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 .
Let , . Then ,
Moreover if , then
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 (, Lemma A.5)
Let satisfying for all and let be a constant. Let be a continuous function on satisfying the following inequalities:
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.
Let and consider the following inequality:
Let be the solutions of (3.8) with replaced of by , respectively. Then the following inequality holds:
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,
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).
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 .
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.
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
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
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).
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.
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.