Luca Boundary Value Problems (2017) 2017:102 DOI 10.1186/s13661-017-0833-6

RESEARCH

Open Access

Positive solutions for a system of Riemann-Liouville fractional differential equations with multi-point fractional boundary conditions Rodica Luca* *

Correspondence: [email protected] Department of Mathematics, Gh. Asachi Technical University, 11 Blvd. Carol I, Iasi, 700506, Romania

Abstract We study the existence and nonexistence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations subject to multi-point boundary conditions which contain fractional derivatives. MSC: 34A08; 45G15 Keywords: Riemann-Liouville fractional differential equations; multi-point boundary conditions; positive solutions; existence; nonexistence

1 Introduction We consider the system of nonlinear ordinary fractional differential equations ⎧ α ⎪ ⎪D+ u(t) + λf (t, u(t), v(t), w(t)) = , ⎨ (S)

β D+ v(t) + μg(t, u(t), v(t), w(t)) = , ⎪ ⎪ ⎩ γ D+ w(t) + νh(t, u(t), v(t), w(t)) = ,

t ∈ (, ), t ∈ (, ), t ∈ (, ),

with the multi-point boundary conditions which contain fractional derivatives

(BC)

⎧ (j) ⎪ ⎪u () = , ⎨

j = , . . . , n – ;

v(j) () = ,

j = , . . . , m – ;

w(j) () = ,

j = , . . . , l – ;

⎪ ⎪ ⎩

p

N

q i= ai D+ u(t)|t=ξi ,  p q D+ v(t)|t= = M i= bi D+ v(t)|t=ηi ,  p q D+ w(t)|t= = Li= ci D+ w(t)|t=ζi ,

D+ u(t)|t= =

where λ, μ, ν > , α, β, γ ∈ R, α ∈ (n – , n], β ∈ (m – , m], γ ∈ (l – , l], n, m, l ∈ N, n, m, l ≥ , p , p , p , q , q , q ∈ R, p ∈ [, n–], p ∈ [, m–], p ∈ [, l –], q ∈ [, p ], q ∈ [, p ], q ∈ [, p ], ξi , ai ∈ R for all i = , . . . , N (N ∈ N),  < ξ < · · · < ξN ≤ , ηi , bi ∈ R for all i = , . . . , M (M ∈ N),  < η < · · · < ηM ≤ , ζi , ci ∈ R for all i = , . . . , L (L ∈ N),  < ζ < · · · < ζL ≤ , and Dk+ denotes the Riemann-Liouville derivative of order k. Under some assumptions on f , g and h, we give intervals for the parameters λ, μ and ν such that positive solutions of (S)-(BC) exist. By a positive solution of problem (S)-(BC), © The Author(s) 2017. 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 mean a triplet of functions (u, v, w) ∈ (C([, ], R+ )) , (R+ = [, ∞)) satisfying (S) and (BC) with u(t) >  for all t ∈ (, ], or v(t) >  for all t ∈ (, ], or w(t) >  for all t ∈ (, ]. The nonexistence of positive solutions for the above problem is also studied. Our results generalize the results from the paper [], where the authors investigated a system with two fractional differential equations and multi-point boundary conditions. Besides, our results improve and extend the results from [], where only a few cases are presented for the existence of positive solutions for a system of integral equations and, as an application, for a system with three fractional equations subject to some boundary conditions in points t =  and t =  (Application . from []). Systems with two fractional differential equations with multi-point or Riemann-Stieltjes integral boundary conditions were also studied in [–], etc. Fractional differential equations describe many phenomena in various fields of engineering and scientific disciplines such as physics, biophysics, chemistry, biology, economics, control theory, signal and image processing, aerodynamics, viscoelasticity, electromagnetics, and so on (see [–]). The paper is organized as follows. In Section , we present some auxiliary results which investigate a nonlocal boundary value problem for fractional differential equations. Section  contains the main existence theorems for positive solutions with respect to a cone for our problem (S)-(BC). In Section , we investigate the nonexistence of positive solutions of (S)-(BC); and in Section , some examples are given to support our results. The main conclusions for our investigations from this paper are presented in Section .

2 Auxiliary results We present firstly some auxiliary results from [] that will be used to prove our main results. We consider the fractional differential equation Dα+ u(t) + x(t) = ,

 < t < ,

()

with the multi-point boundary conditions

u(j) () = ,

j = , . . . , n – ;

p

D+ u(t)|t= =

N 

q

ai D+ u(t)|t=ξi ,

()

i=

where α ∈ (n – , n], n ∈ N, n ≥ , ai , ξi ∈ R, i = , . . . , N (N ∈ N),  < ξ < · · · < ξN ≤ (α) , p , q ∈ R, p ∈ [, n – ], q ∈ [, p ], and x ∈ C[, ]. We denote  = (α–p – ) α–q – (α) N a ξ . i= i i (α–q ) Lemma . ([]) If  = , then the function u ∈ C[, ] given by  u(t) =



G (t, s)x(s) ds,

t ∈ [, ],

()



is solution of problem ()-(), where

G (t, s) = g (t, s) +

N t α–  ai g (ξi , s),

 i=

∀(t, s) ∈ [, ] × [, ],

()

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and ⎧  ⎨t α– ( – s)α–p – – (t – s)α– ,  ≤ s ≤ t ≤ , g (t, s) = (α) ⎩t α– ( – s)α–p – ,  ≤ t ≤ s ≤ , ⎧ ⎨t α–q – ( – s)α–p – – (t – s)α–q – ,  ≤ s ≤ t ≤ ,  g (t, s) = (α – q ) ⎩t α–q – ( – s)α–p – ,  ≤ t ≤ s ≤ .

()

Lemma . ([]) The functions g and g given by () have the properties: (a) g (t, s) ≤ h (s) for all t, s ∈ [, ], where h (s) =



 ( – s)α–p –  – ( – s)p , (α)

s ∈ [, ];

(b) g (t, s) ≥ t α– h (s) for all t, s ∈ [, ]; t α– (c) g (t, s) ≤ (α) for all t, s ∈ [, ]; α–q – (d) g (t, s) ≥ t h (s) for all t, s ∈ [, ], where h (s) =



 ( – s)α–p –  – ( – s)p –q , (α – q )

s ∈ [, ];

 (e) g (t, s) ≤ (α–q t α–q – for all t, s ∈ [, ]; ) (f ) The functions g and g are continuous on [, ] × [, ]; g (t, s) ≥ , g (t, s) ≥  for all t, s ∈ [, ]; g (t, s) > , g (t, s) >  for all t, s ∈ (, ).

Lemma . ([]) Assume that ai ≥  for all i = , . . . , N and  > . Then the function G given by () is a nonnegative continuous function on [, ] × [, ] and satisfies the inequalities:  (a) G (t, s) ≤ J (s) for all t, s ∈ [, ], where J (s) = h (s) +  N i= ai g (ξi , s), s ∈ [, ]; α– (b) G (t, s) ≥ t J (s) for all t, s ∈ [, ]; N α–q –   +  (α–q . (c) G (t, s) ≤ σ t α– , for all t, s ∈ [, ], where σ = (α) i= ai ξi ) Lemma . ([]) Assume that ai ≥  for all i = , . . . , N ,  > , x ∈ C[, ] and x(t) ≥  for all t ∈ [, ]. Then the solution u of problem ()-() given by () satisfies the inequality u(t) ≥ t α– u(t  ) for all t, t  ∈ [, ]. We can also formulate similar results as Lemmas .-. for the fractional boundary value problems β

D+ v(t) + y(t) = , (j)

v () = ,

 < t < ,

j = , . . . , m – ;

() p D+ v(t)|t=

=

M 

q

bi D+ v(t)|t=ηi ,

()

i=

and γ

D+ w(t) + z(t) = ,

 < t < ,

()

Luca Boundary Value Problems (2017) 2017:102

w(j) () = ,

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j = , . . . , l – ;

p

D+ w(t)|t= =

L 

q

ci D+ w(t)|t=ζi ,

()

i=

where β ∈ (m – , m], γ ∈ (l – , l], m, l ∈ N, m, l ≥ , bi , ηi ∈ R, i = , . . . , M (M ∈ N),  < η < · · · < ηM ≤ , ci , ζi ∈ R, i = , . . . , L (L ∈ N),  < ζ < · · · < ζL ≤ , p , q , p , q ∈ R, p ∈ [, m – ], q ∈ [, p ], p ∈ [, l – ], q ∈ [, p ], and y, z ∈ C[, ]. We denote by  , g , g , G , h , h , J and σ , and  , g , g , G , h , h , J and σ the corresponding constants and functions for problem ()-() and problem ()-(), respectively, defined in a similar manner as  , g , g , G , h , h , J and σ , respectively. More precisely, we have M (β)  β–q – (β) – bi ηi , (β – p ) (β – q ) i= ⎧  ⎨t β– ( – s)β–p – – (t – s)β– ,  ≤ s ≤ t ≤ , g (t, s) = (β) ⎩t β– ( – s)β–p – ,  ≤ t ≤ s ≤ , ⎧ ⎨t β–q – ( – s)β–p – – (t – s)β–q – ,  ≤ s ≤ t ≤ ,  g (t, s) = (β – q ) ⎩t β–q – ( – s)β–p – ,  ≤ t ≤ s ≤ ,

 =

G (t, s) = g (t, s) +

M t β–  bi g (ηi , s),

 i=

∀(t, s) ∈ [, ] × [, ],



 ( – s)β–p –  – ( – s)p , s ∈ [, ], (β)

 h (s) = ( – s)β–p –  – ( – s)p –q , s ∈ [, ], (β – q ) h (s) =

J (s) = h (s) +

M   bi g (ηi , s),

 i=

s ∈ [, ],

 β–q –   + bi ηi  , (β)  (β – q ) i= M

σ =

and L (γ )  γ –q – (γ ) – ci ζ , (γ – p ) (γ – q ) i= i ⎧  ⎨t γ – ( – s)γ –p – – (t – s)γ – ,  ≤ s ≤ t ≤ , g (t, s) = (γ ) ⎩t γ – ( – s)γ –p – ,  ≤ t ≤ s ≤ , ⎧ ⎨t γ –q – ( – s)γ –p – – (t – s)γ –q – ,  ≤ s ≤ t ≤ ,  g (t, s) = (γ – q ) ⎩t γ –q – ( – s)γ –p – ,  ≤ t ≤ s ≤ ,

 =

G (t, s) = g (t, s) + h (s) =

L t γ –  ci g (ζi , s),

 i=



 ( – s)γ –p –  – ( – s)p , (γ )

∀(t, s) ∈ [, ] × [, ], s ∈ [, ],

Luca Boundary Value Problems (2017) 2017:102

h (s) =

Page 5 of 35



 ( – s)γ –p –  – ( – s)p –q , (γ – q )

J (s) = h (s) +

L   ci g (ζi , s),

 i=

s ∈ [, ],

s ∈ [, ],

 γ –q –   + ci ζ  . (γ )  (γ – q ) i= i L

σ =

The inequalities from Lemmas . and . for the functions G , G , v and w are the following G (t, s) ≤ J (s), G (t, s) ≥ t β– J (s), G (t, s) ≤ σ t β– , G (t, s) ≤ J (s), G (t, s) ≥ t γ – J (s), G (t, s) ≤ σ t γ – for all t, s ∈ [, ], and v(t) ≥ t β– v(t  ), w(t) ≥ t γ – w(t  ) for all t, t  ∈ [, ]. In the proof of our main existence results, we shall use the following theorem (the GuoKrasnosel’skii fixed point theorem, see []). Theorem . Let X be a Banach space, and let C ⊂ X be a cone in X. Assume  and  are bounded open subsets of X with  ∈  ⊂  ⊂  , and let A : C ∩ (  \  ) → C be a completely continuous operator such that either (i) Au ≤ u , u ∈ C ∩ ∂  , and Au ≥ u , u ∈ C ∩ ∂  , or (ii) Au ≥ u , u ∈ C ∩ ∂  , and Au ≤ u , u ∈ C ∩ ∂  . Then A has a fixed point in C ∩ (  \  ).

3 Existence of positive solutions In this section, we give sufficient conditions on λ, μ, ν, f , g and h such that positive solutions with respect to a cone for our problem (S)-(BC) exist. We present the assumptions that we shall use in the sequel. (H) α, β, γ ∈ R, α ∈ (n – , n], β ∈ (m – , m], γ ∈ (l – , l], n, m, l ∈ N, n, m, l ≥ , p , p , p , q , q , q ∈ R, p ∈ [, n – ], p ∈ [, m – ], p ∈ [, l – ], q ∈ [, p ], q ∈ [, p ], q ∈ [, p ], ξi ∈ R, ai ≥  for all i = , . . . , N (N ∈ N),  < ξ < · · · < ξN ≤ , ηi ∈ R, bi ≥  for all i = , . . . , M (M ∈ N),  < η < · · · < ηM ≤ , and ζi ∈ R, ci ≥  for all i = α–q – (α) (α) N , . . . , L (L ∈ N),  < ζ < · · · < ζL ≤ ; λ, μ, ν > ,  = (α–p – (α–q > i= ai ξi ) )   β–q – γ –q M L (γ ) (γ ) (β) (β)   – > ,  = (γ –p ) – (γ –q ) i= ci ζi > . ,  = (β–p ) – (β–q ) i= bi ηi (H) The functions f , g, h : [, ] × R+ × R+ × R+ → R+ are continuous. For σ ∈ (, ), we introduce the following extreme limits: f (t, u, v, w) , u+v+w→+ t∈[,] u + v + w

gs = lim sup max

hs = lim sup max

h(t, u, v, w) , u+v+w

fi = lim inf min

f (t, u, v, w) , u+v+w

gi = lim inf min

g(t, u, v, w) , u+v+w

hi = lim inf min

h(t, u, v, w) , u+v+w

f (t, u, v, w) , u+v+w→∞ t∈[,] u + v + w

s g∞ = lim sup max

fs = lim sup max

u+v+w→+ t∈[,]

u+v+w→+ t∈[σ ,]

s f∞ = lim sup max

g(t, u, v, w) , u+v+w→+ t∈[,] u + v + w u+v+w→+ t∈[σ ,]

u+v+w→+ t∈[σ ,]

g(t, u, v, w) , u+v+w→∞ t∈[,] u + v + w f (t, u, v, w) , u+v+w→∞ t∈[σ ,] u + v + w

h(t, u, v, w) , u+v+w→∞ t∈[,] u + v + w

i f∞ = lim inf min

g(t, u, v, w) , u+v+w

hi∞ = lim inf min

hs∞ = lim sup max i g∞ = lim inf min

u+v+w→∞ t∈[σ ,]

u+v+w→∞ t∈[σ ,]

h(t, u, v, w) . u+v+w

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In the definition of the extreme limits above, the variables u, v and w are nonnegative. By using the Green functions Gi , i = , , , from Section , we consider the following nonlinear system of integral equations: ⎧  ⎪ ⎪u(t) = λ  G (t, s)f (s, u(s), v(s), w(s)) ds, t ∈ [, ], ⎨  v(t) = μ  G (t, s)g(s, u(s), v(s), w(s)) ds, t ∈ [, ], ⎪ ⎪  ⎩ w(t) = ν  G (t, s)h(s, u(s), v(s), w(s)) ds, t ∈ [, ]. If (u, v, w) is a solution of the above system, then by Lemma . and the corresponding lemmas for problems ()-() and ()-(), we deduce that (u, v, w) is a solution of problem (S)-(BC). We consider the Banach space X = C[, ] with the supremum norm · and the Banach space Y = X × X × X with the norm (u, v, w) Y = u + v + w . We define the cones

P = u ∈ X, u(t) ≥ t α– u , ∀t ∈ [, ] ⊂ X,

P = v ∈ X, v(t) ≥ t β– v , ∀t ∈ [, ] ⊂ X,

P = w ∈ X, w(t) ≥ t γ – w , ∀t ∈ [, ] ⊂ X, and P = P × P × P ⊂ Y . For λ, μ, ν > , we define now the operator Q : P → Y by Q(u, v, w) = (Q (u, v, w), Q (u, v, w), Q (u, v, w)) with 



G (t, s)f s, u(s), v(s), w(s) ds,

t ∈ [, ], (u, v, w) ∈ P,





G (t, s)g s, u(s), v(s), w(s) ds,

t ∈ [, ], (u, v, w) ∈ P,





G (t, s)h s, u(s), v(s), w(s) ds,

t ∈ [, ], (u, v, w) ∈ P.



Q (u, v, w)(t) = λ 

 Q (u, v, w)(t) = μ 



Q (u, v, w)(t) = ν 

Lemma . If (H)-(H) hold, then Q : P → P is a completely continuous operator. Proof Let (u, v, w) ∈ P be an arbitrary element. Because Q (u, v, w), Q (u, v, w) and Q (u, v, w) satisfy problem ()-() for x(t) = λf (t, u(t), v(t), w(t)), t ∈ [, ], problem ()-() for y(t) = μg(t, u(t), v(t), w(t)), t ∈ [, ], and problem ()-() for z(t) = νh(t, u(t), v(t), w(t)), t ∈ [, ], respectively, then by Lemma . and the corresponding ones for problems ()-() and ()-(), we obtain



Q (u, v, w) t  ≥ t α– Q (u, v, w) t  ,



Q (u, v, w) t  ≥ t γ – Q (u, v, w) t  ,





Q (u, v, w) t  ≥ t β– Q (u, v, w) t  , ∀t, t  ∈ [, ], (u, v, w) ∈ P,

and so

Q (u, v, w)(t) ≥ t α– max Q (u, v, w) t  t  ∈[,]

  = t α– Q (u, v, w),

∀t ∈ [, ], (u, v, w) ∈ P,

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Q (u, v, w)(t) ≥ t β– max Q (u, v, w) t  t  ∈[,]

  = t β– Q (u, v, w), Q (u, v, w)(t) ≥ t γ –

∀t ∈ [, ], (u, v, w) ∈ P,

max Q (u, v, w) t 

t  ∈[,]

  = t γ – Q (u, v, w),

∀t ∈ [, ], (u, v, w) ∈ P.

Therefore, Q(u, v, w) = (Q (u, v, w), Q (u, v, w), Q (u, v, w)) ∈ P, and then Q(P) ⊂ P. By using standard arguments, we can easily show that Q , Q and Q are completely continuous (continuous and compact, that is, map bounded sets into relatively compact sets), and then Q is a completely continuous operator.  If (u, v, w) ∈ P is a fixed point of operator Q, then (u, v, w) is a solution of problem (S)(BC). So, we will investigate the existence of fixed points of operator Q.     For σ ∈ (, ), we denote A = σ J (s) ds, B =  J (s) ds, C = σ J (s) ds, D =  J (s) ds, E =   σ J (s) ds, F =  J (s) ds, where J , J and J are defined in Section . i i , g∞ , hi∞ ∈ (, ∞) and numbers α , α , α ≥  with α + α + α = , First, for fs , gs , hs , f∞  α , α , α >  with  α +  α +  α = ,  α , α >  with  α +  α = ,  α , α >  with  α +  α = ,     α >  with  α +  α = , we define the numbers  α , L =

α , α– i A θ σ f∞

L =

α , β– i C θ σ g∞

L =

 α , gs D

L =

 α , hs F

L =

L  =

 α , fs B

L  =

 α , gs D

 L =

 α , gs D 

, fs B

L =

α , γ – θ σ hi∞ E

L =  L =

 α , hs F 

, gs D

L = L =  L =

 α , fs B

 α , fs B

L =

 α , hs F



, hs F

where θ = min{σ α– , σ β– , σ γ – }. Theorem . Assume that (H) and (H) hold, σ ∈ (, ), α , α , α ≥  with α + α + α = α , α >  with  α +  α +  α = ,  α , α >  with  α +  α = ,  α , α >  with  α +  α = , ,  α ,     α >  with  α +  α = .  α , s s s i i , hi∞ ∈ (, ∞), L < L , L < L and L < L , then for each () If f , g , h , f∞ , g∞ λ ∈ (L , L ), μ ∈ (L , L ), ν ∈ (L , L ) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC). i i , g∞ , hi∞ ∈ (, ∞), L < L and L < L , then for each λ ∈ (L , ∞), () If fs = , gs , hs , f∞ μ ∈ (L , L ), ν ∈ (L , L ) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC). i i , g∞ , hi∞ ∈ (, ∞), L < L and L < L , then for each λ ∈ (L , L ), () If gs = , fs , hs , f∞ μ ∈ (L , ∞), ν ∈ (L , L ) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC).   i i () If hs = , fs , gs , f∞ , g∞ , hi∞ ∈ (, ∞), L < L  and L < L , then for each λ ∈ (L , L ), μ ∈ (L , L  ), ν ∈ (L , ∞) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC). i i , g∞ , hi∞ ∈ (, ∞), L <  L , then for each λ ∈ (L , ∞), μ ∈ (L , ∞), () If fs = gs = , hs , f∞ ν ∈ (L ,  L ) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC).

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i i () If fs = hs = , gs , f∞ , g∞ , hi∞ ∈ (, ∞), L <  L , then for each λ ∈ (L , ∞), μ ∈ (L ,  L ), ν ∈ (L , ∞) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC). i i , g∞ , hi∞ ∈ (, ∞), L <  L , then for each λ ∈ (L ,  L ), μ ∈ (L , ∞), () If gs = hs = , fs , f∞ ν ∈ (L , ∞) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC). i i , g∞ , hi∞ ∈ (, ∞), then for each λ ∈ (L , ∞), μ ∈ (L , ∞), () If fs = gs = hs = , f∞ ν ∈ (L , ∞) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC). i i , g∞ , hi∞ is ∞, then for each λ ∈ (, L ), () If fs , gs , hs ∈ (, ∞) and at least one of f∞ μ ∈ (, L ), ν ∈ (, L ) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC). i i , g∞ , hi∞ is ∞, then for each () If fs = , gs , hs ∈ (, ∞) and at least one of f∞   λ ∈ (, ∞), μ ∈ (, L ), ν ∈ (, L ) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC). i i , g∞ , hi∞ is ∞, then for each () If gs = , fs , hs ∈ (, ∞) and at least one of f∞   λ ∈ (, L ), μ ∈ (, ∞), ν ∈ (, L ) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC). i i , g∞ , hi∞ is ∞, then for each () If hs = , fs , gs ∈ (, ∞) and at least one of f∞  λ ∈ (, L  ), μ ∈ (, L ), ν ∈ (, ∞) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC). i i , g∞ , hi∞ is ∞, then for each () If fs = gs = , hs ∈ (, ∞) and at least one of f∞ λ ∈ (, ∞), μ ∈ (, ∞), ν ∈ (,  L ) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC). i i , g∞ , hi∞ is ∞, then for each () If fs = hs = , gs ∈ (, ∞) and at least one of f∞ λ ∈ (, ∞), μ ∈ (,  L ), ν ∈ (, ∞) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC). i i , g∞ , hi∞ is ∞, then for each () If gs = hs = , fs ∈ (, ∞) and at least one of f∞ λ ∈ (,  L ), μ ∈ (, ∞), ν ∈ (, ∞) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC). i i , g∞ , hi∞ is ∞, then for each λ ∈ (, ∞), () If fs = gs = hs =  and at least one of f∞ μ ∈ (, ∞), ν ∈ (, ∞) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC).

Proof We consider the above cone P ⊂ Y and the operators Q , Q , Q and Q. We will prove some illustrative cases of this theorem. i i , g∞ , hi∞ ∈ (, ∞). Let λ ∈ (L , L ), μ ∈ (L , L ) and ν ∈ Case (). We consider fs , gs , hs , f∞ i i , ε < g∞ , ε < hi∞ and (L , L ). We choose ε >  a positive number such that ε < f∞  α  α  α ≥ λ, ≥ μ, ≥ ν, s s + ε)B (g + ε)D (h + ε)F α α α ≤ λ, ≤ μ, ≤ ν. α– i β– i γ – θ σ (f∞ – ε)A θ σ (g∞ – ε)C θ σ (hi∞ – ε)E (fs

By using (H) and the definition of fs , gs and hs , we deduce that there exists R >  such that f (t, u, v, w) ≤ (fs + ε)(u + v + w), g(t, u, v, w) ≤ (gs + ε)(u + v + w), h(t, u, v, w) ≤

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(hs + ε)(u + v + w) for all t ∈ [, ] and u, v, w ≥  with u + v + w ≤ R . We define the set

 = {(u, v, w) ∈ Y , (u, v, w) Y < R }. Now let (u, v, w) ∈ P ∩ ∂  , that is, (u, v, w) Y = R or, equivalently, u + v + w = R . Then u(t) + v(t) + w(t) ≤ R for all t ∈ [, ], and by Lemma ., we obtain  Q (u, v, w)(t) ≤ λ





J (s)f s, u(s), v(s), w(s) ds







J (s) fs + ε u(s) + v(s) + w(s) ds



 ≤λ





≤ λ fs + ε







J (s) u + v + w ds



  

 = λ fs + ε B(u, v, w)Y ≤  α (u, v, w)Y ,  

Q (u, v, w)(t) ≤ μ J (s)g s, u(s), v(s), w(s) ds 

∀t ∈ [, ],

 

≤μ 





J (s) gs + ε u(s) + v(s) + w(s) ds

≤ μ gs + ε







J (s) u + v + w ds



  

 = μ gs + ε D(u, v, w)Y ≤  α (u, v, w)Y ,  

J (s)h s, u(s), v(s), w(s) ds Q (u, v, w)(t) ≤ ν

∀t ∈ [, ],



 ≤ν









J (s) hs + ε u(s) + v(s) + w(s) ds



≤ ν hs + ε







J (s) u + v + w ds



  

 = ν hs + ε F (u, v, w)Y ≤  α (u, v, w)Y ,

∀t ∈ [, ].

α (u, v, w) Y , Q (u, v, w) ≤  α (u, v, w) Y , Q (u, v, w) ≤ Therefore, Q (u, v, w) ≤   α (u, v, w) Y . Then, for (u, v, w) ∈ P ∩ ∂  , we deduce         Q(u, v, w) = Q (u, v, w) + Q (u, v, w) + Q (u, v, w) Y     ≤ ( α +  α +  α )(u, v, w)Y = (u, v, w)Y .

()

i i i , g∞ and hi∞ , there exists R >  such that f (t, u, v, w) ≥ (f∞ – By the definition of f∞ i i ε)(u + v + w), g(t, u, v, w) ≥ (g∞ – ε)(u + v + w), h(t, u, v, w) ≥ (h∞ – ε)(u + v + w) for all u, v, w ≥  with u + v + w ≥ R and t ∈ [σ , ]. We consider R = max{R , R /θ }, and we define  = {(u, v, w) ∈ Y , (u, v, w) Y < R }. Then, for (u, v, w) ∈ P with (u, v, w) Y = R , we obtain



u(t) + v(t) + w(t) ≥ σ α– u + σ β– v + σ γ – w ≥ θ u + v + w   = θ (u, v, w)Y = θ R ≥ R , ∀t ∈ [σ , ].

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Then, by Lemma ., we conclude 



Q (u, v, w)(t) ≥ λ



t α– J (s)f s, u(s), v(s), w(s) ds





≥ λσ α– 





J (s)f s, u(s), v(s), w(s) ds



i



J (s) f∞ – ε u(s) + v(s) + w(s) ds

σ

≥ λσ α– σ

i

≥ λσ α– θ f∞ –ε





σ

  J (s)(u, v, w)Y ds

 i

 = λσ θ f∞ – ε A(u, v, w)Y   ≥ α (u, v, w)Y , ∀t ∈ [σ , ],  

t β– J (s)g s, u(s), v(s), w(s) ds Q (u, v, w)(t) ≥ μ α–





≥ μσ β–





J (s)g s, u(s), v(s), w(s) ds



i



J (s) g∞ – ε u(s) + v(s) + w(s) ds

σ

 ≥ μσ β–

σ

i

≥ μσ β– θ g∞ –ε





σ

  J (s)(u, v, w)Y ds

 i

 = μσ θ g∞ – ε C (u, v, w)Y   ≥ α (u, v, w)Y , ∀t ∈ [σ , ],  

t γ – J (s)h s, u(s), v(s), w(s) ds Q (u, v, w)(t) ≥ ν β–





≥ νσ γ –





J (s)h s, u(s), v(s), w(s) ds







J (s) hi∞ – ε u(s) + v(s) + w(s) ds

σ

 ≥ νσ γ –

σ



≥ νσ γ – θ hi∞ – ε

 σ



  J (s)(u, v, w)Y ds



 – ε F (u, v, w)Y = νσ θ   ≥ α (u, v, w)Y , ∀t ∈ [σ , ]. γ –



hi∞

So Q (u, v, w) ≥ Q (u, v, w)(σ ) ≥ α (u, v, w) Y , Q (u, v, w) ≥ Q (u, v, w)(σ ) ≥ α (u, v, w) Y , Q (u, v, w) ≥ Q (u, v, w)(σ ) ≥ α (u, v, w) Y . Hence, for (u, v, w) ∈ P ∩ ∂  , we obtain         Q(u, v, w) = Q (u, v, w) + Q (u, v, w) + Q (u, v, w) Y     ≥ (α + α + α )(u, v, w)Y = (u, v, w)Y .

()

By using Lemma ., Theorem . i) and relations (), (), we deduce that Q has a fixed point (u, v, w) ∈ P ∩ (  \  ), u(t) ≥ t α– u , v(t) ≥ t β– v , w(t) ≥ t γ – w for all

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t ∈ [, ], and R ≤ u + v + w ≤ R . If u > , then u(t) >  for all t ∈ (, ], if v > , then v(t) >  for all t ∈ (, ], and if w > , then w(t) >  for all t ∈ (, ]. So, (u, v, w) is a positive solution for our problem (S)-(BC). i i Case (). We consider fs = , f∞ = ∞, gs , hs , g∞ , hi∞ ∈ (, ∞). Let λ ∈ (, ∞), μ ∈ (, L )  and ν ∈ (, L ). We choose ε >  a positive number such that ε ≤ λθ σ α– A and ε≤

 – μgs D – νhs F , λB

ε≤

 α – μgs D , μD

ε≤

 α – νhs F . νF

The numerators of the above fractions are positive because μ <  α hs F

 α , gs D

that is,  α > μgs D,

, that is,  α > νhs F, and  – μgs D – νhs F =  α +  α – μgs D – νhs F = ( α – μgs D) + ν< ( α – νhs F) > . By using (H) and the definition of fs , gs , hs , we deduce that there exists R >  such that f (t, u, v, w) ≤ ε(u + v + w), g(t, u, v, w) ≤ (gs + ε)(u + v + w), h(t, u, v, w) ≤ (hs + ε)(u + v + w) for all t ∈ [, ], u, v, w ≥  with u + v + w ≤ R . We define the set  = {(u, v, w) ∈ Y , (u, v, w) Y < R }. Now let (u, v, w) ∈ P ∩ ∂  , that is, (u, v, w) Y = R . Then u(t) + v(t) + w(t) ≤ R for all t ∈ [, ], and by Lemma . we obtain  Q (u, v, w)(t) ≤ λ





J (s)f s, u(s), v(s), w(s) ds





J (s)ε u(s) + v(s) + w(s) ds



 ≤λ







≤ λε



J (s) u + v + w ds



  

  = λεB(u, v, w)Y ≤  – μgs D – νhs F (u, v, w)Y ,   

Q (u, v, w)(t) ≤ μ J (s)g s, u(s), v(s), w(s) ds 

 

≤μ 





J (s) gs + ε u(s) + v(s) + w(s) ds



≤ μ gs + ε







J (s) u + v + w ds





 = μ gs + ε D(u, v, w)Y      α  – μgs D D(u, v, w)Y ≤ μ gs +  μD 

  = μgs D +  α (u, v, w)Y ,   

Q (u, v, w)(t) ≤ ν J (s)h s, u(s), v(s), w(s) ds 

 ≤ν









J (s) hs + ε u(s) + v(s) + w(s) ds



≤ ν hs + ε



 



J (s) u + v + w ds

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    s

  α – νhs F  s   = ν h + ε F (u, v, w) Y ≤ ν h + F (u, v, w)Y νF 

  = νhs F +  α (u, v, w)Y , ∀t ∈ [, ].  Therefore   

 Q (u, v, w) ≤   – μg s D – νhs F (u, v, w) ,   Y    

 Q (u, v, w) ≤  μg s D +  α (u, v, w)Y ,       s

 Q (u, v, w) ≤ νh F +  α (u, v, w)Y .   Then, for (u, v, w) ∈ P ∩ ∂  , we conclude         Q(u, v, w) = Q (u, v, w) + Q (u, v, w) + Q (u, v, w) Y 

   – μgs D – νhs F + μgs D +  α + νhs F +  α (u, v, w)Y    = (u, v, w)Y .

≤

()

i , there exists R >  such that f (t, u, v, w) ≥ ε (u + v + w) for all By the definition of f∞ u, v, w ≥  with u + v + w ≥ R and t ∈ [σ , ]. We consider R = max{R , R /θ }, and we define  = {(u, v, w) ∈ Y , (u, v, w) Y < R }. Then, for (u, v, w) ∈ P with (u, v, w) Y = R , we obtain u(t) + v(t) + w(t) ≥ θ (u, v, w) Y = θ R ≥ R for all t ∈ [σ , ]. Then by Lemma . we deduce





Q (u, v, w)(t) ≥ λ



t α– J (s)f s, u(s), v(s), w(s) ds



≥ λσ





α–



J (s)f s, u(s), v(s), w(s) ds

σ

 ≥ λσ α–

J (s) σ

≥ λσ α– θ



 ε



 σ

 u(s) + v(s) + w(s) ds ε

  J (s)(u, v, w)Y ds

     = λσ α– θ A(u, v, w)Y ≥ (u, v, w)Y , ε

∀t ∈ [σ , ].

Then Q (u, v, w) ≥ Q (u, v, w)(σ ) ≥ (u, v, w) Y , and       Q(u, v, w) ≥ Q (u, v, w) ≥ (u, v, w) . Y Y

()

By using Lemma ., Theorem .(i) and inequalities (), (), we conclude that Q has a fixed point (u, v, w) ∈ P ∩ (  \  ) which is a positive solution of problem (S)-(BC). i i Case (). We consider gs = hs = , g∞ = ∞, fs , f∞ , hi∞ ∈ (, ∞). Let λ ∈ (,  L ), μ ∈ β– (, ∞), ν ∈ (, ∞). We choose ε >  a positive number such that ε ≤ μθ σ C and ε≤

 – λfs B , λB

ε≤

 – λfs B , μD

ε≤

 – λfs B . νF

Luca Boundary Value Problems (2017) 2017:102

Page 13 of 35

The numerator of the above fractions is positive because λ < f sB , that is,  – λfs B > .  By using (H) and the definition of fs , gs , hs , we deduce that there exists R >  such that f (t, u, v, w) ≤ (fs + ε)(u + v + w), g(t, u, v, w) ≤ ε(u + v + w), h(t, u, v, w) ≤ ε(u + v + w) for all t ∈ [, ], u, v, w ≥  with u + v + w ≤ R . We define the set  = {(u, v, w) ∈ Y , (u, v, w) Y < R }. Now let (u, v, w) ∈ P ∩ ∂  , that is, (u, v, w) Y = R . Then u(t) + v(t) + w(t) ≤ R for all t ∈ [, ], and by Lemma ., we obtain  Q (u, v, w)(t) ≤ λ





J (s)f s, u(s), v(s), w(s) ds







J (s) fs + ε u(s) + v(s) + w(s) ds



 ≤λ





≤ λ fs + ε







J (s) u + v + w ds



   

 s  – λfs B  s   B(u, v, w)Y = λ f + ε B (u, v, w) Y ≤ λ f + λB 

  = λfs B +  (u, v, w)Y ,   

J (s)g s, u(s), v(s), w(s) ds Q (u, v, w)(t) ≤ μ 





J (s)ε u(s) + v(s) + w(s) ds



≤μ 





≤ με



J (s) u + v + w ds



    – λfs B  D(u, v, w)Y = μεD(u, v, w)Y ≤ μ μD 

  =  – λfs B (u, v, w)Y ,   

Q (u, v, w)(t) ≤ ν J (s)h s, u(s), v(s), w(s) ds 

 ≤ν





J (s)ε u(s) + v(s) + w(s) ds







≤ νε



J (s) u + v + w ds



    – λfs B  F (u, v, w)Y = νεF (u, v, w)Y ≤ ν νF 

  =  – λfs B (u, v, w)Y , ∀t ∈ [, ].  Therefore   

 Q (u, v, w) ≤  λf s B +  (u, v, w) ,  Y    

 Q (u, v, w) ≤   – λf s B (u, v, w) ,  Y    

 Q (u, v, w) ≤   – λf s B (u, v, w) .  Y 

Luca Boundary Value Problems (2017) 2017:102

Page 14 of 35

Then, for (u, v, w) ∈ P ∩ ∂  , we deduce         Q(u, v, w) = Q (u, v, w) + Q (u, v, w) + Q (u, v, w) Y ≤

  

   + λfs B +  – λfs B +  – λfs B (u, v, w)Y = (u, v, w)Y . 

()

i By the definition of g∞ , there exists R >  such that g(t, u, v, w) ≥ ε (u + v + w) for all u, v, w ≥  with u + v + w ≥ R and t ∈ [σ , ]. We consider R = max{R , R /θ }, and we define  = {(u, v, w) ∈ Y , (u, v, w) Y < R }. Then, for (u, v, w) ∈ P with (u, v, w) Y = R , we obtain u(t) + v(t) + w(t) ≥ θ (u, v, w) Y = θ R ≥ R for all t ∈ [σ , ]. Then, by Lemma ., we conclude





Q (u, v, w)(t) ≥ μ



t β– J (s)g s, u(s), v(s), w(s) ds



≥ μσ





β–



J (s)g s, u(s), v(s), w(s) ds

σ

 ≥ μσ β–



J (s) σ

≥ μσ β– θ

 ε

 σ



 u(s) + v(s) + w(s) ds ε

  J (s)(u, v, w)Y ds

     = μσ β– θ C (u, v, w)Y ≥ (u, v, w)Y , ε

∀t ∈ [σ , ].

Then Q (u, v, w) ≥ Q (u, v, w)(σ ) ≥ (u, v, w) Y , and       Q(u, v, w) ≥ Q (u, v, w) ≥ (u, v, w) . Y Y

()

By using Lemma ., Theorem .(i) and inequalities (), (), we deduce that Q has a fixed point (u, v, w) ∈ P ∩ (  \  ) which is a positive solution of problem (S)-(BC). i i , g∞ ∈ (, ∞). Let λ ∈ (, ∞), μ ∈ Case () We consider fs = gs = hs = , hi∞ = ∞, f∞ (, ∞) and ν ∈ (, ∞). We choose ε >  such that ε≤

ε ≤ νθ σ γ – E,

 , λB

ε≤

 , μD

ε≤

 . νF

By using (H) and the definition of fs , gs , hs , we deduce that there exists R >  such that f (t, u, v, w) ≤ ε(u + v + w), g(t, u, v, w) ≤ ε(u + v + w), h(t, u, v, w) ≤ ε(u + v + w) for all t ∈ [, ], u, v, w ≥  with u + v + w ≤ R . We define the set  = {(u, v, w) ∈ Y , (u, v, w) Y < R }. Now let (u, v, w) ∈ P ∩ ∂  , that is, (u, v, w) Y = R . Then u(t) + v(t) + w(t) ≤ R for all t ∈ [, ], and by Lemma . we obtain  Q (u, v, w)(t) ≤ λ





J (s)f s, u(s), v(s), w(s) ds





J (s)ε u(s) + v(s) + w(s) ds



 ≤λ







≤ λε 



J (s) u + v + w ds

Luca Boundary Value Problems (2017) 2017:102

Page 15 of 35

    = λεB(u, v, w)Y ≤ (u, v, w)Y ,   

Q (u, v, w)(t) ≤ μ J (s)g s, u(s), v(s), w(s) ds 





J (s)ε u(s) + v(s) + w(s) ds



≤μ 





≤ με



J (s) u + v + w ds



    = μεD(u, v, w)Y ≤ (u, v, w)Y ,   

Q (u, v, w)(t) ≤ ν J (s)h s, u(s), v(s), w(s) ds 





≤ν



J (s)ε u(s) + v(s) + w(s) ds







≤ νε



J (s) u + v + w ds



    = νεF (u, v, w)Y ≤ (u, v, w)Y , 

∀t ∈ [, ].

Therefore Q (u, v, w) ≤  (u, v, w) Y , Q (u, v, w) ≤  (u, v, w) Y , Q (u, v, w) ≤  (u, v, w) Y . 

Then, for (u, v, w) ∈ P ∩ ∂  , we conclude     Q(u, v, w) ≤ (u, v, w) . Y Y

()

By the definition of hi∞ , there exists R >  such that h(t, u, v, w) ≥ ε (u + v + w) for all u, v, w ≥  with u + v + w ≥ R and t ∈ [σ , ]. We consider R = max{R , R /θ }, and we define  = {(u, v, w) ∈ Y , (u, v, w) Y < R }. Then, for (u, v, w) ∈ P with (u, v, w) Y = R , we obtain u(t) + v(t) + w(t) ≥ θ (u, v, w) Y = θ R ≥ R for all t ∈ [σ , ]. Then, by Lemma ., we deduce 



Q (u, v, w)(t) ≥ ν



t γ – J (s)h s, u(s), v(s), w(s) ds







≥ νσ γ – 

σ 

≥ νσ γ –

J (s) σ

≥ νσ γ – θ



J (s)h s, u(s), v(s), w(s) ds

 ε



 σ

 u(s) + v(s) + w(s) ds ε

  J (s)(u, v, w)Y ds

     = νσ γ – θ E(u, v, w)Y ≥ (u, v, w)Y , ε

∀t ∈ [σ , ].

Then Q (u, v, w) ≥ Q (u, v, w)(σ ) ≥ (u, v, w) Y , and       Q(u, v, w) ≥ Q (u, v, w) ≥ (u, v, w) . Y Y

()

Luca Boundary Value Problems (2017) 2017:102

Page 16 of 35

By using Lemma ., Theorem .(i) and inequalities (), (), we conclude that Q has a fixed point (u, v, w) ∈ P ∩ (  \  ) which is a positive solution of problem (S)-(BC).  Remark . Each of the cases ()-() of Theorem . contains seven cases as follows: i i i i i i {f∞ = ∞, g∞ , hi∞ ∈ (, ∞)}, or {g∞ = ∞, f∞ , hi∞ ∈ (, ∞)}, or {hi∞ = ∞, f∞ , g∞ ∈ (, ∞)}, i i i i i i i i i or {f∞ = g∞ = ∞, h∞ ∈ (, ∞)}, or {f∞ = h∞ = ∞, g∞ ∈ (, ∞)}, or {g∞ = h∞ = ∞, f∞ ∈ i i i (, ∞)}, or {f∞ = g∞ = h∞ = ∞}. So the total number of cases from Theorem . is , which we grouped in  cases. Each of the cases ()-() contains four subcases because α , α , α ∈ (, ), or α =  and α = α = , or α =  and α = α = , or α =  and α = α = . Remark . In the paper [], the authors present only  cases (Theorems .-. from []) from  cases, namely the first nine cases of our Theorem .. They did not study the cases when some extreme limits are  and other are ∞. Besides, compared to Theorems .-. and .-. from [], our intervals for parameters λ, μ, ν presented in Theorem . (our cases ()-() and ()) are better than the corresponding ones from []. In addition, the cone used in [] implies the existence of nonnegative solutions which satisfy the condition inft∈[ξ ,η] (u(t) + v(t) + w(t)) > , which is different from our definition of positive solutions. Remark . One can formulate existence results for the general case of the system of n fractional differential equations

αj ( S) D+ uj (t) + λj fj t, u (t), . . . , un (t) = ,

j = , . . . , n,

with the boundary conditions

 (BC)

⎧ ⎨u(k) () = , k = , . . . , m – , j = , . . . , n, j j ⎩Dpj uj (t)|t= = Nj ajk Dqj uj (t)|t=ξ , j = , . . . , n, +

k=

+

jk

where αj ∈ (mj – , mj ], mj ∈ N, mj ≥ ; ξjk , ajk ∈ R for all k = , . . . , Nj , (Nj ∈ N);  < ξj < ξj < · · · ≤ ξjNj , pj ∈ [, nj – ], qj ∈ [, pj ], j = , . . . , N . According to the values of fjs = lim supu +···+un →+ supt∈[,] f (t,u ,...,un ) lim infu +···+un →∞ inft∈[σ ,] ju +···+u n n+

grouped in 

fj (t,u ,...,un ) u +···+un n

∈ (, ∞], j = , . . . , n, we have 

i ∈ [, ∞), and fj∞ =

cases, which can be

cases.

s s , g∞ , hs∞ ∈ (, ∞) and numbers α , α , α ≥  with α + In what follows, for fi , gi , hi , f∞ α + α = ,  α , α , α >  with  α +  α +  α = ,  α , α >  with  α +  α = ,  α , α >  with  α +  α = ,  α , α >  with  α +  α = , we define the numbers

M =

α , α– θ σ fi A

M =

 α , s B f∞

M =

M =

 α , hs∞ F

M =

M =

α , β– θ σ gi C

M =

α , γ θ σ – hi E

 α , s D g∞

M =

 α , hs∞ F

M =

 α , s D g∞

 α , s B f∞

M =

 α , hs∞ F

M =

 α , s B f∞

Luca Boundary Value Problems (2017) 2017:102

M =

 α , s D g∞

 = M

Page 17 of 35

 , s B f∞

 = M

 , s D g∞

 = M

 , hs∞ F

where θ = min{σ α– , σ β– , σ γ – }. Theorem . Assume that (H) and (H) hold, σ ∈ (, ), α , α , α ≥  with α + α + α = α , α >  with  α +  α +  α = ,  α , α >  with  α +  α = ,  α , α >  with  α +  α = , ,  α ,     α >  with  α +  α = .  α , i i i s s , hs∞ ∈ (, ∞), M < M , M < M and M < M , then for each () If f , g , h , f∞ , g∞ λ ∈ (M , M ), μ ∈ (M , M ), ν ∈ (M , M ) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC). s s = , g∞ , hs∞ , fi , gi , hi ∈ (, ∞), M < M and M < M , then for each () If f∞ λ ∈ (M , ∞), μ ∈ (M , M ), ν ∈ (M , M ) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC). s s = , f∞ , hs∞ , fi , gi , hi ∈ (, ∞), M < M and M < M , then for each () If g∞ λ ∈ (M , M ), μ ∈ (M , ∞), ν ∈ (M , M ) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC). s s , g∞ , fi , gi , hi ∈ (, ∞), M < M and M < M , then for each () If hs∞ = , f∞  λ ∈ (M , M ), μ ∈ (M , M ), ν ∈ (M , ∞) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC). s s   , then for each λ ∈ (M , ∞), = g∞ = , hs∞ , fi , gi , hi ∈ (, ∞), M < M () If f∞  μ ∈ (M , ∞), ν ∈ (M , M ) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC). s s   , then for each λ ∈ (M , ∞), = hs∞ = , g∞ , fi , gi , hi ∈ (, ∞), M < M () If f∞   ), ν ∈ (M , ∞) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] μ ∈ (M , M for problem (S)-(BC). s s   , then for each λ ∈ (M , M   ), = hs∞ = , f∞ , fi , gi , hi ∈ (, ∞), M < M () If g∞ μ ∈ (M , ∞), ν ∈ (M , ∞) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC). s s = g∞ = hs∞ = , fi , gi , hi ∈ (, ∞), then for each λ ∈ (M , ∞), μ ∈ (M , ∞), () If f∞ ν ∈ (M , ∞) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC). s s , g∞ , hs∞ ∈ (, ∞) and at least one of fi , gi , hi is ∞, then for each λ ∈ (, M ), () If f∞ μ ∈ (, M ), ν ∈ (, M ) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC). s s = , g∞ , hs∞ ∈ (, ∞) and at least one of fi , gi , hi is ∞, then for each () If f∞ λ ∈ (, ∞), μ ∈ (, M ), ν ∈ (, M ) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC). s s = , f∞ , hs∞ ∈ (, ∞) and at least one of fi , gi , hi is ∞, then for each () If g∞  λ ∈ (, M ), μ ∈ (, ∞), ν ∈ (, M ) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC). s s , g∞ ∈ (, ∞) and at least one of fi , gi , hi is ∞, then for each () If hs∞ = , f∞ λ ∈ (, M ), μ ∈ (, M ), ν ∈ (, ∞) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC). s s = g∞ = , hs∞ ∈ (, ∞) and at least one of fi , gi , hi is ∞, then for each () If f∞   ) there exists a positive solution (u(t), v(t), w(t)), λ ∈ (, ∞), μ ∈ (, ∞), ν ∈ (, M t ∈ [, ] for problem (S)-(BC).

Luca Boundary Value Problems (2017) 2017:102

Page 18 of 35

s s () If f∞ = hs∞ = , g∞ ∈ (, ∞) and at least one of fi , gi , hi is ∞, then for each   ), ν ∈ (, ∞) there exists a positive solution (u(t), v(t), w(t)), λ ∈ (, ∞), μ ∈ (, M t ∈ [, ] for problem (S)-(BC). s s = hs∞ = , f∞ ∈ (, ∞) and at least one of fi , gi , hi is ∞, then for each () If g∞   ), μ ∈ (, ∞), ν ∈ (, ∞) there exists a positive solution (u(t), v(t), w(t)), λ ∈ (, M t ∈ [, ] for problem (S)-(BC). s s = g∞ = hs∞ =  and at least one of fi , gi , hi is ∞, then for each λ ∈ (, ∞), () If f∞ μ ∈ (, ∞), ν ∈ (, ∞) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] for problem (S)-(BC).

Proof We consider again the above cone P ⊂ Y and the operators Q , Q , Q and Q. We will also prove for this theorem some illustrative cases. s s , g∞ , hs∞ ∈ (, ∞). Let λ ∈ (M , M ), μ ∈ (M , M ), ν ∈ Case () We consider fi , gi , hi , f∞ (M , M ). We choose ε >  a positive number such that ε < fi , ε < gi , ε < hi and α α– θ σ (fi

– ε)A

α β– θ σ (gi

≤ λ,

– ε)C

 α ≥ μ, s + ε)D (g∞

 α ≥ λ, s + ε)B (f∞

≤ μ,

α γ – θ σ (hi

– ε)E

≤ ν,

 α ≥ ν. (hs∞ + ε)F

By using (H) and the definition of fi , gi , hi , we deduce that there exists R >  such that f (t, u, v, w) ≥ (fi – ε)(u + v + w), g(t, u, v, w) ≥ (gi – ε)(u + v + w), h(t, u, v, w) ≥ (hi – ε)(u + v + w) for all u, v, w ≥  with u + v + w ≤ R and t ∈ [σ , ]. We denote  = {(u, v, w) ∈ Y , (u, v, w) Y < R }. Let (u, v, w) ∈ P ∩ ∂  , that is, (u, v, w) Y = R or, equivalently, u + v + w = R . Because u(t) + v(t) + w(t) ≤ R for all t ∈ [, ], then by Lemma . we obtain for all t ∈ [σ , ] 



Q (u, v, w)(t) ≥ λ



t α– J (s)f s, u(s), v(s), w(s) ds





≥ λσ α– 





J (s)f s, u(s), v(s), w(s) ds







J (s) fi – ε u(s) + v(s) + w(s) ds

σ

≥ λσ α– σ



≥ λσ α– θ fi – ε α–



 σ

  J (s)(u, v, w)Y ds

  

 θ fi – ε A(u, v, w)Y ≥ α (u, v, w)Y ,

= λσ  

t β– J (s)g s, u(s), v(s), w(s) ds Q (u, v, w)(t) ≥ μ 



≥ μσ β–





J (s)g s, u(s), v(s), w(s) ds







J (s) gi – ε u(s) + v(s) + w(s) ds

σ

 ≥ μσ β–

σ



≥ μσ β– θ gi – ε = μσ

β–

 σ



  J (s)(u, v, w)Y ds

  

 θ gi – ε C (u, v, w)Y ≥ α (u, v, w)Y ,

Luca Boundary Value Problems (2017) 2017:102

 Q (u, v, w)(t) ≥ ν



Page 19 of 35



t γ – J (s)h s, u(s), v(s), w(s) ds



≥ νσ







J (s)h s, u(s), v(s), w(s) ds







J (s) hi – ε u(s) + v(s) + w(s) ds

γ – σ

 ≥ νσ γ –

σ



≥ νσ γ – θ hi – ε = νσ

γ –

θ



hi





 σ

  J (s)(u, v, w)Y ds

  

 – ε E(u, v, w)Y ≥ α (u, v, w)Y .

So     Q (u, v, w) ≥ Q (u, v, w)(σ ) ≥ α (u, v, w) , Y     Q (u, v, w) ≥ Q (u, v, w)(σ ) ≥ α (u, v, w) , Y     Q (u, v, w) ≥ Q (u, v, w)(σ ) ≥ α (u, v, w) . Y Then, for an arbitrary element (u, v, w) ∈ P ∩ ∂  , we deduce       Q(u, v, w) ≥ (α + α + α )(u, v, w) = (u, v, w) . Y Y Y

()

Now we define the functions f ∗ , g ∗ , h∗ : [, ] × R+ → R+ , f ∗ (t, x) = max≤u+v+w≤x f (t, u, v, w), g ∗ (t, x) = max≤u+v+w≤x g(t, u, v, w), h∗ (t, x) = max≤u+v+w≤x h(t, u, v, w), t ∈ [, ], x ∈ R+ . Then f (t, u, v, w) ≤ f ∗ (t, x), g(t, u, v, w) ≤ g ∗ (t, x), h(t, u, v, w) ≤ h∗ (t, x) for all t ∈ [, ], u, v, w ≥  and u + v + w ≤ x. The functions f ∗ (t, ·), g ∗ (t, ·), h∗ (t, ·) are nondecreasing for every t ∈ [, ], and they satisfy the conditions f ∗ (t, x) s ≤ f∞ , t∈[,] x

lim sup max x→∞

g ∗ (t, x) s ≤ g∞ , t∈[,] x

lim sup max x→∞

∗

lim sup max

x→∞ t∈[,]

h (t, x) ≤ hs∞ . x

Therefore, for ε > , there exists R¯  >  such that, for all x ≥ R¯  and t ∈ [, ], we have s s + ε)x, g ∗ (t, x) ≤ (g∞ + ε)x, h∗ (t, x) ≤ (hs∞ + ε)x. f (t, x) ≤ (f∞ We consider R = max{R , R¯  }, and we denote  = {(u, v, w) ∈ Y , (u, v, w) Y < R }. Let (u, v, w) ∈ P ∩ ∂  . By the definition of f ∗ , g ∗ , h∗ , we conclude ∗





f t, u(t), v(t), w(t) ≤ f ∗ t, (u, v, w)Y , 



h t, u(t), v(t), w(t) ≤ h∗ t, (u, v, w)Y ,





 g t, u(t), v(t), w(t) ≤ g ∗ t, (u, v, w)Y , ∀t ∈ [, ].

Then, for all t ∈ [, ], we obtain  Q (u, v, w)(t) ≤ λ





J (s)f s, u(s), v(s), w(s) ds ≤ λ



s

≤ λ f∞ +ε

 

 







 J (s)f ∗ s, (u, v, w)Y ds

    J (s)(u, v, w)Y ds ≤  α (u, v, w)Y ,

Luca Boundary Value Problems (2017) 2017:102



Page 20 of 35



Q (u, v, w)(t) ≤ μ



J (s)g s, u(s), v(s), w(s) ds ≤ μ



s

≤ μ g∞ +ε 



Q (u, v, w)(t) ≤ ν 

























 J (s)g ∗ s, (u, v, w)Y ds

    J (s)(u, v, w)Y ds ≤  α (u, v, w)Y ,



J (s)h s, u(s), v(s), w(s) ds ≤ ν

≤ ν hs∞ + ε





 J (s)h∗ s, (u, v, w)Y ds

    J (s)(u, v, w)Y ds ≤  α (u, v, w)Y .

α (u, v, w) Y , Q (u, v, w) ≤  α (u, v, w) Y , Therefore, we deduce Q (u, v, w) ≤  Q (u, v, w) ≤  α (u, v, w) Y . Hence, for (u, v, w) ∈ P ∩ ∂  , we conclude that       Q(u, v, w) ≤ ( α +  α +  α )(u, v, w)Y = (u, v, w)Y . Y

()

By using Lemma ., Theorem .(ii) and relations (), (), we deduce that Q has a fixed point (u, v, w) ∈ P ∩ (  \  ), which is a positive solution for our problem (S)-(BC). s s = , hi = ∞, f∞ , hs∞ , fi , gi ∈ (, ∞). Let λ ∈ (, M ), μ ∈ (, ∞), Case (). We consider g∞  ν ∈ (, M ). We choose ε >  such that ε ≤ νθ σ γ – E and ε≤

s B  α – λf∞ , λB

ε≤

s  – λf∞ B – νhs∞ F , μD

ε≤

 α – νhs∞ F . νF

The numerators of the above fractions are positive because λ <  α 

 α s B, f∞

s that is,  α > λf∞ B,

s s s ν < hs F , that is,  α > νhs∞ F, and  – λf∞ B – νhs∞ F =  α + α – λf∞ B – νhs∞ F = ( α – λf∞ B) + ∞  s ( α – νh∞ F) > . By using (H) and the definition of hi , we deduce that there exists R >  such that h(t, u, v, w) ≥ ε (u + v + w) for all u, v, w ≥  with u + v + w ≤ R and t ∈ [σ , ]. We denote

 = {(u, v, w) ∈ Y , (u, v, w) Y < R }. Let (u, v, w) ∈ P ∩ ∂  , that is, (u, v, w) Y = R . Because u(t) + v(t) + w(t) ≤ R for all t ∈ [, ], then by using Lemma ., we obtain





Q (u, v, w)(t) ≥ ν



t γ – J (s)h s, u(s), v(s), w(s) ds







≥ νσ γ – 

σ 

≥ νσ γ –

J (s) σ

≥ νσ γ – θ



J (s)h s, u(s), v(s), w(s) ds

 ε



 σ

 u(s) + v(s) + w(s) ds ε

  J (s)(u, v, w)Y ds

     = νσ γ – θ E(u, v, w)Y ≥ (u, v, w)Y , ε

∀t ∈ [σ , ].

Then Q (u, v, w) ≥ Q (u, v, w)(σ ) ≥ (u, v, w) Y , and       Q(u, v, w) ≥ Q (u, v, w) ≥ (u, v, w) . Y Y

()

Luca Boundary Value Problems (2017) 2017:102

Page 21 of 35

Now, using the functions f ∗ , g ∗ , h∗ defined in the proof of case (), we have f ∗ (t, x) s , ≤ f∞ t∈[,] x

g ∗ (t, x) = , x→∞ t∈[,] x

lim sup max x→∞

lim max

h∗ (t, x) ≤ hs∞ . t∈[,] x

lim sup max x→∞

Therefore, for ε > , there exists R¯  >  such that, for all x ≥ R¯  and t ∈ [, ], we deduce s f (t, x) ≤ (f∞ + ε)x, g ∗ (t, x) ≤ εx, h∗ (t, x) ≤ (hs∞ + ε)x. We consider R = max{R , R¯  }, and we denote  = {(u, v, w) ∈ Y , (u, v, w) Y < R }. Let (u, v, w) ∈ P ∩ ∂  . Then, for all t ∈ [, ], we obtain ∗

 Q (u, v, w)(t) ≤ λ





J (s)f s, u(s), v(s), w(s) ds





 J (s)f ∗ s, (u, v, w)Y ds



 ≤λ



s

≤ λ f∞ +ε







  J (s)(u, v, w)Y ds

  s   α  – λf∞ B  s ≤ λ f∞ +  B(u, v, w)Y λB 

  s = λf∞ B + α (u, v, w)Y ,   

J (s)g s, u(s), v(s), w(s) ds Q (u, v, w)(t) ≤ μ 





 J (s)g ∗ s, (u, v, w)Y ds



≤μ 





≤ με 

  J (s)(u, v, w)Y ds

s   – λf∞ B – νhs∞ F  ≤μ D(u, v, w)Y μD 

  s =  – λf∞ B – νhs∞ F (u, v, w)Y ,   

J (s)h s, u(s), v(s), w(s) ds Q (u, v, w)(t) ≤ ν 

 ≤ν







 J (s)h∗ s, (u, v, w)Y ds



≤ ν hs∞ + ε

 



  J (s)(u, v, w)Y ds

    α  – νhs∞ F  ≤ ν hs∞ +  F (u, v, w)Y νF 

  = νhs∞ F +  α (u, v, w)Y .  Therefore   

 Q (u, v, w) ≤  λf s B +  α (u, v, w)Y , ∞ 

Luca Boundary Value Problems (2017) 2017:102

Page 22 of 35

  

 Q (u, v, w) ≤   – λf s B – νhs F (u, v, w) , ∞ ∞ Y    

 Q (u, v, w) ≤  νhs F +  α (u, v, w)Y . ∞ Y  Then, for (u, v, w) ∈ P ∩ ∂  , we conclude that   

 s Q(u, v, w) ≤  λf s B +  α +  – λf∞ B – νhs∞ F + νhs∞ F +  α (u, v, w)Y ∞ Y     = (u, v, w)Y .

()

By using Lemma ., Theorem .(ii) and relations (), (), we deduce that Q has a fixed point (u, v, w) ∈ P ∩ (  \  ), which is a positive solution for our problem (S)-(BC). s s Case (). We consider f∞ = hs∞ = , gi = ∞, g∞ , fi , hi ∈ (, ∞). Let λ ∈ (, ∞), μ ∈   ), ν ∈ (, ∞). We choose ε >  such that ε ≤ μθ σ β– C and (, M ε≤

s  – μg∞ D , λB

ε≤

s  – μg∞ D , μD

ε≤

s  – μg∞ D . νF

The numerator of the above fractions is positive because μ <

 , gs D

s that is,  – μg∞ D > .

By using (H) and the definition of gi , we deduce that there exists R >  such that g(t, u, v, w) ≥ ε (u + v + w) for all u, v, w ≥  with u + v + w ≤ R and t ∈ [σ , ]. We denote

 = {(u, v, w) ∈ Y , (u, v, w) Y < R }. Let (u, v, w) ∈ P ∩ ∂  , that is, (u, v, w) Y = R . Because u(t) + v(t) + w(t) ≤ R for all t ∈ [, ], then by using Lemma ., we obtain 



Q (u, v, w)(t) ≥ μ



t β– J (s)g s, u(s), v(s), w(s) ds



≥ μσ





β–



J (s)g s, u(s), v(s), w(s) ds

σ

 ≥ μσ β–

J (s) σ

≥ μσ β– θ



 ε





σ

 u(s) + v(s) + w(s) ds ε

  J (s)(u, v, w)Y ds

     = μσ β– θ C (u, v, w)Y ≥ (u, v, w)Y , ε

∀t ∈ [σ , ].

Then Q (u, v, w) ≥ Q (u, v, w)(σ ) ≥ (u, v, w) Y , and       Q(u, v, w) ≥ Q (u, v, w) ≥ (u, v, w) . Y Y

()

Now, using the functions f ∗ , g ∗ , h∗ defined in the proof of case (), we have f ∗ (t, x) = , x→∞ t∈[,] x lim max

g ∗ (t, x) s ≤ g∞ , t∈[,] x

lim sup max x→∞

h∗ (t, x) = . x→∞ t∈[,] x lim max

Therefore, for ε > , there exists R¯  >  such that, for all x ≥ R¯  and t ∈ [, ], we deduce s f (t, x) ≤ εx, g ∗ (t, x) ≤ (g∞ + ε)x, h∗ (t, x) ≤ εx. ∗

Luca Boundary Value Problems (2017) 2017:102

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We consider R = max{R , R¯  }, and we denote  = {(u, v, w) ∈ Y , (u, v, w) Y < R }. Let (u, v, w) ∈ P ∩ ∂  . Then, for all t ∈ [, ], we obtain  Q (u, v, w)(t) ≤ λ





J (s)f s, u(s), v(s), w(s) ds





 J (s)f ∗ s, (u, v, w)Y ds



 ≤λ







≤ λε 

  J (s)(u, v, w)Y ds

s    D   – μg∞ = λεB(u, v, w)Y ≤ λ B(u, v, w)Y λB 

  s =  – μg∞ D (u, v, w)Y ,   

J (s)g s, u(s), v(s), w(s) ds Q (u, v, w)(t) ≤ μ



 

≤μ 



 J (s)g ∗ s, (u, v, w)Y ds

s +ε ≤ μ g∞

 



  J (s)(u, v, w)Y ds



 s + ε D(u, v, w)Y = μ g∞   s    – μg∞ D s D(u, v, w)Y ≤ μ g∞ + μD 

  s = μg∞ D +  (u, v, w)Y ,   

J (s)h s, u(s), v(s), w(s) ds Q (u, v, w)(t) ≤ ν 

 ≤ν









 J (s)h∗ s, (u, v, w)Y ds 

≤ νε 

  J (s)(u, v, w)Y ds

s     – νg∞ D  F (u, v, w)Y = νεF (u, v, w)Y ≤ ν νF 

  s =  – μg∞ D (u, v, w)Y . 

Therefore   

 Q (u, v, w) ≤   – μg s D (u, v, w) , ∞ Y    

 Q (u, v, w) ≤   + μg s D (u, v, w) , ∞ Y    

 Q (u, v, w) ≤   – μg s D (u, v, w) . ∞ Y  Then, for (u, v, w) ∈ P ∩ ∂  , we conclude that     

 Q(u, v, w) ≤   – μg s D +  + μg s D +  – μg s D (u, v, w) = (u, v, w) . () ∞ ∞ ∞ Y Y Y 

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By using Lemma ., Theorem .(ii) and relations () and (), we deduce that Q has a fixed point (u, v, w) ∈ P ∩ (  \  ), which is a positive solution for our problem (S)-(BC). s s = g∞ = hs∞ = , fi = gi = ∞ and hi ∈ (, ∞). Let λ ∈ (, ∞), Case (). We consider f∞ μ ∈ (, ∞), ν ∈ (, ∞). We choose ε >  such that ε ≤ λθ σ α– A,

ε≤

 , λB

ε≤

 , μD

ε≤

 . νF

By using (H) and the definition of fi , we deduce that there exists R >  such that f (t, u, v, w) ≥ ε (u + v + w) for all u, v, w ≥  with u + v + w ≤ R and t ∈ [σ , ]. We denote

 = {(u, v, w) ∈ Y , (u, v, w) Y < R }. Let (u, v, w) ∈ P ∩ ∂  , that is, (u, v, w) Y = R . Because u(t) + v(t) + w(t) ≤ R for all t ∈ [, ], then by using Lemma ., we obtain 



Q (u, v, w)(t) ≥ λ



t α– J (s)f s, u(s), v(s), w(s) ds





≥ λσ α–





J (s)f s, u(s), v(s), w(s) ds

σ

 ≥ λσ α–

J (s) σ

≥ λσ

α–



 θ ε



 σ

 u(s) + v(s) + w(s) ds ε

  J (s)(u, v, w)Y ds

     = λσ α– θ A(u, v, w)Y ≥ (u, v, w)Y , ε

∀t ∈ [σ , ].

Then Q (u, v, w) ≥ Q (u, v, w)(σ ) ≥ (u, v, w) Y , and       Q(u, v, w) ≥ Q (u, v, w) ≥ (u, v, w) . Y Y

()

Now, using the functions f ∗ , g ∗ , h∗ defined in the proof of case (), we have f ∗ (t, x) = , x→∞ t∈[,] x lim max

g ∗ (t, x) = , x→∞ t∈[,] x

h∗ (t, x) = . x→∞ t∈[,] x

lim max

lim max

Therefore, for ε > , there exists R¯  >  such that f ∗ (t, x) ≤ εx, g ∗ (t, x) ≤ εx, h∗ (t, x) ≤ εx for all x ≥ R¯  and t ∈ [, ]. We consider R = max{R , R¯  }, and we denote  = {(u, v, w) ∈ Y , (u, v, w) Y < R }. Let (u, v, w) ∈ P ∩ ∂  . Then, for all t ∈ [, ], we obtain 



Q (u, v, w)(t) ≤ λ



J (s)f s, u(s), v(s), w(s) ds ≤ λ













 J (s)f ∗ s, (u, v, w)Y ds

      J (s)(u, v, w)Y ds = λεB(u, v, w)Y ≤ (u, v, w)Y ,       



Q (u, v, w)(t) ≤ μ J (s)g s, u(s), v(s), w(s) ds ≤ μ J (s)g ∗ s, (u, v, w)Y ds 

≤ λε



 ≤ με 





      J (s)(u, v, w)Y ds = μεD(u, v, w)Y ≤ (u, v, w)Y , 

Luca Boundary Value Problems (2017) 2017:102

 Q (u, v, w)(t) ≤ ν



Page 25 of 35



J (s)h s, u(s), v(s), w(s) ds ≤ ν



 





≤ νε 





 J (s)h∗ s, (u, v, w)Y ds

      J (s)(u, v, w)Y ds = νεF (u, v, w)Y ≤ (u, v, w)Y . 

Therefore Q (u, v, w) ≤  (u, v, w) Y , Q (u, v, w) ≤  (u, v, w) Y , Q (u, v, w) ≤  (u, v, w) Y . Then, for (u, v, w) ∈ P ∩ ∂  , we conclude that      Q(u, v, w) ≤ (u, v, w) . Y Y

()

By using Lemma ., Theorem .(ii) and relations () and (), we deduce that Q has a ¯  \  ), which is a positive solution for our problem (S)-(BC). fixed point (u, v, w) ∈ P ∩ (  Remark . Each of the cases ()-() of Theorem . contains seven cases as follows: {fi = ∞, gi , hi ∈ (, ∞)}, or {gi = ∞, fi , hi ∈ (, ∞)}, or {hi = ∞, fi , gi ∈ (, ∞)}, or {fi = gi = ∞, hi ∈ (, ∞)}, or {fi = hi = ∞, gi ∈ (, ∞)}, or {gi = hi = ∞, fi ∈ (, ∞)}, or {fi = gi = hi = ∞}. So the total number of cases from Theorem . is , which we grouped in  cases. Each of the cases ()-() contains four subcases because α , α , α ∈ (, ), or α =  and α = α = , or α =  and α = α = , or α =  and α = α = . Remark . In the paper [], the authors present only  cases (Theorems .-. from []) from  cases, namely the first nine cases of our Theorem .. They did not study the cases when some extreme limits are  and other are ∞. Besides, compared to Theorems .-. and .-. from [], our intervals for parameters λ, μ, ν presented in Theorem . (our cases ()-() and ()) are better than the corresponding ones from []. Remark . One can formulate existence results for the general case of the system of n  from Remark .. fractional differential equations ( S) with the boundary conditions (BC) fj (t,u ,...,un ) s According to the values of fj∞ = lim supu +···+un →∞ supt∈[,] u +···+un ∈ [, ∞), and fji = lim infu +···+un → inft∈[σ ,] grouped in n+ cases.

fj (t,u ,...,un ) u +···+un

∈ (, ∞], j = , . . . , n, we have n cases, which can be

4 Nonexistence of positive solutions We present in this section intervals for λ, μ and ν, for which there exist no positive solutions of problem (S)-(BC), viewed as fixed points of operator Q. Theorem . Assume that (H) and (H) hold. If there exist positive numbers A , A , A such that f (t, u, v, w) ≤ A (u + v + w), h(t, u, v, w) ≤ A (u + v + w),

g(t, u, v, w) ≤ A (u + v + w), ∀t ∈ [, ], u, v, w ≥ ,

()

then there exist positive constants λ , μ , ν such that, for every λ ∈ (, λ ), μ ∈ (, μ ), ν ∈ (, ν ) the boundary value problem (S)-(BC) has no positive solution.

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  Proof We define λ = A B , μ = A D , ν = A F , where B =  J (s) ds, D =  J (s) ds, F =   J (s) ds. We will show that for any λ ∈ (, λ ), μ ∈ (, μ ), ν ∈ (, ν ), problem (S)-(BC) has no positive solution. Let λ ∈ (, λ ), μ ∈ (, μ ), ν ∈ (, ν ). We suppose that (S)-(BC) has a positive solution (u(t), v(t), w(t)), t ∈ [, ]. Then we have 



u(t) = Q (u, v, w)(t) = λ



G (t, s)f s, u(s), v(s), w(s) ds







≤λ 



J (s)f s, u(s), v(s), w(s) ds





≤ λA



J (s) u(s) + v(s) + w(s) ds





≤ λA u + v + w





J (s) ds 

  = λA B(u, v, w)Y , ∀t ∈ [, ],  

v(t) = Q (u, v, w)(t) = μ G (t, s)g s, u(s), v(s), w(s) ds 





≤μ 



J (s)g s, u(s), v(s), w(s) ds





≤ μA



J (s) u(s) + v(s) + w(s) ds





≤ μA u + v + w





J (s) ds 

  = μA D(u, v, w)Y , ∀t ∈ [, ],  

G (t, s)h s, u(s), v(s), w(s) ds w(t) = Q (u, v, w)(t) = ν  ≤ν 

 



J (s)h s, u(s), v(s), w(s) ds



≤ νA





J (s) u(s) + v(s) + w(s) ds





≤ νA u + v + w   = νA F (u, v, w)Y ,





J (s) ds 

∀t ∈ [, ].

Therefore we conclude       u ≤ λA B(u, v, w)Y < λ A B(u, v, w)Y = (u, v, w)Y ,        v ≤ μA D(u, v, w)Y < μ A D(u, v, w)Y = (u, v, w)Y ,         w ≤ νA F (u, v, w)Y < ν A F (u, v, w)Y = (u, v, w)Y .  Hence we deduce (u, v, w) Y = u + v + w < (u, v, w) Y , which is a contradiction. So the boundary value problem (S)-(BC) has no positive solution. 

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Remark . In the proof of Theorem . we can also define λ = with α , α , α >  and α + α + α = .

α , A B

μ =

α , A D

ν =

α A F

s s Remark . If fs , gs , hs , f∞ , g∞ , hs∞ < ∞, then there exist positive constants A , A , A such that () holds (see also [] for a system with two equations), and then we obtain the conclusion of Theorem ..

Theorem . Assume that (H) and (H) hold. If there exist positive numbers σ ∈ (, ) and m >  such that ∀t ∈ [σ , ], u, v, w ≥ ,

f (t, u, v, w) ≥ m (u + v + w),

()

λ , μ >  and ν > , the then there exists a positive constant  λ such that, for every λ >  boundary value problem (S)-(BC) has no positive solution.  λ , Proof We define  λ = θσ α– m A , where A = σ J (s) ds. We will show that for every λ >   μ >  and ν > , problem (S)-(BC) has no positive solution. Let λ >  λ , μ >  and ν > . We suppose that (S)-(BC) has a positive solution (u(t), v(t), w(t)), t ∈ [, ]. Then we obtain 



u(t) = Q (u, v, w)(t) = λ



G (t, s)f s, u(s), v(s), w(s) ds







≥ λt α–



J (s)f s, u(s), v(s), w(s) ds

σ

 ≥ λσ α–





J (s)m u(s) + v(s) + w(s) ds

σ





≥ λθ σ α– m



J (s) u + v + w ds

σ

= λθ σ

α–

  m A(u, v, w)Y .

Therefore we deduce       u ≥ u(σ ) ≥ λθ σ α– m A(u, v, w)Y >  λ θ σ α– m A(u, v, w)Y = (u, v, w)Y , and so, (u, v, w) Y = u + v + w > (u, v, w) Y , which is a contradiction. Therefore the boundary value problem (S)-(BC) has no positive solution.  In a similar manner, we obtain the following theorems. Theorem . Assume that (H) and (H) hold. If there exist positive numbers σ ∈ (, ) and m >  such that g(t, u, v, w) ≥ m (u + v + w),

∀t ∈ [σ , ], u, v, w ≥ ,

()

μ and ν > , the then there exists a positive constant  μ such that, for every λ > , μ >  boundary value problem (S)-(BC) has no positive solution.

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In Theorem . we define  μ =

 , θσ β– m C

where C =



σ J (s) ds.

Theorem . Assume that (H) and (H) hold. If there exist positive numbers σ ∈ (, ) and m >  such that ∀t ∈ [σ , ], u, v, w ≥ ,

h(t, u, v, w) ≥ m (u + v + w),

()

ν , the then there exists a positive constant  ν such that, for every λ > , μ >  and ν >  boundary value problem (S)-(BC) has no positive solution. In Theorem . we define  ν =

 , θσ γ – m E

where E =



σ J (s) ds.

Remark . i >  and f (t, u, v, w) >  for all t ∈ [σ , ] and u, v, w ≥  with (a) If for σ ∈ (, ), fi , f∞ u + v + w > , then relation () holds, and we obtain the conclusion of Theorem .. i >  and g(t, u, v, w) >  for all t ∈ [σ , ] and u, v, w ≥  with (b) If for σ ∈ (, ), gi , g∞ u + v + w > , then relation () holds, and we obtain the conclusion of Theorem .. (c) If for σ ∈ (, ), hi , hi∞ >  and h(t, u, v, w) >  for all t ∈ [σ , ] and u, v, w ≥  with u + v + w > , then relation () holds, and we obtain the conclusion of Theorem .. Theorem . Assume that (H) and (H) hold. If there exist positive numbers σ ∈ (, ) and m , m >  such that f (t, u, v, w) ≥ m (u + v + w), g(t, u, v, w) ≥ m (u + v + w),

()

∀t ∈ [σ , ], u, v, w ≥ ,

then there exist positive constants  μ such that, for every λ >  μ and ν > , λ and  λ , μ >  the boundary value problem (S)-(BC) has no positive solution. Proof We define  λ =

Then, for every λ >  λ ,    μ > μ and ν > , problem (S)-(BC) has no positive solution. Indeed, let λ > λ , μ > μ and ν > . We suppose that (S)-(BC) has a positive solution (u(t), v(t), w(t)), t ∈ [, ]. Then, in a similar manner as in the proof of Theorem ., we deduce  θσ α– m A

(=

  u ≥ λθ σ α– m A(u, v, w)Y ,

 λ ) 

and  μ =

 θσ β– m C

(=

 μ ). 

  v ≥ μθ σ β– m C (u, v, w)Y ,

and so   (u, v, w) = u + v + w ≥ u + v Y 

 ≥ λθ σ α– m A + μθ σ β– m C (u, v, w)Y 

 >  λ θ σ α– m A +  μ θ σ β– m C (u, v, w)Y         (u, v, w) = (u, v, w) , + = Y Y   which is a contradiction. Therefore the boundary value problem (S)-(BC) has no positive solution. 

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Remark . In the proof of Theorem . we can also define  λ = with  α , α >  with  α +  α = .

 α , θσ α– m A

 μ =

 α θσ β– m C

In a similar manner we obtain the following theorems. Theorem . Assume that (H) and (H) hold. If there exist positive numbers σ ∈ (, ) and m , m >  such that f (t, u, v, w) ≥ m (u + v + w), h(t, u, v, w) ≥ m (u + v + w),

()

∀t ∈ [σ , ], u, v, w ≥ ,

    ν  such that, for every λ >  ν, then there exist positive constants  λ , μ >  and ν >  λ and  the boundary value problem (S)-(BC) has no positive solution.  In Theorem . we define  λ =

  λ =

 α θσ α– m A



and  ν =

 α θσ γ – m E

 θσ α– m A

(=

 λ ) 



and  ν =

 θσ γ – m E

(= ν ), or in general

with  α , α > ,  α +  α = .

Theorem . Assume that (H) and (H) hold. If there exist positive numbers σ ∈ (, ) and m , m >  such that g(t, u, v, w) ≥ m (u + v + w), h(t, u, v, w) ≥ m (u + v + w),

()

∀t ∈ [σ , ], u, v, w ≥ , 







ν  such that, for every λ > , μ >  μ and ν >  ν, then there exist positive constants  μ and  the boundary value problem (S)-(BC) has no positive solution.   μ  (=  ) and  ν = In Theorem . we define  μ = θσ β–  m C    α α  and  ν  = γ – with  α , α > ,  α +  α = . μ = β– θσ

m C

θσ

 θσ γ – m E

(= ν ), or in general

m E

Remark . i i , gi , g∞ >  and f (t, u, v, w) > , g(t, u, v, w) >  for all t ∈ [σ , ] (a) If for σ ∈ (, ), fi , f∞ and u, v, w ≥  with u + v + w > , then relation () holds, and we obtain the conclusion of Theorem .. i (b) If for σ ∈ (, ), fi , f∞ , hi , hi∞ >  and f (t, u, v, w) > , h(t, u, v, w) >  for all t ∈ [σ , ] and u, v, w ≥  with u + v + w > , then relation () holds, and we obtain the conclusion of Theorem .. i (c) If for σ ∈ (, ), gi , g∞ , hi , hi∞ >  and g(t, u, v, w) > , h(t, u, v, w) >  for all t ∈ [σ , ] and u, v, w ≥  with u + v + w > , then relation () holds, and we obtain the conclusion of Theorem .. Theorem . Assume that (H) and (H) hold. If there exist positive numbers σ ∈ (, ) and m , m , m >  such that f (t, u, v, w) ≥ m (u + v + w), h(t, u, v, w) ≥ m (u + v + w),

g(t, u, v, w) ≥ m (u + v + w), ∀t ∈ [σ , ], u, v, w ≥ ,

()

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then there exist positive constants λˆ  , μˆ  and νˆ  such that, for every λ > λˆ  , μ > μˆ  and ν > νˆ  , the boundary value problem (S)-(BC) has no positive solution. Proof We define λˆ  =

Then, for every λ > λˆ  , μ > μˆ  , ν > νˆ  , problem (S)-(BC) has no positive solution. Indeed, let λ > λˆ  , μ > μˆ  and ν > νˆ  . We suppose that (S)-(BC) has a positive solution (u(t), v(t), w(t)), t ∈ [, ]. Then, in a similar manner as in the proof of Theorem ., we deduce  , θσ α– m A

μˆ  =

  u ≥ λθ σ α– m A(u, v, w)Y ,   w ≥ νθ σ γ – m E(u, v, w)Y ,

 , θσ β– m C

νˆ  =

 . θσ γ – m E

  v ≥ μθ σ β– m C (u, v, w)Y ,

and so   (u, v, w) = u + v + w Y 

 ≥ λθ σ α– m A + μθ σ β– m C + νθ σ γ – m E (u, v, w)Y 

 > λˆ  θ σ α– m A + μˆ  θ σ β– m C + νˆ  θ σ γ – m E (u, v, w)Y   = (u, v, w)Y , which is a contradiction. Therefore, the boundary value problem (S)-(BC) has no positive solution.  Remark . In the proof of Theorem ., we can also define λˆ  = νˆ  =

α γ – θσ m F

, where α , α , α >  with α + α + α = .

α , μˆ  θσ α– m A

=

α , θσ β– m C

i i Remark . If for σ ∈ (, ), fi , f∞ , gi , g∞ , hi , hi∞ >  and f (t, u, v, w) > , g(t, u, v, w) > , h(t, u, v, w) >  for all t ∈ [σ , ], u, v, w ≥ , u + v + w > , then relation () holds, and we have the conclusion of Theorem ..

Remark . The conclusions of Theorems .-. and .-. remain valid for general systems of Hammerstein integral equations of the form ⎧  ⎪ ⎪ ⎨u(t) = λ  G (t, s)f (s, u(s), v(s), w(s)) ds, t ∈ [, ],  v(t) = μ  G (t, s)g(s, u(s), v(s), w(s)) ds, t ∈ [, ], ⎪ ⎪  ⎩ w(t) = ν  G (t, s)h(s, u(s), v(s), w(s)) ds, t ∈ [, ],

()

with positive parameters λ, μ, ν, and instead of assumptions (H)-(H), the following assumptions are satisfied:  The functions G , G , G : [, ] × [, ] → R are continuous, and there exist the con(H) tinuous functions J , J , J : [, ] → R and σ ∈ (, ), α, β, γ >  such that (a)  ≤ Gi (t, s) ≤ Ji (s), ∀t, s ∈ [, ], i = , , ; (b) G (t, s) ≥ t α– J (s), G (t, s) ≥ t β– J (s), G (t, s) ≥ t γ – J (s), ∀t, s ∈ [, ];  (c) σ Ji (s) ds > , i = , , .  The functions f , g, h : [, ] × R+ × R+ × R+ → R+ are continuous. (H)

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5 Examples Let n = , m = , l = , α =  , β =  , γ =  , p = , q =  , p =  , q =  , p =  , q =  ,   N = , M = , L = , ξ =  , ξ =  , a = , a =  , η =  , b = , ζ =  , ζ =  , ζ =  , c = , c = , c = . We consider the system of fractional differential equations ⎧ / ⎪ ⎪D+ u(t) + λf (t, u(t), v(t), w(t)) = , ⎨ (S )

⎪ ⎪ ⎩

t ∈ (, ),

D/ + v(t) + μg(t, u(t), v(t), w(t)) = ,

t ∈ (, ),

D/ + w(t) + νh(t, u(t), v(t), w(t)) = ,

t ∈ (, ),

with the multi-point boundary conditions

(BC )

⎧  /  ⎪ u () = D/ ⎪u() = u () = , + u(t)|t=  +  D+ u(t)|t=  , ⎪ ⎪ ⎪ ⎨v() = v () = v () = v () = , D/ v(t)| = D/ v(t)| t=

+

+

⎪ ⎪ w() = w () = w () = , ⎪ ⎪ ⎪ ⎩ / / / D+ w(t)|t= = D/ + w(t)|t=  + D+ w(t)|t=  + D+ w(t)|t=  . 

√



/

t=  ,



(/) (/) –  (/) ≈ . > ,  = We have  = – π ≈ . > ,  = (/) (/) (/) / / – ( +  +  ) ≈ . > . So assumption (H) is satisfied. (/) / (/) Besides we deduce

⎧  ⎨t / ( – s)/ – (t – s)/ ,  ≤ s ≤ t ≤ , g (t, s) = (/) ⎩t / ( – s)/ ,  ≤ t ≤ s ≤ , ⎧ ⎨t( – s)/ – (t – s),  ≤ s ≤ t ≤ , g (t, s) = ⎩t( – s)/ ,  ≤ t ≤ s ≤ ,

⎧ ⎨t / ( – s)/ – (t – s)/ ,  ≤ s ≤ t ≤ ,  g (t, s) = (/) ⎩t / ( – s)/ ,  ≤ t ≤ s ≤ , ⎧ ⎨t / ( – s)/ – (t – s)/ ,  ≤ s ≤ t ≤ ,  g (t, s) = (/) ⎩t / ( – s)/ ,  ≤ t ≤ s ≤ ,

⎧ ⎨t / ( – s)/ – (t – s)/ ,  ≤ s ≤ t ≤ ,  g (t, s) = (/) ⎩t / ( – s)/ ,  ≤ t ≤ s ≤ , ⎧  ⎨t / ( – s)/ – (t – s)/ ,  ≤ s ≤ t ≤ , g (t, s) = (/) ⎩t / ( – s)/ ,  ≤ t ≤ s ≤ . Then we obtain         t / g , s + g , s ,

       t / G (t, s) = g (t, s) + g , s ,

  G (t, s) = g (t, s) +

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          t / g , s + g , s + g ,s ,

   

  ( – s)/  – ( – s)/ , h (s) = √ s( – s)/ , h (s) = (/)  π

G (t, s) = g (t, s) +



 ( – s)/  – ( – s)/ , (/)           J (s) = √ s( – s)/ + g , s + g , s

     π ⎧  √ ⎪ s( – s)/ +    [( – s)/ + s – ],  ≤ s <  , ⎪ ⎨ π = √ π s( – s)/ +   [( – s)/ + s – ],  ≤ s <  ,  ⎪ ⎪ ⎩ √   / s( – s) + ( – s)/ , ≤ s ≤ ,

   π  

   ( – s)/  – ( – s)/ + g , s J (s) = (/)

  ⎧ ⎨  ( – s)/ ( – ( – s)/ ) + / [( – s)/ – ( – s)/ ],  ≤ s <  ,

 (/)  = (/)  ⎩  ( – s)/ ( – ( – s)/ ) + / ( – s)/ , ≤ s ≤ , (/)

 (/)        

     / / g  – ( – s) + ( – s) , s + g , s + g ,s J (s) = (/)

    ⎧   / / ⎪ ⎪ ⎪ (/) ( – s) ( – ( – s) ) + /  (/) ⎪ ⎪ ⎪ × [( + / + / )( – s)/ – ( – s)/ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ / / / ⎪  ≤ s <  , ⎪ –  ( – s) – ( – s) ], ⎪ ⎪ ⎪  ⎪ ⎪ ( – s)/ ( – ( – s)/ ) + /  (/) ⎪ ⎨ (/)  = × [( + / + / )( – s)/ – / ( – s)/ – ( – s)/ ],  ≤ s <  , ⎪ ⎪ ⎪  ⎪ ( – s)/ ( – ( – s)/ ) + /  (/) ⎪ ⎪ (/)  ⎪ ⎪ ⎪  / / / ⎪ × [( +  +  )( – s) – ( – s)/ ], ≤ s <  , ⎪ ⎪  ⎪ ⎪ ⎪  ⎪ ( – s)/ ( – ( – s)/ ) ⎪ (/) ⎪ ⎪ ⎪ ⎩ +   ( + / + / )( – s)/ , ≤ s ≤ .

h (s) =

/  (/)



Now we choose σ =  ∈ (, ) and then θ = –/ ≈ .. We also obtain A =    J (s) ds ≈ ., B =  J (s) ds ≈ ., C = / J (s) ds ≈ ., D =   /     J (s) ds ≈ ., E = / J (s) ds ≈ ., F =  J (s) ds ≈ .. Example  We consider the functions q + sin v) (t + )[ p (u + v + w) + ](u + v + w)( , u+v+w+ √ q + cos w) t + [ p (u + v + w) + ](u + v + w)( g(t, u, v, w) = , u+v+w+

f (t, u, v, w) =

h(t, u, v, w) =

t  [ p (u + v + w) + ](u + v + w)( q + sin u) , u+v+w+

p , p > ,  q , q , q > . for t ∈ [, ], u, v, w ≥ , where  p ,

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√ √ i i We have fs =  q , gs = ( q + ), hs =  q , f∞ =  q – ), g∞ =  q – ), hi∞ = p ( p (   /  q – ). For α = α = α =  α =  α =  α =  , we obtain L = p (q –)A , L = q B , L = p (     √  ,   p ( q –)C

/

L = √(q +)D , L = p(q –)E , and L = q F .  The conditions L < L , L < L and L < L become  p ( q – ) / B , >  A q

 p ( q – ) / D > / ,   C q + 

 p ( q – ) / F . >  E q

q –)  –) For example, if p (qq–) ≥ , pq(q+ ≥  and p ( ≥ , then the above con q ditions are satisfied. As an example, we consider  q = ,  q = ,  q = ,  p = ,  p = ,  p = , and then the inequalities L < L , L < L and L < L are satisfied. In this case, L ≈ ., L ≈ ., L ≈ ., L ≈ ., L ≈ ., L ≈ .. By Theorem .() we deduce that for every λ ∈ (L , L ), μ ∈ (L , L ) and ν ∈ (L , L ) there exists a positive solution (u(t), v(t), w(t)), t ∈ [, ] of problem (S )(BC ). √ √ s s Because fs =  q , f∞ =  p ( q + ), gs = ( =  q + ), hs =  q , hs∞ = q + ), g∞ p (  p ( q + ), then by Theorem . and Remark ., we conclude that for any λ ∈ (, λ ), μ ∈ (, μ ) and ν ∈ (, ν ), problem (S )-(BC ) has no positive solution, where λ = A B , μ = A D , ν = A F . If we consider as above  p = ,  q = ,  p = ,  q = ,  p = √ ,  q = , then A = , A =   ≈ , A = . Therefore we obtain λ ≈ . × – , μ ≈ . × – , ν ≈ . × – . i i Because fi , f∞ , gi , g∞ , hi , hi∞ >  and f (t, u, v, w) > , g(t, u, v, w) > , h(t, u, v, w) >  for all t ∈ [/, ] and u, v, w ≥  with u+v+w > , we can also apply Theorem . and Remark ..    Here λˆ  = θσ α– , μˆ  = θσ β– and νˆ  = θσ γ – . For the functions f , g, h presented m A m E m C √  ˆ above, we have m = , m =  , m = , λ ≈ ., μˆ  ≈ ., νˆ  ≈ 

.. So, if λ > ., μ > . and ν > ., problem (S )-(BC ) has no positive solution. Example  We consider the functions

f (t, u, v, w) = t a u + v + w , h(t, u, v, w) = (u + v + w)c ,



g(t, u, v, w) = ( – t)b eu+v+w –  ,

t ∈ [, ], u, v, w ≥ ,

i i = ∞, gs = b , g∞ = ∞, hs = , hi∞ = ∞. where a, b > , c > . We have fs = , f∞ By Theorem .(), for any λ ∈ (, ∞), μ ∈ (,  L ) and ν ∈ (, ∞), with  L = bD , prob  lem (S )-(BC ) has a positive solution. Here D =  J (s) ds ≈ .. For example, if  b = , we obtain  L = D ≈ .. We can also use Theorem ., because g(t, u, v, w) ≥ u + v + w for all t ∈ [/, ] and u, v, w ≥ , that is, m = . Because  μ = θσ β– m C ≈ ., we deduce that for every  λ > , μ > . and ν > , the boundary value problem (S )-(BC ) has no positive solution.

6 Conclusion By using the Guo-Krasnosel’skii fixed point theorem, in this paper, we present conditions for the nonlinearities f , g and h, and intervals for the positive parameters λ, μ and ν such

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that problem (S)-(BC) has positive solutions. In addition, we investigate the nonexistence of positive solutions for this problem. The novelties of our paper are the system (S) (a system with three fractional differential equations, unlike the well-studied case of a system with two equations) and the boundary conditions (BC) which, in contrast with other recent papers, contain fractional derivatives in t =  and in various intermediate points. The obtained theorems improve and extend the results from paper [], where only a few cases are presented for the existence of positive solutions. Our results remain valid, with similar proofs, for general systems of Hammerstein integral equations of the form () under  and (H).  assumptions (H)

Funding Own funds. Availability of data and materials Not applicable. Ethics approval and consent to participate Not applicable. Competing interests The author declares that he has no competing interests. Consent for publication Not applicable. Author’s contributions All authors read and approved the final manuscript.

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 27 April 2017 Accepted: 21 June 2017 References 1. Henderson, J, Luca, R: Positive solutions for a system of nonlocal fractional boundary value problems. Fract. Calc. Appl. Anal. 16(4), 985-1008 (2013) 2. Shen, C, Zhou, H, Yang, L: Positive solution of a system of integral equations with applications to boundary value problems of differential equations. Adv. Differ. Equ. 2016, 260 (2016) 3. Henderson, J, Luca, R: Positive solutions for a system of fractional differential equations with coupled integral boundary conditions. Appl. Math. Comput. 249, 182-197 (2014) 4. Henderson, J, Luca, R: Nonexistence of positive solutions for a system of coupled fractional boundary value problems. Bound. Value Probl. 2015, 138 (2015) 5. Henderson, J, Luca, R: Positive solutions for a system of semipositone coupled fractional boundary value problems. Bound. Value Probl. 2016, 61 (2016) 6. Henderson, J, Luca, R, Tudorache, A: Positive solutions for a fractional boundary value problem. Nonlinear Stud. 22(1), 139-151 (2015) 7. Henderson, J, Luca, R, Tudorache, A: On a system of fractional differential equations with coupled integral boundary conditions. Fract. Calc. Appl. Anal. 18(2), 361-386 (2015) 8. Jiang, J, Liu, L, Wu, Y: Symmetric positive solutions to singular system with multi-point coupled boundary conditions. Appl. Math. Comput. 220(1), 536-548 (2013) 9. Jiang, J, Liu, L, Wu, Y: Positive solutions to singular fractional differential system with coupled boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 18(11), 3061-3074 (2013) 10. Luca, R, Tudorache, A: Positive solutions to a system of semipositone fractional boundary value problems. Adv. Differ. Equ. 2014, 179 (2014) 11. Wang, Y, Liu, L, Wu, Y: Positive solutions for a class of higher-order singular semipositone fractional differential systems with coupled integral boundary conditions and parameters. Adv. Differ. Equ. 2014, 268 (2014) 12. Yuan, C: Two positive solutions for (n – 1, 1)-type semipositone integral boundary value problems for coupled systems of nonlinear fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 17(2), 930-942 (2012) 13. Yuan, C, Jiang, D, O’Regan, D, Agarwal, RP: Multiple positive solutions to systems of nonlinear semipositone fractional differential equations with coupled boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2012, 13 (2012) 14. Caballero, J, Cabrera, I, Sadarangani, K: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. Abstr. Appl. Anal. 2012, Article ID 303545 (2012) 15. Das, S: Functional Fractional Calculus for System Identification and Controls. Springer, New York (2008) 16. Graef, JR, Kong, L, Kong, Q, Wang, M: Uniqueness of positive solutions of fractional boundary value problems with non-homogeneous integral boundary conditions. Fract. Calc. Appl. Anal. 15(3), 509-528 (2012)

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