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Regularity for nonlinear evolution variational inequalities with delay terms
Journal of Inequalities and Applications volume 2014, Article number: 387 (2014)
In this paper, we deal with the regularity and a variation of constant formula for solutions of the nonlinear differential equation with delay nonlinear terms governed by the variational inequality in Hilbert spaces. Without the conditions of the uniform boundedness of the nonlinear terms and the compactness of the principal operators, we obtain the wellposedness and the norm estimate of the given equation by converting the problem into the contraction mapping principle on state space.
Let H and V be two complex Hilbert spaces. Assume that V is dense subspace in H and the injection of V into H is continuous. The norm on V and H will be denoted by and , respectively. Let A be a continuous linear operator from V into which is assumed to satisfy Gårding’s inequality, and let be a lower semicontinuous, proper convex function. Then we study the following variational inequality problem with nonlinear term:
Here, , a forcing term and is a nonlinear mapping.
By the definition of the subdifferential operator ∂ϕ, the problem (NVE) is represented by the following nonlinear functional differential problem:
The theory of variational evolution inequalities is one of the most important domains of application of the ideas and techniques of differential equations associated with maximal monotone operators and semigroups of nonlinear contractions. There is extensive literature on parabolic variational inequalities and the Stefan problems (see Lions [1, 2], Brézis [3, 4], Friedman , Elliott and Ockendon , Barbu [7, 8]). For more details on the applications of the theory we refer to the survey of Lions  and the book by Duvaut and Lions .
In this paper we are primarily interested in the regular problem that arise as direct consequences of the general theory developed previously, and we consider to put in perspective those models of initial value problems which can be formulated as nonlinear differential equations of variational inequalities. When the nonlinear mapping h is a Lipschitz continuous from into H, we will find that the most part of the regularity for parabolic variational inequalities can also be applicable to (NDE) with nonlinear perturbations. The above operator h is the semilinear case of the nonlinear part of quasilinear equations considered by Yong and Pan . The approach used here is similar to that developed in [12–14] on the general semilinear evolution equations.
Without conditions of the uniform boundedness of the nonlinear terms and the compactness of the principal operators, we obtain the wellposedness of (NDE) by converting the problem into the contraction mapping principle and the norm estimate of a solution of the above nonlinear equation on . Consequently, in view of the monotonicity of ∂ϕ and using the interpolation theory, we show that the solution mapping
is continuous and the mapping is compact from to , which is useful for physical applications for the equations related with forcing terms containing control part.
2 Parabolic variational inequalities
If H is identified with its dual space we may write densely and the corresponding injections are continuous. The norm on V, H, and will be denoted by , , and , respectively. The duality pairing between the element of and the element of V is denoted by , which is the ordinary inner product in H if . For the sake of simplicity, we may consider
For we denote by the value of l at . The norm of l as an element of is given by
Therefore, we assume that V has a stronger topology than H and, for brevity, we may regard
Let be a bounded sesquilinear form defined in and satisfying Gårding’s inequality,
where and is a real number.
Let A be the operator associated with the sesquilinear form :
Then A is a bounded linear operator from V to by the Lax-Milgram theorem. The realization for the operator A in H which is the restriction of A to
can also be denoted by A. We also assume that there exists a constant such that
for every , where
is the graph norm of . Thus, in terms of the intermediate theory we may assume that
where denotes the real interpolation space between and H.
If X is a Banach space, is the collection of all strongly measurable square integrable functions from into X and is the set of all absolutely continuous functions on such that their derivative belongs to . will denote the set of all continuous functions from into X with the supremum norm.
Lemma 2.1 Let . Then
Proof Put for . From
it follows that
Integrating over t yields
Hence, we obtain
Conversely, suppose that and . Put . Then since A is an isomorphism operator from V to there exists a constant such that
From the assumptions and it follows that
Therefore, . □
By Lemma 2.1, from Theorem 3.5.3 of Butzer and Berens , we can see that
Using the regularity for the variational inequality of parabolic type (i.e., in case where ) as seen in Section 4.3 of  we have the following result on (VE). We denote the closure in H of the set by .
Proposition 2.1 (1) Let and . Then (NDE) has a unique solution
where is some positive constant and .
(2) Let A be symmetric and let us assume that there exists such that for every and any
where . Then for and , (NDE) has a unique solution,
Remark 2.1 In terms of Lemma 2.1, the inclusion
3 Regularity for nonlinear variational inequalities
Let ℒ and ℬ be the Lebesgue σ-field on and the Borel σ-field on , respectively. Let μ be a Borel measure on and be a nonlinear mapping satisfying the following:
for any the mapping is strongly -measurable;
there exist positive constants () such that
for all and .
Remark 3.1 The above operator h is the semilinear case of the nonlinear part of quasilinear equations considered by Yong and Pan .
For any , we set
Here as in  we consider the Borel measurable corrections of .
Lemma 3.1 Let , . Then the nonlinear term defined by (3.1) belongs to and
Moreover, if , then
Proof From (G1) and (G2) it is easily seen that
The proof of (3.3) is similar. □
We denote the closure in H of the set by . In what follows this paper, we assume that for the sake of simplicity. First, we are going to give the following result on a local solvability of (NDE). The following lemma is due to Brézis [, Lemma A.5].
Lemma 3.2 Let satisfying for all and be a constant. Let b be a continuous function on satisfying the following inequalities:
Let . Then invoking Proposition 2.1, we see that the problem
has a unique solution .
Lemma 3.3 Let , be the solutions of (3.4) with x replaced by , respectively. Then the following inequalities hold:
Proof For , we consider the following equation:
From (3.7) it follows that
Multiplying on both sides of and using the monotonicity of ∂ϕ, we get
Noting that since for ,
by (2.1) and (2.2), we have
Hence, integrating (3.8) over , this yields
From (3.9) it follows that
Integrating (3.10) over we have
thus, we get
From (3.9) and (3.11), (3.5) holds, which implies
By using Lemma 3.1, we obtain (3.6), as claimed. □
Theorem 3.1 Let the assumptions (G1) and (G2) be satisfied. Assume that and . Then there exists a time such that the functional differential equation (NDE) admits a unique solution x in .
Proof Let us fix such that
We are going to show that is strictly contractive from to itself if the condition (3.12) is satisfied. The norm in is given by
Let , be the solutions of (3.7) with x replaced by , respectively. From (3.5) and (3.6) it follows that
from (3.13) it follows that
Starting from initial value , for , consider a sequence satisfying
Then from (3.14) it follows that
So by virtue of the condition (3.12) the contraction principle shows that there exists such that
Now we give a norm estimation of the solution of (NDE) and establish the global existence of solutions with the aid of norm estimations.
Theorem 3.2 Let the assumptions (G1) and (G2) be satisfied. Then for any () and , the solution x of (NDE) exists and is unique in , and there exists a constant depending on T such that
Proof Let y be the solution of
by multiplying by and using the monotonicity of ∂ϕ, (2.1), and (2.2), we obtain
By integrating on (3.16) over we have
By a procedure similar to (3.13) regarding
Then we have
and hence, from (2.4) in Proposition 2.1, we have
for some positive constant . Acting on both side of (NDE) by and by using
for all , we have
thus, we obtain the norm estimate of x in satisfying (3.15). Since the condition (3.12) is independent of initial values we can derive from (3.19) that , and the solution of (NDE) can be extended to the internal for the natural number n, i.e., for the initial in the interval , an analogous estimate (3.18) holds for the solution in . Furthermore, the estimate (3.15) is easily obtained from (3.18) and (3.19). □
Theorem 3.3 Suppose that the assumptions (G1) and (G2) are satisfied. Let A be symmetric and let us assume that there exists such that for every and any
If and , then
and the mapping is continuous.
Proof It is easy to show that if and , then from Proposition 2.1 it follows that u belongs to . Let , and be the solution of (NDE) with in place of for . Then in view of Proposition 2.1 and Lemma 2.1 we have
Hence, arguing as in (2.3) we get
Combining (3.20) and (3.21) we obtain
Suppose that in , and let and x be the solutions (SLE) with and , respectively. Let be such that
Then by virtue of (3.22) with T replaced by we see that in . This implies that in . Hence the same argument shows that in
Repeating this process we conclude that in . □
Theorem 3.4 For let be the solution of (NDE). Let us assume the natural assumption that the embedding is compact. Then the mapping is compact from to .
Proof If , noting , then in view of Theorem 3.2
Since , . Consequently and with aid of Proposition 2.1, Lemma 3.1, and (3.23),
Hence if k is bounded in , then so is in . Since is compactly embedded in V by assumption, the embedding
is compact in view of Theorem 2 of Aubin . □
Remark 3.2 Since the operator is an unbounded operator, we will make use of the hypothesis (G2). If A is a bounded operator from H into itself, we may assume that is a nonlinear mapping satisfying the following: there exist positive constants () such that
for all and , then our results can be obtained directly.
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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 (2014045161).
The authors declare that they have no competing interests.
H-HR drafted the manuscript and corrected the main results, J-MJ carried out the main proof of this paper and participated in its design and coordination. Both authors read and approved the final manuscript.