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 H:
(VIP)(x′(t)+Ax(t),x(t)-z)+ϕ(x(t))-ϕ(z) ≤(∫0tk(t-s)g(s,x(s))ds+h(t),x(t)-z), a.e., 0<t≤T, z∈H,x(0)=x0,
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.

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:
(NDE)x′(t)+Ax(t)+∂ϕ(x(t))∋∫0tk(t-s)g(s,x(s))ds+h(t), 0<t≤T,x(0)=x0.

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 ϕϵ:H→H(ϵ>0) such that the function ϕϵ is Fréchet differentiable on H and its Frećhet differential ∂ϕϵ is a single valued and Lipschitz continuous on H with Lipschitz constant ϵ-1, where ∂ϕϵ=ϵ-1(I-(I+ϵ∂ϕ)-1) as is seen in Corollary 2.2 in [1, Chapter II]. It is also well-known results that lim ϵ→0ϕϵ=ϕ and lim ϵ→0∂ϕϵ(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:
(SDE 1)x′(t)+Ax(t)+∂ϕϵ(x(t))=∫0tk(t-s)g(s,x(s))ds+h(t), 0<t≤T,x(0)=x0.

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):
(1.1)x′(t)+Ax(t)+∂ϕϵ(x(t))=h(t)
in a Hilbert space H.

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 L2-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.

2. 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
(2.1)V↪H↪V*
are continuous. Then the following inequality easily follows:
(2.2)∥u∥*≤|u|≤∥u∥, ∀u∈V.

Let a(·,·) be a bounded sesquilinear form defined in V×V and satisfying Gårding’s inequality
(2.3)Re a(u,u)≥ω1∥u∥2-ω2|u|2, ω1>0, ω2≥0.
Let A be the operator associated with the sesquilinear form a(·,·):
(2.4)(Au,v)=a(u,v), u,v∈V.
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
(2.5)D(A)={u∈V; Au∈H}
is also denoted by A. From the following inequalities:
(2.6)ω1∥u∥2≤Re a(u,u)+ω2|u|2≤C|Au| |u|+ω2|u|2≤max {C,ω2}∥u∥D(A)|u|,
where
(2.7)∥u∥D(A)=(|Au|2+|u|2)1/2
is the graph norm of D(A), it follows that there exists a constant C1>0 such that
(2.8)∥u∥≤C1∥u∥D(A)1/2|u|1/2.
Thus, we have the following sequence:
(2.9)D(A)⊂V⊂H⊂V*⊂D(A)*,
where each space is dense in the next one and continuous injection.

Lemma 2.1.
With the notations (2.8), (2.9), one has
(2.10)(D(A),H)1/2,2=V,
where (D(A),H)1/2,2 denotes the real interpolation space between D(A) and H (Section 2.4 of [10] or [11]).

The following abstract linear parabolic equation:
(LE)x′(t)+Ax(t)=h(t), 0<t≤T,x(0)=x0,
has a unique solution x∈L2(0,T;D(A))∩W1,2(0,T,H) for each T>0 if x0∈V≡(D(A),H)1/2,2 and h∈L2(0,T;H). Moreover, one has
(2.11)∥x∥L2(0,T;D(A))∩W1,2(0,T,H)≤C2(∥x0∥(D(A),H)1/2,2+∥h∥L2(0,T;H)),
where C2 depends on T and M (see [12, Theorem 2.3], [13]).

In order to substitute H for the intermediate space V considering A as an operator in B(V,V*) instead of B(D(A),H) one proves the following result.

Lemma 2.2.
Let T>0. Then
(2.12)H={x∈V*:∫0T∥AetAx∥*2dt<∞}.
Hence, it implies that H=(V,V*)1/2,2 in the sense of intermediate spaces generated by an analytic semigroup.

Proof.
Put u(t)=etAx for x∈H. From the result of Theorem 2.3 in [12] it follows
(2.13)u∈L2(0,T;V)∩W1,2(0,T;V*),
hence
(2.14)∫0T∥AetAx∥*2dt=∫0T∥u′(t)∥*2dt<∞.

Conversely, suppose that x∈V* and ∫0T||AetAx||*2dt<∞. Put u(t)=etAx. Then since A is an isomorphism from V to V* there exists a constant c>0 such that
(2.15)∫0T∥u(t)∥2dt≤c∫0T∥Au(t)∥*2dt=c∫0T∥AetAx∥*2dt.
Thus, we have u∈L2(0,T;V)∩W1,2(0,T;V*). By using the definition of real interpolation spaces by trace method, it is known that the embedding L2(0,T;V)∩W1,2(0,T;V*)↪C([0,T];H) is continuous. Hence, it follows x=u(0)∈H.

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

Proposition 2.3.
Let x0∈H and h∈L2(0,T;V*), T>0. Then there exists a unique solution x of (LE) belonging to
(2.16)L2(0,T;V)∩W1,2(0,T;V*)↪C([0,T];H)
and satisfying
(2.17)∥x∥L2(0,T;V)∩W1,2(0,T;V*)≤C2(|x0|+∥h∥L2(0,T;V*)),
where C2 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
(2.18)∂ϕ(x)={x*∈V*;ϕ(x)≤ϕ(y)+(x*,x-y), y∈V}.
First, let us concern with the following perturbation of subdifferential operator:
(VE)x′(t)+Ax(t)+∂ϕ(x(t))∋h(t), 0<t≤T,x(0)=x0.

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).

Proposition 2.4.
(1) Let h∈L2(0,T;V*) and x0∈V satisfying that ϕ(x0)<∞. Then (VE) has a unique solution:
(2.19)x∈L2(0,T;V)∩W1,2(0,T;V*)↪C([0,T];H),
which satisfies
(2.20)x′(t)=(h(t)-Ax(t)-∂ϕ(x(t)))0,∥x∥L2∩W1,2∩C≤C3(1+∥x0∥+∥h∥L2(0,T;V*)),
where C3 is a constant and L2∩W1,2∩C=L2(0,T;V)∩W1,2(0,T;V*)∩C([0,T];H).

(2) Let A be symmetric and let us assume that there exist g∈H such that for every ϵ>0 and any y∈D(ϕ)(2.21)Jϵ(y+ϵg)∈D(ϕ), ϕ(Jϵ(y+ϵg))≤ϕ(y).
Then for h∈L2(0,T;H) and x0∈D(ϕ)¯∩V, (VE) has a unique solution:
(2.22)x∈L2(0,T;D(A))∩W1,2(0,T;H)↪C([0,T];H),
which satisfies
(2.23)∥x∥L2∩W1,2∩C≤C3(1+∥x0∥+∥h∥L2(0,T;H)).

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:
(2.24)|g(t,x1)-g(t,x2)|≤L|x1-x2|, ∀x1,x2∈H.
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∈L2(0,T;V)) (or the semigroup operator S(t) is compact), the following embedding:
(2.25)L2(0,T;V)∩W1,2(0,T;V*)↪L2(0,T;H)
is compact in view of Theorem 2 of Aubin [14]. Hence, the mapping (x0,f)↦x is compact from V×L2(0,T;V*) to L2(0,T;H), which is also applicable to optimal control problem.

3. Regularity for Solutions
We start with the following assumption.

Assumption (<italic>F</italic>).
Let g:[0,T]×V→H be a nonlinear mapping satisfying the following:
(3.1)|g(t,x)-g(t,y)| ≤L∥x-y∥, g(t,0)=0 ∀x,y∈V
for a positive constant L.

For x∈L2(0,T;V) we set
(3.2)f(t,x)=∫0tk(t-s)g(s,x(s))ds,
where k belongs to L2(0,T).

Lemma 3.1.
Let x∈L2(0,T;V), T>0. Then f(·,x)∈L2(0,T;H). And
(3.3)∥f(·,x)∥L2(0,T;H)≤L∥k∥L2T∥x∥L2(0,T;V).
Moreover, if x1,x2∈L2(0,T;H), then
(3.4)∥f(·,x1)-f(·,x2)∥L2(0,T;H)≤L∥k∥T∥x1-x2∥L2(0,T;V).

The proof is immediately obtained from Assumption (F).

For every ϵ>0, define
(3.5)ϕϵ(x)=inf {∥x-Jϵx∥*22ϵ+ϕ(Jϵx):x∈H},
where Jϵ=(I+ϵ∂ϕ)-1. Then the function ϕϵ is Frećhet differentiable on H and its Frećhet differential ∂ϕϵ is Lipschitz continuous on H with Lipschitz constant ϵ-1 where ∂ϕϵ=ϵ-1(I-(I+ϵ∂ϕ)-1) as is seen in Corollary 2.2 in [1, Chapter II]. It is also well-known results that lim ϵ→0 ϕϵ=ϕ and lim ϵ→0 ∂ϕϵ(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:
(SDE 2)x′(t)+Ax(t)+∂ϕϵ(x(t))=f(t,x)+h(t), 0<t≤T,x(0) =x0.
Since -A generates a semigroup S(t) on H, the mild solution of (SDE 2) can be represented by
(3.6)xϵ(t)=S(t)x0+∫0tS(t-s){f(s,xϵ)+h(s)-∂ϕϵ(xϵ(s))}ds.

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 (<italic>A</italic>).
(
∂
ϕ
)
0
is uniformly bounded, that is,
(3.7)|(∂ϕ)0x|≤M1, x∈V.

Lemma 3.2.
For given ϵ, λ>0, let xϵ and xλ be the solutions of (SDE 2) corresponding to ϵ and λ, respectively. Then there exists a constant C independent of ϵ and λ such that
(3.8)∥xϵ-xλ∥C([0,T];H)∩L2(0,T;V)≤C(ϵ+λ), 0<T.

Proof.
From (SDE 2) we have
(3.9)xϵ′(t)-xλ′(t)+A(xϵ(t)-xλ(t))+∂ϕϵ(xϵ(t))-∂ϕλ(xλ(t))=f(t,xϵ)-f(t,xλ),
and hence, from (2.3) and multiplying by xϵ(t)-xλ(t), it follows that
(3.10)12ddt|xϵ(t)-xλ(t)|2+ω1∥xϵ(t)-xλ(t)∥2+(∂ϕϵ(xϵ(t))-∂ϕλ(xλ(t)),xϵ(t)-xλ(t)) ≤(f(t,xϵ)-f(t,xλ),xϵ(t)-xλ(t))+ω2|xϵ(t)-xλ(t)|2.
Here, we note
(3.11)|f(t,xϵ)-f(t,xλ)|≤L∥k∥L2∥xϵ(·)-xλ(·)∥L2(0,t;V)∫0T∥xϵ(·)-xλ(·)∥L2(0,t;V)2dt=T∫0T∥xϵ(t)-xλ(t)∥2dt.
Thus, we have
(3.12)(f(t,xϵ)-f(t,xλ),xϵ(t)-xλ(t)) ≤|f(t,xϵ)-f(t,xλ)|·|xϵ(t)-xλ(t)| ≤ω12T(L∥k∥L2)2|f(t,xϵ)-f(t,xλ)|2+T(L∥k∥L2)22ω1|xϵ(t)-xλ(t)|2 ≤ω12T∥xϵ(·)-xλ(·)∥L2(0,t;H)2+T(L∥k∥L2)22ω1|xϵ(t)-xλ(t)|2.
Therefore, by using the monotonicity of ∂ϕ and integrating (3.10) over [0,T] it holds
(3.13)12|xϵ(t)-xλ(t)|2+ω12∫0T∥xϵ(t)-xλ(t)∥2dt ≤∫0T(∂ϕϵ(xϵ(t))-∂ϕλ(xλ(t)),λ∂ϕλ(xλ(t))-ϵ∂ϕϵ(xϵ(t)))dt +{T(L||k||L2)22ω1+ω2}∫0T|xϵ(t)-xλ(t)|2dt.
Here, we used that
(3.14)∂ϕϵ(xϵ(t))=ϵ-1(xϵ(t)-(I+ϵ∂ϕ)-1xϵ(t)).
Since |∂ϕϵ(x)|≤|(∂ϕ)0x| for every x∈D(∂ϕ) it follows from Assumption (A) and using Gronwall’s inequality that
(3.15)∥xϵ-xλ∥C([0,T];H)∩L2(0,T;V)≤C(ϵ+λ), 0<T.

Let x∈L1(0,T;V). Then it is well known that
(3.16)lim h→0h-1∫0h∥x(t+s)-x(t)∥ds=0
for almost all point of t∈(0,T).

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

We establish the following results on the solvability of (NDE).

Theorem 3.4.
Let Assumptions (F) and (A) be satisfied. Then for every (x0,h)∈V×L2(0,T;V*), (NDE) has a unique solution:
(3.17)x∈L2(0,T;V)∩W1,2(0,T;V*)∩C([0,T];H),
and there exists a constant C4 depending on T such that
(3.18)∥x∥L2∩W1,2∩C≤C4(1+∥x0∥+∥h∥L2(0,T;V*)).

Proof.
Let us fix T0>0 such that
(3.19)C1C2(ϵ-1+T0 L∥k∥L2)(T02)1/2<1.
Let y∈L2(0,T0;V). Then f(·,y(·))∈L2(0,T0;H) from Assumption (F). Set
(3.20)(Fx)(t)=f(t,x(t))-∂ϕϵ(x(t)), 0≤t≤T0.
Then from Lemma 3.1 it follows that
(3.21)|(Fx1)(t)-(Fx2)(t)|≤(ϵ-1+T0L∥k∥L2)∥x1(t)-x2(t)∥.
For i=1,2, we consider the following equation:
(3.22)xi′(t)+Axi(t)=(Fyi)(t)+h(t), 0<t≤T0,xi(0)=x0.
Then
(3.23)ddt(x1(t)-x2(t))+A(x1(t)-x2(t))=(Fy1)(t)-(Fy2)(t), t>0,x1(0)-x2(0)=0.
From (2.11) it follows that
(3.24)∥x1-x2∥L2(0,T0;D(A0))∩W1,2(0,T0;H)≤C2∥Fy1-Fy2∥L2(0,T0;H).
Using the Hölder inequality we also obtain that
(3.25)∥x1-x2∥L2(0,T0;H)={∫0T0|x1(t)-x2(t)|2dt}1/2={∫0T0|∫0t(x˙1(τ)-x˙2(τ))dτ|2dt}1/2≤{∫0T0t∫0t|x˙1(τ)-x˙2(τ)|2dτdt}1/2≤T02∥x1-x2∥W1,2(0,T0;H).
Therefore, in terms of (2.8) and (3.25) we have
(3.26)∥x1-x2∥L2(0,T0;V)≤C1∥x1-x2∥L2(0,T0;D(A0))1/2∥x1-x2∥L2(0,T0;H)1/2 ≤C1∥x1-x2∥L2(0,T0;D(A0))1/2(T02)1/2∥x1-x2∥W1,2(0,T0;H)1/2 ≤C1(T02)1/2∥x1-x2∥L2(0,T0;D(A0))∩W1,2(0,T0;H) ≤C1C2(T02)1/2∥Fy1-Fy2∥L2(0,T0:H) ≤C1C2(ϵ-1+T0L∥k∥L2)(T02)1/2∥y1-y2∥L2(0,T0;V).
So by virtue of the condition (3.19) the contraction principle gives that (SDE 2) has a unique solution in [0,T0]. Thus, letting λ→0 in Lemma 3.1 we can see that there exists a constant C independent of ϵ such that
(3.27)∥xϵ-x∥C([0,T0];H)∩L2(0,T0;V)≤Cϵ, 0<T0,
and hence, lim ϵ→0xϵ(t)=x(t) exists in H. From Assumption (F) and (3.27) it follows that
(3.28)f(·,xϵ)→f(·,x), strongly in L2(0,T0;H),Axn→Ax, strongly in L2(0,T0;V*).
Since ∂ϕϵ(xϵ) is uniformly bounded by Assumption (A), from (3.27), (3.28) we have that
(3.29)ddtxϵ→ddtx, weakly in L2(0,T0;V*),
therefore
(3.30)∂ϕϵ(xϵ)→f(·,x)+h-x′-Ax, weakly in L2(0,T0;V*).
Since (I+ϵ∂ϕ)-1xϵ→x strongly and ∂ϕ is demiclosed, we have that
(3.31)f(·,x)+h-x'-Ax∈∂ϕ(x) in L2(0,T0;V*).
Thus we have proved that x(t) satisfies a.e. on (0,T0) the equation (NDE).

Let y be the solution of
(3.32)y′(t)+Ay(t)+∂ϕ(y(t))∋0, 0<t≤T0,y(0)=x0,
then, it implies
(3.33)ddt(x(t)-y(t))+A(x(t)-y(t))+∂ϕ(x(t))-∂ϕ(y(t))∋f(t,x)+h(t).
Noting that ||·||≤|·|≤||·||, by multiplying by x(t)-y(t) and using the monotonicity of ∂ϕ and (2.3), we obtain
(3.34)12ddt|x(t)-y(t)|2+ω1∥x(t)-y(t)∥2 ≤ω2|x(t)-y(t)|2+|f(t,x)+h(t)|·∥x(t)-y(t)∥.
Since
(3.35)|f(t,x)+h(t)|·∥x(t)-y(t)∥≤12ω1|f(t,x)+h(t))|2+ω12∥x(t)-y(t)∥2
for every c>0 and by integrating on (3.34) over (0,t) we have
(3.36)|x(t)-y(t)|2+ω1∫0t∥x(s)-y(s)∥2ds ≤1ω1∥f(·,x)+h∥L2(0,T0;V*)+2ω2∫0t|x(s)-y(s)|2ds
and by Gronwall’s inequality:
(3.37)|x(t)-y(t)|2+ω1∫0t∥x(s)-y(s)∥2ds≤ω1-1e2ω2T0∥f(·,x)+h∥L2(0,T0;V*)2.
Let us fix T0>T1>0 so that T1 is a Lebesgue point of x, ϕ(x(T1))<∞, and
(3.38)ω1-1e2ω2T1T1L∥k∥L2<ω1.
Put
(3.39)N=ω1-2eω2T1,
then from Assumption (F) it follows
(3.40)∥x-y∥L2(0,T1;V)≤N∥f(·,x)+h∥L2(0,T1;V*)∥x-y∥L2(0,T1;V) ≤NT1 L∥k∥L2∥x∥L2(0,T1;V)+N∥h∥L2(0,T1:V*)
and hence, from (2.17) in Proposition 2.3, we have that
(3.41)∥x∥L2(0,T1;V) ≤11-NT1L∥k∥L2(∥y∥L2(0,T1;V)+N∥h∥L2(0,T1:V*)) ≤11-NT1L∥k∥L2{C2(1+∥x0∥)+N∥h∥L2(0,T1:V*)} ≤C4(1+∥x0∥+∥h∥L2(0,T1:V*))
for some positive constant C4. 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,nT1] for natural number n, that is, for the initial x(nT1) in the interval [nT1,(n+1)T1], as analogous estimate (3.41) holds for the solution in [0,(n+1)T1]. The norm estimate of x in W1,2(0,T;H) can be obtained by acting on both side of (NDE) by x'(t) and by using
(3.42)ddtϕ(x(t))=(g(t),ddtx(t)), a.e., 0<t,
for all g(t)∈∂ϕ(x(t)). Furthermore, the estimate (3.18) is immediately obtained from (3.41).

Theorem 3.5.
Let Assumptions (F) and (A) be satisfied and (x0,h)∈V×L2(0,T;V*), then the solution x of (NDE) belongs to x∈L2(0,T;V)∩W1,2(0,T;V*) and the mapping:
(3.43)V×L2(0,T;V*)∋(x0,h)↦x∈L2(0,T;V)∩C([0,T];H)
is continuous.

Proof.
If (x0,h)∈V×L2(0,T;V*) then x belongs to L2(0,T;V)∩W1,2(0,T;V*) form Theorem 3.4. Let (x0i,hi)∈V×L2(0,T;V*) and xi be the solution of (NDE) with (x0i,hi) in place of (x0,h) for i=1,2. Multiplying on (NDE) by x1(t)-x2(t), we have
(3.44)12ddt|x1(t)-x2(t)|2+ω1∥x1(t)-x2(t)∥2 ≤ω2|x1(t)-x2(t)|2+|f(t,x1)-f(t,x2)|∥x1(t)-x2(t)∥ +∥h1(t)-h2(t)∥*∥x1(t)-x2(t)∥.
Let us fix T1>T2>0 so that T2 is a Lebesgue point of x, ϕ(x(T2)<∞, and
(3.45)ω1-ω1-1e2ω2T2T2 L∥K∥L2>0.
Since
(3.46)∥h1(t)-h2(t))∥*∥x1(t)-x2(t)∥≤1ω1∥h1(t)-h2(t)∥*2+ω14∥x1(t)-x2(t)∥2,
by integrating on (3.44) over [0,T2] where T2<T and as is seen in (3.37), it follows
(3.47)∥x1-x2∥C([0,T2];H)2+ω12∥x1-x2∥L2(0,T2;V)2 ≤∥x01-x02∥2+1ω1∥f(t,x1)-f(t,x2)∥L2(0,T2;H)2+2ω1∥h1-h2∥L2(0,T2;V*) ≤∥x01-x02∥2+ω1-1T2 L∥K∥L2∥x1-x2∥L2(0,T2;V)2+2ω1∥h1-h2∥L2(0,T2;V*).
Putting that
(3.48)N1≡min [1,{ω12-ω1-1T2L∥K∥L2}]1/2, N2≡max {1,2ω1},
we have
(3.49)∥x1-x2∥L2∩C≤2N21-N1(∥x01-x02∥+∥h1-h2∥).
Suppose (x0n,hn)→(x0,h) in V×L2(0,T;V*), and let xn and x be the solutions (SDE 2) with (x0n,hn) and (x0,h), respectively. Then, by virtue of (3.44) and (3.49), we see that xn→x in L2(0,T2,V)∩W1,2(0,T2,V*)↪C([0,T2];H). This implies that xn(T2)→x(T2) in H. Therefore the same argument shows that xn→x in
(3.50)L2(T2,min {2T2,T};V)∩C([T2,min {2T2,T}];H).
Repeating this process, we conclude that xn→x in L2(0,T;V)∩W1,2(0,T2,V*)↪C([0,T2];H).

4. Example
Let Ω be bounded domain in ℝn with smooth boundary ∂Ω. We define the following spaces:
(4.1)H1(Ω)={u:u,∂u∂xi∈L2(Ω), i=1,2,…,n},H2(Ω)={u:u,∂x∂xi,∂2u∂xi∂xj∈L2(Ω), i,j=1,2,…,n},H01(Ω)={u:u∈H1(Ω),u|∂Ω=0}=the closure of C0∞(Ω) in H1(Ω),
where ∂/∂xiu and ∂2/∂xi∂xju are the derivative of u in the distribution sense. The norm of H01(Ω) is defined by
(4.2)∥u∥={∫Ω∑i=1n(∂u(x)∂xi)2dx}1/2.
Hence H01(Ω) is a Hilbert space. Let H-1(Ω) = H01(Ω)* be a dual space of H01(Ω). For any l∈H-1(Ω) and v∈H01(Ω), the notation (l,v) denotes the value l at v. In what follows, we consider the regularity for given equations in the spaces:
(4.3)V=H01(Ω)={u∈H1(Ω); u=0 on ∂Ω}, H=L2(Ω),V*=H-1(Ω)
as introduced in Section 2. We deal with the Dirichlet condition’s case as follows.

Assume that aij=aji are continuous and bounded on Ω¯ and {aij(x)} is positive definite uniformly in Ω, that is, there exists a positive number δ such that
(4.4)∑i,j=1naij(x)ξiξj≥δ|ξ|2, ∀ξ∈Ω-.
Let
(4.5)bi∈L∞(Ω), c∈L∞(Ω), βi=∑j=1n∂aij∂xj+bi.
For each u,v∈H01(Ω), let us consider the following sesquilinear form:
(4.6)a(u,v)=∫Ω{∑i,j=1naij∂u∂xi∂v¯∂xj+∑j=1nβi∂u∂xiv¯+cuv¯}dx.
Since {aij} is real symmetric, by (4.4) the inequality:
(4.7)∑i,j=1naij(x)ξiξ-j≥δ|ξ|2
holds for all complex vectors ξ=(ξ1,…,ξn). By hypothesis, there exists a constant K such that |βi(x)|≤K and c(x)≤K hold a.e., hence
(4.8)Re a(u,u)≥∫Ωδ∑i=1n|∂u∂xi|2dx-K∫Ω∑i=1n|∂u∂xi||u|dx-K∫Ω|u|2dx≥δ∫Ω∑i=1n|∂u∂xi|2dx-K∫Ω∑i=1n(ϵ2|∂u∂xi|2+12ϵ|u|2)dx-K∫Ω|u|2dx=(δ-ϵ2K)∑i=1n∫Ω|∂u∂xi|2dx-(nK2ϵ+K)∫Ω|u|2dx.
By choosing ϵ=δK-1, we have
(4.9)Re a(u,u)≥δ2∑i=1n∫Ω|∂u∂xi|2dx-(nK22δ+K)∫Ω|u|2dx=δ2∥u∥12-(nK22δ+K+δ2)∥u∥2.
By virtue of Lax-Milgram theorem, we know that for any v∈V there exists f∈V* such that
(4.10)a(u,v)=(f,v).
Therefore, we know that the associated operator A:V→V* defined by
(4.11)(Au,v)=-a(u,v), u,v∈V
is bounded and satisfies conditions (2.3) in Section 2.

Let g:[0,T]×V→H be a nonlinear mapping defined by
(4.12)g(t,u(t,x))=∫0t∑i=1n∂∂xiσi(s,∇u(s,x))ds.

We assume the following.

Assumption (<italic>F1</italic>).
The partial derivatives σi(s,ξ), ∂/∂t σi(s,ξ) and ∂/∂ξjσi(s,ξ), exist and continuous for i=1,2, j=1,2,…,n, and σi(s,ξ) satisfies an uniform Lipschitz condition with respect to ξ, that is, there exists a constant L>0 such that
(4.13)|σi(s,ξ)-σi(s,ξ^)|≤L|ξ-ξ^|,
where |·| denotes the norm of L2(Ω).

Lemma 4.1.
If Assumption (F1) is satisfied, then the mapping t↦g(t,·) is continuously differentiable on [0,T] and u↦g(·,u) is Lipschitz continuous on V.

Proof.
Put
(4.14)g1(s,u)=∑i=1n∂∂xiσi(s,∇u),
then we have g1(s,u)∈H-1(Ω). For each w∈H01(Ω), we satisfy the following that
(4.15)(g1(s,u),w)=-∑i=1n(σi(s,∇u),∂∂xiw).
The nonlinear term is given by
(4.16)g(t,u)=∫0tg1(s,u)ds.
For any w∈H01(Ω), if u and u^ belong to H01(Ω), by Assumption (F1) we obtain
(4.17)|(g(t,u)-g(t,u^)),w|≤LT∥u-u^∥∥w∥.

We set
(4.18)f(t,u)=∫0tk(t-s)∫0s∑i=1n∂∂xiσi(τ,∇u(τ,x))dτds,
where k belongs to L2(0,T). Let ϕ:H01(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.

Theorem 4.2.
Let Assumption (F1) be satisfied. Then for any u0∈H01(Ω) and h∈L2(0,T;H-1(Ω)), the following nonlinear problem:
(4.19)(u′(t)+Au(t),u(t)-z)+ϕ(u(t))-ϕ(z) ≤(f(t,u)+h(t),u(t)-z), a.e., 0<t≤T, z∈L2(Ω),u(0)=u0
has a unique solution:
(4.20)u∈L2(0,T;H01(Ω))∩W1,2(0,T;H-1(Ω))↪C([0,T];L2(Ω)).

Furthermore, the following energy inequality holds: there exists a constant CT depending on T such that
(4.21)∥u∥L2∩W1,2≤CT(1+∥u0∥+∥h∥L2(0,T;H-1(Ω))).