Journal of the Egyptian Mathematical Society (2014) xxx, xxx-xxx

Egyptian Mathematical Society Journal of the Egyptian Mathematical Society

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ORIGINAL ARTICLE

Continuous and integrable solutions of a nonlinear Cauchy problem of fractional order with nonlocal conditions

F.M. Gaafar

Faculty of Science, Damanhour University, Damanhour, Egypt

Received 11 August 2013; revised 25 November 2013; accepted 17 December 2013

KEYWORDS

Nonlinear fractional problem; Riemann-Liouville fractional

derivative;

Weighted Cauchy problem; Nonlocal condition; Schauder fixed point theorem

Abstract In this article, we discuss the existence of at least one solution as well as uniqueness for a nonlinear fractional differential equation with weighted initial data and nonlocal conditions. The existence of at least one L1 and continuous solution will be proved under the Caratheodory conditions via a classical fixed point theorem of Schauder. An example is also given to illustrate the efficiency of the main theorems.

MATHEMATICS SUBJECT CLASSIFICATION: 26A33; 34K37; 34A08; 45E10; 47H10

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1. Introduction

Fractional differential equations have gained considerable importance due to their application in various sciences, such as physics, mechanics, chemistry, engineering, etc. In the recent years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives, see the monographs of Kilbas et al. [1], Miller and Ross [2], Podlubny [3], and the papers [4-16] and the references therein.

Let I — (0, T], L1 = L1(0, T] be the space of Lebesgue integrable functions on I. and C(0, T] be the space of continuous functions defined on I.

E-mail address: fatmagaafar2@yahoo.com

Peer review under responsibility of Egyptian Mathematical Society.

Consider the weighted nonlocal Cauchy type fractional problem

D"(p(t)u(t)) — f(t, u(t)) a.e. t 2(0, T], T < 1 (1)

lim t1_"p(t)u(t) — aju(sj), s 2 (0, T). (2)

t!0 j—1

where D" denoted the Riemann-Liouville derivative of order a 2 (0,1].

Problems with non-local conditions have been extensively studied by several authors in the last two decades. The reader is referred to [7-9,17-19] and references therein.

Nonlinear fractional differential equation with weighted initial data has been carried out by various researchers. In present, there are some papers which deal with the existence and multiplicity of solutions for weighted nonlinear fractional differential equations.

In [12] Khaled et al. studied the weighted Cauchy-type problem

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'Dau(t)—f(t, u), ( )1 t1-au(i)|i=0 = b,

t 2 (0, T]

where Da is the fractional derivative (in the sense of Riemann-Liouville) of order 0 < a < 1, f is a continuous nonlinear function.

In [10] Furati et al. studied the weighted Cauchy-type problem (I) where f(t, u) is assumed to be continuous on R+ x R and f(t,u)| 6 t1e-rtW(t)|u|m.

Also in [5] El-Sayed et al. studied the problem (I) where the function f satisfies Caratheodory conditions with growth condition. In [19] the existence and uniqueness of the solution of the problem (I) was discussed by using the method of upper and lower solutions and its associated monotone iterative.

In [16] Weia et al. studied the existence and uniqueness of the solution of the periodic boundary value problem for a fractional differential equation involving a Riemann-Liouville fractional derivative

Dau(t) — f(t, u), t 2(0, T t1-au(t)|t=0 — t1-au(t)|t—T-

by using the monotone iterative method. In [11] Jankowski discussed the existence of solutions of fractional equations ofVol-terra type with the Riemann-Liouville derivative,

iDax(t) —f(t, x(t), Rjk(t, s)x(s)ds), t 2 (0, T]

\ ^«WLo = r,

existence results are obtained by using a Banach fixed point theorem with weighted norms and by a monotone iterative method.

In [4] Belmekki et al. studied the existence and uniqueness of the solution for a class of fractional differential equations

^Dau(t)-ku(t)—f(t, u(t)), t 2(0,1]

lim t1-au(t) — u(1)

. t!0+

by using the fixed point theorem of Schaeffer and the Banach contraction principle.

In this paper we will study the existence of solutions for problem (1) and (2) with certain nonlinearities, using the equivalence of the fractional differ-integral problem with the corresponding Volterra integral equation. We prove the existence of at least L1 and continuous solutions of the problem (1) and (2) such that the function f satisfies Caratheodory conditions and

f(t,u)| 6 h(t), a.e t 2(0, T]

where h(t) is a Lebesgue function on (0, T]. Also the uniqueness of the solution will be studied.

Our problem (1) and (2) includes as a special case when p(t) — 1, the nonlocal fractional differential equation

Dau(t) — f(t, u(t)) a.e. t 2(0, T], T < 1

lim t'~a u(t) — ^^aju(xj), xj 2 (0, T).

t!0 j—1

2. Preliminaries

Definition 2.1 (see [2,3,13,14]). The Riemann-Liouville fractional integral of order a > 0 of a Lebesgue-measurable function f: R+ ! R is defined by

" f(s)ds,

ifW —FiS Î,(t -s)

when a — 0 we write ft) — ft). And we have, for a, b 2 R+,

(n) : ¿1 >¿1,

(r2) f(t)2L1, /a if(t)—cbf(t).

Definition 2.2 (see [2,3,13,14]). The Riemann-Liouville fractional derivative of order a 2 (0,1] of a Lebesgue-measurable function f: R+ ! R is defined by

Daf(t)= — 11-af t) =-1-—

DJ(t) dt f(t) C(1 - a) dt J0

f (t - s)-af(s)ds. 0

Theorem 2.1 (Schauder fixed point Theorem). Let S be a nonempty, closed , convex and bounded subset of the Banach space X and let Q : S ! S be a continuous and compact operator. Then the operator equation Qu — u has at least one fixed-point in S.

Theorem 2.2 (Kolmogorov compactness criterion [20]). Let X # Lp (0, T), 1 6 p < i. If

(i) X is bounded in Lp(0, T) and

(ii) uh ! u as h ! 0 uniformly with respect to u 2 X, then X is relatively compact in Lp(0, T) where

1 /■ t+h uh (t) — - u(s)ds. ht

Definition 2.3. A function f: I x R ! R is called Caratheodory function if:

(i) t ! f (t, u) is measurable for all u 2 R, and,

(ii) u ! f (t, u) is continuous for all t 2 I.

(iii) There exists a Lebesgue function h(t) on I, and

3. Integral equation representation

We investigate in our paper the Cauchy problem for the nonlinear fractional differential equation with the nonlocal condition with the following assumptions.

(h1) The function f : (0, T ]xR ! R is Caratheodory function.

(h2) p(t) > 0 for all 12 i and is continuous with

inf(0,T]|P(t)| — p.

(h3)Yj—1pj-1.

In this section, we present some definitions, lemmas and notation which will be used in our theorems.

Lemma 3.1. The solution of the nonlocal problem (1) and (2) can be expressed by the fractional-order integral equation

u(t) -

EaJ pi тЛ

n (s, - s)"-1

P(t) L-(P(sj) Jo Г(й

■f(s, u(s))ds

(t - s)"

■f(s, u(s))ds

P{t) Jo Г(а

where A — - prm=l pj^j

Proof. From the properties of Riemann-Liouville fractional derivative, Eq. (1) can be written as,

d-rpmt)f u(tj),

integrating from 0 to t both sides, we get

I1-ap(t)u(t) - I1-ap(t)u(t)\t=o — f'f(s, u(s))ds,

I1-ap(t)u(t) - C — i fis, u(s))ds,

operating by Iх on both sides, we have

FI1-"p(t)u(t)- FC — F+1f(t, u(t))

Ip(t)u(t)--

- F+1f(t, u(t)),

Г (a + 1) differentiate both sides, then

P(t)u(t)

"-1 r-t

t - s)"

Г (o) Jo Г (о. - 1)

t^pmt)--^+11-" í'(t - srl

Г (") Jo

and from (2) we have,

-f(s, u(s))ds,

■f(s, u(s))ds,

Г(" - 1)"

hm t1-"p(t)u(t) - — -J2aLu(sj).

Now from (4), putting t — sj, we obtain

u(Sj) - + ~Т"Т

- (Sj - s)"-1 P(sL) Г (") ' P(sJ) Jo Г(° - 1)

-f(s, u(s))ds,

Eaju(sL) - Е

1 P^s^n".)

^ Z^ nts л

(sJ- s)1

j—1 pis') Jo Г(х - 1)

From (5),we have

— -T

+ E „(sA

-f(s, u(s))ds.

H (sj - s)"-1

Г(") jtrPL}-"^) j-1 p(sj) Jo Г(" - 1)

■f(s, u(s))ds,

1 P(sL)s1

Г (s - s)"-1 -(") j-í p(s) Jo г (i - 1)

f(s; u(s))ds.

п (sj - s)"-1

C = A V aL Г(") j-1 p(sj) Jo Г(" - 1)

■f(s, u(s))ds. □

Substituting in (4) we get (3).

Now we want to prove that if u(t) satisfied (3), then limt!0+ tl~"p(t)u(t) — Yj=iaju(?j), from (3) we have,

m a s j s s a - 1 limt1-"p(t)u(t)- AJ2-T-J (}-( ( f(s,u(s))ds

t!0+ L-f p(sL) Jo Г(")

+ lim t1-" í' (t s? f(s, u(s))ds t!0+ Jo Г (o)

Up(sj) .

(sL - s)

-f(s, u(s))ds

Also from (3),

(av -,-l + и

V j-1 ^"p^j) )

-{j ^pL

y- aj I \sj

UP(sj) Jo

(sL - s)"

-f(s, u(s))ds

1 E sypj \ L-1

aj j-1 sL P(sL)

EaL PÍ sA

H (sj - s)"-1

^Е-ОЧ

j-1 P(sj)

1 P(sL) Jo Г(")

(sL - s)"-

/(s, u(s))ds

-f(s, u(s))ds.

Then the integral Eq. (3) is equivalent to the nonlocal problem (1) and (2).

4. Existence of L1 solutions

Here we study the existence of at least one L1 solution of the nonlocal Cauchy problem (1) and (2).

Definition 4.1. By a solutions of the nonlocal Cauchy problem (1) and (2) we mean a functions u 2 L1(0, T] on the interval (0, T] and this functions satisfies (1) and (2).

Theorem 4.1. Assume that the hypothesis (h\) — (h3) holds, then the nonlocal problem (1) and (2) has at least one L\ solution.

Proof. Let T be an operator defined by:

^(„-A^t'L Г (s - s)

lj_ VL_

P(t) L-1 P(sL) Jo Г(°

-f(s, u(s))ds

(t - s)"

-f(s, u(s))ds.

P(t) Jo Г("

Then from Definition 2.1 we can write it as Ata-1 m a 1

(Tu)(t)-K)ï E pL) fL, u(j)+ P1)ft,u(t)).

Let ß < a, then we can write

Ata-1 m a 1

(Tu)(t)-J{- Epj ^Asj, uj+p1) F-ßIßf(t, u(t))

1(^)16 jf X u(Xj))l

+m^ f(t,u(t)) 1

and from assumption (hi), we have

\ (T»)(o |6 ^ X i-'"h(xj)+¿a r-'m

Let M — maxfI'h(t)}.

Then from assumption (h2) we get

UTum 6 Jj^ Y—M r (Xj - s)a-'-1 ds \ (Tu)(t) 1 6 inf |p(t) 111inf |p(t) | J0 C(a - ') ds

t (f_ „\a-b-1

(t - s)

inf |p(t)| J0 C(a - b) 6 | A | ta-1j^| a| M_-

M ta-b

1 p C(a - b + 1) p C(a - b + 1)

| A | ta

| MTa-b M r-b

and ||Tu||L —

p2 j—1 C(a - b + 1) p C(a - b + 1)

Tu(t) dt

6 | a | ,| MTa-b rT 1

6 p2 j—1 C(a - b +1^0 t dt M T-b rT + 7 C(a - b + 1)/, dt | A | a, | MT2a-b M Ta-b+1

6 n2 a 1 . - N

p2 j—1 aC(a-' + 1) p C(a-' + 1)'

The last estimate shows that the operator T maps L1 into itself. Let

_| A | a, | MT2a-b M Ta-b+1

p2 1 aC(a - b + 1) p C(a - b + 1):

define the subset Br c L1(I) by Br — fu(t), t 2 I: ||u

L 6 r},

the set Br is nonempty, closed and convex.

Now let u 2 dBr, that is ||u|| — r, then T(@Br) c Br (closureofBr) if

-jMT2a-b M Ta

IITII 6 JAlv

| u 1 6 p2 j—1 aC(a - ' + 1) ' p C(a - ' + 1) and ||Tu||L 6 r, where r is given by (9). Moreover,

|f| — i Tf(s, u(s)) | ds 6 f h(s) ds —Ml ,

thus f is in L1(0, T]. Further, f is continuous in u (assumption (ii)) and Ia maps L1 (0, T] continuously into itself, then Iaf(t, u(t)) is continuous in u 2 Br, and we have Tmaps Br continuously into L1(0, T].

To prove that T is compact, we apply Theorem 2.2. So let X be a bounded set of Br. Then T(X) is bounded in L1(0, T], i.e. condition (i) of Theorem 2.2 is satisfied. To prove that (Tu)h ! Tu in L1(0, T] as h ! 0 uniformly with respect to u 2 X, we have from (7),

ll(Tu)h - Tu||l1 — / |(Tu)h(t)-(Tu)(t)|dt

(Tu)(s)ds -(Tu)(t)

|(Tu)(s)-(Tu)(t)|dsdt

j A jj a-

6 Ç |p(s )|

T 1 t+h

n (s - y)a

o C(a) sa-1 ta-1

ps)- pit)

-fy, u^dy

ds dt +

p^syfu(s))- W)f,u(t))

ds dt.

Since f 2 ¿1(0, T], then Iaf(.) 2 ¿1(0, T] and ^Iaf(t, u(t)) 2

¿1(0, T]. Moreover 2 L (0, T], so we have, (see [21])

sa-1 ta-1

ps)-W)

pis) f u(s))- pt) fu(t))

for a.e. t 2 (0, T]. Therefore by Theorem 2.2, we have T(X) is relatively compact, that is T is compact operator. Now applying Theorem 2.1, then T has a fixed point.

5. Existence of continuous solutions

Here we study the existence of unique and at least one continuous solution of the nonlocal Cauchy problem (1) and (2).

Definition 5.1. By a solutions of the nonlocal Cauchy problem (1) and (2) we mean a functions fu : t1-au(t) is continuous on the interval (0, T]} and this functions satisfies (1) and (2).

C(0, T] — fu : u(t) is continuous on (0, T] : ||u||C — niax|u(t)|}

C!-a(0, T] — {u : t1-au(t) is continuous on (0, T] with the weighted norm ||u||Cj ^ — ||t1-au(t)||C j

Theorem 5.1. Assume that the hypothesis (h1 )-(h3) holds. Then the nonlocal problem (1) and (2) has at least one solution u 2 C1-a(0, T].

Proof. Define the subset Qr c C1-a(0, T] by

Qr — fu(t)2 C1-a(0, T] : ||u(t)||c1-a(0,r] < r} where

|aj|MTа-' M T1-'

p2 j—1 C(a - b + 1) p C(a - b + 1) The set Qr is nonempty, closed and convex. □

Let T : Qr ! Qr be an operator defined by (7). For u 2 Qr, then Tis a continuous operator, i.e, if fun(t)} is a sequence in Qr converges to u(t), 8t 2(0, T], for

lim Tun (t) = —— —Í— lim

p{t) P[Sj) n!1

n (t, - s)"-' 0 C(«)

f(s, Un(s)) ds

+ —;-: lim

p(t)n!i.

(t - s)a C(«)

-f(s, Un(s)) ds,

by assumption (h1) and the Lebesgue dominated convergence Theorem we deduce that

lim Tun (t) — Tu(t).

Then T is continuous. Now from Eq. (8), let u 2 Qr, then

|t1-a(Tu)(t)| 6 HX|)|^|

< 1—1 JajM

(sj - s))

inf |p(t)| j-'inf |p(t)| J0 C(« - b)

(t - s)a

inf |p(t)| J0 C(« - b)

< ja g |a|M s

M t'-b

p j—1 p C(a - ' + 1) p C(a - ' + 1)

|A|MTa-' ^ M T1-' _ 6 p2C(a - ' + 1) |aj|+ 7 C(a - ' + 1) — r.

Then f Tu(t)} is uniformly bounded in Qr.

In what follows we show that T is a completely continuous operator.

For t1, t2 2 (0, T], t1 < t2 such that |t2 - t11 < d, from (7) we have

|t2-a (Tu)(t2)-t'-a(Tu)(t')| <

1 11 ^ Aa, n (s - s)"-'

Lp(t2) p(t')J

V- Aaj I

Up(Tj) Jo

f(s, u(s)) ds

(t2 - s)" C(a)

■ f(s, u(s)) ds

(t' - s)a

p(t') Jo 1

■ f(s, u(s)) ds

' 1 Aa, <-T<

Lp(t2) p(t')J

h psi) jo

(T- s)" C(a)

-f(s, u(s))ds

t'r [h (t2 - s)a

p( t2) Jo C(a)

T2 (t2 - s)"

(t' - s)"-

-f(s, u(s)) ds

-f(s, u(s)) ds

p(t') 7o ' !

f(s, u(s)) ds

Lp(ta) p(t')J

ÇpN Jo W

(sj - s)a

t2-a r (t' - s)a

p(t2) Jo C(a) + t2-a f2 (t2 - s)

p(t2) Jh C(a)

f(s, u(s)) ds f(s, u(s)) ds

-f(s, u(s))ds

(t' - s)a

p(t')| 7o 1

-f(s, u(s)) ds

Lp(ta) p(t')J

(sj- s)a

*'-a *'-a

+ t'-a

Lp(ia) p(t2)J 1

'p(si) Jo C(a)

p (t' - s)a-' o

-f(s, u(s)) ds

p(t2) p(t')

C(a) ' (t' - s)

f(s, u(s)) ds f(s, u(s))ds

2-a r (t2 - s)a-',, , „ ,

p(W I ~^arf(s,u(s)) ds

p(t() p(t') t

1-a 1-a t1 2 - t1 |

|A||a| iTJ (s - s)a-' p Jo C(a)

h(s) ds

|p(t2)|

(t' - s)a C(a)

-h(s) ds

+ t'-a|

p(t() p(t')

|p(t()|

(t2 - s)" C(a)

" (t' - s)a C(a)

-h(s) ds,

-h(s) ds

p(t() p(t')

'-a _ t'-"I ft'

f, j—m n (s, - s)" j-f p Jo C(")

-h(s) ds

+ t'-"

(t2 - s)"

(t' - s)" -h(s)ds

h(s) ds

p(t() p(t' )

(t' - s)"

-h(s) ds,

which can be written as

t2-a(Tu)(t2)-t'-a(Tu)(t')| < jp!)_|AijajjF'-bIbh(zj)

I A-" _ A-A-"

+---t^-j I"-b Ibh(t')+t^ Ft;bIbti h(t()

" J___L

p(t() p(t')

p(t2)-p(t') j-f p Jo - b)

11'-" - t'-"| ft' (t' -s)

o - b)

II-bIbh(t'),

^ |A||flj|M p (T, -s)"-b-' Z.^ p

'-" - "| /•tl (^ - t(-"M /'t2 (t2 -s)"-b-'

p •/t' - b)

pfe)-p(T) i i p(t2)- pcT)

t1 - s)

Jo c ( " - b) \a<

^ |A||aj|MTI-b MTz-b|t2 - t'|

+ T'-"M (t2 - t')""^ MT'-b

pC ( " - b+ ') pC(" - b+ ')

pC(" - b + ') C (" - b + ')

p(t() p(t')

Hence the class fTu (t)g is equi-continuous, by Arzela-Ascolis Theorem then fTu ( t)g is relatively compact. Since all conditions of Schauder fixed point Theorem are hold, then T has a fixed point in Qr. Therefor the nonlocal problem (1) and (2) has at least one solution u 2 C1-a( 0, T].

' " t'

' " t2

' " t'

Theorem 5.2. Let f : (0, T]x R ! R is continuous and satisfies the Lipschitz condition

Example 6.1. Consider the nonlinear fractional differential problem

f(t,ui)—f(t,u2)| 6 L|u! — u2|,L > 0, for all ui,u2 2 R. If the conditions (h2), (h3) are satisfied and

+ T < 1,

21—2appLÎ imi 1|aj|T2a—1 C(' + i)pl

then the nonlocal problem (1) and (2) has a unique solution

u 2 C1—a(0, T].

Proof. Let T be an operator defined by (7), then

T : C1—a(0, T] ! C1—a(0, T]

D^i^tiu(t)^ — sinu(t)(1 + cos2 u(t))t2 + et a.e. t 2 (0,1], limti^^u(t) — 3u(|) - 2u(!).

Observe, the above problem is a special case of (1) and (2). indeed if we put f(t, u(t)) — sinu(t)(1 + cos2 u(t))t2 + et, a — 2,P(t) — 1+?. Then we can easy check that the assumptions of Theorem 4.1 are satisfied.

Then the problem has at least one positive solution u 2 Li(0,1].

^(Tum — t^^Tvm

A ^ J| P (Sj — s)a

w^ j-i WÏÏ 1 m

-f(s, u(s)) — f(s, v(s))ds

t1—' r (t — s)'—1

W! J0 r(a)

f(s, u(s)) — f(s, v(s))|ds

LA ^ ^ [s (Sj — s)a—1

p j-1 p Jo C(a

_ ' (t — s)' p J0 C(')

|u(s) — v(s)|ds

■ |u(s) — v(s)|ds

LA vS , H s'—1(Sj — s)

j — s) 1 —'

s1—''Hs) — v(s)d

Lt1—' ft f-—1(t — s)'—1 1

Lt1—'

s1—'^u(s) — v(s)^s

Lt1—'

„£|аjl /

j-1 J0

S s'—1 (Sj — s)'

s'—1(t — s)'

IIC1_,

u — v\\

U C(2')

t2'—1r(a) ' r(2')

21—2'^kL

r(' + 2)p

This means that

\t1—'(Tu)(t) —t1—'(Tv)(t)\\

6 .......-v

C1—'-

Then by using Banach fixed point Theorem, the operator T has a unique fixed point u(t) 2 C1—a.

6. Example

In this section we provide an example illustrating our result obtained in Theorem 4.1.

Acknowledgment

The author are extending their heartfelt thanks to the reviewer for the remarks and corrections that helped improve the quality of this paper. I appreciate the time he has taken to read the manuscript and also the suggested modifications.

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