Ru et al. Advances in Difference Equations (2017) 2017:320 DOI 10.1186/s13662-017-1377-x

RESEARCH

Open Access

Some results on the fractional order Sturm-Liouville problems Yuanfang Ru1 , Fanglei Wang1* , Tianqing An1 and Yukun An2 *

Correspondence: [email protected] Department of Mathematics, College of Science, Hohai University, Nanjing, 210098, China Full list of author information is available at the end of the article 1

Abstract In this work, we introduce some new results on the Lyapunov inequality, uniqueness and multiplicity results of nontrivial solutions of the nonlinear fractional Sturm-Liouville problems 

D0+ (p(t)u (t)) + (t)f (u(t)) = 0, 1 < q ≤ 2, t ∈ (0, 1), α u(0) – β p(0)u (0) = 0, γ u(1) + δ p(1)u (1) = 0, q

1 where α , β , γ , δ are constants satisfying 0 = |βγ + αγ 0 p(1τ ) dτ + αδ | < +∞, p(·) is positive and continuous on [0, 1]. In addition, some existence results are given for the problem 

D0+ (p(t)u (t)) + (t)f (u(t), λ) = 0, 1 < q ≤ 2, t ∈ (0, 1), α u(0) – β p(0)u (0) = 0, γ u(1) + δ p(1)u (1) = 0, q

where λ ≥ 0 is a parameter. The proof is based on the fixed point theorems and the Leray-Schauder nonlinear alternative for single-valued maps. MSC: Primary 26A33; 34A08 Keywords: fractional differential equations; Sturm-Liouville problems; Lyapunov inequality; fixed point theorem

1 Introduction On the one hand, since a Lyapunov-type inequality has found many applications in the study of various properties of solutions of differential equations, such as oscillation theory, disconjugacy and eigenvalues problems, there have been many extensions and generalizations as well as improvements in this field, e.g., to nonlinear second order equations, to delay differential equations, to higher order differential equations, to difference equations and to differential and difference systems. We refer the readers to [–] (integer order). Fractional differential equations have gained considerable popularity and importance due to their numerous applications in many fields of science and engineering including physics, population dynamics, chemical technology, biotechnology, aerodynamics, electrodynamics of complex medium, polymer rheology, control of dynamical systems. With the rapid development of the theory of fractional differential equation, there are many © 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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papers which are concerned with the Lyapunov type inequality for a certain fractional order differential equations, see [–] and the references therein. Recently, Ghanbari and Gholami [] introduced the Lyapunov type inequality for a certain fractional order SturmLiouville problem in sense of Riemann-Liouville ⎧ ⎨Dα+ (p(t)u (t)) + q(t)u(t) = ,

 < α ≤ , t ∈ (a, b), b = ,

a

⎩u(a) = u (a) = ,

u(b) = 

like this 

  

(α)  q(s)  ds dω > .  p(ω)  (b – a)α–

b  b

a

a

On the other hand, many authors have studied the existence, uniqueness and multiplicity of solutions for nonlinear boundary value problems involving fractional differential equations, see [–]. But Lan and Lin [] pointed out that the continuity assumptions on nonlinearities used previously are not sufficient and obtained some new results on the existence of multiple positive solutions of systems of nonlinear Caputo fractional differential equations with some of general separated boundary conditions ⎧ ⎨–c Dq z (t) = f (t, z(t)), i

i

t ∈ (, ),

⎩αzi () – βz () = , i

γ zi () + δzi () = ,

where z(t) = (z (t), . . . , zn (t)), fi : [, ] × Rn+ → R+ is continuous on [, ] × Rn+ , c Dq is the Caputo differential operator of order q ∈ (, ). The α, β, γ , δ are positive real numbers. The relations between the linear Caputo fractional differential equations and the corresponding linear Hammerstein integral equations are studied, which shows that suitable Lipschitz type conditions are needed when one studies the nonlinear Caputo fractional differential equations. Motivated by these excellent works, in this paper we focus on the representation of the Lyapunov type inequality and the existence of solutions for a certain fractional order Sturm-Liouville problem ⎧ ⎨Dq + (p(t)u (t)) + (t)f (u(t)) = , 

⎩αu() – βp()u () = ,

 < q ≤ , t ∈ (, ),

γ u() + δp()u () = ,

(.)

 where α, β, γ , δ are constants satisfying  = |βγ + αγ  p(τ ) dτ + αδ| < +∞, p(·) is a positive continuous function on [, ], (t) : [, ] → R is a nontrivial Lebesgue integrable function, f : R → R is continuous. In addition, some existence results are given for the problem ⎧ ⎨Dq + (p(t)u (t)) + (t)f (u(t), λ) = , 

⎩αu() – βp()u () = ,

 < q ≤ , t ∈ (, ),

γ u() + δp()u () = ,

(.)

where λ ≥  is a parameter, f : R × R+ → R is continuous. For the Sturm-Liouville problems, there are many literature works on the studies of the existence and behavior of so-

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lutions to nonlinear Sturm-Liouville equations, for example, [, ] (integer order) and [, ] (fractional order). The discussion of this manuscript is based on the fixed point theorems and the LeraySchauder nonlinear alternative for single-valued maps. For convenience, we list the crucial lemmas as follows. Lemma . ([]) Let ν be a positive measure and be a measurable set with ν( ) = . Let I be an interval and suppose that u is a real function in L(dν) with u(t) ∈ I for all t ∈ . If f is convex on I, then  f

 u(t) dν(t) ≥ f ◦)u(t) dν(t).



(.)



If f is concave on I, then inequality (.) holds with ‘≥’ substituted by ‘≤’. Lemma . ([]) Let E be a Banach space, E be a closed, convex subset of E, be an open subset of E , and  ∈ . Suppose that T : → E is completely continuous. Then either (i) T has a fixed point in , or (ii) there are u ∈ ∂ (the boundary of in E ) and λ ∈ (, ) with u = λTu. Lemma . ([]) Let E be a Banach space and K ⊂ E be a cone in E. Assume that  ,  are open subsets of E with  ∈  ,  ⊂  , and let T : K ∩ (  \  ) → K be a completely continuous operator such that either (i) Tu ≤ u , u ∈ K ∩ ∂  and Tu ≥ u , u ∈ K ∩ ∂  ; or (ii) Tu ≥ u , u ∈ K ∩ ∂  and Tu ≤ u , u ∈ K ∩ ∂  . Then T has a fixed point in K ∩ (  \  ). Lemma . ([]) Let E be a Banach space and K ⊂ E be a cone in E. Assume that  ,  are open subsets of E with  ∩ K = ∅,  ∩ K ⊂  ∩ K . Let T :  ∩ K → K be a completely continuous operator such that: (A) Tu ≤ u , ∀u ∈ ∂(  ∩ K), and (B) there exists e ∈ K \ {} such that u = Tu + μe,

for u ∈ ∂(  ∩ K) and μ > .

Then T has a fixed point in  ∩ K \  ∩ K . The same conclusion remains valid if (A) holds on ∂(  ∩ K) and (B) holds on ∂(  ∩ K).

2 Preliminaries Definition . ([]) For a function u given on the interval [a,b], the Riemann-Liouville derivative of fractional order q is defined as q

Da+ u(t) =

dn 

(n – q) dt n

where n = [q] + .



t

(t – s)n–q– u(s) ds, a

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Definition . ([]) The Riemann-Liouville fractional integral of order q for a function u is defined as q Ia+ u(t) =



(q)



t

(t – s)q– u(s) ds,

q>

a

provided that such integral exists. Lemma . ([]) Let q > . Then q

q

Ia+ Da+ u(t) = u(t) +

n 

ck t q–k ,

n = [q] + .

k=

Lemma . Let h(t) ∈ AC[, ]. Then the fractional Sturm-Liouville problem ⎧ ⎨Dq + (p(t)u (t)) + h(t) = , 

⎩αu() – βp()u () = ,

 < q ≤ , t ∈ (, ), γ u() + δp()u () = 

has a unique solution u(t) in the form  u(t) =



G(t, s)h(s) ds, 

where ⎧   t dτ q– ][δ( – s)q– + γ t (τ –s)p(τ ) dτ ] – H(t, s),  ≤ s ≤ t ≤ ;  ⎨[β + α  p(τ ) G(t, s) =   ρ (q) ⎩[β + α t dτ ][δ( – s)q– + γ  (τ –s)q– dτ ],  ≤ t ≤ s ≤ ;  p(τ ) s p(τ )

 t      dτ (τ – s)q– dτ + αδ, H(t, s) = α δ + γ dτ . ρ = βγ + αγ p(τ )  p(τ ) t p(τ ) s Proof From Definitions ., . and Lemma ., it follows that  t  c – (t – s)q– h(s) ds, p(t) (q)p(t)   t  t τ c (τ – s)q– q(s) u(t) = c + dτ – ds dτ .

(q)p(τ )  p(τ )  

u (t) =

Furthermore, we have u() = c , u () =

c , p()

  τ c (τ – s)q– q(s) u() = c + dτ – ds dτ ,

(q)p(τ )  p(τ )      c  – ( – s)q– q(s) ds. u () = p() (q)p()  



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Combining the boundary conditions, we directly get τ

αγ



c =

(τ –s)q– h(s)

(q)p(τ )



ds dω + αδ

 

(–s)q– h(s)

(q)

ds

(–s)q– h(s)

(q)

ds

ρ βγ

τ 

c =

(τ –s)q– h(s)

(q)p(τ )



ds dτ + βδ

 

ρ

,

.

Finally, substituting c and c , we obtain τ

βγ



u(t) =

(τ –s)q– h(s)

(q)p(τ )

ds dτ + βδ

 

(–s)q– h(s)

(q)

ds

ρ ω

 q– h(s) q– ds dω + αδ  (–s) (q)h(s) ds αγ   (ω–s) 

(q)p(τ ) dτ + ρ  p(ω)  t τ (τ – s)q– h(s) – ds dτ

(q)p(τ )       (τ –s)q–  q– βγ  [ s (q)p(τ dτ ]h(s) ds + βδ  (–s) h(s) ds )

(q) 

=



t

ρ 

  αγ  [ s

q–

(τ –s) dτ ]h(s) ds + αδ 

(q)p(τ ) + dτ ρ  p(τ )

 t  t q– (τ – s) – dτ h(s) ds  s (q)p(τ )   = G(t, s)h(s) ds. t

 

(–s)q– h(s) ds

(q)



For  ≤ t ≤ s ≤ ,   t [ s

βγ u(t) =

(τ –s)q–

(q)p(τ )

dτ ]h(s) ds + βδ

 t

(–s)q– h(s) ds

(q)

ρ   αγ t [ s

 q– (τ –s)q– dτ ]h(s) ds + αδ t (–s) h(s) ds 

(q)p(τ )

(q) + dτ ρ  p(τ )



 t      dτ (τ – s)q– dτ β +α δ( – s)q– + γ h(s) ds. = p(τ ) t ρ  p(τ ) s 

t

For  ≤ s ≤ t ≤ , βγ u(t) =

t  [ s

(τ –s)q–

(q)p(τ )

dτ ]h(s) ds + βδ

t 

(–s)q– h(s) ds

(q)

ρ 

t  ac  [ s

q–

(τ –s) dτ ]h(s) ds + αδ 

(q)p(τ ) + dτ ρ  p(τ )

 t  t (τ – s)q– – dτ h(s) ds  s (q)p(τ ) t

t 

(–s)q– h(s) ds

(q)

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=

  t β +α

Page 6 of 19



  dτ (τ – s)q– dτ δ( – s)q– + γ p(τ )   p(τ ) t

 t   q– dτ (τ – s) dτ h(s) ds. –α δ+γ p(τ ) t p(τ ) s

 ρ

t



Lemma . Assume that α, β, γ , δ > , and p(·) : [, ] → (, +∞). The Green function G(t, s) satisfies the following properties: (i) G(t, s) ≥  for  ≤ t, s ≤ ; (ii) For  ≤ t, s ≤ , there exists C(t) >  such that G(t, s) satisfies the inequalities C(t)G(s, s) ≤ G(t, s) and min C(t) < 

t∈[θ,–θ]



 for θ ∈ , . 

(iii) The maximum value estimate of G(t, s) G = max G(t, s) ≤t,s≤

   = max max G(s, s), max G t (s), s , s∈[,]

s∈[,]

where

t (s) = s +

αδ( – s)q– + αγ   s

(τ –s)q– p(τ )

dτ q– 

ρ

.

 Proof (i) On the one hand, since α, β, γ , δ > , and βγ + αγ  p(τ ) dτ + αδ > , it is clear that G(t, s) ≥  for  ≤ t ≤ s ≤ . On the other hand, for  ≤ s ≤ t ≤ , we can verify the following inequalities:  αδ αγ αγ

t

( – s)q– dτ – αδ p(τ )   t dτ   (τ –s)q– dτ  p(τ )



dτ t p(τ )

t

t

t

(τ – s)q– dτ ≥ , p(τ ) s  t dτ   (t–s)q– dτ  p(τ )

p(τ )

(τ –s)q– s p(τ )



dτ

≥ 

dτ t p(τ )

t

t

p(τ )

(t–s)q– s p(τ )

dτ

≥ .

Then we get G(t, s) ≥  for  ≤ s ≤ t ≤ . (ii) For  ≤ t ≤ s ≤ ,

  ∂G(t, s) α (τ – s)q– dτ = δ( – s)q– + γ ≥ . ∂t ρ (q)p(t) p(τ ) s Then it is easy to obtain G(t, s) ≤ G(s, s) for  ≤ t ≤ s ≤ .

(.)

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For  ≤ s ≤ t ≤ ,     (t – s)q– αδ( – s)q–  (τ – s)q– ∂G(t, s) = –βγ + + αγ dτ ∂t  p(t) p(t) p(t) t p(τ )  t  t (t – s)q–  dτ (t – s)q– (τ – s)q– – αδ + αγ dτ –αγ p(t) p(t) p(t) s p(τ )  p(τ )    dτ (t – s)q– – αγ p(t) t p(τ )

   (τ – s)q– q– q– = –ρ(t – s) + αδ( – s) + αγ dτ . ρ (q)p(t) p(τ ) s

(.)

Let 



F(t) = –ρ(t – s)q– + αδ( – s)q– + αγ s

(τ – s)q– dτ . p(τ )

It is clear that F  (t) = –ρ(q–)(t –s)q– < , which implies that F(·) is decreasing on t ∈ (s, ]. Since F(s) >  and F() < , there exists unique t (s) ∈ (s, ) such that F(t ) = , namely,

t (s) = s +

αδ( – s)q– + αγ   s

(τ –s)q– p(τ )

dτ q– 

ρ

.

From the above discussion, we get the conclusions ∂G(t, s) ≥ , ∂t ∂G(t, s) ≤ , ∂t

for t ∈ [s, t ], and G(s, s) ≤ G(t, s) ≤ G(t , s), for t ∈ [t , ], and G(, s) ≤ G(t, s) ≤ G(t , s).

Furthermore, we obtain the estimate   G(t, s) ≤ G t (s), s ,

for  ≤ s ≤ t ≤ .

For  ≤ t ≤ s ≤ , t G(t, s) β + α   = G(s, s) β + α s

dτ p(τ ) dτ p(τ )

≥

β

t

dτ  p(τ )   dτ + α  p(τ )

β +α

= C (t).

For  ≤ s ≤ t ≤ ,   (τ –s)q– dτ  t dτ q– ] – H(t, s) G(t, s) [β + α  p(τ ) ][δ( – s) + γ t p(τ ) =  s dτ   (τ –s)q– dτ G(s, s) [β + α  p(τ ) ][δ( – s)q– + γ s ] p(τ ) ≥

[β + α

 q– dτ q– + γ t (τ –s)p(τ ) dτ ] – H(t, s)  p(τ ) ][δ( – s)  t dτ   dτ [β + α  p(τ ][δ + γ  p(τ ] ) )

t

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=

≥

[β + α

t

dτ q–  p(τ ) ][δ( – s)

[β βδ( – t)q– + βγ

–

=

Page 8 of 19

t

(t–s)q– dτ ] – α[δ + γ t p(τ )  t dτ   dτ + α  p(τ ) ][δ + γ  p(τ ] )



 t (t–s)q– dτ t p(τ ) ] s p(τ )

t  t dτ + αδ( – s)q–  p(τ + αγ  )  t dτ   dτ [β + α  p(τ ][δ + γ  p(τ ] ) )

(t–s)q– dτ p(τ )

  dτ  t (t–s)q– dτ s p(τ ) + αγ t p(τ ) s p(τ )  t dτ   dτ + α  p(τ ][δ + γ ]  p(τ ) )

αδ( – s)q– [β





+γ

t

[β



–α



dτ t p(τ )

 t dτ  t dτ – s p(τ ) ] αδ( – s)q– [  p(τ ) +  t dτ   dτ [β + α  p(τ ) ][δ + γ  p(τ ] )   t dτ   dτ   dτ βδ( – t)q– + γ (t – s)q– [β t p(τ + α  p(τ –α t ) ) t p(τ ) ≥  t dτ   dτ [β + α  p(τ ][δ + γ  p(τ ] ) )  s dτ αδ( – s)q–  p(τ ) +  t dτ   dτ [β + α  p(τ ) ][δ + γ  p(τ ] ) ≥

βδ( – t)q–  t dτ  [β + α  p(τ ) ][δ + γ 

dτ ] p(τ )

 t

(t–s)q– dτ p(τ )

dτ

 t dτ   dτ dτ t p(τ ) + α  p(τ ) t p(τ )  t dτ   dτ + α  p(τ ][δ + γ  p(τ ] ) )

βδ( – t)q– + γ (t – s)q– [β

dτ p(τ )

dτ

dτ p(τ )

t

dτ s p(τ ) ]

t

dτ  p(τ ) ]

= C (t).

Choosing C(t) = min{C (t), C (t)}, we get C(t)G(s, s) ≤ G(t, s).



3 Existence results I Theorem . (Lyapunov type inequality) Assume that α, β, γ , δ > , p(·) : [, ] → (, +∞), and let (t) : [, ] → R be a nontrivial Lebesgue integrable function. Then, for any nontrivial solution of the fractional Sturm-Liouville problem ⎧ ⎨Dq + (p(t)u (t)) + (t)u(t) = , 

⎩αu() – βp()u () = ,

 < q ≤ , t ∈ (, ),

γ u() + δp()u () = ,

the following so-called Lyapunov type inequality will be satisfied:  



 (s) ds >  , G

where G is defined in (iii) of Lemma .. Proof From Lemma . and the triangular inequality, we get            u(t) =  G(t, s)(s)u(s) ds ≤ G(t, s)(s)u(s) ds.   



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Let E denote the Banach space C[, ] with the norm defined by u = maxt∈[,] |u(t)|. Via some simple computations, we can obtain 

  u(t) ≤ max



t∈[,] 

  G(t, s)(s)u(s) ds

  ≤ u(t) max





t∈[,] 

  G(t, s)(s) ds

      max G(t, s) (s) ds, ≤ u(t) t∈[,]



namely, 



 (s) ds >  . G





Theorem . (Generalized Lyapunov type inequality) Assume that α, β, γ , δ > , p(·) : [, ] → (, +∞), and let (t) : [, ] → R be a nontrivial Lebesgue integrable function, f (u) is a positive function on R. Then, for any nontrivial solution of the fractional SturmLiouville problem (.), the following so-called Lyapunov type inequality will be satisfied: 



 (s) ds >

u∗ G maxu∈[u∗ ,u∗ ] f (u)



,

where u∗ = max u(t).

u∗ = min u(t), t∈[,]

t∈[,]

Proof From the similar proof of Theorem ., we get   u(t) ≤



    G(t, s)(s)f u(s) ds.





Since f is continuous and concave, then using Jensen”s inequality (.), we obtain   u(t) ≤ max



t∈[,] 

≤

  



    G(t, s)(s)f u(s) ds

    max G(t, s) (s)f u(s) ds

t∈[,]

  ≤ G(t)L



 |(s)|  f u(s) ds  |(t)|L   ≤ G max∗ f (u)(t)L , 

u∈[u∗ ,u ]

namely,  



 (s) ds >

u∗ G maxu∈[u∗ ,u∗ ] f (u)

.



Ru et al. Advances in Difference Equations (2017) 2017:320

Page 10 of 19

For convenience, we give some notations:   = max

t∈[,]

t



 G t (s), s (s) ds +





G(s, s)(s) ds ;

t





ς = min C(t) · t∈[θ,–θ]



G(s, s)(s) ds. 

Theorem . Let (t) : [, ] → R+ be a nontrivial Lebesgue integrable function and f : R → R be a continuous function satisfying the Lipschitz condition   f (x) – f (y) ≤ L|x – y|,

∀x, y ∈ R, L > .

Then problem (.) has a unique solution if L < . Proof By Lemma ., the solution of problem (.) is equivalent to a fixed point of the  operator T : E → E defined by T(u(t)) =  G(t, s)(s)f (u(s)) ds. Let supt∈[,] |f ()| = ν. Now we show that T : Br ⊂ Br , where Br = {u ∈ C[, ] : u < r} ν . For u ∈ Br , one has |f (u)| = |f (u) – f () + f ()| ≤ L|u| + ν ≤ Lr + ν. Furtherwith r > –L more, we have          T(u)(t) =  G(t, s)(s)f u(s) ds     t       = G(t, s)(s)f u(s) ds + G(t, s)(s)f u(s) ds 

t

  t            G t (s), s (s)f u(s) ds + G(s, s)(s)f u(s) ds ≤     t

 t     G t (s), s (s) ds + G(s, s)(s) ds ≤ (Lr + ν) max t∈[,]



t

= (Lr + ν) ≤ r, which yields T : Br ⊂ Br . For any x, y ∈ E, we have              T(x) – T(y) =  G(t, s)(s)f x(s) ds –  G(t, s)(s)f y(s) ds      t        ≤ sup G t (s), s (s)f x(s) – f y(s)  ds t∈[,]

 +





     G(s, s)(s)f x(s) – f y(s)  ds

t



≤ L max

t∈[,]

t

  G t (s), s (s) ds +









G(s, s)(s) ds x – y

t

= L x – y . Since L < , from the Banach’s contraction mapping principle it follows that there exists a unique fixed point for the operator T which corresponds to the unique solution for problem (.). This completes the proof. 

Ru et al. Advances in Difference Equations (2017) 2017:320

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Theorem . Let (t) : [, ] → R+ be a nontrivial Lebesgue integrable function and f : R → R be a continuous function satisfying the following: (F) There exists a positive constant K such that |f (u)| ≤ K for u ∈ R. Then problem (.) has at least one solution. Proof First, since the function p : [, ] → (, +∞) is continuous, we get p∗ = mint∈[,] p(t) > . Further, from (.) and (.), we get the following estimates respectively: for  ≤ t ≤ s ≤ ,

  (τ – s)q– dτ α ∂G(t, s) q– = δ( – s) + γ < ∂t ρ (q)p(t) p(τ ) s

  α dτ ; δ+γ ≤ ρ (q)p∗ p(τ )  for  ≤ s ≤ t ≤ ,   

    ∂G(t, s)     (τ – s)q– q– q–  =  + αδ( – s) + αγ –ρ(t – s) dτ  ∂t   ρ (q)p(t)  p(τ ) s

   dτ ρ + αδ + αγ ≤ ; ρ (q)p∗ p(τ )  | is bounded for  ≤ s, t ≤ , namely, there exists S >  such which implies that | ∂G(t,s) ∂t ∂G(t,s) that | ∂t | ≤ S. Combining with |f (t, u)| ≤ K for t ∈ [, ], t ∈ R, we obtain     (Tu) (t) =  

 

      ∂G(t, s) (s)f u(s) ds ≤ SK (t)L . ∂t

Hence, for any t , t ∈ [, ], we have     (Tu)(t ) – (Tu)(t ) =  

t

t

    (Tu) (t) dt  ≤ SK (t)L |t – t |.

This means that T is equicontinuous on [,]. Thus, by the Arzelà-Ascoli theorem, the operator T is completely continuous. Finally, let Br = {u ∈ E : u < r} with r = K + . If u is a solution for the given problem, then, for λ ∈ (, ), we obtain             u = λ Tu(t) = λ G(t, s)(s)f u(s) ds  

  t          G(t, s)(s)f u(s) ds = λ G(t, s)(s)f u(s) ds +   t  t           < max G t (s), s (s)f u(s)  ds + G(s, s)(s)f u(s)  ds t∈[,] 



t

≤ K max

t∈[,]

≤ K ,





 G t (s), s (s) ds +

t





G(s, s)(s) ds t

Ru et al. Advances in Difference Equations (2017) 2017:320

Page 12 of 19

which yields a contradiction. Therefore, by Lemma ., the operator T has a fixed point in E.  Theorem . Let (t) : [, ] → R+ be a nontrivial Lebesgue integrable function and f : R+ → R+ be a continuous function satisfying (F). In addition, the following assumption holds: (F) There exists a positive constant r such that for u ∈ [, r ].

f (u) ≥ ς – r

Then problem (.) has at least one solution. Proof Define a cone P of the Banach space E as P = {u ∈ E : u ≥ }. From the proof of Theorem ., we know that T : P → P is completely continuous. Set Pri = {u ∈ P : u < ri }. For u ∈ ∂Pr , one has  ≤ u ≤ r . For t ∈ [θ ,  – θ ], we have   T u(t) =





  G(t, s)(s)f u(s) ds



  C(t)G(s, s)(s)f u(s) ds



 ≥





≥ min C(t) · t∈[θ,–θ]



  G(s, s)(s)f u(s) ds



 ≥ min C(t) · t∈[θ,–θ]



G(s, s)(s) ds · r



> r = u . Choosing r > K . Then, for u ∈ ∂Pr , we have           T u(t)  =  G(t, s)(s)f u(s) ds      t          G(t, s)(s)f u(s) ds + G(t, s)(s)f u(s) ds =    t  t

    ≤ max G t (s), s (s) ds + G(s, s)(s) ds K t∈[,]



t

< r = u . Then, by Lemma ., problem (.) has at least one positive solution u(t) belonging to E such that r ≤ u ≤ r .  Theorem . Let (t) : [, ] → R+ be a nontrivial Lebesgue integrable function, f : R → R be a continuous function and satisfy the following assumptions: (F) There exists a nondecreasing function ϕ : R+ → R+ such that     f (u) ≤ ϕ u ,

∀u ∈ R;

(F) There exists a constant R >  such that Then problem (.) has at least one solution.

R  ϕ(R)

> .

Ru et al. Advances in Difference Equations (2017) 2017:320

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Proof From the proof of Theorem ., we know that T is completely continuous. Now we show that (ii) of Lemma . does not hold. If u is a solution of (.), then, for λ ∈ (, ), we obtain              u = λ T u(t) = λ G(t, s)(s)f u(s) ds  

  t          G(t, s)(s)f u(s) ds + G(t, s)(s)f u(s) ds = λ    t  t

          < max G t (s), s (s)f u(s)  ds + G(s, s)(s)f u(s)  ds t∈[,]





≤ max

t∈[,]



t

  G t (s), s (s) ds +





t



  G(s, s)(s) ds ϕ u

t

 ≤  ϕ u .

Let BR = {u ∈ E : u < R}. From the above inequality and (F), it yields a contradiction. Therefore, by Lemma ., the operator T has a fixed point in BR .  Theorem . Let (t) : [, ] → R+ be a nontrivial Lebesgue integrable function and f : [, ] × R+ → R+ be a continuous function. Suppose that (F) and (F) hold. In addition, the following assumption holds: (F) There exists a positive constant r with r < R and a function ψ : R+ → R+ satisfying   f (u) ≥ ψ u ,

for u ∈ [, ςr],

ψ(ςr) ≥ r. If ς < , then (.) has at least one positive solution u(t). Proof Let Br = {u ∈ E : u < r}. Part (I). For any u ∈ ∂(BR ∩ P), from (F) and (F) it follows that           T u(t)  =  G(t, s)(s)f u(s) ds      t          G(t, s)(s)f u(s) ds + G(t, s)(s)f u(s) ds =   



t

t

< max

t∈[,] 



t

≤ max

t∈[,]

    G t (s), s (s)f u(s) ds +





 G t (s), s (s) ds +







  G(s, s)(s)f u(s) ds

t



  G s, s(s) ds ϕ u 



t

=  ϕ(R) ≤ R = u , which implies that (A) of Lemma . holds. Now we prove that u = T(u) + μ for u ∈ ∂(Bς r ∩ P) and μ > . On the contrary, if there exists u ∈ ∂(Bς r ∩ P) and μ >  such that u = T(u ) + μ , then, for t ∈ [θ ,  – θ ], one has

Ru et al. Advances in Difference Equations (2017) 2017:320

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mint∈[θ,–θ] C(t) > . Furthermore, from (F) it follows that   u (t) = T u (t) + μ     = G(t, s)(s)f u (s) ds + μ 

 ≥



  C(t)G(s, s)(s)f u (s) ds + μ





≥ min C(t) t∈[θ,–θ]



  G(s, s)(s)f u (s) ds + μ



 ≥ min C(t) t∈[θ,–θ]



G(s, s)(s)ψ(ςr) ds + μ 

= ςr + μ . Furthermore, we get ςr > min u (t) ≥ ςr + μ > ςr, t∈[θ,–θ]

which yields a contradiction. So (B) of Lemma . holds. Therefore, Lemma . guarantees that T has at least one fixed point.



Theorem . Let (t) : [, ] → R+ be a nontrivial Lebesgue integrable function and f : R+ → R+ be a continuous function satisfying (F). In addition, the following assumptions hold: = ; (F) limu→+ f (u) u (F) There exists R >  such that minu∈[ϑR,R] f (u) > σ R, where  <ϑ =η

 min C(t) < ,

t∈[θ,–θ]

G(t (s), s) –  < η = max ≤ , ≤s≤ G(s, s)

–  –θ G(s, s)(s) ds . σ = min C(t)

t∈[θ,–θ]

θ

Then problem (.) has at least two solutions. Proof From Lemma ., we can derive the following inequalities:           T u(t)  =  G(t, s)(s)f u(s) ds   

 t           = G(t, s)(s)f u(s) ds + G(t, s)(s)f u(s) ds    t  t

        G t (s), s (s)f u(s) ds + G(s, s)(s)f u(s) ds ≤ max t∈[,]



t

Ru et al. Advances in Difference Equations (2017) 2017:320



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  G(t (s), s) G(s, s)(s)f u(s) ds + t∈[,] G(s, s)   

  G(t (s), s) G(s, s)(s)f u(s) ds ≤ max ≤s≤ G(s, s)  t





= max

  G(s, s)(s)f u(s) ds

t

and 







  G(t, s)(s)f u(s) ds



  C(t)G(s, s)(s)f u(s) ds.

T u(t) = 

 ≥



Combining the two inequalities, we have      T u(t) ≥ C(t)ηT u(t) . Define a subcone  P of the Banach space E as  P = {u ∈ E : u ≥ C(t)η u(t) }. From the stan  P : u < r}. dard process, we know that T : P → P is completely continuous. Set  Pr = {u ∈  f (u) Since limu→+ u = , there exist  >  and r >  such that f (u) < u, for  ≤ u ≤ r, where  satisfies  < . For u ∈ ∂ Pr , we have           T u(t)  =  G(t, s)(s)f u(s) ds      t          G(t, s)(s)f u(s) ds =  G(t, s)(s)f u(s) ds +  

t



t

≤  max

t∈[,]





 G t (s), s (s) ds +





G(s, s)(s) ds u

t

< u . In a similar way, we choose R > K . Then, for u ∈ ∂ PR , we have           T u(t)  =  G(t, s)(s)f u(s) ds      t          G(t, s)(s)f u(s) ds + G(t, s)(s)f u(s) ds =   

t



t

≤ max

t∈[,]



 G t (s), s (s) ds +







G(s, s)(s) ds K

t

< R = u . For any u ∈ ∂PR , choosing t ∗ ∈ (θ ,  – θ ), it is easy to verify that u(t ∗ ) ∈ [ϑR, R]. Furthermore, we have    T u t∗ =





    G t ∗ , s (s)f u(s) ds



  ≥ C t∗

 θ

–θ

  G(s, s)(s)f u(s) ds

Ru et al. Advances in Difference Equations (2017) 2017:320

  ≥ C t∗



–θ

  G(s, s)(s) min f u(s) ds u∈[ϑR,R]

θ

≥

Page 16 of 19





–θ

min C(t)

t∈[θ,–θ]

G(s, s)(s)σ R ds θ

= R = u . Then by Lemma ., problem (.) has at least two positive solutions r ≤ u (t) ≤ R and R ≤ u (t) ≤ R.  Example  Let us consider the problem ⎧ ⎨Dq + (p(t)u (t)) + (t) arctan u = ,

 < q ≤ , t ∈ (, ),



⎩αu() – βp()u () = ,

γ u() + δp()u () = .

Since |f (u)| = | arctan u| < π , this problem has a solution by Theorem .. If (t) satisfies 

t

 = max

t∈[,]

  G t (s), s (s) ds +







G(s, s)(s) ds < .

t

It is easy to get that f  (u) = (arctan u) =

 ≤  = L.  + u

Therefore, this problem has a unique solution by Theorem .. Example  Let us consider the problem ⎧ ⎨Dq + (p(t)u (t)) + (t)e–u = ,  < q ≤ , t ∈ (, ),  ⎩αu() – βp()u () = , γ u() + δp()u () = . 

Since f (u) = e–u

≤ , we can choose r =  + . Then it is clear that

f (u) ≤  <  – r

for u ∈ [, r ],

which implies that (F) holds. Finally, for any r > , we have f (u) ≥ e–r 

–r limr→+ eς – r



for u ∈ [, r].

Since = +∞, there exists r < r such that f (u) ≥ ς r for u ∈ [, r ], which implies that (F) holds. Therefore, this problem has a unique solution by Theorem .. –

Example  Let us consider the problem ⎧ ⎨Dq + (p(t)u (t)) + (t)e–u (arctan u + sin u + ) = , 

⎩αu() – βp()u () = , 

 < q ≤ , t ∈ (, ),



γ u() + δp()u () = . 







It is clear that |f (u)| = |e–u (arctan u  + sin u  + )| ≤ u  + u  +  = ϕ( u ), ∀u ∈ R. R >  obviThen (F) holds. Furthermore, for sufficiently large R > , the inequality  ϕ(R) ously holds, namely, (F) holds. Then this problem has at least one solution by Theorem ..

Ru et al. Advances in Difference Equations (2017) 2017:320

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









For u ∈ R+ , since f (u) = e–u (arctan u  + sin u  + ) ≥ e–u ≥ e– u = ψ( u ), we have f (u) ≥ ψ( u ) for u ∈ [, ςr], for any r > . Via some simple computations, we get limr→+ ψ(ςr r) = +∞. Then there exists sufficiently small r >  such that ψ(ςr) ≥ r. From the above discussions, we have that (F) holds. Therefore, this problem has at least one positive solution u(t) for ς <  by Theorem .. Example  Let us consider the problem ⎧ ⎨Dq + (p(t)u (t)) + (t) 

σ (ϑ) e–ϑ

⎩αu() – βp()u () = ,

u e–u = ,

 < q ≤ , t ∈ (, ),

γ u() + δp()u () = .

σ +  –u Since f (u) = (ϑ)  e–ϑ u e , via some simple computations, we can verify that (F) and (F) σ + σ + –u  –u hold. In addition, since f  (u) = (ϑ)  e–ϑ e (u – u ) = (ϑ) e–ϑ e u( – u), it is clear that f  (u) >  for u ∈ (, ); f  (u) <  for u ∈ (, +∞). Let R = , then for any u ∈ [ϑ, ], we have σ +  –ϑ minu∈[ϑ,] f (u) = (ϑ) > σ . Therefore, this problem has at least two positive  e–ϑ (ϑ) e solutions u(t) by Theorem ..

4 Existence results II Theorem . Let (t) : [, ] → R+ be a nontrivial Lebesgue integrable function and f : R × [, +∞) → R be a continuous function satisfying the following: (H) There exists a positive constant K such that |f (u, λ)| ≤ K for u ∈ R, λ ∈ R+ . Then problem (.) has at least one solution. This result can be directly derived from the proof of Theorem .. Now define a cone P of the Banach space E as P = {u ∈ E : u ≥ }. Let Pri = {u ∈ P : u < ri }. Define T by 







T u(t) =

  G(t, s)(s)f u(s), λ ds.



From the proof of Theorem ., we know that T : P → P is completely continuous. Theorem . Let (t) : [, ] → R+ be a nontrivial Lebesgue integrable function and f be a nonnegative continuous function satisfying (H). If f (, ) > , then there exists λ∗ >  such that problem (.) has at least one solution for  ≤ λ < λ∗ . Proof Since f (u, λ) is continuous and f (, ) > , for any given  >  (sufficiently small), there exists δ >  such that f (u, λ) > f (, ) –  if  ≤ u < δ,  ≤ λ < δ. Choosing r < min{δ, ς(f (, ) – )} and λ∗ = δ. Then, for any u ∈ ∂Pr and t ∈ [θ ,  – θ ], we have   T u(t) =





  G(t, s)(s)f u(s), λ ds



  C(t)G(s, s)(s)f u(s), λ ds



 ≥





≥ min C(t) · t∈[θ,–θ]





  G(s, s)(s)f u(s), λ ds

Ru et al. Advances in Difference Equations (2017) 2017:320

 ≥ min C(t) · t∈[θ,–θ]



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  G(s, s)(s) ds · f (, ) – 



> r = u . Choosing r > K . Then, for u ∈ ∂Pr , we have           T u(t)  =  G(t, s)(s)f u(s) ds       t       G t (s), s (s)f u(s) ds + G(s, s)(s)f u(s) ds ≤ max t∈[,] 



t

≤ max

t∈[,]



 G t (s), s (s) ds +





t



G(s, s)(s) ds K

t

< r = u . Then, by Lemma ., problem (.) has at least one positive solution u(t) belonging to E  such that r ≤ u ≤ r . Corollary . Let (t) : [, ] → R+ be a nontrivial Lebesgue integrable function and f be a nonnegative continuous function satisfying (H). If limu→+ f (u, λ) = f (, ) > , then problem (.) has at least one solution for any λ ≥ . Example  Let us consider the problem ⎧ ⎨Dq + (p(t)u (t)) + (t)(arctan u + e–λ ) = , 

⎩αu() – βp()u () = , 

 < q ≤ , t ∈ (, ), 

γ u() + δp()u () = .

It is clear that (H) holds and f (, ) > . Then there exists λ∗ >  such that this problem has at least one solution for  ≤ λ < λ∗ . Example  Let us consider the problem ⎧ ⎨Dq + (p(t)u (t)) + (t)e–λu = ,  < q ≤ , t ∈ (, ),  ⎩αu() – βp()u () = , γ u() + δp()u () = . It is clear that (H) holds and limu→+ f (u, λ) = f (, ) > . Then this problem has at least one solution for any λ > .

5 Conclusion In this manuscript, the authors prove some new existence results as well as uniqueness and multiplicity results on fractional boundary value problems. Acknowledgements The authors would like to thank the referees for the helpful suggestions. The second author is supported by NNSF of China (No. 11501165), the Fundamental Research Funds for the Central Universities (2015B19414). Competing interests The authors declare that they have no competing interests.

Ru et al. Advances in Difference Equations (2017) 2017:320

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Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details 1 Department of Mathematics, College of Science, Hohai University, Nanjing, 210098, China. 2 Department of Mathematics, College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 210098, China.

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