In this paper, a class of fractional functional differential equations is investigated. Using differential inequalities and Lyapunov-like functions, Lipschitz stability, uniform Lipschitz stability, and global uniform Lipschitz stability criteria are proved. Since the problem of Lipschitz stability of dynamic systems is relevant in various contexts, including many inverse and control problem, our results can be applied in the qualitative investigations of many practical problems of diverse interest.

1.
V.
Bacchelli
and
S.
Vessella
, “
Lipschitz stability for a stationary 2D inverse problem with unknown polygonal boundary
,”
Inverse Probl.
22
,
1627
1658
(
2006
).
2.
M.
Bellassoued
and
M.
Yamamoto
, “
Lipschitz stability in determining density and two Lam
$\rm {\acute{e}}$
é
coefficients
,”
J. Math. Anal. Appl.
329
,
1240
1259
(
2007
).
3.
M.
Benchohra
,
J.
Henderson
,
S. K.
Ntouyas
, and
A.
Ouahab
, “
Existence results for fractional order functional differential equations with infinite delay
,”
J. Math. Anal. Appl.
338
,
1340
1350
(
2008
).
4.
A.
Chauhan
and
J.
Dabas
, “
Existence of mild solutions for impulsive fractional-order semilinear evolution equations with nonlocal conditions
,”
Electron. J. Differ. Equations
2011
,
1
10
(
2011
).
5.
F. M.
Dannan
and
S.
Elaydi
, “
Lipschitz stability of nonlinear systems of differential equations
,”
J. Math. Anal. Appl.
113
,
562
577
(
1986
).
6.
A. M. A.
El-Sayed
,
F. M.
Gaafar
, and
E. M. A.
Hamadalla
, “
Stability for a non-local non-autonomous system of fractional order differential equations with delays
,”
Electron. J. Differ. Equations
2010
,
1
10
(
2010
).
7.
H. I.
Freedman
and
S.
Ruan
, “
Uniform persistence in functional differential equations
,”
J. Differ. Equations
115
,
173
192
(
1995
).
8.
B.
Guo
and
D.
Huang
, “
Existence and stability of standing waves for nonlinear fractional Schrödinger equations
,”
J. Math. Phys.
53
,
083702
(
2012
).
9.
X.
Guo
and
M.
Xu
, “
Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation
,”
J. Math. Phys.
47
,
082104
(
2006
).
10.
J. K.
Hale
,
Theory of Functional Differential Equations
(
Springer-Verlag
,
New York
,
1977
).
11.
J.
Henderson
and
A.
Ouahab
, “
Fractional functional differential inclusions with finite delay
,”
Nonlinear Anal.
70
,
2091
2105
(
2009
).
12.
R.
Hilfer
,
Application of Fractional Calculus in Physics
(
World Scientific
,
Singapore
,
2000
).
13.
O.
Imanuvilov
and
M.
Yamamoto
, “
Global Lipschitz stability in an inverse hyperbolic problem by interior observations
,”
Inverse Probl.
17
,
717
728
(
2001
).
14.
A.
Kilbas
,
H. M.
Srivastava
, and
J. J.
Trujillo
,
Theory and Applications of Fractional Differential Equations
(
North-Holland Mathematics Studies
,
Elsevier
,
2006
), Vol.
204
.
15.
G. K.
Kulev
and
D. D.
Bainov
, “
Lipschitz stability of impulsive systems of differential equations
,”
Int. J. Theor. Phys.
30
,
737
756
(
1991
).
16.
V.
Lakshmikantham
, “
Theory of fractional functional differential equations
,”
Nonlinear Anal. Theory, Methods Appl.
69
,
3337
3343
(
2008
).
17.
V.
Lakshmikantham
,
S.
Leela
, and
J.
Vasundhara Devi
,
Theory of Fractional Dynamic Systems
(
Cambridge Scientific Publishers
,
2009
).
18.
N.
Laskin
, “
Fractional quantum mechanics
,”
Phys. Rev. E
62
,
3135
(
2000
).
19.
N.
Laskin
, “
Fractional Schrödinger equations
,”
Phys. Rev. E
66
,
056108
(
2002
).
20.
R.
Metzler
and
J.
Klafter
, “
The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics
,”
J. Phys. A
37
,
R161
R208
(
2004
).
21.
B.
Pachpatte
, “
On generalizations of Bihari's inequality
,”
Soochow J. Math.
31
,
261
271
(
2005
).
22.
I.
Podlubny
,
Fractional Differential Equations
(
Academic Press
,
San Diego
,
1999
).
23.
B. S.
Razumikhin
,
Stability of Hereditary Systems
(
Nauka
,
Moscow
,
1988
) (in Russian).
24.
G. T.
Stamov
,
Almost Periodic Solutions of Impulsive Differential Equations
(
Springer
,
Berlin
,
2012
).
25.
G. T.
Stamov
,
J.
Alzabut
,
P.
Atanasov
, and
A.
Stamov
, “
Almost periodic solutions for an impulsive delay model of price fluctuations in commodity markets
,”
Nonlinear Anal.: Real World Appl.
12
,
3170
3176
(
2011
).
26.
I. M.
Stamova
,
Stability Analysis of Impulsive Functional Differential Equations
(
Walter de Gruyter
,
Berlin/New York
,
2009
).
27.
I. M.
Stamova
and
A.
Stamov
, “
Impulsive control on the asymptotic stability of the solutions of a Solow model with endogenous labor growth
,”
J. Franklin Inst.
349
,
2704
2716
(
2012
).
28.
I. M.
Stamova
,
T.
Stamov
, and
N.
Simeonova
, “
Impulsive control on global exponential stability for cellular neural networks with supremums
,”
J. Vib. Control
19
,
483
490
(
2013
).
29.
J. R.
Wang
and
Y.
Zhou
, “
A class of fractional evolution equations and optimal controls
,”
Nonlinear Anal.: Real World Appl.
12
,
262
272
(
2011
).
You do not currently have access to this content.