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.