Attractivity for a k dimensional system of fractional functional differential equations and global attractivity for a k dimensional system of nonlinear fractional differential equations

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R E S E A R C H

Open Access

Attractivity for a

k

-dimensional system of

fractional functional differential equations

and global attractivity for a

k

-dimensional

system of nonlinear fractional differential

equations

Dumitru Baleanu

1,2,3*

_{, Sayyedeh Zahra Nazemi}

4

_{and Shahram Rezapour}

4

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

Cankaya University, Ogretmenler Cad. 14, Balgat, Ankara, 06530, Turkey

2_{Institute of Space Sciences,}

Magurele, Bucharest, Romania Full list of author information is available at the end of the article

Abstract

In this paper, we present some results for the attractivity of solutions for a

k-dimensional system of fractional functional diﬀerential equations involving the

Caputo fractional derivative by using the classical Schauder’s ﬁxed-point theorem.

Also, the global attractivity of solutions for ak-dimensional system of fractional

diﬀerential equations involving Riemann-Liouville fractional derivative are obtained by using Krasnoselskii’s ﬁxed-point theorem. We give two examples to illustrate our main results.

1 Introduction

In recent years, many researchers have been focused on investigation of fractional dif-ferential equations which has played an important role in different areas of science (see for example, [–] and the references therein). As you know, there are many practical applications of fractional differential equations in different fields of science such as econ-omy, biology, and the study of forced van der Pol oscillators (see for example, [–] and the references therein). On the other hand, there are a few papers on the attractiv-ity of solutions for fractional differential equations and fractional functional differential equations (see for example, [] and []). For the details of basic notions of this paper such the standard Caputo fractional derivative, the standard Riemann-Liouville fractional derivative, and the fractional integral of orderqfor a functionf see []. In , Chen and Zhou reviewed the attractivity of solutions for the fractional functional differential equationc_Dα_x₍_t_{) =}_f₍_t_,_x

t) fort∈(t,∞) via the boundary value conditionx(t) =φ(t) for

t–r≤t≤t, wheret≥,r> ,  <α< ,cDis the standard Caputo fractional derivative, φ∈C([t–r,t],R) andf : (t,∞)×C([–r, ],R)→Ra function with some properties []. In , Chenet al.reviewed the global attractivity of solutions for the nonlinear fractional diﬀerential equationDα_x₍_t_{) =}_g₍_t_,_x₍_t_{)) for}_t_∈₍_t_,_∞_{) via the boundary value}

problem [Dα–_x₍_t_)]

t=t=x, where t≥,  <α< ,xis a constant,Dis the standard

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problemx(t) =x, wheret≥,  <α< ,xis a constant,cDis the standard Caputo frac-tional derivative, andg: (t,∞)×R→Ra function with some properties [].

In this paper, we investigate the attractivity of solutions for ak-dimensional system of fractional diﬀerential equations. Also, we investigate the global attractivity of solutions for anotherk-dimensional system of nonlinear fractional diﬀerential equations.

2 Preliminaries

In this paper, we investigate the attractivity of solutions fork-dimensional system of frac-tional funcfrac-tional diﬀerential equations

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

c_Dα_x_(_t_{) =}_f_(_t_,_x

t,xt, . . . ,xkt),

c_Dα_x_(_t_{) =}_f_(_t_,_x

t,xt, . . . ,xkt), ..

.

c_Dα_k_x

k(t) =fk(t,xt,xt, . . . ,xkt),

(.)

via the boundary value problems x(t) =φ(t),x(t) =φ(t), . . . ,xk(t) =φk(t) for t–r≤

t≤t, where  <αi<  fori= , , . . . ,k,t≥,t∈(t,∞), cDis the standard Caputo fractional derivative,J= (t,∞),r> ,φi∈C([t–r,t],Rn) fori= , , . . . ,kandfi:J×

C([–r, ],Rn₎_×_C_([–_r_{, ],}_Rn₎_{× · · · ×}_C_([–_r_{, ],}_Rn₎_→_Rn _{is a function satisfying some}

assumptions that will be speciﬁed later fori= , , . . . ,k. Ifx∈C([t–r,∞),Rn), thenxt

is deﬁned byxt(θ) =x(t+θ) for –r≤θ≤ andt∈[t,∞). Also, we investigate the global

attractivity of solutions for thek-dimensional system of nonlinear fractional diﬀerential equations

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

Dα_x_(_t_{) =}_g_(_t_,_x_(_t_),_x_(_t_{), . . . ,}_x

k(t)),

Dα_x_(_t_{) =}_g_(_t_,_x

(t),x(t), . . . ,xk(t)),

.. .

Dαk_x

k(t) =gk(t,x(t),x(t), . . . ,xk(t)),

(.)

via the boundary value problems [Dα–_x

(t)]t=t =x



, [Dα–x(t)]t=t=x



, . . . , [Dαk–×

xk(t)]t=t =x



k, where  <αi<  fori= , , . . . ,k,t≥,t∈(t,∞),Dis the Riemann-Liouville fractional derivative,J= (t,∞),x, . . . ,xk are constants, andgi:J×Rn×Rn×

· · · ×Rn_→_Rn_{is an integrable function satisfying some assumptions that will be speciﬁed}

later fori= , , . . . ,k. In fact, we say that the solution (x(t),x(t), . . . ,xk(t)) of the problem

(.) is attractive whenever if there exists a constantb

i(t) >  such that|φi(s)| ≤bi for all

i= , , . . . ,kands∈[t–r,t], thenlimt→∞xi(t,t,φi)→. Also, the zero solutionx(t) of

the problem (.) is said to be globally attractive whenever each solution tends to zero as

t→ ∞. LetX=C(J,Rn_{) be the Banach space of all continuous functions from}_J_into_Rn

with the normx=sup_t_∈_J|x(t)|, where| · |denotes a suitable complete norm onRn_{. It is}

clear that the product space (Xk₌_X_×_X_{× · · · ×}_X

k

, · ∗) is also a Banach space, where

(x,x, . . . ,xk)∗=x+x+· · ·+xk. We need the following Schauder ﬁxed-point

theorem and improvement of a ﬁxed-point theorem of Krasnoselskii due to Burton, which one can ﬁnd in [, ] and [].

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Theorem . Let S be a nonempty,closed,convex,and bounded subset of the Banach space X,A:X→X a contraction with constant l< ,B:S→X a continuous map which B(S)

resides in a compact subset of X and x=Ax+By and y∈S implies x∈S.Then the operator equation Ax+Bx=x has a solution in S.

3 Main results

First, we investigate attractive solutions of the problem (.). In this way, suppose that

xt=sup–r≤θ≤|x(t+θ)|fort∈J. We assume thatfi(t,xt,xt, . . . ,xkt) is Lebesgue mea-surable with respect toton [t,∞) andfi(t,ϕ,ϕ, . . . ,ϕk) is continuous with respect toϕj

onC([–r, ],Rn_{) for}_i_,_j_{= , , . . . ,}_k_{. Note that the problem (.) is equivalent to the system}

of equations

xi(t) =

φi(t) +₍_α

i) t

t(t–s) αi–_f

i(s,xs,xs, . . . ,xks)ds, t>t,

φi(t), t∈[t–r,t]

or

xi(t) =



(αi) t

t(t–s)

αi–_[ φi(t)

(–αi)(s–t)–

αi₊_f

i(s,xs,xs, . . . ,xks)]ds, t>t,

φi(t), t∈[t–r,t]

fori= , , . . . ,k. Deﬁne the operatorT:Xk_→_Xk_by

T(x,x, . . . ,xk)(t) =

⎛ ⎜ ⎜ ⎜ ⎜ ⎝

T(x,x, . . . ,xk)(t)

T(x,x, . . . ,xk)(t)

.. .

Tk(x,x, . . . ,xk)(t)

⎞ ⎟ ⎟ ⎟ ⎟ ⎠,

where

Ti(x,x, . . . ,xk)(t) =

φi(t) +(αi) t

t(t–s) αi–_f

i(s,xs,xs, . . . ,xks)ds, t>t,

φi(t), t∈[t–r,t]

fori= , , . . . ,k. It is easy to check that (x(t),x(t), . . . ,xk(t)) is a solution of the problem

(.) if and only if (x(t),x(t), . . . ,xk(t)) is a ﬁxed point of the operatorT.

Theorem . Suppose that for each i∈ {, , . . . ,k}there existγi> andαi∈(,αi)such

that

φi(t) +

 (αi)

t t

(t–s)αi–_f

i(s,xs,xs, . . . ,xks)ds

≤(t–t)–γi

for all t∈J and fi∈L



α_i_₍_J×_C_([–_r_{, ],}Rn₎_×_C_([–_r_{, ],}_Rn₎_{× · · · ×}_C_([–_r_{, ],}_Rn_),_Rn_)._Then

the problem(.)has at least one attractive solution(x,x, . . . ,xk)such that xi∈C([t–

r,∞),Rn₎_{for all i}_{= , , . . . ,}_k_.

Proof Consider the set

S=

(x,x, . . . ,xk) :xi∈C

[t–r,∞),Rn

,xi(t)≤(t–t)–γi

for alli= , , . . . ,kandt≥ ˜t>t

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where t˜ is a constant. It is easy to check that S is a closed, bounded, and convex subset of Rn×Rn× · · · ×Rn

k

. We show that the operator T has a ﬁxed point in S.

This implies that the problem (.) has a solution. Note that |Ti(x,x, . . . ,xk)(t)| ≤(t–

t)–γi _{for all} _i_{= , , . . . ,}_k _{and so} _T₍_S_)⊂_S_{. Now, we show that} _T _{is continuous. Let}

(xm

,xm, . . . ,xmk), (x,x, . . . ,xk)∈ S for all m≥ and limm→∞|xim(t) –xi(t)|=  for all

i= , , . . . ,k. Then, we have limm→∞fi(t,xmt,xmt, . . . ,xmkt) =fi(t,xt,xt, . . . ,xkt) for all i= , , . . . ,k and t>t. Let >  be given. Choose T˜ >t such that t≥ ˜T implies that (t–t)–γi_<

. Letνi=

αi–

–αi and note that  +νi>  fori= , , . . . ,k. Also, we have

Ti

xm_,xm_, . . . ,xm_k(t) –Ti(x,x, . . . ,xk)(t)

≤ 

(αi)

t t

(t–s)αi–fi

s,xm_s,xm_s, . . . ,xm_k_s–fi(s,xs,xs, . . . ,xks)ds

≤ 

(αi)

t

t

(t–s)αi––αi_ds

–αi

× t

t

fi

s,xm_s,xm_s, . . . ,xm_k_s–fi(s,xs,xs, . . . ,xks)



α_i__ds αi

≤ 

(αi)

  +νi

(t–t)

+νi–αi

×

˜

T t

fi

s,xm_s,xm_s, . . . ,xm_k_s–fi(s,xs,xs, . . . ,xks)



α_i__ds αi

≤ 

(αi)

  +νi

(T˜ –t)

+νi–αi

(T˜ –t)αi

× sup

t≤s≤ ˜T

fi

s,xm_s,xm_s, . . . ,xm_k_s–fi(s,xs,xs, . . . ,xks)

for t<t≤ ˜T. Thus,limm→∞|Ti(xm,xm, . . . ,xmk)(t) –Ti(x,x, . . . ,xk)(t)|=  for all t<

t≤ ˜T. Also, we have _T_i

xm_,xm_, . . . ,xm_k(t) –Ti(x,x, . . . ,xk)(t)

= 

(αi)

t t

(t–s)αi–_f

i

s,xm_s,xm_s, . . . ,xm_k_sds

– 

(αi)

t t

(t–s)αi–fi(s,xs,xs, . . . ,xks)ds

≤(t–t)–γi≤

fort>T˜. Hence,limm→∞|Ti(xm,xm, . . . ,xkm)(t) –Ti(x,x, . . . ,xk)(t)|=  fort>t. This

im-plies thatTiis continuous fori= , , . . . ,kand soTis continuous. Now, we show that the

setT(S) is equi-continuous. Let> . Sincelimt→∞(t–t)–γi=  fori= , , . . . ,k, there is aT˜>tsuch that (t–t)–γi<_ for allt>T˜andi= , , . . . ,k. Lett,t>tandt>t. Ift,t∈(t,T˜], then

Ti(x,x, . . . ,xk)(t) –Ti(x,x, . . . ,xk)(t)

≤ 

(αi)

t

t

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–  (αi)

t

t

(t–s)αi–fi(s,xs,xs, . . . ,xks)ds

≤ 

(αi)

t

t

(t–s)αi–– (t–s)αi–fi(s,xs,xs, . . . ,xks)ds

+ 

(αi)

t

t

(t–s)αi–fi(s,xs,xs, . . . ,xks)ds

≤ 

(αi)

t

t

(t–s)αi–– (t–s)αi– 

–αi_ds

–αi

×

t

t

fi(s,xs,xs, . . . ,xks)



αi_ds

αi

+ 

(αi)

t

t

(t–s) α_i– –αi_ds

–αi t

t

fi(s,xs,xs, . . . ,xks)



α_i__ds αi

≤ 

(αi)

  +νi

–αi

(t–t) αi–

–α_i_+_{+ (}_t –t)

αi–

–α_i_+_{– (}_t –t)

αi–

–α_i_+–αi

×

T˜

t

fi(s,xs,xs, . . . ,xks)



αi_ds

αi

+ 

(αi)

  +νi

–αi

(t–t) α_i– –αi+–αi

T˜

t

fi(s,xs,xs, . . . ,xks)



αi_ds

αi

≤ 

(αi)

  +νi

–αi T˜ t

fi(s,xs,xs, . . . ,xks)



αi _ds

αi

(t–t)αi–αi

and solimt→t|Ti(x,x, . . . ,xk)(t) –Ti(x,x, . . . ,xk)(t)|= . Ift,t>T˜, then

Ti(x,x, . . . ,xk)(t) –Ti(x,x, . . . ,xk)(t)

= 

(αi)

t

t

(t–s)αi–fi(s,xs,xs, . . . ,xks)ds

– 

(αi)

t

t

(t–s)αi–fi(s,xs,xs, . . . ,xks)ds

≤(t–t)–γi+ (t–t)–γi≤.

Now, lett<t<T˜<t. Since

Ti(x,x, . . . ,xk)(t) –Ti(x,x, . . . ,xk)(t)

≤Ti(x,x, . . . ,xk)(t) –Ti(x,x, . . . ,xk)

_˜

T

+Ti(x,x, . . . ,xk)

_˜

T–Ti(x,x, . . . ,xk)(t),

we getlimt→t|Ti(x,x, . . . ,xk)(t) –Ti(x,x, . . . ,xk)(t)|=  in all cases. This implies that

the setT(S) is equi-continuous. SinceT(S)⊂Sis uniformly bounded,T(S) is relatively compact. Now by using Theorem .,T has a ﬁxed point inSwhich is a solution of the problem (.). Sincex(t) = (x(t),x(t), . . . ,xk(t))∈S,limt→∞x(t) = . Thus,x(t) is an

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Theorem . Suppose that for each i∈ {, , . . . ,k} there existγi> ,αi∈(,αi)and

li∈L



αi₍_J_,R+₎_{such that}  (αi)

t t(t–s)

αi–_l

i(s)(s–t)–γids≤(t–t)–γi and

φi(t)

( –αi)

(t–t)–αi₊_f

i(t,xt,xt, . . . ,xkt)

≤li(t)xit

for all i= , , . . . ,k,t∈J and xi∈C([t–r,∞),Rn).Then the problem(.)has at least one

attractive solution(x,x, . . . ,xk)such that xi∈C([t–r,∞),Rn)for all i= , , . . . ,k.

Proof It is suﬃcient we consider the set

S=

(x,x, . . . ,xk) :xi∈C

[t–r,∞),Rn

,xit ≤(t–t) –γi

,

where˜tis a constant. By using a similar techniques and proof in Theorem ., one can show that T(S)⊂S, T is continuous andT(S) is relatively compact. Now by using Theorem ., T has a ﬁxed point inS which is a solution of the problem (.). Since

x(t) = (x(t),x(t), . . . ,xk(t))∈S,limt→∞x(t) = . Thus,x(t) is an attractive solution for

the problem (.).

Theorem . Suppose that for each i∈ {, , . . . ,k}there existsβi∈(αi, )such that

φi(t)

( –αi)

(t–t)–αi₊_f

i(t,xt,xt, . . . ,xkt)

≤( +αi–βi)

( –βi)

(t–t)–βi

for all t∈J.Then the problem(.)has at least one attractive solution(x,x, . . . ,xk)such

that xi∈C([t–r,∞),Rn)for all i= , , . . . ,k.

Proof It is suﬃcient we consider the set

S=

(x,x, . . . ,xk) :xi∈C

[t–r,∞),Rn

,xi(t)≤(t–t)βi–αi

,

where˜tis a constant. By using a similar techniques and proof in Theorem ., one can show that T(S)⊂S,T is continuous and T(S) is relatively compact. Now, by using Theorem ., T has a ﬁxed point inS which is a solution of the problem (.). Since

x(t) = (x(t),x(t), . . . ,xk(t))∈S,limt→∞x(t) = . Thus,x(t) is an attractive solution for

the problem (.).

Here, we are going to investigate global attractivity of solutions of the problem (.). We assume thatgi(t,x(t),x(t), . . . ,xk(t)) is Lebesgue measurable with respect toton [t,∞) and there exists a constantα_i_∈(,αi) such thatgi∈L



α_i_₍_J_×_Rn_×_Rn_{× · · · ×}_Rn_,_Rn_{) and}

gi(t,x(t),x(t), . . . ,xk(t)) is continuous with respect toxjon [t,∞) for alli,j= , , . . . ,k.

Note that the problem (.) is equivalent to the system of equations

xi(t) =

x_i

(αi)

(t–t)αi–+  (αi)

t t

(t–s)αi–gi

s,x(s),x(s), . . . ,xk(s)

(7)

for allt>tandi= , , . . . ,k. Deﬁne the operatorT:Xk→Xkby

T(x,x, . . . ,xk)(t) =

⎛ ⎜ ⎜ ⎜ ⎜ ⎝

T(x,x, . . . ,xk)(t)

T(x,x, . . . ,xk)(t)

.. .

Tk(x,x, . . . ,xk)(t)

⎞ ⎟ ⎟ ⎟ ⎟ ⎠,

where

Ti(x,x, . . . ,xk)(t) =

x

i

(αi)

(t–t)αi–+  (αi)

t t

s,x(s),x(s), . . . ,xk(s)

ds

for alli= , , . . . ,k. Now, deﬁne

A(x,x, . . . ,xk)(t) =

⎛ ⎜ ⎜ ⎜ ⎜ ⎝

A(x,x, . . . ,xk)(t)

A(x,x, . . . ,xk)(t)

.. .

Ak(x,x, . . . ,xk)(t)

⎞ ⎟ ⎟ ⎟ ⎟ ⎠,

whereAi(x,x, . . . ,xk)(t) = x_i

(αi)(t–t)

αi–_{for all}_i_{= , , . . . ,}_k_{. Finally, deﬁne}

B(x,x, . . . ,xk)(t) =

⎛ ⎜ ⎜ ⎜ ⎜ ⎝

B(x,x, . . . ,xk)(t)

B(x,x, . . . ,xk)(t)

.. .

Bk(x,x, . . . ,xk)(t)

⎞ ⎟ ⎟ ⎟ ⎟ ⎠,

whereBi(x,x, . . . ,xk)(t) =(αi) t

t(t–s) αi–_g

i(s,x(s),x(s), . . . ,xk(s))dsfor alli= , , . . . ,k.

It is easy to check that (x(t),x(t), . . . ,xk(t)) is a solution of the problem (.) if and only if

it is a ﬁxed point of the operatorT. Note thatAis a contraction with constant .

Theorem . Suppose that for each i∈ {, , . . . ,k}there existαi<βi< and Mi≥such

that|gi(t,x(t),x(t), . . . ,xk(t))| ≤Mi(t–t)–β

i_{for all t}∈_{J and x}_{, . . . ,}_x_k∈_C₍₍_t_,∞_),Rn_). Then the zero solution of the problem(.)is globally attractive.

Proof Consider the set

S_=(x,x, . . . ,xk) :xi∈C

(t,∞),Rn,xi(t)≤(t–t)–γ

i

for alli= , , . . . ,kandt≥t+T˜

,

whereγ_i_=_(β_i_–αi) andT˜is chosen such that

|x_i|

(αi)T˜

 (αi–)

 +

Mi(–βi) (+αi–β_i_)T˜

–_(β_i_–αi)

 ≤ for

alli= , , . . . ,k. First, we show thatBmapsS_intoS_. It is easy to check thatS_is a closed, bounded, and convex subset ofRn_×_Rn_{× · · · ×}_Rn_{. Note that}

Bi(y,y, . . . ,yk)(t)

≤ 

(αi)

t t

(8)

≤ 

(αi)

t t

(t–s)αi–Mi(s–t)–β

i_ds

≤ Mi( –βi) ( +αi–βi)

(t–t)–(βi–αi)

and Mi(–βi) (+αi–βi)(t–t)

–_(β_i_–αi)_≤ Mi(–βi) (+αi–βi) ˜ T–

 (βi–αi)

 ≤ for alli= , , . . . ,kandt≥t+T˜. Thus,

Bi(y,y, . . . ,yk)(t)≤

Mi( –βi) ( +αi–βi)

(t–t)–(βi–αi)

(t–t)–(βi–αi)≤₍_t_–_t_)–γi

for alli= , , . . . ,kandt≥t+T˜. Hence,B(S)⊂S. Now, we show thatBis continuous on [t+T˜,∞). Let (ym,ym, . . . ,ymk), (y,y, . . . ,yk)∈S for allm≥ andlimm→∞|ymi (t) –

yi(t)|= . Then, one can getlimm→∞gi(t,ym(t),ym(t), . . . ,ymk(t)) =gi(t,y(t),y(t), . . . ,yk(t))

for allt≥t+T˜. Let>  be given. ChooseT˜ >t+T˜such that Mi

(–β_i_)

(+αi–β_i_)(T˜–t)

–(β_i_–αi)_<

 for allt>T˜. Letνi=

αi–

–α_i_ fori= , , . . . ,k. Then, we have

Bi

ym_,ym_, . . . ,ym_k(t) –Bi(y,y, . . . ,yk)(t)

≤ 

(αi)

t t

s,ym_(s),ym_(s), . . . ,ym_k(s)–gi

s,y(s),y(s), . . . ,yk(s)ds

≤ 

(αi)

t

t

(t–s)αi–

 –α_i_

ds

–α_i_

×

t

t

gi

s,ym_(s),ym_(s), . . . ,ym_k(s)–gi

s,y(s),y(s), . . . ,yk(s)



α_i__ds α_i_

≤ 

(αi)



 +ν_i_(t–t) +ν_i_

–α_i_

×

T˜

t

gi

s,ym_(s),ym_(s), . . . ,ym_k(s)–gi

s,y(s),y(s), . . . ,yk(s)



α_i__ds α_i_

≤ 

(αi)



 +ν_i_(T˜ –t) +ν_i_

–α_i_

(T˜ –t)αi

× sup

t≤s≤ ˜T

gi

s,ym_(s),ym_(s), . . . ,ym_k(s)–gi

s,y(s),y(s), . . . ,yk(s)

for allt+T˜≤t≤ ˜T. Hence,limm→∞|Bi(ym,ym, . . . ,ymk)(t) –Bi(y,y, . . . ,yk)(t)|=  for all

t+T˜≤t≤ ˜T. Also,

Bi

ym_,ym_, . . . ,ym_k(t) –Bi(y,y, . . . ,yk)(t)

≤ 

(αi)

t t

s,ym_(s),ym_(s), . . . ,ym_k(s)–gi

s,y(s),y(s), . . . ,yk(s)ds

≤ 

(αi)

t t

s,ym_(s),ym_(s), . . . ,ym_k(s)+gi

s,y(s),y(s), . . . ,yk(s)ds

≤ 

(αi)

t t

(t–s)αi–Mi(s–t)–β

(9)

≤ Mi( –βi) ( +αi–βi)

(t–t)–(βi–αi)

≤ Mi( –βi) ( +αi–βi)

(T˜ –t)–(β

i–αi)≤

for allt>T˜. Thus,lim_m_→∞|Bi(ym,ym, . . . ,ymk)(t) –Bi(y,y, . . . ,yk)(t)|=  for allt≥t+T˜. This implies thatBiis continuous on [t+T˜,∞) fori= , , . . . ,kand soBis continuous on [t+T˜,∞). Now, we show thatB(S) is equi-continuous. Let >  be given. Since limt→∞(t–t)–γi_{= , there exists}_T˜_>_t_₊_T˜_ _{such that (}_t_–_t_)–γi _<

 fort>T˜. Let

t,t≥t+T˜andt>t. Ift,t∈[t+T˜,T˜], then we have Bi(y,y, . . . ,yk)(t) –Bi(y,y, . . . ,yk)(t)

= 

(αi)

t

t

(t–s)αi–gi

s,y(s),y(s), . . . ,yk(s)

ds

– 

(αi)

t

t

(t–s)αi–gi

s,y(s),y(s), . . . ,yk(s)

ds

≤ 

(αi)

t

t

(t–s)αi–– (t–s)αi–gi

s,y(s),y(s), . . . ,yk(s)ds

+ 

(αi)

t

t

s,y(s),y(s), . . . ,yk(s)ds

≤ 

(αi)

t

t

(t–s)αi–– (t–s)αi– 

–α_i_

ds

–α_i_

× t

t

gi

s,y(s),y(s), . . . ,yk(s)



α_i_

ds

α_i_

+ 

(αi)

t

t

(t–s) α_i– –α_i_

ds

–α_i_ t

t

gi

s,y(s),y(s), . . . ,yk(s)



α_i_

ds

α_i_

≤ 

(αi)

  +ν_i_

–α_i_

(t–t)+ν

i_{– (}_t__–_t_)+νi_{+ (}_t__–_t_)+νi–α

i

×

T˜

t

gi

s,y(s),y(s), . . . ,yk(s)



α_i_

ds

α_i_

+ 

(αi)

  +ν_i_

–α_i_

(t–t)+ν

i–α

i

T˜

t

gi

s,y(s),y(s), . . . ,yk(s)



α_i_

ds

α_i_

≤ 

(αi)

  +ν_i_

–α_i_ T˜ t

gi

s,y(s),y(s), . . . ,yk(s)



α_i_

ds

α_i_

(t–t)αi–α

i

and solim_t__→_t_|Bi(y,y, . . . ,yk)(t) –Bi(y,y, . . . ,yk)(t)|= . Ift,t>T˜, then Bi(y,y, . . . ,yk)(t) –Bi(y,y, . . . ,yk)(t)

≤ 

(αi)

t

t

s,y(s),y(s), . . . ,yk(s)ds

+ 

(αi)

t

t

(t–s)αi–gi

s,y(s),y(s), . . . ,yk(s)ds

≤(t–t)–γ

(10)

Ift+T˜≤t<T˜<t, then we have

Bi(y,y, . . . ,yk)(t) –Bi(y,y, . . . ,yk)(t)

≤Bi(y,y, . . . ,yk)(t) –Bi(y,y, . . . ,yk)

_˜

T

+Bi(y,y, . . . ,yk)

_˜

T–Bi(y,y, . . . ,yk)(t)

and solimt→t|Bi(y,y, . . . ,yk)(t) –Bi(y,y, . . . ,yk)(t)|= . Thus,B(S) is equi-contin-uous. Since B(S_)⊂S_ is uniformly bounded, B(S_) is relatively compact. Now, sup-pose that x= (x,x, . . . ,xk)∈C((t,∞),Rn)×C((t,∞),Rn)× · · · ×C((t,∞),Rn),y=

(y,y, . . . ,yk)∈Sandx=Ax+By. Then,

xi(t)≤Ai(x,x, . . . ,xk)(t)+Bi(y,y, . . . ,yk)(t)

≤ |xi|

(αi)

(t–t)αi–+  (αi)

t t

s,y(s),y(s), . . . ,yk(s)ds

≤ |xi|

(αi)

(t–t)αi–+ Mi( –β

i) ( +αi–βi)

(t–t)–(βi–αi)

for alli= , , . . . ,k. Since  <αi<βi<  fori= , , . . . ,k, we get

|x

i|

(αi)

(t–t)(αi–)₊ Mi( –β

i) ( +αi–βi)

(t–t)–(βi–αi)

≤ |xi|

(αi)

˜ T  (αi–)

 +

Mi( –βi) ( +αi–βi)

˜ T–

 (βi–αi)

 ≤.

Thus, |xi(t)| ≤[

|x_i|

(αi)(t–t)



(αi–)+ Mi(–β

i) (+αi–β_i_)(t–t)

–_(β_i_–αi)_](_t_–_t )–γ

i ≤(t–t_)–γi for

allt≥t+T˜ andi= , , . . . ,k. This implies thatx(t) = (x(t),x(t), . . . ,xk(t))∈S for all

t≥t+T˜. Therefore, by using Theorem .Thas a ﬁxed point inSwhich is a solution of the problem (.). Since all elements of the setS_tend to  ast→ ∞, the zero solution

of the problem (.) is globally attractive.

Theorem . Suppose that for each i∈ {, , . . . ,k} there existαi<βi < ( +αi)and

li≥such that|gi(t,x(t),x(t), . . . ,xk(t))| ≤li(t–t)–β

i|x_i₍_t₎|_{for all t}∈_{J and x}_{, . . . ,}_x_k∈ C((t,∞),Rn_)._{Then the zero solution of the problem}_(.)_{is globally attractive}_.

Proof It is suﬃcient to consider the set

S_=(x,x, . . . ,xk) :xi∈C

(t,∞),Rn,xi(t)≤(t–t)–γ

i

for alli= , , . . . ,kandt≥t+T˜

,

whereγ_i_=_( –αi) andT˜is chosen such that |

x_i|

(αi)T˜

 (αi–)

 +

li(–βi–γi) (+αi–β_i_–γ_i_)T˜

(11)

· · · ×C((t,∞),Rn_),_y_{= (}_y_,_y_{, . . . ,}_y

k)∈Sandx=Ax+By. Then, xi(t)≤Ai(x,x, . . . ,xk)(t)+Bi(y,y, . . . ,yk)(t)

≤ |xi|

(αi)

(t–t)αi–+  (αi)

t t

(t–s)αi–_g

i

s,y(s),y(s), . . . ,yk(s)ds

≤ |xi|

(αi)

(t–t)αi–+  (αi)

t t

(t–s)αi–_l

i(s–t)–β

i_y

i(s)ds

≤ |xi|

(αi)

(t–t)αi–+  (αi)

t t

(t–s)αi–li(s–t)–β

i–γi_ds

≤ |xi|

(αi)

(t–t)αi–+ li( –β

i–γi) ( +αi–βi–γi)

(t–t)–(βi–γi–αi)

for alli= , , . . . ,k. Since  <αi<βi<( +αi) <  fori= , , . . . ,k, we get

|x

i|

(αi)

(t–t)



(αi–)₊ li( –β

i–γi) ( +αi–βi–γi)

(t–t)–(β

i–αi)

≤ |xi|

(αi)

˜ T  (αi–)

 +

li( –βi–γi) ( +αi–βi–γi)

˜ T–(βi–αi)

 ≤.

Thus,|xi(t)| ≤[

|x_i|

(αi)(t–t)



(αi–)+ li(–β

i–γi) (+αi–β_i_–γ_i_)(t–t)

–(β_i_–αi)_](_t_–_t )–γ

i ≤(t–t_)–γi for

allt≥t+T˜andi= , , . . . ,k. This implies thatx(t) = (x(t),x(t), . . . ,xk(t))∈S, fort≥

t+T˜. Since all elements of the setStend to  ast→ ∞, the zero solution of the problem

(.) is globally attractive.

4 Examples

Here, we give an example to illustrate our results.

Example . Consider the -dimensional system of fractional functional diﬀerential

equations

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

c_D__x_(_t_{) =} () (_)(t+ )

–  sin

₍_x (t–))

+(x(t–)) ×

x(t–)

+|x(t–)|, t> ,

c_D__x

(t) =

(_)

()

(t+ )

–  cos

₍_x (t–))

+sin(x(t–))+|x(t–)|, t> ,

c_D_x_(_t_{) =}(  )

√

π (t+ ) –

 (x(t–)) 

+(x(t–))+|x(t–)|, t> , xi(t) =t, i= , , ,t∈[–, ].

Deﬁne the maps

f(t,xt,xt,xt) = (_)

( )

(t+ )– sin

₍_x_(_t_{– ))}

 + (x(t– )) ×

x(t– )  +|x(t– )|,

f(t,xt,xt,xt) = (_)

(_)

t+ 

–

 _cos₍_x_(_t_{– ))}

 +sin(x(t– )) +|x(t– )| ,

f(t,xt,xt,xt) = (_)

√

π (t+ )

–

 (x(t– ))



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and put m(t) = (

 ) (_)(t+ )

–

_,_m_(_t_{) =} (  ) (_)(t+

 )

–

 _and_m_(_t_{) =} (  )

√

π (t+ ) –

 _{. It is easy}

m(t). Since

 (α)

t t

(t–s)α–_m_(_s₎_ds₌ 

(_) t



(t–s)– ×(

 ) (_)(s+ )

–__ds

≤ (  ) (_)(_) t 

(t–s)–_s–_ds₌_t–_,

 (α)

t t

(t–s)α–_m

(s)ds=  (

) t



(t–s)–×(

 ) (

)

s+  –  ds ≤ (  ) (_)(_) t 

(t–s)–_s–_ds₌_t–

and

 (α)

t t

(t–s)α–_m_(_s₎_ds₌ 

(_) t



(t–s)– ×(

 )

√

π (s+ ) –__ds

≤ () (_)√π

t



(t–s)–_s–_ds₌_t–_,

we get| 

(α)

t

(t–s) –__f

(s,xs,xs,xs)ds| ≤t–

 ,| 

(α)

t

(t–s) –__f

(s,xs,xs,xs)ds| ≤t–

 

and|₍_α

)

t

(t–s)

–__f_(_s_,_x_s,_x

s,xs)ds| ≤t–



_{. Now, let}_α_₌

,α= 

 andα=  . Then,

∞

t

f(t,xt,xt,xt)



α_ _dt≤

∞

t

m(t)



α__dt

= ∞  (_) ( )

(t+ )–

 dt =   (_) (_)  , ∞ t

f(t,xt,xt,xt)



α _dt≤

∞

t

m(t)



α _dt

= ∞  (_) ( )

t+  –  dt =   (_) (_)  and ∞ t

f(t,xt,xt,xt)



α_ _dt≤

∞

t

m(t)



α_ _dt

= ∞  (_) √

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Thus, all conditions of Theorem . hold and so this system of fractional functional dif-ferential equations has an attractive solution.

Example . Let  <αi< ,Mi> ,αi<βi<  andxi be a constant fori= , , . Consider

the -dimensional system of fractional diﬀerential equations

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Dα_x_(_t_{) =}Mx(t)sin(x(t))



+|x(t)|+|x(t)|(t–a)

–β_ _, _t_>_a_,

Dα_x_(_t_{) =} Mt(x(t))

(+t_)(+(_x

(t))+(x(t)))(t–a)

–β_ _, _t_>_a_,

Dα_x_(_t_{) =} Mcos(x(t))(x(t))

+(x(t))+|x(t)| (t–a)

–β_ _, _t_>_a_,

[Dαi–_x

i(t)]t=a=xi, i= , , .

Deﬁne the maps

g

t,x(t),x(t), . . . ,xk(t)

=Mx(t)sin ₍_x_(_t₎₎



+|x(t)|+|x(t)|

(t–a)–β _,

g

t,x(t),x(t), . . . ,xk(t)

= Mt

₍_x_(_t₎₎

( + t_{)( + (}_x

(t))+ (x(t)))

(t–a)–β

and

g

t,x(t),x(t), . . . ,xk(t)

=Mcos

₍_x_(_t₎₎₍_x_(_t₎₎

 + (x(t))₊|x_(_t₎|(t–a) –β_ _.

Thus, one can check that all conditions of Theorem . hold and so this system of frac-tional diﬀerential equations has a globally attractive solution.

5 Conclusions

Investigating the attractive solutions of the problems is an interesting topic within the fractional calculus. In this manuscript, we focus on the attractivity of solutions for two

k-dimensional systems of fractional differential equations. Two illustrative examples show the applicability of the proposed methods. The techniques of the reported results can be applied for investigating the attractivity and global attractivity of solutions of different systems of (singular) fractional differential equations. Also, it is an interesting issue to investigate the attractivity and global attractivity of solutions of some systems of fractional differential inclusions.

Competing interests

The authors declare that they have no competing interests regarding the publication of this article.

Authors’ contributions

All authors read and approved the ﬁnal version of this manuscript.

Author details

1_{Department of Mathematics, Cankaya University, Ogretmenler Cad. 14, Balgat, Ankara, 06530, Turkey.}2_{Institute of Space}

Sciences, Magurele, Bucharest, Romania. 3_{Department of Chemical and Materials Engineering, Faculty of Engineering,}

King Abdulaziz University, P.O. Box 80204, Jeddah, 21589, Saudi Arabia. 4_{Department of Mathematics, Azarbaijan Shahid}

Madani University, Azarshahr, Tabriz, Iran.

Acknowledgements

Research of the second and third authors was supported by Azarbaijan Shahid Madani University. Also, the authors express their gratitude to the referees for their helpful suggestions which improved the ﬁnal version of this paper.

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References

1. Ahmad, B: Existence of solutions for irregular boundary value problems of nonlinear fractional diﬀerential equations. Appl. Math. Lett.23, 390-394 (2010)

2. Ahmad, B, Nieto, JJ: Existence results for a coupled system of nonlinear fractional diﬀerential equations with three-point boundary conditions. Comput. Math. Appl.58, 1838-1843 (2009)

3. Ahmad, B, Sivasundaram, S: Existence results for nonlinear impulsive hybrid boundary value problems involving fractional diﬀerential equations. Nonlinear Anal.3, 251-258 (2009)

4. Ahmad, B, Sivasundaram, S: Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Anal.4, 134-141 (2010)

5. Agarwal, RP, Andrade, B, Cuevas, C: Weighted pseudo-almost periodic solutions of a class of semilinear fractional diﬀerential equations. Nonlinear Anal.11, 3532-3554 (2010)

6. Agarwal, RP, Benchohra, M, Hamani, S: A survey on existence results for boundary value problems of nonlinear fractional diﬀerential equations and inclusions. Acta Appl. Math.109, 973-1033 (2010)

7. Agarwal, RP, Lakshmikantham, V, Nieto, JJ: On the concept of solution for fractional diﬀerential equations with uncertainty. Nonlinear Anal.72, 2859-2862 (2010)

8. Balachandran, K, Kiruthika, S: Existence of solutions of abstract fractional impulsive semilinear evolution equations. Electron. J. Qual. Theory Diﬀer. Equ.2010, 4 (2010)

9. Bai, C: Existence of positive solutions for boundary value problems of fractional functional diﬀerential equations. Electron. J. Qual. Theory Diﬀer. Equ.2010, 30 (2010)

10. Baleanu, D, Agarwal, RP, Mohammadi, H, Rezapour, S: Some existence results for a nonlinear fractional diﬀerential equation on partially ordered Banach spaces. Bound. Value Probl.2013, 112 (2013)

11. Baleanu, D, Mohammadi, H, Rezapour, S: Positive solutions of an initial value problem for nonlinear fractional diﬀerential equations. Abstr. Appl. Anal.2012Article ID 837437 (2012)

12. Baleanu, D, Mohammadi, H, Rezapour, S: Some existence results on nonlinear fractional diﬀerential equations. Philos. Trans. R. Soc. Lond. A371, 20120144 (2013)

13. Baleanu, D, Mohammadi, H, Rezapour, S: On a nonlinear fractional diﬀerential equation on partially ordered metric spaces. Adv. Diﬀer. Equ.2013, 83 (2013)

14. Burton, TA: A ﬁxed point theorem of Kranoselskii. Appl. Math. Lett.11, 85-88 (1998)

15. Chen, F, Nieto, JJ, Zhou, Y: Global attractivity for nonlinear fractional diﬀerential equations. Nonlinear Anal.13, 287-298 (2012)

16. Chen, F, Zhou, Y: Attractivity of fractional functional diﬀerential equations. Comput. Math. Appl.62, 1359-1369 (2011) 17. Hale, JK: Theory of Functional Diﬀerential Equations. Springer, Berlin (1977)

18. Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Diﬀerential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

19. Kranoselskii, MA: Topological Method in the Theory of Nonlinear Integral Equations. Macmillan Co., New York (1964) 20. Tian, Y, Bai, Z: Existence results for the three-point impulsive boundary value problem involving fractional diﬀerential

equations. Comput. Math. Appl.59, 2601-2609 (2010)

21. Wang, G, Ahmad, B, Zhang, L: Impulsive anti-periodic boundary value problem for nonlinear diﬀerential equations of fractional order. Nonlinear Anal.74, 792-804 (2011)

22. Zhou, Y, Jiao, F, Li, J: Existence and uniqueness for fractional neutral diﬀerential equations with inﬁnite delay. Nonlinear Anal.71, 3249-3256 (2009)

23. Pinto, C, Machado, JAT: Forced van der Pol oscillator of complex order. In: ENOC 2011, Rome, Italy, 24-29 July (2011) 24. Rihan, FA: Numerical modeling of fractional order biological systems. Abstr. Appl. Anal.2013, Article ID 816803 (2013) 25. Suchorsky, MK, Rand, RH: A pair of van der Pol oscillators coupled by fractional derivatives. Nonlinear Dyn.69,

313-324 (2012)

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