Let
be a nonlinear mapping such that
is measurable and
for a positive constant
. We assume that
for the sake of simplicity.
For
we set
where
belongs to
.
Lemma.
Let
,
. Then
,
Moreover if
, then
Proof.
By Hölder inequality we obtain
From (F) and using the Hölder inequality, it is easily seen that
The proof of the second paragraph is similar.
The following lemma is from Brézis ([9], Lemma A.5)
Lemma 3.2.
Let
satisfying
for all
and let
be a constant. Let
be a continuous function on
satisfying the following inequalities:
Then,
Let
. Then invoking Proposition 2.2, we obtain that the problem
has a unique solution
.
The following Lemma is one of the useful integral inequalities.
Lemma.
Let
and consider the following inequality:
Then,
Lemma.
Let
be the solutions of (3.8) with
replaced of by
, respectively. Then the following inequality holds:
where
and
Proof.
For
we consider the following equation:
Then, we have that
for
. Acting on both sides of (3.15) by
, we have
By integration of parts, it holds that
Consequently, since
is self-adjoint, we have
and by Lemma 3.1,
where
By integrating over
, (3.16) implies that
which yields that
From (3.22) and Schwarz's inequality it follows that
Integrating (3.23) over
we have
By Lemma 3.2, we get
where
, that is,
Hence, inequality (3.11) is obtained from (3.22) and (3.26), which implies that
So, by using Lemma 3.1, we obtain (3.12).
Theorem.
Let the assumption (F) be satisfied. Assume that
and
. Then there exists a time
such that the functional differential equation (1.1) admits a unique solution
in
.
Proof.
Let us fix
such that
We are going to show that
is strictly contractive from
to itself if condition (3.28) is satisfied. The norm in
is given by
From (3.11) and (3.12) it follows that
Starting from initial value
, consider a sequence
satisfying
Then from (3.30) it follows that
Hence, we obtain that
So by virtue of condition (3.28) the contraction principle gives that there exists
such that
Since
and the operator
is bounded linear from
to
, it follows from (1.1) that the solution
belongs to
. Thus, it completes the proof of theorem.
From now on, we give a norm estimation of the solution of (1.1) and establish the global existence of solutions with the aid of norm estimations.
Theorem.
Let the assumption (F) be satisfied. Assume that
and
. Then the solution
of (1.1) exists and is unique in
, and there exists a constant
depending on
such that
Proof.
We establish the estimates of solution. Let
be the solution of
Then if
is a solution of (1.1), since
by multiplying
and using the monotonicity of
, then we obtain
By the procedure similar to (3.33) we have
Put
Then from Proposition 2.2, we have that
for some positive constant
. Noting that by Lemma 3.1
and by Proposition 2.2
it is easy to obtain the norm estimate of
in
satisfying (3.35).
Now from (3.41) it follows that
So, we can solve the equation in
and obtain an analogous estimate to (3.41). Since condition (3.28) is independent of initial values, the solution of (1.1) can be extended to the internal
for natural number
, that is, for the initial
in the interval
, as analogous estimate (3.41) holds for the solution in
. Furthermore, the estimate (3.35) is easily obtained from (3.41) and (3.43).
Example.
Let
Let
be another Hilbert space and let
be a bounded linear operator from
to
. We consider the following semilinear control equation of second-order:
where
belongs to
. Let the assumption (F) be satisfied. Assume that
. Then the solution
of (3.46) exists and is unique in
, and the variation of constant formula (3.35) is established.
Remark.
Since the operator
is an unbounded operator, we will make use of the hypothesis (F). If
is a bounded operator from
into itself, then we may assume that
is a nonlinear mapping satisfying
for a positive constant
; therefore, our results can be obtained directly.