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Regularity for Solutions of Second-Order Nonlinear Integrodifferential Functional Equations
Journal of Inequalities and Applications volume 2010, Article number: 280463 (2010)
Abstract
We deal with the well-posedness for solutions of nonlinear integrodifferential equations of second-order in Hilbert spaces by converting the problem into the contraction mapping principle with more general conditions on the principal operators and the nonlinear terms and obtain a variation of constant formula of solutions of the given nonlinear equations.
1. Introduction
Let and
be two complex Hilbert spaces. Assume that
is a dense subspace in
and the injection of
into
is continuous. If
is identified with its dual space then we may write
densely and the corresponding injections are continuous. The norm on
,
, and
will be denoted by
,
and
, respectively.
The subset of this paper is to consider the initial value problem of the following second-order nonlinear integrodifferential control system on Hilbert spaces:

where the nonlinear term is given by

Let be a continuous linear operator from
into
which is assumed to satisfy GÃ¥rding's inequality. Here
stands for the dual space of
. The nonlinear mapping
is Lipschitz continuous from
into
.
The regularity results of the evolution equations of second-order in case are given by [1–3]. We also call attention to the method of Baoicchi [4] who has obtained the results directly, without duality, under slightly different hypotheses, which is valid for first- and second-order equations. The well-posedness of solutions for delay evolution equations of second-order in time was referred to in [5]. As for the equation of first order, it is well known as the quasi-autonomous differential equation (see Theorem 2.6 in [1, 11] and [6, 7]).
The paper is a generalization of a particular case of the equation by Brézis [8, 9] based on the Lipschitz continuity of nonlinear term where the nonlinear differential equation of hyperbolic type

has a unique solution which satisfies what follows: if
,
, and
, then

By different methods, results similar to those in result mentioned above were established by Browder [10], Barbu [11], Lions and Strauss [12], and many others. Tanabe [13] proved the existence of local solution of (1.3) when the nonlinear mapping is locally Lipschitz continuous. Even under the weakest assumption on nonlinear term, the existence of weak solution can be shown though the uniqueness is not vertain. For example, it is referred to in a work by Strauss [14]. [15] dealt with an
-approach to second-order nonlinear functional evolutions involving
-accretive operators in Banach spaces.
In this paper, under the assumption that and the local Lipschitz continuity of the nonlinear mapping from
into
(not from
into itself), we deal with regularity for the solution of the given equation (1.1) which will enable us to obtain a global existence theorem for the strict solution
belonging to
; namely, assuming the Lipschitz continuity of nonlinear terms, we show that the well-posedness and stability properties for a class with nonlinear perturbation of second-order are similar to those of its corresponding linear system.
we begin to study well-posedness and stability properties for a class with nonlinear perturbation of second-order.
We will develop and apply the existence theory for first-order differential equations (see [16, 17]) to study certain second-order differential equations associated with nonlinear maximal monotone operators in Hilbert spaces.
2. Linear Hyperbolic Equations
The duality pairing between the element of
and the element
of
is denoted by
, which is the ordinary inner product in
if
. By considering
, we may write
where
and
denote the dual spaces of
and
, respectively. For
we denoted
by the value
of
at
. The norm of
as element of
is given by

Therefore, we assume that has a stronger topology than that of
and, for the brevity, we may regard that

Definition.
Let and
be complex Banach spaces. An operator
from
to
is called antilinear if
and
for
and for
.
Let be a quadratic form defined on
which is linear in
and antilinear in
.
We make the following assumptions:
(i) is bounded, that is,

(ii) is symmetric, that is,

(iii) satisfies the GÃ¥rding's inequality, that is,

Let be the operator such that
. Then, as seen in [13, Theorem 2.2.3], the operator
is positive, definite, and self-adjoint,
, and

It is also known that the operator is a bounded linear from
to
. The realization of
in
which is the restriction of
to
is also denoted by
, which is structured as a Hilbert space with the norm
. Then the operator
generates an analytic semigroup in both of
and
. Thus we have the following sequence:

where each space is dense in the next one which is continuous injection.
If is a Banach space and
, then
is the collection of all strongly measurable functions from
into
the
th powers whose norms are integrable and
is the set of all functions
whose derivatives
up to degree
in the distribution sense belong to
, and
is the set of all
-times continuously differentiable functions from
into
. Let
and
be complex Banach spaces. Denote by
(resp.,
the set of all bounded linear (resp., antilinear) operators from
and
. Let
.
First, consider the following linear hyperbolic equation:

By virtue of Theorem 8.2 of [18], we have the following result on the corresponding linear equation of (2.8) in case .
Proposition.
Suppose that the assumptions for the principal operator stated above are satisfied. Then the following properties hold: For
and
,
, there exists a unique solution
of (2.8) belonging to

and satisfying

where is a constant depending on
. Moreover, the mapping

is a linear continuous map of .
As for the solution which belongs to spaces, we refer to Chapter IV of [17]. The applications of these problems are mixed problems in the sense of Ladyzenskaya [16].
3. Hyperbolic Equations with Nonlinear Perturbations
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.
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Acknowledgment
This work was supported by the Research Foundation funded by Dong-A University.
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Park, DG., Jeong, JM. & Kim, HG. Regularity for Solutions of Second-Order Nonlinear Integrodifferential Functional Equations. J Inequal Appl 2010, 280463 (2010). https://doi.org/10.1155/2010/280463
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DOI: https://doi.org/10.1155/2010/280463
Keywords
- Hilbert Space
- Monotone Operator
- Maximal Monotone
- Integral Inequality
- Maximal Monotone Operator