Research Article Some Existence Results for Impulsive Nonlinear Fractional Differential Equations with Closed Boundary Conditions

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Volume 2012, Article ID 387629,15pages doi:10.1155/2012/387629

Research Article

Some Existence Results for Impulsive

Nonlinear Fractional Differential Equations with

Closed Boundary Conditions

Hilmi Erg ¨oren

1

_{and Adem Kilic¸man}

2

1_{Department of Mathematics, Faculty of Sciences, Yuzuncu Yil University, 65080 Van, Turkey} 2_{Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia,}

43400 Serdang, Malaysia

Correspondence should be addressed to Adem Kilic¸man,[email protected]

Received 30 September 2012; Accepted 22 October 2012 Academic Editor: Beata Rzepka

Copyrightq2012 H. Erg ¨oren and A. Kilic¸man. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We investigate some existence results for the solutions to impulsive fractional diﬀerential equations having closed boundary conditions. Our results are based on contracting mapping principle and Burton-Kirk fixed point theorem.

1. Introduction

This paper considers the existence and uniqueness of the solutions to the closed boundary value problemBVP, for the following impulsive fractional diﬀerential equation:

C_Dα_x_t _f_{t, x}_t_, _t_∈_J_: ₀_{, T}_{, t /}_t k,1< α≤2, Δxtk Ik xt−_k, Δxtk I_k∗xt−_k, k1,2, . . . , p, xT ax0 bTx0, TxT cx0 dTx0, 1.1

whereC_Dα_{is Caputo fractional derivative,}_f_∈_C_J_×_{R, R}_,_I

k, I_k∗∈CR, R,

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with xt_k lim h→0xtk h, x t−_k lim h→0−xtk h, 1.3

andΔxtkhas a similar meaning forxt, where

0t0< t1< t2<· · ·< tp< tp 1T, 1.4

a,b,c, anddare real constants withΔ:c1−b 1−a1−d/0.

The boundary value problems for nonlinear fractional diﬀerential equations have been addressed by several researchers during last decades. That is why, the fractional derivatives serve an excellent tool for the description of hereditary properties of various materials and processes. Actually, fractional diﬀerential equations arise in many engineering and scientific disciplines such as, physics, chemistry, biology, electrochemistry, electromagnetic, control theory, economics, signal and image processing, aerodynamics, and porous mediasee1–

7. For some recent development, see, for example,8–14.

On the other hand, theory of impulsive diﬀerential equations for integer order has become important and found its extensive applications in mathematical modeling of phenomena and practical situations in both physical and social sciences in recent years. One can see a noticeable development in impulsive theory. For instance, for the general theory and applications of impulsive diﬀerential equations we refer the readers to15–17.

Moreover, boundary value problems for impulsive fractional diﬀerential equations have been studied by some authorssee 18–20 and references therein. However, to the

best of our knowledge, there is no study considering closed boundary value problems for impulsive fractional diﬀerential equations.

Here, we notice that the closed boundary conditions in1.1include quasi-periodic boundary conditionsbc0and interpolate between periodicad1, bc0and antiperiodicad−1, bc0boundary conditions.

Motivated by the mentioned recent work above, in this study, we investigate the existence and uniqueness of solutions to the closed boundary value problem for impulsive fractional diﬀerential equation1.1. Throughout this paper, inSection 2, we present some notations and preliminary results about fractional calculus and diﬀerential equations to be used in the following sections. InSection 3, we discuss some existence and uniqueness results for solutions of BVP1.1, that is, the first one is based on Banach’s fixed point theorem,

the second one is based on the Burton-Kirk fixed point theorem. At the end, we give an illustrative example for our results.

2. Preliminaries

Let us setJ0 0, t1,J1 t1, t2, . . . , Jk−1 tk−1, tk,Jk tk, tk 1,J: 0, T\ {t1, t2, . . . , tp} and introduce the set of functions:

P CJ, R {x:J → R:x∈Ctk, tk 1, R,k0,1,2, . . . , pand there existxt_kand

xt−_k,k1,2, . . . , pwithxt−_k xtk}and

P C1_{J, R} _{_x _∈_{P C}_{J, R}_,_x _∈_C_t

k, tk 1, R,k 0,1,2, . . . , pand there existxt_k

andxt−_k,k 1,2, . . . , pwithxt−_k xtk}which is a Banach space with the normx

sup_t_∈_J{xP C,xP C}wherexP C:sup{|xt|:t∈J}. The following definitions and lemmas were given in4.

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Definition 2.1. The fractionalarbitraryorder integral of the functionh∈L1_{J, R} _{of order} α∈R is defined by I₀αht 1 Γα _t 0 t−sα−1hsds, 2.1

whereΓ·is the Euler gamma function.

Definition 2.2. For a functionhgiven on the intervalJ, Caputo fractional derivative of order

α >0 is defined by C_Dα 0 ht 1 Γn−α t 0 t−sn−α−1hnsds, n α 1, 2.2

where the functionhthas absolutely continuous derivatives up to ordern−1.

Lemma 2.3. Letα >0, then the diﬀerential equation

C_Dα_h_t ₀ _2.3

has solutions

ht c0 c1t c2t2 · · · cn−1tn−1, ci∈R, i0,1,2, . . . , n−1, n α 1. 2.4 The following lemma was given in4,10.

Lemma 2.4. Letα >0, then

IαCDαht ht c0 c1t c2t2 · · · cn−1tn−1, 2.5

for someci∈R,i0,1,2, . . . , n−1,n α 1.

The following theorem is known as Burton-Kirk fixed point theorem and proved in 21.

Theorem 2.5. LetXbe a Banach space andA,D:X → Xtwo operators satisfying:

aAis a contraction, and

bDis completely continuous.

Then either

ithe operator equationxAx Dxhas a solution, or

iithe setε{x∈X:xλAx/λ λDx}is unbounded forλ∈0,1.

Theorem 2.6see22, Banach’s fixed point theorem. LetS be a nonempty closed subset of a

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Next we prove the following lemma.

Lemma 2.7. Let 1< α ≤2 and leth :J → Rbe continuous. A functionxtis a solution of the

fractional integral equation:

xt ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ t 0 t−sα−1 Γα hsds Ω1tΛ1 Ω2tΛ2, t∈J0 _t tk t−sα−1 Γα hsds 1 Ω1t k i1 _t_i ti−1 ti−sα−1 Γα hsds T−tkΩ1t Ω2t t−tk k i1 ti ti−1 ti−sα−2 Γα−1 hsds 1 Ω1t k−1 i1 tk−ti _t_i ti−1 ti−sα−2 Γα−1 hsds Ω1t _T tk T−sα−1 Γα hsds Ω2t T tk T−sα−2 Γα−1 hsds 1 Ω1t _k i1 Ii xt−_i I_k∗xt−_k Ω1t k−1 i1 T−tiI_i∗xt−_i Ω2t k−1 i1 I_i∗xt−_i k−1 i1 t−tiIi∗ xt−i , t∈Jk, k1,2, . . . , p 2.6

if and only ifxtis a solution of the fractional BVP

C_Dα_x_t _h_t_, _t_∈_J_, Δxtk Ik xt−_k, Δxtk I_k∗xt−_k, xT ax0 bTx0, TxT cx0 dTx0, 2.7 where Λ1: _T tk T−sα−1 Γα hsds k i1 _t_i ti−1 ti−sα−1 Γα hsds k i1 T−tk _t_i ti−1 ti−sα−2 Γα−1 hsds k−1 i1 tk−ti _t_i ti−1 ti−sα−2 Γα−1 hsds k i1 Ii xt−_i k−1 i1 T−tiIi∗ xt−_i I_k∗xt−_k,

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Λ2: T tk T−sα−2 Γα−1 hsds k i1 ti ti−1 ti−sα−2 Γα−1 hsds k−1 i1 Ii∗ xt−_i, Ω1t:−1−_Δd− ct TΔ, Ω2t: 1−_ΔbT −1−_Δat. 2.8

Proof. Letxbe the solution of2.7. Ift∈J0, thenLemma 2.4implies that

xt Iαht−c0−c1t _t 0 t−sα−1 Γα hsds−c0−c1t, xt t 0 t−sα−2 Γα−1hsds−c1, 2.9 for somec0, c1 ∈R.

Ift∈J1, thenLemma 2.4implies that

xt t t1 t−sα−1 Γα hsds−d0−d1t−t1, xt t t1 t−sα−2 Γα−1hsds−d1, 2.10

for somed0, d1∈R. Thus we have

xt−₁ _t₁ t0 t1−sα−1 Γα hsds−c0−c1t1, x t₁−d0, xt−₁ _t₁ t0 t1−sα−2 Γα−1 hsds−c1, x _t 1 −d1. 2.11 Observing that Δxt1 x t₁−xt−₁I1 xt−₁, Δxt1 x t₁−xt−₁I₁∗xt−₁, 2.12

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then we have −d0 t1 t0 t1−sα−1 Γα hsds−c0−c1t1 I1 xt−₁, −d1 t1 t0 t1−sα−2 Γα−1 hsds−c1 I ∗ 1 xt−₁, 2.13 hence, fort∈t1, t2, xt t t1 t−sα−1 Γα hsds t1 t0 t1−sα−1 Γα hsds t−t1 t1 t0 t1−sα−2 Γα−1 hsds I1 xt−₁ t−t1I1∗ xt−₁−c0−c1t, xt t t1 t−sα−2 Γα−1hsds t1 t0 t1−sα−2 Γα−1 hsds I ∗ 1 xt−₁−c1. 2.14

Ift∈J2, thenLemma 2.4implies that

xt t t2 t−sα−1 Γα hsds−e0−e1t−t2, xt t t2 t−sα−2 Γα−1hsds−e1, 2.15

for somee0, e1∈R. Thus we have

xt−₂ t2 t1 t2−sα−1 Γα hsds t1 t0 t1−sα−1 Γα hsds t2−t1 t1 t0 t1−sα−2 Γα−1 hsds I1 xt−₁ t2−t1I₁∗ xt−₁−c0−c1t2, xt₂ −e0, xt−₂ _t₂ t1 t2−sα−2 Γα−1 hsds _t₁ t0 t1−sα−2 Γα−1 hsds I ∗ 1 xt−₁−c1, xt₂ −e1. 2.16

Similarly we observe that Δxt2 x t₂−xt−₂I2 xt−₂, Δxt2 x t₂−xt−₂I₂∗xt−₂, 2.17

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thus we have −e0x t−₂ I2 xt−₂, −e1x t−2 I2∗ xt−2 . 2.18 Hence, fort∈t2, t3, xt t t2 t−sα−1 Γα hsds t1 t0 t1−sα−1 Γα hsds t2 t1 t2−sα−1 Γα hsds t2−t1 t1 t0 t1−sα−2 Γα−1 hsds t−t2 t1 t0 t1−sα−1 Γα−1 hsds t2 t1 t2−sα−1 Γα−1 hsds I1 xt−₁ I2 xt−₂ t−t1I₁∗ xt−₁ I2∗ xt−₂−c0−c1t. 2.19

By a similar process, ift∈Jk, then again fromLemma 2.4we get

xt ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ _t tk t−sα−1 Γα hsds k i1 _t_i ti−1 ti−sα−1 Γα hsds k i1 t−tk ti ti−1 ti−sα−2 Γα−1 hsds k−1 i1 tk−ti ti ti−1 ti−sα−2 Γα−1 hsds k i1 Ii xt−_i k−1 i1 t−tiI_i∗xt−_i I_k∗xt−_k−c0−c1t, xt t tk t−sα−2 Γα−1hsds k i1 _t_i ti−1 ti−sα−2 Γα−1 hsds k−1 i1 I_i∗xt−_i−c1. 2.20

Now if we apply the conditions:

xT ax0 bTx0, TxT cx0 dTx0, 2.21 we have Λ1 1−ac0 T1−bc1, Λ2−c Tc0 1−dc1, −c0−1−_ΔdΛ1 1−b_ΔTΛ2, −c1−cΛ1 TΔ − 1−aΛ2 Δ . 2.22

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In view of the relations2.8, when the values of−c0and−c1are replaced in2.9and2.20,

the integral equation2.7is obtained.

Conversely, assume that x satisfies the impulsive fractional integral equation2.6,

then by direct computation, it can be seen that the solution given by2.6satisfies2.7. The

proof is complete.

3. Main Results

Definition 3.1. A function x ∈ P C1J, R with its α-derivative existing on J is said to be

a solution of 1.1, if x satisfies the equation C_Dα_x_t _f_{t, x}_t _on _J _{and satisfies the}

conditions:

Δxtk Ik

xt−_k, Δxtk I_k∗xt−_k,

xT ax0 bTx0, TxT cx0 dTx0. 3.1

For the sake of convenience, we define Ω∗ 1 sup t∈J |Ω1t|, Ω∗2sup t∈J |Ω2t|, Ω∗∗1 sup t∈J _Ω 1t, Ω∗∗2 sup t∈J _Ω 2t. 3.2

The followings are main results of this paper.

Theorem 3.2. Assume that

A1the functionf : J×R → Ris continuous and there exists a constantL1 > 0 such that

ft, u−ft, v ≤L1u−v, for allt∈J, andu, v∈R,

A2Ik, I_k∗ : R → R are continuous, and there exist constants L2 > 0 and L3 > 0 such

thatIku−Ikv ≤ L2u−v,I_k∗u−I_k∗v ≤ L3u−vfor eachu, v ∈ Rand

k1,2, . . . , p. Moreover, consider the following:

L1Tα Γα 1 1 Ω∗₁1 p 2pα L1T α−1 Γα Ω ∗ 2 1 p 1 Ω∗₁pL2 L3 Ω∗ 1T Ω∗2 T pL3 <1. 3.3

Then, BVP1.1has a unique solution onJ.

Proof. Define an operatorF :P C1_{J, R} _→ _{P C}1_{J, R}_by

Fxt t tk t−sα−1 Γα fs, xsds 1 Ω1t k i1 ti ti−1 ti−sα−1 Γα fs, xsds T−tkΩ1t Ω2t t−tk k i1 ti ti−1 ti−sα−2 Γα−1 fs, xsds

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1 Ω1t k−1 i1 tk−ti ti ti−1 ti−sα−2 Γα−1 fs, xsds Ω1t T tk T−sα−1 Γα fs, xsds Ω2t T tk T−sα−2 Γα−1 fs, xsds 1 Ω1t _k i1 Ii xt−_i I_k∗xt−_k Ω1t k−1 i1 T−tiIi∗ xt−_i Ω2t k−1 i1 Ii∗ xt−_i k−1 i1 t−tiIi∗ xt−_i. 3.4 Now, forx, y∈P CJ, Rand for eacht∈J, we obtain

_Fx_t₋ Fyt≤ _t tk t−sα−1 Γα fs, xs−f s, ysds |1 Ω1t| k i1 _t_i ti−1 ti−sα−1 Γα fs, xs−f s, ysds |T−tkΩ1t Ω2t t−tk| × k i1 _t_i ti−1 ti−sα−2 Γα−1 fs, xs−f s, ysds |1 Ω1t| k−1 i1 tk−ti ti ti−1 ti−sα−2 Γα−1 fs, xs−f s, ysds |Ω1t| _T tk T−sα−1 Γα fs, xs−f s, ysds |Ω2t| _T tk T−sα−2 Γα−1 fs, xs−f s, ysds |1 Ω1t| _k i1 _I_i xt−_i−Ii yt−_i I_k∗xt−_k−I_k∗yt−_k |Ω1t| k−1 i1 T−tiI_i∗xt−_i−I_i∗yt−_i |Ω2t| k−1 i1 _I∗ i xt−_i−I_i∗yt−_i k−1 i1 |t−ti|Ii∗ xt−_i−Ii∗ yt−_i,

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_Fx_t₋ Fyt≤ L1Tα Γα 1 1 Ω∗₁1 p 2pα L1T α−1 Γα Ω ∗ 2 1 p 1 Ω∗₁pL2 L3 Ω∗ 1T Ω∗2 T pL3 _x_s₋_y_s_. 3.5 Therefore, by3.3, the operatorF is a contraction mapping. In a consequence of Banach’s fixed theorem, the BVP 1.1 has a unique solution. Now, our second result relies on the Burton-Kirk fixed point theorem.

Theorem 3.3. Assume that (A1)-(A2) hold, and

A3there exist constantsM1>0,M2>0,M3>0 such thatft, u ≤M1,Iku ≤M2,

I_k∗u ≤M3for eachu, v∈Randk1,2, . . . , p.

Then the BVP1.1has at least one solution onJ.

Proof. We define the operatorsA, D:P C1_{J, R} _→ _{P C}1_{J, R}_by

Axt 1 Ω1t _k i1 Ii xt−_i I_k∗xt−_k Ω1t k−1 i1 T−tiI_i∗xt−_i Ω2t k−1 i1 I_i∗xt−_i k−1 i1 t−tiI_i∗xt−_i, Dxt _t tk t−sα−1 Γα fs, xsds 1 Ω1t k i1 _t_i ti−1 ti−sα−1 Γα fs, xsds T−tkΩ1t Ω2t t−tk k i1 ti ti−1 ti−sα−2 Γα−1 fs, xsds 1 Ω1t k−1 i1 tk−ti ti ti−1 ti−sα−2 Γα−1 fs, xsds Ω1t _T tk T−sα−1 Γα fs, xsds Ω2t _T tk T−sα−2 Γα−1 fs, xsds. 3.6

It is obvious thatAis contraction mapping for

1 Ω∗₁pL2 L3 Ω∗ 1T Ω∗2 T pL3<1. 3.7

Now, in order to check thatD is completely continuous, let us follow the sequence of the following steps.

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Step 1Dis continuous. Let{xn}be a sequence such thatxn → xinP CJ, R. Then fort∈J, we have |Dxnt−Dxt| ≤ t tk t−sα−1 Γα fs, xns−fs, xsds |1 Ω1t| k i1 _t_i ti−1 ti−sα−1 Γα fs, xns−fs, xsds |T−tkΩ1t Ω2t t−tk| × k i1 _t_i ti−1 ti−sα−2 Γα−1 fs, xns−fs, xsds |1 Ω1t| k−1 i1 tk−ti _t_i ti−1 ti−sα−2 Γα−1 fs, xns−fs, xsds |Ω1t| T tk T−sα−1 Γα fs, xns−fs, xsds |Ω2t| T tk T−sα−2 Γα−1 fs, xns−fs, xsds. 3.8

Sincefis continuous function, we get

Dxnt−Dxt −→0 asn−→ ∞. 3.9

Step 2Dmaps bounded sets into bounded sets inP CJ, R. Indeed, it is enough to show

that for anyr >0, there exists a positive constantlsuch that for eachx∈Br {x∈P CJ, R:

x ≤r}, we haveDx ≤l. ByA3, we have for eacht∈J,

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|Ω1t| _T tk T−sα−1 Γα fs, xsds |Ω2t| _T tk T−sα−2 Γα−1 fs, xsds, Dx ≤ M1T α Γα 1 1 Ω∗₁1 p 2pα M1T α−1 Γα Ω ∗ 2 1 p:l. 3.10

Step 3Dmaps bounded sets into equicontinuous sets inP C1_{J, R}_{. Let}_τ

1, τ2∈Jk,0≤k≤p

withτ1< τ2and letBrbe a bounded set ofP C1J, Ras in Step 2, and letx∈Br. Then

_Dy τ2−Dτ1≤ _τ₂ τ1 Dysds≤Lτ2−τ1, 3.11 where _Dx_t_≤ t tk t−sα−2 Γα−1fs, xsds Ω 1t k i1 ti ti−1 ti−sα−1 Γα fs, xsds _T₋_t_kΩ 1t Ω2t 1 k i1 ti ti−1 ti−sα−2 Γα−1 fs, xsds _Ω 1t k−1 i1 tk−ti _t_i ti−1 ti−sα−2 Γα−1 fs, xsds _Ω 1t T tk T−sα−1 Γα fs, xsds Ω 2t T tk T−sα−2 Γα−1 fs, xsds, ≤ M1Tα Γα 1Ω ∗ 1 1 p 2pα M1T α−1 Γα 1 Ω∗₂1 p:L. 3.12 This implies thatAis equicontinuous on all the subintervalsJk,k 0,1,2, . . . , p. Therefore, by the Arzela-Ascoli Theorem, the operator D : P C1_{J, R} _→ _{P C}1_{J, R} _{is completely}

continuous.

To conclude the existence of a fixed point of the operatorA D, it remains to show that the set

εx∈X :xλAx λ

λDxfor someλ∈0,1 3.13

is bounded.

Letx∈ε, for eacht∈J,

xt λDxt λAx λ

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Hence, fromA3, we have |xt| ≤λ t tk t−sα−1 Γα fs, xsds λ|1 Ω1t| k i1 ti ti−1 ti−sα−1 Γα fs, xsds λ|T−tkΩ1t Ω2t t−tk| k i1 ti ti−1 ti−sα−2 Γα−1 fs, xsds λ|1 Ω1t| k−1 i1 tk−ti ti ti−1 ti−sα−2 Γα−1 fs, xsds λ|Ω1t| T tk T−sα−1 Γα fs, xsds λ|Ω2t| T tk T−sα−2 Γα−1 fs, xsds λ|1 Ω1t| _k i1 Ii xt−_i λ Ik∗ xt−_k λ λ|Ω1t| k−1 i1 T−ti Ii∗ xt−_i λ λ|Ω2t| k−1 i1 Ii∗ xt−_i λ λ k−1 i1 t−tiI_i∗ xt−_i λ , xt ≤ M1Tα Γα 1 1 Ω∗₁1 p 2pα M1T α−1 Γα Ω ∗ 2 1 p 1 Ω∗₁pM2 M3 Ω∗ 1T Ω∗2 T pM3 . 3.15

Consequently, we conclude the result of our theorem based on the Burton-Kirk fixed point theorem.

4. An Example

Consider the following impulsive fractional boundary value problem:

C_D3/2_x_t sin 2t|xt| t 521 |xt|, t∈0,1, t / 1 3, Δx 1 3 |x1−/3| 15 |x1−/3|, Δx 1 3 |x1−/3| 10 |x1−/3|, x1 5x0−2x0, x1 x0 4x0. 4.1

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Here,a 5,b −2,c 1,d 4,α 3/2,T 1,p 1. Obviously,L1 1/25,L2 1/15, L31/10,Ω∗₁4/15,Ω∗₂7/15. Further, L1Tα Γα 1 1 Ω∗₁1 p1 L2 2pα L3 L1Tα−1 Γα Ω ∗ 2 1 p Ω∗₁T Ω∗₂ TpL3 464 1125√π 173 450 <1. 4.2

Since the assumptions ofTheorem 3.2are satisfied, the closed boundary value problem4.1

has a unique solution on0,1. Moreover, it is easy to check the conclusion ofTheorem 3.3.

Acknowledgments

The authors would like to express their sincere thanks and gratitude to the reviewersfor their valuable comments and suggestions for the improvement of this paper. The second author also gratefully acknowledges that this research was partially supported by the University Putra Malaysia under the Research University Grant Scheme 05-01-09-0720RU.

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Research Article Some Existence Results for Impulsive Nonlinear Fractional Differential Equations with Closed Boundary Conditions