KRASNOSELSKII COINCIDENCE TYPE RESULTS FOR GENERAL CLASSES OF MAPS.

1. INTRODUCTION

This paper presents Krasnoselskii compression type theorems for general classes of maps. The approach is elementary and relies on the fact that in an infinite dimensional normed linear space there exists a retraction from the unit ball to the unit sphere (note also in a normed linear space there exists a retraction from the unit ball (in a cone) to the unit sphere (in a cone)). Under appropriate conditions a Krasnoselskii type theorem guarantees the existence of a fixed point in a particular annulus for continuous, compact single valued maps. In this paper we extend this further by establishing the existence of coincidence points in an annulus for maps (which may be multivalued) in a general class.

2. MAIN RESULTS

Let E be a topological space and U an open subset of E.

We consider classes A, B and D of maps.

Definition 2.1. We say F [member of] D([bar.U], E) (respectively F [member of] B([bar.U], E)) if F : [bar.U] [right arrow] [2.sup.E] and F [member of] D(U, E) (respectively F [member of] B([bar.U], E)); here [2.sup.E] denotes the family of nonempty subsets of E and [bar.U] denotes the closure of U in E.

Definition 2.2. We say F [member of] A([bar.U], E) if F : [bar.U] [right arrow] [2.sup.E], F [member of] A([bar.U], E) and there exists a selection [PSI] [member of] D([bar.U], E) of F.

Remark 2.3. Note T is a selection of F (in Definition 2.2) if [PSI](x) [subset or equal to] F(x) for x [member of] U.

Definition 2.4. We say F [member of] M([bar.U], E) if F : [bar.U] [right arrow] [2.sup.E] and F [member of] A([bar.U], E).

Definition 2.5. We say F [member of] D(E, E) (respectively F [member of] M(E, E)) if F : E [right arrow] [2.sup.E] and F [member of] D(E, E) (respectively F [member of] A(E, E)).

In this section we [fix.bar] a [PHI] [member of] B([bar.U], E).

Our first main result is a Krasnoselskii type theorem for A maps. Let E = (E, [parallel] x [parallel]) be an infinite dimensional normed linear space. For p > 0, r> 0, R > r let

[mathematical expression not reproducible].

Theorem 2.6. Let E = (E, [parallel] x [parallel]) be an infinite dimensional normed linear space, and r, R constants with 0 < r < R. Let F [member of] A([bar.[B.sub.R]], E), [PHI] [member of] B([bar.[B.sub.R]], E) fixed, and assume the following conditions hold:

(2.1) [mathematical expression not reproducible]

(2.2) for any map T [member of] D(E, E) there exists x [member of] E with [PHI](x) [intersection] T(x) [not equal to] 0

(2.3) [mathematical expression not reproducible]

and

(2.4) [mathematical expression not reproducible].

Then there exists x [member of] [B.sub.r,R] with [PHI](x) [intersection] F(x) [not equal to] 0.

Proof. Let [r.sub.0]: [bar.[B.sub.r]] [right arrow] [S.sub.r] be a continuous retraction (see [2]) and let

[mathematical expression not reproducible].

Note g : E [right arrow] [bar.[B.sub.r] is a continuous map. Now since F [member of] A([bar.[B.sub.r]], E) there exists a selection 'F [member of] D([bar.[B.sub.r]], E) of F and from (2.1), (2.2) there exists an x [member of] E with [PHI](x) [intersection] [PSI](g(x)) = 0. If x [member of] [B.sub.r] then [PHI](x) [intersection] [PSI]([r.sub.0](x)) [not equal to] 0 and this contradicts (2.3) (note [r.sub.0](x) [member of] [S.sub.r] since x [member of] [B.sub.r]). If [parallel]x[parallel] > R then [PHI](x) [intersection] [PSI]([R.sub.x/[parallel]x[parallel]]) [not equal to] 0 so if g = [R.sub.x/[parallel]x[parallel]] (note [parallel]g[parallel] = R) then [PHI] ([parallel]x[parallel]/R y) [intersection] [PSI](g) [not equal to] 0, and this contradicts (2.4). Thus x [member of] [B.sub.r,R] and [PHI](x) [intersection] [PSI](x) [not equal to] 0. Now since [PSI](x) [subset or equal to] F(x) we have [PHI](x) [intersection] F(x) [not equal to] 0.

Remark 2.7. In the proof of Theorem 2.6 notice (2.3) can be replaced by

(2.5) [PHI](x) [intersection] F(y) = 0 for x [member of] [B.sub.r] and y [member of] [S.sub.r],

and (2.4) can be replaced by

(2.6) [PHI]([lambda]y) [intersection] F(y) = 0 for y [member of] [S.sub.R] and [lambda] > 1.

Of course one could replace (2.1), (2.2), (2.3) and (2.4) in Theorem 2.6 with more abstract formulations. For example we could replace (2.1) with

for any selection [LAMBDA] [member of] D([bar.[B.sub.R]], E) of F the map [LAMBDA]g [member of] D(E, E),

and we could replace (2.2) with

[mathematical expression not reproducible],

and we could replace (2.3) with

[mathematical expression not reproducible].

Remark 2.8. Let E = (E, [parallel] x [parallel]) be a normed linear space and C [subset or equal to] E a cone (i.e. C is a closed, convex, invariant under multiplication by non-negative real numbers, and C [intersection] (-C) = {0}). For [rho] > 0 let

[B.sub.[rho]] = {x [member of] C : [parallel]x[parallel] < [rho]}, [[bar.B].sub.[rho]] = {x [member of] C : [parallel]x[parallel] [less than or equal to] [rho]},

[S.sub.[rho]] = {x [member of] C : [parallel]x[parallel] = [rho]}, and E[B.sub.[rho]] = {x [member of] C : [parallel]x[parallel] [greater than or equal to] [rho]}.

Let r, R be constants with 0 < r < R. Let F [member of] A([bar.[B.sub.R]], C), [PHI] [member of] B([bar.[B.sub.R]], C) fixed, and assume the following conditions hold:

[mathematical expression not reproducible]

for any map T [member of] D(C, C) there exists x [member of] C with [PHI](x) [intersection] T(x) [not equal to] 0

[mathematical expression not reproducible]

and

[mathematical expression not reproducible].

Then there exists x [member of] [B.sub.r,R] = {x [member of] C : r [less than or equal to] [parallel]x[parallel] [less than or equal to] R} with [PHI](x) [intersection] F(x) [not equal to] 0. The proof is similar to that in Theorem 2.6 once one notes that there exists a continuous retraction [r.sub.1] : [bar.[B.sub.r]] [right arrow] [S.sub.r] (see [7]).

One can easily generalize Theorem 2.6 to open convex sets.

Theorem 2.9. Let E = (E, [parallel] x [parallel]) be an infinite dimensional normed linear space, and [U.sub.1] and [U.sub.2] are open convex subsets of E until 0 [member of] [U.sub.1] and [bar.[U.sub.1]] [subset] [U.sub.2]. Let F [member of] A([bar.[U.sub.2]], E), [PHI] [member of] B([bar.[U.sub.2]], E) fixed, and assume (2.2) and the following conditions hold:

(2.7) [mathematical expression not reproducible]

(2.8) [mathematical expression not reproducible]

and

(2.9) [mathematical expression not reproducible].

Then there exists x [member of] [bar.[U.sub.2]]\[[U.sub.1]] with [PHI](x) [intersection] F(x) [not equal to] 0.

Proof. It is easy to see [1] that there exists a continuous retraction [r.sub.2]: [bar.[U.sub.1]] [right arrow] [partial derivative][U.sub.1].

Let

[mathematical expression not reproducible]

where [mu] is the Minkowski functional on [[bar.U].sub.2]. Note g : E [right arrow] [[bar.U].sub.2] is a continuous map. Now since F [member of] A([[bar.U].sub.2], E) there exists a selection T [member of] D([bar.[U.sub.2]], E) of F and (2.2), (2.7) guarantees that there exists an x [member of] E with [PHI](x) [intersection] [PSI](y(x)) [not equal to] 0. If x [member of] [U.sub.1] then [PHI](x) [intersection] [PSI]([r.sub.2](x)) [not equal to] 0 and this contradicts (2.8) (note [r.sub.2](x) [member of] [partial derivative][U.sub.1]). If x [member of] E\[bar.[U.sub.2]] then [PHI](x) [intersection] [PSI] (x/[mu](x)) [not equal to] 0 so if y = x/[mu](x) (note [mu](y) = 1 so y [member of] [partial derivative][U.sub.2]) then [PHI] ([mu](x)y) [intersection] [PSI](y) 0, and this contradicts (2.9) (note x [member of] E\[bar.[U.sub.2]] so [mu](x) > 1). Thus x [member of] [bar.[U.sub.2]]\[U.sub.1] and [PHI](x) [intersection] [PSI](x) [not equal to] 0, so [PHI](x) [intersection] F(x) [not equal to] 0.

Remark 2.10. In the proof of Theorem 2.9 notice (2.8) can be replaced by

(2.10) [PHI](x) [intersection] F(y) = 0 for x [member of] [U.sub.1] and y [member of] [partial derivative][U.sub.1],

and (2.9) can be replaced by

(2.11) [PHI]([lambda]y) [intersection] F(y) = 0 for y [member of] [partial derivative][U.sub.2] and [lambda] > 1.

Theorem 2.11. Let E = (E, [parallel] x [parallel]) be an infinite dimensional normed linear space, and r, R constants with 0 < r < R. Let F [member of] M([bar.[B.sub.R]], E), [PHI] [member of] B([bar.[B.sub.R]], E) fixed, and assume the following conditions hold:

(2.12) for any continuous map [eta] : E [right arrow] [bar.[B.sub.R]] the map F [eta] [member of] M(E, E) and

(2.13) for any map T [member of] M(E, E) there exists x [member of] E with T(x) [intersection] T(x) [not equal to] 0.

Finally assume (2.5) and (2.6) hold. Then there exists x [member of] [B.sub.r,R] with [PHI](x) [intersection] F(x) [not equal to] 0.

Proof. Let [r.sub.0] and g be as in Theorem 2.6. Now F [member of] M([bar.[B.sub.R]], E) so (2.12), (2.13) guarantee that there exists x [member of] E with [PHI](x) [intersection] F(g(x)) = 0. As in Theorem 2.6 we obtain x [member of] [B.sub.r,R].

Remark 2.12. There is an obvious analogue of Remark 2.8 and Theorem 2.9 with A replaced by M. Also in Theorem 2.11 one could replace (2.12) with the assumption that Fg [member of] M(E, E).

We now present Corollaries of Theorem 2.6 and Theorem 2.11 when [PHI] = I (the identity map) (there are also analogues of Remark 2.8 and Theorem 2.9 with A replaced by M).

Corollary 2.13. Let E = (E, [parallel] x [parallel]) be an infinite dimensional normed linear space, and r, R constants with 0 < r < R. Let F [member of] A([bar.[B.sub.R]], E) and suppose (2.1) holds. In addition assume the following conditions hold:

(2.14) for any map T [member of] D(E, E) there exists x [member of] E with x [member of] T(x)

(2.15) [mathematical expression not reproducible]

and

[mathematical expression not reproducible].

Then there exists x [member of] [B.sub.r,R] with x [member of] F(x).

Remark 2.14. Note (2.15) can be replaced by

(2.17) x [not member of] F(y) for x [member of] [B.sub.r] and y [member of] [S.sub.r],

and (2.16) can be replaced by

(2.18) y [not member of] [mu]F(y) for y [member of] [S.sub.R] and [mu] [member of] (0,1).

Proof. The result follows from Theorem 2.6 with [PHI] = I. We note that if for any selection [psi] [member of] D([bar.[B.sub.R], E) of F and y [member of] [S.sub.R], [lambda] > 1 we had [lambda]y [member of] [PSI](g) then y [member of] [mu][PSI](g) with [mu] = 1/[lambda] [member of]. (0, 1), and this contradicts (2.16).

Corollary 2.15. Let E = (E, [parallel] x [parallel]) be an infinite dimensional normed linear space, and r, R constants with 0 < r < R. Let F [member of] M([bar.[B.sub.R]], E) and suppose (2.12), (2.17) and (2.18) hold. In addition assume

(2.19) for any map T [member of] M(E, E) there exists x [member of] E with x [member of] T(x).

Then there exists x [member of] [B.sub.r,R] with x [member of] F(x).

Now we consider a special case of Corollary 2.13. We first recall the PK maps from the literature. Let Z and W be subsets of Hausdorff topological vector spaces [Y.sub.1] and [Y.sub.2] and F a multifunction. We say F [member of] PK(Z, W) if W is convex and there exists a map S : Z [right arrow] W with Z = [union]{int [S.sup.-1](w) : w [member of] W}, co(S(x)) [subset or equal to] F(x) for x [member of] Z and S(x) [not equal to] 0 for each x [member of] Z; here [S.sup.-1](w) = {z : w [member of] S(z)}.

Corollary 2.16. Let E = (E, [parallel] x [parallel]) be an infinite dimensional normed linear space, and r, R constants with 0 < r < R. Let F [member of] PK(BR, E) be a compact map and assume (2.17) and (2.18) hold. Then there exists x [member of] [B.sub.r,R] with x [member of] F(x).

Proof. In this case we let D = D and A = A. We say Q [member of] D([bar.[B.sub.R]], E) if Q : [bar.[B.sub.R]] [right arrow] E is a continuous compact map. We say G [member of] A([bar.[B.sub.R]], E) if G [member of] PK([bar.[B.sub.R]], E) and G is a compact map (the existence of a continuous selection [PSI] of G is guaranteed from [Theorem 1.3, 6] and note [PSI] is compact since T is a selection of G and G is compact). Note (2.1) and (2.14) (Schauder's fixed point theorem) hold. The result follows from Corollary 2.13 (and Remark 2.14).

Next we consider a special case of Corollary 2.15. We first recall the [U.sup.[kappa].sub.c] maps from the literature. Suppose X and Y are Hausdorff topological spaces. Given a class X of maps, X(X, Y) denotes the set of maps F : X [right arrow] [2.sup.Y] belonging to X, and [X.sub.c] the set of finite compositions of maps in X. We let

F(X) = {Z : Fix F [not equal to] 0 for all F [where Fix F denotes the set member of] X(Z, Z)},

where Fix F denotes the set of fixed points of F.

The class U of maps is defined by the following properties:

(i). U contains the class C of single valued continuous functions;

(ii). each F [member of] [U.sub.c] is upper semicontinuous and compact valued; and

(iii). [B.sup.n] [member of] F([U.sub.c]) for all n [member of] {1, 2, ...}; here [B.sup.n] = {x [member of] [R.sub.n]: [parallel]x[parallel] < 1}.

We say F [member of] [U.sup.[kappa].sub.c] (X, Y) if for any compact subset K of X there is a G [member of] [U.sub.c] (K, Y) with G(x) [subset or equal to] F(x) for each x [member of] K. Examples of [U.sup.k.sub.c] (X, Y) maps are the Kakutani maps, the acyclic maps, the O'Neill maps, the maps admissible in the sense of Gorniewicz and the permissible maps; see [4]. Recall U is closed under compositions [8].

Corollary 2.17. Let E = (E, [parallel] x [parallel]) be an infinite dimensional normed linear space, and r, R constants with 0 < r < R. Let F [member of] [U.sup.[kappa].sub.c] ([bar.[B.sub.R]], E) be a compact map and assume (2.17) and (2.18) hold. Then there exists x [member of] [B.sub.r,R] with x [member of] F(x).

Proof. In this case we let M = A and say F [member of] M([bar.[B.sub.R]], E) if F [member of] [U.sup.[kappa].sub.c]{ [bar.[B.sub.R]], E) is a compact map. Note (2.12) is immediate since U is closed under compositions and F[eta] : E [right arrow] [2.sup.E] is compact (here [eta] : E [right arrow] [B.sup.R] is a continuous map). Finally we note that (2.19) holds (see [8, 9]). The result follows from Corollary 2.15.

We now show that the ideas in this section can be applied to other natural situations. Let E be a 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 single valued map; here dom L is a vector subspace of E. Finally T : E [right arrow] Y will be a linear single valued map with L + T : dom L [right arrow] Y a bijection; for convenience we say T [member of] [H.sub.L](E, Y).

Definition 2.18. We say F [member of] D([bar.U], Y; L, T) (respectively F [member of] B([bar.U], Y; L, T)) if F : [bar.U] [right arrow] [2.sup.Y] and [(L + T).sup.-1] (F + T) [member of] D([bar.U], E) (respectively [(L + T).sup.-1](F + T) [member of] B([bar.U], E)).

Definition 2.19. We say F [member of] A([bar.U], Y; L, T) if F : [bar.U] [right arrow] [2.sup.Y] and [(L + T).sup.-1] (F + T) [member of] A([bar.U], E) and there exists a selection [PSI] [member of] D([bar.U], Y; L, T) of F.

Definition 2.20. We say F [member of] D(E, Y; L, T) if F : E [right arrow] [2.sup.Y] and [(L + T).sup.-1] (F + T) [member of] D(E, E).

Remark 2.21. One could also define the class M([bar.U], Y; L, T) (i.e. F [member of] M([bar.U], Y; L, T) if F : [bar.U] [right arrow] [2.sup.Y] and [(L + T).sup.-1] (F + T) [member of] A([bar.U], E)) and M(E, Y; L, T).

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

We obtain an analogue of Theorem 2.6 in this setting (it is also easy to obtain an analogue of Theorem 2.11 using the class M in this setting).

Theorem 2.22. Let E = (E, [parallel] x [parallel]) be an infinite dimensional normed linear space, Y a topological vector space, 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 r, R constants with 0 < r < R. Let F [member of] A([bar.[B.sub.R]], Y; L, T), [PHI] [member of] B([bar.[B.sub.R]], Y; L,T) fixed, and assume the following conditions hold:

(2.20) [mathematical expression not reproducible]

(2.21) [mathematical expression not reproducible]

(2.22) [mathematical expression not reproducible]

and

(2.23) [mathematical expression not reproducible].

Then there exists x [member of] [B.sub.r,R] with [(L + T).sup.-1] ([phi] + T)(x) [intersection] [(L + T).sup.-1](F + T)(x) [not equal to] 0.

Proof. Let [r.sub.0] and g be as in Theorem 2.6. Now since F [member of] A([bar.[B.sub.R]], Y; L, T) there exists a selection T [member of] D([bar.[B.sub.R]], Y; L, T) of F and from (2.20), (2.21) there exists x [member of] E with

[(L + T).sup.-1] ([PHI] + T) (x) [intersection] [(L + T).sup.-1]([PSI] [omicron] g + T)(x) [not equal to] 0.

If x [member of] [B.sub.r] then

[(L + T).sup.-1]([PHI] + T)(x) [intersection] [(L + T).sup.-1]([PSI] [omicron] [r.sub.0] + T)(x) [not equal to] 0,

and this contradicts (2.22). If [parallel]x[parallel] > R then

[(L + T).sup.-1]([PHI] + T)(x) [intersection] [(L + T).sup.-1] ([PSI](R/[parallel]x[parallel] x) + T(x)) [not equal to] 0,

so if y = Rx/[parallel]x[parallel] then

[mathematical expression not reproducible].

and this contradicts (2.23). Thus x [member of] [B.sub.r,R] with [(L + T).sup.- 1](T + T)(x) [intersection] [(L + T).sup.-1]([PSI] + T)(x) [not equal to] 0.

Remark 2.23. There are analogues of Remark 2.7, Remark 2.8, Theorem 2.9 and Theorem 2.11 in this setting (we leave the obvious statements to the reader).

Finally in this paper we discuss briefly a different strategy in Theorem 2.11 (a similar strategy can be applied in Theorem 2.6 and Theorem 2.22). Let E = (E, [parallel] * [parallel]) be an infinite dimensional normed linear space, and r, R constants with 0 < r < R. Let F [member of] M([bar.[B.sub.R]], E) and [PHI] [member of] B([bar.[B.sub.R]], E) fixed. Assume

(2.24) F([S.sub.r]) [subset or equal to] [bar.[B.sub.r]] and F([S.sub.R]) [subset or equal to] E[B.sub.R].

Let [r.sub.0] and g be as in Theorem 2.6. Note if x [member of] [bar.[B.sub.r]] then F(g(x)) = F([r.sub.0](x)) [subset or equal to] F([S.sub.r]) [subset or equal to] [bar.[B.sub.r]] and for x [member of] [EB.sub.R] then F(g(x)) = F (R/[parallel]x[parallel] x) [subset or equal to] F([S.sub.R]) [subset or equal to] E[B.sub.R]. Thus [mathematical expression not reproducible]. Let [OMEGA] = [bar.[B.sub.r]] [union] E[B.sub.R] and note Fg : [OMEGA] [right arrow] [2.sup.[OMEGA]]. Next assume

(2.25) [mathematical expression not reproducible].

If (2.25) is true then automatically T(x) [intersection] F(x) [not equal to] 0 for the x in (2.25).

Note [OMEGA] is an ANR and [OMEGA] is the disjoint union of two contractible components [bar.[B.sub.r]] and E[B.sub.R]. Condition (2.25) arises naturally in applications. For example if [PHI] = I (the identity map) and one has an index theory for the M maps (with appropriate properties) then one can deduce immediately the existence of a fixed point of Fg in [B.sub.r,R] (of course one needs the usual Bowszyc theorem [5] for the class M). If M denotes the maps admissible in the sense of Gorniewicz [4] or the permissible maps [3, 4] then one can deduce immediately (see [5] or [section 57, 4]) that there exists x [member of] [B.sub.r,R] with x [member of] Fg(x) i.e. (2.25) holds (thus the strategy to establish (2.25) here is to obtain an analogue of a theorem of C. Bowszyc for the class of maps considered).

REFERENCES

[1] R. P. Agarwal and D. O'Regan, A note on the topological transversality theorem for acyclic maps, Applied Math. Letters 18 (2005), 17-22.

[2] Y. Benyamini and Y. Sternfeld, Spheres in infinite dimensional normed spaces and Lipschitz contractibility, Proc. Amer. Math. Soc. 88 (1983), 439-445.

[3] Z. Dzedzej, Fixed point index for a class of nonacyclic multivalued maps, Dissert. Math. 253 (1985), 1-55.

[4] L. Gorniewicz, Topological fixed point theory of multivalued mappings, Kluwer Acad. Publishers, Dordrecht, 1999.

[5] M. Izydorek and Z. Kucharski, The Krasnoselskii theorem for permissible multivalued maps, Bull. Polish Acad. Sci. Math. 37 (1989), 145-149.

[6] D. O'Regan, Fixed point theorems for the BK-admissible maps of Park, Applicable Analysis 79 (2001), 173-185.

[7] D. O'Regan, A Krasnoselskii cone compressiom theorem for UK maps, Math. Proc. Royal Irish Acad. 103A (2003), 55-99.

[8] D. O'Regan, Fixed point theory on extension-type spaces and essential maps on topological spaces, Fixed Point Theory Appl. 2004 (2004), 13-20.

[9] S. Park, A unified fixed point theory of multimaps on topological vector spaces, J. Korean Math. Soc. 35 (1998), 803-829.

DONAL O'REGAN

School of Mathematics, Statistics and Applied Mathematics, National University

of Ireland, Galway Ireland

donal.oregan@nuigalway.ie