Regularity for Variational Evolution Integrodifferential Inequalities

We deal with the regularity for solutions of nonlinear functional integrodi ﬀ erential 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.


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 H: k t − s g s, x s ds h t , x t − z , a.e., 0 < t ≤ T, z ∈ H, x 0 x 0 , VIP where A is a unbounded linear operator associated with a sesquilinear form satisfying Gårding's inequality and φ : H → −∞, ∞ is a lower semicontinuous, proper convex function. The nonlinear mapping g is a Lipschitz continuous from R × V into H in the second coordinate, where V is a dense subspace of H.

Abstract and Applied Analysis
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 ∂φ, the control system VIP is represented by the following nonlinear functional differential equation on H: x t Ax t ∂φ x t t 0 k t − s g s, x s ds h t , 0 < t ≤ T, x 0 x 0 .
NDE 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 g ≡ 0. Recently, the regular problem for solutions of the nonlinear functional differential equations with a nonlinear hemicontinuous and coercive operator A 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 φ : H → H > 0 such that the function φ is Fréchet differentiable on H and its Frećhet differential ∂φ is a single valued and Lipschitz continuous on H with Lipschitz constant −1 , where ∂φ −1 I − I ∂φ −1 as is seen in Corollary 2.2 in 1, Chapter II . It is also well-known results that lim → 0 φ φ and lim → 0 ∂φ x ∂φ 0 x for every x ∈ D ∂φ , where ∂φ 0 is the minimal segment of ∂φ. Now, we introduce the smoothing system corresponding to NDE as follows: x 0 x 0 .

SDE 1
First we recall some regularity results and a variation of constant formula for solutions of the semilinear functional differential equation in the case g ≡ 0 in SDE 1 : Next, based on the regularity results for 1.1 , we intend to establish the regularity for solutions of NDE . Here, our approach is that results of a class of semilinear equations as 1.1 on L 2 -regularity remain valid under the above formulation perturbed of nonlinear terms. Here, we note that sine A is not bounded operator H 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. Abstract and Applied Analysis 3

Preliminaries
Let V and H be complex Hilbert spaces forming Gelfand triple V ⊂ H ⊂ V * with pivot space H. The norms of V , H and V * are denoted by || · ||, | · |, and || · || * , respectively. The inner product in H is defined by ·, · . The embeddings are continuous. Then the following inequality easily follows: Let a ·, · be a bounded sesquilinear form defined in V × V and satisfying Gårding's inequality Let A be the operator associated with the sesquilinear form a ·, · : Then A is a bounded linear operator from V to V * and −A generates an analytic semigroup in both of H and V * as is seen in 9, Theorem 6.1 . The realization for the operator A in H which is the restriction of A to is also denoted by A. From the following inequalities: is the graph norm of D A , it follows that there exists a constant C 1 > 0 such that Thus, we have the following sequence: where each space is dense in the next one and continuous injection.  [10] or [11]).
The following abstract linear parabolic equation: Ae tA x 2 * dt.

2.15
Abstract and Applied Analysis 5 Thus, we have u ∈ L 2 0, T; V ∩ W 1,2 0, T; V * . By using the definition of real interpolation spaces by trace method, it is known that the embedding In view of Lemma 2.2 we can apply 2.11 to LE in the space V * as follows.
Then there exists a unique solution x of LE belonging to and satisfying where C 2 is a constant depending on T .
Let φ : V → −∞, ∞ be a lower semicontinuous, proper convex function. Then the subdifferential operator ∂φ of φ is defined by First, let us concern with the following perturbation of subdifferential operator:

VE
Using the regularity for the variational inequality of parabolic type in case where φ : V → −∞, ∞ is a lower semicontinuous, proper convex function as is seen in 1, Section 4.3 one has the following result on VE .

2.20
where C 3 is a constant and

Abstract and Applied Analysis
(2) Let A be symmetric and let us assume that there exist g ∈ H such that for every > 0 and any y ∈ D φ J y g ∈ D φ , φ J y g ≤ φ y .

2.21
Then for h ∈ L 2 0, T; H and x 0 ∈ D φ ∩ V , VE has a unique solution: which satisfies Remark 2.5. When the principal operator A is bounded from H to itself, we assume that φ : H → −∞, ∞ is a lower semicontinuous, proper convex function and g : 0, T × H → H 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 V is compactly embedded in H and x ∈ L 2 0, T; V or the semigroup operator S t is compact , the following embedding: is compact in view of Theorem 2 of Aubin 14 . Hence, the mapping x 0 , f → x is compact from V × L 2 0, T; V * to L 2 0, T; H , which is also applicable to optimal control problem.

Regularity for Solutions
We start with the following assumption.
Assumption F . Let g : 0, T × V → H be a nonlinear mapping satisfying the following: where k belongs to L 2 0, T .
Abstract and Applied Analysis 7 The proof is immediately obtained from Assumption F . For every > 0, define where J I ∂φ −1 . Then the function φ is Frećhet differentiable on H and its Frećhet differential ∂φ is Lipschitz continuous on H with Lipschitz constant −1 where ∂φ −1 I − I ∂φ −1 as is seen in Corollary 2.2 in 1, Chapter II . It is also well-known results that lim → 0 φ φ and lim → 0 ∂φ x ∂φ 0 x for every x ∈ D ∂φ , where ∂φ 0 is the minimal segment of ∂φ. Now, one introduces the smoothing system corresponding to NDE as follows: x t Ax t ∂φ x t f t, x h t , 0 < t ≤ T, x 0 x 0 .

SDE 2
Since −A generates a semigroup S t on H, the mild solution of SDE 2 can be represented by One will use a fixed point theorem and a step and step method to get the global solution for NDE . Then one needs the following hypothesis.
Assumption A . ∂φ 0 is uniformly bounded, that is, Lemma 3.2. For given , λ > 0, let x and x λ be the solutions of SDE 2 corresponding to and λ, respectively. Then there exists a constant C independent of and λ such that x − x λ C 0,T ;H ∩L 2 0,T ;V ≤ C λ , 0 < T.

8 Abstract and Applied Analysis
Proof. From SDE 2 we have and hence, from 2.3 and multiplying by x t − x λ t , it follows that

3.10
Here, we note

3.11
Thus, we have

3.12
Therefore, by using the monotonicity of ∂φ and integrating 3.10 over 0, T it holds

3.15
Let x ∈ L 1 0, T; V . Then it is well known that for almost all point of t ∈ 0, T . We establish the following results on the solvability of NDE .

Theorem 3.4. Let Assumptions (F) and (A) be satisfied. Then for every
NDE has a unique solution: x ∈ L 2 0, T; V ∩ W 1,2 0, T; V * ∩ C 0, T ; H , 3.17 and there exists a constant C 4 depending on T such that Proof. Let us fix T 0 > 0 such that Let y ∈ L 2 0, T 0 ; V . Then f ·, y · ∈ L 2 0, T 0 ; H from Assumption F . Set Then from Lemma 3.1 it follows that

3.21
For i 1, 2, we consider the following equation: x i t Ax i t Fy i t h t , 0 < t ≤ T 0 ,

3.22
Then d dt

3.23
From 2.11 it follows that Using the Hölder inequality we also obtain that

3.25
Therefore, in terms of 2.8 and 3.25 we have

3.26
So by virtue of the condition 3.19 the contraction principle gives that SDE 2 has a unique solution in 0, T 0 . Thus, letting λ → 0 in Lemma 3.1 we can see that there exists a constant C independent of such that x − x C 0,T 0 ;H ∩L 2 0,T 0 ;V ≤ C , 0 < T 0 , 3.27 and hence, lim → 0 x t x t exists in H. From Assumption F and 3.27 it follows that

3.28
Since ∂φ x is uniformly bounded by Assumption A , from 3.27 , 3.28 we have that

3.30
Since I ∂φ −1 x → x strongly and ∂φ is demiclosed, we have that Thus we have proved that x t satisfies a.e. on 0, T 0 the equation NDE . Let y be the solution of y t Ay t ∂φ y t 0, 0 < t ≤ T 0 , Noting that || · || ≤ | · | ≤ || · ||, by multiplying by x t − y t and using the monotonicity of ∂φ and 2.3 , we obtain

3.34
Since for every c > 0 and by integrating on 3.34 over 0, t we have and by Gronwall's inequality: Let us fix T 0 > T 1 > 0 so that T 1 is a Lebesgue point of x, φ x T 1 < ∞, and then from Assumption F it follows x − y L 2 0,T 1 ;V ≤ N f ·, x h L 2 0,T 1 ;V * ≤ N T 1 L k L 2 x L 2 0,T 1 ;V N h L 2 0,T 1 :V *

3.40
and hence, from 2.17 in Proposition 2.3, we have that for some positive constant C 4 . Since the condition 3.38 is independent of initial values, noting the Assumption A , the solution of NDE can be extended to the internal 0, nT 1 for natural number n, that is, for the initial x nT 1 in the interval nT 1 , n 1 T 1 , as analogous Abstract and Applied Analysis 13 estimate 3.41 holds for the solution in 0, n 1 T 1 . The norm estimate of x in W 1,2 0, T; H can be obtained by acting on both side of NDE by x t and by using a.e., 0 < t, 3.42 for all g t ∈ ∂φ x t . Furthermore, the estimate 3.18 is immediately obtained from 3.41 .

3.44
Let us fix T 1 > T 2 > 0 so that T 2 is a Lebesgue point of x, φ x T 2 < ∞, and Since by integrating on 3.44 over 0, T 2 where T 2 < T and as is seen in 3.37 , it follows

14
Abstract and Applied Analysis Putting that we have Suppose x 0n , h n → x 0 , h in V × L 2 0, T; V * , and let x n and x be the solutions SDE 2 with x 0n , h n and x 0 , h , respectively. Then, by virtue of 3.44 and 3.49 , we see that Therefore the same argument shows that x n → x in Repeating this process, we conclude that

Example
Let Ω be bounded domain in R n with smooth boundary ∂Ω. We define the following spaces: where ∂/∂x i u and ∂ 2 /∂x i ∂x j u are the derivative of u in the distribution sense. The norm of

4.2
Hence H 1 0 Ω is a Hilbert space. Let H −1 Ω H 1 0 Ω * be a dual space of H 1 0 Ω . For any l ∈ H −1 Ω and v ∈ H 1 0 Ω , the notation l, v denotes the value l at v. 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 a ij a ji are continuous and bounded on Ω and {a ij x } is positive definite uniformly in Ω, that is, there exists a positive number δ such that For each u, v ∈ H 1 0 Ω , let us consider the following sesquilinear form:  We assume the following.
Assumption F1 . The partial derivatives σ i s, ξ , ∂/∂t σ i s, ξ and ∂/∂ξ j σ i s, ξ , exist and continuous for i 1, 2, j 1, 2, . . . , n, and σ i s, ξ satisfies an uniform Lipschitz condition with respect to ξ, that is, there exists a constant L > 0 such that where | · | denotes the norm of L 2 Ω . where k belongs to L 2 0, T . Let φ : H 1 0 Ω → −∞, ∞ 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.

4.20
Furthermore, the following energy inequality holds: there exists a constant C T depending on T such that u L 2 ∩W 1,2 ≤ C T 1 u 0 h L 2 0,T ;H −1 Ω . 4.21