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Coincidence points for multivalued maps based on [PHI]-EPI and [PHI]-essential maps.

1. INTRODUCTION

The 0-epi maps were introduced by Furi, Martelli and Vignoli [1] and essential maps were introduced by Granas [3]. The notion of [PHI]-epi maps is presented in Section 2 and the notion of [PHI]-essential maps is presented in Section 3. Both approaches allow us to study coincidence points (i.e. F(x) [intersection] [PHI](x) [not equal to] 0) of the maps F and [PHI]. This new theory presents a unified theory for establishing coincidence points for general classes of maps. Our results are more general than those in the literature (see [1-8] and the references therein).

2. [PHI]-EPI MAPS

Let E be a Hausdorff topological space and U an open subset of E. We will consider classes A and B of maps.

Definition 2.1. We say F [member of] A([bar.U],E) if F [member of] A([bar.U],E) and F : [bar.U] [right arrow] K(E) is an upper semicontinuous map; here [bar.U] denotes the closure of U in E and K(E) denotes the family of nonempty compact subsets of E.

Definition 2.2. We say F [member of] B([bar.U],E) if F [member of] B([bar.U],E) and F : [bar.U] [right arrow] K(E) is an upper semicontinuous map.

In this section we fix a [PHI] [member of] B(U, E) in the first three results.

Definition 2.3. We say F [member of] [A.sub.[partial derivative]U] ([bar.U],E) if F [member of] A([bar.U],E) with F(x) [intersection] [PHI](x) = 0 for x [member of] [partial derivative]U; here [partial derivative]U denotes the boundary of U in E.

Definition 2.4. We say F [member of] [B.sub.[PHI]]([bar.U], E) if F [member of] B([bar.U], E) and F(x) [subset or equal to] [PHI](x) for x [member of] [partial derivative]U.

Definition 2.5. A map F [member of] [A.sub.[partial derivative]U] ([bar.U], E) is [PHI]-epi if for every map [member of] [member of] [B.sub.[PHI]]([bar.U], E) there exists x [member of] U with F(x) [intersection] G(x) [not equal to] 0.

Remark 2.6. Suppose F [member of] [A.sub.[partial derivative]U]([bar.U], E) is [PHI]-epi. Then there exists x [member of] U with F(x) [intersection] [PHI](x) [not equal to] 0 (take [member of] = [PHI] in Definition 2.5).

Our next result can be called the "homotopy property" for [PHI]-epi maps. In our result E will be a topological vector space so automatically a completely regular space. For convenience we state the result if E is a normal space and we remark on the general case after the theorem.

Theorem 2.7. Let E be a normal topological vector space and U an open subset of E. Suppose F [member of] [A.sub.[partial derivative]U]([bar.U], E) is [PHI]-epi and H : [bar.U] x [0,1] [right arrow] K(E) is an upper semicontinuous map with H(x, 0) = {0} for x [member of] [partial derivative]U. In addition assume the following conditions hold:

(2.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

(2.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then F(*) - H(*, 1) : [bar.U] [right arrow] K(E) is [PHI]-epi

Proof. Let [member of] G [B.sub.[PHI]]([bar.U], E). We must show that there exists x [member of] U with [F(x) - H(x, 1)] [intersection] G(x) [not equal to] 0. Let

D = {x [member of] [bar.U] : F(x) [intersection] [G(x) + H(x,t)] [not equal to] 0 for some t [member of] [0,1]}.

When t = 0 we have G(*) + H(*, 0) [member of] [B.sub.[PHI]] (U, E) since from (2.1) we have G(*)+H(*, 0) [member of] B([bar.U],E) and for x [member of] [partial derivative]U we have G(x) + H(x, 0) = G(x) [subset or equal to] [PHI](x) and this together with the fact that F is [PHI]-epi yields D = 0. Next we show D is closed. To see this let ([x.sub.[alpha]]) be a net in D (i.e. F([x.sub.[alpha]]) [intersection] [G([x.sub.[alpha]]) + H([x.sub.[alpha]], [t.sub.[alpha]])] [not equal to] 0 for some [t.sub.[alpha]] [member of] [0,1]) with [x.sub.[alpha]] [right arrow] [x.sub.0] [member of] [bar.U]. Without loss of generality assume [t.sub.[alpha]] [t.sub.0] [member of] [0,1]. Suppose [y.sub.[alpha]] [member of] F([x.sub.[alpha]]) with [y.sub.[alpha]] [member of] G([x.sub.[alpha]]) + H([x.sub.[alpha]], [t.sub.[alpha]]). Since F is upper semicontinuous then [9] implies that there exists [y.sub.0] [member of] F([x.sub.0]) and a subnet ([y.sub.[beta]]) of ([y.sub.[alpha]]) with [y.sub.[beta]] [right arrow] [y.sub.0]. The upper semicontinuity of the maps [member of] and H together with [y.sub.[beta]]] [right arrow] [y.sub.0] and [y.sub.[beta]] [member of] G([x.sub.[beta]]) + H([x.sub.[beta]], [t.sub.[beta]]) implies [y.sub.0] [member of] G([x.sub.0]) + H([x.sub.0], [t.sub.0]). Thus F([x.sub.0]) [intersection] [G([x.sub.0]) + H([x.sub.0], [t.sub.0])] [not equal to] 0 i.e. [x.sub.0] [member of] D, so D is closed.

Next we note (2.2) guarantees that D [intersection] [partial derivative]U = 0 (note if x [member of] [partial derivative]U then F(x) [intersection] [G(x) + H(x,t)] [subset or equal to] F(x) [intersection] [[PHI](x) + H(x,t)]). Now Urysohn's lemma guarantees that there exists a continuous map [mu] : [bar.U] [right arrow] [0,1] with [mu]([partial derivative]U) = 0 and [mu](D) = 1.

Define a map J : [bar.U] [right arrow] K(E) by

J (x) = G(x) + H (x, [mu](x)).

Note J [member of] B([bar.U],E) from (2.1) and for x [less than or equal to] [partial derivative]U we have J(x) = G(x) + H(x, [mu](x)) = G(x) + H(x,0) = G(x) [subset or equal to] [PHI](x). Thus J [member of] [B.sub.[PHI]]([bar.U], E). Now since F is [PHI]- epi there exists x [member of] U with F(x) [intersection] J(x) [not equal to] 0 i.e. F(x) [intersection] [G(x) + H(x, [mu](x))] = 0. Thus x [member of] D and as a result [PHI](x) = 1. Consequently F(x) [intersection] [G(x) + H(x, 1)] [not equal to] 0 so [F(x) - H(x, 1)] [intersection] G(x) [not equal to] 0.

Remark 2.8. We can remove the assumption that E is normal in the statement of Theorem 2.7 provided we put conditions on the maps so that D is compact (the existence of the map [mu] in the proof above is then guaranteed since topological vector spaces are completely regular).

Our next result can be called the "coincidence property" for [PHI]-epi maps.

Theorem 2.9. Let E be a normal topological vector space and U an open subset of E. Suppose F [member of] [A.sub.[partial derivative]U]([bar.U],E) is [PHI]-epi, [member of] G B([bar.U], E) and assume the following conditions hold:

(2.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

(2.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then there exists x [member of] [bar.U] with F(x) [intersection] G(x) [not equal to] 0. Proof. Let

D = {x [member of] U : F(x) n [tG(x) + (1 - t)[PHI](x)] [+ or -] 0 for some t e [0,1]}.

When t = 0 note F(x) [intersection] [PHI](x) [not equal to] 0 for some x [member of] U since F [member of] [A.sub.[partial derivative]U] ([bar.U], E) is [PHI]-epi, so D [not equal to] 0. The same reasoning as in Theorem 2.7 guarantees that D is closed. Also D fl dU = 0 from (2.4). Thus there exists a continuous map [mu] : [bar.U] [right arrow] [0,1] with [mu]([partial derivative]U) = 0 and [mu](D) = 1.

Define a map J : [bar.U] [right arrow] K(E) by

J(x) = [mu](x)G(x) + (1 - [mu](x))[PHI](x).

Now (2.3) guarantees that J [member of] B([bar.U], E) and for x [member of] [partial derivative]U we have J(x) = 0 + [PHI](x) = [PHI](x), so J [member of] [B.sub.[PHI]]([bar.U], E). Now since F is [PHI]-epi there exists x [member of] U with F(x)[intersection]J(x) [not equal to] 0. Thus x [member of] D and as a result [mu](x) = 1. Consequently F(x) [intersection] G(x) [not equal to] 0.

Remark 2.10. We also have an analogue of Remark 2.8 in this case also. Finally we restate Theorem 2.9 as a result of Leray-Schauder type.

Theorem 2.11. Let E be a normal topological vector space and U an open subset of E. Suppose F [member of] [A.sub.[partial derivative]U]([bar.U], E) is [section]-epi and G [member of] B([bar.U], E). In addition assume (2.3) holds. Then either

(Al). there exists x [member of] [bar.U] with F(x) [intersection] G(x) [not equal to] 0, or

(A2). there exists x [member of] [partial derivative]U and [lambda] [member of] (0,1) with F(x) [intersection] [[lambda]G(x) + (1 - [lambda])[PHI](x)] [not equal to] 0, holds.

Proof. Suppose (A2) does not hold and F(x) [intersection] G(x) = 0 for x [member of] [partial derivative]U (since otherwise (A1) holds). Also note F(x) [intersection] [PHI](x) = 0 for x [member of] [partial derivative]U since F [member of] [A.sub.[partial derivative]U]([bar.U], E). Thus

there exists x [member of] [partial derivative]U and [lambda] [member of] [0,1] with F(x) [intersection] [[lambda]G(x) + (1 - [lambda])[PHI](x)] [not equal to] 0

cannot occur, so (2.4) holds. Now Theorem 2.9 guarantees that there exists x [member of] U with F(x) [intersection] G(x) [not equal to] 0.

We now show that the ideas in this section can be applied to other natural situations. Let E be a Hausdorff topological vector space, Y a topological vector space, and U an open subset of E. Also let L : dom L [subset or equal to] E [right arrow] Y be a linear (not necessarily continuous) single valued map; here domL is a vector subspace of E. Finally T : E [right arrow] Y will be a linear, continuous single valued map with L + T : dom L [right arrow] Y an isomorphism (i.e. a linear homeomorphism); for convenience we say T [member of] [H.sub.L] (E,Y).

Definition 2.12. We say F [member of] A([bar.U], Y; L, T) if F : [bar.U] [right arrow] [2.sup.y] with [(L + T).sup.-1] (F + T) [member of] A([bar.U], E).

Definition 2.13. We say F [member of] B([bar.U], Y; L, T) if F : [bar.U] [right arrow] [2.sup.Y] with [(L + T).sup.-1] (F + T) [member of] B([bar.U], E).

In our next two results we [fix.bar]a [PHI] [member of] B([bar.U], Y] L, T).

Definition 2.14. We say F [member of] [A.sub.[partial derivative]U]([bar.U], Y; L, T) if F [member of] A([bar.U], Y; L, T) with [(L + T).sup.-1] (F + T)(x) [intersection] [(L + T).sup.-1] ([PHI] + T)(x) = 0 for x [member of] [partial derivative]U.

Definition 2.15. We say F [member of] [B.sub.[PHI]] ([bar.U], Y; L, T) if F [member of] B([bar.U], Y; L, T) and [(L + T).sup.-1] (F + T)(x) [subset or equal to] [(L + T).sup.-1] ([PHI] + T)(x) for x [member of] [partial derivative]U.

Definition 2.16. A map F [member of] [A.sub.[partial derivative]U]([bar.U], Y; L, T) is (L,T)[PHI]-epi if for every map [member of] [member of] [B.sub.[PHI]]([bar.U], Y; L, T) there exists x [member of] U with [(L + T).sup.-1] (F + T)(x)[intersection][(L+T).sup.- 1] (G + T)(x) [not equal to] 0.

Remark 2.17. Suppose F [member of] [A.sub.[partial derivative]U]([bar.U], Y; L, T) is (L, T)[PHI]-epi. Then there exists x [member of] U with ([L + T).sup.-1] (F + T)(x) [intersection] [(L + T).sup.-1]([PHI] + T)(x) = 0 (take [member of] = [PHI] in Definition 2.16).

Theorem 2.18. Let E be a normal topological vector space, Y a topological vector space, U an open subset of E, L : dom L [subset or equal to] E [PHI] Y a linear single valued map and T [member of] [H.sub.L](E, Y). Suppose F [member of] [A.sub.[partial derivative]U] ([bar.U], Y; L, T) is (L, T)[PHI]- epi and H :U x [0,1] [right arrow] [2.sup.Y] with [(L + T).sup.-1] H : [bar.U] x [0,1] [right arrow] K(E) an upper semicontinuous map and [(L + T).sup.-1] H(x, 0) = {0} for x [member of] [partial derivative]U. In addition assume the following conditions hold:

(2.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

(2.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then F(*) - H(*, 1) is (L,T) [PHI]-epi.

Proof. Let [member of] G [B.sub.[PHI]]([bar.U], Y; L, T) and

D = {x [member of] [bar.U]: [(L + T).sup.-1] F + T)(x) [intersection] [(L + T).sup.-1] [G(x) + H(x,t) + T(x)} [not equal to] 0 for some t [member of] [0,1]}.

When t = 0 we have G(*) + H(*,0) [member of] B([bar.U], Y; L, T) and for x [member of] [partial derivative]U we have [(L + T).sup.-1] [G(x) + H(x, 0) + T(x)] = [(L + T).sup.-1] (G + T)(x) [subset or equal to] [(L + T).sup.-1] ([PHI] + T)(x) so G(*) + H(*, 0) [member of] [B.sub.[PHI]]([bar.U], Y; L, T) and this together with the fact that F is (L, T)[PHI]-epi yields D [not equal to] 0. Similar reasoning as in Theorem 2.7 guarantees that D is closed. Also (2.6) guarantees that D [intersection] [partial derivative]U = 0 so there exists a continuous map [mu] : [bar.U] [right arrow] [0,1] with [mu]([partial derivative]U) = 0 and [mu](D) = 1. Define a map J : [bar.U] [right arrow] [2.sup.Y] by

J(x) = G(x) + H (x, [mu](x)).

Now J [member of] B([bar.U], Y; L, T) and for x [member of] [partial derivative]U we have [(L + T).sup.-1](J + T)(x) = [(L + T).sup.-1][G(x) + H(x, 0) + T(x)] = [(L + T).sup.-1](G + T)(x) [subset or equal to] [(L + T).sup.-1] ([PHI] + T)(x). Thus J [member of] [B.sub.[PHI]],([bar.U], Y] L, T) so since F is (L, T)[PHI]-epi there exists x [member of] U with [(L + T).sup.-1] (F + T)(x) [intersection] [(L + T).sup.-1] (J + T)(x) [not equal to] 0. Thus x [member of] D so [mu](x) = 1 and we are finished.

Remark 2.19. We also have an analogue of Remark 2.8 in this case also.

Remark 2.20. If we change Definition 2.12 (respectively Definition 2.13) to F [member of] A([bar.U], Y; L, T) (respectively F [member of] B ([bar.U], Y; L, T)) if F : [bar.U] [right arrow] [2.sup.Y] with [(L + T).sup.-1] F [member of] A([bar.U], F) (respectively [(L+T).sup.-1] F [member of] B([bar.U], E)), Definition 2.14 to F [member of] [A.sub.[partial derivative]U] ([bar.U], Y; L, T) if F [member of] A([bar.U], Y; L, T) with [(L + T).sup.-1] F(x) [intersection] [(L + T).sup.-1] [PHI](x) = 0 for x [member of] [partial derivative]U, Definition 2.15 to F [member of] [B.sub.[PHI]] (U, Y; L, T) if F [member of] B ([bar.U], Y; L, T) and [(L + T).sup.-1]F(x) [subset or equal to] [(L + T).sup.-1] [PHI](x) for x [member of] [partial derivative]U, Definition 2.16 to F [member of] [A.sub.[partial derivative]U] (U, Y; L, T) is (L, T)[PHI]-epi if for every map G [member of] [B.sub.[PHI]](U, Y; L, T) there exists x [member of] U with [(L + T).sup.-1] F(x) [intersection] [(L + T).sup.-1] G(x) [not equal to] 0, then we have an analogue of Theorem 2.18 if (2.6) is replaced by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(L,T) [PHI]-epi maps of this type when [PHI] = 0 were discussed in [5].

Theorem 2.21. Let E be a normal topological vector space, Y a topological vector space, U an open subset of E, L : dom L [subset or equal to] E [right arrow] Y a linear single valued map and T [member of] [H.sub.L] (E, Y). Suppose F [member of] [A.sub.[partial derivative]U] ([bar.U], Y; L, T) is (L, T)[PHI]- epi, [member of] [member of] B([bar.U], Y; L, T) and assume the following conditions hold:

(2.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

(2.8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then there exists x [member of] [bar.U] with [(L + T).sup.-1] (F + T)(x) [intersection] [(L + T).sup.-1] (G + T)(x) [not equal to] 0.

Proof. Let

D = {x [member of] [bar.U]: [(L + T).sup.-1] (F + T)(x) [intersection] [(L + T).sup.-1][tG(x) + (1 - t)[PHI](x) + T(x)] [not equal to] 0 for some t [member of] [0, 1]}.

Now D [not equal to] 0 is closed and D [intersection] [partial derivative]U = 0. Thus there exists a continuous map [mu] : [bar.U] [right arrow] [s0,1] with [mu]([partial derivative]U) = 0 and [mu](D) = 1. Define a map J : [bar.U] [right arrow] [2.sup.y] by

J(x) = [mu](x)G(x) + (1 - [mu](x))[PHI](x).

Now J [member of] B([bar.U], Y; L, T) and for x [member of] [partial derivative]U we have [(L + T).sup.-1](J + T)(a;) = [(L + T).sup.-1] [0 + ([PHI] + T)(x)], so J [member of] [B.sub.[PHI]],([bar.U], Y; L, T). Now since F is (L,T)[PHI]-epi there exists x eU with [(L + T).sup.-1](F + T)(x) [intersection] [(L + T).sup.-1] (J + T)(x) [not equal to] 0. Thus x [member of] D and as a result [mu](x) = 1, so we are finished. ?

Remark 2.22. We also have an analogue of Remark 2.8 in this case also.

3. [PHI]-ESSENTIAL MAPS

Let E be a completely regular topological space and U an open subset of E. As in Section 2 we will consider classes A and B of maps.

Definition 3.1. We say F [member of] A([bar.U], E) (respectively F [member of] B([bar.U], E)) if F [member of] A([bar.U], E) (respectively F [member of] B([bar.U], E)) and F : [bar.U] [right arrow] K(E) is an upper semicontinuous map.

In this section we [fix.bar] a [PHI] [member of] B([bar.U], E) in the first two results.

Definition 3.2. We say F [member of] [A.sub.[partial derivative]U] ([bar.U], E) if F [member of] A([bar.U], F) with F(x) [intersection] [PHI](x) = 0 for x [member of] [partial derivative]U.

Definition 3.3. Let F, G [member of] [A.sub.[partial derivative]U] ([bar.U], E). We say F [congruent to] G in [A.sub.[partial derivative]U] ([bar.U], E) if there exists an upper semicontinuous map [PSI]: U x [0,1] [right arrow] K(E) with [PSI] (*, [eta](*)) [member of] A([bar.U],E) for any continuous function [eta]: [bar.U] [right arrow] [0,1] with [eta]([partial derivative]U) = 0, [[PSI].sub.t](x) [intersection] [PHI](x) = 0 for any x [member of] [partial derivative]U and t [member of] [0,1], = F, [[PSI].sub.0]= G and {x [member of] [bar.U]: [PHI](x) [intersection] [PHI](x,t) [PSI] (x,t) [not equal to] 0 for some t [member of] [0,1]} is relatively compact (here [[PSI].sub.t] (x) = [PSI](x,t)).

Remark 3.4. We note if H: [bar.U] x [0,1] [right arrow] K(E) is an upper semicontinuous map then (similar reasoning as in Section 2) M = {x [member of] [bar.U]: [PHI](x) [intersection] H(x,t) [not equal to] 0 for some t [member of] [0,1]} is closed so that if M is relatively compact then M is compact.

The following condition will be assumed in our next two results:

(3.1) [congruent to] is an equivalence relation in [A.sub.[partial derivative]U] ([bar.U], E).

Definition 3.5. Let F [member of] [A.sub.[partial derivative]U] ([bar.U],E). We say F : [bar.U] [right arrow] K(E) is [PHI]-essential in [A.sub.[partial derivative]U] ([bar.U], E) if for every map J [member of] [A.sub.[partial derivative]U] ([bar.U], E) with [J|.sub.[partial derivative]U] = [F|.sub.[partial derivative]U] and J [congruent to] F in [A.sub.[partial derivative]U] ([bar.U], E) there exists x [member of] U with J{x) [intersection] [PHI](x) [not equal to] 0. Otherwise F is [PHI]-inessential in [A.sub.[partial derivative]U] ([bar.U],E) i.e. there exists a map J [member of] [A.sub.[partial derivative]U] ([bar.U],E) with [J|.sub.[partial derivative]U] = [F|.sub.[partial derivative]U] and J [congruent to] F in [A.sub.[partial derivative]U] ([bar.U],E) with J{x) [intersection] [PHI](x) = 0 for all x [member of] U.

Theorem 3.6. Let E be a completely regular topological space, U an open subset of E and assume (3.1) holds. Suppose F [member of] [A.sub.[partial derivative]U] ([bar.U],E). Then the following are equivalent:

(i). F is inessential in [A.sub.[partial derivative]U] ([bar.U],E);

(ii). there exists a map G [member of] [A.sub.[partial derivative]U] ([bar.U], E) with [member of] = F in [A.sub.[partial derivative]U] ([bar.U], E) and G(x) [intersection] [PHI](x) = 0 for all x [member of] [bar.U].

Proof. (i) implies (ii) is immediate. Next we prove (ii) implies (i). Suppose there exists a map G [member of] [A.sub.[partial derivative]U] ([bar.U], E) with G [congruent to] F in [A.sub.[partial derivative]U]([bar.U], E) and G(x) [intersection] [PHI](x) = 0 for all x [member of] [bar.U]. Let H: [bar.U] x [0,1] [right arrow] K(E) be a upper semicontinuous map with H(*, [eta](*)) [member of] A([bar.U], E) for any continuous function [eta] : [bar.U] [right arrow] [0,1] with [eta]([partial derivative]U) = 0, [H.sub.t](x) [intersection][PHI](x) = 0 for any x [member of] [partial derivative]U and t [member of] [0,1], [H.sub.0] = F, [H.sub.1] = G (here [H.sub.t](x) = H(x,t)) and {x [member of] [bar.U]: [PHI](x) [intersection] H(x,t) [not equal to] 0 for some t [member of] [0,1]} is relatively compact. Consider

D = {x [member of] [bar.U]: [PHI](x) [intersection] H(x,t) [not equal to] 0 for some t [member of] [0,1]}.

If D = 0 then in particular 0 = [PHI](x) [intersection] H(x,0) = [phi] F(x) for x [member of] [bar.U] so F is [PHI]-inessential in [A.sub.[partial derivative]U]([bar.U],E) (take J = F in Definition 3.5). Next suppose D [not equal to] 0. Essentially the same reasoning as in Theorem 2.7 guarantees that D is closed in E so D is compact from Remark 3.4. Also D [intersection] [partial derivative]U = 0. Thus (note E is a completely regular topological space) there exists a continuous map [mu] : [bar.U] [right arrow] [0,1] with [mu]([partial derivative]U) = 0 and [mu](D) = 1. Define J:[bar.U] [right arrow] K(E) by J(x) = H(x, [mu](x)). Note J [member of] A([bar.U], E) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Also note if there exists a x [member of] [bar.U] with J(x) [intersection] [PHI](x) [intersection] 0 then

x [member of] D so [mu](x) = 1 i.e. G(x) [intersection] [PHI](x) [PHI] 0, a contradiction. Thus J [member of] [A.sub.[partial derivative]U] ([bar.U],E) and [J|.sub.[partial derivative]U]= [F|.sub.[partial derivative]U] and J(x) [intersection] [PHI](x) = 0 for x [member of] [bar.U]. We now claim

(3.2) J [intersection] F in [A.sub.[partial derivative]U]([bar.U], E).

If (3.2) is true then F is [PHI]-inessential in [A.sub.[partial derivative]U]([bar.U], E).

It remains to show (3.2). Let Q: [bar.U] x [0,1] [right arrow] K(E) be given by Q(x,t) = H(x, t[mu](x)). Note Q: [bar.U] x [0,1] [right arrow] K(E) is an upper semicontinuous map, Q(*, [eta](*)) G A([bar.U], E) for any continuous function [eta]: [bar.U] [0,1] with [eta]([partial derivative]U) = 0 and

{x [member of] [bar.U]: 0 [PHI](x) [intersection] Q (x,t) = [PHI](x) [intersection] H(x, t[mu](x)) for some t [member of] [0,1]}

is closed and compact. Note [Q.sub.0] = F and [Q.sub.1] = J. Finally if there exists a t [member of] [0,1] and x [member of] [partial derivative]U with [PHI](x) n [Q.sub.t](x) [intersection] 0 than [PHI](x) [intersection] [H.sub.t[mu](x)](x) [not equal to] 0 so x [member of] D and so [mu](x) = 1 i.e. [PHI](x) [intersection] [H.sub.t] (x) [intersection] 0, a contradiction. Thus (3.2) holds.

Theorem 3.7. Let E be a completely regular topological space, U an open subset of E and assume (3.1) holds. Suppose F and [member of] are two maps in [A.sub.[partial derivative]U]([bar.U], E) with F [congruent to] G in [A.sub.[partial derivative]U] ([bar.U], E). Then F is [PHI]-essential in [A.sub.[partial derivative]U]([bar.U],E) if and only if G is [PHI]-essential in [A.sub.[partial derivative]U] ([bar.U], E).

Proof. F is [PHI]-inessential in [A.sub.[partial derivative]U]([bar.U],E) iff there exists a map [PHI] [member of] [A.sub.[partial derivative]U]([bar.U],E) with F [congruent to] [PHI] in [A.sub.[partial derivative]U]([bar.U], E) and [PHI](x) [intersection] [PSI](x) = 0 for x [member of] [bar.U] iff (since (3.1) holds) there exists a map [PHI] [member of] [A.sub.[partial derivative]U]([bar.U], E) with G [congruent to] [PSI] in [A.sub.[partial derivative]U]([bar.U], E) and n [PHI](x) [intersection] [PSI](x) = 0 for x [member of] [bar.U] iff G is [PHI]-inessential in [A.sub.[partial derivative]U]([bar.U],E).

Remark 3.8. If E is a normal topological space then the assumption that

{x [member of] [bar.:] [PHI](x) [intersection] [PSI](x,t) [not equal to] 0 for some t [member of] [0,1]}

is relatively compact can be removed in Definition 3.3 and we still obtain Theorem 3.6 and Theorem 3.7.

Remark 3.9. A result of Theorem 3.7 type was established in [4, Theorem 2.8] but however an assumption was omitted. In [4, Theorem 2.8] the result will work for subclasses of the admissible maps in [4] where [congruent to] is an equivalence relation in that class (for example the Kakutani and acyclic maps [6, 8]).

We next present a result where (3.1) is not needed.

Theorem 3.10. Let E be a completely regular topological space, U an open subset of E and let F [member of] [A.sub.[partial derivative]U]([bar.U],E) be essential in [A.sub.[partial derivative]U]([bar.U],E). Suppose there exists an upper semicontinuous map H : [bar.U] x [0,1] [right arrow] K(E) with H(*, [eta](*)) [member of] A([bar.U],E) for any continuous function [eta] : [bar.U] [right arrow] [0,1] with [eta]([partial derivative]U) = 0; [PHI](x) [intersection] [H.sub.t] (x) = 0 for any x [member of] [partial derivative]U and t [member of] (0,1], [H.sub.0] = F and {x [member of] [bar.U]: [PHI](x) [intersection] H(x,t) [not equal to] 0 for some t [member of] [0,1]} is relatively compact. Then there exists x [member of] U with [PHI](x) [intersection] [H.sub.1](x) = 0.

Proof. Let

D = {x [member of] [bar.U]: [PHI](x) [intersection] H(x,t)[not equal to] 0 for some t [member of] [0,1]}.

Note D [not equal to] 0 since F is [PHI]-essential in [A.sub.[partial derivative]U]([bar.U],E) (note F [congruent to] F in [A.sub.[partial derivative]U]([bar.U],E)). Essentially the same reasoning as in Theorem 2.7 guarantees that D is closed in E so D is compact from Remark 3.4. Also D [intersection] [partial derivative]U = 0 (note [H.sub.0] = F so for t = 0 we have [PHI](x) [intersection] [H.sub.0] (x) = 0 for x [member of] [partial derivative]U since F [member of] [A.sub.[partial derivative]U]([bar.U],E)). Thus there exists a continuous map [mu]: [bar.U] [right arrow] [0,1] with [mu]([partial derivative]U) = 0 and [mu](D) = 1. Define J : [bar.U] [right arrow] K(E) by J{x) = H(x, [mu](x)). Note J [member of] [A.sub.[partial derivative]U]([bar.U],E) with [J|.sub.[partial derivative]U] = [F|.sub.[partial derivative]U] (note if x [member of] [bar.U] then J(x) = [H.sub.0](x) = F(x) and J(x) [intersection] [PHI](x) = F(x) [intersection] [PHI](x) = 0). Also as in Theorem 3.6, J [congruent to] F in [A.sub.[partial derivative]U]([bar.U],E) (take as before Q : [bar.U] x [0,1] [right arrow] K(E) given by Q(x,t) = H(x, t[mu](x))). Now since F is [PHI]-essential in [A.sub.[partial derivative]U]([bar.U],E) then there exists a x [member of] U with J(x) [intersection] [PHI](x) [not equal to] 0 (i.e. [H.sub.[mu](x)](x) [intersection] [PHI](x) [not equal to] 0), and thus x [member of] D so [mu](x) = 1 and as a result [H.sub.1](x) [intersection] [PHI](x) [not equal to] 0.

Remark 3.11. If E is a normal topological space then the assumption that

{x [member of] [bar.U]: [PHI](x) [intersection] H(x,t) [not equal to] 0 for some t [member of] [0,1]}

is relatively compact can be removed in the statement of Theorem 3.10 and we still obtain Theorem 3.10.

Remark 3.12. The result in Theorem 3.10 also holds (proof is easier also) if we change Definition 3.5 as follows: Let F [member of] [A.sub.[partial derivative]U]([bar.U], E). We say F : [bar.U] [right arrow] K(E) is [PHI]-essential in [A.sub.[partial derivative]U]([bar.U],E) if for every map J [member of] [A.sub.[partial derivative]U]([bar.U],E) with [J|.sub.[partial derivative]U] = [F|.sub.[partial derivative]U] there exists x [member of] U with J(x) [intersection] [PHI](x) [not equal to] 0.

Let E be a Hausdorff topological vector space, Y a topological vector space, and U an open subset of E. Also let L : dom L [subset or equal to] E [right arrow] Y be a linear (not necessarily continuous) single valued map; here domL is a vector subspace of E. Finally T: E [right arrow] Y will be a linear, continuous single valued map with L + T : domL [right arrow] Y an isomorphism (i.e. a linear homeomorphism); for convenience we say T [member of] [H.sub.L](E, Y).

Definition 3.13. We say F [member of] A([bar.U], Y; ,L,T) (respectively F [member of] B([bar.U], Y; L, T)) if [(L + T).sup.-1] (F + T) [member of] A([bar.U], E) [(respectively L + T).sup.-1] F + T) [member of] B([bar.U], E)).

We now fix a [PHI] [member of] B([bar.U], Y; L, T).

Definition 3.14. We say F [member of] [A.sub.[partial derivative]U] ([bar.U], Y; L,T) if F [member of] A([bar.U], Y; L, T) with [(L + T).sup.-1] (F + T)(x) [intersection] [(L + T).sup.-1] ([PHI] + T)(x) = 0 for x [member of] [partial derivative]U.

Definition 3.15. Let F,G [member of] [A.sub.[partial derivative]U] ([bar.U],Y; L,T). We say F [congruent to] [member of] in [A.sub.[partial derivative]U]([bar.U], Y; L, T) if there exists a map [PSI]: [bar.U] x [0,1] [right arrow] [2.sup.Y] with [(L + T).sup.-1] ([PSI] + T): [bar.U] x [0,1] [right arrow] K(E) a upper semi continuous map, [(L + T).sup.-1] ([PSI](*, [eta](*)) + T(*)) [member of] A([bar.U],E) for any continuous function [eta]: [bar.U] [right arrow] [0,1] with [eta]([partial derivative]U) = 0, [(L + T).sup.-1] ([[PSI].sub.t] + T] (x) [intersection][ (L + T).sup.-1] ([PHI] + T)(x) = 0 for any x [member of] [partial derivative]U and t [member of] [0,1], [PSI] = F, [[PSI].sub.0] = G and {x [member of] [bar.U]: [(L + T).sup.-1]([PHI] + T)(x) [intersection] [(L + T).sup.-1] ([[PSI].sub.t] + T)(x) [not equal to] 0 for some t e [0,1]} is relatively compact (here [[PSI].sub.t] (x) = [PSI](x,t)).

The following condition will be assumed in our next two results:

(3.3) [congruent to] is an equivalence relation in [A.sub.[partial derivative]U] ([bar.U], Y; L,T).

Definition 3.16. Let F [member of] [A.sub.[partial derivative]U] ([bar.U], Y; L, T). We say F is (L, T) [PHI]-essential in [A.sub.[partial derivative]U]([bar.U], Y; L, T) if for every map J E [A.sub.[partial derivative]U]([bar.U], Y] L, T) with [J|.sub.[partial derivative]U] = [F|.sub.[partial derivative]U] and J [congruent to] F in [A.sub.[partial derivative]U]([bar.U], Y] L, T) there exists x [member of] U with [(L + T).sup.-1] (J + T)(x) [intersection] [(L + T).sup.-1] ([PHI] + T)(x) [not equal to] 0. Otherwise F is (L, T) [PHI]-inessential in [A.sub.[partial derivative]U]([bar.U],E) i.e. there exists a map J [member of] [A.sub.[partial derivative]U]([bar.U], Y; L, T) with [J|.sub.[partial derivative]U] = [F|.sub.[partial derivative]U] and J [congruent to] F in [A.sub.[partial derivative]U]([bar.U], Y; L, T) with [(L + T).sup.-1] J + T)(x) [intersection] [(L + T).sup.-1] ([PHI] + T)(x) = 0 for all x [member of] [bar.U].

Theorem 3.17. Let E be a completely regular topological vector space, Y a topological vector space, U an open subset of E, L : dom L [subset] E [right arrow] Y a linear single valued map, T [member of] [H.sub.L] (E,Y), and assume (3.3) holds. Suppose F [member of] [A.sub.[partial derivative]U]([bar.U],Y; L,T). Then the following are equivalent:

(i). F is (L,T) [PHI]-inessential in [A.sub.[partial derivative]U]([bar.U], Y; L, T);

(ii). there exists a map G [member of] [A.sub.[partial derivative]U]([bar.U], Y; L, T) with G [congruent to] F in [A.sub.[partial derivative]U]([bar.U], Y; L, T) and [(L + T).sup.-1] G + T)(x) [intersection] [(L + T).sup.-1] ([PHI] + T)(x) = 0 for all x [member of] [bar.U].

Proof. (i) implies (ii) is immediate. Next we prove (ii) implies (i). Suppose there exists a map G [member of] [A.sub.[partial derivative]U]([bar.U], Y; L, T) with G [congruent to] F in [A.sub.[partial derivative]U]([bar.U], Y; L, T) and [(L+T).sup.-1] (G+T)(x) [intersection] [(L+T).sup.-1] ([PHI]+T)(x) = 0 for all x [member of] [bar.U]. Let H: [bar.U] x [0,1] [right arrow] [2.sup.Y] with [(L+T).sup.-1] (H+T): [bar.U] x [0,1] [right arrow] K(E) an upper semi continuous map, [(L + T).sup.-1] (H(*, [eta](*)) + T(*)) [member of] A([bar.U], E) for any continuous function [eta]: [bar.U] [right arrow] [0,1] with [eta]([partial derivative]U) = 0, [(L + T).sup.-1] ([H.sub.t] + T)(x) [intersection] [(L + T).sup.-1] ([PHI] + T)(x) = 0 for any x [member of] [partial derivative]U and t [member of] [0,1], [H.sub.1] = F, [H.sub.0] = G (here [H.sub.t](x) = H(x,t)) and

{x [member of] [bar.U]: [(L + T).sup.-1] ([PHI] + T)(x) n [(L + T).sup.-1] ([H.sub.t] + T)(x) [not equal to] 0 for some t [member of] [0,1]}

is relatively compact. Let

D = {x [member of] [bar.U]:[(L + T).sup.-1] ([PHI] + T)(x) [intersection] [(L + T).sup.-1] ([H.sub.t] + T)(x) [not equal to] 0 for some t [member of] [0,1]}.

If D = 0 we are finished. If D [not equal to] 0 then note D is compact, D [intersection] [partial derivative]U = 0 so there exists a continuous map [mu]: [bar.U] [right arrow] [0,1] with [mu]([partial derivative]U) = 0 and [mu](D) = 1. Define J : [bar.U] [right arrow] [2.sup.Y] by J(x) = H(x, [mu](x)). It is easy to check (a slight modification of the argument in Theorem 3.6) that J [member of] [A.sub.[partial derivative]U]([bar.U], Y; L, T), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [(L + T).sup.-1]([PHI] + T) (x) [intersection] [(L + T).sup.-1] (J + T)(x) = 0 for x [member of] [bar.U] and J [congruent to] F in [A.sub.[partial derivative]U]([bar.U], Y; L, T). Thus F is (L, T)[PHI]-inessential in [A.sub.[partial derivative]U]([bar.U], Y; L, T).

Theorem 3.18. Let E be a completely regular topological vector space, Y a topological vector space, U an open subset of E, L : dom L [subset or equal to] E [right arrow] Y a linear single valued map, T [member of] [H.sub.L](E,Y), and assume (3.3) holds. Suppose F and [member of] are two maps in [A.sub.[partial derivative]U]([bar.U], Y; L, T) with F [congruent to] G in [A.sub.[partial derivative]U]([bar.U]L,T). Then F is (L,T)[PHI]-essential in [A.sub.[partial derivative]U]([bar.U], Y] L, T) if and only if [member of] is (L, T)[PHI]-essential in [A.sub.[partial derivative]U]([bar.U], Y; L, T).

Remark 3.19. If E is a normal topological space then the assumption that

{x [member of] [bar.U]: [(L + T).sup.-1] ([PHI] + T)(x) [intersection] [(L + T).sup.-1] ([[PSI].sub.t] + T)(x) [not equal to] 0 for some t [member of] [0,1]}

is relatively compact can be removed in Definition 3.15 and we still obtain Theorem 3.17 and Theorem 3.18.

Theorem 3.20. Let E be a completely regular topological vector space, Y a topological vector space, U an open subset of E, L : dom L [subset or equal to] E [right arrow] Y a linear single valued map and T [member of] [H.sub.L] (E,Y). Let F [member of] [A.sub.[partial derivative]U]([bar.U]L,T) be (L,T) [PHI]-essential in [A.sub.[partial derivative]U]([bar.U], Y; L, T).

Suppose there exists a map H: [bar.U] x [0,1] [right arrow] [2.sup.Y] with [(L + T).sup.-1] (H + T): [bar.U] x [0,1] [right arrow] K(E) an upper semi continuous map, [(L + T).sup.-1] (H(*, [eta](*)) + T(*)) [member of] A([bar.U], E) for any continuous function [eta]: [bar.U] [right arrow] [0,1] with [eta]([partial derivative]U) = 0, [(L + T).sup.-1] ([H.sub.t] + T)(x) [intersection][ (L + T).sup.-1] ([PHI] + T)(x) = 0 for any x [member of] [partial derivative]U and t [member of] (0,1], [H.sub.0] = F (here [H.sub.t](x) = H(x,t)) and

{x [member of] [bar.U]: [(L + T).sup.-1] ([PHI] + T)(x) [intersection] [(L + T).sup.-1]([H.sub.t] + T)(x) [not equal to] 0 for some t [member of] [0,1]}

is relatively compact. Then there exists x [member of] U with [(L + T).sup.-1] ([H.sub.1] + T)(x) [intersection][ (L + T).sup.-1] ([PHI] + T)(x) [not equal to] 0.

Proof. Let

D= {x [member of] [bar.U]:[(L + T).sup.-1] ([PHI] + T)(x) [intersection] [(L + T).sup.-1]([H.sub.t] + T)(x) [not equal to] 0 for some t [member of] [0,1]}.

Note D [not equal to] 0 and D is compact, D [intersection] [partial derivative]U = 0 so there exists a continuous map [mu]: [bar.U] [right arrow] [0,1] with [mu]([partial derivative]U) = 0 and [mu](D) = 1. Define J: [bar.U] [right arrow] [2.sup.Y] by J(x) = H(x, [mu](x)). Note J [member of] [A.sub.[partial derivative]U]([bar.U], Y; L, T), [J|.sub.[partial derivative]U] = [F|.sub.[partial derivative]U] and J = F in [A.sub.[partial derivative]U]([bar.U],Y] L,T). Now since F is (L,T)[PHI]-essential in [A.sub.[partial derivative]U]([bar.U], Y; L, T) there exists x [member of] U with [(L + T).sup.-1] (J + T)(x) [intersection] [(L + T).sup.-1] ([PHI] + T)(x) [not equal to] 0 (i.e. [(L + T).sup.-1] ([H.sub.[mu](x)] + T)(x) [intersection] ([L + T).sup.-1] ([PHI] + T)(x) [not equal to] 0), and thus x <E D so fi(x) = 1 and we are finished.

Remark 3.21. If E is a normal topological space then the assumption that

{x [member of] U: [(L + T).sup.-1] ([PHI] + T)(x) [intersection] [(L + T).sup.-1] ([H.sub.t] + T)(x) [not equal to] 0 for some t [member of] [0,1]} is relatively compact can be removed in the statement of Theorem 3.20 and we still obtain Theorem 3.20.

Received July 11, 2014

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DONAL O'REGAN

School of Mathematics, Statistics and Applied Mathematics

National University of Ireland

Galway Ireland

donal.oregan@nuigalway.ie
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