**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 *E*be 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 in*E*are introduced (condition of strictly transfer positive hemiconti-
nuity is stronger than that of transfer positive hemicontinuity) and for maps*F*:*C**→*2* ^{E}*
and

*G*:

*C*

*→*2

*defined on a nonempty compact convex subset*

^{E}*C*of

*E, 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}

_{∈}*(F(c)*

_{C}*×*

*G(c))*

*→*

*E*has a zero, and, at the same time, we give various char- acterizations of the class of those pairs (F,

*G) and mapsF*that possess coincidences and fixed points, respectively. Transfer positive hemicontinuity and strictly transfer positive hemicontinuity generalize the famous Fan upper demicontinuity which generalizes up- per semicontinuity. Furthermore, a new type of continuity defined here essentially gen- eralizes 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 demi- continuity 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]

Copyright©2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:3 (2005) 389–407 DOI:10.1155/FPTA.2005.389

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).

Let*C*be a nonempty compact convex subset of a real Hausdorﬀtopological vector
space*E, letF*:*C**→*2* ^{E}*and

*G*:

*C*

*→*2

*be set-valued maps and letΦ:*

^{E}^{}

_{c}

_{∈}*(F(c)*

_{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 in

*E*and prove various new results concerning the existence of zeros ofΦ, coincidences of

*F*and

*G*and fixed points of

*F*in which transfer positive hemicontinu- ity 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 posi- tive hemicontinuity and strictly transfer positive hemicontinuity generalize the Fan upper demicontinuity. Furthermore, a new type of continuity defined here essentially gener- alizes upper hemicontinuity (every upper demicontinuous map is upper hemicontinu- ous). Comparisons of transfer hemicontinuity and strictly transfer positive hemiconti- nuity 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 re- sults 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**

Let*E*be a real Hausdorﬀtopological vector space and let*E** ^{}*denote the vector space of all
continuous linear forms on

*E*.

Let*C*be a nonempty subset of*E. A set-valued mapF*:*C**→*2* ^{E}*is a map which assigns
a unique nonempty subset

*F(c)*

*∈*2

*to each*

^{E}*c*

*∈*

*C*(here 2

*denotes the family of all nonempty subsets of*

^{E}*E*).

*Definition 2.1.* Let*C*be a nonempty subset of*E*, let*F*:*C**→*2* ^{E}*and let

*G*:

*C*

*→*2

*. Let Φ:*

^{E}^{}

_{c}

_{∈}*(F(c)*

_{C}*×*

*G(c))*

*→*

*E*be a single-valued map.

(a) We say that a pair (F,G) isΦ-transfer positive hemicontinuous(Φ-t.p.h.c.) on*C*if,
whenever (*c*,*ϕ**c*,*λ**c*)*∈**C**×**E*^{}*×*Rand*ε**c**>*0 are such that

*λ**c*

*ϕ**c**◦*Φ^{}(u,v)*−*
1 +*ε**c*

*λ**c*

*>*0 for any (u,*v)**∈**F(c)**×**G(c),* (2.1)

there exists a neighbourhood*N*(c) of*c*in*C*such that
*λ**c*

*ϕ**c**◦*Φ^{}(*u*,*v*)*−**λ**c*

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

(b) We say that a pair (F,*G) is*Φ-transfer hemicontinuous(Φ-t.h.c.) on*C*if, whenever
(*c*,*ϕ**c*,*λ**c*)*∈**C**×**E*^{}*×*Ris such that

*λ**c*

*ϕ**c**◦*Φ^{}(u,v)*−**λ**c*

*>*0 for any (u,v)*∈**F(c)**×**G(c),* (2.3)

there exists a neighbourhood*N*(*c*) of*c*in*C*such that
*λ**c*

*ϕ**c**◦*Φ^{}(u,v)*−**λ**c*

*>*0 for any*x**∈**N*(c) and any (u,v)*∈**F(x)**×**G(x).* (2.4)
(c) We say that a map*F*isΦ-t.p.h.c.orΦ-t.h.c.on*C*if a pair (F,*I**E*) isΦ-t.p.h.c. or
Φ-t.h.c. on*C, respectively.*

(d) We say that a pair (*F*,*G*) is*transfer positive hemicontinuous (t.p.h.c.)* or*transfer*
*hemicontinuous (t.h.c.)*on*C*if (F,G) isΦ-t.p.h.c. orΦ-t.h.c. on*C, respectively, for*Φof
the formΦ(u,*v)**=**u**−**v*where (u,*v)**∈**F(c)**×**G(c) andc**∈**C.*

(e) We say that a map*F*is*t.p.h.c.*or*t.h.c.*on*C*if a pair (*F*,*I**E*) is t.p.h.c. or t.h.c. on*C*,
respectively.

Recall that an*open half-spaceH*in*E*is a set of the form*H**= {**x**∈**E*:*ϕ*(*x*)*< t**}*where
*ϕ**∈**E*^{}*\ {*0*}*and*t**∈*R.

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

Really define
*H**c*,*ϕ**c*,*λ**c*,*ε**c**=*

*w**∈**E*:*ϕ**c*(w)*<*^{}1 +*ε**c*
*λ**c*

, *ε**c**≥*0,
*W**c,ϕ**c*,λ*c*,Φ*=*

*x**∈**C*:^{}*ϕ**c**◦*Φ^{}(*u*,*v*)*< λ**c*for any (*u*,*v*)*∈**F*(*x*)*×**G*(*x*)^{},
*U**c*,*ϕ**c*,*λ**c*,Φ*=*

*x**∈**C*: sup

(u,v)*∈**F(x)**×**G(x)*

*ϕ**c**◦*Φ^{}(u,*v)**≤**λ**c*

(2.5)

when*λ**c**<*0,

*H**c,ϕ**c*,λ*c*,ε*c**=*

*w**∈**E*:*ϕ**c*(w)*>*^{}1 +*ε**c*
*λ**c*

, *ε**c**≥*0,
*W**c*,*ϕ**c*,*λ**c*,Φ*=*

*x**∈**C*:^{}*ϕ**c**◦*Φ^{}(u,v)*> λ**c*for any (u,*v)**∈**F(x)**×**G(x)*^{},
*U**c,ϕ**c*,λ*c*,Φ*=*

*x**∈**C*: inf

(u,v)*∈**F(x)**×**G(x)*

*ϕ**c**◦*Φ^{}(*u*,*v*)*≥**λ**c*

(2.6)

when*λ**c**>*0.

ByDefinition 2.1, we see that the pair (F,G) isΦ-t.p.h.c. orΦ-t.h.c. on*C*if, when-
ever (*c*,*ϕ**c*,*λ**c*)*∈**C**×**E*^{}*×*Rand*ε**c**≥*0 are such that the setΦ(*F*(*c*)*×**G*(*c*)) is contained

in open half-space*H*(c,ϕ*c*,λ*c*,*ε**c*) (here*ε**c**>*0 in the case ofΦ-transfer positive hemicon-
tinuity and*ε**c**=*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-spaceH**c,ϕ**c*,λ*c*,0; (ii)*c*is an interior point of the
sets*W**c,ϕ**c*,λ*c*,Φand*U**c,ϕ**c*,λ*c*,Φ. Indeed, then*λ**c*[(*ϕ**c**◦*Φ)(*u*,*v*)*−**λ**c*]*>*0 for any*x**∈**N*(*c*) and
any (u,v)*∈**F(x)**×**G(x).*

*Definition 2.3.* Let*C*be a nonempty subset of*E, letF*:*C**→*2* ^{E}*and let

*G*:

*C*

*→*2

*. Let Φ:*

^{E}^{}

_{c}

_{∈}*(F(c)*

_{C}*×*

*G(c))*

*→*

*E*be a single-valued map.

(a) We say that a pair (*F*,*G*) isΦ-strictly transfer positive hemicontinuous(Φ-s.t.p.h.c.)
on*C*if, whenever (c,ϕ*c*,*λ**c*)*∈**C**×**E*^{}*×*Rand*ε**c**>*0 are such that

*λ**c*

*ϕ**c**◦*Φ^{}(*u*,*v*)*−*
1 +*ε**c*

*λ**c*

*>*0 for any (*u*,*v*)*∈**F*(*c*)*×**G*(*c*), (2.7)
then*c*is an interior point of the set*V**c,ϕ**c*,λ*c*,Φ, where

*V**c*,*ϕ**c*,*λ**c*,Φ*=*

*x**∈**C*: sup

(u,v)*∈**F(x)**×**G(x)*

*ϕ**c**◦*Φ^{}(u,v)*< λ**c* if*λ**c**<*0,

*V**c*,*ϕ**c*,*λ**c*,Φ*=*

*x**∈**C*: inf

(*u*,*v*)_{∈}*F*(*x*)_{×}*G*(*x*)

*ϕ**c**◦*Φ^{}(u,v)*> λ**c*

if*λ**c**>*0.

(2.8)

(b) We say that a pair (*F*,*G*) isΦ-strictly transfer hemicontinuous(Φ-s.t.h.c.) on*C*if,
whenever (c,ϕ*c*,λ*c*)*∈**C**×**E*^{}*×*Ris such that

*λ**c*

*ϕ**c**◦*Φ^{}(u,v)*−**λ**c*

*>*0 for any (u,v)*∈**F(c)**×**G(c),* (2.9)
then*c*is an interior point of the set*V**c,ϕ**c*,λ*c*,Φ.

(c) We say that a map*F*isΦ-s.t.p.h.c.orΦ-s.t.h.c.on*C*if a pair (F,*I**E*) isΦ-s.t.p.h.c.

orΦ-s.t.h.c. on*C, 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.) on*C*if (F,G) isΦ-s.t.p.h.c. orΦ-s.t.h.c. on*C,*
respectively, forΦof the formΦ(u,v)*=**u**−**v*where (u,v)*∈**F(c)**×**G(c) andc**∈**C.*

(e) We say that a map*F*is*s.t.p.h.c.*or*s.t.h.c.*on*C*if a pair (F,I*E*) is s.t.p.h.c. or s.t.h.c.

on*C*, 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)*is*Φ-t.h.c. on*C, then*(F,G)*is*Φ-t.p.h.c. on*C.*

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

(iii)*If*(F,*G)is*Φ-s.t.h.c. on*C, then*(F,G)*is*Φ-s.t.p.h.c. on*C.*

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

(v)*If*(F,*G)is*Φ-s.t.p.h.c. (Φ-s.t.h.c., resp.) on*C, then*(F,G)*is*Φ-t.p.h.c. (Φ-t.h.c., resp.)
*onC.*

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

*Proof.* (i) Let (*F*,*G*) beΦ-t.h.c. on*C*and assume that there exist (*c*,*ϕ**c*,*λ**c*)*∈**C**×**E*^{}*×*R
and *ε**c**>*0 such that*λ**c*[(ϕ*c**◦*Φ)(u,v)*−*(1 +*ε**c*)λ*c*]*>*0 or, equivalently, (1 +*ε**c*)λ*c*[(ϕ*c**◦*
Φ)(u,v)*−*(1 +*ε**c*)λ*c*]*>*0 for any (u,v)*∈**F(c)**×**G(c). Then, by*Φ-transfer hemicontinu-
ity, there exists a neighbourhood*N*(*c*) of*c*in*C*such that (1 +*ε**c*)*λ**c*[(*ϕ**c**◦*Φ)(*u*,*v*)*−*(1 +
*ε**c*)λ*c*]*>*0 for any*x**∈**N(c) and any (u,v)**∈**F(x)**×**G(x). This implies, in particular, that*
*λ**c*[(ϕ*c**◦*Φ)(u,v)*−**λ**c*]*>*0 for any*x**∈**N(c) and any (u,v)**∈**F(x)**×**G(x), that is, (F*,G) is
Φ-t.p.h.c. on*C*.

(ii) Let (F,*G) be*Φ-t.p.h.c. on*C*and let there exists (c,ϕ*c*,λ*c*)*∈**C**×**E*^{}*×*Rsuch that,
for any (u,v)*∈**F(c)**×**G(c),* *λ**c*[(ϕ*c**◦*Φ)(u,*v)**−**λ**c*]*>*0 or, equivalently, for any (u,*v)**∈*
*F(c)**×**G(c), (ϕ**c**◦*Φ)(u,v)*< λ**c* if *λ**c**<*0 and (ϕ*c**◦*Φ)(u,v)*> λ**c* if *λ**c* *>*0. Since, for
each *x**∈**C,* Φ(F(x)*×**G(x)) is compact, thus sup*(*u*,*v*)_{∈}*F*(*c*)_{×}*G*(*c*)(ϕ*c**◦*Φ)(u,v)*< λ**c* if
*λ**c**<*0 and inf(*u*,*v*)_{∈}*F*(*c*)_{×}*G*(*c*)(ϕ*c**◦*Φ)(u,*v)> λ**c* if*λ**c**>*0, so there is some*ε**c**>*0 such that
sup_{(u,v)}_{∈}_{F(c)}_{×}* _{G(c)}*(ϕ

*c*

*◦*Φ)(u,v)

*<*(1 +

*ε*

*c*)λ

*c*if

*λ*

*c*

*<*0 and inf(u,v)

*∈*

*F(c)*

*×*

*G(c)*(ϕ

*c*

*◦*Φ)(u,v)

*>*

(1 +*ε**c*)*λ**c*if*λ**c**>*0. Therefore, for any (*u*,*v*)*∈**F*(*c*)*×**G*(*c*), (*ϕ**c**◦*Φ)(*u*,*v*)*<*(1 +*ε**c*)*λ**c*if*λ**c**<*

0 and (ϕ*c**◦*Φ)(u,*v)>*(1 +*ε**c*)λ*c*if*λ**c**>*0 or, equivalently,*λ**c*[(ϕ*c**◦*Φ)(u,v)*−*(1 +*ε**c*)λ*c*]*>*0
for any (u,*v)**∈**F(c)**×**G(c). Then, by*Φ-transfer positive hemicontinuity, there exists a
neighbourhood*N*(*c*) of*c*in*C*such that*λ**c*[(*ϕ**c**◦*Φ)(*u*,*v*)*−**λ**c*]*>*0 for any*x**∈**N*(*c*) and
any (u,v)*∈**F(x)**×**G(x), that is, (F*,G) isΦ-t.h.c. on*C.*

(iii) Let (F,G) beΦ-s.t.h.c. on*C*and assume that there exist (c,ϕ*c*,λ*c*)*∈**C**×**E*^{}*×*R
and *ε**c**>*0 such that*λ**c*[(*ϕ**c**◦*Φ)(*u*,*v*)*−*(1 +*ε**c*)*λ**c*]*>*0 or, equivalently, (1 +*ε**c*)*λ**c*[(*ϕ**c**◦*
Φ)(u,v)*−*(1 +*ε**c*)λ*c*]*>*0 for any (u,v)*∈**F(c)**×**G(c). Then, by*Φ-strictly transfer hemi-
continuity, *c* is an interior point of the set*V**c,ϕ**c*,(1+ε*c*)λ*c*,Φ. But *V**c,ϕ**c*,(1+ε*c*)λ*c*,Φ*⊂**V**c,ϕ**c*,λ*c*,Φ.
This implies, in particular, that*c*is an interior point of the set*V**c,ϕ**c*,λ*c*,Φ, that is, (*F*,*G*) is
Φ-s.t.p.h.c. on*C.*

(iv) Let (F,*G) be*Φ-s.t.p.h.c. on*C*and let there exists (c,ϕ*c*,λ*c*)*∈**C**×**E*^{}*×*Rsuch that,
for any (*u*,*v*)*∈**F*(*c*)*×**G*(*c*), *λ**c*[(*ϕ**c**◦*Φ)(*u*,*v*)*−**λ**c*]*>*0 or, equivalently, for any (*u*,*v*)*∈*
*F(c)**×**G(c), (ϕ**c**◦*Φ)(u,v)*< λ**c* if *λ**c**<*0 and (ϕ*c**◦*Φ)(u,v)*> λ**c* if *λ**c* *>*0. Since, for
each *x**∈**C,* Φ(F(x)*×**G(x)) is compact, thus sup*_{(u,v)}_{∈}_{F(c)}_{×}* _{G(c)}*(ϕ

*c*

*◦*Φ)(u,v)

*< λ*

*c*if

*λ*

*c*

*<*0 and inf

_{(u,v)}

_{∈}

_{F(c)}

_{×}*(*

_{G(c)}*ϕ*

*c*

*◦*Φ)(

*u*,

*v*)

*> λ*

*c*if

*λ*

*c*

*>*0, so there is some

*ε*

*c*

*>*0 such that sup

_{(}

_{u}_{,}

_{v}_{)}

_{∈}

_{F}_{(}

_{c}_{)}

_{×}

_{G}_{(}

_{c}_{)}(ϕ

*c*

*◦*Φ)(u,v)

*<*(1 +

*ε*

*c*)λ

*c*if

*λ*

*c*

*<*0 and inf(

*u*,

*v*)

_{∈}*F*(

*c*)

_{×}*G*(

*c*)(ϕ

*c*

*◦*Φ)(u,v)

*>*

(1 +*ε**c*)λ*c*if*λ**c**>*0. Therefore, for any (u,v)*∈**F(c)**×**G(c), (ϕ**c**◦*Φ)(u,v)*<*(1 +*ε**c*)λ*c*if*λ**c**<*

0 and (*ϕ**c**◦*Φ)(*u*,*v*)*>*(1 +*ε**c*)*λ**c*if*λ**c**>*0 or, equivalently,*λ**c*[(*ϕ**c**◦*Φ)(*u*,*v*)*−*(1 +*ε**c*)*λ**c*]*>*0
for any (u,*v)**∈**F(c)**×**G(c). Then, by*Φ-strictly transfer positive hemicontinuity,*c*is an
interior point of the set*V**c*,*ϕ**c*,*λ**c*,Φ, that is, (F,G) isΦ-s.t.p.h.c. on*C.*

(v) By Definitions2.1and2.3andRemark 2.2, we see that*V**c,ϕ**c*,λ*c*,Φ*⊂**W**c,ϕ**c*,λ*c*,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,ϕ*c*,λ*c*)*∈**C**×**E*^{}*×*Rand *ε**c**≥*0 are such that the setΦ(F(c)*×**G(c)) is contained in*
open half-space*H(c,ϕ**c*,λ*c*,ε*c*) (here*ε**c**>*0 in the case ofΦ-transfer positive hemicon-
tinuity and*ε**c**=*0 in the case ofΦ-transfer hemicontinuity), then there exists a neigh-
bourhood*N(c) ofc*in*C*such that, for any*x**∈**N(c) and any (u,v)**∈**F(x)**×**G(x), (ϕ**c**◦*
Φ)(u,v)*< λ**c* if*λ**c**<*0 and (ϕ*c**◦*Φ)(u,v)*> λ**c* if*λ**c**>*0. Since, for each*x**∈**C,*Φ(F(x)*×*
*G*(*x*)) is compact, thus, for each*x**∈**N*(*c*), sup_{(u,v)}_{∈}_{F(x)}_{×}* _{G(x)}*(

*ϕ*

*c*

*◦*Φ)(

*u*,

*v*)

*< λ*

*c*if

*λ*

*c*

*<*0

and inf(u,v)*∈**F(x)**×**G(x)*(ϕ*c**◦*Φ)(u,*v)> λ**c*if*λ**c**>*0. Consequently,*N*(c)*⊂**V**c,ϕ**c*,λ*c*,Φ, that is,*c*

is an interior point of the set*V**c,ϕ**c*,λ*c*,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.* Let*C*be a nonempty compact convex subset of*E. We say that (c,ϕ)**∈*
*C**×*(E^{}*\ {*0*}*) is*admissible*if*ϕ(c)**=*min_{x}_{∈}_{C}*ϕ(x); thus if (c,ϕ) is admissible, then this*
means that the*closed hyperplane*determined by*ϕ*of the form*{**x**∈**E*:*ϕ*(*x*)*=**ϕ*(*c*)*}*is a
*supporting hyperplane*of*C*at*c.*

*Definition 2.7.* Let*C*be a nonempty subset of*E, letF*:*C**→*2* ^{E}*and let

*G*:

*C*

*→*2

*. Let Φ:*

^{E}^{}

_{c}

_{∈}*(*

_{C}*F*(

*c*)

*×*

*G*(

*c*))

*→*

*E*be 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 map*F*is calledΦ-inward(Φ-outward, resp.) if the pair (F,I*E*) isΦ-inward (Φ-
outward, resp.).

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

(d) A map*F*is called*inward*(outward, resp.) (see Fan [11, page 108]) if a pair (F,I*E*)
is inward (outward, resp.).

*Definition 2.8.* Let*C*be a nonempty subset of*E, letF*:*C**→*2* ^{E}*and let

*G*:

*C*

*→*2

*. Let Φ:*

^{E}^{}

_{c}

_{∈}*(F(c)*

_{C}*×*

*G(c))*

*→*

*E*be a single-valued map.

(a) We say that a pair (*F*,*G*) has aΦ-coincidenceif there exist*c**∈**C*and (*u*,*v*)*∈**F*(*c*)*×*
*G(c), such that*Φ(u,v)*=*0, that is, (u,v)*∈**F(c)**×**G(c) is a zero of*Φ; this point*c*is called
aΦ-coincidence pointfor (F,G).

(b) We say that a map*F*has aΦ-fixed point(a pair (*F*,*I**E*) has aΦ-coincidence) if there
exist*c**∈**C*and*u**∈**F(c), such that*Φ(u,c)*=*0; this point*c*is called aΦ-fixed pointfor*F.*

(c) We say that a pair (F,G) has a*coincidence*if there exist*c**∈**C*and (u,v)*∈**F(c)**×*
*G*(*c*), such that*u**=**v*; this point*c*is called a*coincidence point*for (*F*,*G*).

(d) We say that*F*has a*fixed point*if there exists*c**∈**C*such that*c**∈**F(c); this pointc*
is called a*fixed point*for*F.*

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**→*2^{E}*and letG*:*C**→*2^{E}*. Let*Φ:^{}_{c}_{∈}* _{C}*(F(c)

*×*

*G(c))*

*→*

*Ebe a single-valued map.*

(i)*Let the pair*(F,G)*be*Φ-t.p.h.c. on*C. If*(F,G)*is*Φ-inward orΦ-outward, then there
*existsc*0*∈**Csuch that, for anyϕ**∈**E*^{}*, there is noλ**∈*R*such thatλ[(ϕ**◦*Φ)(u,v)*−**λ]>*0
*for all*(*u*,*v*)*∈**F*(*c*0)*×**G*(*c*0).

(ii)*LetFbe*Φ-t.p.h.c. on*C. IfFis*Φ-inward orΦ-outward, then there exists*c*0*∈**Csuch*
*that, for anyϕ**∈**E*^{}*, there is noλ**∈*R*such thatλ[(ϕ**◦*Φ)(u,c0)*−**λ]>*0*for allu**∈**F(c*0).

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

(iv)*LetFbe t.p.h.c. onC. IfFis inward or outward, then there existsc*0*∈**Csuch that,*
*for anyϕ**∈**E*^{}*, there is noλ**∈*R*such thatλ*[*ϕ*(*u**−**c*0)*−**λ*]*>*0*for allu**∈**F*(*c*0).

*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 existϕ**c**∈**E*^{}*\ {*0*}*,*λ**c**<*0 and*ε**c**≥*0, such that

*ϕ**c**◦*Φ^{}(*u*,*v*)*<*^{}1 +*ε**c*

*λ**c* *∀*(*u*,*v*)*∈**F*(*c*)*×**G*(*c*)*.* (2.11)
ByDefinition 2.1(a), there exists a neighbourhood*N*(*c*) of*c*in*C*such that

*ϕ**c**◦*Φ^{}(*u*,*v*)*< λ**c* for any*x**∈**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*{**c*1,*...,c**n**}* of*C* such that the family*{**N(c**j*) : *j**=*1, 2,...,n*}*covers *C. Let*
*{**β*1,*...*,*β**n**}*be a partition of unity with respect to this cover, that is, a finite family of real-
valued nonnegative continuous maps*β**j*on*C*such that*β**j*vanish outside*N(c**j*) and are
less than or equal to one everywhere, 1*≤**j**≤**n, and*^{}^{n}_{j}* _{=}*1

*β*

*j*(c)

*=*1 for all

*c*

*∈*

*C.*

Define*η(c)**=*_{n}

*j**=*1*β**j*(c)ϕ*c**j* for*c**∈**C. Thenη(c)**∈**E** ^{}*for each

*c*

*∈*

*C. Therefore*

*η(c)*^{}*◦*Φ^{}(u,v)*< λ* (2.13)

for any*c**∈**C*and (u,v)*∈**F(c)**×**G(c), whereλ**=*max1_{≤}*j**≤**n**λ**c**j**<*0 since
*η(c)*^{}*◦*Φ^{}(u,v)*=*

*n*
*j**=*1

*β**j*(c)^{}*ϕ**c**j**◦*Φ^{}(u,v)*<*^{}^{n}

*j**=*1

*β**j*(c)λ*c**j**.* (2.14)

Let now*k*:*C**×**C**→*Rbe a continuous map of the form*k(c,x)**=*[η(c)](c*−**x) for (c,x)**∈*
*C**×**C*. Since, for each*c**∈**C*, the map*k*(*c*,*·*) is quasi-concave on*C*, therefore, by [11, page
103], the following minimax inequality

min*c**∈**C*max

*x**∈**C**k(c,x)**≤*max

*c**∈**C* *k(c,c)* (2.15)

holds. But*k(c,c)**=*0 for each*c**∈**C, so there is somec*0*∈**C*such that*k(c*0,x)*≤*0 for all
*x**∈**C*. Since

*η*^{}*c*0

*c*0

*=*min

*x**∈**C*

*η*^{}*c*0

(*x*), (2.16)

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

*◦*Φ^{}(*u*,*v*)*< λ* for any (*u*,*v*)*∈**F*^{}*c*0

*×**G*^{}*c*0

, (2.17)

which is impossible by (2.10).

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

Two sets*X* and*Y* in *E*can be*strictly separated by a closed hyperplane*if there exist
*ϕ**∈**E** ^{}*and

*λ*

*∈*R, such that

*ϕ*(

*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**→*2^{E}*and letG*:*C**→*2^{E}*. Let*Φ:^{}_{c}_{∈}* _{C}*(F(c)

*×*

*G(c))*

*→*

*E*

*be a single-valued map.*

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

(ii)*LetF* *be*Φ-t.p.h.c. on*Cand inward or outward. Then there existsc*0*∈**Csuch that*
Φ(F(c0)*× {**c*0*}*)*and**{*0*}**cannot be strictly separated by any closed hyperplane inE. If, ad-*
*ditionally,Eis locally convex and, for eachc**∈**C, the set*Φ(*F*(*c*)*× {**c**}*)*is closed and convex,*
*then a mapFhas a*Φ-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*
*existsc*0*∈**C* *such thatF(c*0)*andG(c*0)*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 existsc*0*∈**Csuch that*
*F(c*0)*andG(c*0)*have a nonempty intersection.*

(iv)*LetF*:*C**→*2^{E}*be t.p.h.c. onCand inward or outward. Then, the following hold:*

(iv1)*there existsc*0*∈**Csuch thatF(c*0)*and**{**c*0*}**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 existsc*0*∈**Csuch thatc*0*∈**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 some*λ**∈*R,*ϕ**∈**E** ^{}*and

*ε*

*≥*0, and for all (u,v)

*∈*

*F(c*0)

*×*

*G(c*0), then we obtain that, for all (

*u*,

*v*)

*∈*

*F*(

*c*0)

*×*

*G*(

*c*0), (

*ϕ*

*◦*Φ)(

*u*,

*v*)

*<*(1 +

*ε*)

*λ*

*≤*

*λ < ϕ*(0) if

*λ <*0 and (

*ϕ*

*◦*Φ)(

*u*,

*v*)

*>*

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

Otherwise, assume that, for all (*u*,*v*)*∈**F*(*c*0)*×**G*(*c*0), (*ϕ**◦*Φ)(*u*,*v*)*< t*1*< ϕ*(0) for some
*t*1*∈*Ror (ϕ*◦*Φ)(u,v)*> t*2*> ϕ(0) for somet*2*∈*R. Then we obtain that, for all (u,v)*∈*
*F(c*0)*×**G(c*0), (ϕ*◦*Φ)(u,*v)<*(1 +*ε)λ*1*<*0 where (1 +*ε)λ*1*=**t*1 or (ϕ*◦*Φ)(u,v)*>*(1 +
*ε*)*λ*2*>*0 where (1 +*ε*)*λ*2*=**t*2. Therefore condition (2.18) is then satisfied.

The above considerations,Theorem 2.9(i) and the separation theorem yield the asser- tion.

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

(iii) Assume, without loss of generality, that*G(c*0) is compact.

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

(1 +*ε)λ*^{}*ϕ(u**−**v)**−*(1 +*ε)λ*^{}*>*0 (2.19)
for some*λ**∈*Rand*ε**≥*0 and for all (u,v)*∈**F(c*0)*×**G(c*0), then we obtain that, for all
(u,v)*∈**F(c*0)*×**G(c*0),*ϕ(u)< t*2*< ϕ(v) wheret*2*=*(1 +*ε)λ*+ min*w**∈**G(c*0)*ϕ(w) ifλ <*0 and
*ϕ*(*u*)*> t*1*> ϕ*(*v*) where*t*1*=*(1 +*ε*)*λ*+ max_{w}_{∈}* _{G(c}*0)

*ϕ*(

*w*) if

*λ >*0, that is, the sets

*F*(

*c*0) and

*G(c*0) are strictly separated by a closed hyperplane in

*E.*

Otherwise, assume that, for all (u,v)*∈**F(c*0)*×**G(c*0),*ϕ(u)> t*1*> ϕ(v) for somet*1*∈*R
or*ϕ*(*u*)*< t*2*< ϕ*(*v*) for some*t*2*∈*R. Then we obtain that, for all (*u*,*v*)*∈**F*(*c*0)*×**G*(*c*0),
*ϕ(u**−**v)>*(1 +*ε)λ*1*>*0 where (1 +*ε)λ*1*=**t*1*−*max_{w}_{∈}* _{G}*(

*c*0)

*ϕ(w) orϕ(u*

*−*

*v)<*(1 +

*ε)λ*2

*<*0 where (1 +

*ε)λ*2

*=*

*t*2

*−*min

*w*

*∈*

*G(c*0)

*ϕ(w), respectively. Therefore condition (2.19) is then*satisfied.

The above considerations,Theorem 2.9(iii) and the separation theorem yield the as- sertion.

(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**→*2^{E}*andG*:*C**→*2^{E}*.*

(i) *Denote by*Φ*a 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*)

*be*Φ

*-t.h.c. onC. Then*

*the following hold:*(i1)

*either*(F,

*G)has a*Φ-coincidence or there exists

*λ*

*∈*R

*and, for any*

*c*

*∈*

*C, there existsϕ*

*c*

*∈*

*E*

^{}*such thatλ[(ϕ*

*c*

*◦*Φ)(u,

*v)*

*−*

*λ]>*0

*for all*(u,v)

*∈*

*F(c)*

*×*

*G(c);*

(i2)*if the pair*(*F*,*G*)*is*Φ*-inward or*Φ*-outward, then*(*F*,*G*)*has a*Φ*-coincidence.*

(ii) *Denote by*Φ*a 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*

*is*Φ-t.h.c. on

*C. Then the*

*following hold:*(ii1)

*eitherF*

*has a*Φ

*-fixed point or there existsλ*

*∈*R

*and, for anyc*

*∈*

*C,*

*there existsϕ*

*c*

*∈*

*E*

^{}*such thatλ*[(

*ϕ*

*c*

*◦*Φ)(

*u*,

*c*)

*−*

*λ*]

*>*0

*for allu*

*∈*

*F*(

*c*);(ii2)

*ifFis*Φ

*-inward*

*or*Φ-outward, then

*Fhas a*Φ-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**∈*R*and, for anyc**∈**C, there existsϕ**c**∈**E*^{}*such thatλ[ϕ**c*(u*−**v)**−**λ]>*0*for 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**∈*R*and,*
*for anyc**∈**C, there existsϕ**c**∈**E*^{}*such thatλ[ϕ**c*(u*−**c)**−**λ]>*0*for 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.*

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

For (c,w)*∈**C**×**D**c*, there exists*ϕ**c*,*w**∈**E** ^{}*such that

*ϕ*

*c*,

*w*(w)

*=*0 and we assume, with- out loss of generality, that,

*ϕ*

*c,w*(w)

*>*0 for each (c,w)

*∈*

*C*

*×*

*D*

*c*.

First, let us observe that:

(a)*for eachc**∈**C,there existϕ**c**∈**E*^{}*andλ**c**>*0,*such that*

*ϕ**c**◦*Φ^{}(u,v)*> λ**c* *for any*(u,v)*∈**F(c)**×**G(c).* (2.20)
Indeed, by the continuity of*ϕ**c,w*, we define a neighbourhood*M**c*(w) of*w*in*D**c*such that

*M**c*(*w*)*⊂*

*x**∈**D**c*:*ϕ**c,w*(*x*)*> ϕ**c,w*(*w*)*/*2^{}*.* (2.21)
Clearly, there exists a finite subset *{**w*1,...,*w**m**}*of *D**c* such that*M**c*(w*i*) are nonempty,
1*≤**i**≤**n, andD**c**=*_{m}

*i**=*1*M**c*(w*i*). Let*{**α*1,...,*α**m**}*be a partition of unity with respect to
this cover, that is, a finite family of real-valued nonnegative continuous maps*α**i*on*D**c*

such that*α**i*vanish outside*M**c*(w*i*) and are less than or equal to one everywhere, 1*≤**i**≤**m,*
and^{}^{m}_{i}_{=}_{1}*α**i*(w)*=*1 for all*w**∈**D**c*. Define

*ψ**c*(w)*=*
*m*
*i**=*1

*α**i*(w)ϕ*c*,*w**i* for*w**∈**D**c**.* (2.22)
Then*ψ**c*(w)*∈**E** ^{}*for each

*w*

*∈*

*D*

*c*.

Now, let*h**c*:*D**c**×**D**c**→*Rbe of the form
*h**c*(w,*y)**=*

*ψ**c*(w)^{}(w*−**y)* for (w,*y)**∈**D**c**×**D**c**.* (2.23)
Thus*h**c*is continuous on*D**c**×**D**c*and, for each*w**∈**D**c*, the map*h**c*(w,*·*) is quasi-concave
on*D**c*. By [11, page 103], the following minimax inequality

min*w**∈**D**c*max

*y**∈**D**c**h**c*(*w*,*y*)*≤*max

*w**∈**D**c**h**c*(*w*,*w*) (2.24)
holds. But*h**c*(w,w)*=*0 for each*w**∈**D**c*, so there is some*w**c**∈**D**c*such that*h**c*(w*c*,*y)**≤*0
for all*y**∈**D**c*. Then

*ψ*^{}*w**c*
*w**c*

*=*min

*y**∈**D**c*

*ψ*^{}*w**c*

(y). (2.25)

Since*w**c**∈**M**c*(*w**i*) for some 1*≤**i**≤**m*, therefore*α**i*(*w**c*)*>*0 and
*ψ**c*

*w**c*
*w**c*

*=**α**i*
*w**c*

*ϕ**c*,*w**i*

*w**c*

*≥**α**i*
*w**c*

*ϕ**c*,*w**i*

*w**i*

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

*ϕ**c**=**ψ**c*
*w**c*

, *λ**c**=**α**i*
*w**c*

*ϕ**c,w**i*

*w**i*

*/4,* (2.27)

where*λ**c**>*0. Thus (a) is proved.