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

R E S E A R C H

Open Access

Monotone iterative procedure and

systems of a finite number of nonlinear

fractional differential equations

Dariusz Wardowski

*_{Correspondence:} wardd@math.uni.lodz.pl 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 proper-ties. 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 im-portant 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 equa-tions: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Dα_u (t) = f(t, u(t), u(t), . . . , un(t)), t∈ (, T], Dα_u (t) = f(t, u(t), u(t), . . . , un(t)), t∈ (, T], . . . , Dα_u n(t) = fn(t, u(t), u(t), . . . , un(t)), t∈ (, T], t–α_u (t)|t== x, t–αu(t)|t== x, . . . , t–αun(t)|t== xn, (.)

where Dα_{is the standard Riemann-Liouville fractional derivative of order α, ≤ α ≤ ,}

T> , fi_{∈ C([, T] × R}n_{,R),  ≤ i ≤ n, and x} , . . . , xn∈ R satisfy n  i= xi_– x_≥ . (.)

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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 con-venience 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–αu ∈

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∈ (, T],}

t–α_u(t)| t== u,

i, j∈ N, (.)

has a unique solution, given by the following formula:

u(t) = (α)utα–Eα,α  –Mtα ₊ t  (t – s)α–Eα,α  –M(t – s)α _{σ(s) ds,}

where Eα,βis 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 ∈ (, T],}

t–α_w(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 ⎧ ⎪ ⎨ ⎪ ⎩ Dα_{u(t)≥ –Mu(t) + Nv(t), t ∈ (, T],} Dα_{v(t)≥ –Mv(t) + Nu(t), t ∈ (, T],} t–α_u(t)| t=≥ , t–αv(t)|t=≥ .

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])ndenotes 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 ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ Dα_u (t) = σ(t) – Mu(t) – n i,j=Mjδjiui(t), t∈ (, T], Dα_u j(t) = σj(t) + (Mj– n i=Mi)uj(t) – Mj( n i=ui(t) – uj(t)), t∈ (, T],  ≤ j ≤ n, t–α_u i(t)|t== xi, ≤ 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+· · · + un= p, . . . , 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_p n(t) = (σ(t) + σ(t) +· · · – σn(t)) – (M+ M+· · · – Mn)pn(t), t–α_p n(t)|t== x+ x+· · · – xn.

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

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])

sat-isfy ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ Dα_u (t)≥ –Mu(t) + n i,j=Mjδjiui(t), t∈ (, T], Dα_u s(t)≥ –Mus(t) + ( n i=Mi– Ms)us(t) + Ms( n i=ui(t) – us(t)), ≤ s ≤ n, t ∈ (, T], t–αus(t)|t=≥ ,  ≤ s ≤ n. (.) Then n  i= ui(t)≥ , t ∈ (, T], (.) us(t)≥ , t ∈ (, T],  ≤ s ≤ n, (.) –us(t) + n  i= ui(t)≥ , t ∈ (, T],  ≤ s ≤ n. (.)

Proof Put r(t) =n_s=us(t). Using (.) we obtain

Dαr(t) = n  s= Dαus(t) ≥ –Mu(t) + n  i,j= Mjδjiui(t) – M n  s= us(t) –  n  s= Msus(t) + n  s= n  i= Mius(t) + n  s= n  i= Msui(t) = –Mr(t) + n  i,j=  Mjδjiui(t) + Miuj(t) –  n  s= Msus(t) + n  s= Msr(t). Observe that n  i,j=,i=j  Mjδjiui(t) + Miuj(t) = . (.) Hence, we obtain Dα_{r(t)≥ –}  M– n  s= Ms  r(t) + n  i,j=,i=j  Mjδjiui(t) + Miuj(t) + n  i,j=,i=j  Mjδjiui(t) + Miuj(t) –  n  s= Msus(t) = –  M– n  s= Ms  r(t) +  n  i= Miui(t) –  n  s= Msus(t) = –  M– n  s= Ms  r(t). (.)

Moreover, observe that t–αr(t) = n  s= t–αus(t)≥ . (.)

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

rs(t) = n  i= ui(t) – us(t), t∈ (, T]. By (.) we have Dαrs(t) = n  i= Dαui(t) – Dαus(t) = Dαu(t) + n  i= Dαui(t) – Dαus(t) ≥ –Mu(t) + n  i,j= Mjδjiui(t) – n  i= Mui(t) + Mus(t) + n  i=  _n  j= Mj– Mi  ui(t) –  _n  j= Mj– Ms  us(t) + n  i= Mi  _n  j= uj(t) – ui(t)  – Ms  _n  j= uj(t) – us(t)  = –Mu(t) + n  i,j=  Mjδjiui(t) + Mjui(t) – M n  i= ui(t) + Mus(t) + Msus(t) –  n  i= Miui(t) – us(t) n  j= Mj+ n  i= n  j= Miuj(t) – Msrs(t).

Again, using (.), we obtain

Dαrs(t)≥ –M n  i= ui(t) + Mus(t) + Msus(t) – us(t) n  j= Mj + n  i= Mirs(t) + us(t) n  i= Mi– Msrs(t) = –  M– n  i= Mi+ Ms  rs(t) + Msus(t). (.)

Moreover, observe that (.) implies

Dαus(t)≥ –  M– n  i= Mi+ Ms  us(t) + Msrs(t). (.)

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

Theorem . Suppose that there exist u , u, . . . , un∈ C–α([, T]), u≤ n i=ui, satisfy-ing ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ Dα_u (t)≤ f(t, u(t), u(t), . . . , un(t)), t∈ (, T], Dα_us (t)≥ fs(t, u(t), u(t), . . . , un(t)), t∈ (, T],  ≤ s ≤ n, t–α_u (t)|t=≤ x, t–α_us (t)|t=≥ xs, ≤ s ≤ n, (.)

and there exist M∈ R, M, . . . , Mn>  such that

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

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

u_(t) –  _n  i= ui_(t) – us_(t)  ≤ β–  _n  i= βi– βs  ≤ α–  _n  i= αi– αs  ≤ us (t), u_(t) –  _n  i= ui_(t) – us_(t)  ≤ αs≤ βs≤ us(t), (iii) n  s= fs  t, u(t), u(t), . . . , un(t) – f  t, u(t), u(t), . . . , un(t) ≥  –M+ n  s= Ms  _n  s= us(t) – u(t)  , (.) where u_–  _n  i= ui_– us_  ≤ u_–  _n  i= ui– us  ≤ us_{≤ u}s , ≤ s ≤ n.

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  i= ui_+ us_≤ ¯us≤ us_, ≤ s ≤ n.

Moreover, there exist iterative sequences (u_k), (u_k), . . . , (un_k) such that ui_k→ ¯ui_{, k→ ∞, i =}

Proof Let us first consider the linear system of the form ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Dα_u_{(t) = f} (t, u(t), u(t), . . . , un(t)) + Mu(t) + n i,j=Mjδjiui(t) – Mu(t) – n i,j=Mjδjiui(t), t∈ (, T], Dα_us_{(t) = f} s(t, u(t), u(t), . . . , un(t)) + Mus(t) + ( n i=Mi– Ms)us(t) + Ms(u(t) + n i=ui(t) – us(t)) – Mus(t) – (ni=Mi– Ms)us(t) – Ms(u(t) + n i=ui(t) – us(t)), t∈ (, T],  ≤ s ≤ n, t–αus(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 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Dα_u k(t) = f(t, uk–(t), uk–(t), . . . , unk–(t)) + Muk–(t) + n i,j=Mjδjiuik–(t) – Muk(t) – n i,j=Mjδjiuik(t), t∈ (, T], Dα_us k(t) = fs(t, uk–(t), uk–(t), . . . , unk–(t)) + Musk–(t) + (n_i=Mi– Ms)us_k(t) + Ms(u_k–(t) + n i=uik(t) – usk(t)) – Musk(t) – ( n i=Mi– Ms)usk–(t) – Ms(uk(t) + n i=uik–(t) – usk–(t)), t∈ (, T],  ≤ s ≤ n, t–α_us k(t)|t== xs, ≤ s ≤ n. (.) Now, put p

= u– u, ps= us– us, ≤ s ≤ n. From (.) and (.), for all t ∈ (, T], we

obtain Dαp_(t) = Dαu_(t) – Dαu_(t) = f  t, u_(t), u_(t), . . . , u_n(t) + Mu(t) + n  i,j= Mjδjiui(t) – Mu(t) – n  i,j= Mjδjiui(t) – D α u_(t) ≥ –Mp(t) + n  i,j= Mjδjipi(t), Dα_ps (t) = Dαus(t) – Dαus(t) = Dα_us (t) – fs  t, u_(t), u_(t), . . . , u_n(t) – Mus(t) –  _n  i= Mi– Ms  us_(t) – Ms  u_(t) + n  i= ui_(t) – us_(t)  + Mus(t) +  _n  i= Mi– Ms  us_(t) + Ms  u_(t) + n  i= ui_(t) – us_(t)  ≥ –Mps(t) +  _n  i= Mi– Ms  ps_(t) + Ms  _n  i= pi_(t) – ps_(t)  for all ≤ s ≤ n, t–αp_(t)|t== t–αu(t)|t=– t–αu(t)|t=≥ x– x= , t–αps_(t)|t== t–αus(t)|t=– t–αus(t)|t=≥ xs– xs= , ≤ s ≤ n.

Hence, using Theorem ., we have us_≤ us_, ≤ s ≤ n (.) and u_– u_+ n  i=  ui_– ui_ ≥ us_– us_, ≤ s ≤ n. (.) Consider now q= n

i=ui– u. Using (.) and (.) we have

Dαq(t) = n  s= us_(t) – u_(t) = n  s= Dαus_(t) – Dαu_(t) = n  s= fs  t, u_(t), u_(t), . . . , un_(t) + n  s= Mus(t) + n  s=  _n  i= Mi– Ms  us (t) + n  s= Ms  u_(t) + n  i= ui (t) – us(t)  – n  s= Mus(t) – n  s=  _n  i= Mi– Ms  us_(t) – n  s= Ms  u_(t) + n  i= ui_(t) – us_(t)  – f  t, u_(t), u_(t), . . . , un_(t) – Mu(t) – n  i,j= Mjδjiui(t) + Mu(t) + n  i,j= Mjδjiui(t) = n  s= fs  t, u (t), u(t), . . . , un(t) – f  t, u (t), u(t), . . . , un(t) –  M– n  s= Ms  q(t) +  M– n  s= Ms  _n  s= us_(t) – u_(t)  ≥ –  M– n  s= Ms  q(t). Moreover, (.) implies t–αq(t)|t== n  i= t–αui_(t)|t=– t–αu(t)|t== n  i= xi_– x_≥ . Now, from Lemma . we conclude

u_(t)≤

n



i=

ui_(t) for all t∈ [, T]. (.)

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

u_–  _n  i= ui_– us_  ≤ u –  _n  i= ui_– us_  ≤ us ≤ us.

Let ≤ s ≤ n be fixed and suppose now that for some k ∈ N the following inequalities hold: u_k––  _n  i= ui_k–– us_k–  ≤ u k–  _n  i= ui_k– us_k  ≤ us k≤ usk–. (.) Denote p

k+= uk+– uk, pk+s = usk– usk+, ≤ s ≤ n. From (.), (.), and (.) we obtain

Dαp_k+(t) = Dα u_k+(t) – Dα u_k(t) = f  t, u_k(t), u_k(t), . . . , u_kn(t) + Muk(t) + n  i,j= Mjδjiuik(t) – Muk+(t) – n  i,j= Mjδjiuik+(t) – f  t, u_k–(t), u_k–(t), . . . , u_k–n (t) – Muk–(t) – n  i,j= Mjδjiuik–(t) + Muk(t) + n  i,j= Mjδjiuik(t) ≥ –M  u_k(t) – u_k–(t) – n  i,j= Mjδji  ui_k(t) – ui_k–(t) + Muk(t) + n  i,j= Mjδjiuik(t) – Muk+(t) – n  i,j= Mjδjiuik+(t) – Muk–(t) – n  i,j= Mjδjiuik–(t) + Muk(t) + n  i,j= Mjδjiuik(t) = –Mpk+(t) + n  i,j= Mjδjipik+(t), Dαps_k+(t) = Dαus_k(t) – Dαus_k+(t) ≥  –M+ n  i= Mi– Ms   us_k–(t) – us_k(t) – Ms  u_k–(t) – u_k(t) + us_k–(t) – us_k(t) – n  i=  ui_k–– ui_k  + Musk–(t) +  _n  i= Mi– Ms  us_k(t) + Ms  u_k–(t) + n  i= ui_k(t) – us_k(t)  – Musk(t) –  _n  i= Mi– Ms  us_k–(t) – Ms  u_k(t) + n  i= ui_k–(t) – us_k–(t)  – Musk(t) –  _n  i= Mi– Ms  us_k+(t) – Ms  u_k(t) + n  i= ui_k+(t) – us_k+(t)  + Musk+(t) +  _n  i= Mi– Ms  us_k(t) + Ms  u_k+(t) + n  i= ui_k(t) – us_k(t)  = –Mpsk+(t) +  _n  i= Mi– Ms  ps_k+(t) + Ms  _n  i= pi_k+(t) – ps_k+(t)  .

Also observe that t–α_p

k+(t)|t== t–αpsk+(t)|t== , which, together with the above, due

to Theorem ., gives us_k+≤ us_k, ≤ s ≤ n, (.) us_k– us_k+≤ n  i=  ui_k– ui_k+ + u_k+– u_k. (.) Consider now qk= n

i=uik– uk. Using the same arguments as with qwe obtain

Dαqk(t)≥ –  M– n  s= Ms  qk(t) and t–αqk(t)|t=≥ ,

which, due to Lemma ., gives

u_k≤

n



s=

ui_k. (.)

Summarizing, by (.)-(.) and induction, we obtain the following inequalities de-scribing the sequences (us_k)k∈N∪{}:

u_–  _n  i= ui_– us_  ≤ u –  _n  i= ui_– us_  ≤ · · · ≤ u k–  _n  i= ui_k– us_k  ≤ us k≤ · · · ≤ us≤ us, (.)

where ≤ s ≤ n. The inequalities (.) imply

lim k→∞u s k(t) =¯us(t), s= , . . . , n. Observe that u –  _n  i= ui – us  ≤ ¯us_{≤ u}s , s= , . . . , n.

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= ui_k(t)  = x∗(t).

Hence, putting¯u_{= x}∗₊n– s= ¯us, we have lim k→∞  u_k(t) –¯u(t) = lim k→∞  u_k(t) – x∗(t) – n–  s= ¯us_{(t) +} n–  s= us_k(t) – n–  s= us_k(t)  = lim k→∞  u_k(t) – n–  s= us_k(t) – x∗(t) + n–  s=  us_k(t) –¯us(t)  = lim k→∞  u_k(t) – n–  s= us k(t) – x∗(t)  + n–  s= lim k→∞  us k(t) –¯us(t) = .

In order to show the uniform convergence of sequences (u

k), (uk), . . . , (unk), observe that

from (.) and from the fact that us

k→ ¯us, s = , , . . . , n, we have

¯us_{≤ u}s

k≤ · · · ≤ us≤ us for all k∈ N.

Then, the uniform convergence of sequences (us

k), s = , , . . . , n, on a compact subset of

(, T] is a straightforward consequence of Dini’s theorem, which states that if a mono-tone 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  i= ui_k– us_k  , k∈ N ∪ {}.

From (.) and the convergence of (u

k), . . . , (unk) we have h≤ h≤ · · · ≤ hk≤ ¯u–  _n  i= ¯ui_–¯us  .

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  i= ui_k– us_k  , k∈ N, and thus (u

k) is uniformly convergent on a compact subset of (, T] to ¯uas a linear

com-bination of sequences uniformly convergent.

Moreover, observe that the limit functions satisfy the properties

(n – )u_– (n – ) n  i= ui_≤ ¯u≤ n  i= ui_, u_– n  i= ui_+ us_≤ ¯us≤ us_, ≤ s ≤ n.

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  i= ¯ui_–¯us  ≤ ¯us_{, ≤ s ≤ n,}

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}_{, . . . , u}n₎

were any other solution such that

u_–  _n  i= ui_– us_  ≤ u_–  _n  i= ui– us  ≤ us , u–  _n  i= ui_– us_  ≤ us_{≤ u}s 

for any ≤ s ≤ n, then we would have ¯u_–  _n  i= ¯ui_–¯us  ≤ u_–  _n  i= ui– us  , us≤ ¯us, ≤ s ≤ n.

4 The system of three fractional differential equations

In order to see the nature of the iterative procedure introduced in the proof of Theo-rem ., 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], t–αu(t)|t=≤ x, t–α_v (t)|t=≥ y, t–α_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

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) , D.v(t) = √ t (.)≥ t (.)= g  t, u(t), v(t), w(t) , D.w(t) = √ 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,

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 pro-cedure (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._u

k(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, rkare the solutions of the following systems:

⎧ ⎪ ⎨ ⎪ ⎩ D._p k= (f + g + h)(t, uk–, vk–, wk–) + (.)–uk–– (.)–vk–+ (.)–wk–– (.)–pk, t._p k(t)|t== , ⎧ ⎪ ⎨ ⎪ ⎩ D._q k= (f – g + h)(t, uk–, vk–, wk–) + (.)–uk–– (.)–vk–+ (.)–wk–– (.)–qk, t._q k(t)|t== , ⎧ ⎪ ⎨ ⎪ ⎩ D.rk= (f + g – h)(t, uk–, vk–, wk–) – (.)–_u k–– (.)–vk–+ (.)–wk–+ (.)–rk, t._r k(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, qk(t) = t  (t – s)–.E.,.  –(.)–(t – s). (f – g + h)s, uk–(s), vk–(s), wk–(s) + (.)–uk–(s) – (.)–vk–(s) + (.)–wk–(s) ds,

rk(t) = t  (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, u} k–(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) + (.)–_u k–(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)

5. Belmekki, M, Nieto, JJ, Rodriguez-Lopez, R: Existence of periodic solution for a nonlinear fractional differential equation. Bound. Value Probl. 2009, Article ID 324561 (2009)

6. El-Shahed, M: Positive solutions for boundary value problem of nonlinear fractional differential equation. Abstr. Appl. Anal. 2007, Article ID 10368 (2007)

7. Lakshmikantham, V, Vatsala, AS: Basic theory of fractional differential equations. Nonlinear Anal. 69(8), 2677-2682 (2008)

8. Lakshmikanthan, V, Vatsala, AS: General uniqueness and monotone iterative technique for fractional differential equations. Appl. Math. Lett. 21, 828-834 (2008)

9. McRae, FA: Monotone iterative technique and existence results for fractional differential equations. Nonlinear Anal. 71, 6093-6096 (2009)

10. Wang, G, Agarwal, RP, Cabada, A: Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations. Appl. Math. Lett. 25, 1019-1024 (2012)

11. Wei, Z, Li, Q, Che, J: Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative. J. Math. Anal. Appl. 367, 260-272 (2010)

12. Zhang, S: Monotone iterative method for initial value problem involving Riemann-Liouville fractional derivatives. Nonlinear Anal. 71, 2087-2093 (2009)

13. Zhang, S: The existence of a positive solution for a nonlinear fractional differential equation. J. Math. Anal. Appl. 252, 804-812 (2000)

14. 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)

15. Bai, C, Fang, J: The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations. Appl. Math. Comput. 150, 611-621 (2004)

16. Bhalekar, S, Daftardar-Gejji, V: Fractional ordered Liu system with time-delay. Commun. Nonlinear Sci. Numer. Simul. 15, 2178-2191 (2010)

17. Chen, Y, An, H: Numerical solutions of coupled Burgers equations with time and space fractional derivatives. Appl. Math. Comput. 200, 87-95 (2008)

18. Daftardar-Gejji, V: Positive solutions of a system of non-autonomous fractional differential equations. J. Math. Anal. Appl. 302, 56-64 (2005)

19. Gafiychuk, V, Datsko, B, Meleshko, V: Mathematical modeling of time fractional reaction-diffusion systems. J. Comput. Appl. Math. 220, 215-225 (2008)

20. Jafari, H, Daftardar-Gejji, V: Solving a system of nonlinear fractional differential equations using Adomian decomposition. J. Comput. Appl. Math. 196, 644-651 (2006)

21. Lazarevi, MP: Finite time stability analysis of PDαfractional control of robotic time-delay systems. Mech. Res. Commun. 33, 269-279 (2006)

22. Su, X: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett. 22, 64-69 (2009)

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

24. Podlubny, I: Fractional Differential Equations. Mathematics in Science and Engineering. Academic Press, New York (1999)

25. Samko, SG, Kilbas, AA, Marichev, OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Yverdon (1993)