Wardowski Advances in Difference Equations (2015) 2015:167 DOI 10.1186/s13662-015-0504-9

RESEARCH

Open Access

Monotone iterative procedure and systems of a finite number of nonlinear fractional differential equations Dariusz Wardowski* *

Correspondence: [email protected] Department of Nonlinear Analysis, Faculty of Mathematics and Computer Science, University of Łód´z, Banacha 22, Łód´z, 90-238, Poland

Abstract The aim of the paper is to present a nontrivial and natural extension of the comparison result and the monotone iterative procedure based on upper and lower solutions, which were recently established in (Wang et al. in Appl. Math. Lett. 25:1019-1024, 2012), to the case of any finite number of nonlinear fractional differential equations. MSC: 26A33; 34A08; 34B15 Keywords: monotone iterative procedure; system of fractional differential equations; upper and lower solution

1 Introduction Fractional derivatives and integrals are used for a better description of material properties. In the literature we can find many interesting papers concerning this theory; see e.g., [–]. The study of systems involving fractional differential/integral equations is also important as such systems occur in various problems of applied nature; for example, see [– ]. Some basic theory of fractional differential equations involving the Riemann-Liouville differential operator can be found in [–]. In the paper we consider the following system of nonlinear fractional differential equations: ⎧ ⎪ Dα u (t) = f (t, u (t), u (t), . . . , un (t)), t ∈ (, T], ⎪ ⎪ ⎪ α ⎪ ⎪ ⎨ D u (t) = f (t, u (t), u (t), . . . , un (t)), t ∈ (, T], ..., ⎪ ⎪ ⎪ Dα un (t) = fn (t, u (t), u (t), . . . , un (t)), t ∈ (, T], ⎪ ⎪ ⎪ ⎩ t –α u (t)| = x , t –α u (t)|t= = x , ...,  t= 

(.) t –α un (t)|t= = xn ,

where Dα is the standard Riemann-Liouville fractional derivative of order α,  ≤ α ≤ , T > , f i ∈ C([, T] × Rn , R),  ≤ i ≤ n, and x , . . . , xn ∈ R satisfy n 

xi – x ≥ .

(.)

i=

© 2015 Wardowski. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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We investigate system (.) with respect to the existence of a solution via the method of upper and lower solutions. There is also presented the concept of an iterative procedure, where the appropriately constructed sequences are convergent to the extreme solution. The paper is a continuation of the investigations in [] of Wang et al., where the authors examined system (.) in the case n = . After proving the main results we state, for convenience of the reader, the introduced techniques in the case of three nonlinear fractional differential equations and also present a concrete example.

2 Preliminaries First, let us recall the needed notations and crucial results which will be needed in the next sections of the article. Denote by C–α ([, T]) the family of all functions u ∈ C((, T]) such that t –α u ∈ C([, T]). A basic theorem concerning the existence of the result and its uniqueness for the linear fractional equation is as follows. Lemma . ([]) Let  < α ≤ , M ∈ R, and σ ∈ C–α ([, T]) be fixed. Then the linear initial value problem 

Dα u(t) = σ (t) – Mu(t), t –α u(t)|t= = u ,

t ∈ (, T],

i, j ∈ N,

(.)

has a unique solution, given by the following formula:  u(t) = (α)u t α– Eα,α –Mt α +



t

 (t – s)α– Eα,α –M(t – s)α σ (s) ds,



where Eα,β is the Mittag-Leffler function, i.e. the function of the form Eα,β (z) =

∞  k=

zk , (αk + β)

α, β > , z ∈ R.

The comparison result for the initial value problem (.) due to Wang et al. is as follows. Lemma . ([]) Let  < α ≤  and M ∈ R be given. Then, if w ∈ C–α ([, T]) satisfies 

Dα w(t) + Mw(t) ≥ , t –α w(t)|t= ≥ ,

t ∈ (, T],

then w(t) ≥  for all t ∈ (, T]. The same authors also proved the following result, which will be needed in the sequel. Lemma . ([]) Let  < α ≤ , M ∈ R, and N ≥  be given. Assume that u, v ∈ C–α ([, T]) satisfy ⎧ α ⎪ ⎨ D u(t) ≥ –Mu(t) + Nv(t), t ∈ (, T], Dα v(t) ≥ –Mv(t) + Nu(t), t ∈ (, T], ⎪ ⎩ –α t –α v(t)|t= ≥ . t u(t)|t= ≥ , Then u(t) ≥ , v(t) ≥  for all t ∈ (, T].

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3 The results In the sequel we will use the following notation:  δij =

 –

if i = j, if i = j,

i, j ∈ N.

C–α ([, T])n denotes C–α ([, T]) × C–α ([, T]) × · · · × C–α ([, T]) (n times). Lemma . Let  < α ≤  be fixed, Mi ∈ R, σi ∈ C–α ([, T]), i = , , . . . , n. Then the linear problem of n equations ⎧ α  D u (t) = σ (t) – M u (t) – ni,j= Mj δji ui (t), t ∈ (, T], ⎪ ⎪ ⎪ ⎨ Dα u (t) = σ (t) + (M – n M )u (t) j j j i j i= n ⎪ ( u (t) – u (t)), t ∈ (, T],  ≤ j ≤ n, – M j i j ⎪ i= ⎪ ⎩ –α i t ui (t)|t= = x ,  ≤ i ≤ n,

(.)

has a unique solution in C–α ([, T])n . Proof First observe that for any p , p , . . . , pn ∈ C–α ([, T]) the system ⎧ u + u + · · · + un = p , ⎪ ⎪ ⎪ ⎨u – u + ··· + u = p ,   n  ⎪ ..., ⎪ ⎪ ⎩ u + u + · · · – un = pn

(.)

has exactly one solution, which is a consequence of the fact that ⎡

  ⎢ ⎢ – ⎢ ⎢ det ⎢  ⎢. . ⎢. . ⎣. .  

  – .. . 

··· ··· ··· .. . ···

⎤  ⎥ ⎥ ⎥ ⎥ = (–)n– = . ⎥ ⎥ .. ⎥ . ⎦ – n×n

Next, observe that system (.) can be transformed to system (.), where p , p , . . . , pn solve the following n problems:  

.. . 

Dα p (t) = (σ (t) + σ (t) + · · · + σn (t)) – (M + M + · · · + Mn )p (t), t –α p (t)|t= = x + x + · · · + xn , Dα p (t) = (σ (t) – σ (t) + · · · + σn (t)) – (M – M + · · · + Mn )p (t), t –α p (t)|t= = x – x + · · · + xn ,

Dα pn (t) = (σ (t) + σ (t) + · · · – σn (t)) – (M + M + · · · – Mn )pn (t), t –α pn (t)|t= = x + x + · · · – xn .

Finally, observe that the solutions of the above equations are unique due to Lemma ., which ends the proof. 

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Now we can state and proof the comparison result for system (.). Theorem . Let  < α ≤ , M ∈ R, M , . . . , Mn ≥ , and let u , . . . , un ∈ C–α ([, T]) satisfy ⎧ α  D u (t) ≥ –M u (t) + ni,j= Mj δji ui (t), t ∈ (, T], ⎪ ⎪ ⎪ ⎨ Dα u (t) ≥ –M u (t) + (n M – M )u (t) s  s i s s i= n ⎪ ( u (t) – u (t)),  ≤ s ≤ n, t ∈ (, T], + M s s ⎪ i= i ⎪ ⎩ –α t us (t)|t= ≥ ,  ≤ s ≤ n.

(.)

Then n 

ui (t) ≥ ,

t ∈ (, T],

(.)

i=

us (t) ≥ , –us (t) +

n 

t ∈ (, T],  ≤ s ≤ n,

(.)

ui (t) ≥ ,

(.)

t ∈ (, T],  ≤ s ≤ n.

i=

Proof Put r(t) = Dα r(t) =

n

s= us (t).

n 

Using (.) we obtain

Dα us (t)

s=

≥ –M u (t) +

n 

Mj δji ui (t) – M

n  n 

us (t) – 

s=

i,j=

+

n 

Mi us (t) +

s= i=

n  n 

n 

Ms us (t)

s=

Ms ui (t)

s= i=

= –M r(t) +

n  

n n   Mj δji ui (t) + Mi uj (t) –  Ms us (t) + Ms r(t). s=

i,j=

s=

Observe that n  

Mj δji ui (t) + Mi uj (t) = .

(.)

i,j=,i=j

Hence, we obtain  D r(t) ≥ – M – α

n 

 Ms r(t) +

s=

+

n  

n   Mj δji ui (t) + Mi uj (t) i,j=,i=j

n  Mj δji ui (t) + Mi uj (t) –  Ms us (t)

i,j=,i=j



= – M –  = – M –

n  s= n  s=

 Ms r(t) +   Ms r(t).

s= n  i=

Mi ui (t) – 

n 

Ms us (t)

s=

(.)

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Moreover, observe that t –α r(t) =

n 

t –α us (t) ≥ .

(.)

s=

Applying (.) and (.) to Lemma . we get (.). Now, consider any  ≤ s ≤ n and denote rs (t) =

n 

t ∈ (, T].

ui (t) – us (t),

i=

By (.) we have Dα rs (t) =

n 

Dα ui (t) – Dα us (t) = Dα u (t) +

i=

 n n   i=

+

Dα ui (t) – Dα us (t)

i= n 

≥ –M u (t) +

+

n 

i,j=

Mj δji ui (t) –







Mj – Mi ui (t) –

Mi

n 

i=



n  j=

uj (t) – ui (t) – Ms

j=

= –M u (t) +

M ui (t) + M us (t)

i=

j=

n 

n 

 Mj – Ms us (t)



n 

 uj (t) – us (t)

j=

n n    Mj δji ui (t) + Mj ui (t) – M ui (t) + M us (t) + Ms us (t) i,j=

–

n 

i=

Mi ui (t) – us (t)

n 

i=

j=

Mj +

n  n 

Mi uj (t) – Ms rs (t).

i= j=

Again, using (.), we obtain Dα rs (t) ≥ –M

n 

ui (t) + M us (t) + Ms us (t) – us (t)

i=

+

n  i=

= – M –

Mj

j=

Mi rs (t) + us (t)



n 

n 

Mi – Ms rs (t)

i=

n 



Mi + Ms rs (t) + Ms us (t).

(.)

i=

Moreover, observe that (.) implies  D us (t) ≥ – M – α

n 

 Mi + Ms us (t) + Ms rs (t).

(.)

i=

Finally, note that (.) and (.) applied to Lemma . give (.) and (.). Now, we are in a position to enunciate the main result.



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Theorem . Suppose that there exist u , u , . . . , un ∈ C–α ([, T]), u ≤ ing ⎧ α  D u (t) ≤ f (t, u (t), u (t), . . . , un (t)), ⎪ ⎪ ⎪ ⎨ Dα us (t) ≥ f (t, u (t), u (t), . . . , un (t)), s     ⎪ t –α u (t)|t= ≤ x , ⎪ ⎪ ⎩ –α s t u (t)|t= ≥ xs ,  ≤ s ≤ n,

n

i i= u ,

t ∈ (, T], t ∈ (, T],  ≤ s ≤ n,

satisfy-

(.)

and there exist M ∈ R, M , . . . , Mn >  such that (i) f (t, α , . . . , αn ) – f (t, β , . . . , βn ) ≥ –M (α – β ) –

n 

Mj δji (αi – βi ),

(.)

i,j=

(ii) fs (t, α , . . . , αn ) – fs (t, β , . . . , βn )     n n   ≥ –M + Mi – Ms (αs – βs ) – Ms α – β + αs – βs – (αi – βi ) , i=

i=

where αi , βi ∈ R,  ≤ i ≤ n satisfy for all t ∈ [, T] and  ≤ s ≤ n,  u (t) –  u (t) –

n 



i= n 

 ≤ β –

ui (t) – us (t)

n 





βi – βs ≤ α  –

i=



n 

 αi – αs ≤ us (t),

i=

ui (t) – us (t) ≤ αs ≤ βs ≤ us (t),

i=

(iii) n    fs t, u (t), u (t), . . . , un (t) – f t, u (t), u (t), . . . , un (t) s=



≥ –M +

n 

 Ms

s=

n 

 s



u (t) – u (t) ,

(.)

s=

where  u –

n 





ui – us ≤ u –

n 

i=

 ui – us ≤ us ≤ us ,

 ≤ s ≤ n.

i=

Then there exists a solution (u¯  , u¯  , . . . , u¯ n ) of system (.) such that (n – )u

– (n – )

n  i=

ui

≤ u¯ ≤ 

n  i=

ui ,

u

–

n 

ui + us ≤ u¯ s ≤ us ,

 ≤ s ≤ n.

i=

Moreover, there exist iterative sequences (uk ), (uk ), . . . , (unk ) such that uik → u¯ i , k → ∞, i = , , . . . , n, uniformly on compact subsets of (, T].

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Proof Let us first consider the linear system of the form ⎧ α   D u (t) = f (t, u (t), u (t), . . . , un (t)) + M u (t) + ni,j= Mj δji ui (t) ⎪ ⎪ ⎪  ⎪ ⎪ – M u (t) – ni,j= Mj δji ui (t), t ∈ (, T], ⎪ ⎪ ⎪ ⎨ Dα us (t) = f (t, u (t), u (t), . . . , un (t)) + M us (t) + (n M – M )us (t) s   i s   i= n i  n s  s s ⎪ (u (t) + u (t) – u (t)) – M u (t) – ( M + M s  i – Ms )u (t) ⎪  i= i= ⎪  ⎪ n ⎪ ⎪ – Ms (u (t) + i= ui (t) – us (t)), t ∈ (, T],  ≤ s ≤ n, ⎪ ⎪ ⎩ –α s t u (t)|t= = xs ,  ≤ s ≤ n,

(.)

where u , u , . . . , un ∈ C–α ([, T]). Due to Lemma . there exists a system of solutions (u , u , . . . , un ) ∈ C([, T])n for system (.). Using induction we obtain the sequence (uk , uk , . . . , unk ) ∈ C([, T])n , k ∈ N, satisfying ⎧ n n i    α  ⎪ ⎪ D uk (t) = f (t, uk– (t), uk– (t), . . . , uk– (t)) + M uk– (t) + i,j= Mj δji uk– (t) ⎪  ⎪ ⎪ ⎪ – M uk (t) – ni,j= Mj δji uik (t), t ∈ (, T], ⎪ ⎪ ⎪ s α ⎪ ⎪ uk– (t), uk– (t), . . . , unk– (t)) + M usk– (t) ⎨ D uk (t) = fs (t,  + ( ni= Mi – Ms )usk (t) + Ms (uk– (t) + ni= uik (t) – usk (t)) ⎪  ⎪ ⎪ – M usk (t) – ( ni= Mi – Ms )usk– (t) ⎪ ⎪ ⎪  ⎪ ⎪ – Ms (uk (t) + ni= uik– (t) – usk– (t)), t ∈ (, T],  ≤ s ≤ n, ⎪ ⎪ ⎪ –α s ⎩ t uk (t)|t= = xs ,  ≤ s ≤ n.

(.)

Now, put p = u – u , ps = us – us ,  ≤ s ≤ n. From (.) and (.), for all t ∈ (, T], we obtain Dα p (t) = Dα u (t) – Dα u (t) n   = f t, u (t), u (t), . . . , un (t) + M u (t) + Mj δji ui (t) i,j=

– M u (t) –

n 

Mj δji ui (t) – Dα u (t)

i,j=

≥ –M p (t) +

n 

Mj δji pi (t),

i,j=

Dα ps (t) = Dα us (t) – Dα us (t)  = Dα us (t) – fs t, u (t), u (t), . . . , un (t) – M us (t)     n n   s  i s Mi – Ms u (t) – Ms u (t) + u (t) – u (t) + M us (t) –  +

i= n 

 Mi – Ms

i=

≥ –M ps (t) +



n  i=

i=

 us (t) + Ms  Mi – Ms

u (t) +

n 

 ui (t) – us (t)

i=



ps (t) + Ms

n 

 pi (t) – ps (t)

i=

t –α p (t)|t= = t –α u (t)|t= – t –α u (t)|t= ≥ x – x = , t –α ps (t)|t= = t –α us (t)|t= – t –α us (t)|t= ≥ xs – xs = ,

 ≤ s ≤ n.

for all  ≤ s ≤ n,

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Hence, using Theorem ., we have us ≤ us ,

≤s≤n

(.)

and u – u +

n   i u – ui ≥ us – us ,

 ≤ s ≤ n.

(.)

i=

Consider now q = n 

Dα q (t) =

n

i i= u

– u . Using (.) and (.) we have n 

us (t) – u (t) =

s= n 

=

s= n   fs t, u (t), u (t), . . . , un (t) + M us (t)

s=

+

–



n 

n 

s=

i=

n 

n 

us (t) +

Mi – Ms



 Ms

s=



M us (t) –

s=

–

Dα us (t) – Dα u (t)

n 

s=

i=

n 

s=



Ms

s=

n 

u (t) +

n 

u (t) +

i=



Mi – Ms us (t) 

ui (t) – us (t)

 – f t, u (t), u (t), . . . , un (t) n 

Mj δji ui (t) + M u (t) +

i,j=

=

 ui (t) – us (t)

i= n 

– M u (t) – n 

n 

Mj δji ui (t)

i,j=

  fs t, u (t), u (t), . . . , un (t) – f t, u (t), u (t), . . . , un (t)

s=



– M –

n 

Ms q (t) + M –

s=

 ≥ – M –

n 





n 

 Ms

s=



n 

 us (t) – u (t)

s=

Ms q (t).

s=

Moreover, (.) implies t

–α

q (t)|t= =

n 

t –α ui (t)|t=

– t –α u (t)|t=

i=

=

n 

xi – x ≥ .

i=

Now, from Lemma . we conclude u (t) ≤

n 

ui (t) for all t ∈ [, T].

(.)

i=

Combining (.) and (.) with (.) we obtain for all  ≤ s ≤ n the inequalities  u

–

n  i=

 ui

– us

 ≤ u

–

n  i=

 ui

– us

≤ us ≤ us .

Wardowski Advances in Difference Equations (2015) 2015:167

Page 9 of 16

Let  ≤ s ≤ n be fixed and suppose now that for some k ∈ N the following inequalities hold:  n  n      i s  i s uk– – uk– – uk– ≤ uk – uk – uk ≤ usk ≤ usk– . (.) i=

i=

Denote pk+ = uk+ – uk , psk+ = usk – usk+ ,  ≤ s ≤ n. From (.), (.), and (.) we obtain Dα pk+ (t) = Dα uk+ (t) – Dα uk (t) n   = f t, uk (t), uk (t), . . . , unk (t) + M uk (t) + Mj δji uik (t) – M uk+ (t) i,j=

–

n 

 Mj δji uik+ (t) – f t, uk– (t), uk– (t), . . . , unk– (t) – M uk– (t)

i,j=

–

n 

Mj δji uik– (t) + M uk (t) +

i,j=

n 

Mj δji uik (t)

i,j=

 ≥ –M uk (t) – uk– (t) –

n 

 Mj δji uik (t) – uik– (t) + M uk (t)

i,j=

+

n 

Mj δji uik (t) – M uk+ (t) –

i,j=

–

n 

n 

Mj δji uik+ (t) – M uk– (t)

i,j=

Mj δji uik– (t) + M uk (t) +

i,j=

n 

Mj δji uik (t)

i,j=

= –M pk+ (t) +

n 

Mj δji pik+ (t),

i,j=

Dα psk+ (t) = Dα usk (t) – Dα usk+ (t)    n   s ≥ –M + Mi – Ms uk– (t) – usk (t) – Ms uk– (t) – uk (t) i=

+ usk– (t) – usk (t) – + Ms uk– (t) +

n 





uik (t) – usk (t) – M usk (t) –

i=

 uk (t) +

n 

uk (t) + 

n 

 = –M psk+ (t) +

– M usk (t) –

uik+ (t) – usk+ (t)

n 

i=

 Mi – Ms usk+ (t)

i=

 + M usk+ (t) +

n 

 Mi – Ms usk (t)

i=

 uik (t) – usk (t)

i= n 



Mi – Ms usk– (t)

 n 



i=

+ Ms uk+ (t) +

i= n  i=



uik– (t) – usk– (t)

i=

 – Ms

   n  uik– – uik + M usk– (t) + Mi – Ms usk (t)

i=



– Ms

n  

 Mi – Ms

psk+ (t) + Ms



n  i=

 pik+ (t) – psk+ (t)

.

Wardowski Advances in Difference Equations (2015) 2015:167

Page 10 of 16

Also observe that t –α pk+ (t)|t= = t –α psk+ (t)|t= = , which, together with the above, due to Theorem ., gives usk+ ≤ usk ,

 ≤ s ≤ n,

usk – usk+ ≤

n 



(.)

uik – uik+ + uk+ – uk .

(.)

i=

Consider now qk =

n

i i= uk

 D qk (t) ≥ – M – α

n 

– uk . Using the same arguments as with q we obtain 

Ms qk (t)

s=

and t –α qk (t)|t= ≥ , which, due to Lemma ., gives uk ≤

n 

uik .

(.)

s=

Summarizing, by (.)-(.) and induction, we obtain the following inequalities describing the sequences (usk )k∈N∪{} :  u –

n 





ui – us ≤ u –

i=

n  i=

≤

· · · ≤ uk

≤

usk

–

 ui – us 

n 

 uik

– usk

i=

≤ · · · ≤ us

≤ us ,

(.)

where  ≤ s ≤ n. The inequalities (.) imply lim usk (t) = u¯ s (t),

s = , . . . , n.

k→∞

Observe that  u

–

n 

 ui

– us

≤ u¯ s ≤ us ,

s = , . . . , n.

i=

In order to show that the sequence (uk ) is convergent observe first that from (.) there exists a function x∗ such that  lim

k→∞

uk (t) –

n–  i=

 uik (t)

= x∗ (t).

Wardowski Advances in Difference Equations (2015) 2015:167

Hence, putting u¯  = x∗ +

lim



k→∞

uk (t) – u¯  (t)



n–

Page 11 of 16

u¯ s , we have

s=

 uk (t) – x∗ (t) –

= lim

k→∞

uk (t) –

k→∞

u¯ (t) + s

s=

 = lim

n– 

n– 

= lim uk (t) – k→∞

n– 

usk (t) –

s=

usk (t) – x∗ (t) +

s=



n– 



usk (t) – u¯ s (t)



s=

usk (t) – x∗ (t) +

s=

 usk (t)

s=

n– 



n– 

n–  s=

 lim usk (t) – u¯ s (t) = .

k→∞

In order to show the uniform convergence of sequences (uk ), (uk ), . . . , (unk ), observe that from (.) and from the fact that usk → u¯ s , s = , , . . . , n, we have u¯ s ≤ usk ≤ · · · ≤ us ≤ us

for all k ∈ N.

Then, the uniform convergence of sequences (usk ), s = , , . . . , n, on a compact subset of (, T] is a straightforward consequence of Dini’s theorem, which states that if a monotone sequence of continuous functions is convergent on a compact set, then it converges uniformly. Showing a uniform convergence of (uk ) requires some observations. Take any  ≤ s ≤ n and denote  hk = uk

–

n 

 uik

– usk

k ∈ N ∪ {}.

,

i=

From (.) and the convergence of (uk ), . . . , (unk ) we have  h ≤ h ≤ · · · ≤ hk ≤ u¯ – 

n 

 u¯ – u¯ . i

s

i=

Applying again Dini’s result we get the uniform convergence of (hk ) on every compact subset of (, T]. Finally note that

uk

= hk +

 n 

 uik

– usk

k ∈ N,

,

i=

and thus (uk ) is uniformly convergent on a compact subset of (, T] to u¯  as a linear combination of sequences uniformly convergent. Moreover, observe that the limit functions satisfy the properties (n – )u

– (n – )

n 

ui

≤ u¯ ≤

i=

u –

n  i=

ui + us ≤ u¯ s ≤ us ,



n 

ui ,

i=

 ≤ s ≤ n.

Wardowski Advances in Difference Equations (2015) 2015:167

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Taking k to ∞ in (.) we see that (u¯  , u¯  , . . . , u¯ n ) is a system of solutions of system (.). Also observe that from (.) we have the following relations between the limit functions:  u¯ – 

n 

 u¯ – u¯ i

≤ u¯ s ,

s

 ≤ s ≤ n,

i=



which ends the proof.

Remark . Observe that using the same methods as in the proof of Theorem . we can see that (u¯  , u¯  , . . . , u¯ n ) is an extremal solution of system (.) in the sense that if (u , . . . , un ) were any other solution such that  u

–

n 



 ui

– us

≤u – 

i=

n 



 i

s

u –u

≤ us ,

i=

u

–

n 

 ui

– us

≤ us ≤ us

i=

for any  ≤ s ≤ n, then we would have  u¯ – 

n  i=

 u¯ – u¯ i

s

 ≤u – 

n 

 i

s

u –u ,

us ≤ u¯ s ,  ≤ s ≤ n.

i=

4 The system of three fractional differential equations In order to see the nature of the iterative procedure introduced in the proof of Theorem ., we consider the case n = . Corollary . If there exist u , v , w ∈ C–α ([, T]), u ≤ v + w such that ⎧ Dα u (t) ≤ f (t, u (t), v (t), w (t)), t ∈ (, T], ⎪ ⎪ ⎪ ⎪ ⎪ Dα v (t) ≥ g(t, u (t), v (t), w (t)), t ∈ (, T], ⎪ ⎪ ⎪ ⎨ Dα w (t) ≥ h(t, u (t), v (t), w (t)), t ∈ (, T],     –α ⎪ u (t)| ≤ x , t  t=  ⎪ ⎪ ⎪ ⎪ ⎪ t –α v (t)|t= ≥ y , ⎪ ⎪ ⎩ –α t w (t)|t= ≥ z ,

(.)

and there exist M ∈ R, N, S ≥  satisfying f (t, α , α , α ) – f (t, β , β , βn ) ≥ –M(α – β ) + (–N + S)(α – β ) + (N – S)(α – β ), g(t, α , α , α ) – g(t, β , β , β ) ≥ –N(α – β ) + (–M + S)(α – β ) + N(α – β ), h(t, α , α , α ) – h(t, β , β , β ) ≥ –S(α – β ) + S(α – β ) + (–M + N)(α – β ), where αi , βi ∈ R,  ≤ i ≤  satisfy, for all t ∈ [, T], u (t) – w (t) ≤ β – β ≤ α – α ≤ v (t),

u (t) – w (t) ≤ α ≤ β ≤ v (t),

u (t) – v (t) ≤ β – β ≤ α – α ≤ w (t),

u (t) – v (t) ≤ α ≤ β ≤ w (t)

and (g + h – f )(t, u, v, w) ≥ (–M + N + S)(v + w – u),

(.)

Wardowski Advances in Difference Equations (2015) 2015:167

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where u (t) – w (t) ≤ u – w ≤ v ≤ v (t), u (t) – v (t) ≤ u – v ≤ w ≤ w (t). Then there exists a solution 

u∗ , v∗ , w∗ ∈ [u – v – w , v + w ] × [u – w , v ] × [u – v , w ]

of (.) and the sequences (un ) ⊆ [u – v – w , v + w ], (vn ) ⊆ [u – w , v ], (wn ) ⊆ [u – v , w ] such that un → u∗ , vn → v∗ , wn → w∗ uniformly on compact subsets of (, T]. Moreover, the following inequalities hold: u – v ≤ u – v ≤ · · · ≤ un – vn ≤ · · · ≤ u∗ – v∗ ≤ w∗ ≤ · · · ≤ wn ≤ · · · ≤ w ≤ w , u – w ≤ u – w ≤ · · · ≤ un – wn ≤ · · · ≤ u∗ – w∗ ≤ v∗ ≤ · · · ≤ vn ≤ · · · ≤ v ≤ v .

4.1 Example Consider the nonlinear problem of the form ⎧ . D u(t) = (.)– v(t) – (.)– w(t) + (v(t) – t) + (t – w(t) + u(t)) , ⎪ ⎪ ⎪ ⎨ D. v(t) = (.)– v(t) + (v(t) – t) + (t – w(t) + u(t)) , ⎪ D. w(t) = –(.)– u(t) + (.)– v(t) + (t – w(t) + u(t)) , ⎪ ⎪ ⎩ . t u(t)|t= = t . v(t)|t= = t . w(t)|t= = , where t ∈ [, ]. Taking f (t, u, v, w) = (.)– v – (.)– w + (v – t) + (t – w + u) , g(t, u, v, w) = (.)– v + (v – t) + (t – w + u) , h(t, u, v, w) = –(.)– u + (.)– v + (t – w + u) and u (t) = ,

v (t) = w (t) = t,

t ∈ [, ],

we obtain, for all t ∈ [, ],  D. u (t) =  = f t, u (t), v (t), w (t) , √  t t . D v (t) = ≥ = g t, u (t), v (t), w (t) , (.) (.) √  t . D w (t) = ≥  = h t, u (t), v (t), w (t) . (.) Next, for all αi , βi ∈ R,  ≤ i ≤  such that – t ≤ β – β ≤ α – α ≤ t,

–t ≤ α ≤ β ≤ t,

– t ≤ β – β ≤ α – α ≤ t,

–t ≤ α ≤ β ≤ t,

(.)

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Page 14 of 16

one can calculate that f (t, α , α , α ) – f (t, β , β , β ) ≥ (.)– (α – β ) – (.)– (α – β ), g(t, α , α , α ) – g(t, β , β , β ) ≥ (.)– (α – β ), h(t, α , α , α ) – h(t, β , β , β ) ≥ –(.)– (α – β ) + (.)– (α – β ). Therefore it is sufficient to take in Corollary . M = , N = , S = (.)– . Finally observe that condition (.) also holds. Thus, the system of fractional differential equations (.) has a solution (u∗ , v∗ , w∗ ) ∈ [–t, t] × [–t, t] × [–t, t]. Now, using the proof of Theorem . and Lemma ., we can derive the iterative procedure (uk , vk , wk ) convergent to the solution (u∗ , v∗ , w∗ ). First observe that the sequences (uk ), (vk ), (wk ) satisfy the following system of linear equations: D. uk = f (t, uk– , vk– , wk– ) – (.)– vk– + (.)– wk– + (.)– vk – (.)– wk , D. vk = g(t, uk– , vk– , wk– ) – (.)– vk– + (.)– vk , D. wk = h(t, uk– , vk– , wk– ) + (.)– uk– – (.)– vk– – (.)– uk + (.)– vk , t . uk (t)|t= = t . vk (t)|t= = t . wk (t)|t= = , which can be equivalently transformed to the system ⎧ ⎪ ⎨ uk + vk + wk = pk , uk – vk + wk = qk , ⎪ ⎩ uk + vk – wk = rk , where pk , qk , rk are the solutions of the following systems: ⎧ . ⎪ ⎨ D pk = (f + g + h)(t, uk– , vk– , wk– ) + (.)– uk– – (.)– vk– + (.)– wk– – (.)– pk , ⎪ ⎩ . t pk (t)|t= = , ⎧ . ⎪ ⎨ D qk = (f – g + h)(t, uk– , vk– , wk– ) + (.)– uk– – (.)– vk– + (.)– wk– – (.)– qk , ⎪ ⎩ . t qk (t)|t= = , ⎧ . ⎪ ⎨ D rk = (f + g – h)(t, uk– , vk– , wk– ) – (.)– uk– – (.)– vk– + (.)– wk– + (.)– rk , ⎪ ⎩ . t rk (t)|t= = . The solutions of the above systems, due to Lemma ., are given by the formulas

pk (t) =

t

   (t – s)–. E.,. –(.)– (t – s). (f + g + h) s, uk– (s), vk– (s), wk– (s)



+ (.)– uk– (s) – (.)– vk– (s) + (.)– wk– (s) ds,

t    qk (t) = (t – s)–. E.,. –(.)– (t – s). (f – g + h) s, uk– (s), vk– (s), wk– (s) 

+ (.)– uk– (s) – (.)– vk– (s) + (.)– wk– (s) ds,

Wardowski Advances in Difference Equations (2015) 2015:167

rk (t) =

t

Page 15 of 16

   (t – s)–. E.,. (.)– (t – s). (f + g – h) s, uk– (s), vk– (s), wk– (s)



– (.)– uk– (s) – (.)– vk– (s) + (.)– wk– (s) ds. In consequence, the iterative sequences are of the form  uk (t) = (qk + rk ) 

     t = (t – s)–. E.,. –(.)– (t – s). (f – g + h) s, uk– (s), vk– (s), wk– (s)   + (.)– uk– (s) – (.)– vk– (s) + (.)– wk– (s)    + Eα,α .– (t – s)α (f + g – h) s, uk– (s), vk– (s), wk– (s)  – (.)– uk– (s) – (.)– vk– (s) + (.)– wk– (s) ds,  vk (t) = (pk – qk ) 

     t = (t – s)–. E.,. –(.)– (t – s). (f + g + h) s, uk– (s), vk– (s), wk– (s)   + (.)– uk– (s) – (.)– vk– (s) + (.)– wk– (s)    – E.,. –(.)– (t – s). (f – g + h) s, uk– (s), vk– (s), wk– (s)  + (.)– uk– (s) – (.)– vk– (s) + (.)– wk– (s) ds,  wk (t) = (pk – rk ) 

     t = (t – s)–. E.,. –(.)– (t – s). (f + g + h) s, uk– (s), vk– (s), wk– (s)   + (.)– uk– (s) – (.)– vk– (s) + (.)– wk– (s)    – E.,. (.)– (t – s). (f + g – h) s, uk– (s), vk– (s), wk– (s)  – (.)– uk– (s) – (.)– vk– (s) + (.)– wk– (s) ds.

Competing interests The author declares that he has no competing interests. Author’s contributions The author formulated and proved all the results in the article, produced the illustrative example, wrote the manuscript, and read and approved it. Acknowledgements The author is very grateful to the reviewers for the remarks, which improved the final version of the manuscript. This article was financially supported by University of Łód´z as a part of donation for the research activities aimed at the development of young scientists, grant no. 545/1117. Received: 7 January 2015 Accepted: 17 May 2015 References 1. Babakhani, A, Daftardar-Gejji, V: Existence of positive solutions of nonlinear fractional differential equations. J. Math. Anal. Appl. 278, 434-442 (2003) 2. Bai, ZB: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. 72, 916-924 (2010) 3. Bai, ZB, Lu, HS: Positive solutions of boundary value problems of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, 495-505 (2005) 4. Belmekki, M, Benchohra, M: Existence results for fractional order semilinear functional differential equations with nondense domain. Nonlinear Anal. 72, 925-932 (2010)

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