R E S E A R C H
Open Access
Feasibility of set-valued implicit
complementarity problems
Ren-you Zhong, Xiao-guo Wang and Jiang-hua Fan
**Correspondence:
[email protected] Department of Mathematics, Guangxi Normal University, Guilin, Guangxi, 541004, P.R. China
Abstract
In this paper, we introduce new concepts of (
α
,β
,γ
)-exceptional family of elements and (α
,γ
)-exceptional family of elements for the set-valued implicit complementarity problems inRnand infinite-dimensional Hilbert spaces, respectively. By utilizing these notions and the Leray-Schauder type fixed point theorem, we study the feasibility and strict feasibility of the set-valued implicit complementarity problems. Our results generalize some corresponding previously known results in the literature.MSC: 49J40; 90C31
Keywords: exceptional family of elements; set-valued implicit complementarity problems; Leray-Schauder type fixed point theorem; feasibility; strict feasibility
1 Introduction
Let (H,·,·) be a Hilbert space,f,g:H→Hbe two set-valued mappings andKbe a cone
with its dual coneK∗. The set-valued implicit complementarity problem defined by the
(ordered) pair of mappings (f,g) andKis
SICP(f,g,K) :
⎧ ⎪ ⎨ ⎪ ⎩
findx∗∈Hsuch that there exist
u∗∈f(x∗)∩K∗andv∗∈g(x∗)∩Ksatisfying u∗,v∗= .
Iff,gare single-valued mappings,SICP(f,g,K) reduces to the implicit complementarity
problem
ICP(f,g,K) :
findx∗∈Hsuch that
f(x∗)∈K∗,g(x∗)∈Kandf(x∗),g(x∗)= .
Ifg=I(the identity mapping),SICP(f,g,K) reduces to the complementarity problem
CP(f,K) :
findx∗∈Ksuch that there exists
u∗∈f(x∗)∩K∗satisfyingu∗,x∗= .
SICP(f,g,K) is said to be feasible if
x∈H:f(x)∩K∗=∅,g(x)∩K=∅=∅;
SICP(f,g,K) is said to be strictly feasible if
x∈H:f(x)∩intK∗=∅,g(x)∩K=∅=∅.
Complementarity theory has been intensively considered due to its various applications in operations research, economic equilibrium and engineering design. The reader is re-ferred to [, ] and the reference therein. The implicit complementarity problem was in-troduced into the complementarity theory in [] as a mathematical tool in the study of some stochastic optimal control problems.
Strict feasibility plays an important role in the development of the theory and algorithms of complementarity problems. It is closely related to the solvability of the
complementar-ity problems. For example, whenf is a quasi(pseudo)monotone map, or more generally, a
quasi-P∗-map, then the strict feasibility is sufficient for the solvability of theCP; for more details, see [–]. An important method in studying the feasibility of the complementarity problems is based on the concept of an exceptional family of elements for a continuous function. In recent years, several authors have been dedicated to the feasibility of the (im-plicit) complementarity problems by using the exceptional family of elements method; for example, see [–].
At the end of the paper [], Isac proposed three open problems and two of them can be extracted as follows.
(Q) Are Theorem . and Theorem . true without the assumptionK∗⊆K?
(Q) Can the method presented in this paper be adapted to the study of strict feasibility?
Huanget al.[] and Yoelet al.[] considered the solvability of problems (Q) and (Q),
respectively. In [], they introduced new concepts of α-exceptional family of elements
and (α,β)-exceptional family of elements for continuous functions and studied the
feasi-bility for nonlinear complementarity problems inRnand an infinite-dimensional Hilbert
spaceHwithout the assumptionK∗⊆K. In [], based on their new concepts of (α,γ
)-exceptional family of elements and (α,β,γ)-exceptional family of elements and the
topo-logical degree theory, they studied the feasibility and strict feasibility of ICP(f,g,K) in
Rnand an infinite-dimensional Hilbert spaceH, which partly answered the open
prob-lem (Q).
Isacet al.[] introduced a new notion of exceptional family of elements for the pair
of (f,g) involved in the implicit complementarity problems. By employing the
Leray-Schauder alternative, they gave more general existence theorems for ICP(f,g,K) and
SICP(f,g,K) whenf,gare set-valued lower semicontinuous mappings with closed convex
values. Whenf is a set-valued upper semicontinuous mapping with closed convex values,
gis a one-to-one mapping, [] established some new existence theorems.
Many of the well-known existence theorems for the problemICP(f,g,K) demand that
the mappingsf andgbe subject to some strong restrictions. For example, Isac [, ]
required thatf be strongly monotone and Lipschitz continuous with respect tog.
Motivated by the works mentioned above, we introduce new concepts of (α,β,γ
)-exceptional family of elements and (α,γ)-exceptional family of elements forSICP(f,g,K)
under weaker restrictions on the mappingsf andg. By utilizing these notions and the
Leray-Schauder type fixed point theorem proposed in [], we investigate the (strict)
results presented in this paper not only answer the above open problems (Q) and (Q) proposed in [], but also generalize some corresponding previously known results in [, , , ].
The paper is arranged in the following way. In Section , we recall some required concepts and basic results for the later use. In Section , we introduce new concepts of (α,β,γ)-exceptional family of elements and (α,γ)-exceptional family of elements for SICP(f,g,K) and discuss the (strict) feasibility inRn. In Section , by using the new no-tion of (α,β,γ)-exceptional family of elements, we consider the (strict) feasibility for SICP(f,g,K) in infinite-dimensional Hilbert spaces.
2 Notations and fundamental results
LetXandYbe topological spaces, the collection of all nonempty compact subsets ofXis
denoted byc(X). For any subsetAofX, the interior, closure and boundary ofAare denoted
byintA,A¯ and∂A, respectively. The relative boundary ofUinKis denoted by∂KU.
Definition . The set-valued mappingF:X→Y is said to be upper semicontinuous
onXif{x∈X:f(x)⊂V}is open inXwheneverVis an open subset ofY.
Definition . The set-valued mappingF:X→Y is said to be compact ifF(X) is
rela-tively compact inY.
Definition . An upper semicontinuous set-valued mappingF:X→c(Y) is said to be
admissible if there exist a topological spaceZand continuous functionsp:Z→Xand
q:Z→Ysatisfying
() ∅ =q(p–x)⊂F(x)for eachx∈X;
() pis proper; that is, the inverse imagep–(A)of any compact setA⊂Xis compact; () for eachx∈X,p–(x)is an acyclic subset ofZ.
A nonempty topological space Xis said to be acyclic provided that all of its reduced
˘
Cech homology groups over rational vanish. For a nonempty subset in a topological vector space, we have the following implications:
convex ⇒ star-shaped ⇒ contractible ⇒ acyclic ⇒ connected,
but not conversely.
It is well known that any upper semicontinuous set-valued mapping with compact acyclic values is admissible, and the composition of two admissible mappings is also ad-missible; see []. The admissible mapping is a large class of set-valued mappings, such a class contains composites of a lot of well-known set-valued mappings, which appear in nonlinear analysis and algebraic topology; for more details, see [].
The following property of admissible maps can be found in [].
Lemma . Let X,Y be two topological spaces,E be a topological vector space,F,G:X→
c(E)be admissible,and let f:Y→R be continuous.Then the mappings
S:X→c(E), S(x) :=F(x) +G(x) for x∈X,
T:X×Y→c(E), T(x,y) :=f(y)F(x) for(x,y)∈X×Y
LetEbe a Banach space andA⊂E. The Kuratowski measure of noncompactness ofA is defined by
α(A) =inf
> :A⊂
n i=
Aianddiam(Ai)≤fori= , , . . . ,n
,
wherediam(A) =sup{x–y:x,y∈A}. It is known thatα(A) = if and only ifAis relatively compact.
An upper semi-continuous mapF:E→Eis said to be condensing if for any subset
B⊂Ewithα(B)= , we haveα(F(B)) <α(B).
LetHbe a Hilbert space,K⊂H is a closed pointed convex cone if and only ifK is a
closed subset ofHsatisfying
(i) K+K⊂K; (ii) λK⊂K,∀λ≥; (iii) K∩(–K) ={}.
The dual cone ofKis defined by
K∗=y∈H:y,x ≥,∀x∈K.
The following Leray-Schauder type fixed point theorem is a particular form of Corol-lary . in [], which is the basis of our arguments in this paper.
Theorem . Let H be a Hilbert space,C⊂H be closed and convex and U be a relatively
open subset of C with∈U.Suppose that F:U→c(H)is a condensing admissible mapping such that
F(x)∩ {λx:λ> }=∅, ∀x∈∂KU.
Then F has a fixed point in U.
The projection operator ontoKis denoted byPK, for everyx∈H,PK(x) is the unique
element inKsatisfying
x–PK(x)=min y∈Kx–y.
It is well known that, for eachx∈H, the projectionPK(x) ofxis characterized by the
following properties:
(P) PK(x) –x,y ≥for ally∈K;
(P) PK(x) –x,PK(x)= .
3 Feasibility and strict feasibility inRn
Definition . LetKbe a closed pointed convex cone inRnwithK∗⊆KandintK∗=∅.
Letε:Rn→c(intK∗),f,g:Rn→c(Rn) be admissible mappings. Givenα,β,γ ≥ with
≤α<β, we say that the family of elements{xr}r>⊂Rnis an (α,β,γ)-EFE for the pair (f,g) with respect toKif the following conditions are satisfied:
() xr →+∞asr→+∞;
() for anyr> , there existμr> and elementsfr∈f(xr),gr∈g(xr),εr∈ε(xr)such
thatsr=μrxr+ (β–α)fr–γ εr∈K∗,wr=μrxr+gr–αfr∈Kandsr,wr= .
Remark . Ifg=I,f is single-valued,γ = andμr=tr – , then Definition . reduces
to Definition . in [].
Theorem . Let K be a closed pointed convex cone in Rnwith K∗⊆K andintK∗=∅.
Letε:Rn→c(intK∗),f,g:Rn→c(Rn)be admissible mappings.Then eitherSICP(f,g,K)
is feasible,or for anyα,β,γ ≥with≤α<β,there exists an(α,β,γ)-EFE(in the sense of Definition.)for the pair(f,g)with respect to K.
Moreover,ifγ > ,then eitherSICP(f,g,K)is strictly feasible,or for anyα≥,there exists an(α,β,γ)-EFE(in the sense of Definition.)for the pair(f,g)with respect to K.
Proof Defineφ:Rn→c(Rn) by
φ(x) =g(x) –αf(x) –PK
g(x) –βf(x) +γ ε(x), ∀x∈H.
Consider the equation
∈φ(x). (.)
We have the following two cases to discuss.
Case . If equation (.) has a solution inRn, denoted byx∗, then there existu∗∈f(x∗),
v∗∈g(x∗) and∗∈ε(x∗) such that
=v∗–αu∗–PK
v∗–βu∗+γ ∗,
that is,
v∗–αu∗=PK
v∗–βu∗+γ ∗. (.)
By the property (P) ofPK, we have
v∗–αu∗–v∗–βu∗+γ ∗,y≥, ∀y∈K,
that is,
(β–α)u∗–γ ∗,y≥, ∀y∈K. (.)
Case . If equation (.) does not have a solution, set ψ=I–φ, then the mappingψ
has no fixed point inRn. Thus, for anyr> , letU
r={x∈Rn:x<r}, the mappingψis
fixed-point free with respect to the setUr.
By the continuity ofPK, the upper semicontinuity off,gandε, it is clear to see thatψ
is upper semicontinuous. Since any upper semicontinuous mapping with compact values
is compact inRn, thusψ is compact.
SinceI,PK,f,gandεall are admissible, it follows from Lemma . thatψis admissible.
Applying Theorem . with the mappingψ and the setU=Ur, there existxr∈∂Ur=
{x∈H:x=r}andλr> such that
λrxr∈xr–φ(xr).
Settingμr=λr– , we have
μrxr∈–φ(xr). (.)
From (.), it can be deduced that there existfr∈f(x
r),gr∈g(xr) andεr∈ε(xr) such
that
μrxr+gr–αfr=PK
gr–βfr+γ εr. (.)
Then the properties (P) and (P) ofPKand (.) jointly yield that
μrxr+gr–αfr–
gr–βfr+γ εr,y≥, ∀y∈K
and
μrxr+gr–αfr–
gr–βfr+γ εr,μrxr+gr–αfr
= .
Or equivalently,
μrxr+ (β–α)fr–γ εr,y
≥, ∀y∈K (.)
and
μrxr+ (β–α)fr–γ εr,μrxr+gr–αfr
= . (.)
Lettingsr=μ
rxr+ (β–α)fr–γ εr,wr=μrxr+gr–αfr, then from (.) and (.), we
obtain thatwr∈Kandsr∈K∗. Thus, (.) implies thatsr,wr= .
Sincexr∈∂Ur={x∈H:x=r}, we havexr →+∞asr→+∞.
Then{xr}r>is an (α,β,γ)-EFE for the pair (f,g) with respect toK, thus the first assertion of the theorem is proved.
Ifγ> , then it follows from (.) that (β–γ)u∗–γ ∗∈K∗, thus (β–γ)u∗∈K∗+γ ∗⊂
intK∗. This finishes the proof of the second assertion of the theorem, which completes the
Remark . Since Definition . is a generalization of Definition . in [], then Theo-rem . in [] is a special case of TheoTheo-rem ..
We now introduce a new notion of (α,γ)-exceptional family of elements (for short,
(α,γ)-EFE) for the pair (f,g) with respect toK.
Definition . LetKbe a closed pointed convex cone inRnwithintK∗=∅. Letε:Rn→
c(intK∗),f,g:Rn→c(Rn) be admissible mappings. Givenα,γ ≥, we say that the family
of elements{xr}r>⊂Rnis an (α,γ)-EFE for the pair (f,g) with respect toKif the following conditions are satisfied:
() xr →+∞asr→+∞;
() for anyr> , there existμr> and elementsfr∈f(xr),gr∈g(xr),εr∈ε(xr)such
thatsr=μ
rxr+fr–γ εr∈K∗,wr=μrxr+gr–αPK(fr)∈Kandsr,wr= .
Remark .
() Ifg=Iandf is single-valued, Definition . reduces to Definition . in []; () Ifg=I,f is single-valued andγ= , Definition . reduces to Definition . in []; () Ifg=I,f is single-valued andα=γ = , Definition . reduces to Definition .
in [] or Definition in [].
The following theorem shows us that the conclusion is true without the assumption K∗⊆K, which answers the open problem in []. And the assumption onf andgis weaker than that in [].
Theorem . Let K be a closed pointed convex cone in RnwithintK∗= ∅.Letε:Rn→
c(intK∗),f,g:Rn→c(Rn)be admissible mappings.Then eitherSICP(f,g,K)is feasible,or for anyα,γ ≥,there exists an(α,γ)-EFE(in the sense of Definition.)for the pair(f,g) with respect to K.
Moreover, if γ > , then either SICP(f,g,K) is strictly feasible, or for any α ≥ , there exists an(α,γ)-EFE(in the sense of Definition.)for the pair(f,g)with respect to K.
Proof Defineφ:Rn→c(Rn) by
φ(x) =g(x) –αPK
f(x)–PK
g(x) –f(x) +γ ε(x) –αPK
f(x), ∀x∈H.
Consider the equation
∈φ(x). (.)
We have the following two cases.
Case . If equation (.) has a solution inRn, denoted byx∗∈Rn, then there existu∗∈
f(x∗),v∗∈g(x∗) and∗∈ε(x∗) such that
=v∗–αPK
u∗–PK
v∗–u∗+γ ∗–αPK
that is,
v∗–αPK
u∗=PK
v∗–u∗+γ ∗–αPK
u∗. (.)
It follows from (.) thatv∗∈Kand thusv∗∈g(x∗)∩K.
By the property (P) ofPK, we have
v∗–αPK
u∗–v∗–u∗+γ ∗–αPK
u∗,y≥, ∀y∈K,
which is
u∗–γ ∗,y≥, ∀y∈K. (.)
It follows from (.) thatu∗–γ ∗∈K∗. Sinceγ ≥, it is clear thatu∗∈K∗. Thenu∗∈ f(x∗)∩K∗. Hence the problemSICP(f,g,K) is feasible.
Case . If equation (.) does not have a solution, setψ=I–φ, then the mappingψhas
no fixed point inRn. For anyr> , letU
r={x∈Rn:x<r}, the mappingψis fixed-point
free with respect to the setUr.
By the continuity of the projection operatorPK, and the upper semicontinuity off,g
andε, it is clear to see thatψis upper semicontinuous. Since any upper semicontinuous
mapping is compact inRn, thusψ is compact.
SinceI,PK,f,gandεall are admissible, it follows from Lemma . thatψis admissible.
Applying Theorem . with the mappingψand the setU=Ur, we obtain that there exist
xr∈∂Ur={x∈H:x=r}andλr> such that
λrxr∈xr–φ(xr).
Settingμr=λr– , we have
μrxr∈–φ(xr). (.)
From (.), we deduce that there existfr∈f(x
r),gr∈g(xr) andεr∈ε(xr) such that
μrxr+gr–αPK
fr=PK
gr–fr+γ εr–αPK
fr. (.)
From the properties (P) and (P) ofPKand (.), we have
μrxr+gr–αPK
fr–gr–fr+γ εr–αPK
fr,y≥, ∀y∈K and
μrxr+gr–αPK
fr–gr–fr+γ εr–αPK
fr,μrxr+gr
= .
Or equivalently,
μrxr+fr–γ εr,y
and
μrxr+fr–γ εr,μrxr+gr–αPK
fr= . (.)
Settingsr=μ
rxr+fr–γ εrandwr=μrxr+gr–αPK(fr), from (.) and (.) we have
thatwr∈Kandsr∈K∗. Thus, (.) implies thatsr,wr= .
Sincexr∈∂Ur={x∈H:x=r}, thus we havexr →+∞asr→+∞.
Then{xr}r>is an (α,γ)-EFE for the pair (f,g) with respect toKand so the first assertion of the theorem is proved.
Ifγ > , then it follows from (.) thatu∗–γ ∗∈K∗and sou∗∈K∗+γ ∗⊂intK∗. This
finishes the proof of the second assertion of the theorem, which completes the proof.
4 Feasibility and strict feasibility in Hilbert spaces
In this section, we study the feasibility and strict feasibility of the problemSICPin Hilbert spaces. The following (α,β,γ)-EFE is new.
Definition . LetHbe a Hilbert space,K⊂Hbe a closed convex cone withintK∗=∅
andK∗⊆K. Letε:H→c(intK∗) be compact admissible mappings, and letf,g:H→
c(H) be admissible mappings such thatf(x) =βx–S(x),g(x) =x–T(x), whereβ> , and S,T :H→c(H) are compact. Givenα,γ ≥ with ≤α<β, we say that the family of elements{xr}r>⊂His an (α,β,γ)-EFE for the pair (f,g) with respect toK, if the following conditions are satisfied:
() xr →+∞asr→+∞;
() for anyr> , there existμr> and elementsfr∈f(xr),gr∈g(xr)andεr∈ε(xr)
such thatsr=μr βxr+f
r–
β–αγ εr∈K∗,wr= (β–α)μrxr+βgr–αβfr∈Kand sr,wr= .
Remark . Ifg=I,f is single-valued,γ = andμr=tr – , then Definition . reduces
to Definition . in [].
Theorem . Let H be a Hilbert space,K⊂H be a closed convex cone withintK∗=∅and
K∗⊆K.Letε:H→c(intK∗)be a compact admissible mapping,and let f,g:H→c(H) be admissible mappings such that f(x) = βx–S(x)and g(x) =x–T(x),whereβ> and S,T :H→c(H)are compact.Then eitherSICP(f,g,K)is feasible,or for anyγ ≥and
α≥with≤α<β,there exists an(α,β,γ)-EFE(in the sense of Definition.)for the pair(f,g)with respect to K.
Moreover,ifγ > ,then eitherSICP(f,g,K)is strictly feasible,or for anyα≥,there exists an(α,β,γ)-EFE(in the sense of Definition.)for the pair(f,g)with respect to K.
Proof Defineφ:H→c(H) by
φ(x) =g(x) –αf(x) –PK
g(x) –βf(x) +γ ε(x), ∀x∈H.
Consider the equation
∈φ(x). (.)
Case . If the mapping φ has a zero point inH, denoted byx∗ ∈H, then there exist u∗∈f(x∗),v∗∈g(x∗) and∗∈ε(x∗) such that
=v∗–αu∗–PK
v∗–βu∗+γ ∗,
that is,
v∗–αu∗=PK
v∗–βu∗+γ ∗. (.)
Using (.), as in the proof of Theorem ., we obtain thatSICP(f,g,K) is feasible if
γ ≥. Moreover, ifγ > , thenSICP(f,g,K) is strictly feasible.
Case . Equation (.) does not have a solution, setψ=I–β–αβ φ, then the mappingψhas no fixed point inH. For anyr> , letUr={x∈H:x<r}, the mappingψis fixed-point
free with respect to the setUr.
Sincef(x) =βx–S(x) andg(x) =x–T(x), thus, for anyx∈H, we have
ψ(x) = β
β–αT(x) – αβ
β–αS(x) +PK
β β–α
βS(x) –T(x) +γ ε(x).
By the compactness of the mappingsS,Tandε, it is easy to see thatψis compact.
SinceI,PK,f,gandεall are admissible, it follows from Lemma . thatψis admissible.
Applying Theorem . with the restriction of the mappingψ and the setU=Ur, there
existxr∈∂Ur={x∈H:x=r}andλr> such that
λrxr∈ψ(xr) =xr–
β
β–αφ(xr). (.)
Settingμr=λr– , it can be deduced from (.) that there existfr∈f(xr),gr∈g(xr) and
εr∈ε(xr) such that
μrxr+
β β–αg
r– αβ
β–αf
r= β
β–αPK
gr–βfr+γ εr.
Setting sr=μr βxr+f
r– β–αγ ε
r andwr= (β–α)μ
rxr+βgr–αβfr, as in the proof of
Theorem ., we obtain that{xr}r>is an (α,β,γ)-EFE for the pair (f,g) with respect toK.
Remark . Since Definition . is a generalization of Definition . in [], then
Theo-rem . in [] is a special case of TheoTheo-rem ..
Definition . LetKbe a closed pointed convex cone in a Hilbert spaceHwithintK∗=∅.
Let f,g:H→c(H) be two admissible mappings such thatf(x) =βx–S(x) andg(x) =
βx–T(x), whereβ> ,S,T:H→c(H) are compact. Letε:H→c(intK∗) be a compact admissible mapping. Givenα,γ ≥, we say that the family of elements{xr}r>⊂His an (α,β,γ)-EFE for the pair (f,g) with respect toK, if the following conditions are satisfied:
() xr →+∞asr→+∞;
() for anyr> , there existμr> and elementsfr∈f(xr),gr∈g(xr),εr∈ε(xr)such
thatsr=μ
rxr+βfr–γ εr∈K∗,wr=βμrxr+βgr–αPK(xr–βfr)∈Kand
Remark .
() Ifg=Iandf is single-valued, Definition . reduces to Definition . in []; () Ifg=I,f is single-valued andγ= , Definition . reduces to Definition . in []; () Ifg=I,f is single-valued andα=γ = , Definition . reduces to Definition .
in [].
Theorem . Let H be a Hilbert space, and let K ⊂H be a closed convex cone with
intK∗=∅.Letε:H→c(intK∗)be a compact admissible mapping,and let f,g:H→c(H) be two admissible mappings such that f(x) =βx–S(x)and g(x) =βx–T(x),whereβ> , S,T :H→c(H)are compact.Then eitherSICP(f,g,K)is feasible,or for anyα≥and
γ ≥,there exists an(α,β,γ)-EFE(in the sense of Definition.)for the pair(f,g)with respect to K.
Moreover,ifγ > ,then eitherSICP(f,g,K)is strictly feasible,or for anyα≥,there exists an(α,β,γ)-EFE(in the sense of Definition.)for the pair(f,g)with respect to K.
Proof Defineφ:H→c(H) by
φ(x) =βg(x) –αPK
x–βf(x)
–PK
βg(x) –βf(x) +γ ε(x) –αPK
x–βf(x), ∀x∈H.
Consider the equation
x∈x–φ(x). (.)
We consider the following two cases.
Case . If equation (.) has a solution inH, denoted byx∗∈H, then there existu∗∈
βf(x∗),v∗∈βg(x∗) and∗∈ε(x∗) such that
=v∗–αPK
x∗–u∗–PK
v∗–u∗+γ ∗–αPK
x∗–u∗,
that is,
v∗–αPK
x∗–u∗=PK
v∗–u∗+γ ∗–αPK
x∗–u∗. (.)
Using (.), as in the proof of Theorem ., we obtain that the problemSICP(f,g,K) is
feasible ifγ ≥. Moreover, ifγ > , thenSICP(f,g,K) is strictly feasible.
Case . If equation (.) does not have a solution, setψ=I–φ, then the mappingψhas
no fixed point inH. From the representations off andg, we have
ψ(x) =T(x) +αPK
S(x)
–PK
S(x) –T(x) +γ ε(x) –αPK
S(x), ∀x∈H.
For anyr> , letUr={x∈H:x<r}, then the mappingψis fixed-point free with respect
to the setUr.
By the compactness of the mappingsS,Tandε, it is clear thatψ is compact.
Applying Theorem . with the restriction of the mappingψ and the setU=Ur, we
obtain that there existxr∈∂Ur={x∈H:x=r}andλr> such that
λrxr∈xr–βg(xr) +αPK
xr–βf(xr)
–PK
g(xr) –f(xr) +γ ε(xr) –αPK
xr–βf(xr)
. (.)
Setμr=λrβ–. Then it follows from (.) that there existfr∈f(xr),gr∈g(xr) andεr∈
ε(xr) such that
βμrxr+βgr–αPK
xr–βfr
=PK
βgr–βfr+γ εr–αPK
xr–βfr
. (.)
Using (.), as in the proof of Theorem ., we obtain that{xr}r>is an (α,β,γ)-EFE for the pair (f,g) with respect toK.
The following theorem presents a sufficient condition which ensures that the problem SICP(f,g,K) does not have an (α,β,γ)-EFE for the pair (f,g) with respect toK.
Theorem . Let H be a Hilbert space,K⊂H be a closed convex cone,and f,g:H→H
be two set-valued mappings satisfying the following condition.
(Condition(θg))If there exist e∈K∗and a real numberρ> satisfying that for any x∈K
withx>ρ,there exists x∈K such that for all y∈f(x)and z∈g(x),we have
z–x,y–e ≥ and z–x,x> .
ThenSICP(f,g,K)does not have an(α,β,γ)-EFE(in the sense of Definition.)for the pair (f,g)with respect to K.Thus,SICP(f,g,K)is strictly feasible.
Proof Defineε:H→c(intK∗) by
ε(x) ={e}, ∀x∈H.
We show thatSICP(f,g,K) does not have an (, , )-EFE (in the sense of Definition .)
for the pair (f,g).
Suppose on the contrary that there exists an (, , )-EFE{xr}r>⊂Kfor the pair (f,g), that is,xr →+∞asr→+∞and for anyr> , there existμr> ,yr∈f(xr) andzr∈g(xr)
such thatsr=μrxr+yr–e∈K∗,wr=μrxr+zr∈Kandsr,wr= .
For anyr>ρ,xr=rimplies thatxr>ρ, thus there exists an elementxr∈Ksatisfying
zr–xr,yr
≥ and zr–xr,xr
> . (.)
Sincesr=μrxr+yr, it can be deduced from (.) thatzr–xr,sr> .
Sincewr=μrxr+zr,xr,xr∈Kandsr∈K∗, it is obvious that
<zr–xr,sr
=wr–μrxr–xr,sr
=wr,sr–
μrxr+xr,sr
≤,
which is a contradiction.
Therefore,SICP(f,g,K) does not have an (α,β,γ)-EFE for the pair (f,g) with respect
Remark . The condition (θg) is a generalization of the Karamardian’s condition. We
refer the reader to [, , ] for more details.
As a direct consequence of Theorem ., we have the following corollary.
Corollary . Let H be a Hilbert space,K⊂H be a closed convex cone,and f,g:H→H
be two set-valued mappings satisfying the following condition.
(Condition(θ))If there existsρ> such that for any x∈K withx>ρ,there exists x∈K such that for all y∈f(x)and z∈g(x),we have
z–x,y ≥ and z–x,x> .
ThenSICP(f,g,K)is strictly feasible.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors completed the paper together. All authors read and approved the final manuscript.
Acknowledgements
The author thanks the anonymous referees for their valuable remarks and suggestions, which helped to improve the article considerably. This work was supported by the National Natural Science Foundation of China (11061006), the National Natural Science Foundation of China (11226224), the Program for Excellent Talents in Guangxi Higher Education Institutions, the Guangxi Natural Science Foundation (2012GXNSFBA053008) and the Initial Scientific Research Foundation for PHD of Guangxi Normal University.
Received: 25 March 2013 Accepted: 20 May 2013 Published: 5 June 2013
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doi:10.1186/1029-242X-2013-284
