A Note on Positivity of One-Dimensional Elliptic Differential
Operators
Allaberen Ashyralyev
1and Sema Akturk
2,∗1Department of Elementary Mathematics Education, Fatih University, 34500 Buyukcekmece, Istan-bul, Turkey,
Department of Mathematics, ITTU, 74400 Gerogly Street,Ashgabat, Turkmenistan 2Department of Mathematics, Fatih University, 34500 Buyukcekmece, Istanbul, Turkey
Abstract. We consider the structure of fractional spaces Eα(C R+
,A)generated by the positive dif-ferential operatorAdefined by the formulaAu(t) =−ut t(t) +u(t)with domain
D(A) ={u:ut t,u∈C R +
,u(0) =0,u(∞) =0},
whereR+= [0,∞). It is established that for any 0< α <1/2, the norms in the spacesEα(C R+,A)
andC2α R+are equivalent. The positivity of the differential operatorAinC2α R+is established. 2010 Mathematics Subject Classifications: 35J08, 35J58, 47B65
Key Words and Phrases: Positive operator, Fractional spaces, Green’s function, Hölder spaces
1. Introduction
It is well-known that various local and nonlocal boundary value problems for partial ential equation can be considered as an abstract boundary value problem for ordinary differ-ential equation in a Banach spaceEwith a densely defined unbounded operatorA. Therefore, the study of various properties of partial differential equations is based on the positivity prop-erty of the differential operator in a Banach space[6–8]. Many researcher have studied the positivity of wider class of differential operators (see[12]through[23]).
An differential operatorAdensely defined in a Banach spaceEwith domainD(A)is called positive in E, if its spectrum σA lies in the interior of the sector of angle ϕ, 0 < ϕ < π,
symmetric with respect to the real axis, and moreover on the edges of this sector
S1 ϕ
={ρeiϕ: 0≤ρ≤ ∞}
∗Corresponding author.
Email addresses:[email protected](A. Ashyralyev),[email protected](S. Akturk)
http://www.ejpam.com 165 c 2016 EJPAM All rights reserved.
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 9, No. 2, 2016, 165-174
S2 ϕ={ρe−iϕ: 0≤ρ≤ ∞},
and outside of the sector the resolvent(A−λ)−1is a subject to the bound (see,[6])
(A−λ)−1
E→E ≤
M 1+|λ|.
The infimum of all such anglesϕ is called thespectral angleof the positive operatorAand is denoted byϕ(A) =ϕ(E,A). The operatorAis said to bestrongly positivein a Banach spaceE, ifϕ(E,A)< π2.
Throughout the paper, M will denote positive constants which can be different from time to time and we are not interested to precise. To stress the fact that the constant depends only onα,β, . . . , we will writeM(α,β, . . .).
For a positive operatorAin the Banach space E, let us define the fractional spaces Eα=Eα(E,A)(0< α <1)consisting of thosev∈Efor which the norm
kvkEα=sup
λ>0
λαkA(λ+A)−1vkE+kvkE
is finite.
It is well-known that from the positivity of operatorAin the Banach spaceEit follows the positivity of this operator in fractional spaces Eα=Eα(E,A)(0< α <1).
In this study, we consider the second order differential operator
Au(t) =−ut t(t) +u(t) (1)
with domain
D(A) ={u:ut t,u∈C R+,u(0) =0,u(∞) =0},
whereR+= [0,∞).
The Green’s function of Ais constructed. The positivity of the operator Ain the Banach spaceE=C R+with norm
ϕ
C(R+)=sup t≥0|
ϕ(t)|
is proved. Moreover, the structure of the fractional spacesEα(E,A),α∈(0, 1/2)are established
and the positivity ofAin the Hölder spacesC2α R+,α∈(0, 1/2)is established.
2. Green’s Function of
A
and Positivity of
A
in
C
R
+To find the Green’s function of operatorAwe need to solve the resolvent equation
Au(t) +λu(t) =ϕ(t), 0<t<∞
or
¨
−ut t(t) + (1+λ)u(t) =ϕ(t), 0<t <∞,
u(0) =0, u(∞) =0 . (2)
Lemma 1. Forλ≥0, equation(2)is uniquely solvable and the following formula holds:
u(t) = (A+λ)−1ϕ(t) = Z ∞
0
G(t,s)ϕ(s)ds (3)
where
G(t,s) = e
−p1+λ|t−s|−e−p1+λ(t+s)
2p1+λ , t,s≥0.
Now, we will prove the positivity ofAin the Banach spaceC R+.
Theorem 1. Forλin the sectorΣϕ
0={λ=̺e
iθ;|θ| ≤ϕ
0< π/2}, the following estimate holds:
(A+λ)−1
C(R+)→C(R+)≤
M(ϕ0)
1+|λ| (4)
where the resolvent(A+λ)−1defined by formula(3).
Proof. For λ = |λ|eiϕ ∈ Σϕ0, we have 1+λ = |1+λ|e
iψ, ψ ≤ ϕ < ϕ
0 <
π
2. Then, p
1+λ=|1+λ|1/2eiψ2 withψ < π
4. Clearly, we have
p 1+λ
=
4 q
1+2|λ|cosϕ+|λ|2≥M(ϕ0) Æ
1+|λ|. (5)
Using formula (3), estimate (5) and the triangle inequality, we get
(A+λ)−1f(t)≤ f
C(R+)
M(ϕ0) p
1+|λ|
∞ Z
0
e−M(ϕ0)p1+|λ||t−s|ds
≤ f
C(R+)
M(ϕ0) p
1+|λ|
t
Z
0
e−M(ϕ0)p1+|λ|(t−s)ds+ ∞ Z
t
e−M(ϕ0)p1+|λ|(s−t)ds
≤M(ϕ0) 1+|λ|.
This finishes the proof of Theorem 1.
Now, we will introduce the Banach spaceC2α R+(0< α <1) of all continuous functions
ϕ(x)defined onR+and satisfying a Hölder condition for which the following norm is finite:
kϕkC2α(R+) =kϕkC(R+) + sup t16=t2
t1,t2∈R+
|ϕ(t1)−ϕ(t2)| |t1−t2|2α
3. The Structure of Fractional Spaces
E
α(
C
R
+,
A
)
Theorem 2. Forα∈(0, 1/2), the Banach spaces Eα(C R+
,A)and C2α R+are equivalent.
Proof. Letλ >0 andt≥0. From formula (3) it follows that
A(A+λ)−1f(t) =λ
1
λf(t)−(A+λ)
−1f(t)
= 1
λ+1f(t) +λ
1
λ+1f(t)−(A+λ)
−1f(t)
= 1
λ+1f(t) +λ 1
2p1+λ
∞ Z 0 e− p
1+λ|t−s|−e−p1+λ(t+s) f(t)−f(s) ds.
Then, by this formula, the triangle inequality, and the definition ofC2α R+−norm, we have
λαA(A+λ)−1f(t) ≤M f
C2α
λα
1+λ+ λα+1
2p1+λ
∞ Z 0 e− p
1+λ|t−s|−e−p1+λ(t+s) |t−s|
2α
ds
≤Mf
C2α
λα
1+λ+λ
α+1 1 p
1+λ
∞ Z
0
e−p1+λ|t−s||t−s|2αds
. (6)
The substitutionp1+λ|t−s|=pyields that
∞ Z
0 e−
p
1+λ|t−s||t−s|2α
ds= t Z 0 e− p
1+λ|t−s||t−s|2α
ds+ ∞ Z
t
e−p1+λ(s−t) 2p1+λ |s−t|
2α ds =− 0 Z p
1+λt
e−p p 2α
(1+λ)α+12d p+ ∞ Z
0
e−p p 2α
(1+λ)α+12d p
≤ 2
(1+λ)α+12Γ(2α+1) (7)
whereΓ(·)is the gamma function.
Thus, from estimate (7) it follows that estimate (6) becomes
λαA(A+λ)−1f(t) ≤M f
C2α
λα
1+λ+λ
α+1 1
2(1+λ)1+αM(α)
≤M(α)f
C2α.
Hence, we get
sup
λ>0 sup
t∈[0,∞)
λαA(A+λ)−1f(t)
≤M(α) f
or
f
Eα(A,C)≤M(α)
f
C2α.
Therefore, we prove
C2α R+⊂Eα(C R+,A).
Next, let us prove that Eα(C R+,A)⊂ C2α R+. Clearly, for a positive operator Ain a Banach spaceE, we have
v= ∞ Z
0
A(λ+A)−2vdλ.
By this fact, for t+τ >t ≥0, we have
f(t) = ∞ Z
0
A(λ+A)−2f(t)dλ= ∞ Z
0
(λ+A)−1A(λ+A)−1f(t)dλ
= ∞ Z 0 ∞ Z 0 1
2p1+λ
e−p1+λ|t−s|−e−p1+λ(t+s)A(λ+A)−1f(s)dsdλ, (8)
and
f(t+τ) = ∞ Z 0 ∞ Z 0 1
2p1+λ
e−p1+λ|t+τ−s|−e−p1+λ(t+τ+s)A(λ+A)−1f(s)dsdλ. (9)
Clearly,
kfkC(R+)≤M(α). (10)
From equations (8) and (9) it follows that
f(t+τ)−f(t)
τ2α =
∞ Z
0
λ−α
2p1+λ
1
τ2α t
Z
0
e−p1+λ|t+τ−s|−e−p1+λ(t+τ+s)−e−p1+λ|t−s|−e−p1+λ(t+s)
×λαA(λ+A)−1f(s)dsdλ
+ ∞ Z
0
λ−α
2p1+λ
1
τ2α t+τ
Z
t
e−p1+λ|t+τ−s|−e−p1+λ(t+τ+s)−e−p1+λ(s−t)−e−p1+λ(s+t)
×λαA(λ+A)−1f(s)dsdλ
+ ∞ Z
0
λ−α
2p1+λ
1
τ2α
∞ Z
t+τ
e−
p
×λαA(λ+A)−1f(s)dsdλ
=J1+J2+J3.
Clearly, we have
1−e−p1+λτ≤(1+λ)ατ2α. (11) Using estimate (11), the triangle inequality, and the definition of Eα−norm, we obtain
|J1| ≤kfkC(Eα) ∞ Z
0
λ−α
2p1+λ×
1
τ2α t
Z
0 e−
p
1+λ|t+τ−s|−e−p1+λ(t+τ+s)−e−p1+λ|t−s|−e−p1+λ(t+s) dsdλ
≤kfkC(Eα) ∞ Z
0
λ−α
2(1+λ) 1
τ2α
1−e−p1+λτdλ
=kfkC(E
α)
1 Z
0
λ−α
2(1+λ) 1
τ2α
1−e−p1+λτdλ+ ∞ Z
1
λ−α
2(1+λ) 1
τ2α
1−e−p1+λτdλ
≤M(α)kfkC(Eα). (12)
In the same manner, we get
|J2| ≤M(α)kfkC(Eα), (13)
|J3| ≤M(α)kfkC(Eα). (14) Estimates (12)-(14) yield that
sup 0≤t<t+τ<∞
|f(t+τ)−f(t)|
τ2α ≤M(α)kfkC(Eα). (15)
Therefore, estimates (10) and (15) finish the proof of Theorem 2.
From the positivity of an elliptic operator Ain the Banach space C R+and estimate (9) it follows the positivity of this operator in Banach spacesC2α R+.
4. Applications
In this section, we will consider some applications of Theorems 1-2. First, we will consider the boundary value problem for the elliptic equation
−∂2∂u(tt2,x)−
∂2u(t,x)
∂x2 +δu(t,x) = f(t,x), 0<t<T, x ∈R+,
u(0,x) =ϕ(x), u(T,x) =ψ(x), x ∈R+, u(t, 0) =0, 0≤t ≤T
. (16)
Theorem 3. Let0<2α <1. Then for the solution of boundary value problem(16), we have the following coercive stability inequality
kut tkC(C2α(R+))+kukC(C2+2α(R+))≤M(α)
kϕkC2+2α(R+)+kψkC2+2α((R+)+kfkC(C2α(R+))
.
The proof of Theorem 3 is based on Theorem 2 on the structure of the fractional spaces Eα(C(R+),A), Theorem 1 on the positivity of the operatorA, on the following theorems on
coercive stability of boundary value for the abstract elliptic equation and on the structure of the fractional space Eα′ = Eα(E,A1/2) which is the Banach space consists of those v ∈ E for
which the norm
||v||E′ α=sup
λ>0 λα
A
1/2 λ+A1/2−1 v
E+||v||E
is finite.
Theorem 4 ([5]). The spaces Eα(E,A)and E2′α(A
1/2,E)coincide for any0< α < 1
2, and their norms are equivalent.
Theorem 5([7]). Let A be positive operator in a Banach space E and f ∈C([0,T],Eα′) (0<
α <1). Then, for the solution of boundary value problem
¨
−u′′(t) +Au(t) = f(t), 0<t<T,
u(0) =ϕ, u(T) =ψ (17)
in a Banach space E with positive operator A the coercive inequality
ku′′kC([0,T],E′
α)+kAukC([0,T],E′α)≤M
kAϕkE′
α+kAψkEα′ +
M
α(1−α)kfkC([0,T],Eα′)
holds.
Second, we will consider the nonlocal-boundary value problem for the elliptic equation
−∂2u(t,x) ∂t2 −
∂2u(t,x)
∂x2 +δu(t,x) = f(t,x), 0<t<T, x ∈R+,
u(0,x) =u(T,x), ut(0,x) =ut(T,x), x ∈R+, u(t, 0) =0, 0≤t≤T
. (18)
Here, f(t,x) is a sufficiently smooth function and they satisfies every compatibility conditions which guarantee the problem(18)has a smooth solution u(t,x). Assume that the assumption of the uniform ellipticity holds.
Theorem 6. Let0<2mα <1. Then for the solution of boundary value problem(18), we have
the following coercive stability inequality
kut tkC(C2α(R+))+kukC(C2+2α(R+))≤M(α)kfkC(C2α(R+)).
The proof of Theorem 6 is based on Theorem 2 on the structure of the fractional spaces Eα(C(R+),A), Theorem 1 on the positivity of the operatorA, Theorem 4 on the structure of
Theorem 7([7]). Let A be positive operator in a Banach space E and f ∈C([0,T],Eα′) (0< α <1). Then, for the solution of the nonlocal boundary value problem
¨
−u′′(t) +Au(t) = f(t), 0<t<T,
u(0) =u(T),u′(0) =u′(T) (19)
in a Banach space E with positive operator A the coercive inequality
ku′′kC([0,T],E′
α)+kAukC([0,T],E′α)≤
M
α(1−α)kfkC([0,T],Eα′)
holds.
5. Conclusion
In the present article, the structure of the fractional spaces Eα(C(R+),A)generated by the
one-dimensional elliptic differential operator Ais investigated. The positivity of this opera-torAin Banach spaces is established. Of course, the difference operatorAh approximates to the operatorAcan be presented. The positivity of this operatorAh in Banach spaces can be
established.
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