#### Abstract

We deal with the regularity for solutions of nonlinear functional integrodifferential equations governed by the variational inequality in a Hilbert space. Moreover, by using the simplest definition of interpolation spaces and the known regularity result, we also prove that the solution mapping from the set of initial and forcing data to the state space of solutions is continuous, which very often arises in application. Finally, an example is also given to illustrate our main result.

#### 1. Introduction

In this paper, we deal with the regularity for solutions of nonlinear functional integrodifferential equations governed by the variational inequality in a Hilbert space : where is a unbounded linear operator associated with a sesquilinear form satisfying Gårding’s inequality and is a lower semicontinuous, proper convex function. The nonlinear mapping is a Lipschitz continuous from into in the second coordinate, where is a dense subspace of .

The background of these problems has emerged vigorously in such applied fields as automatic control theory, network theory, and the dynamic systems.

By using the subdifferential operator , the control system is represented by the following nonlinear functional differential equation on :

In Section 4.3.2 of Barbu [1] (also see Section 4.3.1 in [2]) is widely developed the existence of solutions for the case . Recently, the regular problem for solutions of the nonlinear functional differential equations with a nonlinear hemicontinuous and coercive operator was studied in [3]. Some results for solutions of a class of semilinear equations with the nonlinear terms have been dealt with in [3–7]. As for nontrivial physical examples from the field of visco-elastic materials modeled by integrodifferential equations on Banach spaces, we refer to [8].

In this paper, we will define such that the function is Fréchet differentiable on and its Frećhet differential is a single valued and Lipschitz continuous on with Lipschitz constant , where as is seen in Corollary 2.2 in [1, Chapter II]. It is also well-known results that and for every , where is the minimal segment of . Now, we introduce the smoothing system corresponding to as follows:

First we recall some regularity results and a variation of constant formula for solutions of the semilinear functional differential equation (in the case in : in a Hilbert space .

Next, based on the regularity results for (1.1), we intend to establish the regularity for solutions of . Here, our approach is that results of a class of semilinear equations as (1.1) on -regularity remain valid under the above formulation perturbed of nonlinear terms. Here, we note that sine is not bounded operator into itself, the Lipschitz continuity of nonlinear terms must be defined on some adjusted spaces (see Section 3). Moreover, using the simplest definition of interpolation spaces and known regularity, we have that the solution mapping from the set of initial and forcing data to the state space of solutions is continuous, which very often arises in application. Finally, an example is also given to illustrate our main result.

#### 2. Preliminaries

Let and be complex Hilbert spaces forming Gelfand triple with pivot space . The norms of , and are denoted by , , and , respectively. The inner product in is defined by . The embeddings are continuous. Then the following inequality easily follows:

Let be a bounded sesquilinear form defined in and satisfying Gårding’s inequality Let be the operator associated with the sesquilinear form : Then is a bounded linear operator from to and generates an analytic semigroup in both of and as is seen in [9, Theorem 6.1]. The realization for the operator in which is the restriction of to is also denoted by . From the following inequalities: where is the graph norm of , it follows that there exists a constant such that Thus, we have the following sequence: where each space is dense in the next one and continuous injection.

Lemma 2.1. *With the notations (2.8), (2.9), one has
**
where denotes the real interpolation space between and (Section 2.4 of [10] or [11]).*

The following abstract linear parabolic equation: has a unique solution for each if and . Moreover, one has where depends on and (see [12, Theorem 2.3], [13]).

In order to substitute for the intermediate space considering as an operator in instead of one proves the following result.

Lemma 2.2. *Let . Then
**
Hence, it implies that in the sense of intermediate spaces generated by an analytic semigroup.*

* Proof. *Put for . From the result of Theorem 2.3 in [12] it follows
hence

Conversely, suppose that and . Put . Then since is an isomorphism from to there exists a constant such that
Thus, we have . By using the definition of real interpolation spaces by trace method, it is known that the embedding is continuous. Hence, it follows .

In view of Lemma 2.2 we can apply (2.11) to in the space as follows.

Proposition 2.3. *Let and , . Then there exists a unique solution of belonging to
**
and satisfying
**
where is a constant depending on .*

Let be a lower semicontinuous, proper convex function. Then the subdifferential operator of is defined by First, let us concern with the following perturbation of subdifferential operator:

Using the regularity for the variational inequality of parabolic type in case where is a lower semicontinuous, proper convex function as is seen in [1, Section 4.3] one has the following result on .

Proposition 2.4. *(1) Let and satisfying that . Then has a unique solution:
**
which satisfies
**
where is a constant and .**(2) Let be symmetric and let us assume that there exist such that for every and any **
Then for and , has a unique solution:
**
which satisfies
*

*Remark 2.5. *When the principal operator is bounded from to itself, we assume that is a lower semicontinuous, proper convex function and be a nonlinear mapping satisfying the following:
Then it is easily seen that the result of (2) of Proposition 2.4. is immediately obtained.

*Remark 2.6. *Here, we remark that if is compactly embedded in and (or the semigroup operator is compact), the following embedding:
is compact in view of Theorem 2 of Aubin [14]. Hence, the mapping is compact from to , which is also applicable to optimal control problem.

#### 3. Regularity for Solutions

We start with the following assumption.

*Assumption ( F). *Let be a nonlinear mapping satisfying the following:
for a positive constant .

For we set where belongs to .

Lemma 3.1. *Let , . Then . And
**
Moreover, if , then
*

The proof is immediately obtained from Assumption (F).

For every , define where . Then the function is Frećhet differentiable on and its Frećhet differential is Lipschitz continuous on with Lipschitz constant where as is seen in Corollary 2.2 in [1, Chapter II]. It is also well-known results that and for every , where is the minimal segment of .

Now, one introduces the smoothing system corresponding to as follows: Since generates a semigroup on , the mild solution of can be represented by

One will use a fixed point theorem and a step and step method to get the global solution for . Then one needs the following hypothesis.

*Assumption ( A). * is uniformly bounded, that is,

Lemma 3.2. *For given , , let and be the solutions of corresponding to and , respectively. Then there exists a constant independent of and such that
*

*Proof. *From we have
and hence, from (2.3) and multiplying by , it follows that
Here, we note
Thus, we have
Therefore, by using the monotonicity of and integrating (3.10) over it holds
Here, we used that
Since for every it follows from Assumption (A) and using Gronwall’s inequality that

Let . Then it is well known that for almost all point of .

*Definition 3.3. *The point which permits (3.16) to hold is called the Lebesgue point of .

We establish the following results on the solvability of .

Theorem 3.4. *Let Assumptions (F) and (A) be satisfied. Then for every , has a unique solution:
**
and there exists a constant depending on such that
*

*Proof. *Let us fix such that
Let . Then from Assumption (F). Set
Then from Lemma 3.1 it follows that
For , we consider the following equation:
Then
From (2.11) it follows that
Using the Hölder inequality we also obtain that
Therefore, in terms of (2.8) and (3.25) we have
So by virtue of the condition (3.19) the contraction principle gives that has a unique solution in . Thus, letting in Lemma 3.1 we can see that there exists a constant independent of such that
and hence, exists in . From Assumption (F) and (3.27) it follows that
Since is uniformly bounded by Assumption (A), from (3.27), (3.28) we have that
therefore
Since and is demiclosed, we have that
Thus we have proved that satisfies a.e. on the equation .

Let be the solution of
then, it implies
Noting that , by multiplying by and using the monotonicity of and (2.3), we obtain
Since
for every and by integrating on (3.34) over we have
and by Gronwall’s inequality:
Let us fix so that is a Lebesgue point of , , and
Put
then from Assumption (F) it follows
and hence, from (2.17) in Proposition 2.3, we have that
for some positive constant . Since the condition (3.38) is independent of initial values, noting the Assumption (A), the solution of 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 . The norm estimate of in can be obtained by acting on both side of by and by using
for all . Furthermore, the estimate (3.18) is immediately obtained from (3.41).

Theorem 3.5. *Let Assumptions (F) and (A) be satisfied and , then the solution of belongs to and the mapping:
**
is continuous.*

*Proof. *If then belongs to form Theorem 3.4. Let and be the solution of with in place of for . Multiplying on by , we have
Let us fix so that is a Lebesgue point of , , and
Since
by integrating on (3.44) over where and as is seen in (3.37), it follows
Putting that
we have
Suppose in , and let and be the solutions with and , respectively. Then, by virtue of (3.44) and (3.49), we see that in . This implies that in . Therefore the same argument shows that in
Repeating this process, we conclude that in .

#### 4. Example

Let be bounded domain in with smooth boundary . We define the following spaces: where and are the derivative of in the distribution sense. The norm of is defined by Hence is a Hilbert space. Let = be a dual space of . For any and , the notation denotes the value at . In what follows, we consider the regularity for given equations in the spaces: as introduced in Section 2. We deal with the Dirichlet condition’s case as follows.

Assume that are continuous and bounded on and is positive definite uniformly in , that is, there exists a positive number such that Let For each , let us consider the following sesquilinear form: Since is real symmetric, by (4.4) the inequality: holds for all complex vectors . By hypothesis, there exists a constant such that and hold a.e., hence By choosing , we have By virtue of Lax-Milgram theorem, we know that for any there exists such that Therefore, we know that the associated operator defined by is bounded and satisfies conditions (2.3) in Section 2.

Let be a nonlinear mapping defined by

We assume the following.

*Assumption ( F1). *The partial derivatives , and , exist and continuous for , , and satisfies an uniform Lipschitz condition with respect to , that is, there exists a constant such that
where denotes the norm of .

Lemma 4.1. *If Assumption (F1) is satisfied, then the mapping is continuously differentiable on and is Lipschitz continuous on .*

*Proof. *Put
then we have . For each , we satisfy the following that
The nonlinear term is given by
For any , if and belong to , by Assumption (F1) we obtain

We set where belongs to . Let be a lower semicontinuous, proper convex function. Now in virtue of Lemma 4.1, we can apply the results of Theorem 3.4 as follows.

Theorem 4.2. *Let Assumption (F1) be satisfied. Then for any and , the following nonlinear problem:
**
has a unique solution:
**Furthermore, the following energy inequality holds: there exists a constant depending on such that
*

#### Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0026609).