Nonlinear Analysis 71 (2009) 2391–2396

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Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

Boundary value problems for differential equations with fractional orderand nonlocal conditionsM. Benchohra a,∗, S. Hamani a, S.K. Ntouyas ba Laboratoire de Mathématiques, Université de Sidi Bel-Abbès, B.P. 89, 22000, Sidi Bel-Abbès, Algérieb Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

a r t i c l e i n f o

Article history:Received 10 November 2007Accepted 15 January 2009

MSC:26A3334B15

Keywords:Boundary value problemCaputo fractional derivativeFractional integralExistenceUniquenessFixed pointNonlocal boundary conditions

a b s t r a c t

In this paper, we establish sufficient conditions for the existence of solutions for a classof boundary value problem for fractional differential equations involving the Caputofractional derivative and nonlocal conditions.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

This paper deals with the existence and uniqueness of solutions for the boundary value problems (BVP for short), forfractional order differential equations and nonlocal boundary condition of the form

cDαy(t) = f (t, y(t)), for each t ∈ J = [0, T ], 1 < α ≤ 2, (1)y(0) = g(y), y(T ) = yT , (2)

where cDα is the Caputo fractional derivative, f : [0, T ] × R → R, is a continuous function, g : C(J,R) → R acontinuous function and yT ∈ R. Differential equations of fractional order have recently proved to be valuable tools inthe modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applicationsin viscoelasticity, electrochemistry, control, porous media, electromagnetic, etc. (see [1–7]). For three noteworthy papersdealing with the integral operator and the arbitrary fractional order differential operator, see [8–10]. There has been asignificant development in fractional differential equations in recent years; see the monographs of Kilbas et al. [11], Millerand Ross [12], Podlubny [13], Samko et al. [14] and the papers of Delbosco and Rodino [15], Diethelm et al. [1,16,17], El-Sayed [18], Kilbas and Marzan [19], Mainardi [5], Momani and Hadid [20], Momani et al. [21], Podlubny et al. [22], Yu andGao [23] and the references therein.

∗ Corresponding author. Fax: +213 48 54 43 44.E-mail addresses: benchohra@univ-sba.dz, benchohra@yahoo.com (M. Benchohra), hamani_samira@yahoo.fr (S. Hamani), sntouyas@cc.uoi.gr

(S.K. Ntouyas).

0362-546X/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2009.01.073

2392 M. Benchohra et al. / Nonlinear Analysis 71 (2009) 2391–2396

Very recently some basic theory for the initial value problems of fractional differential equations involvingRiemann–Liouville differential operator of order 0 < α ≤ 1 has been discussed by Lakshmikantham and Vatsala [24–26].Nonlocal conditions were initiated by Byszewski [27] when he proved the existence and uniqueness of mild and classical

solutions of nonlocal Cauchy problems. As remarked by Byszewski [28,29], the nonlocal condition can be more useful thanthe standard initial condition to describe some physical phenomena. For example, g(y)may be given by

g(y) =p∑i=1

ciy(ti)

where ci, i = 1, . . . , p, are given constants and 0 < t1 < · · · < tp ≤ T .Applied problems require definitions of fractional derivatives allowing the utilization of physically interpretable initial

conditions, which contain y(0), y′(0), etc. the same requirements of boundary conditions. Caputo’s fractional derivativesatisfies these demands. For more details on the geometric and physical interpretation for fractional derivatives of both theRiemann–Liouville and Caputo types see [30]. In this paper, we present existence results for the problem (1)–(2) involvingCaputo’s fractional derivative. We give two results, one based on Banach fixed point theorem (Theorem 3.5) and anotherone based on Schaefer’s fixed point theorem (Theorem 3.6). Finally we present an example. These results can be consideredas a contribution to this emerging field.

2. Preliminaries

In this section,we introduce notations, definitions, and preliminary factswhich are used throughout this paper. By C(J,R)we denote the Banach space of all continuous functions from J into Rwith the norm

‖y‖∞ := sup{|y(t)| : t ∈ J}.

Definition 2.1 ([8,9,18,10]). The fractional (arbitrary) order integral of the function h ∈ L1([a, b],R+) of order α ∈ R+ isdefined by

Iαa h(t) =∫ t

a

(t − s)α−1

0(α)h(s)ds,

where 0 is the Gamma function. When a = 0, we write Iαh(t) = h(t) ∗ ϕα(t), where ϕα(t) = tα−10(α)

for t > 0, and ϕα(t) = 0for t ≤ 0, and ϕα → δ(t) as α→ 0, where δ is the delta function.

Definition 2.2 ([8,9,18,10]). For a function h given on the interval [a, b], the αth Riemann–Liouville fractional orderderivative of h, is defined by

(Dαa+h)(t) =1

0(n− α)

(ddt

)n ∫ t

a(t − s)n−α−1h(s)ds.

Here n = [α] + 1 and [α] denotes the integer part of α.

Definition 2.3 ([19]). For a function h given on the interval [a, b], the Caputo fractional order derivative of order α of h, isdefined by

(cDαa+h)(t) =1

0(n− α)

∫ t

a(t − s)n−α−1h(n)(s)ds,

where n = [α] + 1.

3. Existence of solutions

Let us start by defining what we mean by a solution of the problem (1)–(2).

Definition 3.1. A function y ∈ C2([0, T ],R) with its α-derivative exists on [0, T ] is said to be a solution of (1)–(2) if ysatisfies the equation cDαy(t) = f (t, y(t)) on J , and conditions y(0) = g(y) and y(T ) = yT .

For the existence of solutions for the problem (1)–(2), we need the following auxiliary lemmas:

Lemma 3.2 ([31]). Let α > 0, then the fractional differential equation

Dαh(t) = 0

has solutions

h(t) = c0 + c1t + c2t2 + · · · + cn−1tn−1, ci ∈ R, i = 0, 1, . . . , n− 1, n = [α] + 1.

M. Benchohra et al. / Nonlinear Analysis 71 (2009) 2391–2396 2393

Lemma 3.3 ([31]). Let α > 0, then

IαDαh(t) = h(t)+ c0 + c1t + c2t2 + · · · + cn−1tn−1

for some ci ∈ R, i = 0, 1, . . . , n− 1, n = [α] + 1.

As a consequence of Lemmas 3.2 and 3.3 we have the following result which is useful in what follows.

Lemma 3.4. Let 1 < α ≤ 2 and let h : [0, T ] → R be continuous. A function y is a solution of the fractional integral equation

y(t) =1

0(α)

∫ t

0(t − s)α−1h(s)ds−

tT0(α)

∫ T

0(T − s)α−1h(s)ds−

(tT− 1

)g(y)+

tTyT (3)

if and only if y is a solution of the fractional BVPcDαy(t) = h(t), t ∈ [0, T ], (4)y(0) = g(y), y(T ) = yT . (5)

Our first result is based on Banach fixed point theorem.

Theorem 3.5. Assume that:(H1) There exists a constant k > 0 such that

|f (t, u)− f (t, u)| ≤ k|u− u|, for each t ∈ J, and all u, u ∈ R.

(H2) There exists a constant k∗ > 0 such that

|g(u)− g(u)| ≤ k∗|u− u|, for each t ∈ J, and all u, u ∈ C([0, T ],R).

If

2kTα

0(α + 1)+ k∗ < 1 (6)

then the BVP (1)–(2) has a unique solution on [0, T ].

Proof. Transform the problem (1)–(2) into a fixed point problem. Consider the operator

F : C([0, T ],R)→ C([0, T ],R)

defined by

F(y)(t) =1

0(α)

∫ t

0(t − s)α−1f (s, y(s))ds−

tT0(α)

∫ T

0(T − s)α−1f (s, y(s))ds−

(tT− 1

)g(y)+

tTyT .

Clearly, the fixed points of the operator F are solution of the problem (1)–(2). We shall use the Banach contraction principleto prove that F has a fixed point. We shall show that F is a contraction.Let x, y ∈ C([0, T ],R). Then, for each t ∈ J we have

|F(x)(t)− F(y)(t)| ≤1

0(α)

∫ t

0(t − s)α−1|f (s, x(s))− f (s, y(s))|

+1

0(α)

∫ T

0(T − s)α−1|f (s, x(s))− f (s, y(s))|ds+ |g(x)− g(y)|

≤k‖x− y‖∞0(α)

∫ t

0(t − s)α−1ds+

k‖x− y‖∞0(α)

∫ T

0(T − s)α−1ds+ k∗‖x− y‖∞

≤2kTα

α0(α)‖x− y‖∞ + k∗‖x− y‖∞.

Thus

‖F(x)− F(y)‖∞ ≤[2kTα

0(α + 1)+ k∗

]‖x− y‖∞.

Consequently F is a contraction. As a consequence of Banach fixed point theorem, we deduce that F has a fixed point whichis a solution of the problem (1)–(2). �

The second result is based on Schaefer’s fixed point theorem.

2394 M. Benchohra et al. / Nonlinear Analysis 71 (2009) 2391–2396

Theorem 3.6. Assume that:(H3) The function f : [0, T ] × R→ R is continuous.(H4) There exists a constant M > 0 such that

|f (t, u)| ≤ M for each t ∈ J and all u ∈ R.

(H5) There exists a constant M1 > 0 such that

|g(y)| ≤ M1 for all y ∈ C([0, T ],R).

Then the BVP (1)–(2) has at least one solution on [0, T ].

Proof. We shall use Schaefer’s fixed point theorem to prove that F has a fixed point. The proof will be given in several steps.Step 1: F is continuous.Let {yn} be a sequence such that yn → y in C([0, T ],R). Then for each t ∈ [0, T ]

|F(yn)(t)− F(y)(t)| ≤1

0(α)

∫ t

0(t − s)α−1|f (s, yn(s))− f (s, y(s))|ds

+1

0(α)

∫ T

0(T − s)α−1|f (s, yn(s))− f (s, y(s))|ds+ |g(yn)− g(y)|

≤1

0(α)

∫ t

0(t − s)α−1 sup

s∈[0,T ]|f (s, yn(s))− f (s, y(s))|ds

+1

0(α)

∫ T

0(T − s)α−1 sup

s∈[0,T ]|f (s, yn(s))− f (s, y(s))|ds+ |g(yn)− g(y)|.

Since f and g are continuous functions, then we have

‖F(yn)− F(y)‖∞ → 0 as n→∞.

Step 2: F maps bounded sets into bounded sets in C([0, T ],R).Indeed, it is enough to show that for any η∗ > 0, there exists a positive constant ` such that for each y ∈ Bη∗ = {y ∈

C([0, T ],R) : ‖y‖∞ ≤ η∗}, we have ‖F(y)‖∞ ≤ `. By (H4) and (H5) we have for each t ∈ [0, T ],

|F(y)(t)| ≤1

0(α)

∫ t

0(t − s)α−1|f (s, y(s))|ds+

10(α)

∫ T

0(T − s)α−1|f (s, y(s))|ds+ 2|g(y)| + |yT |

≤M0(α)

∫ t

0(t − s)α−1ds+

M0(α)

∫ T

0(T − s)α−1ds+ 2M1 + yT

≤M

α0(α)Tα +

Mα0(α)

Tα + 2M1 + |yT |.

Thus

‖F(y)‖∞ ≤2MTα

0(α + 1)+ 2M1 + |yT | := `.

Step 3: F maps bounded sets into equicontinuous sets of C([0, T ],R).Let t1, t2 ∈ (0, T ], t1 < t2, Bη∗ be a bounded set of C([0, T ],R) as in Step 2, and let y ∈ Bη∗ . Then

|F(y)(t2)− F(y)(t1)| =∣∣∣∣ 10(α)

∫ t1

0[(t2 − s)α−1 − (t1 − s)α−1]f (s, y(s))ds+

10(α)

∫ t2

t1(t2 − s)α−1f (s, y(s))ds

∣∣∣∣+(t2 − t1)T0(α)

∫ T

0(T − s)α−1|f (s, y(s))|ds+

(t2 − t1)T|g(y)| +

t2 − t1T|yT |

≤M0(α)

∫ t1

0[(t2 − s)α−1 − (t1 − s)α−1]ds

+M0(α)

∫ t2

t1(t2 − s)α−1ds+M

t2 − t1T0(α)

∫ T

0(T − s)α−1ds+

t2 − t1TM1 +

t2 − t1T|yT |

≤M

0(α + 1)[(t2 − t1)α + tα1 − t

α2 ] +

M0(α + 1)

(t2 − t1)α

+Mt2 − t1T0(α)

Tα +t2 − t1TM1 +

t2 − t1T|yT |.

M. Benchohra et al. / Nonlinear Analysis 71 (2009) 2391–2396 2395

As t1 −→ t2, the right-hand side of the above inequality tends to zero. As a consequence of Steps 1 to 3 together with theArzelá–Ascoli theorem, we can conclude that F : C([0, T ],R) −→ C([0, T ],R) is completely continuous.Step 4: A priori bounds.Now it remains to show that the set

E = {y ∈ C(J,R) : y = λF(y) for some 0 < λ < 1}

is bounded.Let y ∈ E , then y = λF(y) for some 0 < λ < 1. Thus, for each t ∈ J we have

F(y)(t) =λ

0(α)

∫ t

0(t − s)α−1f (s, y(s))ds−

λtT0(α)

∫ T

0(T − s)α−1f (s, y(s))ds− λ

(tT− 1

)g(y)+ λ

tTyT .

This implies by (H4) and (H5) that for each t ∈ J we have

|F(y)(t)| ≤M0(α)

∫ t

0(t − s)α−1ds+

M0(α)

∫ T

0(T − s)α−1ds+ 2M1 + |yT |

≤M

α0(α)Tα +

Mα0(α)

Tα + 2M1 + |yT |.

Thus for every t ∈ [0, T ], we have

‖F(y)‖∞ ≤2MTα

0(α + 1)+ 2M1 + |yT | := R.

This shows that the set E is bounded. As a consequence of Schaefer’s fixed point theorem, we deduce that F has a fixedpoint which is a solution of the problem (1)–(2). �

4. An example

In this section we give an example to illustrate the usefulness of our main results. Let us consider the following fractionalboundary value problem,

cDαy(t) =e−t |y(t)|

(9+ et)(1+ |y(t)|), t ∈ J := [0, 1], 1 < α ≤ 2, (7)

y(0) =n∑i=1

ciy(ti), y(1) = 0, (8)

where 0 < t1 < t2 < · · · < tn < 1ci, i = 1, . . . , n are given positive constants with∑ni=1 ci <

45 . Set

f (t, x) =e−tx

(9+ et)(1+ x), (t, x) ∈ J × [0,∞),

and

g(y) =n∑i=1

ciy(ti).

Let x, y ∈ [0,∞) and t ∈ J . Then we have

|f (t, x)− f (t, y)| =e−t

(9+ et)

∣∣∣∣ x1+ x

−y1+ y

∣∣∣∣=

e−t |x− y|(9+ et)(1+ x)(1+ y)

≤e−t

(9+ et)|x− y|

≤110|x− y|.

Hence the condition (H1) holds with k = 110 . Also we have

|g(x)− g(y)| ≤n∑i=1

ci|x− y|.

2396 M. Benchohra et al. / Nonlinear Analysis 71 (2009) 2391–2396

Hence (H2) is satisfied with k∗ =∑ni=1 ci. We shall check that condition (6) is satisfied with T = 1. Indeed

2kTα

0(α + 1)+ k∗ =

150(α + 1)

+

n∑i=1

ci < 1⇔ 0(α + 1) > 1, (9)

which is satisfied for each α ∈ (1, 2]. Then by Theorem 3.5 the problem (7)–(8) has a unique solution on [0, 1]

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