Three-point boundary value problems with delta Riemann-Liouville fractional derivative on time scales

(1)

Differential

Calculus

Volume 6, Number 1 (2016), 1–16 doi:10.7153/fdc-06-01

THREE–POINT BOUNDARY VALUE PROBLEMS WITH DELTA RIEMANN–LIOUVILLE FRACTIONAL DERIVATIVE ON TIME SCALES

.

ISMA.ILYASLAN ANDONURL.ICEL.I

Abstract. In this paper, we establish the criteria for the existence and uniqueness of solutions of a three-point boundary value problem for a class of fractional differential equations on time scales. By using some well known fixed point theorems, sufficient conditions for the existence of solutions are established. An illustrative example is also presented.

1. Introduction

Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary (noninteger) order. Fractional differential equations arise in many engineer-ing and scientific disciplines as the mathematical models of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, [18,23, 24,25]. Among all the researches on the theory of the fractional differential equations, the study of the boundary value problems for frac-tional differential equations recently has attracted a great deal of attention from many researchers. Some results have been obtained on the existence of positive solutions of the boundary value problems for some specific fractional differential equations (see [11,17,20,22,27,28,31,32] and references therein).

Miller and Ross [21] has been done a pioneering work in discrete fractional cal-culus. Particularly, Atici and Eloe [5,7,8,9] contributed to the improvement of the discrete fractional calculus. The existence problems of discrete fractional difference equations have been investigated by a several authors (see [1,7,8,14,15,16] and refer-ences therein). Then, to unify the fractional differential equations with both continuous and discrete forms, fractional calculus on time scales was used, see [3,4,6,10,29]. Some basic definitions and theorems on time scales can be found in the books [12,13], which are excellent references for calculus of time scales.

Recently, existence problems of initial value problems for fractional differential equations on time scales has been studied by a few authors [2,30,33]. However, to the best of our knowledge, there exists no literature devoted to three-point boundary value problems of fractional differential equations on time scales. Boundary value problems are an important class of dynamic equations, because of their striking applications to almost all area of science, engineering and technology. By researching boundary value

Mathematics subject classification(2010): 34B15, 34A08, 34N05.

Keywords and phrases: Boundary value problems, existence of solutions, fixed point theorems, frac-tional differential equations, time scales.

c

D l , Zagreb

(2)

problems of fractional differential equations on time scales, the results unify the the-ory of fractional differential and fractional difference equations (and removes obscurity from both areas) and provide accurate information of phenomena that manifest them-selves partly in continuous time and partly in discrete time.

We shall consider the following nonlinear three-point boundary value problem with delta Riemann-Liouville fractional derivative on time scales of orderα− 1 :

∆α−1

a∗ x(t) = f (t, x(t)), t ∈ J := [a, b] ∩ T, 2 <α<3

x(a) = x∆_{(b) = 0, x}∆_{(a) = x}∆_(c), (1)

where T is any time scale, c∈ (a, b) ∩ T, f ∈ C ([a, b] × R, R) and ∆α−1_a∗ denotes the

delta fractional derivative on time scale T of orderα− 1 which will be defined later. 2. Preliminaries

To state the main results of this paper, we will give some definitions of delta Riemann-Liouville type fractional integral and delta fractional derivative on time scales and auxiliary lemmas which are needed later.

Let us consider the rd-continuous functions hα: T× T → R,α > 0 , such that

hα+1(t, s) =

t

Z

s

hα(τ,s)∆τ, h0(t, s) = 1 ∀s,t ∈ T, (2)

where T is a time scale such that Tk= T. Also, we suppose t

Z

σ (u)

hα−1(t,σ(τ))hβ −1(τ,σ(u))∆τ= hα+β −1(t,σ(u)), α,β >1, u < t, u,t∈ T,

(3) whereσ is the forward jump operator.

If T= R, thenσ(t) = t and hk(t, s) =(t−s) k k! ,∀k ∈ N0= N ∪ {0} and we define hα(t, s) = (t − s) α Γ(α+ 1),α>0 which satisfies the properties in (2) and (3) (see [3]).

If T= Z, then σ(t) = t + 1 and hk(t, s) = (t−s) (k) k! , ∀k ∈ N0, where t(0)= 1 , t(k)=k∏−1 i=0 (t − i). We define hα(t, s) =(t − s) (α) Γ(α+ 1), α>0

(3)

DEFINITION1. [3] For α > 1 , a time scale delta Riemann-Liouville type frac-tional integral is defined by

K_aαf(t) = t

Z

a

hα−1(t,σ(τ)) f (τ)∆τ, K_a0f = f ,

where f∈ L1([a, b] ∩ T) and t ∈ [a, b] ∩ T.

If α= 1 , then we have K1 af(t) = t R a f(τ)∆τ.

DEFINITION2. [3] Forα> 2 , m− 1 <α6 m∈ N, i.e., m = pαq (ceiling of the

number) and v= m −α, the ∆− fractional derivative on time scale T of orderα− 1 is defined by ∆α−1_a_∗ f(t) = (Kv+1 a f∆ m )(t) = t Z a hv(t,σ(τ)) f∆ m (τ)∆τ, ∀t ∈ [a, b] ∩ T, where f∈ Cm rd([a, b] ∩ T) and f∆ m

is a Lebesgue∆ -integrable function. If we takeα= m, then we have ∆α−1

a∗ f(t) = (Ka1f∆

m

)(t) = f∆m−1_.

LEMMA1. [3] Letα>2 , m− 1 <α<m∈ N, v = m−α, f∈ Cm_rd(T), a, b ∈ T, Tk_{= T. Suppose h}

α−2(s,σ(t)), hv(s,σ(t)) to be continuous on ([a, b] ∩ T)2. Then, we have K_aα−1∆α−1_a_∗ f(t) = f (t) + E( f∆m,α− 1, v + 1, T,t) − m−1

∑

k=0 hk(t, a) f∆ k (a), where E( f∆m,α−1, v+1, T,t) = t R a

f∆m(u)µ(u)hα−2(t,σ(u))hv(u,σ(u))∆u andµ(t) = σ(t) − t .

LEMMA2. Assume that 2 <α<3 ,β= 3 −α, x∈ C3

rd(T), g ∈ Crd([a, b] ∩ T), a, b∈ T, a < b and Tk_{= T, h}

α−1(t,σ(s)) is continuous on J × J . Then, a function

x∈ C3

rd(T) is a solution of the boundary value problem ∆α−1

a∗ x(t) = g(t), t ∈ [a, b] ∩ T, 2 <α<3

(4)

if and only if x is a solution of the following integral equation x(t) = t Z a hα−2(t,σ(τ)) g(τ) − x∆3(τ)µ(τ)hβ(τ,σ(τ)) ∆τ +(t − a)(b − a) − h2(t, a) c− a c Z a h∆_α−2(c,σ(τ))g(τ) − x∆3(τ)µ(τ)hβ(τ,σ(τ)) ∆τ −(t − a) b Z a h∆_α−2(b,σ(τ))g(τ) − x∆3(τ)µ(τ)hβ(τ,σ(τ)) ∆τ. (5)

Proof. Let x be a solution of the BVP (4). By Lemma1, we have K_aα−1g(t) = Kaα−1∆α−1a∗ x(t) = x(t) + t Z a x∆3(τ)µ(τ)hα−2(t,σ(τ))hβ(τ,σ(τ))∆τ− 2

∑

k=0 h_k(t, a)x∆k(a). Then, we get x(t) = t Z a hα−2(t,σ(τ))g(τ)∆τ− t Z a hα−2(t,σ(τ))x∆ 3 (τ)µ(τ)hβ(τ,σ(τ))∆τ

+x(a) + h1(t, a)x∆(a) + h2(t, a)x∆ 2

(a) (6)

and using the differentiation formula [12, Theorem 1.117], we find

x∆(t) = t Z a h∆_α−2(t,σ(τ))g(τ) − x∆3(τ)µ(τ)hβ(τ,σ(τ))

∆τ+ x∆(a) + (t − a)x∆2(a).

From x∆(a) = x∆(c), we have

x∆2(a) = − 1 c− a c Z a h∆_α−2(c,σ(τ))g(τ) − x∆3(τ)µ(τ)hβ(τ,σ(τ)) ∆τ. From x∆(b) = 0 , we get x∆(a) = b− a c− a c Z a h∆_α−2(c,σ(τ))g(τ) − x∆3(τ)µ(τ)hβ(τ,σ(τ)) ∆τ − b Z a h∆_α−2(b,σ(τ))g(τ) − x∆3(τ)µ(τ)hβ(τ,σ(τ)) ∆τ.

(5)

Thus, we obtain (5) by using (6).

The converse of the lemma follows by a direct computation. This completes the proof.

C3_rd(T) is a Banach space with the norm kxk = max

t∈J |x(t)| + maxt∈J |x ∆3

(t)|. The so-lutions of the BVP (1) are the fixed points of the operator A : C_rd3(T) → C3

rd(T) defined by Ax(t) = t Z a hα−2(t,σ(τ)) f(τ,x(τ)) − x∆3(τ)µ(τ)hβ(τ,σ(τ)) ∆τ +(t − a)(b − a) − h2(t, a) c− a × c Z a h∆_α−2(c,σ(τ))f(τ,x(τ))−x∆3(τ)µ(τ)hβ(τ,σ(τ)) ∆τ −(t − a) b Z a h∆_α−2(b,σ(τ))f(τ,x(τ)) − x∆3(τ)µ(τ)hβ(τ,σ(τ)) ∆τ. (7)

For the sake of convenience, we set

(6)

LEMMA3. Assume the following two conditions hold: (H1) | f (t, x)| 6φ(t)ψ(|x|) for all t ∈ J , x ∈ C3

rd(T), where φ: J→ [0, ∞) and ψ:[0, ∞) → [0, ∞) are continuous and nondecreasing.

(H2) The functions hα−2(t,σ(τ)), h∆ 3

α−2(t,σ(τ)), h2(t, a) and µ(t)hβ(t,σ(t))

are continuous for t∈ J andτ∈ J . Then, the operator A: C_rd3(T) → C3

rd(T) is completely continuous. Proof. We divide the proof into two steps.

Step1. We show that A is continuous. Let(xn) be a sequence such that xn→ x ∈ C_rd3(T). Then, we obtain that

From f ∈ C ([a, b] × R, R), (H2) and kxn− xk → 0 as n → ∞, it follows that kAxn− Axk → 0 as n → ∞. Hence, A is continuous.

(7)

Step 2. We show that the image of any bounded subset Ω of C3

rd(T) under A is relatively compact in C_rd3(T). For each x ∈ Ω =x ∈ C3

(8)

6φ(b)ψ(r) t Z a h ∆3 α−2(t,σ(τ)) ∆τ +kxk t Z a h ∆3 α−2(t,σ(τ))µ(τ)hβ(τ,σ(τ)) ∆τ. Therefore, kAxk 6φ(b)ψ(r)M + kxkN (10) 6φ(b)ψ(r)M + rN,

that is AΩ is a bounded set.

Now we show that AΩ is equicontinuous on J . For each t1,t2∈ J , without loss

of generality we may assume that t1<t2, and for all x∈ Ω one can see that

|Ax(t2) − Ax(t1)| 6 t2 Z a hα−2(t2,σ(τ)) f(τ,x(τ)) − x∆3(τ)µ(τ)hβ(τ,σ(τ)) ∆τ − t1 Z a hα−2(t1,σ(τ)) f(τ,x(τ)) − x∆3(τ)µ(τ)hβ(τ,σ(τ)) ∆τ + (t2−a)(b−a)−h2(t2,a) c−a c Z a h∆_α−2(c,σ(τ))f(τ,x(τ))−x∆3(τ)µ(τ)hβ(τ,σ(τ)) ∆τ −(t1−a)(b−a)−h2(t1,a) c−a c Z a h∆_α−2(c,σ(τ))f(τ,x(τ))−x∆ 3 (τ)µ(τ)hβ(τ,σ(τ)) ∆τ + (t2− a) b Z a h∆_α−2(b,σ(τ))f(τ,x(τ)) − x∆3(τ)µ(τ)hβ(τ,σ(τ)) ∆τ −(t1− a) b Z a h∆_α−2(b,σ(τ))f(τ,x(τ)) − x∆ 3 (τ)µ(τ)hβ(τ,σ(τ)) ∆τ 6 t1 Z a |hα−2(t2,σ(τ)) − hα−2(t1,σ(τ))| f(τ,x(τ)) − x ∆3 (τ)µ(τ)hβ(τ,σ(τ)) ∆τ + t2 Z t1 |hα−2(t2,σ(τ))| f(τ,x(τ)) − x ∆3 (τ)µ(τ)hβ(τ,σ(τ)) ∆τ +|t2− t1|(b − a) + |h2(t2,a) − h2(t1,a)| c− a c Z a h∆_α−2(c,σ(τ))

(9)

×f(τ,x(τ)) − x∆ 3 (τ)µ(τ)hβ(τ,σ(τ)) ∆τ +(t2− t1) b Z a h ∆ α−2(b,σ(τ)) f(τ,x(τ)) − x∆3(τ)µ(τ)hβ(τ,σ(τ)) ∆τ → 0 (t1→ t2) and (Ax) ∆3 (t2) − (Ax)∆ 3 (t1) 6 t2 Z a h∆_α−23 (t2,σ(τ)) f(τ,x(τ)) − x∆ 3 (τ)µ(τ)hβ(τ,σ(τ)) ∆τ − t1 Z a h∆_α−23 (t1,σ(τ)) f(τ,x(τ)) − x∆3(τ)µ(τ)hβ(τ,σ(τ)) ∆τ 6 t1 Z a h ∆3 α−2(t2,σ(τ)) − h∆ 3 α−2(t1,σ(τ)) f(τ,x(τ)) − x ∆3 (τ)µ(τ)hβ(τ,σ(τ)) ∆τ + t2 Z t1 h ∆3 α−2(t2,σ(τ)) f(τ,x(τ)) − x ∆3 (τ)µ(τ)hβ(τ,σ(τ)) ∆τ → 0 (t1→ t2)

by using(H2). It yields that AΩ is equicontinuous in C3

rd(T).

As a consequence of this steps, we obtain that A is completely continuous opera-tor.

3. Existence and uniqueness of solutions

In this section, we will use the following well-known contraction mapping theorem named also as the Banach fixed point theorem: Let B be a Banach space and S a nonempty closed subset of B . Assume A : S→ S is a contraction, i.e., there is a λ (0 <λ<1) such that kAx − Ayk 6λkx − yk for all x , y in S . Then A has a unique fixed point in S .

THEOREM1. Suppose that(H2) holds. Also, we assume that

(H3) Let the function f (t, x) satisfy the following Lipschitz condition: there is a constant L >0 such that

| f (t, x) − f (t, y)| 6 L|x − y| for each t ∈ J, (11) for all x and y in C_rd3(T). Moreover, LM + N < 1 , where M and N are defined in (8) and (9), respectively.

(10)

(11)

(Ax) ∆3 (t) − (Ay)∆3(t) 6 t Z a h ∆3 α−2(t,σ(τ)) ( f (τ,x(τ)) − f (τ,y(τ))) ∆τ + t Z a h ∆3 α−2(t,σ(τ))µ(τ)hβ(τ,σ(τ)) x∆3(τ) − y∆3(τ) ∆τ 6 L t Z a h ∆3 α−2(t,σ(τ)) |x(τ) − y(τ)| ∆τ + t Z a h ∆3 α−2(t,σ(τ))µ(τ)hβ(τ,σ(τ)) x ∆3 (τ) − y∆3(τ) ∆τ 6 Lkx − yk t Z a h ∆3 α−2(t,σ(τ)) ∆τ +kx − yk t Z a h ∆3 α−2(t,σ(τ))µ(τ)hβ(τ,σ(τ)) ∆τ. Then, we obtain

kAx − Ayk = max

t∈J |(Ax)(t) − (Ay)(t)| + maxt∈J

(Ax) ∆3 (t) − (Ay)∆3(t) 6(LM + N)kx − yk =λkx − yk,

whereλ = LM + N ∈ (0, 1). Hence, A is a contraction mapping and the theorem is proved.

In the next theorem, the function f(t, x) satisfies a Lipschitz condition on a subset of C_rd3(T).

THEOREM2. Suppose that(H2) holds. Besides, we assume that (H4) Let there a number r > 0 such that

| f (t, x) − f (t, y)| 6 L|x − y| for each t ∈ J, (12) for all x and y in S= {x ∈ C3

rd(T) : kxk 6 r} , where L > 0 is a constant which may depend on r . Moreover, LM+ N < 1 , where M and N are defined in (8) and (9), respectively.

(H5) lim x→0

f(t,x)

x = 0 .

Then, the BVP (1) has a unique solution x∈ C3

rd(T) with max_t_∈J |x(t)|+max_t_∈J |x∆

3

(t)| 6 r .

(12)

Proof. Since lim x→0

f(t,x)

x = 0 from (H5), there exists a constant r > 0 such that | f (t, x)| 6δ|x| for 0 < |x| 6 r , where δ >0 is a constant satisfying δM+ N 6 1 . Let us take S= {x ∈ C3

rd(T) : kxk 6 r} . Obviously, S is a closed subset of C3rd(T). Let A : C_rd3(T) → C3

rd(T) be the operator defined by (7). For x and y in S , taking into account (H4), in exactly the same way in the proof of Theorem1 we can get kAx − Ayk 6λkx − yk , where 0 <λ<1 .

It remains to show that A maps S into itself. If x∈ S , then we obtain

(13)

|(Ax)∆3(t)| 6 t Z a h ∆3 α−2(t,σ(τ)) |f(τ,x(τ))| ∆τ + t Z a h ∆3 α−2(t,σ(τ))µ(τ)hβ(τ,σ(τ)) x ∆3 (τ)) ∆τ 6δr   t Z a h ∆3 α−2(t,σ(τ)) ∆τ   +r   t Z a h ∆3 α−2(t,σ(τ))µ(τ)hβ(τ,σ(τ)) ∆τ   .

SincekAxk = max

t∈J |(Ax)(t)| + maxt∈J |(Ax) ∆3

(t)| 6δrM+ rN 6 r , we have A : S → S . From the contraction mapping theorem, the BVP (1) has a unique solution in C_rd3(T).

4. Existence of solutions

THEOREM3. [19,26] Let E be a Banach space. Assume that A : E→ E is com-pletely continuous operator and the set V= {u ∈ E : u =λAu,0 <λ<1} is bounded. Then A has a fixed point in E.

THEOREM4. If the conditions(H1) and (H2) satisfy, then the BVP (1) has at least one solution in C_rd3(T).

Proof. From Lemma3, A : C_rd3(T) → C3

rd(T) is completely continuous operator. Now, we will show that the set V =x ∈ C3

rd(T) : x =λAxfor some 0 <λ<1

is bounded. For all x∈ V , we get

kxk = kλAxk

6λ φ(b)ψ(r)M +λkxkN

by using (10). Then we obtain kxk 6λ φ (b)ψ(r)M_{1−λ N} . From Theorem3, the BVP (1) has at least one solution in C_rd3(T).

THEOREM5. [19, 26] Let E be a Banach space. Assume that Ω is an open bounded subset of E withθ∈ Ω and let A : Ω → E be a completely continuous operator such that

kAuk 6 kuk ∀x ∈∂Ω, (13)

(14)

THEOREM6. If the conditions(H2) and (H5) satisfy, then the BVP (1) has at least one solution.

Proof. Since lim x→0

f(t,x)

x = 0 , there exists a constant r > 0 such that | f (t, x)| 6δ|x| for 0 <|x| < r , where δ >0 is a constant satisfying δM+ N < 1 . Let us take Ω = x ∈ C3

rd(T) : kxk < r . Since the function f satisfies the condition (H1) by taking φ(t) =δ and ψ(|x|) = |x|, A : Ω → C3

rd(T) is completely continuous operator from Lemma3. If we take x∈∂Ω , then we obtain kAxk 6 r as in the proof of Theorem2. It follows thatkAxk 6 kxk , ∀x ∈∂Ω . Therefore, by means of Theorem5the operator A has at least one fixed point inΩ . Thus, the BVP (1) has at least one solution u∈ Ω. COROLLARY1. Assume that(H1) and (H2) hold. Ifφ(b)M +N 6 1 andψ(z) 6 z ,∀z ∈ [0, ∞), then the BVP (1) has at least one solution.

EXAMPLE1. Let T= qZ = {qk : k ∈ Z} and define

hα(t, s) =

qαt−s_q (α) Γ(α+ 1) ,

where t(α)=_Γ(t−α+1)Γ(t+1) which satisfies the properties in (2) and (3). Consider the fol-lowing boundary value problem

( ∆α−1 0∗ x(t) = tx4 5+x4, t∈ J := [0, 20] ∩ T, 2 <α<3 x(0) = x∆(20) = 0, x∆(0) = x∆(10). (14) Since f(t, x) = tx4

5+x4 ∈ C ([0, 20] × R, R) holds f (t, x) 6 20 and the condition (H2) is

satisfied, the BVP (14) has at least one solution by using Theorem4.

R E F E R E N C E S

[1] T. ABDELJAVAD ANDD. BALEANU, Caputo q-fractional initial value problems and a q-analogue

Mittag-Leffler function, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), 4682–4688.

[2] A. AHMADKHANLU ANDM. JAHANSHAHI, On the existence and uniqueness of solution of initial

value problem for fractional order differential equations on time scales, Bulletin of the Iranian Math-ematical Society 38 (2012), 241–252.

[3] G. A. ANASTASSIOU, Principles of delta fractional calculus on time scales and inequalities, Math. Comput. Model. 52 (2010), 556–566.

[4] G. A. ANASTASSIOU, Foundations of nabla fractional calculus on time scales and inequalities, Com-put. Math. Appl. 59 (2010), 3750–3762.

[5] F. M. ATICI ANDP. W. ELOE, A transform method in discrete fractional calculus, Int. J. Differ. Equ.

2 (2007), 165–176.

[6] F. M. ATICI ANDP. W. ELOE, Fractional q-calculus on a time scale, J. Nonlinear Math. Phys. 14 (2007), 341–352.

[7] F. M. ATICI ANDP. W. ELOE, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc. 137 (2009), 981–989.

(15)

[8] F. M. ATICI ANDP. W. ELOE, Two-point boundary value problems for finite fractional difference

equations, J. Difference Equ. Appl. 17 (2011), 445–456.

[9] F. M. ATICI ANDS. SENGUL¨ , Modeling with fractional difference equations, J. Math. Anal. Appl.

369 (2010), 1–9.

[10] N. R. O. BASTOS, D. MOZYRSKA ANDD. F. M. TORRES, Fractional derivatives and integrals on

time scales via the inverse generalized Laplace transform, Int. J. Math. Comput. 11 (2011), 1–9. [11] A. S. BERDYSHEV, A. CABADA ANDB. KH. TURMETOV, On solvability of a boundary value

prob-lem for a nonhomogeneous biharmonic equation with a boundary operator of a fractional order, Acta Math. Sci. Ser. B Engl. Ed. 34 (2014), 1695–1706.

[12] M. BOHNER ANDA. PETERSON, Dynamic Equations on Time Scales: An Introduction with

Applica-tions, Birkh¨auser, Boston, 2001.

[13] M. BOHNER AND A. PETERSON (editors), Advances in Dynamic Equations on Time Scales, Birkh¨auser, Boston, 2003.

[14] R. A. C. FERREIRA, Positive solutions for a class of boundary value problems with fractional

q-differences, Comput. Math. Appl. 61 (2011), 367–373.

[15] C. S. GOODRICH, Existence of positive solution to a systemof discrete fractional boundary value

problems, Appl. Math. Comput. 217 (2011), 4740–4753.

[16] C. S. GOODRICH, On discrete sequential fractional boundary value problems, J. Math. Anal. Appl.

385 (2012), 111–124.

[17] J. R. GRAEF, L. KONG ANDQ. KONG, Application of the mixed monotone operator method to

fractional boundary value problems, Fractional Differential Calculus 2, 1 (2012), 87–98.

[18] A. A. KILBAS, H. M. SRIVASTAVA ANDJ. J. TRUJILLO, Theory and Applications of Fractional

Differential Equations, Elsevier, Amsterdam, 2006.

[19] M. A. KRASNOSELSKII, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964. [20] Y. LIU, Existence of three positive solutions for boundary value problems of singular fractional

dif-ferential equations, Fractional Differential Calculus 2, 1 (2012), 55–69.

[21] K. S. MILLER ANDB. ROSS, Fractional difference calculus, Proceedings of the International Sym-posium on Univalent Functions, Fractional Calculus and Their Applications, Nihon University, Ko-riyama, May 1988, Ellis Horwood Series in Mathematics and Its Applications, Horwood, Chichester, 1989, 139–152.

[22] N. NYAMORADI ANDT. BASHIRI, Multiple positive solutions for nonlinear fractional differential

systems, Fractional Differential Calculus 2, 2 (2012), 119–128.

[23] K. B. OLDHAM ANDJ. SPANIER, Fractional Calculus: Theory and Applications, Differentiation and

Integration to Arbitrary Order, Academic Press, New York, NY, USA, 1974.

[24] O. P. SABATIER, J. A. AGRAWAL ANDT. MACHADO, Advances in Fractional Calculus, Springer, Dordrecht, The Netherlands, 2007.

[25] S. G. SAMKO, A. A. KILBAS ANDO. I. MARICHEV, Fractional Integral and Derivatives: Theory

and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.

[26] J. X. SUN, Nonlinear Functional Analysis and Its Applications, Science Press, Beijing, 2008. [27] G. T. WANG, B. AHMAD ANDL. H. ZHANG, Some existence results for impulsive nonlinear

frac-tional differential equations with mixed boundary conditions, Comput. Math. Appl. 62 (2011), 1389– 1397.

[28] G. T. WANG, B. AHMAD ANDL. H. ZHANG, On nonlocal integral boundary value problems for

impulsive nonlinear differential equations of fractional order, Fixed Point Theory 15 (2014), 265– 284.

[29] P. A. WILLIAMS, Fractional calculus on time scales with Taylor’s theorem, Frac. Calc. Appl. Anal.

15 (2007), 616–638.

[30] X. ZHANG ANDC. ZHU, Cauchy problem for a class of fractional differential equations on time

scales, International Journal of Computer Mathematics 91 (2014), 527–538.

[31] C. ZHOU, Existence and uniqueness of positive solutions to higher-order nonlinear fractional

differen-tial equation with integral boundary conditions, Electron. J. Differential Equations 2012, 234 (2012), 1–11.

(16)

[32] X. ZHOU ANDW. WU, Uniqueness and asymptotic behavior of positive solutions for a

fractional-order integral boundary-value problem, Electron. J. Differential Equations 2013, 37 (2013), 1–10. [33] J. ZHU ANDY. ZHU, Fractional Cauchy problem with Riemann-Liouville fractional delta derivative

on time scales, Abstract and Applied Analysis 2013, Article ID 401596 (2013), 1–19.

(Received October 10, 2015) .Ismail Yaslan

Pamukkale University, Department of Mathematics 20070 Denizli, Turkey e-mail:iyaslan@pau.edu.tr

Onur Liceli Pamukkale University, Department of Mathematics 20070 Denizli, Turkey e-mail:onurliceli@gmail.com

Fractional Differential Calculus

www.ele-math.com fdc@ele-math.com