Regularity for nonlinear evolution variational inequalities with delay terms

R E S E A R C H

Open Access

Regularity for nonlinear evolution variational

inequalities with delay terms

Hyun-Hee Rho and Jin-Mun Jeong

*_{Correspondence:}

[email protected] Department of Applied Mathematics, Pukyong National University, Busan, 608-737, Korea

Abstract

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.

MSC: Primary 35K85; secondary 35F25

Keywords: variational inequality; subdifferential operator; delay term; regularity; analytic semigroup

1 Introduction

LetHandV be two complex Hilbert spaces. Assume thatVis dense subspace inHand the injection ofVintoHis continuous. The norm onVandHwill be denoted by · and |·|, respectively. LetAbe a continuous linear operator fromVintoV∗which is assumed to satisfy Gårding’s inequality, and letφ:V→(–∞, +∞] be a lower semicontinuous, proper

convex function. Then we study the following variational inequality problem with nonlin-ear term:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

(x(t) +Ax(t),x(t) –z) +φ(x(t)) –φ(z)

≤(_–rh(t,s,x(t),x(t+s),u(t))μ(ds) +k(t),x(t) –z), a.e.∀z∈V,

x() =g_{, x(s) =g}_{(s), –r≤s≤.}

(NVE)

Here,g= (g_,g_)∈H×L_{(,T;V), a forcing termk∈L}_(,T;V∗_{) andh:R}+_×V×H→H

is a nonlinear mapping.

By the definition of the subdifferential operator∂φ, the problem (NVE) is represented by the following nonlinear functional differential problem:

⎧ ⎨ ⎩

x(t) +Ax(t) +∂φ(x(t))_–rh(t,s,x(t),x(t+s),u(t))μ(ds) +k(t),  <t,

x() =g, x(s) =g(s), –r≤s≤. (NDE)

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 max-imal monotone operators and semigroups of nonlinear contractions. There is extensive

literature on parabolic variational inequalities and the Stefan problems (see Lions [, ], Brézis [, ], Friedman [], Elliott and Ockendon [], Barbu [, ]). 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 conse-quences 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 mappinghis a Lipschitz con-tinuous fromR×V intoH, 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 operatorhis the semilinear case of the nonlinear part of quasilinear equations con-sidered by Yong and Pan []. The approach used here is similar to that developed in [– ] on the general semilinear evolution equations.

Without conditions of the uniform boundedness of the nonlinear terms and the com-pactness 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 onL_{(,T;V)∩W},_(,T;V∗_{)∩C([,T];H). Consequently,}

in view of the monotonicity of∂φand using the interpolation theory, we show that the so-lution mapping

H×L(–r, ;V)×L(,T;H)g,g,k→x

∈L,T;D(A)∩W,(,T;H)

is continuous and the mappingk→xkis compact fromL_{(,T;H) toL}_{(,T;V), which}

is useful for physical applications for the equations related with forcing terms containing control part.

2 Parabolic variational inequalities

IfH is identified with its dual space we may writeV ⊂H⊂V∗ densely and the corre-sponding injections are continuous. The norm onV,H, andV∗will be denoted by · , | · |, and · ∗, respectively. The duality pairing between the elementvofV∗and the ele-mentvofVis denoted by (v,v), which is the ordinary inner product inHifv,v∈H. For the sake of simplicity, we may consider

u∗≤ |u| ≤ u, u∈V.

Forl∈V∗we denote (l,v) by the valuel(v) oflatv∈V. The norm oflas an element of

V∗is given by

l∗=sup

v∈V

|(l,v)|

v .

Therefore, we assume that V has a stronger topology thanH and, for brevity, we may regard

Leta(·,·) be a bounded sesquilinear form defined in V×V and satisfying Gårding’s inequality,

Rea(u,u)≥ωu–ω|u|, (.)

whereω>  andωis a real number.

LetAbe the operator associated with the sesquilinear forma(·,·):

(Au,v) =a(u,v), u,v∈V. (.)

ThenAis a bounded linear operator fromV toV∗by the Lax-Milgram theorem. The realization for the operatorAinHwhich is the restriction ofAto

D(A) ={u∈V;Au∈H}

can also be denoted byA. We also assume that there exists a constantCsuch that

u ≤Cu/_D(A)|u|/ (.)

for everyu∈D(A), where

uD(A)=

|Au|+|u|/

is the graph norm ofD(A). Thus, in terms of the intermediate theory we may assume that

D(A),H_/,=V,

where (D(A),H)/,denotes the real interpolation space betweenD(A) andH.

IfXis a Banach space,L_{(,T;X) is the collection of all strongly measurable square}

inte-grable functions from (,T) intoXandW,_{(,T;X) is the set of all absolutely continuous}

functions on [,T] such that their derivative belongs toL_{(,T;X).C([,T];X) will denote}

the set of all continuous functions from [,T] intoXwith the supremum norm.

Lemma . Let T> .Then

H= x∈V∗:

T



AetAx_∗dt<∞

.

Proof Putu(t) =etA_{xforx∈H. From}

 

d dtu(t)



=Reu˙(t),u(t)=ReAu(t),u(t)

= –Reau(t),u(t)≤–cu(t) 

,

it follows that

 

d dtu(t)



Integrating overtyields

 u(t)



+c t



u(s)ds≤

|x|

_.

Hence, we obtain

T



AetAx_∗dt≤ T



AB(V,V∗)u(s) 

ds<∞.

Conversely, suppose thatx∈V∗and_TAetA_x

∗dt<∞. Putu(t) =etAx. Then sinceAis an isomorphism operator fromVtoV∗there exists a constantc>  such that

T



u(t)dt≤c

T



Au(t)_∗dt=c

T



AetAx_∗dt.

From the assumptions andu˙(t) =AetA_{xit follows that}

u∈L(,T;V)∩W,,T;V∗⊂C[,T];H.

Therefore,x=u()∈H.

By Lemma ., from Theorem .. of Butzer and Berens [], we can see that

V,V∗_/,=H.

Using the regularity for the variational inequality of parabolic type (i.e., in case where

g≡) as seen in Section . of [] we have the following result on (VE). We denote the closure inHof the setD(φ) ={u∈V:φ(u) <∞}byD(φ).

Proposition . ()Let k∈L_(,T;V∗_)and(g_,g_)∈D(φ)×L_{(–r, ;V).Then(NDE)} has a unique solution

x∈L(,T;V)∩W,,T;V∗∩C[,T];H,

which satisfies

x(t) =k(t) –Ax(t) –∂φx(t)

and

x_L_∩W,_∩C≤C_

 +g+g_L_(–r,;V)+kL_(,T;V∗₎, (.)

where Cis some positive constant and L∩C=L(,T;V)∩C([,T];H).

()Let A be symmetric and let us assume that there exists h∈H such that for every> 

and any y∈D(φ)

J(y+h)∈D(φ) and φ

J(y+h)

where J= (I+A)–.Then for k∈L(,T;H)and(g,g)∈D(φ)∩V×L(–r, ;V), (NDE)

has a unique solution,

x∈L,T;D(A)∩W,(,T;H)∩C[,T];H,

which satisfies

x_L_∩W,_∩C≤C_

 +g+g_L_{(–r,;D(A))}+kL_(,T;H)

. (.)

Remark . In terms of Lemma ., the inclusion

L(,T;V)∩W,,T;V∗⊂C[,T];H

is well known and is an easy consequence of the definition of real interpolation spaces by the trace method (see [, ]).

3 Regularity for nonlinear variational inequalities

LetLandBbe the Lebesgueσ-field on [,∞) and the Borelσ-field on [–r, ], respectively. Letμbe a Borel measure on [–r, ] andh: [,∞)×[–r, ]×V×H→Hbe a nonlinear mapping satisfying the following:

(G) for anyx,y∈Vthe mappingh(·,·,x,y)is stronglyL×B-measurable; (G) there exist positive constantsLi(i= , , ) such that

h(t,s,x,y) –h(t,s,xˆ,yˆ)≤Lx–xˆ+L|y–yˆ|, h(t,s, , )≤L

for all(t,s)∈[,∞)×[–r, ]andx,xˆ,y,yˆ∈V.

Remark . The above operatorhis the semilinear case of the nonlinear part of quasilin-ear equations considered by Yong and Pan [].

For anyx∈L_{(–r,T;V),T>  we set}

G(t,x) =



–r

ht,s,x(t),x(t+s)μ(ds). (.)

Here as in [] we consider the Borel measurable corrections ofx(·).

Lemma . Let x∈L_{(–r,T;V),T > .Then the nonlinear term G(·,x)defined by(.)} belongs to L_(,T;H)and

G(·,x)_L_(,T;H)≤μ

[–r, ]L√T+ (L+L)xL_(,T;V)+Lx_L_(–r,;V). (.)

Moreover,if x,x∈L_{(–r,T;V),then}

G(·,x) –G(·,x)_L_(,T;H)

≤μ[–r, ](L+L)x–xL_(,T;V)+L_x_–x_L_(–r,;V)

Proof From (G) and (G) it is easily seen that

_G(·,x) L_(,T;H)

≤μ[–r, ]L

√

T+LxL_(,T;V)+L_x_L_(–r,T;V)

≤μ[–r, ]L√T+ (L+L)xL_(,T;V)+Lx_L_(–r,;V)

.

The proof of (.) is similar.

We denote the closure inHof the setD(φ) ={u∈V:φ(u) <∞}byD(φ). In what follows this paper, we assume thatD(φ) =Hfor 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.].

Lemma . Let m∈L_{(,T;R)satisfying m(t)≥for all t∈(,T)and a≥be a} con-stant.Let b be a continuous function on[,T]⊂Rsatisfying the following inequalities:

 b

_(t)≤

a

₊ t



m(s)b(s)ds, t∈[,T].

Then

b(t)≤a+

t



m(s)ds, t∈[,T].

Letx∈L_{(,T;V)∩C([,T];H). Then invoking Proposition ., we see that the problem} ⎧

⎨ ⎩

y(t) +Ay(t) +∂φ(y(t))G(t,x) +k(t),  <t≤T,

y() =g_{, y(s) =g}_{(s), –r≤s≤,} (.)

has a unique solutiony∈L_{(,T;V)∩W},_(,T;V∗_{)∩C([,T];H).}

Lemma . Let y,y be the solutions of(.)with x replaced by x,x∈L(,T;V), re-spectively.Then the following inequalities hold:



y(t) –y(t)



+ω t



y(s) –y(s) 

ds

≤

t



eω(t–s)H(s)y(s) –y(s)ds (.)

and

_y(t) –y(t)≤

t



eω(t–s)H(s)ds, (.)

where

Proof Fori= , , we consider the following equation:

⎧ ⎨ ⎩

y_i(t) +Ayi(t) +∂φ(yi(t))G(t,xi) +k(t),  <t≤T,

yi() =g, yi(s) =g(s), –r≤s≤.

(.)

From (.) it follows that

d dt

y(t) –y(t)+Ay(t) –y(t)+∂φy(t)–∂φy(t) G(t,x) –G(t,x), t> .

Multiplying on both sides ofy(t) –y(t) and using the monotonicity of∂φ, we get

 

d

dty(t) –y(t) 

+ay(t) –y(t),y(t) –y(t)

≤G(t,x) –G(t,x),y(t) –y(t)

.

Noting that sincex(s) –x(s) =  fors∈[–r, ),

G(t,x) –G(t,x)≤μ[–r, ]Lx(t) –x(t)+μ[–r, ]Lx–xC([,t];H),

and putting

H(t) =μ[–r, ]Lx(t) –x(t)+μ[–r, ]Lx–xC([,t];H),

by (.) and (.), we have

 

d

dty(t) –y(t) 

+ωy(t) –y(t) 

≤ωy(t) –y(t) 

+H(t)y(t) –y(t). (.)

Hence, integrating (.) over (,t), this yields



y(t) –y(t)



+ω t



y(s) –y(s)ds

≤ω t



y(s) –y(s)ds+

t



H(s)y(s) –y(s)ds. (.)

From (.) it follows that

d dt e

–ωt t



y(s) –y(s)ds

= e–ωt 

y(t) –y(t)



–ω t



y(s) –y(s)ds

≤e–ωt t



Integrating (.) over (,t) we have

e–ωt t



y(s) –y(s)ds

≤

t

 e–ωτ

τ



H(s)y(s) –y(s)ds dτ

= 

ω t



e–ωs–e–ωt_{H(s)y(s) –y(s)ds,}

thus, we get

ω t



y(s) –y(s)ds≤ t



eω(t–s)– H(s)y(s) –y(s)ds. (.)

From (.) and (.), (.) holds, which implies

 

e–ωt_y

(t) –y(t) 

+ωe–ωt t



y(s) –y(s) 

ds

≤ t



e–ωsH(s)e–ωs_{y(s) –y(s)ds.}

By using Lemma ., we obtain (.), as claimed.

Theorem . Let the assumptions(G)and(G)be satisfied.Assume that k∈L_(,T;V∗₎ and(g_,g_)∈H×L_{(–r, ;V).Then there exists a time T}

> such that the functional differential equation(NDE)admits a unique solution x in L_{(–r,T;V)∩C([,T];V}∗_).

Proof Let us fixT>  such that

M(eωT_{– )}

ωmin{/,ω}

< , (.)

where

M≡Lμ[–r, ]+ LLμ[–r, ]+Lμ[–r, ].

We are going to show thatx→yis strictly contractive fromL_(,T

;V)∩C([,T];H) to

itself if the condition (.) is satisfied. The norm inL(,T;V)∩C([,T];H) is given by

· L_{(,T;V)∩C([,T];H)}=max

· L_(,T;V), · _{C([,T];H)}

.

Lety, y be the solutions of (.) withxreplaced by x,x∈L_{(,T;V), respectively.}

From (.) and (.) it follows that



y(t) –y(t)



+ω t



y(s) –y(s) 

ds

≤

t



eω(t–s)H(s)

s



=eωt t



e–ωs_H(s) s



e–ωτ_H(τ)dτds

=eωt t    d ds s 

e–ωτ_H(τ)dτ

 ds=

e

ωt t



e–ωτ_H(τ)dτ



≤ e

ωt t



e–ωτ_dτ t



H(τ)dτ= e

ωt –e–ωt

ω t



H(τ)dτ

=  ω

eωt– 

t



H(s)ds. (.)

Here, since

t



H(s)ds=

t



Lμ

[–r, ]x(s) –x(s) 

+ LLμ[–r, ]x(s) –x(s)· x–xC([,s];H)

+Lμ

[–r, ]x–xC([,s];H)

ds

≤Lμ[–r, ]+ LLμ[–r, ] +Lμ[–r, ]x–x_L_{(,t;V)∩C([,t];H)},

taking

M≡Lμ[–r, ]+ LLμ[–r, ]+Lμ[–r, ],

from (.) it follows that

y–y_L_{(,T;V)∩C([,T];H)}≤

M(eωT_{– )}

ωmin{/,ω}

x–x_L_{(,T;V)∩C([,T];H)}. (.)

Starting from initial valuex(t) =g_{,x(s) =g}_{(s) for –r≤s≤, consider a sequence}

{xn(·)}satisfying ⎧

⎨ ⎩

xn+(t) +Axn+(t) +∂φ(xn+(t))G(t,xn) +k(t),  <t≤T, xn+() =g_{, x}

n+(s) =g(s), –r≤s≤.

Then from (.) it follows that

xn+–xn_L_{(,T;V)∩C([,T];H)}≤

M(eωT_{– )}

ωmin{/,ω}

xn–xn–_L_{(,T;V)∩C([,T];H)}.

So by virtue of the condition (.) the contraction principle shows that there existsx(·)∈

L_{(,T;V)∩C([,T];H) such that}

xn(·)→x(·) inL(,T;V)∩C[,T];H.

Theorem . Let the assumptions(G)and(G)be satisfied.Then for any(g_,g_)∈H× L_{(–r, ;V) (T> )and k∈L}_(,T;V∗_{),the solution x of(NDE)exists and is unique in}

W(T)≡L_{(–r,T;V)∩W},_(,T;V∗_{),and there exists a constant Cdepending on T such} that

xW(T)≤C

 +g+g_L_(–r,;V)+kL_(,T;V∗₎. (.)

Proof Letybe the solution of

⎧ ⎨ ⎩

dy(t)

dt +Ay(t) +∂φ(y(t))k(t),  <t≤T, y() =g_.

Then, since

d dt

x(t) –y(t)+Ax(t) –y(t)+∂φx(t)–∂φy(t)G(t,x),

by multiplying byx(t) –y(t) and using the monotonicity of∂φ, (.), and (.), we obtain 



d

dtx(t) –y(t) 

+ωx(t) –y(t)

≤ωx(t) –y(t)+G(t,x)·x(t) –y(t). (.)

By integrating on (.) over (,t) we have



x(t) –y(t)



+ω t



x(s) –y(s)ds

≤ω t



x(s) –y(s)ds+

t



G(s,x)·x(s) –y(s)ds. (.)

By a procedure similar to (.) regarding

H(t) =μ[–r, ]Lx(t)+μ[–r, ]LxC([,t];H),

we have

x–y_L_{(,T;V)∩C([,T];H)}≤

M(eωT_{– )}

ωmin{/,ω}

x_L_{(,T;V)∩C([,T];H)}.

Put

N= M(e

ωT_{– )}

ωmin{/,ω}

.

Then we have

and hence, from (.) in Proposition ., we have

xL_{(,T;V)∩C([,T];H)}

≤ 

 –N/yL(,T;V)∩C([,T];H)

≤ C

 –N/

 +g+g_L_(,T;V)+kL_(,T;V∗₎

≤C

 +g+g_L_(,T;V)+kL_(,T;V∗₎ (.)

for some positive constantC. Acting on both side of (NDE) byx(t) and by using

d dtφ

x(t)=

g(t), d

dtx(t)

, a.e.  <t

for allg(t)∈∂φ(x(t)), we have

t



x_n(s)ds+ 

Axn(t),xn(t)+φxn(t) ≤

(Ax,x) +φ(x) +

t



Gs,xn(s)

+k(s)x_n(s)ds, (.)

thus, we obtain the norm estimate ofxinW,(,T;H) satisfying (.). Since the condi-tion (.) is independent of initial values we can derive from (.) thatφ(x(nT)) <∞, and the solution of (NDE) can be extended to the internal [,nT] for the natural number n,i.e., for the initialx(nT) in the interval [nT, (n+ )T], an analogous estimate (.) holds for the solution in [, (n+ )T]. Furthermore, the estimate (.) is easily obtained

from (.) and (.).

Theorem . Suppose that the assumptions(G)and(G)are satisfied.Let A be symmet-ric and let us assume that there exists h∈H such that for every> and any y∈D(φ)

J(y+h)∈D(φ) and φ

J(y+h)

≤φ(y).

If k∈L(,T;H)and(g,g)∈V×L(–r, ;D(A)),then

x∈W(T)≡L–r,T;D(A)∩W,(,T;H),

and the mapping(g,g,k)→u∈W(T)is continuous.

Proof It is easy to show that if (g,g)∈V×L(–r, ;D(A)) andk∈L(,T;H), then from Proposition . it follows thatubelongs toW(T). Let (g_i,g_i,ki)∈V×L_{(–r, ;D(A))×} L_{(,T;H), anduibe the solution of (NDE) with (g}

i,gi,ki) in place of (g,g,k) fori= , .

Then in view of Proposition . and Lemma . we have

x–xW(T)≤Cg–g+g–gL_{(–r,;D(A))}

≤Cg_–g_+g_–g__L_{(–r,;D(A))}+k–kL_(,T;H)

+μ[–r, ](L+L)x–xL_(,T;V)

+Lg_–g__L_(–r,;V)

. (.)

Since

x(t) –x(t) =g–g+ t



˙

x(s) –x˙(s)

ds,

we get

x–xL_(,T;H)≤

√

Tg_–g_+√T

x–xW,(,T;H).

Hence, arguing as in (.) we get

x–xL_(,T;V)≤C_x_–x_/

L_(,T;D(A))x–x/_L_(,T;H)

≤Cx–x/_L_(,T;D(A))

× T/g_–g_/+

T

√ 

/

x–x/_W,_(,T;H)

≤CT/g_–g_/x–x/_L_(,T;D(A))+C

T

√ 

/

x–x_W(T)

≤–/Cg_–g_+ C

T

√ 

/

x–xW(T). (.)

Combining (.) and (.) we obtain

x–xW(T)≤Cg–g+g–gL_{(–r,;D(A))}

+k–kL_(,T;H)+μ

[–r, ]Lg_–g__L_(–r,;V)

+ –/CCμ[–r, ](L+L)g_–g_+ CC

T

√ 

/

×μ[–r, ](L+L)x–xW(T). (.)

Suppose that (g

n,gn,kn)→(g,g,k) inX×L(–r, ;D(A))×L(,T;H), and letxnandx

be the solutions (SLE) with (g

n,gn,kn) and (g,g,k), respectively. Let  <T≤Tbe such

that

CC(T/

√ )/_(L

+LLT/

√ ) < .

Then by virtue of (.) withTreplaced byTwe see thatun→uinW(T). This implies that (xn(T), (xn)T)→(x(T),xT) inV×L(–r, ;D(A)). Hence the same argument shows

thatun→uin

LT,min{T,T};D(A)

∩W,T,min{T,T};H

.

Theorem . For k∈L_{(,T;H)let xkbe the solution of(NDE).Let us assume the natural} assumption that the embedding D(A)⊂V is compact.Then the mapping k→xkis compact from L_{(,T;H)to L}_(,T;V).

Proof Ifk∈L_{(,T;H), notingk}

L_(,T;V∗₎≤ k_L_(,T;H), then in view of Theorem .

xkW(T)≤C

 +g+g_L_(–r,;V)+kL_(,T;H)

. (.)

Sincexk∈L(,T;V),G(·,xk)∈L(,T;H). Consequentlyxk∈L(,T;D(A))∩W,(,T; H) and with aid of Proposition ., Lemma ., and (.),

xkL_{(,T;D(A))∩W},_(,T;V)

≤Cg+g_L_{(–r,;D(A))}+G(·,xk) +k_L_(,T;H)

≤Cg+g_L_{(–r,;D(A))}+μ

[–r, ]L√T

+μ[–r, ](L+L)xL_(,T;V)+μ[–r, ]Lg

L_(–r,;V)+kL_(,T;H)

≤Cg+g_L_{(–r,;D(A))}+μ

[–r, ]L√T

+μ[–r, ](L+L)C +g+g_L_(–r,;V)+kL_(,T;H)

+μ[–r, ]Lg_L_(–r,;V)+kL_(,T;H)

.

Hence ifkis bounded inL(,T;H), then so isxkinL(,T;D(A))∩W,(,T;H). Since

D(A) is compactly embedded inVby assumption, the embedding

L,T;D(A)∩W,(,T;H)⊂L(,T;V)

is compact in view of Theorem  of Aubin [].

Remark . Since the operatorA:D(A)⊂H→H is an unbounded operator, we will make use of the hypothesis (G). IfAis a bounded operator fromHinto itself, we may assume thath: [,∞)×[–r, ]×V×H→His a nonlinear mapping satisfying the fol-lowing: there exist positive constantsLi(i= , ) such that

h(t,s,x,y) –h(t,s,xˆ,ˆy)≤L|x–xˆ|+L|y–ˆy|

for all (t,s)∈[,∞)×[–r, ] andx,xˆ,y,yˆ∈V, then our results can be obtained directly.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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.

Acknowledgements

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).

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doi:10.1186/1029-242X-2014-387

Cite this article as:Rho and Jeong:Regularity for nonlinear evolution variational inequalities with delay terms.