Mardanov et al. Advances in Difference Equations (2015) 2015:185 DOI 10.1186/s13662-015-0532-5

RESEARCH

Open Access

Existence and uniqueness results for q-fractional difference equations with p-Laplacian operators Misir J Mardanov1 , Nazim I Mahmudov2* and Yagub A Sharifov3 *

Correspondence: [email protected] 2 Eastern Mediterranean University, Gazimagusa, TRNC, Mersin 10, Turkey Full list of author information is available at the end of the article

Abstract In this paper, we consider the following two-point boundary value problem for q-fractional p-Laplace difference equations. New results on the existence and uniqueness of solutions for q-fractional boundary value problem are obtained. These results extend the corresponding ones of ordinary differential equations of integer order. Finally, an example is presented to illustrate the validity and practicability of our main results. Keywords: fractional differential equations; existence; fixed point

1 Introduction Fractional q-difference (q-fractional difference) equations are regarded as the fractional analog of q-difference equations. The topic of q-fractional equations has attracted the attention of many researchers. The details of some recent development of the subject can be found in [–], whereas the background material on q-fractional calculus can be found in [, ]. The study of boundary value problems of fractional q-difference equations is in its infancy. In , Ferreira [] considered the existence of nontrivial solutions to the fractional q-difference equation   Dαq,+ x(t) = f t, x(t) ,  ≤ t ≤ , x() = x() = , where  < α ≤  and f : [, ] × R → R is a nonnegative continuous function. In , El-Shahed and Al-Askar [] studied the existence of a positive solution for the boundary value problem of the nonlinear factional q-difference equation C

  Dαq,+ x(t) + a(t)f t, x(t) = ,

x() = Dq x() = ,

 ≤ t ≤ ,  < α ≤ ,

γ Dq x() + βDq x() = .

In , Liang and Zhang [] studied the existence and uniqueness of positive solutions for the three-point boundary problem of fractional q-differences C

  Dαq,+ x(t) + f t, x(t) = ,

 ≤ t ≤ ,  < α ≤ ,

© 2015 Mardanov et al. 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.

Mardanov et al. Advances in Difference Equations (2015) 2015:185

x() = Dq x() = ,

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Dq x() – βDq x(η) = ,

where  < βηα– < . In , Zhao et al. [] studied the existence results for fractional q-difference equations with nonlocal q-integral boundary conditions,   Dαq,+ x(t) + f t, x(t) = , x() = ,

 ≤ t ≤ ,  < α ≤ ,  η (η q qs)β– x() = μx(η) = μ u(s) ds. q (β) 

For some recent work on q-difference equations with p-Laplacian, we refer the reader to [–]. In [], Aktuğlu 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 < ,

k = , , . . . , n – ,

x() = a x(),

Dq x() = a Dq x(),

where a , a = , α > , 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: C

     Dβq ϕp C Dαq x(t) = f t, x(t) ,

x() = Dq x() = ,

 < t < ,  < α < ,

Dq x() = ,

Dβq u() = ,

where  < βγ α– < . 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 [], Yang investigated the following fractional q-difference boundary value problem with p-Laplacian operator: C

     Dβq ϕp C Dαq x(t) = f t, x(t) ,

x() = x() = ,

C

 < t < ,  < α < ,

Dαq x() = C Dαq x() = ,

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. Very recently, in [], Yuan and Yang considered a class of four-point boundary value problems of fractional q-difference equations with p-Laplacian operator      Dβq ϕp Dαq x(t) = f t, x(t) ,

 < t < ,  < α < ,

x() = ,

Dαq x() = ,

x() = axξ ,

Dαq x() = bDαq x(η),

Mardanov et al. Advances in Difference Equations (2015) 2015:185

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β

where Dq , Dαq are the fractional q-derivative of the Riemann-Liouville type with  < α, β ≤ . 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. However, the theory of boundary value problems for nonlinear q-difference equations is still in the initial stages and many aspects of this theory need to be explored. To the best of our knowledge, the theory of boundary value problems for nonlinear q-difference equations with p-Laplacian is yet to be developed. Motivated by the previously mentioned works, we will consider the existence of solutions of q-fractional p-Laplacian BVP with two-point boundary conditions. The main difficulty is that, for p = , it is impossible for us to find a Green’s function in the equivalent β integral operator since the differential operator Dq,+ ϕp (Dαq,+ ) is nonlinear. This paper is concerned with the BVP ⎧ β C C α ⎪ ⎨ Dq,+ ϕp ( Dq,+ x)(t) = f (t, x(t)), x() = γ x(), ⎪ ⎩C α Dq x() = η C Dαq,+ x(),

t ∈ [, ], ()

where ϕp (s) := |s|p– s, p > , ϕp– = ϕν , p + ν = ,  < α, β ≤ ,  < α + β ≤ ,  < γ , η < , by using some known fixed point theorems. To make this paper self-contained, below we recall some well-known facts on q-calculus (see [, ] and references therein) and on fractional q-calculus. In what follows, q is a real number satisfying  < q < . We define the q-derivative of a real valued function f as Dq f (t) =

f (t) – f (qt) , ( – q)t

Dq f () = lim Dq f (t). t→

Higher order q-derivatives are given by Dq f (t) = f (t),

Dnq f (t) = Dq Dn– q f (t),

n ∈ N.

The q-integral of a function f defined in the interval [, t] is given by  Iq f (t) =

t

f (s) dq s = 

∞

  t( – q)qn f xqn ,

n=

provided the series converges. If  < a < b and f is defined on the interval [, b], then 



b

f (s) dq s = a



b

a

f (s) dq s – 

f (s) dq s. 

Similarly, we have Iq f (t) = f (t),

Iqn f (t) = Iq Iqn– f (t),

n ∈ N.

Observe that the operators Iq and Dq are inverses of each other, in the sense that Dq Iq f (t) = f (t),

()

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and if f is continuous at t = , then Iq Dq f (t) = f (t) – f (). In q-calculus, the product rule and integration by parts formula are Dq (gh)(t) = Dq g(t)h(t) + g(qt)Dq h(t),  x  x

x f (t)Dq g(t) dq t = f (t)g(t)  – Dq f (t)g(qt) dq t. 

() ()



In the limit q →  the above results correspond to their counterparts in standard calculus. A q-number denoted by [a]q is defined by [a]q :=

 – qa , –q

a ∈ R.

The q-shifted factorial is defined as (a; q) = ,

(a; q)n =

n–    – aqj ,

n ∈ N ∪ {∞}.

j=

The q-analog of the (x – y)n is (x q y) := ;

(x q y)n :=

n– 

 x – yqj ,

n ∈ N, x, y ∈ R,

j= ∞ x – yqj , (x q y) := x x – yqα+j j= α

α

α ∈ R.

The q-gamma function q (x) is defined as q (x) =

( q q)x– , ( – q)x–

y ∈ R\{, –, –, . . .},

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



t

(t q qs)α– f (s) dq s, q (α)

α > , t ∈ [, ].

The q-fractional integration possesses the semigroup property: β

β+α

α Iq,+ Iq, + f (t) = Iq,+ f (t),

α, β ∈ R+ .

Further, β

Iq,+ xσ =

q (σ + ) β+σ x . q (β + σ + )

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Definition  The Caputo fractional q-derivative of order β >  is defined by C

β–β

β

β

Dq,+ f (t) = Iq,+ Dq,+ f (t),

where β is the smallest integer greater than or equal to β. Next we recall some properties involving Riemann-Liouville q-fractional integral and Caputo fractional q-derivative ([], Theorem .):

β– β

β

Iq,+ C Dq,+ f (t) = f (t) –

k=

C

β

β

Dq,+ Iq,+ f (t) = f (t),

 k  +  tk D f  , q (k + ) q

t ∈ (, a], β > .

t ∈ (, a], β > ;

() ()

The following properties of the p-Laplacian operator will be used in the rest of the paper. (L) If  < p < , uv > ; u , v ≥ r > , then

ϕp (u) – ϕp (v) ≤ (p – )rp– |u – v|. (L) If p > , |u|, |v| ≤ R, then

ϕp (u) – ϕp (v) ≤ (p – )Rp– |u – v|. Next we present the fixed point theorems that will be used in the proofs of our main results. Theorem  (Banach fixed point theorem) Let (X, d) be a complete metric space, and let  : X → X be a contraction mapping. Then  admits a unique fixed point X. Theorem  Let E be a Banach space, C a closed, convex subset of E and U ⊂ C an open subset with  ∈ U. Let F : U → C be a continuous function such that F(U) is contained in a compact set. Then either . F has a fixed point in U, or . there exist u ∈ ∂U and λ ∈ (, ) with u = λF(u).

2 Main results As mentioned before, we will discuss the existence (and uniqueness) of solutions for the nonlinear q-fractional p-Laplacian BVP with two-point boundary conditions. In what follows we assume that  < p < , ν > . Lemma  Given f ∈ C[, ], the unique solution of ⎧ β C C α ⎪ ⎨ Dq,+ ϕp ( Dq,+ x)(t) = f (t), x() = γ x(), ⎪ ⎩C α Dq,+ x() = η C Dαq,+ x(),

t ∈ [, ], ()

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is  t  (t q qs)α– x(t) = q (α)     s ϕp (η) β  Iq,+ f () dq s × ϕν (s q qτ )β– f (τ ) dτ + q (β)   – ϕp (η)   γ + ( q qs)α– –γ     s ϕp (η) β  Iq,+ f () dq s. × ϕν (s q qτ )β– f (τ ) dτ + q (β)   – ϕp (η) Proof Assume that x(t) satisfies (). Then from () we have   β ϕp C Dαq,+ x(t) = Iq,+ f (t) + c ,

c ∈ R.

From the boundary condition C Dαq,+ x() = η C Dαq,+ x(), one can see that   ϕp (η) = |η|p– η, ϕp C Dαq,+ x() = c ,   β ϕp C Dαq,+ x() = Iq,+ f () + c ,   β ϕp η C Dαq,+ x() = ϕp (η)Iq,+ f () + c ϕp (η), β

ϕp (η)Iq,+ f () + c ϕp (η) = c , c =

ϕp (η) β I + f ().  – ϕp (η) q,

Thus β  α x(t) = Iq, + ϕν Iq,+ f (·) + c (t) + c , which together with the boundary value condition x() = γ x() yields β   α c = γ Iq, + ϕν Iq,+ f () + c + c , β

c =

α γ Iq, + ϕν (Iq,+ f + c )()

–γ

.

It follows that  ϕp (η) β Iq,+ f () (t)  – ϕp (η)   ϕp (η) β γ α β + Iq,+ ϕν Iq,+ f + + Iq,+ f () (). –γ  – ϕp (η)

 β α ϕ x(t) = Iq, + ν Iq,+ f +

The converse is clear. The proof is completed.



We denote by C[, ] the Banach space of all continuous functions from [, ] to R endowed with a topology of uniform convergence with the norm defined by

  x := max x(t) :  ≤ t ≤  .

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We use Lemma  to define an operator  : C[, ] → C[, ] by  t  (x)(t) = (t q qs)α– q (α)     s ϕp (η) β  β– I + f () dq s × ϕν (s q qτ ) f (τ ) dτ + q (β)   – ϕp (η) q,   γ  + ( q qs)α–  – γ q (α)     s ϕp (η) β  Iq,+ f () dq s. × ϕν (s q qτ )β– f (τ ) dτ + q (β)   – ϕp (η)

()

Observe that the problem () has a (unique) solution if and only if the operator equation x = y has a (unique) fixed point. In the sequel, we need the following operators:  ( x)(s) = ϕν

 q (β)



s

  (s q qτ )β– f τ , x(τ ) dτ



     ϕp (η)  β– + ( q qτ ) f τ , x(τ ) dτ ,  – ϕp (η) q (β)   t  ( h)(t) = (t q qs)α– h(s) dq s q (α)    γ  + ( q qs)α– h(s) dq s.  – γ q (α)  It is obvious that  : C[, ] → C[, ],

 : C[, ] → C[, ],

(x)(t) = ( ◦  )x(t). Set   Br := x ∈ C[, ] : x < r . To state and prove the existence and uniqueness theorem we impose the following assumptions. (A) f : [, ] × R → R is a continuous function such that

 

f (t, x) ≤ a(t), a(t) ∈ C [, ], R+ ,

f (t, x) – f (t, y) ≤ L|x – y|, t ∈ [, ], x, y ∈ R. (A) The following inequality holds:  :=

ν –   a ν– < .  – γ qν– (β + )q (α + ) ( – ηp– )ν–

Lemma  Assume that the assumption (A) holds.

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(i) If ν > , then the operator  satisfies the following conditions:

( x)(t) – ( y)(t) ≤ (ν – )Rν–

( x)(t) ≤ Rν– ,

  x – y , q (β + )  – ηp–

where R :=

  a . q (β + )  – ηp–

(ii) For any x ∈ C[, ] the function ( x)(t) is uniformly continuous on [, ]. Proof (i) Set

M := sup f (s, ) , ≤s≤

  β v(s) := Iq,+ f s, y(s) +

  β u(s) := Iq,+ f s, x(s) +

 ηp– β  I + f , x() ,  – ηp– q,

 ηp– β  I + f , y() .  – ηp– q,

It is clear that  ηp–

β  Iq,+ f , x() p– –η

s   (s q qτ )β– f τ , x(τ )

dq τ

 

u(s) ≤ I β + f s, x(s) + q, ≤

 

q (β) 



   ηp–

 β–

dq τ (  qτ ) f τ , x(τ ) q

q (β)  – ηp–   s  ≤ a (s q qτ )β– dq τ q (β)     ηp– + a ( q qτ )β– dq τ q (β)  – ηp–  +

=

  a := R. q (β + )  – ηp–

It follows that

 

( x)(s) = ϕν u(s) ≤ Rν– and

( x)(t) – ( y)(t)

    = ϕν u(s) – ϕp v(s)

≤ (ν – )Rν– u(s) – v(s)



 s     

s β– β–

≤ (ν – )Rν– (s  qτ ) f τ , x(τ ) dτ – (s  qτ ) f τ , y(τ ) dτ q q

q (β)  

Mardanov et al. Advances in Difference Equations (2015) 2015:185

+ (ν – )Rν–

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ηp–  q (β)  – ηp–

 

 

    β– β–

× ( q qτ ) f τ , x(τ ) dτ – ( q qτ ) f τ , y(τ ) dτ





≤ (ν – )Rν–

  x – y . q (β + )  – ηp–

(ii) We have

( x)(t ) – ( x)(t )

    = ϕν u(t ) – ϕp u(t )

≤ (ν – )Rν– u(t ) – u(t ) ≤ (ν – )Rν–



×

t

 q (β)

  (t q qτ )β– f τ , x(τ ) dτ –



 

t

  (t q qτ )β– f τ , x(τ ) dτ

 = (ν – )Rν– q (β)

 

 

β     β β– β–

× t ( q qs) f t s, x(t s) ds – t ( q qs) f t s, x(t s) ds





   

 β β   ν– β– ≤ (ν – )R ( q qs) f t s, x(t s) ds

t – t q (β)  



     

β  ν– β– + (ν – )R ( q qs) f t s, x(t s) – f t s, x(t s) ds

. t q (β)   It follows that ( x)(t) is uniformly continuous on [, ].



The first theorem that we state follows from the Banach fixed point theorem. Theorem  Under the assumptions (A) and (A) the boundary value problem () has a unique solution in C[, ]. Proof The idea of the proof is to show that  as defined in () admits a unique fixed point in C[, ]. For x, y ∈ C[, ] and for each t ∈ [, ], from the definition of  and the assumptions (A) and (A), we obtain

(x)(t) – (y)(t)

= ( ◦  )x(t) – ( ◦  )y(t)  t

 (t q qs)α– ( x)(s) – ( y)(s) dq s ≤ q (α)   

γ  + ( q qs)α– ( x)(s) – ( y)(s) dq s  – γ q (α)  ≤

   x –  y .  – γ q (α + )

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Consequently, by Lemma  we have

 

(x)(t) – (y)(t) ≤ ν –  a ν– x – y . ν–  – γ q (β + )q (α + ) ( – ηp– )ν– Taking into account that, by our assumption (A),  < , we conclude that the operator  is a contraction. Therefore, by the Banach contraction principle, the problem () has a unique solution. This completes the proof of Theorem .  For our next result, we use the Leray-Schauder alternative to ensure the existence of a solution for (). (A) f : [, ] × R → R is a continuous function and there exist a function l ∈ C([, ], R+ ) and nondecreasing functions ψ : R+ → R+ such that

 

f (s, x) ≤ l(s)ψ |x| ,

(s, x) ∈ [, ] × R.

(A) There exists a constant ω >  such that ω>

   l ν– ψ ν– (ω).  – γ qν– (β + )q (α + ) ( – ηp– )ν–

Theorem  Under conditions (A) and (A), the boundary value problem () has at least one solution in C[, ]. Proof Consider the operator  : C[, ] → C[, ] defined by (). It is easy to show that  is continuous. We complete the proof in the following steps. Step :  maps bounded sets into bounded sets in C[, ]. Indeed, for x ∈ Br from (A) and Lemma  (a(s) = l(s)ψ(r)) we have



(x)(t) = ( ◦  )x(t)  t

 ≤ (t q qs)α–  x(s) dq s q (α)   

γ  ( q qs)α–  x(s) dq s +  – γ q (α)  ≤

   x  – γ q (α + )

≤

  Rν–  – γ q (α + )

≤

   l ν– ψ ν– (r) ν–  – γ q (β + )q (α + ) ( – ηp– )ν–

and the result follows. Step :  maps bounded sets into equicontinuous sets of C[, ]. Let t , t ∈ [, ] with t < t and x ∈ Br . Then we can write

(x)(t ) – (x)(t )

 t  t

  (t q qs)α– ( x)(s) dq s – (t q qs)α– ( x)(s) dq s

=

q (α)  q (α) 

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 



α  α– α α– t = ( q qτ ) ( x)(t τ ) dq τ – t ( q qτ ) ( x)(t τ ) dq τ

q (α)  

 



 α α  α– ≤ ( q qτ ) ( x)(t τ ) dq τ

t – t

q (α)   

 α t ( q qτ )α– ( x)(t τ ) – ( x)(t τ ) dq τ := J. + q (α)  By Lemma  the function ( x)(t) is uniformly continuous on [, ] and uniformly bounded on Br , it follows that limt →t J = . Thus (Br ) is equicontinuous. It follows from the Arzelá-Ascoli theorem that the operator  : C[, ] → C[, ] is compact. Step :  has a fixed point in Bω . Let x be a solution and x = λx,  < λ < . Using the arguments of the proof of boundedness of , for  ≤ t ≤  we can write



x(t) = λ(x)(t) ≤

     l ν– ψ ν– x . ν– p– ν–  – γ q (β + )q (α + ) ( – η )

Consequently, in view of (A), there exists ω >  such that x = ω. We observe that the operator  : Bω → C[, ] is continuous and completely continuous. From the choice of Bω , there is no x ∈ ∂Bω such that x = λx for some λ ∈ (, ). Consequently, we can apply a nonlinear Leray-Schauder type alternative, to conclude that  has a fixed point x ∈ Bω which is a solution of the problem (). This completes the proof of Theorem . 

3 Examples Example  Consider a two-point boundary value problem of nonlinear fractional q-difference equations given by C

C /  – D/ q,+ ϕ/ Dq,+ x(t) = tan x(t) + sin t,

x() = γ x(), C

()

C / D/ q,+ x() = η Dq,+ x().

Corresponding to (), we get β = /, p = /, ν = , α = /, and f (t, x) = tan– x + sin t, a(t) =  + π/. It is obvious that

f (t, x) ≤  + π , 



f (t, x) – f (t, y) ≤ |x – y|.

We choose γ , η such that

 

ϕp (η) ν–  ν –

+ = a ν–  – γ qν– (β + )q (α + )  – ϕp (η)     π     + < . =  – γ q (π/)q (/)  – η/  The above facts imply that the BVP () satisfies all assumptions of Theorem  and has a unique solution.

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Example  Consider () with a different right-hand side f (t, x): C



C / D/ q,+ ϕ/ Dq,+ x(t)



 

|x(t)| cos(t  + )



x(t) + , = √ +  + |x(t)|  +t

x() = γ x(), C

()

C / D/ q,+ x() = η Dq,+ x().

Choosing f (t, x) =

  |x| cos(t  + )  |x| + + √  + |x|  +t

one can see that

 

f (t, x) ≤ l(t)ψ |x| ,

(t, x) ∈ [, ] × R,

with l(t) =

cos(t  + ) , √ +t

   ψ |x| = |x| + . 

We may choose γ , η such that  < ρ :=

    l ν– < .  – γ qν– (β + )q (α + ) ( – ηp– )ν– 

Using this one can see that there is ω >  such that     ω>ρ ω+ .  Thus all the conditions of Theorem  are satisfied. Hence there exists a solution of the problem ().

Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the manuscript and typed, read, and approved the final manuscript. Author details 1 Institute of Mathematics and Mechanics of NAS of Azerbaijan, B. Vahabzade, Baku, 1141, Azerbaijan. 2 Eastern Mediterranean University, Gazimagusa, TRNC, Mersin 10, Turkey. 3 Institute of Control Systems of NASA, Baku State University, Baku, Azerbaijan. Acknowledgements The authors would like to thank the anonymous referees for their valuable comments and suggestions. Received: 26 January 2015 Accepted: 8 June 2015 References 1. Ferreira, R: Nontrivial solutions for fractional q-difference boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2010, 70 (2010) 2. El-Shahed, M, Al-Askar, F: Positive solution for boundary value problem of nonlinear fractional q-difference equation. ISRN Math. Anal. 2011, Article ID 385459 (2011)

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