1)}\int_{0}^{\tau}\big{|}f\big{(}s,x(s),{}^{C}\mathfrak{D}_{0}^{\beta}x(s)\big{)}\big{|}ds$
(5.6)
$\displaystyle\leq|b_{1}|+C_{2}\tau^{\alpha}+C_{3}\tau^{\alpha}\int_{0}^{\tau}\Big{(}F_{1}(s,|x(s)|)+F_{2}\big{(}s,s^{\beta}|^{C}\mathfrak{D}_{0}^{\beta}x(s)|\big{)}\Big{)}ds,$

for $\tau>0$. By Lemma 2.9, we see that

${}^{C}\mathfrak{D}_{0}^{\beta}x(\tau)=\mathfrak{I}_{0}^{\alpha-\beta}(^{C}\mathfrak{D}_{0}^{\alpha}x(\tau)).$
(5.7)

Let us insert the expression (5.4) into (5.7), using Lemmas 2.6 and 2.5, we
have

${}^{C}\mathfrak{D}_{0}^{\beta}x(\tau)$
$\displaystyle=\mathfrak{I}_{0}^{\alpha-\beta}\Big{(}b_{2}+\mathfrak{I}_{0}^{1}f\big{(}s,x(s),{}^{C}\mathfrak{D}_{0}^{\beta}x(s)\big{)}\Big{)}(\tau)$
$\displaystyle=\frac{b_{2}}{\Gamma(\alpha-\beta+1)}\tau^{\alpha-\beta}+\mathfrak{I}_{0}^{\alpha-\beta+1}f\big{(}\tau,x(\tau),{}^{C}\mathfrak{D}_{0}^{\beta}x(\tau)\big{)}$
$\displaystyle=\frac{b_{2}}{\Gamma(\alpha-\beta+1)}\tau^{\alpha-\beta}$
$\displaystyle\quad+\frac{1}{\Gamma(\alpha-\beta+1)}\int_{0}^{\tau}(\tau-s)^{\alpha-\beta}f(s,x(s),{}^{C}\mathfrak{D}_{0^{+}}^{\beta}x(s))ds,\quad\tau>0.$

Then from this and (5.1) we obtain the bound

$\displaystyle\tau^{\beta}|^{C}\mathfrak{D}_{0}^{\beta}x(\tau)|$ (5.8)
$\displaystyle\leq
C_{3}|b_{2}|\tau^{\alpha}+C_{3}\tau^{\alpha}\int_{0}^{\tau}|f(s,x(s),{}^{C}\mathfrak{D}_{0^{+}}^{\beta}x(s))|ds$
$\displaystyle\leq
C_{2}\tau^{\alpha}+C_{3}\tau^{\alpha}\int_{0}^{\tau}\big{(}F_{1}(s,|x(s)|)+F_{2}(s,s^{\beta}|^{C}\mathfrak{D}_{0}^{\beta}x(s)|)\big{)}ds,\quad\tau>0.$

Relation (5.2) follows directly from (5.3), (5.6) and (5.8). ∎

######  Theorem 5.2.

Suppose that $f$ satisfies (C3)-(C4) and

$\int_{0}^{\infty}s^{\alpha}N_{i}(s)ds<\infty,\quad\int_{0}^{\infty}F_{i}(s,|b_{1}|)ds<\infty\text{,
}i=1,2.$ (5.9)

Then, every solution $x(\tau)$ of problem (1.1) has the following property

$\lim_{\tau\to\infty}\frac{x(\tau)}{\tau^{\alpha}}=a,\quad a\in\mathbb{R}.$

###### Proof.

By using Lemma 5.1 we have

$\displaystyle F_{1}(\tau,|x(\tau)|)\leq
F_{1}(\tau,|b_{1}|+z(\tau)),\quad\tau>0,$ (5.10) $\displaystyle
F_{2}\big{(}\tau,\tau^{\beta}|^{C}\mathfrak{D}_{0}^{\beta}x(\tau)|\big{)}\leq
F_{2}(\tau,z(\tau)+|b_{1}|),\quad\tau>0.$ (5.11)

Taking into account (5.3), (5.10) and (5.11) we arrive at

$z(\tau)\leq
C_{2}\tau^{\alpha}+C_{3}\tau^{\alpha}\int_{0}^{\tau}[F_{1}(s,|b_{1}|+z(s))+F_{2}(s,z(\tau)+|b_{1}|)]ds,\quad\tau>0.$

Then, by Lemma 3.2, we find that

$z(\tau)\leq C\tau^{\alpha},\quad\tau>0$ (5.12)

where

$\displaystyle C$
$\displaystyle=\Big{(}C_{2}+C_{3}\int_{0}^{\infty}[F_{1}(s,|b_{1}|)+F_{2}(s,|b_{1}|)]ds\Big{)}$
$\displaystyle\quad\times\exp(C_{3}\int_{0}^{\infty}s^{\gamma}[N_{1}(s)+N_{2}(s)]ds)<\infty.$

It follows from Lemma 5.1 and (5.12) that

$|x(\tau)|\leq|b_{1}|+C\tau^{\alpha},\quad\tau^{\beta}|^{C}\mathfrak{D}_{0}^{\beta}x(\tau)|\leq|b_{1}|+C\tau^{\alpha},\quad\tau>0.$
(5.13)

Again by hypothesis (5.1) we have

$\displaystyle|\int_{0}^{\tau}f(s,x(s),{}^{C}\mathfrak{D}_{0}^{\beta}x(s))ds|$
$\displaystyle\leq\int_{0}^{\tau}|f(s,x(s),{}^{C}\mathfrak{D}_{0}^{\beta}x(s))|ds$
$\displaystyle\leq\int_{0}^{\tau}[F_{1}(s,|x(s)|)+F_{2}(s,s^{\beta}|^{C}\mathfrak{D}_{0}^{\beta}x(s)|)]ds,$

for $\tau>0$. From this inequality and (5.13), we obtain

$\displaystyle|\int_{0}^{\tau}f(s,x(s),{}^{C}\mathfrak{D}_{0}^{\beta}x(s))ds|$
$\displaystyle\leq\int_{0}^{\tau}[F_{1}(s,|b_{1}|+Cs^{\alpha})+F_{2}(s,|b_{1}|+Cs^{\alpha})]ds$
$\displaystyle=\int_{0}^{\tau}\Big{\\{}F_{1}(s,|b_{1}|+Cs^{\alpha})-F_{1}(s,|b_{1}|)+F_{1}(s,|b_{1}|)$
$\displaystyle\quad+F_{2}(s,|b_{1}|+Cs^{\alpha})-F_{2}(s,|b_{1}|)+F_{2}(s,|b_{1}|)\Big{\\}}ds,\quad\tau>0.$

As the functions $F_{i}$, $i=1,2$ are in $M$, we obtain

$\displaystyle\big{|}\int_{0}^{\tau}f(s,x(s),{}^{C}\mathfrak{D}_{0}^{\beta}x(s))ds\big{|}$
(5.14) $\displaystyle\leq
C\int_{0}^{\tau}s^{\alpha}[N_{1}(s)+N_{s}(s)]ds+\int_{0}^{\tau}[F_{1}(s,|b_{1}|)+F_{2}(s,|b_{1}|)]ds<\infty,$

where we have used (5.9). Then

$\lim_{\tau\to\infty}\int_{0}^{\tau}f(s,x(s),{}^{C}\mathfrak{D}_{0}^{\beta}x(s))ds<\infty.$

By (5.4) we conclude that there is $b\in\mathbb{R}$ such that
$\lim_{\tau\to\infty}{}^{C}\mathfrak{D}_{0}^{\alpha}x(\tau)=b$. Further, by
Lemma 2.11, we deduce that

$\lim_{\tau\to\infty}\frac{x(\tau)}{\tau^{\alpha}}=\lim_{\tau\to\infty}\frac{{}^{C}\mathfrak{D}_{0}^{\alpha}x(\tau)}{\Gamma(\alpha+1)}=a,$

and the proof is now complete. ∎

##  6\. Boundedness

We consider the fractional differential problem (1.2) in the space

$C^{\alpha}[0,\infty)=\left\\{x\in
AC[0,\infty):{}^{C}\mathfrak{D}_{0}^{\alpha}x\in C[0,\infty)\right\\}.$ (6.1)

We assume the following conditions:

  * (C5)

$f:[0,\infty)\times\mathbb{R}^{2}\to\mathbb{R}$ is so that
$f(\cdot,v(\cdot),w(\cdot))\in C[0,\infty)$ for every $v$, $w$ in
$C[0,\infty)$.

  * (C6)

$|f(\tau,u,v)|\leq\tau^{\gamma}h(\tau)\varphi_{1}(|u(\tau)|)\varphi_{2}(|v(\tau)|),\quad\tau>0,$
(6.2)

where $h$, $\varphi_{1}$, $\varphi_{2}:\mathbb{R}_{+}\to\mathbb{R}_{+}$ are
continuous functions with $\varphi_{i}$, $i=1,2$, are nondecreasing functions
and $h\in L_{q}(0,\infty)$ for some $q>\frac{1}{\alpha-\beta}$,
$\gamma=\frac{1}{q}-\alpha$.

######  Lemma 6.1.

: Suppose that $f$ satisfies (C5), (C6) and $x\in AC[0,\infty)$ is a solution
of (1.2). Then

$\max\big{\\{}|x(\tau)|,|^{C}\mathfrak{D}_{0}^{\beta}x(\tau)|\big{\\}}\leq
z(\tau),\quad\tau\geq\tau_{0}>0,$ (6.3)

where

$z(\tau)=|b|+K_{1}\Big{(}\int_{0}^{\tau}h^{q}(s)\varphi_{1}^{q}(|x(s)|)\varphi_{2}^{q}(|^{C}\mathfrak{D}_{0}^{\beta}x(s)|)ds\Big{)}^{1/q},\quad\tau>0,$
(6.4)

and

$\displaystyle
K_{1}=\max\big{\\{}\frac{K_{1+p(\alpha-1),p\gamma}^{1/p}}{\Gamma(\alpha)},\frac{K_{1+p(\alpha-\beta-1),p\gamma}^{1/p}}{\Gamma(\alpha-\beta)\tau_{0}^{\beta}}\big{\\}},$
$\displaystyle
K_{\alpha,\beta}=\frac{\Gamma(\beta+1)\Gamma(\alpha)}{\Gamma(\alpha+\beta+1)},\quad\frac{1}{p}+\frac{1}{q}=1.$

###### Proof.

Applying $\mathfrak{I}_{0}^{\alpha}$ to (1.2) and taking into account Lemma
2.8, we have

$x(\tau)=b+\frac{1}{\Gamma(\alpha)}\int_{0}^{\tau}(\tau-s)^{\alpha-1}f(s,x(s),{}^{C}\mathfrak{D}_{0}^{\beta}x(s))ds,\quad\tau>0.$
(6.5)

Using the inequality (6.2), we obtain

$|x(\tau)|\leq|b|+\frac{1}{\Gamma(\alpha)}\int_{0}^{\tau}(\tau-s)^{\alpha-1}s^{\gamma}h(s)\varphi_{1}(|x(s)|)\varphi_{2}(|^{C}\mathfrak{D}_{0}^{\beta}x(s)|)ds,$
(6.6)

for $\tau>0$. It follows from the assumptions $\beta<\alpha$,
$q>\frac{1}{\alpha-\beta}$ and $\gamma=\frac{1}{q}-\alpha$ that
$p(\alpha-1)+1\geq p(\alpha-\beta-1)+1>0$ and
$p\gamma+1=p(\frac{1}{q}-\alpha)+1=p(1-\alpha)>0$. Then, we apply Lemma 3.4,
to obtain

$\displaystyle|x(\tau)|$
$\displaystyle\leq|b|+\frac{1}{\Gamma(\alpha)}K_{p(\alpha-1)+1,p\gamma}^{1/p}\tau^{\alpha+\gamma-1/q}\Big{(}\int_{0}^{\tau}h^{q}(s)\varphi_{1}^{q}(|x(s)|)\varphi_{2}^{q}(|^{C}\mathfrak{D}_{0}^{\beta}x(s)|)ds\Big{)}^{1/q}$
(6.7)
$\displaystyle\leq|b|+K_{1}\Big{(}\int_{0}^{\tau}h^{q}(s)\varphi_{1}^{q}(|x(s)|)\varphi_{2}^{q}(|^{C}\mathfrak{D}_{0}^{\beta}x(s)|)ds\Big{)}^{1/q},\quad\tau>0.$

Also, by Lemma 2.9, we conclude that

$\displaystyle{}^{C}\mathfrak{D}_{0}^{\beta}x(\tau)$
$\displaystyle=\mathfrak{I}_{0}^{\alpha-\beta}~{}^{C}\mathfrak{D}^{\alpha}x(\tau)=\frac{1}{\Gamma(\alpha-\beta)}\int_{0}^{\tau}(\tau-s)^{\alpha-\beta-1}~{}^{C}\mathfrak{D}_{0}^{\alpha}x(s)ds$
(6.8)
$\displaystyle=\frac{1}{\Gamma(\alpha-\beta)}\int_{0}^{\tau}(\tau-s)^{\alpha-\beta-1}f(s,x(s),{}^{C}\mathfrak{D}_{0}^{\beta}x(s))ds,\quad\tau>0.$

In view of (6.2), we have

$|^{C}\mathfrak{D}_{0}^{\beta}x(\tau)|\leq\frac{1}{\Gamma(\alpha-\beta)}\int_{0}^{\tau}(\tau-s)^{\alpha-\beta-1}s^{\gamma}h(s)\varphi_{1}(|x(s)|)\varphi_{2}(|^{C}\mathfrak{D}_{0}^{\beta}x(s)|)ds,$

for $\tau>0$. Again, from Lemma 3.4, we find

$\displaystyle|^{C}\mathfrak{D}_{0}^{\beta}x(\tau)|$ (6.9)
$\displaystyle\leq\frac{K_{p(\alpha-\beta-1)+1,p\gamma}^{1/p}}{\Gamma(\alpha-\beta)}\tau^{\alpha-\beta+\gamma-1/q}\Big{(}\int_{0}^{\tau}h^{q}(s)\varphi_{1}^{q}(|x(s)|)\varphi_{2}^{q}(|^{C}\mathfrak{D}_{0}^{\beta}x(s)|)ds\Big{)}^{1/q}$
$\displaystyle\leq\frac{K_{p(\alpha-\beta-1)+1,p\gamma}^{1/p}}{\Gamma(\alpha-\beta)}\tau^{-\beta}\big{(}\int_{0}^{\tau}h^{q}(s)\varphi_{1}^{q}(|x(s)|)\varphi_{2}^{q}(|^{C}\mathfrak{D}_{0}^{\beta}x(s)|)ds\Big{)}^{1/q}$
$\displaystyle\leq
K_{1}(\int_{0}^{\tau}h^{q}(s)\varphi_{1}^{q}(|x(s)|)\varphi_{2}^{q}(|^{C}\mathfrak{D}_{0}^{\beta}x(s)|)ds)^{1/q},\quad\tau\geq\tau_{0}>0.$

Therefore (6.3) follows from (6.4), (6.7) and (6.9). ∎

######  Theorem 6.2.

Assume that $f$ satisfies (C5), (C6). Then, any solution $x$ of (1.2)
satisfies

$|x(\tau)|\leq C,\quad|^{C}\mathfrak{D}_{0}^{\beta}x(\tau)|<C,$

for some positive constant $C,\tau>0$, provided that

$\int_{\mathfrak{\xi}_{0}}^{\infty}\frac{ds}{\varphi_{1}^{q}(s^{1/q})\varphi_{2}^{q}(s^{1/q})}=\infty,\quad\mathfrak{\xi}_{0}>0.$

###### Proof.

In view of Lemma 6.1 we have

$\varphi_{1}(|x(\tau)|)\leq\varphi_{1}(z(\tau)),\quad_{2}(|^{C}\mathfrak{D}_{0}^{\beta}x(\tau)|)\leq\varphi_{2}(z(\tau)),\quad\tau>0.$
(6.10)

From this inequality and (6.4), we obtain

$z(\tau)\leq|b|+K_{1}\Big{(}\int_{0}^{\tau}h^{q}(s)\varphi_{1}^{q}(z(s))\varphi_{2}^{q}(z(s))ds\Big{)}^{1/q},\quad\tau>0.$
(6.11)

Therefore, Lemma 3.5 implies

$z(\tau)\leq\Big{[}E^{-1}\Big{(}E(2^{q-1}K_{1})+2^{q-1}K_{2}\int_{0}^{\tau}h^{q}(s)ds\Big{)}\Big{]}^{1/q}<\infty,$

because $h\in L_{q}(0,\infty)$. This completes the proof. ∎

######  Example 6.3.

Consider the problem

$\begin{gathered}{}^{C}\mathfrak{D}_{0}^{2/3}x(\tau)=\tau^{1/q-2/3}e^{-\lambda\tau}(x(\tau))^{3/5}\big{(}^{C}\mathfrak{D}_{0}^{1/3}x(\tau)\big{)}^{1/3}\big{(}\cos(^{C}\mathfrak{D}_{0}^{1/3}x)\big{)}\quad\tau>0,\\\
x(0)=b,\quad q>3,\quad\lambda>0.\end{gathered}$ (6.12)

Let $\varphi_{1}(\tau)=\tau^{3/5}$, $\varphi_{2}(\tau)=\tau^{1/3}$ and
$h(\tau)=e^{-\lambda\tau}$, $\gamma=1/q-2/3$. Then $h\in L_{q}(0,\infty)$ and

$\int_{\mathfrak{\xi}_{0}}^{\infty}\frac{ds}{\varphi_{1}^{q}(s^{\frac{1}{q}})\varphi_{2}^{q}(s^{1/q})}=\int_{\mathfrak{\xi}_{0}}^{\infty}\frac{ds}{s^{3/5}s^{1/3}}=\int_{\mathfrak{\xi}_{0}}^{\infty}\frac{ds}{s^{14/15}}=\infty.$

Then, by Theorem 6.2, we deduce that any solution $x$ of (6.12) satisfies

$|x(\tau)|\leq C,\quad|^{C}\mathfrak{D}_{0}^{\beta}x(\tau)|<C,$

for $\alpha=2/3$, $\beta=1/3$, and $\tau>0$.

### Acknowledgements

M. D. Kassim wants to thank Imam Abdulrahman Bin Faisal University for its
support and facilities. N. E. Tatar is grateful for the financial support and
the facilities provided by King Fahd University of Petroleum and Minerals
through project number IN181008.

## References

  * [1] D. Băleanu, O. G. Mustafa; _On the asymptotic integration of a class of sublinear fractional differential equations_ , J. Math. Phys., 50 (2009) 123520.
  * [2] D. Băleanu, O. G. Mustafa, R. P. Agarwal; _On the solution set for a class of sequential fractional differential equations_ , J. Phys. A: Math. Theor. 43 (2010) 385209.
  * [3] D. Băleanu, O. G. Mustafa, R. P. Agarwal; _Asymptotically linear solutions for some linear fractional differential equations_ , Abstr. Appl. Anal., (2010). Article ID 865139. http://dx.doi.org/10.1155/2010/865139.
  * [4] D. Băleanu, O. G. Mustafa, R. P. Agarwal; _Asymptotic integration of $(1+\alpha)$-order fractional differential equations_, Computers Math. Appl., 62 (2011), 1492–1500.
  * [5] D. Băleanu, R. P. Agarwal, O. G. Mustafa, M. Coşulschi; _Asymptotic integration of some nonlinear differential equations with fractional time derivative_ , J. Phys. A: Math. Theor., 44 (2011), 055203 9 pp.
  * [6] E. Brestovanská M. Medveď; _Asymptotic behavior of solutions to second-order differential equations with fractional derivative perturbations_ , Electronic Journal of Differential Equations, 2014 (2014) no. 201, 1-10.
  * [7] D. S. Cohen; _The asymptotic behavior of a class of nonlinear differential equations_ , Proc. Amer. Math. Soc. 18 (1967), 607–609.
  * [8] A. Constantin; _On the asymptotic behavior of second order nonlinear differential equations_ , Rend. Math. Appl., 13 (7) (1993), 627–634.
  * [9] A. Constantin; _On the existence of positive solutions of second order differential equations_ , Annali di Matematica, Vol. 184 (2005), 131-138.
  * [10] F. M. Dannan; _Integral inequalities of Gronwall-Bellman-Bihari type and asymptotic behavior of certain second order nonlinear differential equations_ , J. Math. Anal. Appl., 108 (1985), 151–164.
  * [11] M. D. Kassim; _Well-posedness for a Cauchy fractional differential problem with Hilfer type fractional derivative_ , Ph D thesis, King Fahd University of Petroleum and Minerals, Saudi Arabia (2011).
  * [12] M. Kassim, K. Furati, N.-E. Tatar; _Asymptotic behavior of solutions to nonlinear fractional differential equations_ , Math. Model Anal., 21:5 (2016), 610-629.
  * [13] M. D. Kassim, K. M. Furati, N.-E. Tatar; _Asymptotic behavior of solutions to nonlinear initial-value fractional differential problems_ , Electronic Journal of Differential Equations, 2016 (2016) no. 291, 1-14.
  * [14] M. D. Kassim, N.-E. Tatar; _Well-posedness and stability for a differential problem with Hilfer-Hadamard fractional derivative_ , Abstract and Applied Analysis. Vol. 2013, 1-12.
  * [15] M. D. Kassim, N.-E. Tatar; _Stability of logarithmic type for a Hadamard fractional differential problem_ , J. Pseudo-Differ. Oper. Appl., 11 (2020), 447–466.
  * [16] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo; _Theory and Applications of Fractional Differential Equations_ , North-Holland Mathematics Studies 204, Editor: Jan van Mill, Elsevier, Amsterdam, The Netherlands 2006.
  * [17] T. Kusano, W. F. Trench; _Global existence of second order differential equations with integrable coefficients_ , J. London Math. Soc. 31 (1985), 478–486.
  * [18] T. Kusano, W. F. Trench; _Existence of global solutions with prescribed asymptotic behavior for nonlinear ordinary differential equations_ , Mat. Pura Appl. 142 (1985), 381–392.
  * [19] O. Lipovan; _On the asymptotic behaviour of the solutions to a class of second order nonlinear differential equations_ , Glasg. Math. J. 45 no. 1, (2003), 179–187.
  * [20] M. Medveď; _On the asymptotic behavior of solutions of nonlinear differential equations of integer and also of non-integer order_ , Electron. J. Qual. Th. Diff. Eq., Proc. 9th Coll. QTDE, No. 10 (2012), 1-9.
  * [21] M. Medveď; _Asymptotic integration of some classes of fractional differential equations_ , Tatra Mt. Math. Publ. 54 (2013), 119–132.
  * [22] M. Medveď, E. Pekárková; _Asymptotic integration of differential equations with singular p-laplacian_ , Archivum Mathematicum (Brno), 52 (2016), 13-19.
  * [23] M. Medveď, E. Pekárková; _Large time behavior of solutions to second-order differential equations with p-Laplacian_ , Electronic Journal of Differential Equations, 2008 (2008) no. 108, 1-12.
  * [24] M. Medveď, M. Pospíšil; _Asymptotic integration of fractional differential equations with integrodifferential right-hand side_ , Math. Model. Anal., 20.4 (2015), 471-489.
  * [25] O. G. Mustafa, Y. V. Rogovchenko; _Global existence of solutions with prescribed asymptotic behavior for second-order nonlinear differential equations_ , Nonl. Anal. TMA, 51 (2002), 339–368.
  * [26] I. Podlubny; _Fractional Differential Equations_ , Mathematics in Sciences and Engineering, 198, Academic Press, San-Diego, 1999.
  * [27] Y. V. Rogovchenko; _On asymptotics behavior of solutions for a class of second order nonlinear differential equations_ , Collect. Math., 49 (1) (1998), 113–120.
  * [28] S. P. Rogovchenko, Y. V. Rogovchenko; _Asymptotics of solutions for a class of second order nonlinear differential equations_ , Portugaliae Math., 57 (1) (2000), 17–32.
  * [29] S. G. Samko, A. A. Kilbas, O. I. Marichev; _Fractional Integrals and Derivatives: Theory and Applications_ , Gordon and Breach, 1987. (Trans. from Russian) 1993.
  * [30] J. Tong; _The asymptotic behavior of a class of nonlinear differential equations of second order_ , Proc. Amer. Math. Soc., 84 (1982), 235–236.
  * [31] W. F. Trench; _On the asymptotic behavior of solutions of second order linear differential equations_ , Proc. Amer. Math. Soc., 54 (1963), 12–14.
  * [32] P. Waltman; _On the asymptotic behavior of solutions of a nonlinear equation_ , Proc. Amer. Math. Soc. 15 (1964), 918-923.