Variational methods for fractional $q$-Sturm--Liouville Problems

0.28 MB·

by  Zeinab S. I. Mansour

Variational methods for fractional q-Sturm–Liouville
Problems
Zeinab S.I. Mansour
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1 Department of Mathematics, Faculty of Science, King Saud University,Riyadh
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J

5 Abstract
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Inthis paper,weformulatearegularq-fractionalSturm–Liouvilleproblem(qF-

] SLP) which includes the left-sided Riemann–Liouville and the right-sided Ca-
A
puto q-fractional derivatives of the same order α, α (0,1). We introduce
C the essential q-fractional variational analysis needed in∈ proving the existence
. of a countable set of real eigenvalues and associated orthogonal eigenfunctions
h
t for the regular qFSLP when α > 1/2 asscociated with the boundary condition
a
y(0) = y(a) = 0. A criteria for the first eigenvalue is proved. Examples are
m
included. These results are a generalization of the integer regular q-Sturm–
[
Liouville problem introduced by Annaby and Mansour in[1].

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Keywords: Left and right sided Riemann–Liouville and Caputo q-derivatives,
v
8 eigenvalues and eigenfunctions, q-fractional variational calculus.
9 2000 MSC: 39A13, 26A33, 49R05.
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1. Introduction
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1
In the joint paper of Sturm and Liouville [2], they studied the problem
:
v
i d dy
X p +r(x)y(x)=λwy(x), x [a,b], (1.1)
− dx dx ∈
r (cid:18) (cid:19)
a
with certainboundary conditions at a and b. Here, the functions p, w are posi-
tive on[a,b] and r is a realvalued function on [a,b]. They provedthe existence
ofnon-zerosolutions(eigenfunctions)onlyforspecialvaluesofthe parameterλ
which is called eigenvalues. For a comprehensive study for the contribution of
SturmandLiouvilletothetheory,see[3]. Recently,manymathematicianswere
interestedinafractionalversionof (1.1),i.e. whenthe derivativeisreplacedby
a fractional derivative like Riemann–Liouville derivative or Caputo derivative,
see [4–9]. Iterative methods, variational method, and the fixed point theory
Email address: [email protected], [email protected] (ZeinabS.I.Mansour)
Preprint submitted toElsevier February 5, 2016
are three different approaches used in proving the existence and uniqueness of
solutionsofSturm–Liouvilleproblems,c.f.[3,10,11]. Thecalculusofvariations
has recently developed to calculate extremum of functional contains fractional
derivatives,whichiscalledfractionalcalculusofvariations,seeforexample[12–
19]. In [4], Klimek et al. applied the methods of fractional variational calculus
toprovetheexistenceofacountablesetoforthogonalsolutionsandcorrespond-
ingeigenvalues. In[1]AnnabyandMansourintroducedaq-versionof (1.1),i.e.,
when the derivative is replaced by Jackson q-derivative. Their results are ap-
plied and developed in different aspects, for example, see [20–25]. Throughout
this paper q is a positive number less than 1. The set of non negative integers
is denoted by N , and the set of positive integers is denoted by N. For t>0,
0
A := tqn : n N , A∗ :=A 0 ,
q,t { ∈ 0} q,t q,t∪{ }
and
:= tqn : n N .
q,t 0
A {± ∈ }
When t = 1, we simply use A , A∗, and to denote A , A∗ , and ,
q q Aq q,1 q,1 Aq,1
respectively. We follow [26] for the definitions and notations of the q-shifted
factorial, the q-gamma and q-beta functions, the basic hypergeometric series,
andJacksonq-difference operatorandintegrals. A setA is calleda q-geometric
set if qx A whenever x A. Let X be a q-geometric set containing zero. A
∈ ∈
function f defined on X is called q-regular at zero if
lim f(xqn)=f(0) for allx X.
n→∞ ∈
Let C(X) denote the space of all q-regular at zero functions defined on X with
values in R. C(X) associated with the norm function
f =sup f(xqn) : x X, n N ,
0
k k {| | ∈ ∈ }
is a normed space. The q-integration by parts rule [27] is
b b
f(x)D g(x)=f(x)g(x)b + D f(x)g(qx)d x, a,b X, (1.2)
q |a q q ∈
Za Za
and f, g are q-regular at zero functions.
For p>0, and Y is A or A∗ , the space Lp(Y) is the normed space of all
q,t q,t q
functions defined on Y such that
t 1/p
f := f(u)pd u < .
k kp | | q ∞
(cid:18)Z0 (cid:19)
If p=2, then L2(Y) associated with the inner product
q
t
f,g := f(u)g(u)d u (1.3)
q
h i
Z0
2
is a Hilbert space. By a weighted L2(Y,w) space is the space of all functions f
q
defined on Y such that
t
f(u)2w(u)d u< ,
q
| | ∞
Z0
where w is a positive function defined onY. L2(Y,w) associatedwith the inner
q
product
t
f,g := f(u)g(u)w(u)d u
q
h i
Z0
is a Hilbert space. The space of all q-absolutely functions on A∗ is denoted by
q,t
C (A∗ )anddefinedasthespaceofallq-regularatzerofunctionsf satisfying
A q q,t
∞
f(uqj) f(uqj+1) K for allu A∗ ,
| − |≤ ∈ q,t
j=0
X
and K is a constantdepending on the function f, c.f. [27, Definition 4.3.1]. I.e.
∗ ∗
C (A ) C (A ).
A q q,t ⊆ q q,t
The space C(n)(A∗ ) (n N) is the space of all functions defined on X
such that fA, Dqf, ..q.,,tDn−1∈f are q-regular at zero and Dn−1f C (A∗ ),
q q q ∈ A q q,t
c.f. [27, Definition 4.3.2]. Also it is proved in [27, Theorem 4.6] that a function
f C(n)(A∗ ) if and only if there exists a function φ L1(A∗ ) such that
∈A q q,t ∈ q q,t
n−1 Dkf(0) xn−1 x
f(x)= Γ (qk+1)xk+ Γ (n) (qu/x;q)n−1φ(u)dqu, x∈A∗q,t.
k=0 q q Z0
X
In particular, f C(A∗ ) if and only if f is q-regular at zero such that
D f L1(A∗ ). ∈It Ais wortqh,t noting that in [27], all the definitions and results
q ∈ q q,t
we have just mentioned are defined and proved for functions defined on the
interval [0,a] instead of A∗ . In [28], Mansour studied the problem
q,t
Dα p(x)cDα y(x)+(r(x) λw (x))y(x)=0, x A∗ , (1.4)
q,a− q,0+ − α ∈ q,a
where p(x) = 0 and w > 0 for all x A∗ , p,r, w are real valued functions
defined in A6∗ and theαassociated bou∈ndaqr,ya conditioαns are
q,a
c y(0)+c I1−αpcDα y (0)=0, (1.5)
1 2 q,a− q,0+
h i
a
d y(a)+d I1−αpcDα y ( )=0, (1.6)
1 2 q,a− q,0+ q
h i
withc2+c2 =0andd2+d2 =0. itisprovedthattheeigenvaluesarerealandthe
1 2 6 1 2 6
eigenfunctions associated to different eigenvalues are orthogonal in the Hilbert
space L2(A∗ ,w ). A sufficient condition on the parameter λ to guarantee
q q,a α
3
the existence and uniqueness of the solution is introduced by using the fixed
point theorem, also a condition is imposed on the domain of the problem in
order to prove the existence and uniqueness of solution for any λ. This paper
is organized as follows. Section 2 is on the q-fractional operators and their
properties which we need in the sequel. Cardoso [29] introduced basic Fourier
seriesfor functions defined ona q-lineargridofthe form qn : n N 0 .
0
{± ∈ }∪{ }
In Section 3, we reformulate Cardoso’s results for functions defined on a q-
linear grid of the form aqn : n N 0 . In Section 4, we introduce a
0
{± ∈ }∪{ }
fractional q-analogue for Euler–Lagrange equations for functionals defined in
termsofJacksonq-integrationandthe integrandcontainsthe left sidedCaputo
fractional q-derivative. We also introduce a fractional q-isoperimetric problem.
InSection5,weusethevariationalq-calculusdevelopedinSection4toprovethe
existenceofacountablenumberofeigenvaluesandorthogonaleigenfunctionsfor
the fractional q-Sturm–Liouville problem with the boundary condition y(0) =
y(a) = 0. We also define the Rayleigh quotient and prove a criteria for the
smallest eigenvalue.
2. Fractional q-Calculus
Thissectionincludesthedefinitionsandpropertiesoftheleftsidedandright
sided Riemann–Liouville q-fractional operators which we need in our investiga-
tions.
The left sided Riemann–Liouville q-fractional operator is defined by
xα−1 x
Iqα,a+f(x)= Γ (α) (qt/x;q)α−1f(t)dqt. (2.1)
q Za
This definition is introduced by Agarwal in [30] when a = 0 and by Rajkovi´c
et.al [31] for a=0. The right sided Riemann–Liouville q-fractionaloperatorby
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1 b
Iqα,b−f(x)= Γ (α) tα−1(qx/t;q)α−1f(t)dqt, (2.2)
q Zqx
see [28]. The left sided Riemann–Liouville q-fractional operator satisfies the
semigroup property
Iα Iβ f(x)=Iα+βf(x).
q,a+ q,a+ q,a+
The case a=0 is proved in [30] and the case a>0 is proved in [31].
The right sided Riemann–Liouville q-fractional operator satisfies the semi-
group property [28]
Iα Iβ f(x)=Iα+βf(x), x A∗ , (2.3)
q,b− q,b− q,b− ∈ q,b
for any function defined on A and for any values of α and β.
q,b
Forα>0andpαq=m,theleftandrightsideRiemann–Liouvillefractional
q-derivatives of order α are defined by
m
1
Dqα,a+f(x):=DqmIqm,a−+αf(x), Dqα,b−f(x):= −q Dqm−1Iqm,b−−αf(x),
(cid:18) (cid:19)
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theleftandrightsidedCaputofractionalq-derivativesoforderαaredefinedby
m
1
cDqα,a+f(x):=Iqm,a−+αDqmf(x), cDqα,b− := −q Iqm,b−−αDqm−1f(x).
(cid:18) (cid:19)
see [28]. From now on, we shall consider left sided Riemann–Liouville and
Caputo fractional q-derivatives when the lower point a = 0 and right sided
Riemann–Liouville and Caputo fractional q-derivatives when b = a. According
to [27, pp. 124,148], Dα f(x) exists if
q,0+
f L1(A∗ ) such that Im−αf C(m)(A∗ ),
∈ q q,a q,0+ ∈A q q,a
and cDα f exists if
q,a+
f C(m)(A∗ ).
∈A q q,a
The following proposition is proved in [28]
Proposition 2.1. Let α (0,1).
∈
(i) If f L1(A∗ ) such that Iα f C (A∗ ) then
∈ q q,a q,0+ ∈A q q,a
Iα f(0)
cDα Iα f(x)=f(x) q,0+ x−α. (2.4)
q,0+ q,0+ − Γ (1 α)
q
−
Moreover, if f is bounded on A∗ then
q,a
cDα Iα f(x)=f(x). (2.5)
q,0+ q,0+
(ii) For any function f defined on A∗
q,a
a−α a
cDqα,a−Iqα,a−f(x)=f(x)− Γ (1 α)(qx/a;q)−α Iqα,a−f (q). (2.6)
q
− (cid:16) (cid:17)
(iii) If f L1(A ) then
∈ q q,a Dα Iα f(x)=f(x). (2.7)
q,0+ q,0+
(iv) For any function f defined on A∗
q,a
Dα Iα f(x)=f(x). (2.8)
q,a− q,a−
(v) If f C (A∗ )) then
∈A q q,a
Iα cDα f(x)=f(x) f(0). (2.9)
q,0+ q,0+ −
(vi) If f is a function defined on A∗ then
q,a
aα−1 a
Iqα,a−Dqα,a−f(x)=f(x)− Γ (α)(qx/a;q)α−1 Iq1,−a−αf (q). (2.10)
q
(cid:16) (cid:17)
(v) If f is defined on [0,a] such that D f is continuous on [0,a] then
q
cDα f(x)=Dα [f(x) f(0)]. (2.11)
q,0+ q,0+ −
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Set X =A or A∗ . Then
q,a q,a
C(X) L2(X) L1(X).
⊆ q ⊆ q
Moreover,if f C(X) then
∈
f √a f a f .
k k1 ≤ k k2 ≤ k k
We have also the following inequalities:
1. If f C(A∗ ) then Iα f C(A∗ ) and
∈ q,a q,0+ ∈ q,a
aα
Iα f f . (2.12)
q,0+ ≤ Γ (α+1)k k
q
(cid:13) (cid:13)
(cid:13) (cid:13)
2. If f L1(X) then Iα (cid:13)f L1((cid:13)X) and
∈ q q,0+ ∈ q
aα(1 q)α
Iα f M f , M := − . (2.13)
q,0+ 1 ≤ α,1k k1 α,1 (1 qα)(q;q)∞
(cid:13) (cid:13) −
(cid:13) (cid:13)
3. If f L(cid:13)2(X) t(cid:13)hen Iα f L2(X) and
∈ q q,0+ ∈ q
Iα f M f , (2.14)
q,0+ 2 ≤ α,2k k2
(cid:13) (cid:13)
where (cid:13) (cid:13)
(cid:13) (cid:13)
aα (1 q) 1 1/2
M := − (qξ;q)2 d ξ .
α,2 Γq(α)s(1−q2α)(cid:18)Z0 α−1 q (cid:19)
4. If α> 1 and f L2(X) then Iα f C(X) and
2 ∈ q q,0+ ∈
Iα f M f , M := aα−21 1(qξ;q)2 d ξ 1/2. (2.15)
q,0+ ≤ αk k α Γ (α) α−1 q
(cid:13) (cid:13) q (cid:18)Z0 (cid:19)
5. Since(cid:13)(cid:13) f (cid:13)(cid:13) √aff , we cfonclude that if f C(X) then Iα f L2(X)
k k2 ≤ k k ∈ q,0+ ∈ q
and
Iα f K f , K :=√aM . (2.16)
q,0+ 2 ≤ αk k α α,2
6. If f ∈C(A∗q,a) th(cid:13)(cid:13)(cid:13)en Iqα,a−(cid:13)(cid:13)(cid:13)f ∈C(A∗q,a) and
aα(1 q)α
Iqα,a−f ≤cα,0kfk, cα,0 := (1 qα)−(q;q)∞.
(cid:13) (cid:13) −
(cid:13) (cid:13)
7. If f ∈L1q(X)(cid:13)then Iqα(cid:13),a−f ∈L1q(X) and
(1 q)αaα
− f , ifα<1,
(1 qα)(q;q)∞ k k1
Iα f  −
(cid:13) q,a− (cid:13)1 ≤ (1−q)α−1aα−1 f , ifα 1.
(cid:13)(cid:13) (cid:13)(cid:13) (q;q)∞ k k1 ≥

6
8. If α= 1 and f L2(X) then Iα f L1(X) and
6 2 ∈ q q,a− ∈ q
(1−q)α−12aα f , ifα< 1,
 1 q2α−1(q;q)∞ k k2 2
(cid:13)(cid:13)(cid:13)Iqα,a−f(cid:13)(cid:13)(cid:13)2 ≤ (q;q)∞p (−(11−qq2)αα−a1α)(1 q2α)kfk2, ifα> 12.
The following lemmas are introdupced a−nd proved−in [28]
Lemma 2.2. Let α>0. If
(a) f L1(X) and g is a bounded function on A ,
∈ q q,a
or
(b) α= 1 and f, g are L2(X) functions
6 2 q
then
a a
g(x)Iα f(x)d x= f(x)Iα g(x)d x. (2.17)
q,0+ q q,a− q
Z0 Z0
Lemma 2.3. Let α (0,1).
∈
(a) If g L1(A∗ ) such that I1−αg C (A∗ ), and Dif C(A∗ ) (i =
∈ q q,a q ∈ A q q,a q ∈ q,a
0,1) then
a x a a
f(x)Dα g(x)d x= f( )I1−αg(x) + g(x)cDα f(x)d x.
Z0 q,0+ q − q q,0+ (cid:12)x=0 Z0 q,a− (2.q18)
(cid:12)
(cid:12)
(b) If f C (A∗ ), and g is a bounded function on A∗ such that Dα g
∈A q q,a q,a q,a− ∈
L1(A∗ ) then
q q,a
a x a a
g(x)cDα f(x)d x= I1−αg ( )f(x) + f(x)Dα g(x)d x.
Z0 q,0+ q (cid:16) q,a− (cid:17) q (cid:12)x=0 Z0 q,a− (2.1q9)
(cid:12)
(cid:12)
3. Basic Fourier series on q-Linear grid and some properties
The purpose of this section is to reformulate Cardoso’s results of Fourier
seriesexpansionsforfunctionsdefinedontheq-lineargrid := qn, n N to
q 0
functions defined on q-linear grids :=:= aqn, n AN , {a > 0. ∈Card}oso
q,a 0
A {± ∈ }
in[29]defined the spaceof all q-linearHo¨lder functions onthe q-lineargrid .
q
A
We generalize his definition for functions defined on a q-linear grid of the form
, a>0.
q,a
A
Definition 3.1. Afunctionf definedon ,a>0,iscalledaq-linearHo¨lder
q,a
A
of order λ if there exists a constant M >0 such that
f( aqn−1) f( aqn) Mqnλ,for alln N.
± − ± ≤ ∈
(cid:12) (cid:12)
(cid:12) (cid:12)
7
Definition 3.2. The q-trigonometricfunctions S (z)andC (z) aredefined for
q q
z C by, see [29, 32]
∈
S (z) = ∞ ( 1)nqn(n+12)z2n+1 = z φ 0;q3;q,q3/2z2 ,
q 1 1
− (q;q) 1 q
2n+1
nX=0 − (cid:16) (cid:17)
C (z) = ∞ ( 1)nqn(n−12)z2n = φ 0;q;q,q1/2z2 .
q 1 1
− (q;q)
2n
nX=0 (cid:16) (cid:17)
One can verify that
w
D S (wz) = C (√qz),
q,z q q
1 q
−
w
D C (wz) = S (√qz),
q,z q q
−1 q
−
wherez Candw Cisafixedparameter. Amodificationoftheorthogonality
∈ ∈
relation given in [32, Theorem 4.1] is
Theorem3.3. Letwandw′ berootsofS (z),andµ(w):=(1 q)C (q1/2w)S′(w).
q − q q
Then
a C (q12wx)C (q12w′x)d x = 20a,, ifwifw=6=w′w=′,0,
Z−a q a q a q  aµ(w), ifw =w′ =0,
6
a qwx qw′x 0, ifw =w′,
Sq( a )Sq( a )dqx =  aq−1/2µ(w), ifw =6 w′.
Z−a (cid:26)
Cardosointroducedasufficientconditionforthe uniformconvergenceofthe
basic Fourier series
∞
a
S (f):= 0 + a C (q1/2w x)+b S (qw x),
q k q k k q k
2
k=1
X
1
where a = f(t)d t and for k =1,2,...,
0 −1 q
R1 1 1 1
a = f(t)C (q1/2w t)d t, b = f(t)S (qw t)d t,
k q k q k q k q
µ µ
k Z−1 k Z−1
µ =(1 q)C (q1/2w )S′(w )
k − q k q k
on the q-linear grid , where w : k N is the set of positive zeros of
q k
A { ∈ }
S (z). Cardoso proved that µ =O(q−2k2) as k for any q (0,1). In the
q k
→∞ ∈
followingwegiveamodifiedversionofCardoso’sresultforanyfunctiondefined
on the q-linear grid , a>0.
q,a
A
Theorem 3.4. If f C( ∗ ) is a q-linear Ho¨lder function of order λ > 1,
∈ Aq,a 2
then the q-Fourier series
∞
a w x w x
S (f):= 0 + a C (q1/2 k )+b S (q k ), (3.1)
q k q k q
2 a a
k=1
X
8
where a = 1 a f(t)d t and for k =1,2,...,
0 a −a q
1R a w t √q a w t
a (f)= f(t)C (q1/2 k )d t, b (f)= f(t)S (q k )d t,
k q q k q q
aµ a aµ a
k Z−a k Z−a
converges uniformly to the function f on the q-linear grid .
q,a
A
Proof. The proof is a modification of the proof of [29, Theorem 4.1] and is
omitted.
Remark 3.5. We replaced the condition f(0+)=f(0−) where
f(0+):= lim f(x), f(0−):= lim f(x),
x→0+ x→0−
in [29, Theorem 4.1] by the weakest condition that f is q-regular at zero.
Because he needs this condition only to guarantee that limn→∞f(qn−1/2) =
limn→∞f( qn−1/2) and this holds if fis q-regular at zero. See [27, (1.22)] for
−
a function which is q-regular at zero but not continuous at zero.
A modified version of [29, Theorem 3.5] is
Theorem 3.6. If there exists c>1 such that
a w t a w t
f(t)C (√q k )=O(qck)and f(t)S (q k )=O(qck) ask ,
q q
a a →∞
Z−a Z−a
then the q-Fourier series (3.1) converges uniformly on .
q,a
A
A modified version of [29, Corollary 4.3] is
Corollary 3.7. If f is continuous and piecewise smooth on a neighborhood of
theorigin, then thecorresponding q-Fourier series S (f)converges uniformly to
q
f on the set of points .
q,a
A
Theorem 3.8. If f C( ∗ ) is a q-linear Ho¨lder odd function of order λ> 1
∈ Aq,a 2
and satisfying f(0)=f(a)=0, then the q-Fourier series
∞
w x
k
S (f):= c S ( ),
q k q
a
k=1
X
where
2 a w t
k
c (f)=c = f(t)S ( )d t,
k k q q
a√qµk Z0 a
converges uniformly to the function f on the q-linear grid .
q,a
A
Proof. The proof follows from (3.4) by considering the function g(x) := f(qx),
x . Since, it is odd, we have a =0 for k =0,1,..., and
k
∈A
a qw t
k
b (f)=√qµ g(t)S ( )d t,
k k q q
a
Z−a
making the substitution u= qt and using that g is an odd function, we obtain
the required result.
9
Definition 3.9. Let (f ) be a sequence of functions in C( ∗ ). We say that
n n Aq,a
f converges to a function f in q-mean if
n
a
lim f (x) f(x)2d x=0.
n q
n→∞sZ−a| − |
Proposition 3.10. If g C( ∗ ) is an odd function satisfying Dkg (k =
∈ Aq,a q
0,1,2) is continuous and piecewise smooth function in a neighborhood of zero,
and satisfying the boundary condition
g(0)=g(a)=0 (3.2)
then g can be approximated in the q mean by a linear combination
−
n
w x
g (x)= c(n)S ( k )
n r q a
r=
X
where at the same time Dkg (k=1,2) converges in q-mean to the Dkg. More-
q n q
over, the coefficients c(n) need not depend on n and can be written simply as
r
c .
r
Proof. We consider the q-sine Fourier transform of D2g. Hence
q
∞
qw x
D2g(x)= b S ( k )= lim γ (x), x A , (3.3)
q k q a n→∞ n ∈ q,a
k=1
X
where
n √q a qw x
γ (x)= b S (qw xa), b = D2g(x)S ( k )d x.
n k q k k aµ q q a q
k=1 k Z0
X
Consequently,
a
lim D2g(x) γ (x) 2 d x=0.
n→∞ q − n q
Z0
(cid:12) (cid:12)
Hence (cid:12) (cid:12)
x a(1 q) ∞ b q1/2w x
D g(x) D g(0)= D2g(x)d x= − k C ( k )+1 .
q − q Z0 q q √q k=1wk (cid:18)− q a (cid:19)
X
Applying the q-integration by parts rule (1.2) gives
a(1 q)
a (D g)= − b (D2g).
k q − √qwk k q
∞
q1/2w x
k
I.e. D g(x) D g(0)= a (D g) C ( ) 1 .
q q k q q
− a −
k=1 (cid:18) (cid:19)
X
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