Yuan and Yang Journal of Inequalities and Applications 2014, 2014:481 http://www.journalofinequalitiesandapplications.com/content/2014/1/481

RESEARCH

Open Access

Positive solution for q-fractional four-point boundary value problems with p-Laplacian operator Qiaozhen Yuan and Wengui Yang* * Correspondence: [email protected] Ministry of Public Education, Sanmenxia Polytechnic, Sanmenxia, Henan 472000, China

Abstract This paper investigates a class of four-point boundary value problems of fractional β q-difference equations with p-Laplacian operator Dq (ϕp (Dαq u(t))) = f (t, u(t)), t ∈ (0, 1), β u(0) = 0, u(1) = au(ξ ), Dαq u(0) = 0, and Dαq u(1) = bDαq u(η), where Dαq and Dq are the fractional q-derivative of the Riemann-Liouville type, p-Laplacian operator is defined as ϕp (s) = |s|p–2 s, p > 1, and f (t, u) may be singular at t = 0, 1 or u = 0. By applying the upper and lower solutions method associated with the Schauder fixed point theorem, some sufficient conditions for the existence of at least one positive solution are established. Furthermore, two examples are presented to illustrate the main results. MSC: 39A13; 34B18; 34A08 Keywords: fractional q-difference equations; four-point boundary conditions; p-Laplacian operator; positive solution; upper and lower solutions method

1 Introduction Recently, fractional differential equations with p-Laplacian operator have gained its popularity and importance due to its distinguished applications in numerous diverse fields of science and engineering, such as viscoelasticity mechanics, non-Newtonian mechanics, electrochemistry, fluid mechanics, combustion theory, and material science. There have appeared some results for the existence of solutions or positive solutions of boundary value problems for fractional differential equations with p-Laplacian operator; see [–] and the references therein. For example, under different conditions, Wang et al. [] and Ren and Chen [] established the existence of positive solutions to four-point boundary value problems for nonlinear fractional differential equations with p-Laplacian operator by using the upper and lower solutions method and fixed point theorems, respectively. Since Al-Salam [] and Agarwal [] proposed the fractional q-difference calculus, new developments in this theory of fractional q-difference calculus have been made due to the explosion in research within the fractional differential calculus setting. For example, some researcher obtained q-analogs of the integral and differential fractional operators properties such as the q-Laplace transform, the q-Taylor formula, the Mittag-Leffler function [–], and so on. Recently, the theory of boundary value problems for nonlinear fractional q-difference equations has been addressed extensively by several researchers. There have been some ©2014 Yuan and Yang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Yuan and Yang Journal of Inequalities and Applications 2014, 2014:481 http://www.journalofinequalitiesandapplications.com/content/2014/1/481

Page 2 of 14

papers dealing with the existence and multiplicity of solutions or positive solutions for boundary value problems involving nonlinear fractional q-difference equations by the use of some well-known fixed point theorems and the upper and lower solutions method. For some recent developments on the subject, see [–] and the references therein. El-Shahed and Al-Askar [] studied the existence of multiple positive solutions to the nonlinear q-fractional boundary value problems by using the Guo-Krasnoselskii fixed point theorem in a cone. Under different conditions, Graef and Kong [, ] investigated the existence of positive solutions for the boundary value problem with fractional q-derivatives in terms of different ranges of λ, respectively. Zhao et al. [] showed some existence results of positive solutions to nonlocal q-integral boundary value problem of nonlinear fractional q-derivatives equation using the generalized Banach contraction principle, the monotone iterative method, and the Krasnoselskii fixed point theorem. Ahmad et al. [] considered the existence of solutions for the nonlinear fractional q-difference equation with nonlocal boundary conditions by applying some well-known tools of fixed point theory such as the Banach contraction principle, the Krasnoselskii fixed point theorem, and the Leray-Schauder nonlinear alternative. By applying the nonlinear alternative of Leray-Schauder type and Krasnoselskii fixed point theorems, the author [] established sufficient conditions for the existence of positive solutions for nonlinear semipositone fractional q-difference system with coupled integral boundary conditions. Relying on the standard tools of fixed point theory, Agarwal et al. [] and Ahmad et al. [] discussed the existence and uniqueness of solutions for a new class of sequential q-fractional integrodifferential equations with q-antiperiodic boundary conditions and nonlocal four-point boundary conditions, respectively. In [], Aktuˇglu and Özarslan dealt with the following Caputo q-fractional boundary value problem involving the p-Laplacian operator:      Dq ϕp c Dαq x(t) = f t, x(t) , Dkq x() = ,

t ∈ (, ),

for k = , , . . . , n – ,

x() = a x(),

Dq x() = a Dq x(),

where a , a = ,  < α ∈ R, and f ∈ C([, ] × R, R). Under some conditions, the authors obtained the existence and uniqueness of the solution for the above boundary value problem by using the Banach contraction mapping principle. In [], Miao and Liang studied the following three-point boundary value problem with p-Laplacian operator:      Dγq φp Dαq u(t) + f t, u(t) = , u() = (Dq u)() = ,

 < t < ,  < α < ,

(Dq u)() = ,

γ

D+ u(t)|t= = ,

where  < βηα– < ,  < q < . The authors proved the existence and uniqueness of a positive and nondecreasing solution for the boundary value problems by using a fixed point theorem in partially ordered sets. In [], the author investigated the following fractional q-difference boundary value problem with p-Laplacian operator:      Dβq ϕp Dαq u(t) = f t, u(t) , u() = u() = ,

 < t < ,

Dαq u() = Dαq u() = ,

Yuan and Yang Journal of Inequalities and Applications 2014, 2014:481 http://www.journalofinequalitiesandapplications.com/content/2014/1/481

Page 3 of 14

where  < α, β ≤ . The existence results for the above boundary value problem were obtained by using the upper and lower solutions method associated with the Schauder fixed point theorem. In this paper, motivated greatly by the above mentioned works, we consider the following fractional q-difference boundary value problem with p-Laplacian operator:      Dβq ϕp Dαq u(t) = f t, u(t) , u() = ,

t ∈ (, ), Dαq u() = ,

u() = au(ξ ),

Dαq u() = bDαq u(η),

(.)

β

where Dαq , Dq are the fractional q-derivative of the Riemann-Liouville type with  < α, β ≤ ,  ≤ a, b ≤ ,  < ξ , η < , ϕp (s) = |s|p– s, p > , (φp )– = φr , (/p) + (/r) = , and f (t, u) : (, ) × (, +∞) → [, ∞) is continuous and may be singular at t = ,  or u = . By applying the upper and lower solutions method associated with the Schauder fixed point theorem, the existence results of at least one positive solution for the above fractional q-difference boundary value problem with p-Laplacian operator are established. This work improves essentially the results of []. At the end of this paper, we will give two examples to show the effectiveness of the main results.

2 Preliminaries For the convenience of the reader, we present some necessary definitions and lemmas of fractional q-calculus theory to facilitate the analysis of problem (.). These details can be found in the recent literature; see [] and references therein. Let q ∈ (, ) and define [a]q =

qa –  , q–

a ∈ R.

The q-analog of the Pochhammer symbol (the q-shifted factorial) is defined by

(a; q) = ,

n–    (a; q)n = a – bqk ,

n ∈ N ∪ {∞}.

k=

The q-analog of the power (a – b)n with n ∈ N = {, , , . . .} is

(a – b)() = ,

(a – b)(n) =

n–  

 a – bqk ,

n ∈ N , a, b ∈ R.

k=

The following relation between them holds: (a – b)(n) = an (b/a; q)n ,

a = .

Their natural extensions to the reals are (a; q)γ =

(a; q)∞ (aqγ ; q)∞

and (a – b)(γ ) = aγ

(b/a; q)∞ , (qγ b/a; q)∞

γ ∈ R.

Yuan and Yang Journal of Inequalities and Applications 2014, 2014:481 http://www.journalofinequalitiesandapplications.com/content/2014/1/481

Page 4 of 14

Clearly, (a – b)(γ ) = aγ (b/a; q)γ , a = . Note that, if b =  then a(α) = aα . The q-gamma function is defined by x ∈ R \ {, –, –, . . .},

q (x) = ( – q)(x–) ( – q)–x ,

and satisfies q (x + ) = [x]q q (x). The q-derivative of a function f is here defined by (Dq f )(x) =

f (x) – f (qx) , ( – q)x

(Dq f )() = lim (Dq f )(x), x→

and q-derivatives of higher order by 

 Dq f (x) = f (x) and

 n    Dq f (x) = Dq Dn– q f (x),

n ∈ N.

The q-integral of a function f defined in the interval [, b] is given by 

x

(Iq f )(x) =

f (t) dq t = x( – q) 

∞    f xqn qn ,

x ∈ [, b].

n=

If a ∈ [, b] and f is defined in the interval [, b], its integral from a to b is defined by 



b



b

f (t) dq t =

f (t) dq t.



a

a

f (t) dq t – 

Similarly as done for derivatives, an operator Iqn can be defined, namely, 

     Iq f (x) = f (x) and Iqn f (x) = Iq Iqn– f (x),

n ∈ N.

The fundamental theorem of calculus applies to these operators Iq and Dq , i.e., (Dq Iq f )(x) = f (x), and if f is continuous at x = , then (Iq Dq f )(x) = f (x) – f (). Basic properties of the two operators can be found in the book []. We now point out three formulas that will be used later (i Dq denotes the derivative with respect to variable i): 

(α) = aα (t – s)(α) , a(t – s)

  x Dq f (x, t) dq t (x) = x



t Dq (t

– s)(α) = [α]q (t – s)(α–) ,

x x Dq f (x, t) dq t

+ f (qx, x).



Denote that if α >  and a ≤ b ≤ t, then (t – a)(α) ≥ (t – b)(α) [].

Yuan and Yang Journal of Inequalities and Applications 2014, 2014:481 http://www.journalofinequalitiesandapplications.com/content/2014/1/481

Page 5 of 14

Definition . ([]) Let α ≥  and f be function defined on [, ]. The fractional q-integral of the Riemann-Liouville type is Iq f (x) = f (x) and 

 Iqα f (x) =



q (α)



x

(x – qt)(α–) f (t) dq t,

α > , x ∈ [, ].



Definition . ([]) The fractional q-derivative of the Riemann-Liouville type of order α ≥  is defined by Dq f (x) = f (x) and 

   m–α f (x), Dαq f (x) = Dm q Iq

α > ,

where m is the smallest integer greater than or equal to α. Lemma . ([]) Let α, β ≥  and f be a function defined on [, ]. Then the next formulas hold: β α+β () (Iq Iqα f )(x) = Iq f (x), () (Dαq Iqα f )(x) = f (x). Lemma . ([]) Let α >  and p be a positive integer. Then the following equality holds: 

    Iqα Dpq f (x) = Dpq Iqα f (x) – p–

k=

 k  xα–p+k D f ().

q (α + k – p + ) q

Lemma . Let y ∈ C[, ],  < α ≤ ,  < ξ < , and  ≤ a ≤ . Then the unique solution of the following linear fractional q-difference boundary value problem: Dαq u(t) + y(t) = , u() = ,

t ∈ (, ),

(.)

u() = au(ξ ),

is given by 



u(t) =

(.)

G(t, qs)y(s) dq s, 

where ag(ξ , s)t α– ,  – aξ α–   (t( – s))(α–) – (t – s)(α–) ,  ≤ s ≤ t ≤ , g(t, s) =

q (α) (t( – s))(α–) ,  ≤ t ≤ s ≤ .

(.)

G(t, s) = g(t, s) +

Proof By applying Lemma ., we may reduce (.) to an equivalent integral equation u(t) = –Iqα y(t) + c t α– + c t α– ,

c , c ∈ R.

(.)

From u() =  and (.), we have c = . Consequently the general solution of (.) is  u(t) = –Iqα y(t) + c t α– = –

t 

(t – qs)(α–) y(s) dq s + c t α– .

q (α)

(.)

Yuan and Yang Journal of Inequalities and Applications 2014, 2014:481 http://www.journalofinequalitiesandapplications.com/content/2014/1/481

Page 6 of 14

By (.), one has 



u() = – 



( – qs)(α–) y(s) dq s + c ,

q (α)

ξ

u(ξ ) = – 

(ξ – qs)(α–) y(s) dq s + c ξ α– .

q (α)

And from u() = au(ξ ), then we have 



( – qs)(α–) y(s) dq s – ( – aξ α– ) q (α)

c = 



ξ 

a(ξ – qs)(α–) y(s) dq s. ( – aξ α– ) q (α)

So, the unique solution of problem (.) is 

t

(t – qs)(α–) y(s) dq s +

q (α)

u(t) = – 



ξ



 

t α– ( – qs)(α–) y(s) dq s ( – aξ α– ) q (α)

(α–)

α–

at (ξ – qs) y(s) dq s α– ) (α) q  ( – aξ   α–  t (t – qs)(α–) t ( – qs)(α–) y(s) dq s + y(s) dq s =–

q (α)

q (α)     α– α–  ξ α– aξ t ( – qs)(α–) at (ξ – qs)(α–) y(s) d y(s) dq s + s – q α– ) (α) ( – aξ α– ) q (α) q   ( – aξ   G(t, qs)y(s) dq s, = –

(.)





where G(t, s) is defined in (.). The proof is completed.

Lemma . Let y ∈ C[, ],  < α, β ≤ ,  < ξ , η < , and  ≤ a, b ≤ . Then the following fractional q-difference boundary value problem with p-Laplacian operator:    Dβq ϕp Dαq u(t) = y(t), u() = ,

t ∈ (, ),

u() = au(ξ ),

Dαq u() = ,

Dαq u() = bDαq u(η),

(.)

has unique solution given by 





u(t) =



G(t, qs)ϕr 

H(s, qτ )y(τ ) dq τ dq s,

(.)



where b = bp– , G(t, s) is defined by (.) and b h(ξ , s)t α– ,  – b ξ α–   (t( – s))(β–) – (t – s)(β–) ,  ≤ s ≤ t ≤ , h(t, s) =

q (β) (t( – s))(β–) ,  ≤ t ≤ s ≤ .

H(t, s) = h(t, s) +

(.)

Proof By applying Lemma ., we may reduce (.) to an equivalent integral equation,   ϕp Dαq u(t) = Iqβ y(t) + c t β– + c t β– ,

c , c ∈ R.

(.)

Yuan and Yang Journal of Inequalities and Applications 2014, 2014:481 http://www.journalofinequalitiesandapplications.com/content/2014/1/481

Page 7 of 14

From Dαq u() =  and (.), we have c = . Consequently the general solution of (.) is 





ϕp Dαq u(t)

= Iqβ y(t) + c t β–

= 

t

(t – qs)(β–) y(s) dq s + c t β– .

q (β)

(.)

By (.), one has 

  ϕp Dαq u() = 





( – qs)(β–) y(s) dq s + c ,

q (β)

η

(η – qs)(β–) y(s) dq s + c ηβ– .

q (β)





ϕp Dαq u(η) =



From Dαq u() = bDαq u(η), we have 



c = 

( – qs)(β–) y(s) dq s – ( – b ηβ– ) q (β)

 

η

b (η – qs)(β–) y(s) dq s, ( – b ηβ– ) q (β)

where b = bp– . Similar to Lemma ., we have   ϕp Dαq u(t) = –





H(t, qs)y(s) dq s. 

Consequently, the fractional boundary value problem (.) is equivalent to the following problem:  Dαq u(t) + ϕr u() = ,



H(t, qs)y(s) dq s = ,

t ∈ (, ),



u() = au(ξ ).

Lemma . implies that the fractional boundary value problem (.) has a unique solution 





u(t) =

H(s, qτ )y(τ ) dq τ dq s.

G(t, qs)ϕr 



 



The proof is completed.

Lemma . Let  < α, β ≤ ,  < ξ , η < , and  ≤ a, b ≤ . Then functions G(t, s) and H(t, s) defined by (.) and (.), respectively, are continuous on [, ] × [, ] satisfying (a) G(t, qs) ≥ , H(t, qs) ≥ , for all t, s ∈ [, ]; (b) for all t, s ∈ [, ], σ (qs)t α– ≤ G(t, qs) ≤ σ (qs)t α– , where σ (s) =

ag(ξ , s) ,  – aξ α–

σ (s) =

ag(ξ , s) ( – s)(α–) + .

q (α)  – aξ α–

Proof The proof is obvious, so we omit the proof. From Lemmas . and ., it is easy to obtain the following lemma.



Yuan and Yang Journal of Inequalities and Applications 2014, 2014:481 http://www.journalofinequalitiesandapplications.com/content/2014/1/481

Page 8 of 14

β

Lemma . Let u(t) ∈ C([, ], R) satisfies u() = , u() = ϕp (b)u(η), and Dq u(t) ≥  for any t ∈ (, ), then u(t) ≤ , for t ∈ [, ]. Let E = {u : u, ϕp (Dαq u) ∈ C  [, ]}. Now we introduce the following definitions about the upper and lower solutions of the fractional q-difference boundary value problem (.). Definition . A function φ(t) is called a lower solution of the fractional q-difference boundary value problem (.), if φ(t) ∈ E and φ(t) satisfies      Dβq ϕp Dαq φ(t) ≤ f t, φ(t) , φ() ≤ ,

t ∈ (, ), Dαq φ() ≥ ,

φ() ≤ aφ(ξ ),

Dαq φ() ≥ bDαq φ(η).

Definition . A function ψ(t) is called an upper solution of the fractional q-difference boundary value problem (.), if ψ(t) ∈ E and ψ(t) satisfies      Dβq ϕp Dαq ψ(t) ≥ f t, ψ(t) , ψ() ≤ ,

t ∈ (, ), Dαq ψ() ≥ ,

ψ() ≤ aψ(ξ ),

Dαq ψ() ≥ bDαq ψ(η).

3 Main results For the sake of simplicity, we make the following assumptions throughout this paper. (H ) f (t, u) ∈ C[(, ) × (, +∞), [, +∞)] and f (t, u) is decreasing in u. (H ) Set e(t) = t α– . For any constant ρ > , f (t, ρ) ≡ , and 







σ (qs)ϕr

< 

  H(s, qτ )f τ , ρe(τ ) dq τ dq s < +∞.



We define P ={u ∈ C[, ]: there exist two positive constants  < lu < Lu such that lu e(t) ≤ u(t) ≤ Lu e(t), t ∈ [, ]}. Obviously, e(t) ∈ P. Therefore, P is not empty. For any u ∈ P, define an operator T by 





(Tu)(t) =



G(t, qs)ϕr 

  H(s, qτ )f τ , u(τ ) dq τ dq s,

t ∈ [, ].



Theorem . Suppose that conditions (H )-(H ) are satisfied, then the boundary value problem (.) has at least one positive solution u, and there exist two positive constants  < λ <  < λ such that λ e(t) ≤ u(t) ≤ λ e(t), t ∈ [, ]. Proof We will divide our proof into four steps. Step . We show that T is well defined on P and T(P) ⊂ P, and T is decreasing in u. In fact, for any u ∈ P, by the definition of P, there exist two positive constants  < lu <  < Lu such that lu e(t) ≤ u(t) ≤ Lu e(t) for any t ∈ [, ]. It follows from Lemma . and conditions (H )-(H ) that 





(Tu)(t) =

G(t, qs)ϕr 



  H(s, qτ )f τ , u(τ ) dq τ dq s









≤ e(t)

σ (qs)ϕr 





  H(s, qτ )f τ , lu e(τ ) dq τ dq s < +∞.

(.)

Yuan and Yang Journal of Inequalities and Applications 2014, 2014:481 http://www.journalofinequalitiesandapplications.com/content/2014/1/481

Page 9 of 14

On the other hand, it follows from Lemma . that 





(Tu)(t) =



G(t, qs)ϕr 

  H(s, qτ )f τ , u(τ ) dq τ dq s









≥ e(t)



σ (qs)ϕr 

  H(s, qτ )f τ , Lu e(τ ) dq τ dq s.

(.)



Take lu

 

    = min , σ (qs)ϕr H(s, qτ )f τ , Lu e(τ ) dq τ dq s , 



 

     Lu = max , σ (qs)ϕr H(s, qτ )f τ , lu e(τ ) dq τ dq s , 



then by (.) and (.), lu e(t) ≤ (Tu)(t) ≤ Lu e(t), which implies that T is well defined and T(P) ⊂ P. It follows from (H ) that the operator T is decreasing in u. By direct computations, we can state that      Dβq ϕp Dαq (Tu)(t) = f t, Tu(t) , (Tu)() = ,

 < t < ,

(Tu)() = a(Tu)(ξ ),

Dαq (Tu)() = ,

(.)

Dαq (Tu)() = bDαq (Tu)(η).

Step . We focus on lower and upper solutions of the fractional q-difference boundary value problem (.). Let   m(t) = min e(t), (Te)(t) ,

  n(t) = max e(t), (Te)(t) ,

(.)

then, if e(t) = (Te)(t), the conclusion of Theorem . holds. If e(t) = (Te)(t), clearly, m(t), n(t) ∈ P, and m(t) ≤ e(t) ≤ n(t).

(.)

We will prove that the functions φ(t) = Tn(t), ψ(t) = Tm(t) are a couple of lower and upper solutions of the fractional q-difference boundary value problem (.), respectively. From (H ), we know that T is nonincreasing relative to u. Thus it follows from (.) and (.) that φ(t) = Tn(t) ≤ Tm(t) = ψ(t), Tn(t) ≤ Te(t) ≤ n(t),

(.)

Tm(t) ≥ Te(t) ≥ m(t),

and φ(t), ψ(t) ∈ P. It follows from (.)-(.) that           Dβq ϕp Dαq φ(t) – f t, φ(t) ≤ Dβq ϕp Dαq (Tn)(t) – f t, n(t) = , φ() = , φ() = aφ(ξ ), Dαq φ() = , Dαq φ() = bDαq φ(η),           Dβq ϕp Dαq ψ(t) – f t, ψ(t) ≥ Dβq ϕp Dαq (Tm)(t) – f t, m(t) = , ψ() = ,

ψ() = aψ(ξ ),

Dαq ψ() = ,

Dαq ψ() = bDαq ψ(η),

(.)

Yuan and Yang Journal of Inequalities and Applications 2014, 2014:481 http://www.journalofinequalitiesandapplications.com/content/2014/1/481

Page 10 of 14

that is, φ(t) and ψ(t) are a couple of lower and upper solutions of the fractional q-difference boundary value problem (.), respectively. Step . We will show that the fractional q-difference boundary value problem      Dβq ϕp Dαq u(t) = g t, u(t) , u() = ,

 < t < , Dαq u() = ,

u() = au(ξ ),

(.)

Dαq u() = bDαq u(η),

has at least one positive solution, where ⎧ ⎪ f (t, φ(t)), if u(t) < φ(t),   ⎨ g t, u(t) = f (t, u(t)), if φ(t) ≤ u(t) ≤ ψ(t), ⎪ ⎩ f (t, ψ(t)), if u(t) > ψ(t).

(.)

It follows from (H ) and (.) that g(t, u) : [, ] × R → R is continuous. To see this, we consider the operator A : C[, ] → C[, ] defined as follows: 





Au(t) =





H(s, qτ )g τ , u(τ ) dq τ dq s,

G(t, qs)ϕr 







where G(t, s) is defined as (.), H(t, s) is defined as (.). It is clear that Au ≥ , for all u ∈ P, and a fixed point of the operator A is a solution of the boundary value problem (.). Noting that φ(t) ∈ P, there exists a positive constant  < lφ <  such that φ(t) ≥ lφ e(t), t ∈ [, ]. It follows from Lemma ., (.), and (H ) that 





Au(t) =



G(t, qs)ϕr 

  H(s, qτ )g τ , u(τ ) dq τ dq s









≤ e(t)



  H(s, qτ )g τ , u(τ ) dq τ dq s



  H(s, qτ )g τ , lφ e(τ ) dq τ dq s < +∞,

σ (qs)ϕr 







≤ e(t)



σ (qs)ϕr 



which implies that the operator A is uniformly bounded. On the other hand, since G(t, s) is continuous on [, ] × [, ], it is uniformly continuous on [, ] × [, ]. So, for fixed s ∈ [, ] and for any ε > , there exists a constant δ > , such that any t , t ∈ [, ] and |t – t | < δ,   G(t , qs) – G(t , qs) <  

ε



 ϕr (

 H(s, qτ )g(τ , lφ e(τ )) dq τ ) dq s

.

Then, for all u(t) ∈ C[, ], we have   Au(t ) – Au(t ) 

 G(t , qs) – G(t , qs)ϕr



= 







  H(s, qτ )g τ , u(τ ) dq τ dq s

 

< 

=  

 



ϕr (



ϕr (







ε H(s, qτ )g(τ , lφ e(τ )) dq τ ) dq s   ε

H(s, qτ )g(τ , lφ e(τ )) dq τ ) dq s





ϕr

  H(s, qτ )f τ , ϕ(τ ) dq τ dq s













H(s, qτ )f τ , ϕ(τ ) dq τ dq s = ε,

ϕr 

Yuan and Yang Journal of Inequalities and Applications 2014, 2014:481 http://www.journalofinequalitiesandapplications.com/content/2014/1/481

Page 11 of 14

that is to say, A is equicontinuous. Thus, from the Arzela-Ascoli theorem, we know that A is a compact operator, by using the Schauder fixed point theorem, the operator A has a fixed point u such that u = Au; i.e., the fractional q-difference boundary value problem (.) has a positive solution. Step . We will prove that the boundary value problem (.) has at least one positive solution. Suppose that u(t) is a solution of (.), we only need to prove that φ(t) ≤ u(t) ≤ ψ(t), t ∈ [, ]. Now we claim that φ(t) ≤ u(t) ≤ ψ(t), t ∈ [, ]. In fact, since u is fixed point of A and (.), we get u() = ,

u() = au(ξ ),

ψ() = ,

ψ() = aψ(ξ ),

Dαq u() = ,

Dαq u() = bDαq u(η),

Dαq ψ() = ,

Dαq ψ() = bDαq ψ(η).

(.)

Suppose by contradiction that u(t) ≥ ψ(t). According to the definition of g, one verifies that        Dβq ϕp Dαq u(t) = g t, u(t) = f t, ψ(t) ,

 < t < .

(.)

On the other hand, since ψ is an upper solution to (.), we obviously have      Dβq ϕp Dαq ψ(t) ≥ f t, ψ(t) ,

 < t < .

(.)

Let z(t) = ϕp (Dαq ψ(t)) – ϕp (Dαq u(t)),  < t < . From (.) and (.), we can get           Dβq z(t) = Dβq ϕp Dαq ψ(t) – Dβq ϕp Dαq u(t) ≥ f t, ψ(t) – f t, ψ(t) = , z() = ,

z() = ϕp (b)z(η).

Thus, by Lemma ., we have z(t) ≤ , t ∈ [, ], which implies that     ϕp Dαq ψ(t) ≤ ϕp Dαq u(t) ,

t ∈ [, ].

Since ϕp is monotone increasing, we obtain Dαq ψ(t) ≤ Dαq u(t), i.e., Dαq (ψ – u)(t) ≤ . Combining Lemma ., we have (ψ – u)(t) ≥ . Therefore, ψ(t) ≥ u(t), t ∈ [, ], a contradiction to the assumption that u(t) > ψ(t). Hence, u(t) > ψ(t) is impossible. Similarly, suppose by contradiction that u(t) ≤ φ(t). According to the definition of g, one verifies that     g t, u(t) = f t, φ(t) ,

 < t < .

Consequently, we obtain      Dβq ϕp Dαq u(t) = f t, φ(t) ,

 < t < .

(.)

On the other hand, since φ is an upper solution to (.), we obviously have      Dβq ϕp Dαq φ(t) ≤ f t, φ(t) ,

 < t < .

(.)

Yuan and Yang Journal of Inequalities and Applications 2014, 2014:481 http://www.journalofinequalitiesandapplications.com/content/2014/1/481

Page 12 of 14

Let z(t) = ϕp (Dαq u(t)) – ϕp (Dαq φ(t)),  < t < . From (.) and (.), we get           Dβq z(t) = Dβq ϕp Dαq u(t) – Dβq ϕp Dαq ϕ(t) ≥ f t, φ(t) – f t, φ(t) = , z() = ϕp (b)z(η).

z() = ,

Thus, by Lemma ., we have z(t) ≤ , t ∈ [, ], which implies that     ϕp Dαq u(t) ≤ ϕp Dαq ϕ(t) ,

t ∈ [, ].

Since ϕp is monotone increasing, we obtain Dαq u(t) ≤ Dαq φ(t), i.e., Dαq (u – φ)(t) ≤ . Combining Lemma ., we have (u – φ)(t) ≥ . Therefore, u(t) ≥ φ(t), t ∈ [, ], a contradiction to the assumption that u(t) < φ(t). Hence, u(t) < φ(t) is impossible. Consequently, we have φ(t) ≤ u(t) ≤ ψ(t), t ∈ [, ], that is, u(t) is a positive solution of the boundary value problem (.). Furthermore, φ(t), ψ(t) ∈ P implies that there exist two positive constants  < λ <  < λ such that λ e(t) ≤ u(t) ≤ λ e(t), t ∈ [, ]. Thus, we have finished the proof of Theorem ..  Theorem . If f (t, u) ∈ C([, ] × [, +∞), [, +∞)) is decreasing in u and f (t, ρ) ≡  for any ρ > , then the boundary value problem (.) has at least one positive solution u, and there exist two positive constants  < λ <  < λ such that λ e(t) ≤ u(t) ≤ λ e(t), t ∈ [, ]. Proof The proof is similar to Theorem ., we omit it here.



4 Two examples Example . Consider the p-Laplacian fractional q-difference boundary value problem √  ( +  t)   / D/ u(t) = ϕ D , √  / / tu(t)

  , u() = , u() = u   D/ / u() = ,

D/ / u() =

 < t < , (.)



 /  D u .  / 

It is easy to check that (H ) holds. For any ρ > , f (t, ρ) ≡ , we have 





σ (qs)ϕ

< 

 ≤

 σ (qs)ϕ



 = √ ρ







  H(, qτ )f τ , ρe(τ ) dq τ dq s

 



σ (qs) dq s 



H(s, qτ )f τ , ρe(τ ) dq τ dq s 









H(, qτ ) 

√ ( +  τ ) dq τ < +∞, τ /

which implies that (H ) holds. Theorem . implies that the boundary value problem (.) has at least one positive solution.

Yuan and Yang Journal of Inequalities and Applications 2014, 2014:481 http://www.journalofinequalitiesandapplications.com/content/2014/1/481

Page 13 of 14

Example . Consider the p-Laplacian fractional q-difference boundary value problem     /  D/ , / ϕp D/ u(t) = t + √ u(t) + 

  , u() = , u() = u   D/ / u() = ,

D/ / u() =

 < t < , (.)



 /  . D u  / 

It is not difficult to check that f (t, u) : [, ] × [, +∞) → [, +∞) is continuous and decreasing in u and f (t, ρ) ≡  for any ρ > . Theorem . implies that the boundary value problem (.) has at least one positive solution.

Competing interests The authors declare that they have no competing interests. Authors’ contributions The authors contributed to each part of this work equally and read and approved the final version of the manuscript. Acknowledgements The authors sincerely thank the editor and reviewers for their valuable suggestions and useful comments, which have led to the present improved version of the original manuscript. Received: 16 October 2014 Accepted: 20 November 2014 Published: 03 Dec 2014 References 1. Chai, G: Positive solutions for boundary value problem of fractional differential equation with p-Laplacian operator. Bound. Value Probl. 2012, 18 (2012) 2. Wu, W, Zhou, X: Eigenvalue of fractional differential equations with p-Laplacian operator. Discrete Dyn. Nat. Soc. 2013, Article ID 137890 (2013) 3. Yao, S, Wang, G, Li, Z, Yu, L: Positive solutions for three-point boundary value problem of fractional differential equation with p-Laplacian operator. Discrete Dyn. Nat. Soc. 2013, Article ID 376938 (2013) 4. Nyamoradi, N, Baleanu, D, Bashiri, T: Positive solutions to fractional boundary value problems with nonlinear boundary conditions. Abstr. Appl. Anal. 2013, Article ID 579740 (2013) 5. Su, Y, Li, Q, Liu, X: Existence criteria for positive solutions of p-Laplacian fractional differential equations with derivative terms. Adv. Differ. Equ. 2013, 119 (2013) 6. Liu, Y, Lu, L: A class of fractional p-Laplacian integrodifferential equations in Banach spaces. Abstr. Appl. Anal. 2013, Article ID 398632 (2013) 7. Liu, X, Jia, M: Multiple solutions for fractional differential equations with nonlinear boundary conditions. Comput. Math. Appl. 59, 2880-2886 (2010) 8. Wang, J, Xiang, H, Liu, Z: Upper and lower solutions method for a class of singular fractional boundary value problems with p-Laplacian operator. Abstr. Appl. Anal. 2010, Article ID 971824 (2010) 9. Ren, T, Chen, X: Positive solutions of fractional differential equation with p-Laplacian operator. Abstr. Appl. Anal. 2013, Article ID 789836 (2010) 10. Al-Salam, WA: Some fractional q-integrals and q-derivatives. Proc. Edinb. Math. Soc. 15(2), 135-140 (1966/1967) 11. Agarwal, RP: Certain fractional q-integrals and q-derivatives. Proc. Camb. Philos. Soc. 66, 365-370 (1969) 12. Atici, FM, Eloe, PW: Fractional q-calculus on a time scale. J. Nonlinear Math. Phys. 14, 333-344 (2007) 13. Rajkovi´c, PM, Marinkovi´c, SD, Stankovi´c, MS: On q-analogues of Caputo derivative and Mittag-Leffler function. Fract. Calc. Appl. Anal. 10, 359-373 (2007) 14. Rajkovi´c, PM, Marinkovi´c, SD, Stankovi´c, MS: Fractional integrals and derivatives in q-calculus. Appl. Anal. Discrete Math. 1, 311-323 (2007) 15. Abdeljawad, T, Benli, B, Baleanu, D: A generalized q-Mittag-Leffler function by q-Caputo fractional linear equations. Abstr. Appl. Anal. 2012, Article ID 546062 (2012) 16. Alsaedi, A, Ahmad, B, Al-Hutami, H: A study of nonlinear fractional q-difference equations with nonlocal integral boundary conditions. Abstr. Appl. Anal. 2013, Article ID 410505 (2013) 17. El-Shahed, M, Al-Askar, FM: On the existence and uniqueness of solutions for q-fractional boundary value problem. Int. J. Math. Anal. 5, 1619-1630 (2011) 18. Ferreira, RAC: Nontrivial solutions for fractional q-difference boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2010, 70 (2010) 19. Ferreira, RAC: Positive solutions for a class of boundary value problems with fractional q-differences. Comput. Math. Appl. 61, 367-373 (2011) 20. Li, X, Han, Z, Sun, S: Existence of positive solutions of nonlinear fractional q-difference equation with parameter. Adv. Differ. Equ. 2013, 140 (2013)

Yuan and Yang Journal of Inequalities and Applications 2014, 2014:481 http://www.journalofinequalitiesandapplications.com/content/2014/1/481

21. Yu, C, Wang, J: Positive solutions of nonlocal boundary value problem for high-order nonlinear fractional q-difference equations. Abstr. Appl. Anal. 2013, Article ID 928147 (2013) 22. Yuan, Q, Yang, W: Positive solutions of nonlinear boundary value problems for delayed fractional q-difference systems. Adv. Differ. Equ. 2014, 51 (2014) 23. Yang, W: Positive solutions for boundary value problems involving nonlinear fractional q-difference equations. Differ. Equ. Appl. 5, 205-219 (2013) 24. Zhao, Y, Chen, H, Zhang, Q: Existence results for fractional q-difference equations with nonlocal q-integral boundary conditions. Adv. Differ. Equ. 2013, 48 (2013) 25. Zhao, Y, Chen, H, Zhang, Q: Existence and multiplicity of positive solutions for nonhomogeneous boundary value problems with fractional q-derivative. Bound. Value Probl. 2013, 103 (2013) 26. El-Shahed, M, Al-Askar, FM: Positive solutions for boundary value problem of nonlinear fractional q-difference equation. ISRN Math. Anal. 2011, Article ID 385459 (2011) 27. Graef, JR, Kong, L: Positive solutions for a class of higher order boundary value problems with fractional q-derivatives. Appl. Math. Comput. 218, 9682-9689 (2012) 28. Graef, JR, Kong, L: Existence of positive solutions to a higher order singular boundary value problem with fractional q-derivatives. Fract. Calc. Appl. Anal. 16, 695-708 (2013) 29. Zhao, Y, Ye, G, Chen, H: Multiple positive solutions of a singular semipositone integral boundary value problem for fractional q-derivatives equation. Abstr. Appl. Anal. 2013, Article ID 643571 (2013) 30. Ahmad, B, Ntouyas, S, Purnaras, I: Existence results for nonlocal boundary value problems of nonlinear fractional q-difference equations. Adv. Differ. Equ. 2012, 140 (2012) 31. Yang, W: Positive solutions for nonlinear semipositone fractional q-difference system with coupled integral boundary conditions. Appl. Math. Comput. 244, 702-725 (2014) 32. Agarwal, RP, Ahmad, B, Alsaedi, A, Al-Hutami, H: Existence theory for q-antiperiodic boundary value problems of sequential q-fractional integrodifferential equations. Abstr. Appl. Anal. 2014, Article ID 207547 (2014) 33. Ahmad, B, Nieto, JJ, Alsaedi, A, Al-Hutami, H: Existence of solutions for nonlinear fractional q-difference integral equations with two fractional orders and nonlocal four-point boundary conditions. J. Franklin Inst. 351, 2890-2909 (2014) ˇ H, Özarslan, M: On the solvability of Caputo q-fractional boundary value problem involving p-Laplacian 34. Aktuglu, operator. Abstr. Appl. Anal. 2013, Article ID 658617 (2013) 35. Miao, F, Liang, S: Uniqueness of positive solutions for fractional q-difference boundary-value problems with p-Laplacian operator. Electron. J. Differ. Equ. 2013, 174 (2013) 36. Yang, W: Positive solution for fractional q-difference boundary value problems with φ -Laplacian operator. Bull. Malays. Math. Soc. 36(4), 1195-1203 (2013) 37. Kac, V, Cheung, P: Quantum Calculus. Springer, New York (2002)

10.1186/1029-242X-2014-481 Cite this article as: Yuan and Yang: Positive solution for q-fractional four-point boundary value problems with p-Laplacian operator. Journal of Inequalities and Applications 2014, 2014:481

Page 14 of 14