Existence and uniqueness of positive and nondecreasing solutions for a class of fractional boundary value problems involving the p-Laplacian operator

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R E S E A R C H

Open Access

Existence and uniqueness of positive and

nondecreasing solutions for a class of

fractional boundary value problems involving

the p-Laplacian operator

Serkan Araci

1*

_{, Erdo ˘gan ¸Sen}

2,3

_{, Mehmet Açikgöz}

4

_{and Hari M Srivastava}

5

*_{Correspondence:}

mtsrkn@hotmail.com

1_{Department of Economics, Faculty}

of Economics, Administrative and Social Science, Hasan Kalyoncu University, Gaziantep, 27410, Turkey Full list of author information is available at the end of the article

Abstract

In this article, we investigate the existence of a solution arising from the following fractional q-diﬀerence boundary value problem by using the p-Laplacian operator:

Dγq(

φ

p(Dδqy(t))) + f (t, y(t)) = 0 (0 < t < 1; 0 <

γ

< 1; 3 <

δ

< 4), y(0) = (Dqy)(0) = (D2qy)(0) = 0, a1(Dqy)(1) + a2(D2qy)(1) = 0, a1+|a2| = 0, D

γ

0+y(t)|t=0= 0. We make use of such a fractional q-diﬀerence boundary value problem in order to show the existence and uniqueness of positive and nondecreasing solutions by means of a familiar ﬁxed point theorem.

MSC: Primary 05A30; 26A33; 34K10; 39A13; secondary 34A08; 34B18

Keywords: positive solutions; ﬁxed point theorem; fractional q-diﬀerence equation; p-Laplacian operator

1 Introduction, deﬁnitions, and preliminaries

Recently, many mathematicians, physicists and engineers have extensively studied various families of fractional differential equations and their applications. The development of the theory of fractional calculus stems from the applications in many widespread disciplines such as engineering, economics and other fields. Jackson [] introduced the q-difference calculus (or the so-called quantum calculus), which is an old subject. New developments in this theory were made. These include (for example) the q-analogs of the fractional integral and the fractional derivative operators, the q-analogs of the Laplace, Fourier, and other integral transforms, and so on (see, for details, [–], and []; see also a very recent work [] dealing with q-calculus).

Throughout our present investigation, we make use of the following notations: N := {, , , . . .} and N:=N ∪ {}.

Moreover, as usual,R denotes the set of real numbers, R+denotes the set of positive real numbers,Z–denotes the set of negative integers, andC denotes the set of complex num-bers.

Al-Salam [] and Agarwal [] investigated several properties and results for some frac-tional q-integrals and fracfrac-tional q-derivatives which are based on the q-analog of the or-©2015 Araci et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribuAttribu-tion, and reproducAttribu-tion in any medium, provided the original work is properly credited.

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dinary integral: x

a

f(t) dt.

Atici and Eloe [] constructed interesting links between the fractional q-calculus in the existing literature and the fractional q-calculus on a time scale given by

Tt=

t: t = tqn(n∈ N; t∈ R;  < q < )

.

They also derived some properties of a q-Laplace transform, which are used to solve frac-tional q-difference equations. Benchohra et al. [] investigated the existence of solutions for fractional-order functional equations by means of the Banach fixed point theorem and its nonlinear alternative of Leray-Schauder type. El-Sayed et al. [] studied the stability, existence, uniqueness, and numerical solution of the fractional-order logistic equation. The work of El-Shahed [] was concerned with the existence and non-existence of pos-itive solutions for some nonlinear fractional boundary value problems. Ferreira (see [] and []) investigated the existence of nontrivial solutions to some nonlinear q-fractional boundary value problems by applying a fixed point theorem in cones. For more infor-mation on the positive solutions (or nontrivial solutions) for a class of boundary value problems with the fractional differential equations (or q-fractional differential equations), we refer the reader to such earlier works as (for example) [, , –], and [].

We now review brieﬂy some concepts of the quantum calculus. For q∈ (, ), the q-integer [λ]qis deﬁned by

[λ]q=  – qλ  – q (λ∈ R). Clearly, we have lim q→–[λ]q= λ,

so we say that [λ]qis a q-analog of the number λ. The q-analog of the binomial formula (a – b)nis given by (a – b)=  and (a – b)n= n– k= a– bqk (a, b∈ R; n ∈ N).

More generally, we have (a – b)(δ)= aδ ∞ n= a– bqn a– bqδ+n (δ∈ R). (.)

Clearly, if we set b =  in Eq. (.), it reduces immediately to

a(δ)= aδ (δ∈ R).

The q-gamma function is deﬁned as follows:

q(x) = ( – q)(x–) ( – q)x– x∈ R \{} ∪ Z–

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and satisﬁes the formula:

q(x + ) = [x]q(x).

The q-derivative of a function f (x) is given by (Dqf)(x) = f(x) – f (qx) ( – q)x and qlim→–(Dqf)(x) = f _{(x) =} d dx f(x). For the q-derivatives of higher order, we have

D_qf(x) = f (x) and Dn_qf(x) = Dq

Dn_q–f(x) (n∈ N).

Suppose now that  < a < b. Then the deﬁnite q-integral is deﬁned as follows:

(Iqf)(x) = x  f(t) dqt= x( – q) ∞ n= fxqnqn x∈ [, b] and b a f(t) dqt= b  f(t) dqt– a  f(t) dqt. The operator In q can be deﬁned by I_qf(x) = f (x) and I_qnf(x) = Iq I_qn–f(x) (n∈ N).

The Fundamental Theorem of Calculus does indeed apply mutatis mutandis to the op-erators Iqand Dq. We thus have

(DqIqf)(x) = f (x),

and if f is continuous at x = , then (IqDqf)(x) = f (x) – f ().

Denoting byxDqthe q-derivative with respect to the variable x, we now recall the fol-lowing three formulas which will be used in the remainder of this paper:

a(t – s)(δ)= aδ(t – s)(δ), (.) xDq(t – s)(δ)= [δ]q(x – s)(δ–), (.) xDq x  f(x, t) dqt (x) = x  xDqf(x, t) dqt+ f (qx, x). (.)

Deﬁnition (see []) Let δ  and f be a function deﬁned on [, ]. The fractional q-integral of the Riemann-Liouville type is given by

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and I_qδf(x) =  q(δ) x  (x – qt)(δ–)f(t) dqt δ> ; x∈ [, ].

Deﬁnition (see [] and []) The fractional q-derivative of the Riemann-Liouville type of order δ (δ ) is deﬁned by D qf (x) = f (x) and Dδ qf (x) =Dm_qI_qm–δf(x) (δ > ),

where m is the smallest integer greater than or equal to δ.

Lemma (see []) Let δ , β , and f be a function deﬁned on [, ]. Then the

fol-lowing two formulas hold true: () (IqβIqδf)(x) = (I

δ+β q f)(x); () (Dδ

qIqδf)(x) = f (x).

Lemma (see [] and []) Let δ >  and p be a positive integer. Then the following

equal-ity holds: I_qδDp_qf(x) =Dp_qI_qδf(x) – p– k= xδ–p+k q(δ + k – p + ) Dk_qf().

Theorem (see [, ], and []) (a) Let (E,) be a partially ordered set and suppose

that there exists a metric d in E such that(E, d) is a complete metric space. Assume that E

satisﬁes the condition that, if{xn} is a nondecreasing sequence in E such that xn→ x, then

xn x (n ∈ N).

Let T: E→ E be a nondecreasing mapping such that

d(Tx, Ty) d(x, y) – ψd(x, y) (x y),

where

ψ: [, +∞) → [, +∞)

is a continuous and nondecreasing function such that ψ is positive in(,∞), ψ() = , and lim

t→∞ψ(t) =∞.

If there exists x∈ E with x T(x), then T has a ﬁxed point.

(b) If we assume that (E,) satisﬁes the condition that, for x ∈ E and y ∈ E, there exists

z∈ E which is comparable to x and y and the hypothesis of (a), then it leads to the uniqueness of the ﬁxed point.

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Mena et al. [] investigated the existence and uniqueness of positive and nondecreasing solutions for the following singular fractional boundary value problem:

Dα_+u(t) + f

t, u(t)=  ( < t < ;  < α < ),

u() = u() = u() = .

Miao and Liang [], on the other hand, studied the existence and uniqueness of a posi-tive and nondecreasing solution for the following fractional q-diﬀerence boundary value problem:

Dγ_qφp

Dα_qu(t)+ ft, u(t)=  ( < t < ;  < α < ),

u() = (Dqu)() = , (Dqu)() = , and Dγ+u(t)|t== .

Motivated essentially by the aforementioned work by Miao and Liang [], we introduce and investigate here the following q-diﬀerence boundary value problem by using the p-Laplacian operator: Dγ q φp Dδ qy(t) + ft, y(t)=  ( < t < ;  < δ < ), (.) y() = (Dqy)() = (Dqy)() = , a(Dqy)() + a(Dqy)() = , and D γ +y(t)|t== . (.)

We prove the existence and uniqueness of a positive and nondecreasing solution for the boundary value problem given by Eqs. (.) and (.) by means of a ﬁxed point theorem involving partially ordered sets.

2 Fractional boundary value problem

Throughout of this paper, we always make use of the usual space E = C[, ] which is known as the space of continuous functions on [, ]. We note that E is a real Banach space with the norm given by

u = max tu(t).

Suppose that x∈ C[, ] and y ∈ C[, ]. Then we have

x y ⇔ x(t) y(t) ∀t ∈ [, ].

We know from the recent work [] that (C[, ],) with the familiar metric:

d(x, y) = sup t

x(t) – y(t)

satisﬁes the hypothesis of Theorem (a). Moreover, for x∈ C[, ] and y ∈ C[, ] such that max{x, y} ∈ C[, ], (C[, ], ) satisﬁes the condition of Theorem (b).

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We ﬁrst demonstrate Lemma .

Lemma  If h∈ C[, ], the following boundary value problem:

Dδ_qy(t) + h(t) =  ( < t < ;  < δ < ), (.) u() = (Dqu)() = (Dqu)() = , a(Dqu)() + a(Dqu)() =  (|a| + |a| = ) (.)

has a unique solution given by

u(t) =   G(t, qs)h(s) dqs, (.) where G(t, s) =  (a+ a[δ – ]q)q(δ) × ⎧ ⎪ ⎨ ⎪ ⎩ (a( – s)(δ–)+ a[δ – ]q( – s)(δ–))tδ– – (a+ a[δ – ]q)(t – s)(δ–) ( s t ), (a( – s)(δ–)+ a[δ – ]q( – s)(δ–))tδ– ( t s ). (.)

Proof By applying Lemma , Lemma  (with p = ) and Eq. (.), we have I_qδD_qI_q–δu(x) = –I_qδft, u(t) and u(t) = ctδ–+ ctδ–+ ctδ–+ ctδ–– t  (t – qs)(δ–) q(δ) h(s) dqs. (.)

From Eq. (.), we get c= . Thus, upon diﬀerentiating both sides of Eq. (.), if we make use of Eqs. (.) and (.), we see that

(Dqu)(t) = [δ – ]qctδ–+ [δ – ]qctδ–+ ctδ–– [δ – ]q q(δ) t  (t – qs)(δ–)h(s) dqs. (.)

Using the boundary condition (.), we have c= . Moreover, by diﬀerentiating both sides of Eq. (.), and using Eqs. (.) and (.), we obtain

D_qu(t) = [δ – ]q[δ – ]qctδ–+ [δ – ]q[δ – ]qctδ– –[δ – ]q[δ – ]q q(δ) t  (t – qs)(δ–)h(s) dqs.

Similarly, by using the boundary condition (.), we have c=  and

c= a  ( – qs)(δ–)h(s) dqs+ a[δ – ]q  ( – qs)(δ–)h(s) dqs (a+ a[δ – ]q)q(δ) .

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Consequently, we have the following unique solution of the boundary value problem given by Eqs. (.) and (.): u(t) = – t  (t – qs)(δ–) q(δ) h(s) dqs +(a  ( – qs) (δ–)_h_{(s) d} qs+ a[δ – ]q  ( – qs) (δ–)_h_{(s) d} qs)tδ– (a+ a[δ – ]q)q(δ) =  q(δ) t  (a( – qs)(δ–)+ a[δ – ]q( – qs)(δ–))tδ– (a+ a[δ – ]q) – (t – qs)(δ–) h(s) dqs +  t (a( – qs)(δ–)+ a[δ – ]q( – qs)(δ–))tδ– (a+ a[δ – ]q)q(δ) h(s) dqs =   G(t, qs)h(s) dqs.

We thus arrive at the desired result asserted by Lemma . By using the method in [] mutatis mutandis, it can easily be proven that, if f ∈

C([, ]× [, +∞), [, +∞)), then the boundary value problem given by Eqs. (.) and (.) is equivalent to the following integral equation:

u(t) =   G(t, qs)φ–_p s  (s – τ )(γ –)_f_{(τ , u(τ ))} (a+ a[δ – ]q)q(γ ) dqτ dqs, (.)

where G(t, s) is deﬁned, as before, by Eq. (.).

Lemma  The function G(t, s) given by Eq. (.) has the following properties: () G(t, s) is a continuous function and G(t, qs) ;

() G(t, s) is strictly increasing in the ﬁrst variable t.

Proof The continuity of G(t, s) can easily be checked. We, therefore, omit the details in-volved. Next, for  s t , we let

g(t, s) = a( – s)δ–+ a[δ – ]q( – s)δ– tδ– –a+ a[δ – ]q (t – s)δ– and, for  t s , we suppose that

g(t, s) =

a( – s)δ–+ a[δ – ]q( – s)δ–

tδ–. Then it is not diﬃcult to see that

g(t, qs) .

Now, for g(, qs) = , δ > , and a b t, we have (t – a)(δ) (t – b)(δ) (t= ).

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We thus ﬁnd that g(t, qs) = a( – qs)δ–+ a[δ – ]q( – qs)δ– tδ– –a+ a[δ – ]q  – qs t tδ– a( – qs)δ–+ a[δ – ]q( – qs)δ– –a+ a[δ – ]q ( – qs)δ–_tδ– a( – qs)δ–+ a[δ – ]q( – qs)δ– –a+ a[δ – ]q ( – qs)δ–tδ–= .

So, clearly, G(t, qs)  for all (t, s) ∈ [, ]×[, ]. This completes the proof of Lemma (). Next, for a ﬁxed s∈ [, ], we see that

tDqg(t, qs) = [δ – ]q a( – qs)(δ–)+ a[δ – ]q( – qs)(δ–) tδ– – [δ – ]q a+ a[δ – ]q (t – qs)δ– = [δ – ]q a( – qs)(δ–)+ a[δ – ]q( – qs)(δ–) tδ– – [δ – ]q a+ a[δ – ]q  – qs t δ– tδ– [δ – ]q( – qs)(δ–) a+ a[δ – ]q tδ– – [δ – ]q( – qs)(δ–) a+ a[δ – ]q tδ– = .

This implies that g(t, qs) is an increasing function of the first argument t. Furthermore, obviously, g(t, qs) is an increasing function of the first argument t. Therefore, G(t, qs) is an increasing function of t for a fixed s∈ [, ]. This completes the proof of Lemma .

3 Uniqueness of positive solutions

For notational convenience, we write

M:= φ– p  q(γ )(a+ a[δ – ]q) sup t   G(t, qs) dqs> . (.)

The main result of this paper is the assertion in Theorem .

Theorem  The boundary value problem given by Eqs. (.) and (.) has a unique positive

and increasing solution u(t) if each of the following two conditions is satisﬁed: (i) the function f : [, ]× [, ∞) → [, ∞) is continuous and nondecreasing with

respect to the second variable;

(ii) there exist λ and M given by Eq. (.) ( < λ +  < M) such that, for u∈ [, ∞) and

v∈ [, ∞) with u v and t ∈ [, ], φp ln(v + ) f (t, v) f (t, u) φp ln(u + )(u – v + )λ.

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Furthermore, if f (t, ) >  for t∈ [, ], then the solution u(t) of the boundary value

prob-lem given by Eqs. (.) and (.) is strictly increasing on [,∞).

Proof First of all, we set

u:= u(t) and v:= v(t).

We then consider the set K (called a cone) given by

K=u: u∈ C[, ] and u(t) .

Since K is a closed set, K is a complete metric space in accordance with the usual metric

d(u, v) = sup t∈[,]

u(t) – v(t).

Let us now consider the operator T as follows:

Tu(t) =   G(t, qs)φ_p–  (a+ a[δ – ]q)q(γ ) s  (s – τ )(γ –)fτ, u(τ )dqτ dqs.

Then, by applying Lemma  and the condition (i) of Theorem , we see that T(K )⊂ K. We now show that all conditions of Theorem  are satisﬁed. Firstly, by the condition (i) of Theorem , for u, v∈ K and u v, we have

Tu(t) =   G(t, qs)φ_p–  (a+ a[δ – ]q)q(γ ) s  (s – τ )(γ –)fτ, u(τ )dqτ dqs   G(t, qs)φ_p–  (a+ a[δ – ]q)q(γ ) s  (s – τ )(γ –)fτ, v(τ )dqτ dqs = Tv(t).

This shows that T is a nondecreasing operator. On the other hand, for u v and by the condition (ii) of Theorem , we have

d(Tu, Tv) = sup t (Tu)(t) – (Tv)(t) = sup t (Tu)(t) – (Tv)(t) sup t   G(t, qs)φ–_p  (a+ a[δ – ]q)q(γ ) × s  (s – τ )(γ –)fτ, u(τ )dqτ dqs –   G(t, qs)φ_p–  (a+ a[δ – ]q)q(γ ) × s  (s – τ )(γ –)fτ, v(τ )dqτ dqs

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ln(u + )(u – v + )λ– ln(v + ) × sup t   G(t, qs)φ_p–  (a+ a[δ – ]q)q(γ ) s  (s – τ )(γ –)dqτ dqs ln(u + )(u – v + )λ v+  φ – p  (a+ a[δ – ]q)q(γ ) sup t   G(t, qs) dqs (λ + ) ln(u – v + )φ– p  (a+ a[δ – ]q)q(γ ) sup t   G(t, qs) dqs.

Since the function h(x) = ln(x + ) is nondecreasing, from the condition (ii) of Theorem , we have d(Tu, Tv) (λ + ) lnu – v + φ_p–  (a+ a[δ – ]q)q(γ ) sup t   G(t, qs) dqs = (λ + ) lnu – v + M u – v –u – v – lnu – v + .

We now let ψ(x) = x – ln(x + ). Then, obviously, the function ψ given by

ψ: [, +∞) → [, +∞)

is continuous, nondecreasing, and positive in (,∞). It is also clearly seen that ψ(x) satis-ﬁes the following conditions:

ψ() =  and lim

x→+∞ψ(x) =∞. Thus, for u v, we have

d(Tu, Tv) d(u, v) – ψd(u, v). As G(t, qs)  and f , we have

(T)(t) = 



G(t, qs)f (s, ) dqs .

Consequently, in view of Theorem , the boundary value problem given by Eqs. (.) and (.) has at least one nonnegative solution. Since (K ,) satisﬁes the condition (ii) of The-orem , TheThe-orem  implies the uniqueness of the solution. Thus, clearly, the proof of the last assertion of Theorem  follows immediately from the proof of a well-known result in [, Theorem .]. Our proof Theorem  is thus completed.

4 Concluding remarks and observations

Our present study was motivated by several aforementioned recent works. Here, we have successfully addressed the problem involving the existence and uniqueness of positive and nondecreasing solutions of a family of fractional q-diﬀerence boundary value problems given by Eqs. (.) and (.). The proof of our main result asserted by Theorem  of the

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preceding section has made use of some familiar ﬁxed point theorems. We have also in-dicated the relevant connections of the results derived in this investigation with those in earlier works on the subject.

Competing interests

The authors declare that they have no competing interests. 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 Economics, Faculty of Economics, Administrative and Social Science, Hasan Kalyoncu University,}

Gaziantep, 27410, Turkey.2_{Department of Mathematics, Faculty of Arts and Science, Namik Kemal University, Tekirda ˘g,}

59030, Turkey.3_{Department of Mathematics Engineering, Istanbul Technical University, Maslak, Istanbul, 34469, Turkey.} 4_{Department of Mathematics, Faculty of Science and Arts, University of Gaziantep, Gaziantep, 27310, Turkey.} 5_{Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada.}

Received: 19 November 2014 Accepted: 13 January 2015 References

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