Positive solutions of a system for nonlinear singular higher-order ...

Xie and Xie Boundary Value Problems (2016) 2016:134 DOI 10.1186/s13661-016-0643-2

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Positive solutions of a system for nonlinear singular higher-order fractional differential equations with fractional multi-point boundary conditions Shengli Xie1* and Yiming Xie2 *

Correspondence: [email protected] 1 School of Mathematics & Physics, Urban Construction College, Anhui Jianzhu University, Hefei, Anhui 230601, P.R. China Full list of author information is available at the end of the article

Abstract This paper deals with the existence and multiplicity of positive solutions for a system of nonlinear singular higher-order fractional differential equations with fractional multi-point boundary conditions. The main tool used in the proof is fixed point index theory. Some limit type conditions for ensuring the existence of positive solutions are given, and our conditions are suitable for more general functions. MSC: 34B10; 34B16; 34B18; 45G10 Keywords: singular fractional differential equations; positive solution; cone; fixed point index

1 Introduction We discuss the following multi-point boundary problem of the system for nonlinear singular higher-order fractional differential equations: ⎧ α D+ u(x) + h (x)f (x, u(x), v(x)) = , x ∈ (, ), ⎪ ⎪ ⎪ ⎨Dβ v(x) + h (x)f (x, u(x), v(x)) = , x ∈ (, ),   + p (i) ⎪ Dμ+ u() = k= ak Dμ+ u(ξk ), u () = ,  ≤ i ≤ n – , ⎪ ⎪ q ⎩ (j) Dν+ v() = k= bk Dν+ v(ηk ), v () = ,  ≤ j ≤ m – ,

()

β

where Dα+ , D+ are the standard Riemann-Liouville fractional derivative of order α ∈ (n – , n], β ∈ (m – , m], μ ∈ [, n – ], ν ∈ [, m – ] for n, m ∈ N+ and n, m ≥ , ai , bj ∈ R+ , i = , , . . . , p, j = , , . . . , q for p, q ∈ N+ , fk ∈ C([, ] × R+ × R+ , R+ ), hk ∈ C((, ), R+ ), R+ = [, +∞), hk (x) (k = , ) is allowed to be singular at x =  and/or x =  and  < ξ < ξ < · · · < ξp < ,

 < η < η < · · · < ηq < .

Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, Bode’s analysis © 2016 Xie and Xie. 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.

Xie and Xie Boundary Value Problems (2016) 2016:134

Page 2 of 18

of feedback amplifiers, capacitor theory, electrical circuits, electro-analytical chemistry, biology, control theory, fitting of experimental data, and so forth. Hence fractional differential equations have attracted great research interest in recent years, and for more details we refer the reader to [–] and the references cited therein. Recently, the existence and multiplicity of positive solutions for the nonlinear fractional differential equations have been researched, see [–] and the references therein. For instance, Zhang et al. [] studied the existence of two positive of following singular fractional boundary value problems: Dα+ u(t) + f (t, u(t)) = ,  < t < , β β β D+ u() = ∞ u() = , D+ u() = , i= ξi D+ u(ηi ), where Dα+ is the standard Riemann-Liouville fractional derivative of order α ∈ (, ], α–β– β ∈ [, ], ξi , ηi ∈ (, ), α – β ≥  with ∞ < . i= ξi ηi In [–], the authors studied the existence of a positive solutions of two types of systems for nonlinear fractional differential equations Dα+ u(t) + λf (t, u(t), v(t)) = , β D+ v(t) + μg(t, u(t), v(t)) = ,

t ∈ (, ), t ∈ (, ),

()

with boundary conditions:

u() = u () = · · · = u(n–) () = , v() = v () = · · · = v(n–) () = , u(i) () = , v(j) () = ,

 ≤ i ≤ n – ,  ≤ j ≤ m – ,

 u() =  v(t) dH(t),  v() =  u(t) dK(t),

()

p u() = k= ak u(ξk ), q v() = k= bk v(ηk ),

and u(i) () = v(i) () = , γ D+ u() = φ (u),

 ≤ i ≤ n – , γ D+ v() = φ (v),

γ

β

where Dα+ , D+ , and D+ are the standard Riemann-Liouville fractional derivative, α, β ∈ (n – , n], γ ∈ [, n – ] for n ≥ , λ, μ > . Equations () with λf (t, u, v) and μg(t, u, v) replaced by f (t, v) and g(t, u), respectively, the existence and multiplicity of positive solutions of the system (), () was investigated in []. The extreme limits fδs =: lim sup max

u+v→δ t∈[,]

f (t, u, v) , t∈[θ,–θ] u + v

fδi =: lim inf min u+v→δ

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

gδs =: lim sup max

u+v→δ t∈[,]

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

g(t, u, v) , t∈[θ,–θ] u + v

gδi =: lim inf min u+v→δ

are used in [, ], where θ ∈ (,  ), δ = + or +∞. Some similar extreme limits are used in [, , –]. However, for equation systems [–, –] and a single equation using the extreme limits, there is no essential difference.


Page 3 of 18

Motivated by the above mentioned work, in this paper, we present some limit type conditions and discuss the existence and multiplicity of positive solutions of the singular fractional multi-point boundary problems () by using fixed point index theory in a cone. The results obtained here are different from those in [–, –], and some examples explain our conditions are applicable for more general functions.

2 Preliminaries Definition . [] The Riemann-Liouville fractional integral of order α >  of a function u : (, +∞) → R is given by  (α)

α u(t) = I+

t

(t – s)α– u(s) ds,

t > ,



provided the right side is pointwise defined on (, +∞), where (α) is the Euler gamma function. The Riemann-Liouville fractional derivative of order α >  of a continuous function u : (, +∞) → R is given by Dα+ u(t) =

dn  (n – α) dt n

t

(t – s)n–α– u(s) ds,

t > ,



where n = [α] + , [α] denotes the integer part of number α, provided the right side is pointwise defined on (, +∞). Lemma . [] Let x ∈ Lp (, ) ( ≤ p ≤ +∞), ρ > σ > . ρ ρ–σ ρ σ ρ+σ σ x(t) = x(t), I+ I+ x(t) = I+ x(t) hold at almost every (i) Dσ+ I+ x(t) = I+ x(t), Dσ+ I+ point t ∈ (, ). If ρ + σ > , then the above third equation holds at any point of [, ]; (ii) Dσ+ t ρ– = (ρ)t ρ–σ – / (ρ – σ ), t > . Lemma . [] Let α > , n = [α] +  for α ∈/ N and n = α for α ∈ N, n is the smallest integer greater than or equal to α. Then, for any y ∈ L (, ), the solution of the fractional differential equation Dα+ u(t) + y (t) =  ( < t < ) is u(t) +

 (α)

t

(t – s)α– y (s) ds = c t α– + c t α– + · · · + cn t α–n ,

 < t < ,



where c , c , . . . , cn are arbitrary real constants. p α–μ– ∈ [, ), α ∈ (n – , n], μ ∈ [, n – ] (n ≥ ) and y ∈ C[, ]. Lemma . Let j= aj ξj Then the solution of the fractional boundary value problem ⎧ α ⎪ ⎨D+ u(t) + y (t) = ,  < t < , u(i) () = ,  ≤ i ≤ n – , ⎪ p ⎩ μ D+ u() = j= aj Dμ+ u(ξj ),

()

is given by



G (t, s)y (s) ds,

u(t) = 

()


G (t, s) = g (t, s) +

where d =  –

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p t α– aj h (ξj , s), d j=

()

p

α–μ– , j= aj ξj

 t α– ( – s)α–μ– – (t – s)α– ,  ≤ s ≤ t ≤ , g (t, s) = (α) t α– ( – s)α–μ– ,  ≤ t ≤ s ≤ ,  t α–μ– ( – s)α–μ– – (t – s)α–μ– ,  ≤ s ≤ t ≤ , h (t, s) = (α) t α–μ– ( – s)α–μ– ,  ≤ t ≤ s ≤ .

()

()

Proof By using Lemma ., the solution for the above equation is u(t) =

– (α)

t

(t – s)α– y (s) ds + c t α– + c t α– + · · · + cn t α–n ,



where c , c , . . . , cn are arbitrary real constants. By u() = , we have cn = . Then – u(t) = (α)

t

(t – s)α– y (s) ds + c t α– + c t α– + · · · + cn– t α–n+ .

()



Differentiating (), we have u (t) =

–α (α)

t

(t – s)α– y (s) ds + c (α – )t α– + · · · + cn– (α – n + )t α–n .



By u () = , we have cn– = . Similarly, we get c = c = · · · = cn– = . Hence – u(t) = (α) By Dμ+ u() =

t

(t – s)α– y (s) ds + c t α– .

()



p

μ j= aj D+ u(ξj )

and Lemma ., we get

t  α–μ– α–μ– c (α)t – (t – s) y (s) ds , (α – μ) 

ξj p   α–μ– α–μ– ( – s) y (s) ds – aj (ξj – s) y (s) ds . c = d (α)   j=

μ D+ u(t) =

Substituting c into (), we see that the unique solution of the problem () is t α– u(t) = d (α)



( – s)α–μ– y (s) ds – 

t

p j=

ξj

aj

(ξj – s)α–μ– y (s) ds



 (t – s)α– y (s) ds (α) 

t  = t α– ( – s)α–μ– – (t – s)α– y (s) ds (α) 

  – d  α– t α– ( – s)α–μ– y (s) ds – t ( – s)α–μ– y (s) ds + d  t  –


+

p



g (t, s)y (s) ds +

= 

+ 



= 

=

α–μ–

y (s) ds



j=

ξj

(ξj – s)

aj

Page 5 of 18

 p t α– aj ξjα–μ– ( – s)α–μ– y (s) ds d j= ξj

ξj

ξjα–μ– ( – s)α–μ– – (ξj – s)α–μ– y (s) ds

 p t α– g (t, s)y (s) ds + aj h (ξj , s)y (s) ds d j= 



G (t, s)y (s) ds, 

i.e. () holds. Conversely, if u ∈ C[, ] is a solution of the integral equation (), from Lemma . we easily see that u satisfies the equation and boundary conditions of (). Lemma . Under the assumptions of Lemma ., the functions g (t, s) and h (ξj , s) defined by () and () have the following properties: (i) g (t, s) ≥  is continuous on [, ] × [, ] and g (t, s) >  for all t, s ∈ (, ). (ii) maxt∈[,] g (t, s) = g (, s) for all s ∈ [, ], where g (, s) =

 ( – s)α–μ– – ( – s)α– . (α)

(iii) g (t, s) ≥ t α– g (, s) for all t, s ∈ [, ], and there are θ ∈ (,  ), γα ∈ (, ) such that mint∈Jθ g (t, s) ≥ γα g (, s) for each s ∈ [, ], where Jθ = [θ ,  – θ ], γα = θ α– . (iv) h (t, s) ≥  is continuous on [, ] × [, ] and h (t, s) >  for all t, s ∈ (, ). Proof For the proof of (i), (ii), and (iv), respectively, see Theorem . in [] and Lemma . in []. It remains to prove (iii). We have by () g (t, s) ≥

 α– s α– ( – s)α–μ– –  – ≥ t α– g (, s), t (α) t

g (t, s) ≥

 α– t ( – s)α–μ– ≥ t α– g (, s), (α)

 ≤ s < t ≤ ,

 ≤ t ≤ s ≤ .

Hence g (t, s) ≥ t α– g (, s) for all t, s ∈ [, ], and so mint∈Jθ g (t, s) ≥ γα g (, s) for all s ∈ [, ]. From Lemma . it is easy to get the following result. Lemma . Under the assumptions of Lemma ., the Green’s function G (t, s) defined by () has the following properties: (i) G (t, s) ≥  is continuous on [, ] × [, ] and G (t, s) >  for all t, s ∈ (, ). (ii) maxt∈[,] G (t, s) = G (, s) for each s ∈ [, ], where G (, s) = g (, s) +

p  ( – s)α–μ– . aj h (ξj , s) ≤ d j= d (α)

()


Page 6 of 18

(iii) G (t, s) ≥ t α– G (, s) for all t, s ∈ [, ], there are θ ∈ (,  ), γα ∈ (, ) such that mint∈Jθ G (t, s) ≥ γα G (, s) for each s ∈ [, ], where Jθ = [θ ,  – θ ], γα = θ α– . We can also formulate similar results as Lemmas .-. above, for the fractional differential equation with fractional multi-point boundary conditions ⎧ β ⎪ ⎨D+ v(t) + y (t) = ,  < t < , v(i) () = ,  ≤ i ≤ m – , ⎪ q ⎩ ν D+ v() = j= bj Dν+ v(ηj ), where m, q ∈ N+ , m ≥ ,  < η < · · · < ηq < , bj ≥  for all j = , , . . . , q and y ∈ C[, ]. We q β–ν– denote by d =  – j= bj ηj , γβ and g (t, s), h (ηj , s), G (t, s), G (, s) the corresponding constants and functions for the problem () defined in a similar manner to d , γα and g (t, s), h (ξj , s), G (t, s), G (, s), respectively. From Lemma . we know that G (t, s) and G (t, s) have the same properties, and there exists γβ = θ β– such that mint∈Jθ G (t, s) ≥ γβ G (, s). Let γ = min{γα , γβ },

–θ

δk =

Gk (, y)hk (y) dy,



μk =

Gk (, y)hk (y) dy

(k = , ).



θ

For convenience we list the following assumptions: (H ) hk ∈ C((, ), R+ ), hk (x) ≡  on any subinterval of (, ) and




 uniformly with respect to a(v) b(u) + (x, u) ∈ Jθ × R and (x, v) ∈ Jθ × R+ , respectively (specifically, f = f = +∞); () limu→+ a(Cb(u)) = +∞ for any constant C > . u (H ) There exists τ ∈ (, +∞) such that f∞ = lim sup v→+∞

f (x, u, v) < +∞, vτ

f∞ = lim sup u→+∞

f (x, u, v) 

=

uτ

uniformly with respect to (x, u) ∈ [, ] × R+ and (x, v) ∈ [, ] × R+ , respectively (specifically, f∞ = f∞ = ). (H ) There exist p, q ∈ C(R+ , R+ ) such that () p is concave and strictly increasing on R+ ; () f∞ = lim infv→+∞ f (x,u,v) > , f∞ = lim infu→+∞ f (x,u,v) >  uniformly with respect to p(v) q(u) + (x, u) ∈ Jθ × R and (x, v) ∈ Jθ × R+ , respectively (specifically, f∞ = f∞ = +∞); () limu→+∞ p(Cq(u)) = +∞ for any constant C > . u (H ) There exists ς ∈ (, +∞) such that f = lim sup v→+

f (x, u, v) < +∞, vς

f = lim sup u→+

f (x, u, v) 

=

uς

uniformly with respect to (x, u) ∈ [, ] × R+ and (x, v) ∈ [, ] × R+ , respectively (specifically, f = f = ).


Page 7 of 18

(H ) There exists r >  such that f (x, u, v) and f (x, u, v) are nondecreasing in the second variable and the third variable u, v ∈ [, r] for all x ∈ [, ], and f (x, γ r, γ r) ≥ (γ δ )– r,

f (x, γ r, γ r) ≥ (γ δ )– r,

∀x ∈ [θ ,  – θ ].

(H ) There exists R > r >  such that f (x, u, v) and f (x, u, v) are nondecreasing in the second variable and the third variable u, v ∈ [, R] for all x ∈ [, ], and f (x, R, R) ≤ (μ )– R,

f (x, R, R) ≤ (μ )– R,

∀x ∈ [, ].

Let E = C[, ], u = maxt∈[,] |u(t)|, the product space E × E be equipped with norm (u, v) = u + v for (u, v) ∈ E × E, and P = u ∈ E : u(t) ≥ , t ∈ [, ], min u(t) ≥ γ u . t∈Jθ

Then E is a real Banach space and P is a cone of E. By (H ), we can define operators Ak : P × P → E as follows:



Ak (u, v)(x) =

Gk (x, y)h (y)fk y, u(y), v(y) dy (k = , ),

()



A(u, v) = (A (u, v), A (u, v)). Clearly (u, v) is a positive solution of the system () if and only if (u, v) ∈ P × P \ {(, )} is a fixed point of A. Let Br = {u ∈ E : u < r} for r > . Lemma . Assume that the condition (H ) is satisfied, then A : P × P → P × P is a completely continuous operator. Proof First of all, we show that A : P × P → P is uniformly bounded continuous operator. For any (u, v) ∈ P × P, it follows from () that A (u, v)(x) ≥ , x ∈ [, ], A (u, v) ≤



G (, y)h (y)f y, u(y), v(y) dy



and



min A (u, v)(x) ≥ γ x∈Jθ

G (, y)h (y)f y, u(y), v(y) dy ≥ γ A (u, v).



Hence A (P × P) ⊂ P. Let ⊂ P × P be a bounded set, we assume that (u, v) ≤ d for any (u, v) ∈ . Let M = maxx∈[,],(u,v)∈ f (x, u, v) + . Equation () and (H ) imply that A (u, v) ≤





G (, y)h (y)f y, u(y), v(y) dy ≤ M



G (, y)h (y) dy < +∞, 

from this we know that A () is a bounded set. We show that A : P × P → P is continuous. Let (un , vn ), (u , v ) ∈ P × P, (un , vn ) – (u , v ) = (un – u , vn – v ) →  (n → ∞). Then {(un , vn )} is a bounded set, we assume


Page 8 of 18

that (un , vn ) ≤ d (n = , , , . . .). From (H ), f ∈ C([, ] × R+ × R+ , R+ ), A (un , vn ) – A (u , v ) ≤



G (, y)h (y)f y, un (y), vn (y) – f y, u (y), v (y) dy



and the Lebesgue control convergent theorem, we know that A is a continuous operator. Now we show that A is equicontinuous on . For any given ε > , taking δ ∈ d (α)ε (, min{ Ml , }), for each (u, v) ∈ , x , x ∈ [, ], x < x , and x – x < δ, we have by  (α–) () and () A (u, v)(x ) – A (u, v)(x )  G (x , y) – G (x , y) h (y)f y, u(y), v(y) dy = 

=

x



≤ =

x

+

+



G (x , y) – G (x , y) h (y)f y, u(y), v(y) dy

x

x

M α– x – xα–  d (α)

x 



x

+

( – y)α–μ– h (y) dy

+ x

x

Ml (α – ) Ml (α – ) Ml α– < x – xα– (x – x ) < δ < ε.  d (α)  d (α) d (α)

By means of the Arzela-Ascoli theorem, A : P × P → P is completely continuous. Similarly, we can prove that A : P × P → P is completely continuous. Hence A : P × P → P × P is a completely continuous operator. Lemma . [] Assume that A : Br ∩ P → P is a completely continuous operator. If there exists u ∈ P \ {} such that u = Au + λu ,

∀λ ≥ , u ∈ ∂Br ∩ P,

then the fixed point index i(A, Br ∩ P, P) = . Lemma . [, ] Assume that A : Br ∩ P → P is a completely continuous operator. () If u Au or Au ≤ u for all u ∈ ∂Br ∩ P, then the fixed point index i(A, Br ∩ P, P) = . () If u Au or Au ≥ u for all u ∈ ∂Br ∩ P, then the fixed point index i(A, Br ∩ P, P) = . In the following, we adopt the convention that C , C , C , . . . stand for different positive constants. Let r = {(u, v) ∈ E × E : (u, v) < r} for r > .

3 Existence of a positive solution Theorem . Assume that the conditions (H )-(H ) are satisfied, then the system () has at least one positive solution. Proof By (H ), there are ξ > , η >  and a sufficiently small ρ >  such that f (x, u, v) ≥ ξ a(v),

∀(x, u) ∈ Jθ × R+ ,  ≤ v ≤ ρ,

f (x, u, v) ≥ η b(u),

∀(x, v) ∈ Jθ × R+ ,  ≤ u ≤ ρ,

()


Page 9 of 18

and a K b(u) ≥

K u, ξ η δ δ γ 

∀u ∈ [, ρ],

()

where K = max{η γ G (, y)h (y) : y ∈ Jθ }. We claim that ∀λ ≥ , (u, v) ∈ ∂ρ ∩ (P × P),

(u, v) = A(u, v) + λ(ϕ, ϕ),

where ϕ ∈ P \ {}. If not, there are λ ≥  and (u, v) ∈ ∂ρ ∩ (P × P) such that (u, v) = A(u, v) + λ(ϕ, ϕ), then u ≥ A (u, v), v ≥ A (u, v). By using the monotonicity and concavity of a(·), Jensen’s inequality and Lemma ., we have by () and ()



u(x) ≥

G (x, y)h (y)f y, u(y), v(y) dy





≥ ξ γα

G (, y)h (y)a v(y) dy





≥ ξ γα



G (, y)h (y)a 

≥ ξ γ



–θ



a η γ G (, z)h (z)b u(z) dz dy



a K– η γ G (, z)h (z)K b u(z) dz dy

G (, y)h (y)



θ

–θ

≥ ξ γ

G (, y)h (y) θ

≥ ξ η γ  K–

–θ

≥ ξ η γ  δ K–

–θ



θ

–θ

G (, y)h (y)G (, z)h (z)a K b u(z) dz dy

θ

 ≥ δ γ

η G (y, z)h (z)b u(z) dz dy

–θ

G (, z)h (z)a K b u(z) dz

θ

G (, z)h (z)u(z) dz ≥ u,

x ∈ Jθ .

()

θ

Consequently, u = . Next, () and () yield a v(x) ≥ a



G (x, y)h (y)f y, u(y), v(y) dy





≥

a η γ G (, y)h (y)b u(y) dy



≥ η γ K–

–θ

G (, y)h (y)a K b u(y) dy

θ

–θ  G (, y)h (y)u(y) dy ≥ ξ δ δ γ  θ



–θ  G (, y)h (y) dy G (, z)h (z)a v(z) dz ≥ δ δ γ θ 

–θ  ≥ G (, z)h (z)a v(z) dz ≥ a v , x ∈ Jθ , δ γ θ

()


Page 10 of 18

this means that a(v) = . It follows from the strict monotonicity of a(v) and a() =  that v = . Hence (u, v) = , which is a contradiction. Lemma . implies that i A, ρ ∩ (P × P), P × P = .

()

On the other hand, by (H ), there exist ζ >  and C > , C >  such that f (x, u, v) ≤ ζ vτ + C ,

∀(x, u, v) ∈ [, ] × R+ × R+ ,



f (x, u, v) ≤ ε u τ + C ,

()

∀(x, u, v) ∈ [, ] × R+ × R+ ,

where ε = min

 

.



,



μ (ζ μ ) τ μ (ζ μ ) τ

Let W = (u, v) ∈ P × P : (u, v) = λA(u, v),  ≤ λ ≤  . We prove that W is bounded. Indeed, for any (u, v) ∈ W , there exists λ ∈ [, ] such that u = λA (u, v), v = λA (u, v). Then () implies that



u(x) ≤ A (u, v)(x) ≤ ζ

G (, y)h (y)vτ (y) dy + C , 



v(x) ≤ A (u, v)(x) ≤ ε



G (, y)h (y)u τ (y) dy + C . 

Consequently,

 τ  G (, y)h (y) dy ε G (, z)h (z)u τ (z) dz + C + C



u(x) ≤ ζ 



 τ  ≤ ζ μ ε G (, z)h (z)u τ dz + C + C 

≤ ζ μ



v(x) ≤ ε

(u, v) ζ μ

τ

τ + C

+ C ,

()

τ  G (, y)h (y) dy ζ G (, z)h (z)vτ (z) dz + C + C





 τ τ ≤ ε μ ζ G (, z)h (z)v dz + C + C 

≤

 

(ζ μ ) τ

τ  ζ μ (u, v) + C τ + C .

()

Since 

lim

w→+∞

ζ μ [( ζwμ ) τ + C ]τ w

 = , 



lim

w→+∞

(ζ μ wτ + C ) τ   τ

(ζ μ ) w

 = , 


Page 11 of 18

there exists r > r, when (u, v) > r , () and () yield u(x) ≤

 (u, v) + C , v(x) ≤  (u, v) + C .  

Hence (u, v) ≤ (C + C ) and W is bounded. Select G > (C + C ). We obtain from the homotopic invariant property of fixed point index that i A, G ∩ (P × P), P × P = i θ , G ∩ (P × P), P × P = .

()

Equations () and () yield i A, (G \ ρ ) ∩ (P × P), P × P = i A, G ∩ (P × P), P × P – i A, ρ ∩ (P × P), P × P = . So A has at least one fixed point on (G \ ρ ) ∩ (P × P). This means that the system () has at least one positive solution. Theorem . Assume that the conditions (H ), (H ), and (H ) are satisfied. Then the system () has at least one positive solution. Proof By (H ), there are ξ > , η > , C > , C > , and C >  such that f (x, u, v) ≥ ξ p(v) – C ,

f (x, u, v) ≥ η q(u) – C ,

(x, u, v) ∈ Jθ × R+ × R+ ,

and p K q(u) ≥

K u – C , ξ η δ δ γ 

u ∈ R+ ,

()

where K = max{η γ G (, y)h (y) : y ∈ Jθ }. Then we have



A (u, v)(x) ≥ ξ

G (x, y)h (y)p v(y) dy – C ,

x ∈ Jθ ,



A (u, v)(x) ≥ η



()

G (x, y)h (y)q u(y) dy – C ,

x ∈ Jθ .



We affirm that the set W = (u, v) ∈ P × P : (u, v) = A(u, v) + λ(ϕ, ϕ), λ ≥  is bounded, where ϕ ∈ P \ {}. Indeed, (u, v) ∈ W implies that u ≥ A (u, v), v ≥ A (u, v) for some λ ≥ . We have by ()

u(x) ≥ ξ



G (x, y)h (y)p v(y) dy – C ,

x ∈ Jθ ,

()

G (x, y)h (y)q u(y) dy – C ,

x ∈ Jθ .

()

 

v(x) ≥ η 


Page 12 of 18

By the monotonicity and concavity of p(·) as well as Jensen’s inequality, () implies that  η G (x, y)h (y)q u(y) dy p v(x) + C ≥ p 



≥

p η γ G (, y)h (y)q u(y) dy



≥ η γ K–

–θ

G (, y)h (y)p K q u(y) dy,

x ∈ Jθ .

()

θ

Since p(v(x)) ≥ p(v(x) + C ) – p(C ), we have by (), (), and ()



u(x) ≥ ξ γ

G (, y)h (y) p v(y) + C – p(C ) dy – C



–θ

≥ ξ γ

G (, y)h (y)p v(y) + C dy – C

θ

≥ ξ η γ



K–

θ

≥ ξ η γ  δ K– ≥ (δ γ )–

–θ

G (, y)h (y)

–θ

G (, z)h (z)p K q u(z) dz dy – C

θ

–θ

–θ

G (, z)h (z)p K q u(z) dz – C

θ

G (, z)h (z)u(z) dz – C ≥ u – C ,

x ∈ Jθ .

()

θ

Hence u ≤ C . Since p(v(x)) ≥ γ p(v) for x ∈ Jθ , v ∈ P, it follows from (), (), and () that p v(x) ≥ p v(x) + C – p(C )

–θ ≥ η γ K– G (, y)h (y)p K q u(y) dy – p(C ) θ

–θ  G (, y)h (y)u(y) dy – C ξ δ δ γ  θ

–θ

  ≥ G (, y)h (y) dy G (, z)h (z)p v(z) dz – C δ δ γ θ 

–θ ≥ δ– G (, z)h (z)p v dz – C ≥

θ

= p v – C ,

x ∈ Jθ .

Hence p(v) ≤ C . By () and () of the condition (H ), we know that limv→+∞ p(v) = +∞, thus there exists C >  such that v ≤ C . This shows W is bounded. Then there exists a sufficiently large K >  such that (u, v) = A(u, v) + λ(ϕ, ϕ),

∀(u, v) ∈ ∂K ∩ (P × P), λ ≥ .

Lemma . yields i A, K ∩ (P × P), P × P = .

()


Page 13 of 18

On the other hand, by (H ), there is a σ >  and sufficiently small ρ >  such that f (x, u, v) ≤ σ vς ,

∀(x, u) ∈ [, ] × R+ , v ∈ [, ρ],



f (x, u, v) ≤ ε u ς ,

()

∀(x, v) ∈ [, ] × R+ , u ∈ [, ρ],

where  ς – ε = min σ μ μ ς , μ–  .

We claim that ∀(u, v) ∈ ∂ρ ∩ (P × P).

(u, v) A(u, v),

()

If not, there exists a (u, v) ∈ ∂ρ ∩ (P × P) such that (u, v) ≤ A(u, v), that is, u ≤ A (u, v), v ≤ A (u, v). Then () implies that

G (x, y)h (y)f y, u(y), v(y) dy



u(x) ≤ 



≤σ

G (, y)h (y)vς (y) dy 



≤σ



G (, y)h (y) 

ς dy





≤σ

G (y, z)h (z)f z, u(z), v(z) dz

ς  G (, y)h (y) dy G (, z)h (z)f z, u(z), v(z) dz







= σ μ

ς

G (, z)h (z)f z, u(z), v(z) dz 



ς

≤ σ μ ε



ς

G (, z)h (z)u ς (z) dz 

 ς ς ≤ σ μ ε μ u ≤ u, 

x ∈ [, ],

()

and

G (x, y)h (y)f y, u(y), v(y) dy



v(x) ≤ 

≤ ε









G (, y)h (y)u τ (y) dy ≤ ε μ u ς ≤ u ς ,

x ∈ [, ].

()



Equations () and () imply that (u, v) = , which contradicts (u, v) = ρ, and the inequality () holds. Lemma . yields i A, ρ ∩ (P × P), P × P = . We have by () and () i A, (K \ ρ ) ∩ (P × P), P × P = i A, K ∩ (P × P), P × P – i A, ρ ∩ (P × P), P × P = –.

()


Page 14 of 18

Hence A has a fixed point on (K \ ρ ) ∩ (P × P). This means that the system () has at least one positive solution. Theorem . Assume that the conditions (H ), (H ), and (H ) are satisfied. Then the system () has at least one positive solution. Proof Since γ r ≤ u(x), v(x) ≤ r for (u, v) ∈ ∂r ∩ (P × P), x ∈ [θ ,  – θ ], we know from (H ) that

A (u, v)(x) ≥

–θ

G (x, y)h (y)f (y, γ r, γ r) dy θ

≥

δ– r

–θ

G (, y)h (y) dy = r,

x ∈ [θ ,  – θ ],

θ

A (u, v)(x) ≥ –θ θ G (x, y)h (y)f (y, γ r, γ r) dy

–c – G (, y)h (y) dy = r, x ∈ [θ ,  – θ ]. ≥ δ r c

Hence A(u, v) > r = (u, v) for any (u, v) ∈ ∂r ∩ (P × P). Lemma . yields i A, r ∩ (P × P), P × P = .

()

On the other hand, for any x ∈ [, ],  ≤ u, v ≤ R, (H ) implies that



A (u, v)(x) ≤

G (, y)h (y)f (y, R, R) dy ≤

R , 

G (, y)h (y)f (y, R, R) dy ≤

R . 



A (u, v)(x) ≤





Hence A(u, v) ≤ R = (u, v) for (u, v) ∈ ∂R ∩ (P × P). Lemma . yields i A, R ∩ (P × P), P × P = .

()

We have by () and () i A, (R \ r ) ∩ (P × P), P × P = i A, R ∩ (P × P), P × P – i A, r ∩ (P × P), P × P = .

()

So A has a fixed point on (R \ r ) ∩ (P × P). This means that the system () has at least one positive solution.

4 Existence of multiple positive solutions Theorem . Assume that the conditions (H ), (H ), (H ), and (H ) hold. Then the system () has at least two positive solutions. Proof We may take G > r > σ such that (), (), and () hold. Then we have i A, (G \ r ) ∩ (P × P), P × P = i A, G ∩ (P × P), P × P – i A, r ∩ (P × P), P × P = ,


i A, (r \ σ ) ∩ (P × P), P × P = i A, r ∩ (P × P), P × P – i A, σ ∩ (P × P), P × P = –.

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()

Hence A has a fixed point on (G \ r ) ∩ (P × P) and (r \ σ ) ∩ (P × P), respectively. This means the system () has at least two positive solutions. Theorem . Assume that the conditions (H ), (H ), (H ), and (H ) hold. Then the system () has at least two positive solutions. Proof We may take K > R > ρ such that (), (), and () hold. Then we have i A, (K \ R ) ∩ (P × P), P × P = i A, K ∩ (P × P), P × P – i A, R ∩ (P × P), P × P = –, i A, (R \ ρ ) ∩ (P × P), P × P = i A, R ∩ (P × P), P × P – i A, ρ ∩ (P × P), P × P = .

()

Hence A has a fixed point on (K \ R ) ∩ (P × P) and (R \ ρ ) ∩ (P × P), respectively. This means the system () has at least two positive solutions. Theorem . Assume that the conditions (H ), (H ), (H ), (H ), and (H ) hold. Then the system () has at least three positive solutions. Proof We may take K > R > r > σ such that (), (), (), and () hold. From the proof of Theorem ., Theorem ., and Theorem . we know that (), (), and () hold. Hence A has a fixed point on (K \ R ) ∩ (P × P), (R \ r ) ∩ (P × P) and (r \ σ ) ∩ (P × P), respectively. Hence the system () has at least three positive solutions. Similar to the proof of Theorem ., we can get the following result. Theorem . Assume that (H ) holds. If there are l positive numbers dk , Dk (k = , , . . . , l) with d < D < d < D < · · · < dl < Dl , such that f (x, u, v) and f (x, u, v) are nondecreasing in the second variable and the third variable u, v ∈ [, Dl ] for all x ∈ [, ], and (H ) f (x, γ dk , γ dk ) ≥ (γ δ )– dk , f (x, γ dk , γ dk ) ≥ (γ δ )– dk for all x ∈ Jθ , k = , , . . . , l, Dk – Dk (H ) f (x, Dk , Dk ) ≤ μ–   , f (x, Dk , Dk ) ≤ μ  for all x ∈ [, ], k = , , . . . , l. Then the system () has at least l positive solutions (uk , vk ) satisfying dk ≤ (uk , vk ) ≤ Dk ,

k = , , . . . , l.

5 Some examples In the following, we give some examples to illustrate our main results. In Examples .-., the meaning of α, β, μ, ν is the same as in the system ().


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Example . Let h (x) = /( – x)α–μ– , h (x) = /( – x)β–ν– , x ∈ (, ), f (x, u, v) = ex ( +   e–(u+v) ), f (x, u, v) =  – e–(u+v) , x ∈ [, ], u, v ∈ R+ , a(v) = v  , b(u) = u  , τ = /. Clearly,





( – y)α–μ– h (y) dy = 

( – y)β–ν– h (y) dy = , 

 but  hk (y) dy = +∞ (k = , ) for α – μ –  ≥ , β – ν –  ≥ . The results of [–, , ] are not suitable for the problem. It is easy to verify that the conditions (H )-(H ) hold, hence Theorem . implies that the system () has at least one positive solution. Here f (x, u, v) and f (x, u, v) are sublinear on u and v at  and +∞. 

Example . Let hk (x) be as in Example ., f (x, u, v) = ex ( + e–(u+v) ), f (x, u, v) = u  , a(v) =  v  , b(u) = u , τ = /. It is easy to verify that the conditions (H )-(H ) hold, Theorem . implies that the system () has at least one positive solution. Here f (x, u, v) is sublinear on u and v at  and +∞, whereas f (x, u, v) is superlinear on u at  and +∞. Example . Let hk (x) be as in Example ., f (x, u, v) = ( + e–u )v , f (x, u, v) = u , p(v) =  v  , q(u) = u , ς = . It is easy to verify that the conditions (H ), (H ), and (H ) hold. Theorem . shows that the system () has at least one positive solution. Here f (x, u, v) is superlinear on v at  and +∞, f (x, u, v) is superlinear on u at  and +∞. 

Example . Let hk (x) be as in Example ., f (x, u, v) = ( + e–u )v  , f (x, u, v) = ( + e–v )u ,  p(v) = v  , q(u) = u , ς = /. It is easy to see that the conditions (H ), (H ), and (H ) hold. Theorem . shows that the system () has at least one positive solution. Here f (x, u, v) is sublinear on v at  and +∞, whereas f (x, u, v) is superlinear on u at  and +∞. Example . Consider the system of nonlinear singular fractional differential equations with fractional three-point boundary conditions: ⎧   ⎪ u(x) + h (x)f (x, u(x), v(x)) = , x ∈ (, ), D+ ⎪ ⎪ ⎪ ⎨  D+ v(x) + h (x)f (x, u(x), v(x)) = , x ∈ (, ), √ ⎪   ⎪ () = , u () = u (  ), u() = u ⎪ ⎪ √ ⎩   v () =  v (  ), v() = v () = , where α = β =  , μ = ν = , a = b = √  f (x, u, v) =  π v  ,

√

 , ξ 

()



= η =  , h (x) = ( – x)–  ,

x ∈ [, ], v ≥ ,

√ u , x, u ∈ [, ], f (x, u, v) =  π  u  , x ∈ [, ], u ≥ .

By a simple calculation, we have d =  , γ =  , δ =  

   

G (, s)h (s) ds =

√  √– .  π

Take r =

, τ = ς = in (H ) and (H ), it is easy to verify that the conditions (H ), (H ), (H ), and (H ) hold. From Theorem . one concludes that the system () has two positive solutions. Example . Consider the singular system (), where √ f (x, u, v) =  π v ,

x ∈ [, ], v ≥ ,

 √ u  , x, u ∈ [, ], f (x, u, v) =  π  u  , x ∈ [, ], u ≥ .


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Take r = , τ = ς =  in (H ) and (H ), it is easy to verify that the conditions (H ), (H ), (H ), and (H ) hold. From Theorem . one concludes that the system () has two positive solutions. Example . Consider the singular system (), where √   π  v  , x, v ∈ [, ], f (x, u, v) = v  , x ∈ [, ], v ≥ ,  √   π u +u , x, u ∈ [, ],  f (x, u, v) = u  , x ∈ [, ], u ≥ . 

By a simple calculation, we get μ = μ =  

 



 G (, s)h (s) ds =

√ (√ –) .  π

Take R = , a(v) =



p(v) = v , b(u) = u , q(u) = u , it is easy to see that the conditions (H ), (H ), (H ), and (H ) hold. From Theorem . one concludes that the system () has two positive solutions. Example . Consider the singular system (), where √ π  f (x, u, v) = v , 

√ x ≥ , v ≥ ,

f (x, u, v) =



π  u , 

x ≥ , u ≥ .



Take R = , a(v) = p(v) = v  , b(u) = q(u) = u  , it is easy to see that the conditions (H ), (H ), (H ), and (H ) hold. From Theorem . one concludes that the system () has two positive solutions. Remark . From Examples .-. we know that the conditions (H )-(H ) are applicable to more general functions and our results are different from those in [–, –]. Remark . If n, m ≥ , n –  < α ≤ n, m –  < β ≤ m, μ = ν =  in the system (), all our conclusion is true because the corresponding Green’s function gk (t, s) (k = , ) satisfies a Harnack-like inequality (see []). Hence our results improve and generalize some corresponding results in [, , ] and [].

Competing interests The authors declare that they have no competing interests. Authors’ contributions The authors, S Xie and Y Xie, contributed to each part of this work equally and read and approved the final version of the manuscript. Author details 1 School of Mathematics & Physics, Urban Construction College, Anhui Jianzhu University, Hefei, Anhui 230601, P.R. China. 2 School of Civil Engineering, Anhui Jianzhu University, Hefei, Anhui 230601, P.R. China. Acknowledgements The work was supported by Natural Science Foundation of Anhui Province and Anhui Provincial Education Department (1508085MA08, KJ2014A043), P.R. China. Received: 29 April 2016 Accepted: 11 July 2016 References 1. 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)


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