R E S E A R C H
Open Access
Existence theorems of generalized
quasi-variational-like inequalities for
pseudo-monotone type II operators
Mohammad SR Chowdhury
1*, Afrah AN Abdou
2and Yeol Je Cho
3**Correspondence:
[email protected]; [email protected]; [email protected]
1Department of Mathematics,
University of Management and Technology (UMT), C-II, Johar Town, Lahore, 54770, Pakistan
3Department of Mathematics
Education and RINS, Gyeongsang National University, Jinju 660-701, Korea
Full list of author information is available at the end of the article
Abstract
In this paper, we prove the existence results of solutions for a new class of generalized quasi-variational-like inequalities (GQVLI) for pseudo-monotone type II operators defined on compact sets in locally convex Hausdorff topological vector spaces. In obtaining our results on GQVLI for pseudo-monotone type II operators, we use Chowdhury and Tan’s generalized version (Chowdhury and Cho in J. Inequal. Appl. 2012:79, 2012) of Ky Fan’s minimax inequality (Fan in Inequalities, vol. III, pp.103-113, 1972) as the main tool.
Keywords: generalized quasi-variational-like inequalities; pseudo-monotone type II operators; locally convex Hausdorff topological vector spaces
1 Introduction
IfXis a nonempty set, then we denote by Xthe family of all nonempty subsets ofXand byF(x) the family of all nonempty finite subsets ofX. LetEbe a topological vector space over,Fbe a vector space overandXbe a nonempty subset ofE. Let·,·:F×E→ be a bilinear functional. Throughout this paper,denotes either the real fieldRor the complex fieldC.
For eachx∈E, each nonempty subsetAofEand each> , letW(x;) :={y∈F:
|y,x|<}andU(A;) :={y∈F:supx∈A|y,x|<}. Let σF,E be the (weak) topol-ogy on F generated by the family {W(x;) :x∈E, > } as a subbase for the neigh-borhood system at andδF,Ebe the (strong) topology onFgenerated by the family
{U(A;) :Ais a nonempty bounded subset ofEand> }as a base for the neighborhood system at . We note then thatF, when equipped with the (weak) topologyσF,Eor the (strong) topologyδF,E, becomes a locally convex topological vector space which is not necessarily Hausdorff. But, if the bilinear functional·,·:F×E→separates points in
F,i.e., for eachy∈Fwithy= , there exists x∈Esuch thaty,x = , thenFalso becomes Hausdorff. Furthermore, for any net{yα}α∈inFandy∈F,
(a) yα→yinσF,Eif and only ifyα,x → y,xfor eachx∈E;
(b) yα→yinδF,Eif and only ifyα,x → y,xuniformly for eachx∈A, whereAis a
nonempty bounded subset ofE.
Suppose that, for the setsX,EandFmentioned above,S:X→XandT:X→F are two set-valued mappings. We now introduce below a slightly modified definition of the
generalized quasi-variational inequalityin infinite dimensional spaces given by Shih and Tan in []:
Findyˆ∈S(yˆ)andwˆ∈T(yˆ)such that
Re ˆw,yˆ–x ≤
for allx∈S(yˆ).
Now, we state the following definition which is a slightly corrected version of the corre-sponding definition given in []. Please note that there were typos in Definition . in [].
Definition . Let the setsX,EandFand the mappingsSandTbe as defined above. Let η:X×X→Ebe a single-valued mapping andh:X×X→Rbe a real-valued function. Then thegeneralized quasi-variational-like inequality problemis defined as follows: Find
ˆ
y∈S(ˆy) andwˆ ∈T(ˆy) such that
Rewˆ,η(yˆ,x)+h(yˆ,x)≤
for allx∈S(yˆ).
For more results related to the generalized quasi-variational-like inequality problems, we refer to [–] and the references therein.
The following definition given in [] is a slight modification of demi-operators defined in [] and of pseudo-monotone type II operators defined in [] (see also []).
Definition . LetXbe a nonempty subset of a topological vector spaceEover,Fbe a vector space overwhich is equipped withσF,E-topology, where·,·:F×E→is a bilinear functional. Leth:X×X→R,η:X×X→EandT:X→F be three mappings. ThenT is said to be:
() an(η,h)-pseudo-monotone type II(respectively, astrongly(η,h)-pseudo-monotone type II)operatorif, for eachy∈Xand every net{yα}α∈inXconverging toy
(respectively, weakly toy) with
lim sup
α
inf u∈T(y)Re
u,η(yα,y)
+h(yα,y)
≤,
we have
lim sup
α
inf u∈T(x)Re
u,η(yα,x)
+h(yα,x)
≥ inf u∈T(x)Re
u,η(y,x)+h(y,x)
for allx∈X;
() anh-pseudo-monotone type II operator(respectively, astronglyh-pseudo-monotone type II operator) ifTis an(η,h)-pseudo-monotone type II operator with
η(x,y) =x–yand, for someh :X→R,h(x,y) =h(x) –h(y)for allx,y∈X.
Pseudo-monotone type II operators were first introduced by Chowdhury in [] with a slight variation in the name of this operator. Later, these operators were renamed as pseudo-monotone type II operators by Chowdhury in [].
Next, we shall state and prove the following lemma which provides a numerous col-lection of (η,h)-pseudo-monotone type II and strongly (η,h)-pseudo-monotone type II operators.
Lemma . Let E be a topological vector space and X be a nonempty bounded subset of E.
Let T:X→E∗be an operator such that each T(x)is strongly compact.Suppose that h:
X×X→Ris a real-valued function such that,for each y∈X,h(·,y)is continuous and h(X×X)is bounded.Letη:X×X→E be a continuous mapping.Suppose further that the operator T is a continuous mapping from the relative weak topology on X to the weak∗ topology on E∗.Then T is both an(η,h)-pseudo-monotone type II and a strongly(η,h) -pseudo-monotone type II operator.
Proof Suppose that{yα}α∈ is a net in Xandy∈X with yα→y(respectively,yα→y
weakly) and that
lim sup
α
inf u∈T(y)Re
u,η(yα,y)
+h(yα,y)
≤.
Letx∈Xbe arbitrarily fixed. Then, using the continuity ofh(·,y),ηandT, we obtain the following:
lim sup
α
inf u∈T(x)Re
u,η(yα,x)
+h(yα,x)
≥lim sup
α
inf u∈T(x)Re
u,η(yα,x)
+lim inf
α h(yα,x)
= inf
u∈T(x)Re
u,η(y,x)+h(y,x)
for allx∈X. Consequently,T is both an (η,h)-pseudo-monotone type II and a strongly
(η,h)-pseudo-monotone type II operator.
The above lemma will, therefore, provide ample examples for our main results in The-orems . and . given in Section .
In this paper, we obtain some general theorems on solutions for a new class of gener-alized quasi-variational-like inequalities for pseudo-monotone type II operators defined on compact sets in topological vector spaces. In obtaining our results, we shall mainly use the following generalized version of Ky Fan’s minimax inequality [] due to Chowdhury and Tan which was stated and proved as Theorem . in [] and is a slight modification of Theorem in [].
Theorem . Let E be a Hausdorff topological vector space and X be a nonempty convex subset of E.Let h:X×X→Randφ:X×X→R∪ {–∞, +∞}be the mappings such that
(a) for eachA∈F(X)and fixedx∈co(A),y→φ(x,y)is lower semi-continuous on
co(A);
(c) for each fixedx∈X,y→h(x,y)is lower semi-continuous and concave onX,and
h(x,x) = ;
(d) for eachA∈F(X)and each pair of pointsx,y∈co(A)such that every net{yα}α∈in Xconverging toywithφ(tx+ ( –t)y,yα) +h(yα,tx+ ( –t)y)≤for allα∈and
allt∈[, ],we haveφ(x,y) +h(y,x)≤;
(e) there exist a nonempty closed and compact subsetKofXandx∈Ksuch that
φ(x,y) +h(y,x) > for ally∈X\K.
Then there existsyˆ∈K such thatφ(x,yˆ) +h(yˆ,x)≤for all x∈X.
Proof For the proof, we refer to [].
Definition . A functionφ:X×X→R∪ {±∞}is said to be -diagonally concave (in short, -DCV) in the second argument [] if, for any finite set{x, . . . ,xn} ⊂Xandλi≥ withni=λi= , we have
n
i=λiφ(y,xi)≤, wherey= n
i=λixi. Now, we state the following definition given in [].
Definition . LetX,E,F be the sets defined before andT :X→F, η:X×X→E,
g:X→Ebe mappings.
() The mappingsT andηare said to have the-diagonally concave relation(in short, -DCVR) if the functionφ:X×X→R∪ {±∞}defined by
φ(x,y) = inf w∈T(x)Re
w,η(x,y)
is -DCV iny.
() The mappingsT andgare said to have the-diagonally concave relationifTand η(x,y) =g(x) –g(y)have the -DCVR.
The following definition of upper hemi-continuity was given in []. For a more general definition, we refer to Definition in [].
Definition . LetE be a topological vector space,X be a nonempty subset ofE and
T :X→E∗. ThenT is said to beupper hemi-continuousonXif and only if, for each
p∈E, the functionfp:X→R∪ {+∞}defined by
fp(z) = sup u∈T(z)
Reu,p
for eachz∈Xis upper semi-continuous onX(if and only if, for eachp∈E, the function
gp:X→R∪ {–∞}defined by
gp(z) = inf
u∈T(z)Reu,p
for eachz∈Xis lower semi-continuous onX).
2 Preliminaries
Lemma . Let X be a nonempty subset of a Hausdorff topological vector space E and S:X→Ebe an upper semi-continuous map such that S(x)is a bounded subset of E for
each x∈X.Then,for each continuous linear functional p on E,the mapping fp:X→R
defined by
fp(y) = sup x∈S(y)
Rep,x
is upper semi-continuous,i.e.,for eachλ∈R,the set{y∈X:fp(y) =supx∈S(y)Rep,x<λ}
is open in X.
The following result is Lemma of Takahashi in [] (see also Lemma in []).
Lemma . Let X and Y be topological spaces,f :X→Rbe non-negative and continuous and g:Y→Rbe lower semi-continuous.Then the mapping F :X×Y →Rdefined by F(x,y) =f(x)g(y)for all(x,y)∈X×Y is lower semi-continuous.
The following result, which was stated and proved as Lemma . in [], follows from slight modification of Lemma of Chowdhury and Tan given in [].
Lemma . Let E be a Hausdorff topological vector space over,A∈F(E)and X=co(A),
whereco(A)denotes the convex hull of A.Let F be a vector space overand·,·:F×E→φ
be a bilinear functional such that·,·separates points in F.We equip F with theσF,
E-topology.Suppose that,for each w∈F,x→Rew,xis continuous.Letη:X×X→E be continuous.Let T:X→F be upper semi-continuous from X intoFsuch that each T(x)
isσF,E-compact.Let f :X×X→Rbe defined by f(x,y) =infw∈T(y)Rew,η(y,x)for all
x,y∈X.Suppose that·,·is continuous on the(compact)subset[y∈XT(y)]×η(X×X)
of F×E.Then,for each fixed x∈X,y→f(x,y)is lower semi-continuous on X.
For the completeness, we include the proof here given in [].
Proof Letλ∈Rbe given and letx∈X=co(A) be arbitrarily fixed. LetAλ={y∈X: f(x,y)≤λ}. Suppose that{yα}α∈ is a net inAλandy∈co(A) =X such thatyα→y.
Then, for eachα∈,
λ≥f(x,yα) = inf
w∈T(yα)Re
w,η(yα,x)
.
SinceFis equipped with theσF,E-topology, for eachx∈E, the functionw→Rew,x
is continuous. Also, η(yα,x)→ η(y,x) because η(·,x) is continuous. By the σF,E
-compactness ofT(yα), there existswα∈T(yα) such that
λ≥ inf w∈T(yα)
Rew,η(yα,x)
=Rewα,η(yα,x)
.
SinceTis upper semi-continuous fromX=co(A) to theσF,E-topology onF,Xis com-pact, and each T(z) is σF,E-compact,z∈XT(z) is alsoσF,E-compact by Proposi-tion .. of Aubin and Ekeland []. Thus there is a subnet {wα}α∈ of{wα}α∈and w∈
z∈XT(z) such thatwα →win theσF,E-topology. Again, asT is upper
Suppose thatA={a,a, . . . ,an}and lett,t, . . . ,tn≥ with
n
i=ti= such thaty=
n
i=tiai. For eachα ∈, lettα,tα, . . . ,tαn ≥ with
n
i=tiα = such thatyα =
n
i=tiαai. SinceEis Hausdorff andyα →y, we must havetiα →tifor eachi= , , . . . ,n. Thus
λ ≥ Rewα,η(yα,x)
=Re
wα,η
n
i=
tαi ai,x
→Re
w,η
n
i=
tiai,x
= Rew,η(y,x)
≥ inf w∈T(y)Re
w,η(y,x)
= f(x,y), (.)
where (.) is true sinceη(·,x) is continuous onXand·,·is continuous on the compact subset [y∈XT(y)]×η(X×X) ofF×E. Hencey∈Aλ. ThusAλis closed inX=co(A)
for eachλ∈R. Thereforey→f(x,y) is lower semi-continuous onX. This completes the
proof.
By the slight modification of Lemma . in [], we obtained the following result given in [] as Lemma ..
Lemma . Let E be a topological vector overφ,X be a nonempty convex subset of E and F be a vector space overφwith theσF,E-topology such that,for each w∈F,the function x→Rew,xis continuous.Let T:X→Fbe upper hemi-continuous along line segments
in X.Letη:X×X→E be such that for each fixed y∈X, η(·,y)is continuous,and let h:X×X→Rbe a mapping such that,for each fixed y∈X,h(·,y)is lower semi-continuous onco(A)for each A∈F(X)and,for each fixed x∈X,h(x,·)is concave and h(x,x) = and T,ηhave the-DCVR.Suppose thatyˆ∈X such thatinfu∈T(x)Reu,η(yˆ,x) ≤h(x,yˆ)for all
x∈X.Then
inf w∈T(ˆy)Re
w,η(ˆy,x)≤h(x,yˆ)
for all x∈X.
We need the following Kneser’s minimax theorem in [] (see also Aubin []).
Theorem . Let X be a nonempty convex subset of a vector space and Y be a nonempty compact convex subset of a Hausdorff topological vector space.Suppose that f is a real-valued function on X×Y such that for each fixed x∈X,the map y→f(x,y),i.e.,f(x,·)is lower semi-continuous and convex on Y and,for each fixed y∈Y,the mapping x→f(x,y),
i.e.,f(·,y)is concave on X.Then
min
y∈Y supx∈Xf(x,y) =supx∈Xminy∈Y f(x,y).
3 Generalized quasi-variational-like inequalities
domain in locally convex Hausdorff topological vector spaces. Our results extend and/or generalize the corresponding results in [].
First, we establish the following result.
Theorem . Let E be a locally convex Hausdorff topological vector space over,X be a nonempty compact convex subset of E and F be a vector space overwithσF,E-topology,
where·,·:F×E→is a bilinear functional separating points on F such that,for each w∈F,the function x→Rew,xis continuous.Let S:X→X,T:X→F,η:X×X→E
and h:E×E→Rbe the mappings such that
(a) Sis upper semi-continuous such that eachS(x)is closed and convex; (b) h(X×X)is bounded;
(c) T is an(η,h)-pseudo-monotone type II(respectively,a strongly
(η,h)-pseudo-monotone type II)operator and is upper hemi-continuous along line segments inXto theσF,E-topology onFsuch that eachT(x)isσF,E-compact and convex andT(X)isδF,E-bounded;
(d) T andηhave the-DCVR andηis continuous;
(e) for each fixedy∈X,x→h(x,y),i.e.,h(·,y)is lower semi-continuous onco(A)for eachA∈F(X)and,for each fixedx∈X,h(x,·)andη(x,·)are concave,η(x,·)is affine,h(x,x) = andη(x,x) = ;
(f ) the set={y∈X:supx∈S(y)[infu∈T(x)Reu,η(y,x)+h(y,x)] > }is open inX;
(g) for eachA∈F(X)and eachy∈co(A),there existx¯∈Aandu¯∈T(x¯)such that
β(y)
Reu¯,η(y,x¯)+h(y,x¯)+ p∈LF(E)
βp(y)Rep,y–x ≤¯
for any family{β,βp:p∈LF(E)}of non-negative real-valued functions fromXinto [, ],whereLF(E)denotes the set of all continuous linear functionals onE;
(h) for eachA∈F(X),the bilinear functional·,·is continuous over the compact subset [y∈co(A)T(y)]×η(co(A)×co(A))ofF×E.
Then there exists a pointˆy∈X such that
() yˆ∈S(yˆ);
() there exists a pointwˆ∈T(yˆ)withRe ˆw,η(yˆ,x)+h(yˆ,x)≤for allx∈S(yˆ).
Proof Step . Let us first show that there exists a pointyˆ∈Xsuch thatyˆ∈S(ˆy) and
sup x∈S(yˆ)
inf u∈T(x)Re
u,η(yˆ,x)+h(yˆ,x)
≤.
Now, we prove this by contradiction. So, we assume that, for eachy∈X, eithery∈/S(y) or there existsx∈S(y) such that
inf u∈T(x)Re
u,η(y,x)+h(y,x) > ,
that is, for eachy∈X, eithery∈/S(y) ory∈. Ify∈/S(y), then, by a slight modification of a separation theorem for convex sets in locally convex Hausdorff topological vector spaces, there exists a continuous linear functionalponEsuch that
Rep,y– sup x∈S(y)
For eachy∈X, set
γ(y) := sup x∈S(y)
inf u∈T(x)Re
u,η(y,x)+h(y,x),
V:==
y∈X:γ(y) >
and, for each continuous linear functionalponE,
Vp:=
y∈X:Rep,y– sup x∈S(y)
Rep,x>
.
Then we have
X=V∪
p∈LF(E)
Vp.
SinceV is open by hypothesis and eachVp is open inX by Lemma . (Lemma in []), {V,Vp :p∈LF(E)} is an open covering forX. Since X is compact, there exist
p,p, . . . ,pn∈LF(E) such thatX=V∪
n
i=Vpi. For the simplicity of notation, letVi=Vpi
fori= , , . . . ,n. Let{β,β, . . . ,βn}be a continuous partition of unity onXsubordinated to the covering{V,V, . . . ,Vn}. Thenβ,β, . . . ,βnare continuous non-negative real-valued functions onXsuch thatβivanishes onX\Vifor eachi= , , . . . ,nand
n
i=βi(x) = for
allx∈X. Note that, for eachy∈XandA∈F(X),x→h(x,y),i.e.,h(·,y) is continuous on co(A) (see [], Corollary ..). Define a functionφ:X×X→Rby
φ(x,y) =β(y)
min u∈T(x)Re
u,η(y,x)+h(y,x)
+ n
i=
βi(y)Repi,y–x
for allx,y∈X. Then we have the following:
(I) SinceEis Hausdorff, for eachA∈F(X) and fixedx∈co(A), the mapping
y→ inf u∈T(x)Re
u,η(y,x)+h(y,x)
is continuous onco(A) by Lemma . and the fact thathis continuous onco(A), and so the mapping
y→β(y)
min u∈T(x)Re
u,η(y,x)+h(y,x)
is lower semi-continuous onco(A) by Lemma .. Also, for each fixedx∈X,
y→
n
i=
βi(y)Repi,y–x
(II) Sinceβ,β, . . . ,βnis a family of continuous non-negative real-valued functions on
Xinto [, ], by the hypothesis, for eachA∈F(X) and eachy∈co(A), there existx¯∈A
andu¯∈T(x¯) such that
β(y)
Reu¯,η(y,x¯)+h(y,x¯)+ n
i=
βi(y)Repi,y–x ≤¯ .
Thus we have
min u∈T(x)
β(y)
Reu,η(y,x¯)+h(y,x¯)+ n
i=
βi(y)Repi,y–x¯ ≤,
i.e.,
β(y)
min u∈T(x)
Reu,η(y,x¯)+h(y,x¯)+ n
i=
βi(y)Repi,y–x¯ ≤.
Therefore, we have
min x∈A
β(y)
min u∈T(x)
Reu,η(y,x)+h(y,x)+ n
i=
βi(y)Repi,y–x
≤
and sominx∈Aφ(x,y)≤ for eachA∈F(X) andy∈co(A).
(III) Suppose thatA∈F(X),x,y∈co(A) and{yα}α∈is a net inXconverging toy
(re-spectively, weakly toy) withφ(tx+ ( –t)y,yα)≤ for allα∈and allt∈[, ].
Case :β(y) = . Sinceβis continuous andyα→y, we haveβ(yα)→β(y) = . Note
thatβ(yα)≥ for eachα∈. SinceT(X) is strongly bounded and{yα}α∈is a bounded
net, it follows that
lim sup
α
β(yα)
min u∈T(x)Re
u,η(yα,x)
+h(yα,x)
= . (.)
Also, we have
β(y)
min u∈T(x)Re
u,η(y,x)+h(y,x)= .
Thus it follows from (.) that
lim sup
α
β(yα)min
u∈T(x)Re
u,η(yα,x)
+h(yα,x)
+ n
i=
βi(y)Repi,y–x
= n
i=
βi(y)Rep,y–x
=β(y)
min u∈T(x)Re
u,η(y,x)+h(y,x)
+ n
i=
Whent= , we haveφ(x,yα)≤ for allα∈,i.e.,
β(yα)
min u∈T(x)Re
u,η(yα,x)
+h(yα,x)
+ n
i=
βi(yα)Rep,yα–x
≤ (.)
for allα∈. Therefore, by (.), we have
lim sup
α
β(yα)
min u∈T(x)Re
u,η(yα,x)
+h(yα,x)
+lim inf
α
n
i=
βi(yα)Rep,yα–x
≤lim sup
α
β(yα)
min u∈T(x)Re
u,η(yα,x)
+h(yα,x)
+ n
i=
βi(yα)Rep,yα–x
≤, and so lim sup α β(yα)
min u∈T(x)Re
u,η(yα,x)
+h(yα,x)
+ n
i=
βi(y)Rep,y–x
≤. (.)
Hence, by (.) and (.), we haveφ(x,y)≤.
Case :β(y) > . Sinceβis continuous,β(yα)→β(y). Again sinceβ(y) > , there
existsλ∈such thatβ(yα) > for allα≥λ.
Whent= , we haveφ(y,yα)≤ for allα∈,i.e.,
β(yα)
min u∈T(y)Re
u,η(yα,y)
+h(yα,y)
+
n
i=
βi(yα)Rep,yα–y ≤
for allα∈, and so
lim sup
α
β(yα)
min u∈T(y)Re
u,η(yα,y)
+h(yα,y)
+ n
i=
βi(yα)Rep,yα–y
Hence, by (.), we have
lim sup
α
β(yα)
min u∈T(y)Re
u,η(yα,y)
+h(yα,y)
+lim inf
α
n
i=
βi(yα)Rep,yα–y
≤lim sup
α
β(yα)
min u∈T(y)Re
u,η(yα,y)
+h(yα,y)
+ n
i=
βi(yα)Rep,yα–y
≤.
Sincelim infα[
n
i=βi(yα)Rep,yα–y] = , we have
lim sup
α
β(yα)
min u∈T(y)Re
u,η(yα,y)
+h(yα,y)
≤. (.)
Sinceβ(yα) > for allα≥λ, it follows that
β(y)lim sup
α
min u∈T(y)Re
u,η(yα,y)
+h(yα,y)
=lim sup
α
β(yα)
min u∈T(y)Re
u,η(yα,y)
+h(yα,y)
. (.)
Sinceβ(y) > , by (.) and (.), we have
lim sup
α
min u∈T(y)Re
u,η(yα,y)
+h(yα,y)
≤.
Since T is an (η,h)-pseudo-monotone type II (respectively, a strongly (η,h )-pseudo-monotone type II) operator, we have
lim sup
α
min u∈T(x)Re
u,η(yα,x)
+h(yα,x)
≥ min u∈T(x)Re
u,η(y,x)+h(y,x)
for allx∈X. Sinceβ(y) > , we have
β(y)
lim sup
α
min u∈T(x)Re
u,η(yα,x)
+h(yα,x)
≥β(y)
min u∈T(x)Re
u,η(y,x)+h(y,x)
,
and so
β(y)
lim sup
α
min u∈T(x)Re
u,η(yα,x)
+h(yα,x)
+ n
i=
≥β(y)
min u∈T(x)Re
u,η(y,x)+h(y,x)
+ n
i=
βi(y)Rep,y–x. (.)
Whent= , we haveφ(x,yα)≤ for allα∈,i.e.,
β(yα)
min u∈T(x)Re
u,η(yα,x)
+h(yα,x)
+
n
i=
βi(yα)Rep,yα–x ≤
for allα∈and so, by (.),
≥lim sup
α
β(yα)
min u∈T(x)Re
u,η(yα,x)
+h(yα,x)
+ n
i=
βi(yα)Rep,yα–x
≥lim sup
α
β(yα)
min u∈T(x)Re
u,η(yα,x)
+h(yα,x)
+lim inf
α
n
i=
βi(yα)Rep,yα–x
=β(y)
lim sup
α
min u∈T(x)Re
u,η(yα,x)
+h(yα,x)
+ n
i=
βi(y)Rep,y–x
≥β(y)
min u∈T(x)Re
u,η(y,x)+h(y,x)
+ n
i=
βi(y)Rep,y–x. (.)
Hence we haveφ(x,y)≤.
(IV) SinceXis a compact (respectively, weakly compact) subset of the Hausdorff topo-logical vector spaceE, it is also closed. Now, if we takeK=X, then, for anyx∈K=X,
we haveφ(x,y) > for ally∈X\K(=X\X=∅). Thus the hypothesis (d) of Theorem .
is satisfied trivially. (IfT is a strongly (η,h)-quasi-pseudo-monotone type II operator, we equipEwith the weak topology.) Thusφsatisfies all the hypotheses of Theorem .. Hence, by Theorem ., there exists a pointyˆ∈K=Xsuch thatφ(x,yˆ)≤ for allx∈X,i.e.,
β(yˆ)
min u∈T(x)Re
u,η(ˆy,x)+h(yˆ,x)
+ n
i=
βi(ˆy)Repi,yˆ–x ≤ (.)
for allx∈X.
Ifβ(ˆy) > , thenyˆ∈V=so thatγ(yˆ) > . Choosexˆ∈S(yˆ)⊂Xsuch that
min u∈T(xˆ)
Then it follows that
β(yˆ)
min u∈T(xˆ)Re
u,η(ˆy,xˆ)+h(yˆ,xˆ)> .
Ifβi(yˆ) > for eachi= , , . . . ,n, thenyˆ∈Viand hence
Repi,yˆ > sup x∈S(ˆy)
Repi,x ≥Repi,xˆ
and soRepi,yˆ–xˆ> . Then we see thatβi(yˆ)Repi,yˆ–xˆ> wheneverβi(yˆ) > for each
i= , , . . . ,n. Sinceβ(yˆ) > orβi(yˆ) > for eachi= , , . . . ,n, it follows that
φ(xˆ,ˆy) =β(yˆ)
min u∈T(xˆ)Re
u,η(yˆ,xˆ)+h(ˆy,xˆ)
+ n
i=
βi(yˆ)Repi,yˆ–xˆ > ,
which contradicts (.). This contradiction proves Step . Hence we have shown that there exists a pointyˆ∈Xsuch thatyˆ∈S(yˆ) and
sup x∈S(yˆ)
inf u∈T(x)Re
u,η(yˆ,x)+h(yˆ,x)
≤.
Step . We need to show that
inf w∈T(ˆy)
Rew,η(ˆy,x)≤h(x,yˆ)
for allx∈S(yˆ).
From Step , we know thatyˆ∈S(ˆy) which is a convex subset ofXand
inf u∈T(x)Re
u,η(yˆ,x)≤h(x,yˆ)
for allx∈S(yˆ). Hence, applying Lemma ., we obtain
inf w∈T(ˆy)
Rew,η(ˆy,x)≤h(x,yˆ)
for allx∈S(yˆ).
Step . There exists a pointwˆ ∈T(yˆ) withRe ˆw,η(yˆ,x) ≤h(x,yˆ) for allx∈S(yˆ). From Step , we have
sup x∈S(yˆ)
inf w∈T(yˆ)Re
w,η(yˆ,x)+h(yˆ,x)
≤, (.)
whereT(yˆ) is aσF,E-compact convex subset of the Hausdorff topological vector space (F,σF,E) andS(yˆ) is a convex subset ofX.
So, we can apply Keneser’s minimax theorem (Theorem .) and obtain the following:
min w∈T(ˆy)
sup x∈S(ˆy)
Rew,η(yˆ,x)+h(yˆ,x)= sup x∈S(yˆ)
min w∈T(yˆ)
Rew,η(yˆ,x)+h(ˆy,x).
Hence, by (.), we obtain
min w∈T(ˆy)xsup∈S(ˆy)
Rew,η(yˆ,x)+h(yˆ,x)≤.
SinceT(yˆ) is compact, there existswˆ ∈T(yˆ) such that
Rewˆ,η(yˆ,x)+h(yˆ,x)≤
for allx∈S(yˆ). This completes the proof.
Note that, if for each open subsetUofXand for eachx,y∈U,η(x,y) =x–yand there exists h :X→Rsuch thath(x,y) =h(x) –h(y); and if the mappingS:X→X is, in addition, lower semi-continuous and, for eachy∈,Tis upper semi-continuous at some pointxinS(y) withinfu∈T(x)Reu,η(y,x)+h(y,x) > , then the setin Theorem . is always open inX, and so we obtain the following result.
Theorem . Let E be a locally convex Hausdorff topological vector space over,X be a nonempty compact convex subset of E and F be a vector space overwithσF,E-topology,
where·,·:F×E→is a bilinear functional separating points on F such that,for each w∈F,the function x→Rew,xis continuous.Let S:X→X,T:X→F,η:X×X→E
and h:E×E→Rbe the mappings such that
(a) Sis continuous such that eachS(x)is closed and convex; (b) h(X×X)is bounded;
(c) T is an(η,h)-pseudo-monotone type II(respectively,a strongly
(η,h)-pseudo-monotone type II)operator and is upper hemi-continuous along line segments inXto theσF,E-topology onFsuch that eachT(x)isσF,E-compact and convex andT(X)isδF,E-bounded;
(d) T andηhave the-DCVR andηis continuous;
(e) for each fixedy∈X,x→h(x,y),i.e.,h(·,y)is lower semi-continuous onco(A)for eachA∈F(X)and,for each fixedx∈X,h(x,·)andη(x,·)are concave,η(x,·)is affine,h(x,x) = andη(x,x) = ;
(f ) for each open subsetUofXandx,y∈U,η(x,y) =x–y,and there existsh :X→R
such thath(x,y) =h(x) –h(y);
(g) for eachy∈={y∈X:supx∈S(y)[infu∈T(x)Reu,η(y,x)+h(y,x)] > },Tis upper
semi-continuous at some pointxinS(y)withinfu∈T(x)Reu,η(y,x)+h(y,x) > ;
(h) for eachA∈F(X)andy∈co(A),there existx¯∈Aandu¯∈T(x¯)such that
β(y)
Reu¯,η(y,x¯)+h(y,x¯)+ p∈LF(E)
βp(y)Rep,y–x ≤¯
(i) for eachA∈F(X),the bilinear functional·,·is continuous over the compact subset [y∈co(A)T(y)]×η(co(A)×co(A))ofF×E.
Then there exists a pointˆy∈X such that
() yˆ∈S(yˆ);
() there exists a pointwˆ∈T(yˆ)withRe ˆw,η(yˆ,x)+h(yˆ,x)≤for allx∈S(yˆ).
The proof is similar to the proof of Theorem . in []. For the completeness, we include the proof here.
Proof The proof follows from Theorem . if we can show that the set
=y∈X: sup x∈S(y)
inf u∈T(x)Re
u,η(y,x)+h(y,x)>
is open inX. To show thatis open inX, we start as follows.
Lety∈be an arbitrary point. We show that there exists an open neighborhoodN
ofyinXsuch thatN⊂. Now, by hypothesis (g),Tis upper semi-continuous at some
pointxinS(y) with
inf u∈T(x)Re
u,η(y,x)
+h(y,x) > .
Let
α:= inf u∈T(x)Re
u,η(y,x)
+h(y,x).
Thusα> . Again, let
W:=
w∈F: sup z,z∈X
w,z–z<α/
.
ThenW is a strongly open neighborhood of inF, and soU:=T(x) +W is an open
neighborhood ofT(x) inF. SinceTis upper semi-continuous atx, there exists an open
neighborhoodV ofxinXsuch thatT(x)⊂U for allx∈V. Since the mappingx→
infu∈T(x)Reu,η(x,x)+h(x,x) is continuous atx, there exists an open neighborhood
VofxinXsuch that
inf
u∈T(x)Re
u,η(x,x)
+h(x,x)<
α
for all x∈V. Let V:=V ∩V. Then Vis an open neighborhood of xin X. Since
x∈V∩S(y)=∅ andS is lower semi-continuous at y, there exists an open
neigh-borhood N ofyin X such thatS(y)∩V=∅for ally∈N. Since the mappingy→
infu∈T(x)Reu,η(y,y)+h(y,y) is continuous aty, there exists an open neighborhood
NofyinXsuch that
inf
u∈T(x)Re
u,η(y,y)
+h(y,y)<
α
LetN:=N∩N. ThenNis an open neighborhood ofyinXsuch that for eachy∈N,
we have the following:
(a) S(y)∩V=∅asy∈N; so we can choose anyx∈S(y)∩V;
(b) |infu∈T(x)Reu,η(y,y)+h(y,y)|<α asy∈N;
(c) T(x)⊂U=T(x) +Wasx∈V;
(d) |infu∈T(x)Reu,η(x,x)+h(x,x)|<α asx∈V.
Hence, using property (f ) and (b)-(d), we can obtain the following by omitting the details:
inf u∈T(x)Re
u,η(y,x)
+h(y,x)
≥ inf
[u∈T(x)+W]Re
u,η(y,x)
+h(y,x)
≥ inf u∈T(x)Re
u,η(y,x)
+h(y,x)
+ inf u∈WRe
u,η(y,x)
≥ inf
u∈T(x)Reu,y–y+h(y) –h(y)
+ inf
u∈T(x)Reu,y–x+h(y) –h(x)
+ inf
u∈T(x)Reu,x–x+h(x) –h(x)
+ inf
u∈WReu,y–x
≥–α +α–
α
–
α
=α > .
Consequently, we have
sup x∈S(y)
inf u∈T(x)Re
u,η(y,x)
+h(y,x)
> ,
sincex∈S(y). Hencey∈for ally∈N. Therefore,y∈N⊂. Butywas arbitrary.
Consequently,is open inX. Thus all the hypotheses of Theorem . are satisfied. Hence,
the conclusion follows from Theorem .. This completes the proof.
Remark .
() Theorems . and . of this paper are further extensions of the results obtained in [, Theorem .] and in [, Theorem .], respectively, into generalized
quasi-variational-like inequalities of(η,h)-pseudo-monotone type II operators on compact sets.
() In , Shih and Tan [] obtained results on generalized quasi-variational inequalities in locally convex topological vector spaces, and their results were obtained on compact sets where the set-valued mappings were either lower
() The results in [] were obtained on non-compact sets where one of the set-valued mappings is a pseudo-monotone type II operator which was defined first in [] and later renamed as pseudo-monotone type II operator in []. Our present results are extensions of the results in [] using an extension of the operators defined in [] (and originally in []).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The first author made the first draft of this paper with substantial contributions to conception and design. The second and third authors have been involved equally in drafting the manuscript or revising it critically for important intellectual content. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, University of Management and Technology (UMT), C-II, Johar Town, Lahore, 54770,
Pakistan. 2Department of Mathematics, King Abdulaziz University, Jeddah, 21589, Saudi Arabia.3Department of
Mathematics Education and RINS, Gyeongsang National University, Jinju 660-701, Korea.
Acknowledgements
The second author was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, under grant No. 31-130-35-HiCi. The third author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2014R1A2A2A01002100).
Received: 12 March 2014 Accepted: 14 October 2014 Published:05 Nov 2014
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10.1186/1029-242X-2014-449
