The method of upper and lower solutions and impulsive fractional differential inclusions

Nonlinear Analysis: Hybrid Systems 3 (2009) 433–440

Contents lists available at ScienceDirect

Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs

The method of upper and lower solutions and impulsive fractional differential inclusions Mouffak Benchohra ∗ , Samira Hamani Laboratoire de Mathématiques, Université de Sidi Bel-Abbès, B.P. 89, 22000, Sidi Bel-Abbès, Algérie

article

info

Article history: Received 21 February 2009 Accepted 23 February 2009 Keywords: Initial value problem Fractional differential inclusions Impulses Caputo fractional derivative Fractional integral Lower and upper solution Fixed point

abstract In this paper the concept of lower and upper solutions combined with the fixed point theorem of Bohnnenblust–Karlin is used to investigate the existence of solutions for a class of the initial value problem for impulsive differential inclusions involving the Caputo fractional derivative. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction This paper is concerned the existence of solutions for the initial value problems (IVP for short), for impulsive fractional order 0 < α ≤ 1 differential inclusions c

Dα y(t ) ∈ F (t , y),

∆y|t =tk = Ik (y(tk )), −

for each , t ∈ J = [0, T ], t 6= tk , k = 1, . . . , m, k = 1, . . . , m,

y(0) = y0 , α

(1) (2) (3)

where D is the Caputo fractional derivative, F : J × R → P (R) is a multivalued map (P (R) is the family of all nonempty subsets of R), Ik : R → R, k = 1, . . . , m and y0 ∈ R, 0 = t0 < t1 < · · · < tm < tm+1 = T , ∆y|t =tk = y(tk+ )− y(tk− ), y(tk+ ) = limh→0+ y(tk + h) and y(tk− ) = limh→0− y(tk + h) represent the right and left limits of y(t ) at t = tk , k = 1, . . . , m. Differential equations of fractional order have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetic, etc. (see [1–7]). There has been a significant development in fractional differential equations in recent years; see the monographs of Lakshmikantham et al. [8], Kilbas et al. [9], Kiryakova [10], Miller and Ross [11], Samko et al. [12] and the papers of Agarwal et al. [13,14], Belarbi et al. [15,16], Benchohra et al. [17–20], Delbosco and Rodino [21], Diethelm et al. [1,22,23], Kilbas and Marzan [24], Mainardi [5], Podlubny et al. [25] and Zhang [26] and the references therein. Applied problems require definitions of fractional derivatives allowing the utilization of physically interpretable initial conditions, which contain y(0), y0 (0), etc. the same requirements of boundary conditions. Caputo’s fractional derivative satisfies these demands. For more details on the geometric and physical interpretation for fractional derivatives of both c

∗

Corresponding author. Fax: +213 48 54 43 44. E-mail addresses: [email protected] (M. Benchohra), [email protected] (S. Hamani).

1751-570X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2009.02.009

434

M. Benchohra, S. Hamani / Nonlinear Analysis: Hybrid Systems 3 (2009) 433–440

the Riemann–Liouville and Caputo types see [27,28]. The web site http://people.tuke.sk/igor.podlubny/, authored by Igor Podlubny contains more information on fractional calculus and its applications, and hence it is very useful for those that are interested in this field. Integer order impulsive differential equations have become important in recent years as mathematical models of phenomena in both the physical and social sciences. There has a significant development in impulsive theory especially in the area of impulsive differential equations with fixed moments; see for instance the monographs by Benchohra et al. [29], Lakshmikantham et al. [30], and Samoilenko and Perestyuk [31] and the references therein. In [32], Benchohra and Slimani have initiated the study of fractional differential equations with impulses. The method of upper and lower solutions plays an important role in the investigation of solutions for differential equations and inclusions. See the monographs by Benchohra et al. [29], Heikkila and Lakshmikantham [33], Ladde et al. [34] and the references therein. By means of the concept of upper and lower solutions combined with fixed point theorem of Bohnnenblust–Karlin, we present an existence result for the problem (1)–(3). This paper initiates the application of the upper and lower solution method for impulsive fractional differential inclusions at fixed moments of impulse. 2. Preliminaries In this section, we introduce notations, definitions, and preliminary facts that will be used in the remainder of this paper. Let [a, b] be a compact interval. C ([a, b], R) be the Banach space of all continuous functions from [a, b] into R with the norm

kyk∞ = sup{|y(t )| : a ≤ t ≤ b}, and we let L1 ([a, b], R) denote the Banach space of functions y : [a, b] −→ R that are Lebesgue integrable with norm b

Z

|y(t )|dt .

kykL1 = a

AC (J , R) is the space of functions y : J → R, which are absolutely continuous. Let (X , k · k) be a Banach space. let Pcl (X ) = {Y ∈ P (X ) : Y closed}, Pb (X ) = {Y ∈ P (X ) : Y bounded}, Pcp (X ) = {Y ∈ P (X ) : Y compact} and Pcp,c (X ) = {Y ∈ P (X ) : Y compact and convex}. A multivalued map G : X → P (X ) is convex (closed) valued if G(x) is convex (closed) for all x ∈ X . G is bounded on bounded sets if G(B) = ∪x∈B G(x) is bounded in X for all B ∈ Pb (X ) (i.e. supx∈B {sup{|y| : y ∈ G(x)}} < ∞). G is called upper semi-continuous (u.s.c.) on X if for each x0 ∈ X , the set G(x0 ) is a nonempty closed subset of X , and if for each open set N of X containing G(x0 ), there exists an open neighborhood N0 of x0 such that G(N0 ) ⊆ N. G is said to be completely continuous if G(B ) is relatively compact for every B ∈ Pb (X ). If the multivalued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph (i.e. xn −→ x∗ , yn −→ y∗ , yn ∈ G(xn ) imply y∗ ∈ G(x∗ )). G has a fixed point if there is x ∈ X such that x ∈ G(x). The fixed point set of the multivalued operator G will be denoted by FixG. A multivalued map G : J → Pcl (R) is said to be measurable if for every y ∈ R, the function t 7−→ d(y, G(t )) = inf{|y − z | : z ∈ G(t )} is measurable. For more details on multivalued maps see the books of Aubin and Frankowska [35], Deimling [36] and Hu and Papageorgiou [37]. Definition 2.1. A multivalued map F : J × R → P (R) is said to be L1 -Carathéodory if (i) t 7−→ F (t , u) is measurable for each u ∈ R; (ii) u − 7 → F (t , u) is upper semicontinuous for almost all t ∈ J . (iii) for each q > 0, there exists ϕq ∈ L1 ([0, 1], R+ ) such that

kF (t , u)kP = sup{|v| : v ∈ F (t , u)} ≤ ϕq (t ) for all |u| ≤ q and for a.e. t ∈ J . For each y ∈ C (J , R), define the set of selections of F by SF ,y = {v ∈ L1 (J , R) : v(t ) ∈ F (t , y(t )) a.e. t ∈ J }. Let (X , d) be a metric space induced from the normed space (X , | · |). Consider Hd : P (X ) × P (X ) −→ R+ ∪ {∞} given by





Hd (A, B) = max sup d(a, B), sup d(A, b) , a∈A

b∈B

where d(A, b) = infa∈A d(a, b), d(a, B) = infb∈B d(a, b). Then (Pb,cl (X ), Hd ) is a metric space and (Pcl (X ), Hd ) is a generalized metric space (see [38]). Lemma 2.2 (Bohnenblust–Karlin [39]). Let X be a Banach space and K ∈ Pcl,c (X ) and suppose that the operator G : K → Pcl,c (K ) is upper semicontinuous and the set G(K ) is relatively compact in X . Then G has a fixed point in K .

M. Benchohra, S. Hamani / Nonlinear Analysis: Hybrid Systems 3 (2009) 433–440

435

Definition 2.3 ([9,40]). The fractional (arbitrary) order integral of the function h ∈ L1 ([a, b], R+ ) of order α ∈ R+ is defined by Iaα h(t ) =

t

Z a

(t − s)α−1 h(s)ds, 0 (α)

where 0 is the gamma function. When a = 0, we write I α h(t ) = h(t ) ∗ ϕα (t ), where ϕα (t ) = for t ≤ 0, and ϕα → δ(t ) as α → 0, where δ is the delta function.

t α−1

0 (α)

for t > 0, and ϕα (t ) = 0

Definition 2.4 ([9,40]). For a function h given on the interval [a, b], the α th Riemann–Liouville fractional-order derivative of h, is defined by

(Dαa+ h)(t )

=

1



0 (n − α)

d

n Z

dt

t

(t − s)n−α−1 h(s)ds. a

Here n = [α] + 1 and [α] denotes the integer part of α . Definition 2.5 ([9]). For a function h given on the interval [a, b], the Caputo fractional order derivative of h, is defined by

(c Dαa+ h)(t ) =

t

Z

1

0 (n − α)

(t − s)n−α−1 h(n) (s)ds,

a

where n = [α] + 1. 3. Main result Consider the following space PC (J , R) = {y : J → R : y ∈ C ((tk , tk+1 ], R), k = 0, . . . , m + 1 and there exist y(tk− ) and y(tk+ ), k = 1, . . . , m with y(tk− ) = y(tk )}. PC (J , R) is a Banach space with norm

kykPC = sup{|y(t )| : 0 ≤ t ≤ T }. Set J := [0, T ] \ {t1 , . . . , tm }. 0

Definition 3.1. A function y ∈ PC (J , R) k=0 AC ((tk , tk+1 ), R) is said to be a solution of (1)–(3) if there exists a function v ∈ L1 (J , R) with v(t ) ∈ F (t , y(t )), for a.e. t ∈ J, such that the differential equation c Dα y(t ) = v(t ) on J 0 , and conditions

T Sm

∆y|t =tk = Ik (y(tk− )),

k = 1, . . . , m,

and y(0) = y0 are satisfied. Definition 3.2. A function u ∈ PC (J , R) k=0 AC ((tk , tk+1 ), R) is said to be a lower solution of (1)–(3) if there exists a function v1 ∈ L1 (J , R) with v1 (t ) ∈ F (t , u(t )), for a.e. t ∈ J, c Dα u(t ) ≤ v1 (t ) on J 0 and ∆u|t =tk ≤ Ik (u(tk− )), k = 1, . . . , m, and u(0) ≤ y0 . T Sm Similarly, a function w ∈ PC (J , R) k=0 AC ((tk , tk+1 ), R) is said to be a upper solution of (1)–(3) if there exists a function v2 ∈ L1 (J , R) with v2 (t ) ∈ F (t , w(t )), for a.e. t ∈ J c Dα w(t ) ≥ v2 (t ) on J 0 and ∆w|t =tk ≥ Ik (w(tk− )), k = 1, . . . , m, and w(0) ≥ y0 .

T Sm

For the existence of solutions for the problem (1)–(3), we need the following auxiliary lemmas: Lemma 3.3 (Lemma 2.22 [9]). Let α > 0, then the differential equation c

D α h( t ) = 0

has solutions h(t ) = c0 + c1 t + c2 t 2 + · · · + cn−1 t n−1 , ci ∈ R, i = 0, 1, 2, . . . , n − 1, n = [α] + 1. Lemma 3.4 (Lemma 2.22 [9]). Let α > 0, then I α c Dα h(t ) = h(t ) + c0 + c1 t + c2 t 2 + · · · + cn−1 t n−1 for some ci ∈ R, i = 0, 1, 2, . . . , n − 1, n = [α] + 1.

436

M. Benchohra, S. Hamani / Nonlinear Analysis: Hybrid Systems 3 (2009) 433–440

As a consequence of Lemmas 3.3 and 3.4 we have the following result which is useful in what follows. The proof may be found in [32]. For completeness we present it. Lemma 3.5. Let 0 < α ≤ 1 and let ρ ∈ PC (J , R). A function y is a solution of the fractional integral equation

Z t  1   (t − s)α−1 ρ(s)ds if t ∈ [0, t1 ], y + 0   0 (α) 0  Z t k k Z X 1 X ti 1 y(t ) = α−1  y + ( t − s ) ρ( s ) ds + (t − s)α−1 ρ(s)ds + Ii (y(ti− )), 0 i   0 (α) 0 (α)  t t k i−1 i=1 i =1  if t ∈ (tk , tk+1 ], k = 1, . . . , m

(4)

if and only if y is a solution of the fractional IVP c

Dα y(t ) = ρ(t ),

for each, t ∈ J 0 ,

∆y|t =tk = Ik (y(tk )),

(5)

k = 1, . . . , m,

−

(6)

y(0) = y0 .

(7)

Proof. Assume y satisfies (5)–(7). If t ∈ [0, t1 ] then c

Dα y(t ) = ρ(t ).

Lemma 3.4 implies y(t ) = y0 +

Z

1

0 (α)

t

(t − s)α−1 ρ(s)ds.

0

If t ∈ (t1 , t2 ] then Lemma 3.4 implies y(t ) = y(t1+ ) +

t

Z

1

0 (α)

(t − s)α−1 ρ(s)ds

t1

= ∆y|t =t1 + y(t1− ) +

1

0 (α) Z

0 (α)

(t − s)α−1 ρ(s)ds

t1 t1

1

= I1 (y(t1− )) + y0 +

t

Z

(t1 − s)α−1 ρ(s)ds +

0

1

0 (α)

Z

t

(t − s)α−1 ρ(s)ds.

t1

If t ∈ (t2 , t3 ] then again Lemma 3.4 we get y(t ) = y(t2+ ) +

Z

1

0 (α)

t

(t − s)α−1 ρ(s)ds

t2

= ∆y|t =t2 + y(t2 ) + −

1

0 (α)

Z

t

(t − s)α−1 ρ(s)ds

t2

Z t1 1 = I2 (y(t2 )) + I1 (y(t1 )) + y0 + (t1 − s)α−1 ρ(s)ds 0 (α) 0 Z t2 Z t 1 1 + (t2 − s)α−1 ρ(s)ds + (t − s)α−1 ρ(s)ds. 0 (α) t1 0 (α) t2 −

−

If t ∈ (tk , tk+1 ] then again from Lemma 3.4 we get (4). Conversely, assume that y satisfies the impulsive fractional integral equation (4). If t ∈ [0, t1 ] then y(0) = y0 and using the fact that c Dα is the left inverse of I α we get c

Dα y(t ) = ρ(t ),

for each t ∈ [0, t1 ].

If t ∈ [tk , tk+1 ), k = 1, . . . , m and using the fact that c Dα C = 0, where C is a constant, we get c

Dα y(t ) = ρ(t ),

for each t ∈ [tk , tk+1 ).

Also, we can easily show that

∆y|t =tk = Ik (y(tk− )),

k = 1, . . . , m.

M. Benchohra, S. Hamani / Nonlinear Analysis: Hybrid Systems 3 (2009) 433–440

437

For the study of this problem we first list the following hypotheses: (H1) The function F : [0, 1] × R → Pcp,c (R) is L1 -Carathéodory; (H2) There exist u and w ∈ PC ∩ AC ((tk , tk+1 ), R), k = 0, . . . , m, lower and upper solutions for the problem (1)–(3) such that u(t ) ≤ w(t ) for each t ∈ [0, T ]; (H3) u(tk+ ) ≤

min

− − y∈[u(tk ),w(tk )]

Ik (y) ≤

max

− − y∈[u(tk ),w(tk )]

Ik (y) ≤ w(tk+ ),

k = 1, . . . , m.

(H4) there exists l ∈ L1 (J , R+ ) such that Hd (F (t , y), F (t , y)) ≤ l(t )|y − y| for every y, y ∈ R, and d(0, F (t , 0)) ≤ l(t ),

a.e. t ∈ J . 

Theorem 3.6. Assume that hypotheses (H1)–(H4) hold. Then the problem (1)–(3) has at least one solution y such that u(t ) ≤ y(t ) ≤ w(t ) for all t ∈ J . Proof. Transform the problem (1)–(3) into a fixed point problem. Consider the following modified problem c

Dα y(t ) ∈ F (t , τ (y(t ))),

t ∈ J , t 6= tk , k = 1, . . . , m, 0 < α ≤ 1,

∆y|t =tk = Ik (τ (tk , y(tk ))), −

−

k = 1, . . . , m,

(8) (9)

y(0) = y0 ,

(10)

where τ : PC (J , R) −→ PC (J , R) be the truncation operator defined by u(t ), y(t ) < u(t ) (τ y)(t ) = y(t ), u(t ) ≤ y(t ) ≤ w(t ) w(t ), y(t ) > w(t ).

(

A solution to (9)–(10) is a fixed point of the operator G : PC (J , R) −→ P (PC (J , R)) defined by:

G(y) =

              

h ∈ PC (J , R) :

1

X Z

tk

  (tk − s)α−1 v(s)ds   0 (α) 0
h(t ) = y0 +

0
where

v ∈ S˜F1,τ y = {v ∈ SF1,τ y : v(t ) ≥ v1 (t ) on A1 and v(t ) ≤ v2 (t ) on A2 }, SF1,τ y = {v ∈ L1 (J , R) : v(t ) ∈ F (t , (τ y)(t )) for t ∈ J }, A1 = {t ∈ J : y(t ) < u(t ) ≤ w(t )},

Remark 3.7.

A2 = {t ∈ J : u(t ) ≤ w(t ) < y(t )}. 

(i) For each y ∈ PC (J , R), the set S˜F1,τ y is nonempty. In fact, (H1 ) implies that there exists v3 ∈ SF1,τ y , so we set

v = v1 χA1 + v2 χA2 + v3 χA3 , where A3 = {t ∈ J : u(t ) ≤ y(t ) ≤ w(t )}. Then, by decomposability, v ∈ S˜F1,τ y . (ii) By the definition of τ it is clear that F (., τ y(.)) is an L1 -Carathéodory multi-valued map with compact convex values and there exists φ1 ∈ L1 (J , R+ ) such that

kF (t , τ y(t ))kP ≤ φ1 (t ) for each y ∈ R.

438

M. Benchohra, S. Hamani / Nonlinear Analysis: Hybrid Systems 3 (2009) 433–440

(iii) By the definition of τ and from (H3) we have u(tk+ ) ≤ Ik (τ (tk , y(tk ))) ≤ w(tk+ ),

k = 1, . . . , m.

(11)

Set

η = |y0 | +

m m kφ1 kL1 X kφ1 kL1 T α X + max{|u(tk+ )|, |w(tk+ )|}, (tk − tk−1 )α + 0 (α + 1) k=1 0 (α + 1) k =1

and D = {y ∈ PC (J , R) : kykPC ≤ η}. Clearly D is a closed convex subset of PC (J , R) and that G maps D into D. We shall show that D satisfies the assumptions of Lemma 2.2. The proof will be given in several steps. Step 1: G(y) is convex for each y ∈ D. Indeed, if h1 , h2 belong to G(y), then there exist v1 , v2 ∈ S˜F1,τ y such that for each t ∈ J we have hi ( t ) = y 0 +

tk

X Z

1

0 (α)

0
(tk − s)α−1 vi (s)ds +

tk−1

t

Z

1

0 (α)

(t − s)α−1 vi (s)ds +

X

Ik (τ (tk− , y(tk− ))),

i = 1, 2.

0
tk

Let 0 ≤ d ≤ 1. Then, for each t ∈ J, we have

(dh1 + (1 − d)h2 )(t ) =

0 (α)

Since ˜

SF1,τ y

0 < tk < t

Z

1

−

tk

X Z

1

0 (α)

t

(tk − s)α−1 [dv1 (s) + (1 − d)v2 (s)]ds

tk−1

(t − s)α−1 [dv1 (s) + (1 − d)v2 (s)]ds +

X

Ik (τ (tk− , y(tk− ))).

0
tk

is convex (because F has convex values), we have

dh1 + (1 − d)h2 ∈ G(y). Step 2: G(D) is bounded. This is clear since G(D) ⊂ D and D is bounded. Step 3: N (D) is equicontinuous. Let ξ1 , ξ2 ∈ J , ξ1 < ξ2 , y ∈ D and h ∈ G(y), then

|h(ξ2 ) − h(ξ1 )| =

X

0 (α)

0
+

1

ξ2

Z

0 (α)

ξ1

tk

Z

1

0 (α)

tk−1

|(ξ2 − s)α−1 kv(s)|ds +

X

ξ1

Z

1

(tk − s)α−1 |v(s)|ds +

|(ξ2 − s)α−1 − (ξ1 − s)α−1 kv(s)|ds

0

|Ik (τ (tk− , y(tk− )))|

0
X Z tk ψ(kyk∞ )T ψ(kyk∞ )p0 [2(ξ2 − ξ1 )α + ξ2α − ξ1α ] ≤ (tk − s)α−1 ds + 0 (α + 1) 0
As ξ1 −→ ξ2 , the right-hand side of the above inequality tends to zero. As a consequence of Steps 1 to 3 together with the Arzelá–Ascoli theorem,we can conclude that G : D −→ P (D) is compact. Step 4: G has a closed graph. Let yn → y∗ , hn ∈ G(yn ) and hn → h∗ . We need to show that h∗ ∈ G(y∗ ). hn ∈ G(yn ) means that there exists vn ∈ S˜F1,τ yn such that, for each t ∈ J, hn (t ) = y0 +

1

0 (α)

tk

X Z 0
(tk − s)α−1 vn (s)ds +

tk−1

We must show that there exists v∗ ∈ ˜

SF1,τ y∗

h∗ (t ) = y0 +

1

0 (α)

X Z 0 < tk < t

tk

1

0 (α)

t

Z

(t − s)α−1 vn (s)ds +

X

Ik (τ (tk− , yn (tk− ))).

0 < tk < t

tk

such that, for each t ∈ J,

(tk − s)α−1 v∗ (s)ds +

tk−1

1

0 (α)

Z

t tk

(t − s)α−1 v∗ (s)ds +

X

Ik (τ (tk− , y∗ (tk− ))).

0
Since F (t , ·) is upper semicontinuous, then for every ε > 0, there exist n0 () ≥ 0 such that for every n ≥ n0 , we have

vn (t ) ∈ S˜F1,τ yn ⊂ F (t , τ y∗ (t )) + ε B(0, 1),

a.e. t ∈ J .

M. Benchohra, S. Hamani / Nonlinear Analysis: Hybrid Systems 3 (2009) 433–440

439

Since F (·, ·) has compact values, then there exists a subsequence vnm (·) such that

vnm (·) → v∗ (·) as m → ∞ and

v∗ (t ) ∈ F (t , τ y∗ (t )),

a.e. t ∈ J .

For every w ∈ F (t , τ y∗ (t )), we have

|vnm (t ) − v∗ (t )| ≤ |vnm (t ) − w| + |w − v∗ (t )|. Then

|vnm (t ) − v∗ (t )| ≤ d(vnm (t ), F (t , τ y∗ (t ))). By an analogous relation, obtained by interchanging the roles of vnm and v∗ , it follows that

|vnm (t ) − v∗ (t )| ≤ Hd (F (t , τ yn (t )), F (t , τ y∗ (t ))) ≤ l(t )kyn − y∗ k∞ . Then

|hn (t ) − h∗ (t )| ≤

X Z

1

tk

1

(tk − s)α−1 |vnm (s) − v∗ (s)|ds −

0 (α) 0
Z

t

(t − s)α−1 |vnm (s) − v∗ (s)|

tk

0
≤

mT α

Z

T

l(s)dskynm − y∗ k∞ +

Tα

Z

T

0 (α + 1) 0 0 (α + 1) 0 X − − + |Ik (τ (tk , ynm (tk ))) − Ik (τ (tk− , y∗ (tk− )))|.

l(s)dskynm − y∗ k∞

0
Hence

khnm − h∗ k∞ ≤

mT α

Z

T

l(s)dskynm − y∗ k∞ +

Tα

Z

T

l(s)dskynm − y∗ k∞ 0 (α + 1) 0 0 (α + 1) 0 m X + |Ik (τ (tk− , ynm (tk− ))) − Ik (τ (tk− , y∗ (tk− )))| → 0 as m → ∞. k=1

Step 5: The solution y of (8)–(10) satisfies u(t ) ≤ y(t ) ≤ w(t ) for all t ∈ J . Let y be the above solution to (8)–(10). We prove that y(t ) ≤ w(t ) for all t ∈ J . Assume that y − w attains a positive maximum on [tk+ , tk−+1 ] at t k ∈ [tk+ , tk−+1 ] for some k = 0, . . . , m; that is,

(y − w)(t k ) = max{y(t ) − w(t ) : t ∈ [tk+ , tk−+1 ]} > 0,

for some k = 0, . . . , m.

We distinguish the following cases. Case 1. If t k ∈ (tk+ , tk−+1 ) there exists tk∗ ∈ (tk+ , tk−+1 ) such that y(tk∗ ) − w(tk∗ ) ≤ 0,

(12)

and y(t ) − w(t ) > 0,

for all t ∈ (tk∗ , t k ].

(13)

By the definition of τ one has c

Dα y(t ) ∈ F (t , w(t ))

for all t ∈ [tk∗ , t k ].

An integration on [tk∗ , t ] for each t ∈ [tk∗ , t k ] yields y(t ) − y(tk∗ ) =

1

0 (α)

Z

t tk∗

(t − s)α−1 v(s)ds,

(14)

where v(t ) ∈ F (t , w(t )). From (14) and using the fact that w is an upper solution to (1)–(3) we get y(t ) − y(tk∗ ) ≤ w(t ) − w(tk∗ ).

(15)

440

M. Benchohra, S. Hamani / Nonlinear Analysis: Hybrid Systems 3 (2009) 433–440

Thus from (12), (13) and (15) we obtain the contradiction 0 < y(t ) − w(t ) ≤ y(tk∗ ) − w(tk∗ ) ≤ 0,

for all t ∈ [tk∗ , t k ].

Case 2. If t k = tk+ , k = 1, . . . , , m. Then

w(tk+ ) < Ik (τ (tk− , y(tk− ))) < w(tk+ ) which is a contradiction. Thus y(t ) ≤ w(t ) for all t ∈ [0, T ]. Analogously, we can prove that y(t ) ≥ u(t ),

for all t ∈ [0, T ].

This shows that the problem (8)–(10) has a solution in the interval [u, v] which is a solution of (1)–(3). References [1] K. Diethelm, A.D. Freed, On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, in: F. Keil, W. Mackens, H. Voss, J. Werther (Eds.), Scientifice Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, Springer-Verlag, Heidelberg, 1999, pp. 217–224. [2] L. Gaul, P. Klein, S. Kempfle, Damping description involving fractional operators, Mech. Syst. Signal Process. 5 (1991) 81–88. [3] W.G. Glockle, T.F. Nonnenmacher, A fractional calculus approach of self-similar protein dynamics, Biophys. J. 68 (1995) 46–53. [4] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. [5] F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien, 1997, pp. 291–348. [6] F. Metzler, W. Schick, H.G. Kilian, T.F. Nonnenmacher, Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phys. 103 (1995) 7180–7186. [7] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, London, 1974. [8] V. Lakshmikantham, S. Leela, J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, 2009. [9] A.A. Kilbas, Hari M. Srivastava, Juan J. Trujillo, Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006. [10] V. Kiryakova, Generalized Fractional Calculus and Applications, in: Pitman Research Notes in Mathematics Series, vol. 301, Longman Scientific & Technical, Harlow, New York, 1994. [11] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993. [12] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993. [13] R.P. Agarwal, M. Benchohra, S. Hamani, Boundary value problems for fractional differential equations, Georgian. Math. J. (in press). [14] R.P. Agarwal, M. Benchohra, S. Hamani, A survey on existence result for boundary value problems of nonlinear fractional differential equations and inclusions, Acta. Appl. Math. (in press). [15] A. Belarbi, M. Benchohra, S. Hamani, S.K. Ntouyas, Perturbed functional differential equations with fractional order, Commun. Appl. Anal. 11 (3–4) (2007) 429–440. [16] A. Belarbi, M. Benchohra, A. Ouahab, Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces, Appl. Anal. 85 (2006) 1459–1470. [17] M. Benchohra, J.R. Graef, S. Hamani, Existence results for boundary value problems of nonlinear fractional differential equations with integral conditions, Appl. Anal. 87 (7) (2008) 851–863. [18] M. Benchohra, S. Hamani, S.K. Ntouyas, Boundary value problems for differential equations with fractional order, Surv. Math. Appl. 3 (2008) 1–12. [19] M. Benchohra, J. Henderson, S.K. Ntouyas, A. Ouahab, Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl. 338 (2) (2008) 1340–1350. [20] M. Benchohra, J. Henderson, S.K. Ntouyas, A. Ouahab, Existence results for fractional functional differential inclusions with infinite delay and application to control theory, Fract. Calc. Appl. Anal. 11 (1) (2008) 35–56. [21] D. Delbosco, L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl. 204 (1996) 609–625. [22] K. Diethelm, N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002) 229–248. [23] K. Diethelm, G. Walz, Numerical solution of fractional order differential equations by extrapolation, Numer. Algorithms 16 (1997) 231–253. [24] A.A. Kilbas, S.A. Marzan, Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, Differential Equations 41 (2005) 84–89. [25] I. Podlubny, I. Petraš, B.M. Vinagre, P. O’Leary, L. Dorčak, Analogue realizations of fractional-order controllers. Fractional order calculus and its applications, Nonlinear Dyn. 29 (2002) 281–296. [26] S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional diffrential equations, Electron. J. Differential Equations 36 (2006) 1–12. [27] N. Heymans, I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann–Liouville fractional derivatives, Rheol. Acta 45 (5) (2006) 765–772. [28] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal. 5 (2002) 367–386. [29] M. Benchohra, J. Henderson, S.K. Ntouyas, Impulsive Differential Equations and Inclusions, vol. 2, Hindawi Publishing Corporation, New York, 2006. [30] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differntial Equations, Worlds Scientific, Singapore, 1989. [31] A.M. Samoilenko, N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. [32] M. Benchohra, B.A. Slimani, Impulsive fractional differential equations, Electron. J. Differential Equations 10 (2009) 1–11. [33] S. Heikkila, V. Lakshmikantham, Monotone Iterative Technique for Nonlinear Discontinuous Differential Equations, Marcel Dekker Inc., New York, 1994. [34] G.S. Ladde, V. Lakshmikanthan, A.S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman Advanced Publishing Program, London, 1985. [35] J.P. Aubin, H. Frankowska, Set-Valued Analysis, Birkhauser, Boston, 1990. [36] K. Deimling, Multivalued Differential Equations, Walter De Gruyter, Berlin-New York, 1992. [37] Sh. Hu, N. Papageorgiou, Handbook of Multivalued Analysis, Theory I, Kluwer, Dordrecht, 1997. [38] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer, Dordrecht, The Netherlands, 1991. [39] H.F. Bohnenblust, S. Karlin, On a theorem of ville. Contribution to the theory of games, in: Annals of Mathematics Studies, vol. 24, Priceton University Press, Princeton. N. G., 1950, pp. 155–160. [40] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.