Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 18 (2017), No. 2, pp. 1073–1083 DOI: 10.18514/MMN.2017.2396

NULL CONTROLLABILITY OF NONLOCAL HILFERFRACTIONAL STOCHASTIC DIFFERENTIAL EQUATIONS

JINRONG WANG AND HAMDY M. AHMED

Received 08 September, 2017

Abstract. In this paper, we study exact null controllability of Hilfer fractional semilinear stochasticdifferential equations in Hilbert spaces. By using fractional calculus and fixed point approach,sufficient conditions of exact null controllability for such fractional systems are established. Anexample is given to show the application of our results.

2010 Mathematics Subject Classification: 26A33; 93B05

Keywords: null controllability, stochastic differential equation, Hilfer fractional derivative

1. INTRODUCTION

The stochastic differential equations arise in many mathematical models [5,14,18,20]. The problem of controllability of nonlinear stochastic or deterministic systemhas been discussed in [3, 4, 6, 8, 9, 16, 19].

Recently, basic theory of differential equations involving Caputo and Riemann-Liouville fractional derivatives can be found in [1, 2, 13, 21–24, 26–30] and the ref-erences cited therein. Beside Caputo and Riemann-Liouville fractional derivatives,there exists a new definition of fractional derivative introduced by Hilfer, which gen-eralized the concept of Riemann-Liouville derivative and has many application inphysics, for more details, see [10–12, 25].

In this paper, we investigate the exact null controllability of Hilfer fractional semi-linear stochastic differential equation of the form8<

:D�;�0C x.t/D Ax.t/CBu.t/

CF.t;x.t//CG.t;x.t//d!.t/dt

; t 2 J D Œ0;b�;

I.1��/.1��/0C x.0/Ch.x/D x0;

(1.1)

where D�;�0C is the Hilfer fractional derivative, 0 � � � 1; 12< � < 1; A is the in-

finitesimal generator of strongly continuous semigroup of bounded linear operators

This work is supported by National Natural Science Foundation of China (11661016), TrainingObject of High Level and Innovative Talents of Guizhou Province ((2016)4006), and Unite Foundationof Guizhou Province ([2015]7640), and Graduate ZDKC ([2015]003).

c 2017 Miskolc University Press

1074 JINRONG WANG AND HAMDY M. AHMED

S.t/; t � 0; on a separable Hilbert space H with inner product h:; :i and norm k : k.There exists aM � 1 such that supt�0 kS.t/k�M: The control function u.�/ is givenin L2.J;U /, the Hilbert space of admissible control functions with U as a separableHilbert space. The symbol B stands for a bounded linear operator from U into H .Here ! is anH -valued Wiener process associated with a positive, nuclear covarianceoperator Q; F is an H -valued map and G is a L.K;H/-valued map both definedon J �H (where K is a real separable Hilbert space with norn k � kK and L.K;H/is the space of all bounded, linear operators from K to H; we write simply L.H/ ifH DK) and h W C.J;H/!H:

2. PRELIMINARIES

In this section, some definitions and results are given which will be used through-out this paper.

Definition 1 (see [15, 17]). The fractional integral operator of order � > 0 for afunction f can be defined as

I�f .t/D1

� .�/

Z t

0

f .s/

.t � s/1��ds; t > 0

where � .�/ is the Gamma function.

Definition 2 (see [11]). The Hilfer fractional derivative of order 0 � � � 1 and0 < � < 1 for a function f is defined by

D�;�0C f .t/D I

�.1��/0C

d

dtI.1��/.1��/0C f .t/:

Let .˝;�;P / be a complete probability space furnished with complete family ofright continuous increasing sub � -algebras f�t W t 2 J g satisfying �t � �: An H -valued random variable is an � - measurable function x.t/ W˝!H and a collectionof random variables D fx.t;!/ W ˝ ! H jt 2 J g is called a stochastic process.Usually we suppress the dependence on ! 2˝ and write x.t/ instead of x.t;!/ andx.t/ W J !H in the place of . Let ˇn.t/ .nD 1;2; :::/ be a sequence of real valuedone-dimensional standard Brownian motions mutually independent over .˝;�;P /.Set

!.t/D

1XnD1

p�nˇn.t/en; t � 0;

where �n; .n D 1;2; :::/ are nonnegative real numbers and feng .n D 1;2; :::/ is acomplete orthonormal basis inK: LetQ 2L.K;K/ be an operator defined byQenD�nen with finite T r.Q/ D

P1nD1�n <1, (Tr denotes the trace of the operator).

Then the above K-valued stochastic process !.t/ is called Q-Wiener process.We assume that �t D �f!.s/ W 0� s � tg is the � -algebra generated by !:

HILFER FRACTIONAL STOCHASTIC DIFFERENTIAL EQUATIONS 1075

For � 2 L.K;H/ we define

k � k2QD T r.�Q��/D

1XnD1

k

p�n�en k

2 :

If k � k2Q<1; then � is called a Q-Hilbert-Schmidt operator. Let LQ.K;H/ de-note the space of all Q-Hilbert-Schmidt operators � W K ! H . The completionLQ.K;H/ of L.K;H/ with respect to the topology induced by the norm k : kQwhere k � k2QD k�;�k is a Hilbert space with the above norm topology. The col-lection of all strongly-measurable, square-integrable, H -valued random variables,denoted by L2.˝;H/; is a Banach space equipped with norm k x.�/ kL2.˝;H/D.E k x.:;!/ k2/

12 ; where the expectation, E is defined by E.x/ D

R˝ x.!/dP: An

important subspace of L2.˝;H/ is given by L02.˝;H/D fx 2 L2.˝;H/; x is �0-measurable g:

Let C.J;L2.˝;H// be the Banach space of all continuous maps from J intoL2.˝;H/ satisfying the condition supt2J Ekx.t/k

2 <1:

Define Y D fx W t .1��/.1��/x.t/ 2 C.J;L2.˝;H//g; with norm k �kY defined byk � kY D .supt2J Ekt

.1��/.1��/x.t/k2/12 : Obviously, Y is a Banach space.

For x 2H;we define two families of operators fS�;�.t/ W t � 0g and fP�.t/ W t � 0gby

S�;�.t/D I�.1��/0C P�.t/; P�.t/D t

��1T�.t/; T�.t/D

Z 10

���.�/S.t��/d�;

where

�.�/D

1XnD1

.��/n�1

.n�1/Š� .1�n�/; 0 < � < 1; � 2 .0;1/

is a function of Wright-type which satisfiesZ 10

��.�/d� D� .1C/

� .1C�/

for � � 0.

Lemma 1 (see [10]). The operator S�;� and P� have the following properties.(i) fP�.t/ W t > 0g is continuous in the uniform operator topology.(ii) For any fixed t > 0; S�;�.t/ and P�.t/ are linear and bounded operators, and

kP�.t/xk �Mt��1

� .�/kxk; kS�;�.t/xk �

Mt .��1/.��1/

� .�.1��/C�/kxk:

(iii) fP�.t/ W t > 0g and fS�;�.t/ W t > 0g are strongly continuous.

1076 JINRONG WANG AND HAMDY M. AHMED

To study the exact null controllability of (1.1) we consider the fractional linearsystem(

D�;�0C y.t/D Ay.t/CBu.t/CF.t/CG.t/

d!.t/dt

; t 2 J D Œ0;b�;

I.1��/.1��/0C y.0/D y0;

(2.1)

associated with the system (1.1).Define the operator

Lb0uD

Z b

0

P�.b� s/Bu.s/ds W L2.J;U /!H;

whereLb0u has a bounded inverse operator .L0/�1 with values inL2.J;U /=ker.Lb0/;and

N b0 .y;F;G/D S�;�.b/yC

Z b

0

P�.b� s/F.s/ds

C

Z b

0

P�.b� s/G.s/d!.s/ WH �L2.J;U /!H:

Definition 3. The system (2.1) is said to be exactly null controllable on J if

Im Lb0 � Im N b0 :

By [7], the system (2.1) is exactly null controllable if there exists > 0 such that

k.Lb0/�yk2 � k.N b

0 /�yk2

for all y 2H .

Lemma 2 (see [16]). Suppose that the linear system (2.1) is exactly null control-lable on J: Then the linear operator

W D .L0/�1N b

0 WH �L2.J;H/! L2.J;U /

is bounded and the control

u.t/D�.L0/�1

"S�;�.b/y0C

Z b

0

P�.b� s/F.s/dsC

Z b

0

P�.b� s/G.s/d!.s/

#D�W.y0;F;G/

transfers the system (2.1) from y0 to 0; whereL0 is the restriction ofLb0 to ŒkerLb0�?;

F 2 L2.J;H/ and G 2 L2.J;L.K;H//:

HILFER FRACTIONAL STOCHASTIC DIFFERENTIAL EQUATIONS 1077

3. EXACT NULL CONTROLLABILITY

In this section, we formulate sufficient conditions for exact null controllabilityfor the system (1.1). First, we give the definitions of mild solution and exact nullcontrollability for it.

Definition 4. We say x 2 C.J;L2.˝;H// is a mild solution to (1.1) if it satisfiesthat

x.t/D S�;�.t/Œx0�h.0/�C

Z t

0

P�.t � s/ŒF.s;x.s//CBu.s/�ds

C

Z t

0

P�.t � s/G.s;x.s//d!.s/; t 2 J:

Definition 5. The system (1.1) is said to be exact null controllable on the intervalJ if there exists a stochastic control u 2 L2.J;U / such that the solution x of thesystem (1.1) satisfies x.b/D 0:

To prove the main result, we need the following hypotheses:.H1/ The fractional linear system (2.1) is exactly null controllable on J:.H2/ The function F W J �H ! H is locally Lipschitz continuous, for all t 2

J; x; x1; x2 2H; there exist constant c1 > 0; such that

kF.t;x2/�F.t;x1/k2� c1kx2�x1k

2; kF.t;x/k2 � c1.1Ckxk2/:

.H3/ The function G W J �H ! L.K;H/ is locally Lipschitz continuous, for allt 2 J; x; x1; x2 2H; there exist constant c2 > 0; such that

kG.t;x2/�G.t;x1/k2Q � c2kx2�x1k

2; kG.t;x/k2Q � c2.1Ckxk2/:

.H4/ The function h WC.J;H/!H is continuous, for any x; x1; x2 2C.J;H/;there exist constant c3 > 0; such that

kh.x2/�h.x1/k2� c3kx2�x1k

2; kh.x/k2 � c3.1Ckxk2/:

Set %1 WD4M2c3

� 2.�.1��/C�/C

M2b1C2�.��1/

.2��1/� 2.�/.c1 C c2T r.Q// and

%2 WD 1C4M2b2��1kW k2kBk2

.2��1/� 2.�/.

Theorem 1. If the hypotheses .H1/-.H4/ are satisfied, then the system (1.1) isexactly null controllable on J provided that

% WD %1%2 < 1: (3.1)

Proof. For an arbitrary x define the operator ˚ on Y as follows

.˚x/.t/D S�;�.t/Œx0�h.x/� (3.2)

C

Z t

0

P�.t � s/ŒF.s;x.s//�BW.x0�h.x/;F;G/�ds

1078 JINRONG WANG AND HAMDY M. AHMED

C

Z t

0

P�.t � s/G.s;x.s//d!.s/; t 2 J;

where

u.t/DW.x0�h.x/;F;G/.t/

D�.L0/�1fS�;�.b/Œx0�h.x/�

C

Z b

0

P�.b� s/F.s;x.s//dsC

Z b

0

P�.b� s/G.s;x.s//d!.s/g:

It will be shown that the operator ˚ from Y into itself has a fixed point.Step 1. The control u.�/D�W.x0�h.x/;F;G/ is bounded on Y:Indeed,

kuk2Y D supt2J

t2.1��/.1��/Ekuk2 (3.3)

� supt2J

t2.1��/.1��/EkW.x0�h.x/;F;G/.s/k2

� kW k2�

M 2

� 2.�.1��/C�/ŒEkx0k

2C c3.1CEkxk

2/�

CM 2b1C2�.��1/

.2��1/� 2.�/.1CEkxk2/.c1C c2T r.Q//

�:

Step 2. We show that ˚ maps Y into itself.From (3.2) and (3.3) for t 2 J; we have

k.˚x/.t/k2Y D supt2J

t2.1��/.1��/Ek.˚x/.t/k2

� 4 supt2J

t2.1��/.1��/�EkS�;�.t/Œx0�h.x/�k

2

CE

Z t

0

P�.t � s/F.s;x.s//ds

2�C4 sup

t2J

t2.1��/.1��/

�

�E

Z t

0

P�.t � s/ŒBW.x0�h.x/;F;G/.s/�ds

2CE

Z t

0

P�.t � s/G.s;x.s//d!.s/

2��

�4M 2

� 2.�.1��/C�/ŒEkx0k

2C c3.1CEkxk

2/�

HILFER FRACTIONAL STOCHASTIC DIFFERENTIAL EQUATIONS 1079

C4M 2b1C2�.��1/

.2��1/� 2.�/.1CEkxk2/.c1C c2T r.Q//

��

�1C

M 2kBk2kW k2kb2��1

.2��1/� 2.�/

�<1:

Therefore ˚ maps Y into itself.Step 3. We prove .˚x/.t/ is continuous on J for any x 2 Y:Let 0 < t � b and � > 0 be sufficiently small, then,

k.˚x/.tC �/� .˚x/.t/k2Y (3.4)

D supt2J

t2.1��/.1��/Ek.˚x/.tC �/� .˚x/.t/k2

� 4 supt2J

t2.1��/.1��/

�Ek.S�;�.tC �/�S�;�.t//Œx0�h.x/�k2

C4 supt2J

t2.1��/.1��/

�E

Z tC�

0

P�.tC �� s/ŒBW.x0�h.x/;F;G/.s/�ds

�

Z t

0

P�.t � s/ŒBW.x0�h.x/;F;G/.s/�ds

2C4 sup

t2J

t2.1��/.1��/

�E

Z tC�

0

P�.tC �� s/F.s;x.s//ds�

Z t

0

P�.t � s/F.s;x.s//ds

2C4 sup

t2J

t2.1��/.1��/

�E

Z tC�

0

P�.tC �� s/G.s;x.s//d!.s/�

Z t

0

P�.t � s/G.s;x.s//d!.s/

2:Clearly, from Lemma 1, .H2/ and .H3/, the right hand side of (3.4) tends to zero

as �! 0: Hence, .˚x/.t/ is continuous on J:Step 4. We show that .˚x/.t/ is a contraction on Y:Let x1; x2 2 Y; for any t 2 .0;b� be fixed, then

k.˚x2/.t/� .˚x1/.t/k2Y

D supt2J

t2.1��/.1��/Ek.˚x2/.t/� .˚x1/.t/k2

� 4 supt2J

t2.1��/.1��/Ek.S�;�.t/Œh.x2/�h.x1/�k2

1080 JINRONG WANG AND HAMDY M. AHMED

C4 supt2J

t2.1��/.1��/E

Z t

0

P�.t � s/ŒBW.x0�h.x2/;F;G/.s/

�BW.x0�h.x1/;F;G/.s/�ds

2C4 sup

t2J

t2.1��/.1��/E

Z t

0

P�.t � s/ŒF.s;x2.s//�F.s;x1.s//�ds

2C4 sup

t2J

t2.1��/.1��/E

Z t

0

P�.t � s/ŒG.s;x2.s//�G.s;x1.s//�d!.s/

2� %Ekx2�x1k

2:

Hence, ˚ is a contraction in Y via (3.1). From the Banach fixed point theorem, ˚ hasa unique fixed point. Therefore the system (1.1) is exact null controllable on J: �

4. AN EXAMPLE

Consider the following Hilfer fractional stochastic partial differential system8<ˆ:D�; 23

0C x.t;´/D@2

@´2x.t;´/Cu.t;´/

Cf .t;x.t;´//Cg.t;x.t;´//d!.t/dt

; t 2 J; 0 < ´ < 1;

x.t;0/D x.t;1/D 0; t 2 J;

I13.1��/

0C .x.0;´//CPpiD1 kix.ti ;´/D x0.´/; 0� ´� 1;

(4.1)

where p is a positive integer, 0 < t0 < t1 < ::: < tp < b and !.t/ is Wiener process,u 2 L2.0;b/; and H D L2.Œ0;1�/: Let f W R�R! R and g W R�R! R are con-tinuous and global Lipschitz continuous in the second variable. Also, let A WH !H

be defined by Ay D @2

@´2y with domain D.A/D fy 2H W y; @y

@´are absolutely con-

tinuous, and @2y

@´22H;y.0/D y.1/D 0g:

It is known that A is self-adjoin and has the eigenvalues �nD�n2�2; n 2N; withthe corresponding normalized eigenvectors en.´/ D

p2 sin.n�´/: Furthermore, A

generates an analytic compact semigroup of bounded linear operator S.t/; t � 0; ona separable Hilbert space H which is given by

S.t/y D

1XnD1

.yn; en/en D

1XnD1

2e�n2�2t sin.n�´/

Z 1

0

sin.n��/y.�/d�; y 2H:

If u 2 L2.J;H/; then B D I; B� D I:

HILFER FRACTIONAL STOCHASTIC DIFFERENTIAL EQUATIONS 1081

Now we consider8<ˆ:D�; 23

0C y.t;´/D@2

@´2y.t;´/Cu.t;´/

Cf .t;´/Cg.t;´/d!.t/; t 2 J; 0 < ´ < 1;

y.t;0/D y.t;1/D 0; t 2 J;

I13.1��/

0C .y.0;´//D y0.´/; 0� ´� 1;

(4.2)

The system (4.2) is exact null controllability if there is a > 0; such thatZ b

0

kB�P ��.b� s/yk2ds �

"kS��;�.b/yk

2C

Z b

0

kP ��.b� s/yk2ds

#;

or equivalentlyZ b

0

kP�.b� s/yk2ds �

"kS�;�.b/yk

2C

Z b

0

kP�.b� s/yk2ds

#:

If f D 0 and g D 0 in (4.2), then the fractional linear system is exactly null con-trollable if Z b

0

kP�.b� s/yk2ds � bkS�;�.b/yk

2:

Therefore,Z b

0

kP�.b� s/yk2ds �

b

1Cb

"kS�;�.b/yk

2C

Z b

0

kP�.b� s/yk2ds

#:

Hence, the linear fractional system (4.2) is exactly null controllable on Œ0;b�. So thehypothesis .H1/ is satisfied.

We define F W J �H ! H; G W J �H ! L.K;H/ and h W C.J;H/! H asfollows: F.t;x/D f .t;x.t;´//; G.t;x/D g.t;x.t;´// and h.x/D

PpiD1kix.ti ;´/

Then h.�/ satisfies .H4/.By choosing the constants ki ; i D 1;2; :::;p;M;c1; c2 and c3 such that % <

1: Hence, all the hypotheses of Theorem 1 are satisfied, so the Hilfer fractionalstochastic partial differential system (4.1) is exact null controllable on Œ0;b�:

REFERENCES

[1] R. Agarwal, S. Hristova, and D. O’Regan, “A survey of Lyapunov functions, stability and impuls-ive Caputo fractional differential equations,” Fract. Calc. Appl. Anal, vol. 19, no. 2, pp. 290–318,2016, doi: 10.1515/fca-2016-0017.

[2] R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary valueproblems of nonlinear fractional differential equations and inclusions,” Acta Appl. Math, vol. 109,no. 3, pp. 973–1033, 2010, doi: 10.1007/s10440-008-9356-6.

[3] H. M. Ahmed, “Controllability of fractional stochastic delay equations,” Lobachevskii J. Math,vol. 30, no. 3, pp. 195–202, 2009, doi: 10.1134/S19950802090300019.

1082 JINRONG WANG AND HAMDY M. AHMED

[4] H. M. Ahmed, “Approximate controllability of impulsive neutral stochastic differential equationswith fractional Brownian motion in a Hilbert space,” Adv. Difference Equ, vol. 2014, no. 113, pp.1–11, 2014, doi: 10.1186/1687-1847-2014-113.

[5] K. Balachandran, P. Balasubramaniam, and J. P. Dauer, “Local null controllability of nonilinearfunctional differential systems in Banach spaces,” J. Optim Theory Appl., vol. 88, no. 1, pp. 61–75,1996, doi: 0022-3239/96/0100.0061509,50/.

[6] K. Balachandran and J. H. Kim, “Sample controllability of nonlinear stochastic integrodif-ferential systems,” Nonlinear Anal.: Hybrid Syst, vol. 4, no. 3, pp. 543–549, 2010, doi:10.1016/j.nahs.2010.02.001.

[7] R. F. Curtain and H. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory. NewYork: World Scientific, 1995. doi: 10.1007/978-1-4612-4224-6.

[8] J. P. Dauer and P. Balasubramaniam, “Null controllability of semilinear integrodifferential systemsin Banach space,” Appl. Math. Lett, vol. 10, no. 6, pp. 117–123, 1997, doi: 10.1016/S0893-9659(97)00114-6.

[9] X. Fu and Y. Zhang, “Exact null controllability of non-autonomous functional evolution sys-tems with nonlocal conditions,” Acta Math. Sci, vol. 33B, no. 3, pp. 747–757, 2013, doi:10.1016/S0252-9602(13)60035-1.

[10] H. B. Gu and J. J. Trujillo, “Existence of mild solution for evolution equation with Hilfer fractionalderivative,” Appl. Math. Comput, vol. 257, pp. 344–354, 2015, doi: 10.1016/j.amc.2014.10.083.

[11] R. Hilfer, Applications of Fractional Calculus in Physics. Singapore: World Scientific, 2000.[12] R. Hilfer, “Experimental evidence for fractional time evolution in glass materials,” Chem. Phys,

vol. 284, no. 1-2, pp. 399–408, 2002, doi: org/10.1016/S0301-0104(02)00670-5.[13] J. W. M. Li, “Finite time stability of fractional delay differential equations,” Appl. Math. Lett,

vol. 64, no. 3, pp. 170–176, 2017, doi: org10.1016/j.aml.2016.09.004.[14] X. Mao, Stochastic differential equations and applications. Horwood: Chichester, 1997.[15] K. S. Miller and B. Ross, An Introduction to the fractional calculus and fractional differential

equations. New York: John Wiley, 1993.[16] J. Y. Park and P. Balasubramaniam, “Exact null controllability of abstract semilinear functional

integrodifferential stochastic evolution equations in Hilbert spaces,” Taiwanese J. Math, vol. 13,no. 6B, pp. 2093–2103, 2009.

[17] I. Podlubny, Fractional differential equations. San Diego: Academic press, 1999.[18] P. Protter, Stochastic integration and differential equations, Applications of Mathematics. New

York: Springer, Berlin, 1990. doi: 10.1007/978-3-662-02619-9.[19] R. Sakthivel, S. Suganya, and S. M. Anthoni, “Approximate controllability of fractional

stochastic evolution equations,” Comput. Math. Appl, vol. 63, no. 3, pp. 660–668, 2012, doi:org/10.1016/j.camwa.2011.11.024.

[20] K. Sobczyk, Stochastic differential equations with applications to Physics and Engineering. Lon-don: Kluwer Academic, 1991. doi: 10.1007/978-94-011-3712-6.

[21] J. Wang, “Approximate mild solutions of fractional stochastic evolution equations in Hilbertspaces,” Appl. Math. Comput, vol. 256, pp. 315–323, 2015, doi: 10.1016/j.amc.2014.12.155.

[22] J. Wang, A. G. Ibrahim, and M. Feckan, “Differential inclusions of arbitrary fractional orderwith anti-periodic conditions in Banach spaces,” Electron. J. Qual. Theory Differ. Equ, vol. 2016,no. 34, pp. 1–22, 2016, doi: 10.14232/ejqtde.2016.1.34.

[23] J. Wang and X. Li, “A Uniformed Method to Ulam-Hyers Stability for Some Linear FractionalEquations,” Mediterr. J. Math, vol. 13, no. 2, pp. 625–635, 2016, doi: org/10.1007/s00009-015-0523-5.

[24] J. Wang, X. Li, M. Feckan, and Y. Zhou, “Hermite-Hadamard-type inequalities for Riemann-Liouville fractional integrals via two kinds of convexity,” Appl. Anal, vol. 92, no. 11, pp. 2241–2253, 2013, doi: 10.1080/00036811.2012.727986.

HILFER FRACTIONAL STOCHASTIC DIFFERENTIAL EQUATIONS 1083

[25] J. Wang and Y. Zhang, “Nonlocal initial value problems for differential equations with Hil-fer fractional derivative,” Appl. Math. Comput, vol. 266, no. 1, pp. 850–859, 2015, doi:org/10.1016/j.amc.2015.05.144.

[26] J. Wang, Y. Zhou, and Z. Lin, “On a new class of impulsive fractional differential equations,”Appl. Math. Comput, vol. 242, no. 1, pp. 649–657, 2014, doi: org/10.1016/j.amc.2014.06.002.

[27] Y. Zhou, J. R. Wang, and L. Zhang, Basic Theory of Fractional Differential Equations. Singapore:World Scientific, 2016.

[28] Y. Zhou and L. Zhang, “Existence and multiplicity results of homoclinic solutions for frac-tional Hamiltonian systems,” Comput. Math. Appl, vol. 73, no. 6, pp. 1325–1345, 2017, doi:org/10.1016/j.camwa.2016.04.041.

[29] Y. Zhou, L. Zhang, and X. H. Shen, “Existence of mild solutions for fractional evolution equa-tions,” J. Int. Equ. Appl, vol. 25, no. 4, pp. 557–586, 2013, doi: 10.1216/JIE-2013-25-4-557.

[30] Y. Zhou, F. Zhou, and J. Pecaric, “Abstract Cauchy problem for fractional functional differentialequations,” Topol. Meth. Nonlinear Anal, vol. 42, no. 1, pp. 119–136, 2013, doi: projecteuc-lid.org/euclid.tmna/1461247296.

Authors’ addresses

JinRong WangGuizhou University, Department of Mathematics, Guiyang 550025, ChinaE-mail address: sci.jrwang@gzu.edu.cn

Hamdy M. AhmedHigher Institute of Engineering, El-Shorouk Academy, El-Shorouk City, Cairo, EgyptE-mail address: hamdy 17eg@yahoo.com