Existence and uniqueness solution of an inverse problems
for degenerate differential equations
Mahmoud M. El-borai & Osama L. Mostafa & Hoda A. Fouad
m−m−[email protected] &moustaf a−[email protected] & hoda−[email protected] Faculty of Science, Alexandria University, Alexandria, Egypt
Khadug S. Sharnana &khadog−[email protected] Faculty of Science, Almerqeb University, Alkhomes, Libya
Abstract
In this paper we concerned with study existence and uniqueness of solutions for a class of inverse problems of degenerate differential equations.The main tool perturba-tion theory for linear operators.
consider the inverse problems for degenerate differential equations of the form
dBu(t)
dt =Au(t) +Bγ(t)f(t),
with the initial condition
u(0) =u0
and the overdetermination condition
(u(t), v) =w(t)
where A and B are closed linear operators in a Hilbert space H,f is a given abstract function with values in H,v is a given element in H,u0is an initial value,and{u, γ}are
the unknown functions.
Key words: Perturbation Theory of Linear Operators; Linear a c0−Semigroup ;
degen-erate differential equations.
1
Introduction
2
Abstract inverse problem
We establish existence and uniqueness of solutions for a class of inverse problems of degen-erate differential equations.The main tool perturbation theory for linear operators.
consider the inverse problems for degenerate differential equations of the form
dBu(t)
dt =Au(t) +Bγ(t)f(t), (1)
with the initial condition
u(0) =u0 (2)
and the overdetermination condition
(u(t), v) =w(t) (3)
where A and B are closed linear operators in a Hilbert space H,f is a given abstract function with values in H,v is a given element in H,u0 is an initial value,and{u, γ}are the unknown
functions.
We introduce some facts a bout the generator of ac0-semigroup (see [4]).
we denote by X a Banach space with norm k.k and A:D(A)→ X is the infinitesimal generator of ac0-semigroup of bounded linear operator T(t),t >0,on X.
It is well known that A is closed and its domain D(A) equipped with the graph norm
kxkA= kxk+ kAxk
becomes a Banach space,which we shall denote byXA.
Theorem 2.1 Let A be a linear operator on X such that A− β I is maximal dissipative with some real number β and I is a Identity operator, i.e A satisfies
Re(γ, u)X ≤ β kuk2X f or all γ ∈ Au (4)
with the rang condition
R(λ0 I −A) = X f or some λ0 > β. (5)
thenρ(A)⊃(β,∞) and A satisfies
k(λ I−A)−1kX ≤
1
(λ−β), λ > β.
By virtue of theorem (2.1), if A is a linear operator on X with a maximal dissipativeA−β I
,β ∈ R, a semigroup T(t) is generated by A on the whole space X.
We will be interested in the case when
[A∗(B∗)−1]∗= B−1A
whereA∗, B∗ are the adjoint operators of A and B.
The following perturbation result for linear operator will be helpful in the sequel ([3],[2]).
Theorem 2.2 Let X be a Banach space, and let M be the infinitesimal generator of a c0
-semigroup T(t) on X. IfL:XM → XM is a continuous linear operator,thenM+L is the
infinitesimal generator of a c0-semigroup on X.
For more details a bout perturbation theory one can (see [5]) .
Consider the identification problem (1), (2) and (3) where A and B are densely linear operators in the Hilbert space X, such that there adjoint operator satisfyD(A∗) ⊂ D(B∗),
f ∈ X,u0 is an initial value and {u, γ} are the unknown functions.
We assume
Re(A∗s−B∗s)X ≤ βksk2X, f or some S ∈ D(A
∗
). (6)
λ0 ∈ ρB∗(A∗), f or some λ0 > β (7)
Let
T = A∗(B∗)−1, if h ∈ T u,
then
h= A∗(B∗)−1u= A∗s
and
B∗s=u f or some s ∈ D(A∗),
so that
(h, u)X = (A∗s, B∗s)X
On the other hand, for any h ∈ X, we have
h= (λ0 B∗−A∗)s f or some s ∈ D(A∗).
If we putu= B∗s, then
According to theorem (2.1), this proves thatT −β I = A∗(B∗)−1−β I is maximal dissi-pative in X.
As a consequence, its adjoint [A∗(B∗)−1]∗ −β I is also maximal dissipative, so that, [A∗(B∗)−1]∗ is the generator of ac0-semigroup on X.
On the other hand, clearly (1), (2) and (3) is written in the form
du(t)
dt =M u(t) +γ(t)f(t), 0 ≤ t ≤ T (8)
with the initial condition
u(0) =u0 ∈ H (9)
and the overdetermination condition
(u(t), v) =w(t) (10)
Theorem 2.3 Let M and L be densely defined linear operators in the Hilbert space X such thatD(L∗)⊂ D(M∗), f ∈ X, if (6), (7) are satisfied and M is a bounded linear operator on X with ρM∗(L∗)T ρM(L)6= 0.
Then, for any u0 ∈ D(L) such that Lu0 ∈ R(M), the inverse problem (1), (2) and (3)
possesses a unique solution {u, γ}.
Proof:
with a coefficient operator M = B−1A. Moreover, we are interested in the case when [A∗(B∗)−1]∗ and B−1A are coincide.
If B is a bounded linear operator on X andρB∗(A∗)∩ρB(A)6= 0, then [A∗(B∗)−1]∗ =B−1A
and so B−1A is the generator of ac0-semigroup on X.
Multiplying both sides of (8) by v scalarly in H, we obtain the relation
d(u(t), v)
dt = (M u(t), v) +γ(t)(f(t), v)
and using (10),we get
dw(t)
dt = (M u(t), v) +γ(t)(f(t), v)
w0(t) = (M u(t), v) +γ(t)(f(t), v)
then
γ(t) = 1 (f(t), v)(w
0
(t)−(M u(t), v)) (11)
substituting (11) in (8),we obtain
u0(t) =M u(t) + 1 (f(t), v)(w
0
(t)−(M u(t), v))f(t) (12)
common practice involves the operator
L=− 1
(f(t), v)(M u(t), v)f(t) (13)
and equation (12) becomes
u0(t) = (M +L)u(t)+
1 (f(t), v)w
0
(t)f(t) (14)
Plain calculation show that L is bounded in XM,
kLkM = supkLkM = supk − 1
(f(t), v)(M u(t), v)f(t)k
≤ sup 1
|(f(t), v)|kM ukkvkkfk.
Theorem (2.2) now implies thatM+Lis the infinitesimal generator of a semigroupS(t), t≥ 0. Sinceu0 ∈ D(M),then the problem (14) with condition (9) has a unique solutionu(t)
u(t) =S(t)u0+
1 (f(t), v)
Z t
0
S(t−s)w0(s)f(s)ds (15)
By (11) and (15) γ(t) is uniquely determined and the reduced problem (8), (9) and (10) possesses a unique solution (u, γ) , see [6],[7],[8],[9].
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