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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 Gårding'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 Gårding'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, D., Jeong, J. & Kim, H. 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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Keywords
- Hilbert Space
- Monotone Operator
- Maximal Monotone
- Integral Inequality
- Maximal Monotone Operator
