Hindawi Publishing Co po a ion
Abs ac and Applied Analysis
Volume 2009, A icle ID 494720, 9pages
doi:10.1155/2009/494720
Resea ch A icle
Exis ence o Solu ions o Nonlocal Bounda y
Value P oblems o Highe -O de
Nonlinea F ac ional Di e en ial Equa ions
Bashi Ahmad1and Juan J. Nie o2
1Depa men o Ma hema ics, Facul y o Science, King Abdulaziz Uni e si y, P.O. Box 80203,
Jeddah 21589, Saudi A abia
2Depa amen o de An´
alisis Ma em´
a ico, Facul ad de Ma em´
a icas, Uni e sidad de San iago de Compos ela,
15782 San iago de Compos ela, Spain
Co espondence should be add essed o Bashi Ahmad, bashi [email p o ec ed]
Recei ed 19 Feb ua y 2009; Accep ed 27 Ap il 2009
Recommended by Paul Eloe
We s udy some exis ence esul s in a Banach space o a nonlocal bounda y alue p oblem
in ol ing a nonlinea diffe en ial equa ion o ac ional o de qgi en by cDqx   , x ,
0< <1, q∈m−1,m,m∈N,m≥2, x00, x00,x
00,...,xm−200,
x1αxη. Ou esul s a e based on he con ac ion mapping p inciple and K asnoselskii’s
ixed poin heo em.
Copy igh q2009 B. Ahmad and J. J. Nie o. This is an open access a icle dis ibu ed unde
he C ea i e Commons A ibu ion License, which pe mi s un es ic ed use, dis ibu ion, and
ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed.
1. In oduc ion
F ac ional diffe en ial equa ions in ol e de i a i es o ac ional o de . They a ise in many
enginee ing and scien i ic disciplines such as he ma hema ical modeling o sys ems and
p ocesses in he ields o physics, chemis y, ae odynamics, elec o-dynamics o complex
medium, and polyme heology. In consequence, he subjec o ac ional diffe en ial
equa ions is gaining much impo ance and a en ion. Fo examples and de ails, see 1–17
and he e e ences he ein. Howe e , he heo y o bounda y alue p oblems o nonlinea
ac ional diffe en ial equa ions is s ill in he ini ial s ages and many aspec s o his heo y
need o be explo ed.
The subjec o mul ipoin nonlocal bounda y alue p oblems, ini ia ed by Ilin and
Moisee 18,19, has been add essed by many au ho s, o ins ance, 20–26. The mul ipoin
bounda y condi ions appea in ce ain p oblems o he modynamics, elas ici y, and wa e
p opaga ion, see 27and he e e ences he ein. The mul ipoin bounda y condi ions may
be unde s ood in he sense ha he con olle s a he end poin s dissipa e o add ene gy
acco ding o censo s loca ed a in e media e posi ions.
2 Abs ac and Applied Analysis
Fo m∈N,m ≥2,and q∈m−1,m,we conside he ollowing nonlinea ac ional
diffe en ial equa ion o o de qwi h nonlocal bounda y condi ions:
cDqx   ,  ,0< <1,
x00,x
00,x
00,...,x
m−200,x
1αxη,
0<η<1,αη
m−1/
1,α∈R,
1.1
whe e cDis he Capu o ac ional de i a i e and :0,1×X→Xis con inuous. He e,
X, ·is a Banach space and CC0,1,Xdeno es he Banach space o all con inuous
unc ions om 0,1→Xendowed wi h a opology o uni o m con e gence wi h he no m
deno ed by ·.
By a solu ion o 1.1, we mean a unc ion x∈Co class Cm0,1which sa is ies he
nonlocal ac ional bounda y alue p oblem 1.1.
2. P elimina ies
Le us ecall some basic de ini ions 12,15,17on ac ional calculus.
De ini ion 2.1. Fo a unc ion g:0,∞→R, he Capu o de i a i e o ac ional o de qis
de ined as
cDqg 1
Γn−q
0
 −sn−q−1gnsds, n −1<q<n, nq1,2.1
whe e qdeno es he in ege pa o he eal numbe q.
De ini ion 2.2. The Riemann-Liou ille ac ional in eg al o o de qis de ined as
Iqg 1
Γq
0
gs
 −s1−qds, q > 0,2.2
p o ided ha he in eg al exis s.
De ini ion 2.3. The Riemann-Liou ille ac ional de i a i e o o de q o a unc ion g is
de ined by
Dqg 1
Γn−qd
d n
0
gs
 −sq−n1ds, n q1,2.3
p o ided he igh hand side is poin wise de ined on 0,∞.
We ema k ha he Capu o de i a i e becomes he con en ional n h de i a i e o
he unc ion as q→nand he ini ial condi ions o ac ional diffe en ial equa ions e ain
he same o m as ha o o dina y diffe en ial equa ions wi h in ege -o de de i a i es.
On he o he hand, he Riemann-Liou ille ac ional de i a i e could ha dly p oduce he
physical in e p e a ion o he ini ial condi ions equi ed o he ini ial alue p oblems
Abs ac and Applied Analysis 3
in ol ing ac ional diffe en ial equa ions  he same applies o he bounda y alue p oblems
o ac ional diffe en ial equa ions. Mo eo e , he Capu o de i a i e o a cons an is ze o
while he Riemann-Liou ille ac ional de i a i e o a cons an is nonze o. Fo mo e de ails,
see 17.
Lemma 2.4 see 28.Fo q>0, he gene al solu ion o he ac ional diffe en ial equa ion
cDqx 0is gi en by
x c0c1 c2 2···cn−1 n−1,2.4
whe e ci∈R,i0,1,2,...,n−1(nq1).
In iew o Lemma 2.4, i ollows ha
IqcDqx x c0c1 c2 2···cn−1 n−1,2.5
o some ci∈R,i0,1,2,...,n−1nq1.
Now, we s a e a known esul due o K asnoselskii 29which is needed o p o e he
exis ence o a leas one solu ion o 1.1.
Theo em 2.5. Le Mbe a closed con ex and nonemp y subse o a Banach space X. Le A, B be he
ope a o s such ha
iAx By ∈Mwhene e x, y ∈M,
iiAis compac and con inuous,
iiiBis a con ac ion mapping.
Then he e exis s z∈Msuch ha zAz Bz.
To s udy he nonlinea p oblem 1.1, we i s conside he associa ed linea p oblem
and ob ain i s solu ion.
Lemma 2.6. Fo a gi en σ∈C0,1, he unique solu ion o he bounda y alue p oblem,
cDqx σ ,0< <1,q∈m−1,m
,m∈N,m≥2,
x00,x
00,x
00,...,x
m−200,x
1αxη,
0<η<1,αη
m−1/
1,α∈R,
2.6
is gi en by
x 
0
 −sq−1
Γqσsds − m−1
1−αηm−11
0
1−sq−1
Γqσsds −αη
0η−sq−1
Γqσsds.
2.7
4 Abs ac and Applied Analysis
P oo . Using 2.5, we ha e
x 
0
 −sq−1
Γqσsds −c0−c1 −c2 2−···−cm−1 m−1,2.8
whe e c0,c
1,c
2,...,c
m−1∈Ra e a bi a y cons an s. In iew o he ela ions cDqIqx x 
and IqIpx Iqpx  o q, p > 0,x∈L0,1,we ob ain
x 
0
 −sq−2
Γq−1σsds −c1−2c2 −···−m−1cm−1 m−2,
x 
0
 −sq−3
Γq−2σsds −2c2−···−m−1m−2cm−1 m−3,....
2.9
Applying he bounda y condi ions o 2.6,we ind ha c00,c
10,...,c
m−20,and
cm−11
1−αηm−11
0
1−sq−1
Γqσsds −αη
0η−sq−1
Γqσsds.2.10
Subs i u ing he alues o c0,c
1,...,c
m−1in 2.8,weob ain
x 
0
 −sq−1
Γqσsds
− m−1
1−αηm−11
0
1−sq−1
Γqσsds −αη
0η−sq−1
Γqσsds.
2.11
This comple es he p oo .
3. Main Resul s
Fo he o hcoming analysis, we need he ollowing assump ions:
A1  , x−  , y≤Lx−y, o all ∈0,1,x,y∈X;
A2  , x≤μ , o all  , x∈0,1×X, μ ∈L10,1,R
.
In ela ion o he nonlocal p oblem 1.1, we de ine he cons an s:
Λ L
Γq1λ, λ L1|α|ηq
Γq11−αηm−1
.3.1
Theo em 3.1. Assume ha :0,1×X→Xis a join ly con inuous unc ion and sa is ies he
assump ion A1.Then he bounda y alue p oblem 1.1has a unique solu ion p o ided Λ<1,whe e
Λis gi en by 3.1.
Abs ac and Applied Analysis 5
P oo . De ine :C→Cby
x 
0
 −sq−1
Γq s, xsds − m−1
1−αηm−1
×1
0
1−sq−1
Γq s, xsds −αη
0η−sq−1
Γq s, xsds, ∈0,1.
3.2
Le us se sup ∈0,1  , 0M, and choose
≥M
1−βΓq111|α|ηq
1−αηm−1,3.3
whe e βis such ha Λ≤β<1.Now we show ha B ⊂B ,whe e B {x∈C:x≤ }.
Fo x∈B ,we ha e
x ≤
0
 −sq−1
Γq
 s, xs
ds
 m−1
1−αηm−11
0
1−sq−1
Γq
 s, xs
ds |α|η
0η−sq−1
Γq s, xsds
≤
0
 −sq−1
Γq
 s, xs − s, 0

 s, 0
ds
 m−1
1−αηm−11
0
1−sq−1
Γq s, xs − s, 0 s, 0ds
|α|η
0η−sq−1
Γq
 s, xs − s, 0
 s, 0ds
≤L M
0
 −sq−1
Γqds  m−1
1−αηm−11
0
1−sq−1
Γqds |α|η
0η−sq−1
Γqds
≤L
Γq1L1|α|ηq
Γq11−αηm−1 M
Γq111|α|ηq
1−αηm−1
Λ M
Γq111|α|ηq
1−αηm−1
≤Λ1−β ≤ .
3.4

6 Abs ac and Applied Analysis
Now, o x, y ∈Cand o each ∈0,1,we ob ain
x −y 
≤
0
 −sq−1
Γq s, xs − s, ysds
 m−1
1−αηm−11
0
1−sq−1
Γq s, xs − s, ysds
|α|η
0η−sq−1
Γq s, xs − s, ysds
≤L
x−y

0
 −sq−1
Γqds  m−1
1−αηm−11
0
1−sq−1
Γqds |α|η
0η−sq−1
Γqds
L
x−y
 q
Γq1 m−1
1−αηm−11|α|ηq
Γq1
≤L
x−y

1
Γq111|α|ηq
1−αηm−1
Λ

x−y
.
3.5
Clea ly Λdepends on he pa ame e s q, m,α,η,Lin ol ed in he p oblem. As Λ<1,
he e o e, is a con ac ion. Thus, he conclusion o he heo em ollows by he con ac ion
mapping p inciple.
Theo em 3.2. Le :0,1×X→Xbe a join ly con inuous unc ion mapping bounded subse s o
0,1×Xin o ela i ely compac subse s o X. Fu he , he assump ions A1−A2hold wi h λ<1,
whe e λis gi en by 3.1. Then he bounda y alue p oblem 1.1has a leas one solu ion on 0,1.
P oo . Le us ix
≥
μ
L1
Γq11|α|ηq−1
1−αηm−1,3.6
and conside B {x∈C:x≤ }.We de ine he ope a o s Φand Ψon B as
Φx 1
Γq
0
 −sq−1 s, xsds,
Ψx − m−1
1−αηm−11
0
1−sq−1
Γq s, xsds −αη
0η−sq−1
Γq s, xsds.
3.7
Abs ac and Applied Analysis 7
Fo x, y ∈B ,we ind ha

ΦxΨy
≤
μ
L1
Γq11|α|ηq−1
1−αηm−1≤ . 3.8
Thus, ΦxΨy∈B .I ollows om he assump ion A1 ha Ψis a con ac ion mapping
o λ<1.Con inui y o implies ha he ope a o Φis con inuous. Also, Φis uni o mly
bounded on B as
Φx≤
μ
L1
Γq.3.9
To show ha he ope a o Φis compac , we use he classical A zela-Ascoli heo em. Le Abe
a bounded subse o C.We ha e o show ha ΦAis equicon inuous and o each , he se
ΦA is ela i ely compac in X. In iew o A1,A2,we de ine sup ,x∈0,1×B   , x
max,and consequen ly we ha e
Φx 1−Φx 2





1
Γq 1
0 2−sq−1− 1−sq−1 s, xsds  2
1
 2−sq−1 s, xsds




≤ max
Γq12 2− 1q q
1− q
2,
3.10
which is independen o x. Thus, Φis equicon inuous. Using he ac ha maps bounded
subse s in o ela i ely compac subse s, we ha e ha ΦA is ela i ely compac in X
o e e y . The e o e, Φis ela i ely compac on B .Hence, By A zela-Ascoli heo em, Φ
is compac on B .Thus all he assump ions o Theo em 2.5 a e sa is ied and he conclusion
o Theo em 2.5 implies ha he bounda y alue p oblem 1.1has a leas one solu ion on
0,1.
Example 3.3. Conside he ollowing bounda y alue p oblem:
cDqx 1
 72
x
1x,2<q≤3, ∈0,1,
x00,x
00,x
1x1
2.
3.11
He e, m3,  , x   1/ 72x/1x,α 1,η 1/2.As   , x−  , y≤
1/49x−y, he e o e, A1is sa is ied wi h L1/49.Fu he ,
Λ L
Γq111|α|ηq
|1−ηm−1|1
49Γq114
311
2q<1,2<q≤3.3.12
Thus, by Theo em 3.1, he bounda y alue p oblem 3.11has a unique solu ion on 0,1.
8 Abs ac and Applied Analysis
Acknowledgmen s
The au ho s hank he e iewe s o hei use ul commen s. The esea ch o J. J. Nie o has been
pa ially suppo ed by Minis e io de Educacion y Ciencia and FEDER, p ojec MTM2007-
61724, and by Xun a de Galicia and FEDER, p ojec PGIDIT06PXIB207023PR.
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