positive solutions for boundary value problem of nonlinear...

International Scholarly Research NetworkISRN Mathematical AnalysisVolume 2011, Article ID 385459, 12 pagesdoi:10.5402/2011/385459

Research ArticlePositive Solutions for Boundary Value Problem ofNonlinear Fractional q-Difference Equation

Moustafa El-Shahed1 and Farah M. Al-Askar2

1 Department of Mathematics, Faculty of Arts and Sciences, Qassim-Unizah 51911,P.O. Box 3771, Saudi Arabia

2 Department of Mathematics, Majmaah University, Al-Majmaah 11952, P.O. Box 566, Saudi Arabia

Correspondence should be addressed to Farah M. Al-Askar, [email protected]

Received 17 January 2011; Accepted 24 February 2011

Academic Editor: G. L. Karakostas

Copyright q 2011 M. El-Shahed and F. M. Al-Askar. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

We investigate the existence of multiple positive solutions to the nonlinear q-fractional boundaryvalue problem cD

aqu(t) + a(t)f(u(t)) = 0, 0 ≤ t ≤ 1, 2 < a ≤ 3, u(0) = D2

qu(0) = 0, γDqu(1) +βD2

qu(1) = 0, by using a fixed point theorem in a cone.

1. IntroductionFractional differential calculus is a discipline to which many researchers are dedicatingtheir time, perhaps because of its demonstrated applications in various fields of science andengineering [1]. Many researchers studied the existence of solutions to fractional boundaryvalue problems, for example, [2–5].

The q-difference calculus or quantum calculus is an old subject that was initiallydeveloped by Jackson [6, 7], basic definitions and properties of q-difference calculus can befound in the book mentioned in [8].

The fractional q-difference calculus had its origin in the works by Al-Salam [9] andAgarwal [10]. More recently, maybe due to the explosion in research within the fractionaldifferential calculus setting, new developments in this theory of fractional q-differencecalculus were made, for example, q-analogues of the integral and differential fractionaloperators properties such as MittageLeffler function [11], just to mention some.

El-Shahed and Hassan [12] studied the existence of positive solutions of the q-difference boundary value problem:

−D2qu(t) = a(t)f(u(t)), 0 ≤ t ≤ 1,

αu(0) − βDqu(0) = 0,

γu(1) + δDqu(1) = 0.

(1.1)

2 ISRN Mathematical Analysis

Ferreira [13] considered the existence of positive solutions to nonlinear q-differenceboundary value problem:

(RLD

αqu)(t) = −f(t, u(t)), 0 < t < 1, 1 < α ≤ 2,

u(0) = u(1) = 0.(1.2)

In other paper, Ferreira [14] studied the existence of positive solutions to nonlinear q-difference boundary value problem:

(RLD

αqu)(t) = −f(t, u(t)), 0 < t < 1, 2 < α ≤ 3,

u(0) =(Dqu

)(0) = 0,

(Dqy

)(1) = β ≥ 0.

(1.3)

In this paper, we consider the existence of positive solutions to nonlinear q-differenceequation:

CDαqu + a(t)f(u(t)) = 0, 0 ≤ t ≤ 1, 2 < α ≤ 3, (1.4)

with the boundary conditions

u(0) = D2qu(0) = 0,

γDqu(1) + βD2qu(1) = 0,

(1.5)

where γ, β ≥ 0 and CDαq is the fractional q-derivative of the Caputo type.

2. Preliminaries of Fractional q-Calculus

Let q ∈ (0, 1) and define [8]

[α]q =qa − 1q − 1

= qa−1 + · · · + 1, a ∈ R. (2.1)

The q-analogue of the power (a − b)n is

(a − b)(0) = 1, (a − b)(n) =n−1∏k=0

(a − bqk

), a, b ∈ R, n ∈ N. (2.2)

If α is not a positive integer, then

(a − b)(α) = aα∞∏i=0

(1 − (b/a)qi

)(1 − (b/a)qα+i

) . (2.3)

ISRN Mathematical Analysis 3

Note that if b = 0, then a(α) = aα. The q-gamma function is defined by

Γq(x) =

(1 − q

)(x−1)(1 − q

)x−1 , x ∈ R \ {0,−1,−2, . . .}, 0 < q < 1 (2.4)

and satisfies Γq(x + 1) = [x]qΓq(x).The q-derivative of a function f is here defined by

Dqf(x) =dqf(x)dqx

=f(qx) − f(x)(

q − 1)x

(2.5)

and q-derivatives of higher order by

Dnqf(x) =

⎧⎨⎩f(x) if n = 0,

DqDn−1q f(x) if n ∈ N.

(2.6)

The q-integral of a function f defined in the interval [0, b] is given by

∫x

0f(t)dqt = x

(1 − q

) ∞∑n=0

f(xqn)qn, 0 ≤ ∣∣q∣∣ < 1, x ∈ [0, b]. (2.7)

If a ∈ [0, b] and f is defined in the interval [0, b], its integral from a to b is defined by

∫b

a

f(t)dqt =∫b

0f(t)dqt −

∫a

0f(t)dqt. (2.8)

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

(I0qf)(x) = f(x),

(Inq f)(x) = Iq

(In−1q f

)(x), n ∈ N. (2.9)

The fundamental theorem of calculus applies to these operators Iq and Dq, that is,

(DqIqf

)(x) = f(x), (2.10)

and if f is continuous at x = 0, then

(IqDqf

)(x) = f(x) − f(0). (2.11)


Basic properties of the two operators can be found in the book mentioned in [8]. We nowpoint out three formulas that will be used later (iDq denotes the derivative with respect tovariable i) [13]

[a(t − s)](α) = aα(t − s)(α), (2.12)

tDq(t − s)(α) = [α]q(t − s)(α−1), (2.13)

(xDq

∫x

0f(x, t)dqt

)(x) =

∫x

0xDqf(x, t)dqt + f

(qx, x

). (2.14)

Remark 2.1. We note that if α > 0 and a ≤ b ≤ t, then (t − a)(α) ≥ (t − b)(α) [13].The following definition was considered first in [10].

Definition 2.2. Let α ≥ 0 and f be a function defined on [0, 1]. The fractional q-integral of theRiemann-Liouville type is (RLI0qf)(x) = f(x) and

(RLI

αqf)(x) =

1Γq(α)

∫x

a

(x − qt

)(α−1)f(t)dqt, α ∈ R+, x ∈ [0, 1]. (2.15)

Definition 2.3 (see [15]). The fractional q-derivative of the Riemann-Liouville type of orderα ≥ 0 is defined by (RLD0

qf)(x) = f(x) and

(RLD

αqf)(x) =

(D

[α]q I

[α]−αq f

)(x), α > 0, (2.16)

where [α] is the smallest integer greater than or equal to α.

Definition 2.4 (see [15]). The fractional q-derivative of the Caputo type of order α ≥ 0 isdefined by

(CD

αqf)(x) =

(I[α]−αq D

[α]q f)(x), α > 0, (2.17)

where [α] is the smallest integer greater than or equal to α.

Lemma 2.5. Let α, β ≥ 0 and let f be a function defined on [0, 1]. Then, the next formulas hold:

(1) (Iβq Iαq f)(x) = (Iα+βq f)(x),

(2) (DαqI

αq f)(x) = f(x).

The next result is important in the sequel. It was proved in a recent work by the authorof [13].

Theorem 2.6. Let α > 0 and n ∈ N. Then, the following equality holds:

(RLI

αq RLD

nqf)(x) = RLD

nq RLI

αqf(x) −

α−1∑k=0

xα−n+k

Γq(α + k − n + 1)

(Dk

qf)(0). (2.18)


Theorem 2.7 (see [15]). Let α > 0 and n ∈ R+ \N. Then, the following equality holds:

(Iαq CD

αqf)(x) = f(x) −

[α]−1∑k=0

xk

Γq(k + 1)

(Dk

qf)(0). (2.19)

3. Fractional Boundary Value Problem

Wewill consider now the question of existence of positive solutions to the following problem:

CDαqu(t) + a(t)f(u(t)) = 0, 0 ≤ t ≤ 1, 2 < α ≤ 3,

u(0) = D2qu(0) = 0,

γDqu(1) + βD2qu(1) = 0,

(3.1)

where γ, β ≥ 0, CDαq is the fractional q-derivative of the Caputo type, and f ∈ C([0,∞),

[0,∞)), a ∈ C([0, 1], [0,∞)) such that∫10 a(s)dqs > 0. To that end, we need the following

theorem.

Theorem 3.1 (see [16, 17]). Let X be a Banach space, and P ⊂ X is a cone in X. Assume that Ω1

and Ω2 are open subsets of X with 0 ∈ Ω1 and Ω1 ⊂ Ω2.Let T : P ∩ (Ω2 \Ω1) → P be completely continuous operator. In addition suppose either:

P1: ‖Tu‖ ≤ ‖u‖, u ∈ P ∩ ∂Ω1 and ‖Tu‖ ≥ ‖u‖, u ∈ P ∩ ∂Ω2 or

P2: ‖Tu‖ ≤ ‖u‖, u ∈ P ∩ ∂Ω2 and ‖Tu‖ ≥ ‖u‖, u ∈ P ∩ ∂Ω1,

holds. Then, T has a fixed point in P ∩ (Ω2 \Ω1).

Lemma 3.2. Let y ∈ C [0, 1]; then the boundary value problem

CDαqu(t) + y(t) = 0, 0 ≤ t ≤ 1, 2 < α ≤ 3,

u(0) = D2qu(0) = 0,

γDqu(1) + βD2qu(1) = 0

(3.2)

has a unique solution

u(t) =∫1

0G(t, qs)y(s)dqs, (3.3)

where

G(t, s) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

t(1 − s)(α−2)

Γq(α − 1)+β

γ

t(1 − s)(α−3)

Γq(α − 2)0 ≤ t ≤ s ≤ 1,

t(1 − s)(α−2)

Γq(α − 1)+β

γ

t(1 − s)(α−3)

Γq(α − 2)− (t − s)(α−1)

Γq(α)0 ≤ s ≤ t ≤ 1.

(3.4)


Proof. We may apply Lemma 2.5 and Theorem 2.7; we see that

u(t) = u(0) +Dqu(0)Γq(2)

t +D2

qu(0)

Γq(3)t2 − Iαq y(t). (3.5)

By using the boundary conditions u(0) = D2qu(0) = 0, we get

u(t) = At −∫ t

0

(t − qs

)(α−1)Γq(α)

y(s)dqs. (3.6)

Differentiating both sides of (3.6), one obtains, with the help of (2.13), and (2.14)

(Dqu

)(t) = A −

∫ t

0

[α − 1]q(t − qs

)(α−2)

Γq(α)y(s)dqs,

(D2

qu)(t) = −

∫ t

0

[α − 1]q[α − 2]q(t − qs

)(α−3)

Γq(α)y(s)dqs.

(3.7)

Then, by the condition γDqu(1) + βD2qu(1) = 0, we have

A =∫1

0

(1 − qs

)(α−2)Γq(α − 1)

y(s)dqs +β

γ

∫1

0

(1 − qs

)(α−3)Γq(α − 2)

y(s)dqs. (3.8)

The proof is complete.

Lemma 3.3. Function G defined above satisfies the following conditions:

G(t, qs) ≥ 0, G

(1, qs

) ≥ G(t, qs), 0 ≤ t, s ≤ 1, (3.9)

G(t, qs) ≥ g(t)G

(1, qs

), 0 ≤ t, s ≤ 1 with g(t) = t. (3.10)

Proof. We start by defining two functions

g1(t, s) =t(1 − s)(α−2)

Γq(α − 1)+β

γ

t(1 − s)(α−3)

Γq(α − 2)− (t − s)(α−1)

Γq(α), 0 ≤ s ≤ t ≤ 1,

g2(t, s) =t(1 − s)(α−2)

Γq(α − 1)+β

γ

t(1 − s)(α−3)

Γq(α − 2), 0 ≤ t ≤ s ≤ 1.

(3.11)


It is clear that g2(t, qs) ≥ 0. Now, g1(0, qs) = 0 and, in view of Remark 2.1, for t /= 0

g1(t, qs)=

t(1 − qs

)(α−2)Γq(α − 1)

+β

γ

t(1 − qs

)(α−3)Γq(α − 2)

−(t − qs

)(α−1)Γq(α)

≥ t(1 − qs

)(α−2)Γq(α − 1)

+β

γ

t(1 − qs

)(α−3)Γq(α − 2)

− tα−1(1 − qs

)(α−1)Γq(α)

≥ t(1 − qs

)(α−2)Γq(α − 1)

+β

γ

t(1 − qs

)(α−3)Γq(α − 2)

− t(1 − qs

)(α−1)Γq(α)

=t

Γq(α)

[[α − 1]q

(1 − qs

)(α−2) − (1 − qs)(α−1)] ≥ 0.

(3.12)

Therefore, G(t, qs) ≥ 0. Moreover, for fixed s ∈ [0, 1],

tDqg1(t, qs)=

(1 − qs

)(α−2)Γq(α − 1)

+β

γ

(1 − qs

)(α−3)Γq(α − 2)

−[α − 1]q

(t − qs

)(α−2)

Γq(α)

≥ 1Γq(α)

[[α − 1]q

(1 − qs

)(α−2) − [α − 1]q(t − qs

)(α−2)]

=1

Γq(α − 1)

[(1 − qs

)(α−2) − (t − qs)(α−2)] ≥ 0,

(3.13)

that is, g1(t, qs) is an increasing function of t. Obviously, g2(t, qs) is increasing in t, therefore;G(t, qs) is an increasing function of t for fixed s ∈ [0, 1]. This concludes the proof of (3.9).

Suppose now that t ≥ qs. Then,

G(t, qs)

G(1, qs

) =γ[α − 1]qt

(1 − qs

)(α−2) + β[α − 1]q[α − 2]qt(1 − qs

)(α−3) − γ(t − qs

)(α−1)

γ[α − 1]q(1 − qs

)(α−2) + β[α − 1]q[α − 2]q(1 − qs

)(α−3) − γ(1 − qs

)(α−1)

≥γ[α − 1]qt

(1 − qs

)(α−2) + β[α − 1]q[α − 2]qt(1 − qs

)(α−3) − γt(1 − qs

)(α−1)

γ[α − 1]q(1 − qs

)(α−2) + β[α − 1]q[α − 2]q(1 − qs

)(α−3) − γ(1 − qs

)(α−1)

= t.

(3.14)

If t ≤ qs, Then

G(t, qs)

G(1, qs

) = t, (3.15)

and this finishes the proof of (3.10).


Remark 3.4. If we let 0 < τ < 1, then

mint∈[τ,1]

G(t, qs) ≥ τG

(1, qs

), for s ∈ [0, 1]. (3.16)

Let X = C[0, 1] be the Banach space endowed with norm ‖u‖ = supt∈[τ,1]|u(t)|. Let τ = qn [18]for a given n ∈ N and define the cone P ⊂ X by

P ={u ∈ X : u(t) ≥ 0, min

t∈[τ,1]u(t) ≥ τ‖u‖

}, (3.17)

and the operator T : P → X is defined by

Tu(t) =∫1

0G(t, qs)a(s)f(u(s))dqs. (3.18)

Remark 3.5. It follows from the nonnegativeness and continuity of G, a, and f that theoperator T : P → X is completely continuous [3]. Moreover, for u ∈ P, Tu(t) ≥ 0 on [0, 1]and

mint∈[τ,1]

Tu(t) = mint∈[τ,1]

∫1

0G(t, qs)a(s)f(u(s))dqs

≥ τ

∫1

0G(1, qs

)a(s)f(u(s))dqs

= τ‖Tu‖

(3.19)

that is, T(P) ⊂ P .Throughout this section, we will use the following notations:

Λ1 =

(∫1

0G(1, qs

)a(s)dqs

)−1, Λ2 =

(τ2∫1

τ

G(1, qs

)a(s)dqs

)−1. (3.20)

It is obvious that Λ1,Λ2 > 0. Also, we define

f0 = limu→ 0+

max0≤t≤1

f(u)u

, f∞ = limu→+∞

max0≤t≤1

f(u)u

. (3.21)

Theorem 3.6. Let τ = qn with n ∈ N. And assume that the following assumptions are satisfied:

(H1) f0 = f∞ = 0;

(H2) There exist constant k > 0 and M ∈ (Λ2,∞) such that

f(u) ≥ Mk, for u ∈ [τk, k]. (3.22)


Then, the BVP (3.1) has at least two positive solutions u1, u2, such that

r < ‖u1‖ < k < ‖u2‖ < R. (3.23)

Proof. First, since f0 = 0, for any ε ∈ (0, Λ1) there exists r ∈ (0, k) such that

f(u) ≤ εu, for u ∈ [0, r]. (3.24)

Letting Ω1 = {u ∈ P : ‖u‖ < r}, for any u ∈ ∂Ω1, we get

‖Tu‖ = maxt∈[0,1]

∫1


≤ εr

∫1

0G(1, qs

)a(s)dqs < r = ‖u‖,

(3.25)

which yields

‖Tu‖ ≤ ‖u‖, ∀u ∈ ∂Ω1. (3.26)

Thus,

i(T,Ω1, P) = 1. (3.27)

Second, in view of f∞ = 0, then for any ε ∈ (0,Λ1), there exists l > k such that

f(u) ≤ εu, for u ∈ [l,∞) (3.28)

and we consider two cases.

Case 1. Suppose that f(u) is unbounded; then from f ∈ C([0,∞), [0,∞)), there is R > l suchthat

f(u) ≤ f(R), for u ∈ [0, R], (3.29)

then, from (3.28) and (3.29), one has

f(u) ≤ f(R) ≤ εR, for u ∈ [0, R]. (3.30)


For u ∈ ∂Ω2, we get

‖Tu‖ = maxt∈[0,1]

∫1


≤∫1

0G(1, qs

)a(s)f(R)dqs

≤ εR

∫1

0G(1, qs

)a(s)dqs

< R = ‖u‖.

(3.31)

Case 2. Suppose that f(u) is bounded and L = max0≤t≤1,0≤u≤R |f(t, u(t))|. Taking R =max{L/ε, k}, for u ∈ ∂Ω2, we get

‖Tu‖ = maxt∈[0,1]

∫1


≤ L

∫1

0G(1, qs

)a(s)dqs

≤ εR

∫1

0G(1, qs

)a(s)dqs

< R = ‖u‖.

(3.32)

Hence, in either case, we always may set Ω2 = {u ∈ P : ‖u‖ < R} such that

‖Tu‖ ≤ ‖u‖, ∀u ∈ ∂Ω2. (3.33)

Thus,

i(T,Ω2, P) = 1. (3.34)

Finally, set Ω = {u ∈ P : ‖u‖ < k}, for u ∈ ∂Ω, since mint∈[τ,1]u(t) ≥ τ‖u‖ = τk for u ∈ P , andhence, for any u ∈ ∂Ω, from (H2), we can get

‖Tu‖ = maxt∈[0,1]

∫1


≥ τ

∫1

0G(1, qs

)a(s)f(u(s))dqs

≥ τ2Mk

∫1

τ

G(1, qs

)a(s)dqs > k = ‖u‖,

(3.35)


which implies

‖Tu‖ ≥ ‖u‖, ∀u ∈ ∂Ω. (3.36)

Thus,

i(T,Ω, P) = 0. (3.37)

Hence, since r < k < R and from (3.27), (3.34), and (3.37), it follows from the additivity of thefixed point index that

i(T,Ω2 \Ω, P

)= 1,

i(T,Ω \Ω1, P

)= −1.

(3.38)

Thus, T has two positive solutions u1, u2 such that r < ‖u1‖ < k < ‖u2‖ < R for t ∈ (0, 1]. So,the proof is complete.

References

[1] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.[2] M. El-Shahed, “Existence of solution for a boundary value problem of fractional order,” Advances in

Applied Mathematical Analysis, vol. 2, no. 1, pp. 1–8, 2007.[3] S. Zhang, “Existence of solution for a boundary value problem of fractional order,” Acta Mathematica

Scientia, vol. 26, no. 2, pp. 220–228, 2006.[4] M. El-Shahed and F. M. Al-Askar, “On the existence of positive solutions for a boundary value

problem of fractional order,” International Journal of Mathematical Analysis, vol. 4, no. 13–16, pp. 671–678, 2010.

[5] Z. Bai and H. Lu, “Positive solutions for boundary value problem of nonlinear fractional differentialequation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005.

[6] F. H. Jackson, “On qfunctions and a certain difference operator,” Transactions of the Royal SocietyEdinburgh, vol. 46, pp. 253–281, 1908.

[7] R. Jackson, “On qdefinite integrals,” Quarterly Journal of Pure and Applied Mathematics, vol. 41, pp.193–203, 1910.

[8] V. Kac and P. Cheung, Quantum Calculus, Springer, New York, NY, USA, 2002.[9] W. A. Al-Salam, “Some fractional qintegrals and qderivatives,” Proceedings of the Edinburgh

Mathematical Society, vol. 15, no. 2, pp. 135–140, 1967.[10] R. P. Agarwal, “Certain fractional qintegrals and qderivatives,” Mathematical Proceedings of the

Cambridge Philosophical Society, vol. 66, pp. 365–370, 1969.[11] P. M. Rajkovic, S. D. Marinkovic, and M. S. Stankovic, “On qanalogues of caputo derivative and

Mittag-Leffler function,” Fractional Calculus & Applied Analisys, vol. 10, no. 4, pp. 359–373, 2007.[12] M. El-Shahed and H. A. Hassan, “Positive solutions of qdifference equation,” Proceedings of the

American Mathematical Society, vol. 138, no. 5, pp. 1733–1738, 2010.[13] R. A.C. Ferreira, “Nontrivial solutions for fractional qdifference boundary value problems,” Electronic

Journal of Qualitative Theory of Differential Equations, vol. 70, pp. 1–10, 2010.[14] R. A.C. Ferreira, “Positive solutions for a class of boundary value problems with fractional

qdifferences,” Computers and Mathematics with Applications, vol. 61, no. 2, pp. 367–373, 2011.


[15] M. S. Stankovic, P. M. Rajkovic, and S. D. Marinkovic, “On q-fractional deravtives of Riemann-Liouville and Caputo type,” 2009, http://arxiv.org/abs/0909.0387.

[16] D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, San Diego,Calif, USA, 1988.

[17] M. A. Krasnoselskii, Positive Solutions of Operators Equations, Noordhoff, Groningen, The Netherlands,1964.

[18] H. Gauchman, “Integral Inequalities in qcalculus,” Computers and Mathematics with Applications, vol.47, no. 2-3, pp. 281–300, 2004.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

positive solutions for boundary value problem of nonlinear...

Documents