R E S E A R C H
Open Access
On stability analysis for generalized Minty
variational-hemivariational inequality in
reflexive Banach spaces
Lu-Chuan Ceng
1, Ravi P. Agarwal
2,3, Jen-Chih Yao
4and Yonghong Yao
5**Correspondence:
[email protected] 5Department of Mathematics,
Tianjin Polytechnic University, Tianjin, China
Full list of author information is available at the end of the article
Abstract
The stability for a class of generalized Minty variational-hemivariational inequalities has been considered in reflexive Banach spaces. We demonstrate the equivalent characterizations of the generalized Minty variational-hemivariational inequality. A stability result is presented for the generalized Minty variational-hemivariational inequality with (f,J)-pseudomonotone mapping.
Keywords: Generalized variational-hemivariational inequality; Stability; Clarke’s generalized directional derivative; Pseudomonotone mapping; Reflexive Banach space
1 Introduction
LetXbe a real Banach space with its dualX∗. LetK⊂Xbe a nonempty, closed, and convex set. LetF:K→2X∗be a set-valued mapping. LetA:K→X∗be a single-valued mapping. Letf :K⊂X→R∪ {+∞}be a proper, convex, and lower semicontinuous functional. Let
J:X→Rbe a locally Lipschitz functional. We useJ◦(·,·) to denote Clarke’s generalized directional derivative ofJ. Recall that the variational-hemivariational inequality [1] can mathematically be formulated as the problem of finding a pointu∈Ksuch that
VHVI(A,J,K) : Au,v–u+J◦(u,v–u) +f(v) –f(u)≥0, ∀v∈K. (1.1)
In particular, if J= 0, then theVHVI(A,J,K) reduces to the following mixed variational inequality of findingu∈Ksuch that
MVI(A,K) : Au,v–u+f(v) –f(u)≥0, ∀v∈K. (1.2)
MVI has been studied extensively in the literature, see, for instance, [2–6].
Under some suitable conditions, (1.2) is equivalent to the following Minty mixed varia-tional inequality [7–15] which is to findu∈Ksuch that
MMVI(A,K) : Av,v–u+f(v) –f(u)≥0, ∀v∈K. (1.3)
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In the present paper, we consider the following generalized Minty variational-hemi-variational inequality of findingu∈Ksuch that
GMVHVI(F,J,K) : sup
v∗∈F(v)
v∗,u–v+J◦(v,u–v) +f(u) –f(v)≤0, ∀v∈K. (1.4)
Special cases: (i) IfJ= 0, then (1.4) reduces to the following generalized Minty mixed vari-ational inequality of findingu∈Ksuch that
GMMVI(F,K) : sup
v∗∈F(v)
v∗,u–v+f(u) –f(v)≤0, ∀v∈K. (1.5)
(ii) IfF=Aandf = 0, then (1.5) reduces to the following classical Minty variational in-equality of findingu∈Ksuch that
MVI(A,K) : Av,u–v ≤0, ∀v∈K. (1.6)
Let (Z1,d1) and (Z2,d2) be two metric spaces.L:Z1→2Xbe a set-valued mapping with nonempty, closed, and convex values. LetF:X×Z2→2X
∗
be a set-valued mapping. Let
f :X→R∪ {+∞}be a proper, convex, and lower semicontinuous functional. Next, we consider the following parameter generalized Minty variational-hemivariational inequal-ity which is to findx∈L(u) such that
GMVHVIF(·,v),J,L(u): sup
y∗∈F(y,v)
y∗,x–y+J◦(y,x–y) +f(x) –f(y)≤0,
∀y∈L(u). (1.7)
In particular, ifJ= 0, then (1.7) reduces to the following parameter generalized Minty mixed variational inequality: findx∈Ksuch that
GMMVIF(·,v),L(u): sup
y∗∈F(y,v)
y∗,x–y+f(x) –f(y)≤0, ∀y∈L(u). (1.8)
It is well known that the variational inequality theory has wide applications in finance, economics, transportation, optimization, operations research, and engineering sciences, see [16–25]. In 2010, Zhong and Huang [19] studied the stability of solution sets for the generalized Minty mixed variational inequality in reflexive Banach spaces.
Inspired and motivated by the above work of Zhong and Huang [19], we investigate the stability of solution sets for the generalized Minty variational-hemivariational inequal-ity in reflexive Banach spaces. We first present several equivalent characterizations for the generalized Minty variational-hemivariational inequality. Consequently, we show the stability of a solution set for the generalized Minty variational-hemivariational inequal-ity with (f,J)-pseudomonotone mapping in reflexive Banach spaces. As an application, we give the stability result for a generalized variational-hemivariational inequality. The results presented in this paper extend the corresponding results of Zhong and Huang [19] from the generalized mixed variational inequalities to the generalized variational-hemivariational inequalities.
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2 Preliminaries
LetXbe a real reflexive Banach space. LetJ:X→Rbe a locally Lipschitz function onX. Clarke’s generalized directional derivative ofJatxin the directiony, denoted byJ◦(x,y), is defined by
J◦(x,y) =lim sup
z→xλ↓0
J(z+λy) –J(z)
λ .
Letf :X→R∪ {+∞}be a proper, convex, and lower semicontinuous function. Denote by
∂f :X→2X∗and∂J:X→2X∗the subgradient off and Clarke’s generalized gradient ofJ (see [26]), respectively. That is,
∂f(x) =z∈X∗:f(y) –f(x)≥ z,y–x,∀y∈X
and
∂J(x) =u∈X∗:J◦(x,y)≥ u,y,∀y∈X. It is known that∂J(x) =∂(J◦(x,·))(0), see [27].
Proposition 2.1([1]) Let X be a Banach space and J be a locally Lipschitz functional on X.
Then we have:
(i) The functiony→J◦(x,y)is finite,convex,positively homogeneous,and subadditive; (ii) J◦(x,y)is upper semicontinuous and is Lipschitz continuous on the second variable; (iii) J◦(x, –y) = (–J)◦(x,y);
(iv) ∂J(x)is a nonempty,convex,bounded,and weak∗-compact subset ofX∗; (v) For everyy∈X,J◦(x,y) =max{ ξ,y:ξ∈∂J(x)};
(vi) The graph of∂J(x)is closed inX×(w∗–X∗)topology,where(w∗–X∗)denotes the spaceX∗equipped with weak∗topology,i.e.,if{xn} ⊂Xand{x∗n} ⊂X∗are sequences
such thatx∗n∈∂J(xn),xn→xinXandxn∗→x∗weakly∗inX∗,thenx∗∈∂J(x). LetK be a nonempty, closed, and convex subset ofX. Let Y be a topological space. We usebarr(K) to denote the barrier cone ofKwhich is defined bybarr(K) :={x∗∈X∗:
supx∈K x∗,x<∞}. The recession cone ofK, denoted byK∞, is defined byK∞:={d∈X:
x0+μd∈K,∀μ> 0,∀x0∈K}. The negative polar coneK–ofK is defined byK–:={x∗∈
X∗: x∗,x ≤0,∀x∈K}. The positive polar cone ofKis defined asK+:={x∗∈X∗: x∗,x ≥ 0,∀x∈K}.
Letf :K→R∪ {+∞}be a proper, convex, and lower semicontinuous function. The recession functionf∞off is defined by
f∞(x) := lim t→+∞
f(x0+tx) –f(x0)
t ,
wherex0∈Domf. It is known that
f(x+y)≤f(x) +f∞(y), ∀x∈Domf,y∈X, (2.1)
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andf∞(·) satisfiesf∞(λx) =λf∞(x) for allx∈X,λ≥0. According to Proposition 2.5 in [28], we deduce
f∞(x)≤lim inf
n→∞
f(tnxn)
tn
, (2.2)
where{xn}is any sequence inXconverging weakly toxandtn→+∞.
Definition 2.2 A set-valued mappingF:K⊂X→2X∗is said to be
(i) upper semicontinuous atx0∈Kiff, for any neighborhoodN(F(x0))ofF(x0), there exists a neighborhoodN(x0)ofx0such that
F(x)⊂NF(x0), ∀x∈N(x0)∩K;
(ii) lower semicontinuous atx0∈Kiff, for anyy0∈F(x0)and any neighborhoodN(y0) ofy0, there exists a neighborhoodN(x0)ofx0such that
F(x)∩N(y0)=∅, ∀x∈N(x0)∩K.
Fis said to be continuous atx0iff it is both upper and lower semicontinuous atx0; and
Fis continuous onKiff it is both upper and lower semicontinuous at every point ofK.
Definition 2.3 The mappingFis said to be
(i) monotone onKiff, for all(x,x∗), (y,y∗)in thegraph(F),
y∗–x∗,y–x≥0;
(ii) pseudomonotone onKiff, for all(x,x∗), (y,y∗)in thegraph(F),
x∗,y–x≥0 implies that y∗,y–x≥0;
(iii) stably pseudomonotone onKwith respect to a setU⊂X∗iffFandF(·) –uare pseudomonotone onKfor everyu∈U;
(iv) f-pseudomonotone onKiff, for all(x,x∗), (y,y∗)in thegraph(F),
x∗,y–x+f(y) –f(x)≥0 ⇒ y∗,x–y+f(x) –f(y)≤0;
(v) (f,J)-pseudomonotone onKiff, for all(x,x∗), (y,y∗)in thegraph(F),
x∗,y–x+J◦(x,y–x) +f(y) –f(x)≥0 ⇒ y∗,x–y+J◦(y,x–y) +f(x) –f(y)≤0.
Definition 2.4 Let{An} ⊂Xbe a sequence. Define
ω–lim sup
n→∞ An:=
x∈X:∃{nk}andxnk∈Anksuch thatxnkx
.
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Definition 2.5 Letψ :X×X→Rbe a function.ψ is said to be bi-sequentially weakly lower semicontinuous iff, for any sequences{xn}and{yn}withxnx0andyny0, one has
ψ(x0,y0)≤lim inf
n→∞ ψ(xn,yn).
Lemma 2.6([29]) Let K⊂X be a nonempty,closed,and convex set withint(barr(K))=∅.
Then there exists no sequence{xn} ⊂K satisfyingxn → ∞andxnxn0.If K is a cone,
then there exists no sequence{dn} ⊂K withdn= 1satisfying dn0.
Lemma 2.7([30]) Let K⊂X be a nonempty,closed,and convex set withint(barr(K))=∅.
Then there exists no sequence{dn} ⊂K∞withdn= 1satisfying dn0.
Lemma 2.8([30]) Let(Z,d)be a metric space and u0∈Z be a given point.Let L:Z→2X
be a set-valued mapping with nonempty values,and let L be upper semicontinuous at u0.
Then there exists a neighborhood U of u0such that(L(u))∞⊂(L(u0))∞for all u∈U. Lemma 2.9([31]) Let E be a Hausdorff topological vector space and K⊂E be a nonempty and convex set.Let G:K→2Ebe a set-valued mapping satisfying the following conditions:
(i) Gis a KKM mapping,i.e.,for every finite subsetAofK,conv(A)⊂x∈AG(x); (ii) G(x)is closed inEfor everyx∈K;
(iii) G(x0)is compact inEfor somex0∈K.
Then x∈KG(x)=∅.
3 Boundedness of solution sets
In this section, we introduce several characterizations for the solution set D of
GMVHVI(F,J,K).
LetK⊂Xbe a nonempty, closed, and convex set. LetF:K→2X∗be a set-valued map-ping with nonempty values,J:X→Rbe a locally Lipschitz functional, andf:K⊂X→R be a convex and lower semicontinuous function.
Theorem 3.1 Suppose D=∅.Then
D∞=K∞∩d∈Rn:y∗,d+J◦(y,d) +f∞(d)≤0,∀y∗∈F(y),y∈K.
Proof Define a functionΦ:X→R∪ {+∞}by
Φ(x) := sup
y∗∈F(y),y∈K
y∗,x–y+J◦(y,x–y) +f(x) –f(y)
ϕ(y,y∗) ,
whereϕ(y,y∗) :=max{y∗, 1}max{y, 1}max{|f(y)|, 1}. Clearly,Φis a proper, convex, and lower semicontinuous function and soΦ∞is well defined onX.
LetD={x∈K:Φ(x)≤0}. It is clear thatDis nonempty. According to formula (2.29) in [32],{x∈X:Φ(x)≤r}∞={d∈X:Φ∞(d)≤0}. Hence
D∞=K∩x∈X:Φ(x)≤0∞=K∞∩d∈X:Φ∞(d)≤0.
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It remains to prove that
d∈X:Φ∞(d)≤0=d∈X:y∗,d+J◦(y,d) +f∞(d)≤0,∀y∗∈F(y),y∈K. Letd∈ {d∈X: y∗,d+J◦(y,d) +f∞(d)≤0,∀y∗∈F(y),y∈K}andx0∈XwithΦ(x0) <∞. By virtue of the subadditivity and positive homogeneousness of the functiony→J◦(x,y), we have
Φ(x0+td) –Φ(x0) = sup
y∗∈F(y),y∈K
y∗,x0+td–y+J◦(y,x0+td–y) +f(x0+td) –f(y)
ϕ(y,y∗) – sup
y∗∈F(y),y∈K
y∗,x0–y+J◦(y,x0–y) +f(x0) –f(y)
ϕ(y,y∗)
≤ sup
y∗∈F(y),y∈K
y∗,x0+td–y+J◦(y,td) +J◦(y,x0–y) +f(x0+td) –f(y)
ϕ(y,y∗) – sup
y∗∈F(y),y∈K
y∗,x0–y+J◦(y,x0–y) +f(x0) –f(y)
ϕ(y,y∗)
≤ sup
y∗∈F(y),y∈K
y∗,td+tJ◦(y,d) +f(x0+td) –f(x0)
ϕ(y,y∗) for anyt> 0. This implies that
Φ(x0+td) –Φ(x0)
t ≤y∗∈F(y),ysup∈K
y∗,d+J◦(y,d) +f(x0+td)–f(x0)
t
ϕ(y,y∗) ,
and so
Φ∞(d) = lim t→∞
Φ(x0+td) –Φ(x0)
t ≤0.
Therefore,
d∈X:y∗,d+J◦(y,d) +f∞(d)≤0,∀y∗∈F(y),y∈K⊂d∈X:Φ∞(d)≤0. Conversely, ifd∈ {/ d∈X: y∗,d+J◦(y,d) +f∞(d)≤0,∀y∗∈F(y),y∈K}, then there exist
y∈Kandy∗∈F(y) such that y∗,d+J◦(y,d) +f∞(d) > 0. Hence,
Φ(x0+td) –Φ(x0)
t
>
y∗,x0+td–y+J◦(y,x0+td–y)+f(x0+td)–f(y)
ϕ(y,y∗) –Φ(x0)
t
≥ y∗,x0–y–J◦(y,y–x0) +f(x0) –f(y) –ϕ(y,y∗)Φ(x0)
ϕ(y,y∗)t
+ y
∗,d+J◦(y,d)
ϕ(y,y∗) +
f(x0+td) –f(x0)
ϕ(y,y∗)t
→ y∗,d+J◦(y,d) +f∞(d)
ϕ(y,y∗) ast→ ∞.
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This yields that
Φ∞(d)≥ y
∗,d+J◦(y,d) +f ∞(d)
ϕ(y,y∗) > 0,
and hence the converse inclusion is true. This completes the proof.
Corollary 3.2 Suppose D=∅.Then
D∞=K∞∩d∈X:y∗,d+f∞(d)≤0,∀y∗∈F(y),y∈K.
Proof IfJ= 0, thenJ◦= 0. In this case,GMVHVI(F,J,K) reduces toGMMVI(F,K). Utiliz-ing Theorem3.1, we immediately deduce Corollary3.2.
Remark3.3 It is known that ifJ= 0 then Theorem3.1reduces to Zhong and Huang’s one [19, Theorem 3.1]. Thus, Theorem3.1generalizes and extends Theorem 3.1 in Zhong and Huang [19] fromGMMVI(F,K) toGMVHVI(F,J,K). Iff = 0 additionally, thenf∞= 0 and so
D∞=K∞∩d∈X:y∗,d≤0,∀y∗∈F(K)=K∞∩F(K)–.
Hence, Zhong and Huang’s Theorem 3.1 in [19] is a generalization of Lemma 3.1 in [29].
Theorem 3.4 Suppose the following statements hold: (i) Dis nonempty and bounded;
(ii) K∞∩ {d∈X: y∗,d+J◦(y,d) +f∞(d)≤0,∀y∗∈F(y),y∈K}={0};
(iii) There exists a bounded setC⊂Ksuch that,for everyx∈K\C,there exists some
y∈Csatisfying
sup
y∗∈F(y)
y∗,x–y+J◦(y,x–y) +f(x) –f(y) > 0.
Then (i)⇒(ii). (ii)⇒(iii) if barr(K)has nonempty interior. (iii)⇒(i)if F is (f,J) -pseudo-monotone on K.
Proof The relationship (i)⇒(ii) can be deduced from Theorem3.1.
Next, we first prove that (ii)⇒(iii). If (iii) does not hold, then there exists a sequence
{xn} ⊂K such that, for eachn,xn ≥nandsupy∗∈F(y) y∗,xn–y+J◦(y,xn–y) +f(xn) –
f(y)≤0 for everyy∈K with y ≤n. Without loss of generality, we may assume that
dn=xn/xnweakly converges tod. Thend∈K∞. By Lemma2.7, we getd= 0. Lety∈K andy∗∈F(y). Then, for alln>y, we have
0≥ y ∗,x
n–y+J◦(y,xn–y)
xn
+f(xndn)
xn
– f(y)
xn
≥ y∗,xn–y+J◦(y,xn) –J◦(y,y)
xn
+f(xndn)
xn
– f(y)
xn = y
∗,x
n–y–J◦(y,y)
xn
+J◦(y,dn) +f(xndn)
xn
– f(y)
xn .
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This together with (2.2) implies that
0≥y∗,d+lim inf
n→∞ J
◦(y,dn) +lim inf
n→∞
f(xndn)
xn ≥
y∗,d+J◦(y,d) +f∞(d), ∀y∗∈F(y),
and so
d∈K∞∩d∈X:y∗,d+J◦(y,d) +f∞(d)≤0,∀y∗∈F(y),y∈K. This implies that
0=d∈K∞∩d∈X:y∗,d+J◦(y,d) +f∞(d)≤0,∀y∗∈F(y),y∈K, a contradiction to (ii).
It remains to prove that (iii) implies (i) under the assumption thatF is (f,J )-pseudo-monotone onK. Indeed, letG:K→2Kbe a set-valued mapping defined by
G(y) :=
x∈K: sup
y∗∈F(y)
y∗,x–y+J◦(y,x–y) +f(x) –f(y)≤0
, ∀y∈K.
Firstly, we show thatG(y) is a closed subset ofK. In fact, for anyxn∈G(y) withxn→x0, we have
sup
y∗∈F(y)
y∗,xn–y
+J◦(y,xn–y) +f(xn) –f(y)≤0.
From the lower semicontinuity off and the Lipschitz continuity ofJ◦(·,·) in the second variable, it follows that
sup
y∗∈F(y)
y∗,x0–y
+J◦(y,x0–y) +f(x0) –f(y)
≤lim inf
n→∞
sup
y∗∈F(y)
y∗,xn–y
+lim inf
n→∞
J◦(y,xn–y) +f(xn) –f(y)
≤0.
This shows thatx0∈G(y) and soG(y) is closed.
Next we prove that G:K →K is a KKM mapping. If it is not so, then there exist
t1,t2, . . . ,tn∈[0, 1],y1,y2, . . . ,yn∈K, andy¯=t1y1+t2y2+· · ·+tnyn∈conv{y1,y2, . . . ,yn} such thaty¯∈/i∈{1,2,...,n}G(yi). Hence,
sup
y∗i∈F(yi)
y∗i,y¯–yi
+J◦(yi,¯y–yi) +f(¯y) –f(yi) > 0, i= 1, 2, . . . ,n.
By the (f,J)-pseudomonotonicity ofF, we get
sup
¯ y∗∈F(y)¯
¯
y∗,y¯–yi
–J◦(y¯,yi–¯y) +f(y¯) –f(yi) > 0, i= 1, 2, . . . ,n.
Sincey→J◦(x,y) is convex, we deduce n
i=1
tiJ◦(y¯,yi–¯y)≥J◦
¯ y, n i=1
tiyi–y¯
=J◦(¯y, 0) = 0,
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which yields
– n
i=1
tiJ◦(y¯,yi–y¯)≤0.
It follows that
f(y¯) – n
i=1
tif(yi)
≥ sup
¯ y∗∈F(y)¯
¯
y∗,y¯– n
i=1
tiyi
– n
i=1
tiJ◦(y¯,yi–y¯) +f(¯y) – n
i=1
tif(yi) > 0,
and hence
f(y¯) > n
i=1
tif(yi),
which is a contradiction. Therefore,Gis a KKM mapping.
Assume that C is a bounded, closed, and convex (otherwise, we can use the closed convex hull ofC instead of C). Let {y1, . . . ,ym} be a finite number of points inK, and let M:=conv(C∪ {y1, . . . ,ym}). It is obvious thatMis weakly compact and convex. Let
G(y) :=G(y)∩Mfor ally∈M. ThenG(y) is a weakly compact and convex subset ofM
andGis a KKM mapping. We claim that
∅ =
y∈M
G(y)⊂C. (3.1)
Indeed, by Lemma2.9, the intersection in (3.1) is nonempty. Moreover, if there exists some
x0∈ y∈MG(y) butx0∈/C, then by (iii) we have
sup
y∗∈F(y)
y∗,x0–y
+J◦(y,x0–y) +f(x0) –f(y) > 0
for somey∈C. Thus,x0∈/G(y) and sox0∈/G(y), which is a contradiction to the choice ofx0.
Letz∈ y∈MG(y). Thenz∈Cby (11) and soz∈ mi=1(G(yi)∩C). This shows that the collection{G(y)∩C:y∈K}has the finite intersection property. For eachy∈K, it follows from the weak compactness ofG(y)∩Cthat y∈K(G(y)∩C) is nonempty, which coincides with the solution set ofGMVHVI(F,J,K). This completes the proof. Corollary 3.5 Suppose the following statements hold:
(i) Dis nonempty and bounded;
(ii) K∞∩ {d∈X: y∗,d+f∞(d)≤0,∀y∗∈F(y),y∈K}={0};
(iii) There exists a bounded setC⊂Ksuch that,for everyx∈K\C,there exists some
y∈Csatisfying sup
y∗∈F(y)
y∗,x–y+f(x) –f(y) > 0.
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Then (i)⇒(ii). (ii)⇒(iii) if barr(K)has nonempty interior. (iii)⇒(i)if F is (f,J) -pseudo-monotone on K.
Remark3.6 It is known that ifJ= 0 then Theorem3.4reduces to Theorem 3.2 in Zhong and Huang [19]. Thus, Theorem3.4generalizes and extends Theorem 3.2 in Zhong and Huang [19] fromGMMVI(F,K) toGMVHVI(F,J,K). If f = 0 additionally, thenf∞= 0. Consequently, statements (i), (ii), and (iii) in [19, Theorem 3.2] reduce to (i), (ii), and (iii) in [29, Theorem 3.1], respectively. Thus, Zhong and Huang’s Theorem 3.2 in [19] is a generalization of Theorem 3.1 in [29].
4 Stability of solution sets
In this section, we will establish the stability of solution sets for the generalized Minty variational-hemivariational inequalityGMVHVI(F,J,K) and the generalized variational-hemivariational inequalityGVHVI(F,J,K) with (f,J)-pseudomonotone mappings.
Let (Z1,d1) and (Z2,d2) be two metric spaces,u0∈Z1andv0∈Z2be given points. Let
L:Z1→2Xbe a continuous set-valued mapping with nonempty, closed, and convex values andint(barrL(u0))=∅. Suppose that there exists a neighborhoodU×V of (u0,v0) such thatM=u∈UL(u),F:M×V→2X∗is a lower semicontinuous set-valued mapping with nonempty values, and letf :M⊂X→Rbe a convex and lower semicontinuous function. LetJ:X→Rbe a locally Lipschitz functional such thatJ◦:M×M⊂X×X→Ris bi-sequentially weakly lower semicontinuous.
Theorem 4.1 If
L(u0)
∞∩
d∈X:y∗,d+J◦(y,d) +f∞(d)≤0,∀y∗∈F(y,v0),y∈L(u0)
={0}, (4.1)
then there exists a neighborhood U×Vof(u0,v0)with U×V⊂U×V such that
L(u)∞∩d∈X:y∗,d+J◦(y,d) +f∞(d)≤0,∀y∗∈F(y,v),y∈L(u)={0} (4.2)
for all(u,v)∈U×V.
Proof Assume that the conclusion does not hold. Then there exists a sequence{(un,vn)} inZ1×Z2with (un,vn)→(u0,v0) such that
L(un)∞∩d∈X:y∗,d+J◦(y,d) +f∞(d)≤0,∀y∗∈F(y,vn),y∈L(un)={0}. Sincef∞(λx) =λf∞(x) for allx∈Xandλ≥0, we deduce that
L(un)∞∩d∈X:y∗,d+J◦(y,d) +f∞(d)≤0,∀y∗∈F(y,vn),y∈L(un) is a cone. Thus, we can select a sequence{dn}such that
dn∈
L(un)∞∩d∈X:y∗,d+J◦(y,d) +f∞(d)≤0,∀y∗∈F(y,vn),y∈L(un) satisfyingdn= 1 for everyn= 1, 2, . . . . Without loss of generality, we can assume that
dnd0= 0 by Lemma2.7. By the upper semicontinuity ofLand Lemma2.8, we have
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(L(un))∞⊂(L(u0))∞ for large enoughnand sodn∈(L(u0))∞ for large enoughn. Since (L(u0))∞is weakly closed, we haved0∈(L(u0))∞. Take any fixedy∈L(u0) andy∗∈F(y,v0). From the lower semicontinuity of L, there existsyn∈L(un) such that yn→y. Hence, (yn,vn)→(y,v0). By the lower semicontinuity of F, there existsy∗n∈F(yn,vn) such that
y∗n→y∗. Since
dn∈
d∈X:y∗,d+J◦(y,d) +f∞(d)≤0,∀y∗∈F(y,vn),y∈L(un)
,
we have
y∗n,dn
+J◦(yn,dn) +f∞(dn)≤0.
Combining withyn→y,y∗n→y∗,dnd0, the bi-sequential weak lower semicontinuity of
J◦and the weak lower semicontinuity off∞, it follows that y∗,d0+J◦(y,d0) +f∞(d0)≤0. Sincey∈L(u0) andy∗∈F(y,v0) are arbitrary, from the above discussion, we have
d0∈
d∈X:y∗,d+J◦(y,d) +f∞(d)≤0,∀y∗∈F(y,v0),y∈L(u0)
,
and so
d0∈
L(u0)
∞∩
d∈X:y∗,d+J◦(y,d) +f∞(d)≤0,∀y∗∈F(y,v0),y∈L(u0)
withd0= 0, which contradicts the assumption. This completes the proof. Corollary 4.2 If
L(u0)∞∩d∈X:y∗,d+f∞(d)≤0,∀y∗∈FL(u0),v0
={0}, (4.3)
then there exists a neighborhood U×Vof(u0,v0)with U×V⊂U×V such that
L(u)∞∩d∈X:y∗,d+f∞(d)≤0,∀y∗∈FL(u),v={0} (4.4)
for all(u,v)∈U×V.
Proof WheneverJ= 0, we know thatJ◦= 0 and henceJ◦is bi-sequentially weakly lower semicontinuous. In this case, (4.1) and (4.2) in Theorem4.1reduce to (4.3) and (4.4), re-spectively. Utilizing Theorem4.1, we immediately deduce Corollary4.2.
Remark4.3 It is known that ifJ= 0 then Theorem4.1reduces to Theorem 4.1 in Zhong and Huang [19]. Thus, Theorem4.1generalizes and extends Zhong and Huang’s Theo-rem 4.1 [19] to the case of Clarke’s generalized directional derivative of a locally Lipschitz functional. Iff = 0 additionally, thenf∞= 0. Thus, (4.3) and (4.4) in Corollary4.2reduce to (3.1) and (3.2) in [30, Theorem 3.1], respectively. Therefore, Zhong and Huang’s Theo-rem 4.1 in [19] is a generalization of Theorem 3.1 in [30].
Theorem 4.4 Assume that all the conditions of Theorem4.1are satisfied.Suppose that
(i) for eachv∈V,the mappingx→F(x,v)is(f,J)-pseudomonotone onM;
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(ii) the solution set ofGMVHVI(F(·,v0),J,L(u0))is nonempty and bounded.
Then there exists a neighborhood U×Vof(u0,v0)with U×V⊂U×V such that,for every(u,v)∈U×V,the solution set ofGMVHVI(F(·,v),J,L(u))is nonempty and bounded.
Moreover,if f is continuous on M=u∈UL(u)and J◦:M×(M–M)→Ris continuous,
thenω-lim sup(u,v)→(u
0,v0)SGM(u,v)⊂SGM(u0,v0),where SGM(u,v)and SGM(u0,v0)are the
solution sets ofGMVHVI(F(·,v),J,L(u))andGMVHVI(F(·,v0),J,L(u0)),respectively.
Proof By Theorem3.1, we get
L(u0)∞∩d∈X:y∗,d+J◦(y,d) +f∞(d)≤0,∀y∗∈F(y,v0),y∈L(u0)={0}.
It follows from Theorem 4.1that there exists a neighborhoodU×Vof (u0,v0) with
U×V⊂U×Vsuch that
L(u)∞∩d∈X:y∗,d+J◦(y,d) +f∞(d)≤0,∀y∗∈F(y,v),y∈L(u)={0}
for all (u,v)∈U×V. SinceF is (f,J)-pseudomonotone, Theorem3.4implies that the solution set ofGMVHVI(F(·,v),J,L(u)) is nonempty and bounded for every (u,v)∈U× V.
Next, we prove thatω-lim sup(u,v)→(u
0,v0)SGM(u,v)⊂SGM(u0,v0). For{(un,vn)} ⊂U
×V with (un,vn)→(u0,v0), we need to prove thatω-lim supn→∞SGM(un,vn)⊂SGM(u0,v0). For anyn= 0, 1, 2, . . . , define a functionΦn:X→Rby
Φn(x) := sup
y∈L(un),y∗∈F(y,vn)
y∗,x–y+J◦(y,x–y) +f(x) –f(y)
ϕ(y,y∗) ,
where
ϕy,y∗:=maxy∗, 1maxy, 1maxf(y), 1.
LetAn:={x∈L(un) :Φn(x)≤0}for every non-negative integern. By the definition ofΦn, it is easy to see thatAn={x∈L(un) :Φn(x)≤0}coincides with the solution setSGM(un,vn)
ofGMVHVI(F(·,v),J,L(u)) for alln= 0, 1, 2, . . . . Thus,Anis nonempty and bounded by
condition (ii) for every non-negative integern. From the above discussion, we need only to prove thatω-lim supn→∞An⊂A0. Letx∈ω-lim supn→∞An. Then there exists a sequence
{xnj}with eachxnj ∈Anj such thatxnj weakly converges tox. We claim that there exists
znj∈L(u0) such thatlimj→∞xnj–znj= 0. Indeed, if the claim does not hold, then there exist a subsequence{xnjk}of{xnj}and someε0> 0 such that
dxnjk,L(u0)
≥ε0, k= 1, 2, . . . .
This implies thatxnjk ∈/L(u0) +ε0B(0, 1) and soL(unjk)⊂L(u0) +ε0B(0, 1), which contra-dicts the upper semicontinuity ofL(·). Moreover, we obtainx∈L(u0) asL(u0) is a closed and convex subset ofXand hence weakly closed. Next we prove thatΦ0(x)≤0 and hence
x∈A0. In fact, for any fixedy∈L(u0) andy∗∈F(y,v0), sinceL is lower semicontinu-ous and un→u0, we know that there existsyn∈L(un) for everyn= 1, 2, . . . such that
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limn→∞yn=y. SinceFis lower semicontinuous, it follows that there exists a sequence of elementsy∗n∈F(yn,vn) such thaty∗n→y∗. Nowxnj∈Anj implies thatΦnj(xnj)≤0 and so
y∗nj,xnj–ynj+J◦(ynj,xnj–ynj) +f(xnj) –f(ynj)
ϕ(ynj,y∗nj)
≤0.
Sincef is continuous onM=u∈UL(u) andJ◦:M×(M–M)→Ris also continuous, lettingj→ ∞, we have
y∗,x–y+J◦(y,x–y) +f(x) –f(y)
ϕ(y,y∗) ≤0.
Sincey∈L(u0) andy∗∈F(y,v0) are arbitrary, we know thatΦ0(x)≤0 and hencex∈A0.
This completes the proof.
Corollary 4.5 Assume that all the conditions of Corollary4.2are satisfied.Suppose that
(i) for eachv∈V,the mappingx→F(x,v)isf-pseudomonotone onM; (ii) the solution set ofGMMVI(F(·,v0),L(u0))is nonempty and bounded.
Then there exists a neighborhood U×Vof(u0,v0)with U×V⊂U×V such that,for every(u,v)∈U×V,the solution set ofGMMVI(F(·,v),L(u))is nonempty and bounded.
Moreover, if f is continuous on M =u∈UL(u), then ω-lim sup(u,v)→(u
0,v0)SM(u,v)⊂
SM(u0,v0),where SM(u,v)and SM(u0,v0)are the solution sets ofGMMVI(F(·,v),L(u))and
GMMVI(F(·,v0),L(u0)),respectively.
Proof WheneverJ= 0, we know thatJ◦= 0,GMVHVI(F(·,v),J,L(u)) (resp.,GMVHVI(F(·,
v0),J,L(u0))) reduces to GMMVI(F(·,v),L(u)) (resp., GMMVI(F(·,v0),L(u0))), SGM(u,v) (resp.,SGM(u0,v0)) reduces toSM(u,v) (resp.,SM(u0,v0)), and the (f,J )-pseudomonotoni-city ofFin the first variable reduces to thef-pseudomonotonicity ofFin the first variable. Utilizing Theorem4.9, we immediately deduce Corollary4.5.
Remark4.6 It is known that ifJ= 0 then Theorem4.4reduces to Theorem 4.2 in Zhong and Huang [19]. Thus, Theorem4.4generalizes and extends Theorem 4.2 in Zhong and Huang [19] from the generalized Minty mixed variational inequality to the generalized Minty variational-hemivariational inequality. Iff= 0 additionally, thenf∞= 0, and so the generalized Minty mixed variational inequalityGMMVI(F,K) reduces to the generalized Minty variational inequality. Hence, Zhong and Huang’s Theorem 4.2 [19] generalizes [30, Theorem 3.2] from the generalized Minty variational inequality to the generalized Minty mixed variational inequality. In addition, for the case ofJ=f = 0, He [29] obtained the corresponding result of Zhong and Huang’s Theorem 4.2 [19] when either the mapping or the constraint set is perturbed (see Theorems 4.1 and 4.4 of [29]). Therefore, Zhong and Huang’s Theorem 4.2 [19] is a generalization of Theorems 4.1 and 4.4 in [29].
In the following, as an application of Theorem 4.4, we will consider the stability be-havior for the following generalized variational-hemivariational inequality, denoted by
GVHVI(F,J,K), which is to findx∈Kandx∗∈F(x) such that
GVHVI(F,J,K) :x∗,y–x+J◦(x,y–x) +f(y) –f(x)≥0, ∀y∈K. (4.5)
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If J= 0, thenGVHVI(F,J,K) reduces to the generalized mixed variational inequality, which is to findx∈Kandx∗∈F(x) such that
GMVI(F,K) :x∗,y–x+f(y) –f(x)≥0, ∀y∈K. (4.6)
IfFis single-valued, then (4.5) reduces to (1.1). Furthermore, iff= 0, then (4.6) reduces to the following generalized variational inequality of findingx∈Kandx∗∈F(x) such that
GVI(F,K) :x∗,y–x≥0, ∀y∈K. (4.7)
Next we consider the parametric generalized variational-hemivariational inequality, de-noted byGVHVI(F(·,v),J,L(u)), which is to findx∈L(u) andx∗∈F(x,v) such that
GVHVIF(·,v),J,L(u):x∗,y–x+J◦(x,y–x) +f(y) –f(x)≥0, ∀y∈L(u). (4.8)
In particular, ifJ= 0, then (4.8) reduces to the following parametric generalized mixed variational inequality, which is to findx∈L(u) andx∗∈F(x,v) such that
GMVIF(·,v),L(u):x∗,y–x+f(y) –f(x)≥0, ∀y∈L(u). (4.9) The following lemma shows that GVHVI(F,J,K) is closely related to its generalized Minty variational-hemivariational inequality.
Lemma 4.7 (i)If F is(f,J)-pseudomonotone on K,then every solution ofGVHVI(F,J,K)
solvesGMVHVI(F,J,K). (ii)If F is upper hemicontinuous on K with nonempty values,then every solution ofGMVHVI(F,J,K)solvesGVHVI(F,J,K).
Proof (i) The conclusion is obvious. Now we prove (ii). Suppose thatxis a solution of
GMVHVI(F,J,K), but it is not a solution ofGVHVI(F,J,K). Then there exists somey∈K
such that
x∗,y–x+J◦(x,y–x) +f(y) –f(x) < 0, ∀x∗∈F(x).
Since the set{x∗∈X∗: x∗,y–x+J◦(x,y–x) +f(y) –f(x) < 0}is a weakly∗open neigh-borhood ofF(x) andF is upper hemicontinuous, settingxt=ty+ (1 –t)xfort> 0 small enough, we deduce from the positive homogeneousness ofJ◦in the second variable
x∗t,y–x+J◦(xt,y–x) +f(y) –f(x) < 0. It follows that, for anyt> 0,
x∗t,t(y–x)+J◦xt,t(y–x)
+tf(y) –f(x)< 0. (4.10)
By the convexity off, we have
f(xt) =fty+ (1 –t)x≤tf(y) + (1 –t)f(x)
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and sof(xt) –f(x)≤t(f(y) –f(x)). Utilizing (4.10) and the subadditivity ofJ◦in the second variable, we obtain that
x∗t,xt–x
–J◦(xt,x–xt) +f(xt) –f(x)
≤x∗t,xt–x
+J◦(xt,xt–x) +f(xt) –f(x)
≤x∗t,xt–x
+J◦(xt,xt–x) +t
f(y) –f(x)< 0, which immediately leads to
x∗t,x–xt
+J◦(xt,x–xt) +f(x) –f(xt) > 0.
This contradicts the fact thatxis a solution ofGMVHVI(F,J,K). Hence, the conclusion
of (ii) holds. This completes the proof.
Corollary 4.8 (i)If F is f -pseudomonotone on K,then every solution ofGMVI(F,K)solves
GMMVI(F,K). (ii)If F is upper hemicontinuous on K with nonempty values,then every
solution ofGMMVI(F,K)solvesGMVI(F,K).
Proof WheneverJ= 0, we know thatJ◦= 0,GMVHVI(F,J,K) (resp.,GVHVI(F,J,K)) re-duces toGMMVI(F,K) (resp.,GMVI(F,K)), and the (f,J)-pseudomonotonicity ofF re-duces to thef-pseudomonotonicity ofF. Utilizing Lemma4.7, we immediately deduce
Corollary4.8.
Lemma 4.9 Let K be a nonempty,closed,and convex subset in a reflexive Banach space X,
f :K⊂X→Rbe a convex and lower semicontinuous function,and J:X→Rbe a locally Lipschitz functional.Suppose that F is upper hemicontinuous and(f,J)-pseudomonotone on K with nonempty values.Consider the following statements:
(i) the solution set ofGVHVI(F,J,K)is nonempty and bounded; (ii) the solution set ofGMVHVI(F,J,K)is nonempty and bounded; (iii) K∞∩ {d∈X: y∗,d+J◦(y,d) +f∞(d)≤0,∀y∗∈F(y),y∈K}={0}.
Then(i)⇔(ii)and(ii)⇒(iii);moreover,ifint(barr(K))=∅,then(iii)⇒(ii)and hence they all are equivalent.
Proof Under the assumptions ofF, the equivalence of (i) and (ii) is stated in Lemma4.7.
Then the conclusion follows from Theorem3.4.
Corollary 4.10 Let K be a nonempty,closed,and convex subset in a reflexive Banach space X and f :K⊂X→Rbe a convex and lower semicontinuous function.Suppose that F is upper hemicontinuous and f -pseudomonotone on K with nonempty values.Consider the following statements:
(i) the solution set ofGMVI(F,K)is nonempty and bounded; (ii) the solution set ofGMMVI(F,J,K)is nonempty and bounded; (iii) K∞∩ {d∈X: y∗,d+f∞(d)≤0,∀y∗∈F(y),y∈K}={0}.
Then(i)⇔(ii)and(ii)⇒(iii);moreover,ifint(barr(K))=∅,then(iii)⇒(ii)and hence they all are equivalent.
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Proof WheneverJ= 0, we know thatJ◦= 0, the (f,J)-pseudomonotonicity ofFreduces to thef-pseudomonotonicity ofF, and statements (i), (ii), and (iii) in Lemma4.9reduce to (i), (ii), and (iii) in Corollary4.10. Utilizing Lemma4.9, we deduce the desired result.
Remark4.11 It is known that ifJ= 0 then Lemmas4.7and4.9reduce to Lemmas 4.1 and 4.2 in [19], respectively. Thus, Lemmas4.7and4.9generalize and extend Lemmas 4.1 and 4.2 in [19] from the generalized mixed variational inequality to the generalized variational-hemivariational inequality. Iff = 0 additionally, then Lemma 4.2 in [19] reduces to The-orem 3.2 of [29]. Therefore, Lemma 4.2 in [19] generalizes Theorem 3.2 of [29] from the generalized variational inequality to the generalized mixed variational inequality.
From Theorem4.4and Lemma4.9, we can easily establish the following stability result for the generalized variational-hemivariational inequality.
Theorem 4.12 Assume that all the conditions of Theorem4.1are satisfied.Suppose that
(i) for eachv∈V,the mappingx→F(x,v)is upper hemicontinuous and
(f,J)-pseudomonotone onM;
(ii) the solution set ofGVHVI(F(·,v0),J,L(u0))is nonempty and bounded.
Then there exists a neighborhood U×Vof(u0,v0)with U×V⊂U×V such that,for
every(u,v)∈U×V,the solution set ofGVHVI(F(·,v),J,L(u))is nonempty and bounded.
Moreover,if f is continuous on M=u∈UL(u)and J◦:M×(M–M)→Ris continuous,
thenω-lim sup(u,v)→(u
0,v0)SG(u,v)⊂SG(u0,v0),where SG(u,v)and SG(u0,v0)are the solution
sets ofGVHVI(F(·,v),J,L(u))andGVHVI(F(·,v0),J,L(u0)),respectively.
Proof SinceFis upper hemicontinuous with nonempty values and (f,J)-pseudomonotone onM, it follows from Lemma4.9that the solution set ofGMVHVI(F(·,v),J,L(u)) coincides with that ofGVHVI(F(·,v),J,L(u)), and so the result follows directly from Theorem4.4.
This completes the proof.
Funding
This research was partially supported by the Grant MOST 106-2923-E-039-001-MY3.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors have made the same contribution and finalized the current version of this article. They read and approved the final manuscript.
Author details
1Department of Mathematics, Shanghai Normal University, Shanghai, China.2Department of Mathematics, Texas A&M University, Kingsville, USA.3Florida Institute of Technology, Melbourne, USA.4Center for General Education, China Medical University, Taichung, Taiwan.5Department of Mathematics, Tianjin Polytechnic University, Tianjin, China.
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Received: 3 August 2018 Accepted: 25 October 2018 References
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