COINCIDENCES, AND FIXED POINTS OF MAPS IN TOPOLOGICAL VECTOR SPACES

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COINCIDENCES, AND FIXED POINTS OF MAPS IN TOPOLOGICAL VECTOR SPACES

K. WŁODARCZYK AND D. KLIM

Received 9 November 2004 and in revised form 13 December 2004

Let Ebe a real Hausdorfftopological vector space. In the present paper, the concepts of the transfer positive hemicontinuity and strictly transfer positive hemicontinuity of set-valued maps inEare introduced (condition of strictly transfer positive hemicontinuity is stronger than that of transfer positive hemicontinuity) and for mapsF:C→2Ê andG:C→2Êdefined on a nonempty compact convex subsetCofE, we describe how some ideas of K. Fan have been used to prove several new, and rather general, conditions (in which transfer positive hemicontinuity plays an important role) that a single-valued mapΦ:_c_∈_C(F(c)×G(c))→Ehas a zero, and, at the same time, we give various char- acterizations of the class of those pairs (F,G) and mapsFthat possess coincidences and fixed points, respectively. Transfer positive hemicontinuity and strictly transfer positive hemicontinuity generalize the famous Fan upper demicontinuity which generalizes upper semicontinuity. Furthermore, a new type of continuity defined here essentially generalizes upper hemicontinuity (the condition of upper demicontinuity is stronger than the upper hemicontinuity). Comparison of transfer positive hemicontinuity and strictly transfer positive hemicontinuity with upper demicontinuity and upper hemicontinuity and relevant connections of the results presented in this paper with those given in earlier works are also considered. Examples and remarks show a fundamental difference between our results and the well-known ones.

1. Introduction

One of the most important tools of investigations in nonlinear and convex analysis is the minimax inequality of Fan [11, Theorem 1]. There are many variations, generalizations, and applications of this result (see, e.g., Hu and Papageorgiou [16,17], Ricceri and Si- mons [19], Yuan [21,22], Zeidler [24] and the references therein). Using the partition of unity, his minimax inequality, introducing in [10, page 236] the concept of upper demicontinuity and giving in [11, page 108] the inwardness and outwardness conditions, Fan initiated a new line of research in coincidence and fixed point theory of set-valued maps in topological vector spaces, proving in [11] the general results ([11, Theorems 3–6]) which extend and unify several well-known theorems (e.g., Browder [7], [5, Theorems 1 and 2]

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and [6, Theorems 3 and 5], Fan [6,9], [10, Theorem 5] and [8, Theorem 1], Glicksberg [14], Kakutani [18], Bohnenblust and Karlin [3], Halpern and Bergman [15], and others) concerning upper semicontinuous maps and, in particular, inward and outward maps (the condition of upper semicontinuity is stronger than that of upper demicontinuity).

LetCbe a nonempty compact convex subset of a real Hausdorfftopological vector spaceE, letF:C→2ÊandG:C→2Êbe set-valued maps and letΦ:_c_∈_C(F(c)×G(c))→ E be a single-valued map. The purpose of our paper is to introduce the concepts of the transfer positive hemicontinuity and strictly transfer positive hemicontinuity of set- valued maps inEand prove various new results concerning the existence of zeros ofΦ, coincidences ofFandGand fixed points ofFin which transfer positive hemicontinuity and strictly transfer positive hemicontinuity plays an important role (seeSection 2).

In particular, our results generalize theorems of Fan type (e.g., [11, Theorems 3–6]) and contain fixed point theorems for set-valued transfer positive hemicontinuous maps with the inwardness and outwardness conditions given by Fan [11, page 108]. Transfer positive hemicontinuity and strictly transfer positive hemicontinuity generalize the Fan upper demicontinuity. Furthermore, a new type of continuity defined here essentially generalizes upper hemicontinuity (every upper demicontinuous map is upper hemicontinuous). Comparisons of transfer hemicontinuity and strictly transfer positive hemicontinuity with upper demicontinuity and upper hemicontinuity are given in Sections3and 4. The remarks, examples and comparisons of our results with Fan’s results and other results concerning coincidences and fixed points of upper hemicontinuous maps given by Yuan et al. [22,23] (see also the references therein) show that our theorems are new and diﬀer from those given by the above-mentioned authors (see Sections2–4).

2. Transfer positive hemicontinuity, strictly transfer positive hemicontinuity, zeros, coincidences, and fixed points

LetEbe a real Hausdorﬀtopological vector space and letEdenote the vector space of all continuous linear forms onE.

LetCbe a nonempty subset ofE. A set-valued mapF:C→2Êis a map which assigns a unique nonempty subsetF(c)∈2Ê to eachc∈C (here 2Ê denotes the family of all nonempty subsets ofE).

Definition 2.1. LetCbe a nonempty subset ofE, letF:C→2^Eand letG:C→2^E. Let Φ:_c_∈_C(F(c)×G(c))→Ebe a single-valued map.

(a) We say that a pair (F,G) isΦ-transfer positive hemicontinuous(Φ-t.p.h.c.) onCif, whenever (c,ϕc,λc)∈C×E×Randεc>0 are such that

λc

ϕc◦Φ(u,v)− 1 +εc

λc

>0 for any (u,v)∈F(c)×G(c), (2.1)

there exists a neighbourhoodN(c) ofcinCsuch that λc

ϕc◦Φ(u,v)−λc

>0 for anyx∈N(c) and any (u,v)∈F(x)×G(x). (2.2)

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(b) We say that a pair (F,G) isΦ-transfer hemicontinuous(Φ-t.h.c.) onCif, whenever (c,ϕc,λc)∈C×E×Ris such that

λc

ϕc◦Φ(u,v)−λc

>0 for any (u,v)∈F(c)×G(c), (2.3)

there exists a neighbourhoodN(c) ofcinCsuch that λc

ϕc◦Φ(u,v)−λc

>0 for anyx∈N(c) and any (u,v)∈F(x)×G(x). (2.4) (c) We say that a mapFisΦ-t.p.h.c.orΦ-t.h.c.onCif a pair (F,IE) isΦ-t.p.h.c. or Φ-t.h.c. onC, respectively.

(d) We say that a pair (F,G) istransfer positive hemicontinuous (t.p.h.c.) ortransfer hemicontinuous (t.h.c.)onCif (F,G) isΦ-t.p.h.c. orΦ-t.h.c. onC, respectively, forΦof the formΦ(u,v)=u−vwhere (u,v)∈F(c)×G(c) andc∈C.

(e) We say that a mapFist.p.h.c.ort.h.c.onCif a pair (F,IE) is t.p.h.c. or t.h.c. onC, respectively.

Recall that anopen half-spaceHinEis a set of the formH= {x∈E:ϕ(x)< t}where ϕ∈E\ {0}andt∈R.

Remark 2.2. The geometric meaning of theΦ-transfer positive hemicontinuity andΦ- transfer hemicontinuity is clear.

Really define Hc,ϕc,λc,εc=

w∈E:ϕc(w)<1 +εc λc

, εc≥0, Wc,ϕc,λc,Φ=

x∈C:ϕc◦Φ(u,v)< λcfor any (u,v)∈F(x)×G(x), Uc,ϕc,λc,Φ=

x∈C: sup

(u,v)∈F(x)×G(x)

ϕc◦Φ(u,v)≤λc

(2.5)

whenλc<0,

Hc,ϕc,λc,εc=

w∈E:ϕc(w)>1 +εc λc

, εc≥0, Wc,ϕc,λc,Φ=

x∈C:ϕc◦Φ(u,v)> λcfor any (u,v)∈F(x)×G(x), Uc,ϕc,λc,Φ=

x∈C: inf

(u,v)∈F(x)×G(x)

ϕc◦Φ(u,v)≥λc

(2.6)

whenλc>0.

ByDefinition 2.1, we see that the pair (F,G) isΦ-t.p.h.c. orΦ-t.h.c. onCif, whenever (c,ϕc,λc)∈C×E×Randεc≥0 are such that the setΦ(F(c)×G(c)) is contained

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in open half-spaceH(c,ϕc,λc,εc) (hereεc>0 in the case ofΦ-transfer positive hemicontinuity andεc=0 in the case of Φ-transfer hemicontinuity), then the following hold:

(i) there exists a neighbourhood N(c) of c in C such that, for any x∈N(c), the set Φ(F(x)×G(x)) is contained in open half-spaceHc,ϕc,λc,0; (ii)cis an interior point of the setsWc,ϕc,λc,ΦandUc,ϕc,λc,Φ. Indeed, thenλc[(ϕc◦Φ)(u,v)−λc]>0 for anyx∈N(c) and any (u,v)∈F(x)×G(x).

(a) We say that a pair (F,G) isΦ-strictly transfer positive hemicontinuous(Φ-s.t.p.h.c.) onCif, whenever (c,ϕc,λc)∈C×E×Randεc>0 are such that

λc

ϕc◦Φ(u,v)− 1 +εc

λc

>0 for any (u,v)∈F(c)×G(c), (2.7) thencis an interior point of the setVc,ϕc,λc,Φ, where

Vc,ϕc,λc,Φ=

x∈C: sup

(u,v)∈F(x)×G(x)

ϕc◦Φ(u,v)< λc ifλc<0,

Vc,ϕc,λc,Φ=

x∈C: inf

(u,v)_∈F(x)_×G(x)

ϕc◦Φ(u,v)> λc

ifλc>0.

(2.8)

(b) We say that a pair (F,G) isΦ-strictly transfer hemicontinuous(Φ-s.t.h.c.) onCif, whenever (c,ϕc,λc)∈C×E×Ris such that

λc

ϕc◦Φ(u,v)−λc

>0 for any (u,v)∈F(c)×G(c), (2.9) thencis an interior point of the setVc,ϕc,λc,Φ.

(c) We say that a mapFisΦ-s.t.p.h.c.orΦ-s.t.h.c.onCif a pair (F,IE) isΦ-s.t.p.h.c.

orΦ-s.t.h.c. onC, respectively.

(d) We say that a pair (F,G) is strictly transfer positive hemicontinuous(s.t.p.h.c.) or strictly transfer hemicontinuous(s.t.h.c.) onCif (F,G) isΦ-s.t.p.h.c. orΦ-s.t.h.c. onC, respectively, forΦof the formΦ(u,v)=u−vwhere (u,v)∈F(c)×G(c) andc∈C.

(e) We say that a mapFiss.t.p.h.c.ors.t.h.c.onCif a pair (F,IE) is s.t.p.h.c. or s.t.h.c.

onC, respectively.

Proposition2.4. LetCbe a nonempty subset ofE, letF:C→2^E and letG:C→2^E. Let Φ:_c_∈_C(F(c)×G(c))→Ebe a single-valued map.

(i)If(F,G)isΦ-t.h.c. onC, then(F,G)isΦ-t.p.h.c. onC.

(ii) If(F,G)isΦ-t.p.h.c. onC and, for eachx∈C,Φ(F(x)×G(x))is compact, then (F,G)isΦ-t.h.c. onC.

(iii)If(F,G)isΦ-s.t.h.c. onC, then(F,G)isΦ-s.t.p.h.c. onC.

(iv)If(F,G)isΦ-s.t.p.h.c. onCand, for eachx∈C,Φ(F(x)×G(x))is compact, then (F,G)isΦ-s.t.h.c. onC.

(v)If(F,G)isΦ-s.t.p.h.c. (Φ-s.t.h.c., resp.) onC, then(F,G)isΦ-t.p.h.c. (Φ-t.h.c., resp.) onC.

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(vi)If(F,G)isΦ-t.p.h.c. (Φ-t.h.c., resp.) onCand, for eachx∈C,Φ(F(x)×G(x))is compact, then(F,G)isΦ-s.t.p.h.c. (Φ-s.t.h.c., resp.) onC.

Proof. (i) Let (F,G) beΦ-t.h.c. onCand assume that there exist (c,ϕc,λc)∈C×E×R and εc>0 such thatλc[(ϕc◦Φ)(u,v)−(1 +εc)λc]>0 or, equivalently, (1 +εc)λc[(ϕc◦ Φ)(u,v)−(1 +εc)λc]>0 for any (u,v)∈F(c)×G(c). Then, byΦ-transfer hemicontinuity, there exists a neighbourhoodN(c) ofcinCsuch that (1 +εc)λc[(ϕc◦Φ)(u,v)−(1 + εc)λc]>0 for anyx∈N(c) and any (u,v)∈F(x)×G(x). This implies, in particular, that λc[(ϕc◦Φ)(u,v)−λc]>0 for anyx∈N(c) and any (u,v)∈F(x)×G(x), that is, (F,G) is Φ-t.p.h.c. onC.

(ii) Let (F,G) beΦ-t.p.h.c. onCand let there exists (c,ϕc,λc)∈C×E×Rsuch that, for any (u,v)∈F(c)×G(c), λc[(ϕc◦Φ)(u,v)−λc]>0 or, equivalently, for any (u,v)∈ F(c)×G(c), (ϕc◦Φ)(u,v)< λc if λc<0 and (ϕc◦Φ)(u,v)> λc if λc >0. Since, for each x∈C, Φ(F(x)×G(x)) is compact, thus sup(u,v)_∈F(c)_×G(c)(ϕc◦Φ)(u,v)< λc if λc<0 and inf(u,v)_∈F(c)_×G(c)(ϕc◦Φ)(u,v)> λc ifλc>0, so there is someεc>0 such that sup_(u,v)_∈_F(c)_×_G(c)(ϕc◦Φ)(u,v)<(1 +εc)λc if λc <0 and inf(u,v)∈F(c)×G(c)(ϕc◦Φ)(u,v)>

(1 +εc)λcifλc>0. Therefore, for any (u,v)∈F(c)×G(c), (ϕc◦Φ)(u,v)<(1 +εc)λcifλc<

0 and (ϕc◦Φ)(u,v)>(1 +εc)λcifλc>0 or, equivalently,λc[(ϕc◦Φ)(u,v)−(1 +εc)λc]>0 for any (u,v)∈F(c)×G(c). Then, byΦ-transfer positive hemicontinuity, there exists a neighbourhoodN(c) ofcinCsuch thatλc[(ϕc◦Φ)(u,v)−λc]>0 for anyx∈N(c) and any (u,v)∈F(x)×G(x), that is, (F,G) isΦ-t.h.c. onC.

(iii) Let (F,G) beΦ-s.t.h.c. onCand assume that there exist (c,ϕc,λc)∈C×E×R and εc>0 such thatλc[(ϕc◦Φ)(u,v)−(1 +εc)λc]>0 or, equivalently, (1 +εc)λc[(ϕc◦ Φ)(u,v)−(1 +εc)λc]>0 for any (u,v)∈F(c)×G(c). Then, byΦ-strictly transfer hemicontinuity, c is an interior point of the setVc,ϕc,(1+εc)λc,Φ. But Vc,ϕc,(1+εc)λc,Φ⊂Vc,ϕc,λc,Φ. This implies, in particular, thatcis an interior point of the setVc,ϕc,λc,Φ, that is, (F,G) is Φ-s.t.p.h.c. onC.

(iv) Let (F,G) beΦ-s.t.p.h.c. onCand let there exists (c,ϕc,λc)∈C×E×Rsuch that, for any (u,v)∈F(c)×G(c), λc[(ϕc◦Φ)(u,v)−λc]>0 or, equivalently, for any (u,v)∈ F(c)×G(c), (ϕc◦Φ)(u,v)< λc if λc<0 and (ϕc◦Φ)(u,v)> λc if λc >0. Since, for each x∈C, Φ(F(x)×G(x)) is compact, thus sup_(u,v)_∈_F(c)_×_G(c)(ϕc◦Φ)(u,v)< λc if λc<0 and inf_(u,v)_∈_F(c)_×_G(c)(ϕc◦Φ)(u,v)> λc ifλc>0, so there is someεc>0 such that sup₍_u_,_v₎_∈_F₍_c₎_×_G₍_c₎(ϕc◦Φ)(u,v)<(1 +εc)λc if λc <0 and inf(u,v)_∈F(c)_×G(c)(ϕc◦Φ)(u,v)>

(1 +εc)λcifλc>0. Therefore, for any (u,v)∈F(c)×G(c), (ϕc◦Φ)(u,v)<(1 +εc)λcifλc<

0 and (ϕc◦Φ)(u,v)>(1 +εc)λcifλc>0 or, equivalently,λc[(ϕc◦Φ)(u,v)−(1 +εc)λc]>0 for any (u,v)∈F(c)×G(c). Then, byΦ-strictly transfer positive hemicontinuity,cis an interior point of the setVc,ϕc,λc,Φ, that is, (F,G) isΦ-s.t.p.h.c. onC.

(v) By Definitions2.1and2.3andRemark 2.2, we see thatVc,ϕc,λc,Φ⊂Wc,ϕc,λc,Φ. (vi) By Definition 2.1, the pair (F,G) is Φ-t.p.h.c. or Φ-t.h.c. on C if, whenever (c,ϕc,λc)∈C×E×Rand εc≥0 are such that the setΦ(F(c)×G(c)) is contained in open half-spaceH(c,ϕc,λc,εc) (hereεc>0 in the case ofΦ-transfer positive hemicontinuity andεc=0 in the case ofΦ-transfer hemicontinuity), then there exists a neigh- bourhoodN(c) ofcinCsuch that, for anyx∈N(c) and any (u,v)∈F(x)×G(x), (ϕc◦ Φ)(u,v)< λc ifλc<0 and (ϕc◦Φ)(u,v)> λc ifλc>0. Since, for eachx∈C,Φ(F(x)× G(x)) is compact, thus, for eachx∈N(c), sup_(u,v)_∈_F(x)_×_G(x)(ϕc◦Φ)(u,v)< λc ifλc<0

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and inf(u,v)∈F(x)×G(x)(ϕc◦Φ)(u,v)> λcifλc>0. Consequently,N(c)⊂Vc,ϕc,λc,Φ, that is,c

is an interior point of the setVc,ϕc,λc,Φ.

Remark 2.5. This proves, in particular, that the condition of strictly transfer positive hemicontinuity is stronger than that of transfer positive hemicontinuity.

Definition 2.6. LetCbe a nonempty compact convex subset ofE. We say that (c,ϕ)∈ C×(E\ {0}) isadmissibleifϕ(c)=min_x_∈_Cϕ(x); thus if (c,ϕ) is admissible, then this means that theclosed hyperplanedetermined byϕof the form{x∈E:ϕ(x)=ϕ(c)}is a supporting hyperplaneofCatc.

Definition 2.7. LetCbe a nonempty subset ofE, letF:C→2^Eand letG:C→2^E. Let Φ:_c_∈_C(F(c)×G(c))→Ebe a single-valued map.

(a) A pair (F,G) is calledΦ-inward(Φ-outward, resp.) if, for any admissible (c,ϕ)∈ C×(E\ {0}) there is a point (u,v)∈F(c)×G(c) such that (ϕ◦Φ)(u,v)≥0 ((ϕ◦Φ)(u,v)

≤0, resp.).

(b) A mapFis calledΦ-inward(Φ-outward, resp.) if the pair (F,IE) isΦ-inward (Φ- outward, resp.).

(c) A pair (F,G) is calledinward(outward, resp.) if the pair (F,G) isΦ-inward (Φ- outward, resp.) forΦof the formΦ(u,v)=u−vwhere (u,v)∈F(c)×G(c) andc∈C.

(d) A mapFis calledinward(outward, resp.) (see Fan [11, page 108]) if a pair (F,IE) is inward (outward, resp.).

(a) We say that a pair (F,G) has aΦ-coincidenceif there existc∈Cand (u,v)∈F(c)× G(c), such thatΦ(u,v)=0, that is, (u,v)∈F(c)×G(c) is a zero ofΦ; this pointcis called aΦ-coincidence pointfor (F,G).

(b) We say that a mapFhas aΦ-fixed point(a pair (F,IE) has aΦ-coincidence) if there existc∈Candu∈F(c), such thatΦ(u,c)=0; this pointcis called aΦ-fixed pointforF.

(c) We say that a pair (F,G) has acoincidenceif there existc∈Cand (u,v)∈F(c)× G(c), such thatu=v; this pointcis called acoincidence pointfor (F,G).

(d) We say thatFhas afixed pointif there existsc∈Csuch thatc∈F(c); this pointc is called afixed pointforF.

With the background given, the first result of our paper can now be presented.

Theorem 2.9. Let Ebe a real Hausdorff topological vector space. Let C be a nonempty compact convex subset ofE, letF:C→2Êand letG:C→2Ê. LetΦ:_c_∈_C(F(c)×G(c))→ Ebe a single-valued map.

(i)Let the pair(F,G)beΦ-t.p.h.c. onC. If(F,G)isΦ-inward orΦ-outward, then there existsc0∈Csuch that, for anyϕ∈E, there is noλ∈Rsuch thatλ[(ϕ◦Φ)(u,v)−λ]>0 for all(u,v)∈F(c0)×G(c0).

(ii)LetFbeΦ-t.p.h.c. onC. IfFisΦ-inward orΦ-outward, then there existsc0∈Csuch that, for anyϕ∈E, there is noλ∈Rsuch thatλ[(ϕ◦Φ)(u,c0)−λ]>0for allu∈F(c0).

(iii)Let the pair(F,G)be t.p.h.c. onC. If(F,G)is inward or outward, then there exists c0∈Csuch that, for anyϕ∈E, there is noλ∈Rsuch thatλ[ϕ(u−v)−λ]>0 for all (u,v)∈F(c0)×G(c0).

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(iv)LetFbe t.p.h.c. onC. IfFis inward or outward, then there existsc0∈Csuch that, for anyϕ∈E, there is noλ∈Rsuch thatλ[ϕ(u−c0)−λ]>0for allu∈F(c0).

Proof. (i) Assume that, for any admissible (c,ϕ)∈C×(E\ {0}), there exists (u,v)∈ F(c)×G(c) such that

(ϕ◦Φ)(u,v)≥0 (2.10) and assume that the assertion does not hold, that is, without loss of generality, for any c∈C, there existϕc∈E\ {0},λc<0 andεc≥0, such that

ϕc◦Φ(u,v)<1 +εc

λc ∀(u,v)∈F(c)×G(c). (2.11) ByDefinition 2.1(a), there exists a neighbourhoodN(c) ofcinCsuch that

ϕc◦Φ(u,v)< λc for anyx∈N(c) and any (u,v)∈F(x)×G(x). (2.12)

Since the family{N(c) :c∈C}is an open cover of a compact set C, there exists a finite subset{c1,...,cn} ofC such that the family{N(cj) : j=1, 2,...,n}covers C. Let {β1,...,βn}be a partition of unity with respect to this cover, that is, a finite family of real- valued nonnegative continuous mapsβjonCsuch thatβjvanish outsideN(cj) and are less than or equal to one everywhere, 1≤j≤n, andⁿ_j₌1βj(c)=1 for allc∈C.

Defineη(c)=_n

j=1βj(c)ϕcj forc∈C. Thenη(c)∈Efor eachc∈C. Therefore

η(c)◦Φ(u,v)< λ (2.13)

for anyc∈Cand (u,v)∈F(c)×G(c), whereλ=max1_≤j≤nλcj<0 since η(c)◦Φ(u,v)=

n j=1

βj(c)ϕcj◦Φ(u,v)<ⁿ

j=1

βj(c)λcj. (2.14)

Let nowk:C×C→Rbe a continuous map of the formk(c,x)=[η(c)](c−x) for (c,x)∈ C×C. Since, for eachc∈C, the mapk(c,·) is quasi-concave onC, therefore, by [11, page 103], the following minimax inequality

minc∈Cmax

x∈Ck(c,x)≤max

c∈C k(c,c) (2.15)

holds. Butk(c,c)=0 for eachc∈C, so there is somec0∈Csuch thatk(c0,x)≤0 for all x∈C. Since

ηc0

c0

=min

x∈C

ηc0

(x), (2.16)

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we have that (c0,η(c0))∈C×(E\ {0}) is admissible and, by (2.13), ηc0

◦Φ(u,v)< λ for any (u,v)∈Fc0

×Gc0

, (2.17)

which is impossible by (2.10).

(ii)–(iv) The argumentation is analogous and will be omitted.

Two setsX andY in Ecan bestrictly separated by a closed hyperplaneif there exist ϕ∈Eandλ∈R, such thatϕ(x)< λ < ϕ(y) for each (x,y)∈X×Y.

Theorem 2.9has the following consequence.

Theorem2.10. Let Ebe a real Hausdorfftopological vector space. LetCbe a nonempty compact convex subset of E, letF:C→2Êand letG:C→2Ê. LetΦ:_c_∈_C(F(c)×G(c))→E be a single-valued map.

(i)Let the pair(F,G)beΦ-t.p.h.c. onCand inward or outward. Then there existsc0∈C such thatΦ(F(c0)×G(c0))and{0}cannot be strictly separated by any closed hyperplane in E. If, additionally,Eis locally convex and, for eachc∈C, the setΦ(F(c)×G(c))is closed and convex, then a pair(F,G)has aΦ-coincidence.

(ii)LetF beΦ-t.p.h.c. onCand inward or outward. Then there existsc0∈Csuch that Φ(F(c0)× {c0})and{0}cannot be strictly separated by any closed hyperplane inE. If, ad- ditionally,Eis locally convex and, for eachc∈C, the setΦ(F(c)× {c})is closed and convex, then a mapFhas aΦ-fixed point.

(iii)Let the pair(F,G)be t.p.h.c. onCand inward or outward. Then, the following hold:

(iii1)if, for eachc∈C, at least one of the setsF(c)orG(c)is compact, then there existsc0∈C such thatF(c0)andG(c0)cannot be strictly separated by any closed hyperplane in E;

(iii2)ifEis locally convex and, for eachc∈C, the setsF(c)andG(c)are convex and closed and at least one of them is compact, then there existsc0∈Csuch that F(c0)andG(c0)have a nonempty intersection.

(iv)LetF:C→2^Ebe t.p.h.c. onCand inward or outward. Then, the following hold:

(iv1)there existsc0∈Csuch thatF(c0)and{c0}cannot be strictly separated by any closed hyperplane inE;

(iv2)ifEis locally convex and, for eachc∈C, the setF(c)is closed and convex, then there existsc0∈Csuch thatc0∈F(c0).

Proof. (i) Let us observe that if we assume that the following condition holds:

(1 +ε)λ(ϕ◦Φ)(u,v)−(1 +ε)λ>0 (2.18) for someλ∈R,ϕ∈Eandε≥0, and for all (u,v)∈F(c0)×G(c0), then we obtain that, for all (u,v)∈F(c0)×G(c0), (ϕ◦Φ)(u,v)<(1 +ε)λ≤λ < ϕ(0) ifλ <0 and (ϕ◦Φ)(u,v)>

(1 +ε)λ≥λ > ϕ(0) ifλ >0, that is, the setsΦ(F(c0)×G(c0)) and{0}are strictly separated by a closed hyperplane inE.

Otherwise, assume that, for all (u,v)∈F(c0)×G(c0), (ϕ◦Φ)(u,v)< t1< ϕ(0) for some t1∈Ror (ϕ◦Φ)(u,v)> t2> ϕ(0) for somet2∈R. Then we obtain that, for all (u,v)∈ F(c0)×G(c0), (ϕ◦Φ)(u,v)<(1 +ε)λ1<0 where (1 +ε)λ1=t1 or (ϕ◦Φ)(u,v)>(1 + ε)λ2>0 where (1 +ε)λ2=t2. Therefore condition (2.18) is then satisfied.

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The above considerations,Theorem 2.9(i) and the separation theorem yield the assertion.

(ii) This is a consequence of (i).

(iii) Assume, without loss of generality, thatG(c0) is compact.

Let us observe that if we assume that the following condition holds:

(1 +ε)λϕ(u−v)−(1 +ε)λ>0 (2.19) for someλ∈Randε≥0 and for all (u,v)∈F(c0)×G(c0), then we obtain that, for all (u,v)∈F(c0)×G(c0),ϕ(u)< t2< ϕ(v) wheret2=(1 +ε)λ+ minw∈G(c0)ϕ(w) ifλ <0 and ϕ(u)> t1> ϕ(v) wheret1=(1 +ε)λ+ max_w_∈_G(c0)ϕ(w) ifλ >0, that is, the setsF(c0) and G(c0) are strictly separated by a closed hyperplane inE.

Otherwise, assume that, for all (u,v)∈F(c0)×G(c0),ϕ(u)> t1> ϕ(v) for somet1∈R orϕ(u)< t2< ϕ(v) for somet2∈R. Then we obtain that, for all (u,v)∈F(c0)×G(c0), ϕ(u−v)>(1 +ε)λ1>0 where (1 +ε)λ1=t1−max_w_∈_G(c0)ϕ(w) orϕ(u−v)<(1 +ε)λ2<0 where (1 +ε)λ2=t2−minw∈G(c0)ϕ(w), respectively. Therefore condition (2.19) is then satisfied.

The above considerations,Theorem 2.9(iii) and the separation theorem yield the assertion.

(iv) This is a consequence of (iii).

We now prove the result under stronger condition.

Theorem 2.11. LetE be a real Hausdorfftopological vector space, letCbe a nonempty compact convex subset of E and suppose thatF:C→2ÊandG:C→2Ê.

(i) Denote byΦa single-valued map of_c_∈_C(F(c)×G(c)) intoE such that, for each c∈C,Φ(F(c)×G(c))is convex and compact and let the pair(F,G)beΦ-t.h.c. onC. Then the following hold:(i1)either(F,G)has aΦ-coincidence or there existsλ∈Rand, for any c∈C, there existsϕc∈Esuch thatλ[(ϕc◦Φ)(u,v)−λ]>0for all(u,v)∈F(c)×G(c);

(i2)if the pair(F,G)isΦ-inward orΦ-outward, then(F,G)has aΦ-coincidence.

(ii) Denote byΦa single-valued map of_c_∈_C(F(c)× {c}) intoEsuch that, for each c∈C,Φ(F(c)× {c})is convex and compact and assume thatF isΦ-t.h.c. onC. Then the following hold:(ii1)eitherF has aΦ-fixed point or there existsλ∈Rand, for anyc∈C, there existsϕc∈Esuch thatλ[(ϕc◦Φ)(u,c)−λ]>0for allu∈F(c);(ii2)ifFisΦ-inward orΦ-outward, thenFhas aΦ-fixed point.

(iii)Suppose thatF(c)andG(c)are compact subsets ofEandF(c)−G(c)is convex for each c∈Cand assume that the pair(F,G)is t.h.c. onC. Then the following hold: (iii1) either(F,G)has a coincidence or there existsλ∈Rand, for anyc∈C, there existsϕc∈E such thatλ[ϕc(u−v)−λ]>0for all(u,v)∈F(c)×G(c);(iii2)if the pair(F,G)is inward or outward, then(F,G)has a coincidence;(iii3)either(F,G)has a coincidence or, for any c∈C, the setsF(c)andG(c)are strictly separated by a closed hyperplane inE.

(iv)Suppose thatF is a t.h.c. map onCsuch that, for eachc∈C,F(c)is convex and compact. Then the following hold:(iv1)eitherFhas a fixed point or there existsλ∈Rand, for anyc∈C, there existsϕc∈Esuch thatλ[ϕc(u−c)−λ]>0for allu∈F(c);(iv2)ifF is inward or outward, thenFhas a fixed point;(iv3)eitherF has a fixed point or, for any c∈C, the setsF(c)and{c}are strictly separated by a closed hyperplane inE.

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Proof. (i1) Assume that (F,G) has noΦ-coincidence inC. Then, for allc∈C, the setDc, Dc=Φ(F(c)×G(c)), is convex, compact and 0∈/ Dc.

For (c,w)∈C×Dc, there existsϕc,w∈Esuch thatϕc,w(w) =0 and we assume, without loss of generality, that,ϕc,w(w)>0 for each (c,w)∈C×Dc.

First, let us observe that:

(a)for eachc∈C,there existϕc∈Eandλc>0,such that

ϕc◦Φ(u,v)> λc for any(u,v)∈F(c)×G(c). (2.20) Indeed, by the continuity ofϕc,w, we define a neighbourhoodMc(w) ofwinDcsuch that

Mc(w)⊂

x∈Dc:ϕc,w(x)> ϕc,w(w)/2. (2.21) Clearly, there exists a finite subset {w1,...,wm}of Dc such thatMc(wi) are nonempty, 1≤i≤n, andDc=_m

i=1Mc(wi). Let{α1,...,αm}be a partition of unity with respect to this cover, that is, a finite family of real-valued nonnegative continuous mapsαionDc

such thatαivanish outsideMc(wi) and are less than or equal to one everywhere, 1≤i≤m, and^m_i₌₁αi(w)=1 for allw∈Dc. Define

ψc(w)= m i=1

αi(w)ϕc,wi forw∈Dc. (2.22) Thenψc(w)∈Efor eachw∈Dc.

Now, lethc:Dc×Dc→Rbe of the form hc(w,y)=

ψc(w)(w−y) for (w,y)∈Dc×Dc. (2.23) Thushcis continuous onDc×Dcand, for eachw∈Dc, the maphc(w,·) is quasi-concave onDc. By [11, page 103], the following minimax inequality

minw∈Dcmax

y∈Dchc(w,y)≤max

w∈Dchc(w,w) (2.24) holds. Buthc(w,w)=0 for eachw∈Dc, so there is somewc∈Dcsuch thathc(wc,y)≤0 for ally∈Dc. Then

ψwc wc

=min

y∈Dc

ψwc

(y). (2.25)

Sincewc∈Mc(wi) for some 1≤i≤m, thereforeαi(wc)>0 and ψc

wc wc

=αi wc

ϕc,wi

wc

≥αi wc

ϕc,wi

wi

/2>0. (2.26) Consequently, we may assume that

ϕc=ψc wc

, λc=αi wc

ϕc,wi

wi

/4, (2.27)

whereλc>0. Thus (a) is proved.