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Coincidence theory for multimaps
Applied Mathematics and Computation 219 (2012) 2026–2034
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Coincidence theory for multimaps Donal O’Regan School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland
a r t i c l e
i n f o
Keywords: Continuation methods Essential maps Continua of fixed points Coincidences
a b s t r a c t Several continuation principles in a variety of settings are presented which guarantee the existence of coincidence points for a general class of multimaps. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Essential maps for single-valued maps were introduced by Granas [2] and extended by Precup [8]. These notions were considered by O’Regan in [3,4] and O’Regan and Precup in [6]. In this paper we present homotopy principles for essential type multimaps. Moreover we present a unified theory for establishing the existence of fixed points and coincidence points for a general class of maps. In addition we present very general results for homotopies H for which the maps Hk may be defined on different domains. Also in this paper we discuss continua of fixed points and coincidence points. Let X and Y be Hausdorff topological spaces. Given a class X of maps, XðX; YÞ denotes the set of maps F : X ! 2Y , (nonempty subsets of Y) belonging to X, and Xc the set of finite compositions of maps in X. We let
FðXÞ ¼ fZ : Fix F – ; for all F 2 XðZ; ZÞg; 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 2 Uc is upper semicontinuous and compact valued; and (iii) Bn 2 FðUc Þ for all n 2 f1; 2; . . .g; here Bn ¼ fx 2 Rn : kxk 6 1g. We say F 2 Ukc ðX; YÞ if for any compact subset K of X there is a G 2 Uc ðK; YÞ with GðxÞ # FðxÞ for each x 2 K. Recall Ukc is closed under compositions. The class Ukc contains almost all the well-known maps in the literature (see [7] and the references therein). It is also possible to consider more general maps (see [5,7]) and in this paper we will consider a class of maps which we will call A. 2. Essential maps Let E be a completely regular topological space and U an open subset of E. We will consider a class A of maps. In some results the following condition will be assumed:
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.08.044
D. O’Regan / Applied Mathematics and Computation 219 (2012) 2026–2034
8 > < for Hausdorff topological spaces X 1 ; X 2 and X 3 ; if F 2 AðX 1 ; X 3 Þ and f 2 CðX 2 ; X 1 Þ; > : then F f 2 AðX 2 ; X 3 Þ:
2027
ð2:1Þ
Definition 2.1. We say F 2 AðU; EÞ if F 2 AðU; EÞ and F : U ! KðEÞ is an upper semicontinuous map; here U denotes the closure of U in E and KðEÞ denotes the family of nonempty compact subsets of E. Definition 2.2. We say F 2 A@U ðU; EÞ if F 2 AðU; EÞ with x R FðxÞ for x 2 @U; here @U denotes the boundary of U in E. Definition 2.3. Let F; G 2 A@U ðU; EÞ. We say F ffi G in A@U ðU; EÞ if there exists a map W : U ½0; 1 ! KðEÞ with W 2 AðU ½0; 1; EÞ, x R Wt ðxÞ for any x 2 @U and t 2 ½0; 1, W1 ¼ F, W0 ¼ G (here Wt ðxÞ ¼ Wðx; tÞ) and x 2 U : x 2 Wðx; tÞ for some t 2 ½0; 1 is relatively compact. Remark 2.1. We note (see the proof in Theorem 2.1) if U : U ½0; 1 ! KðEÞ is an upper semicontinuous map then M ¼ x 2 U : x 2 Uðx; tÞ for some t 2 ½0; 1 is closed so that if M is relatively compact then M is compact. If U : U ½0; 1 ! KðEÞ is an upper semicontinuous compact map then
x 2 U : x 2 Uðx; tÞ for some t 2 ½0; 1 is compact. Remark 2.2. The results below (with (2.1) removed) also hold true if we use the following definition of ffi. Let F; G 2 A@U ðU; EÞ. We say F ffi G in A@U ðU; EÞ if there exists an upper semicontinuous map W : U ½0; 1 ! KðEÞ with Wð; gðÞÞ 2 AðU; EÞ for any continuous function g : U ! ½0; 1 with gð@UÞ ¼ 0; x R Wt ðxÞ for any x 2 @U and t 2 ½0; 1; W1 ¼ F; W0 ¼ G and x 2 U : x 2 Wðx; tÞ for some t 2 ½0; 1 is relatively compact. The following condition will be assumed:
ffi is an equivalence relation in A@U ðU; EÞ:
ð2:2Þ
Definition 2.4. Let F 2 A@U ðU; EÞ. We say F : U ! KðEÞ is essential in A@U ðU; EÞ if for every map J 2 A@U ðU; EÞ with Jj@U ¼ Fj@U , and J ffi F in A@U ðU; EÞ there exists x 2 U with x 2 JðxÞ. Otherwise F is inessential in A@U ðU; EÞ i.e. there exists a fixed point free map J 2 A@U ðU; EÞ with Jj@U ¼ Fj@U and J ffi F in A@U ðU; EÞ. Theorem 2.1. Let E be a completely regular topological space, U an open subset of E and assume (2.1) and (2.2) hold. Suppose F 2 A@U ðU; EÞ. Then the following are equivalent: (i) F is inessential in A@U ðU; EÞ; (ii) there exists a fixed point free map G 2 A@U ðU; EÞ with G ffi F in A@U ðU; EÞ. Proof. (i) implies (ii) is immediate. Next we prove (ii) implies (i). Suppose there exists a fixed point free map G 2 A@U ðU; EÞ with G ffi F in A@U ðU; EÞ. Let H : U ½0; 1 ! KðEÞ be a map with H 2 AðU ½0; 1; EÞ; x R Ht ðxÞ for any x 2 @U and t 2 ½0; 1; H0 ¼ F, H1 ¼ G (here Ht ðxÞ ¼ Hðx; tÞ) and x 2 U : x 2 Hðx; tÞ for some t 2 ½0; 1 is relatively compact. Consider
D ¼ x 2 U : x 2 Hðx; tÞ for some t 2 ½0; 1 : If D ¼ ; then in particular x R Hðx; 0Þ ¼ FðxÞ for x 2 U so F is inessential in A@U ðU; EÞ. Next suppose D – ;. Note D is closed in E. To see this let ðxa Þ be a net in D with xa ! x 2 U. Now there exists ta 2 ½0; 1 with xa 2 Hðxa ; ta Þ. Without loss of generality assume t a ! t 2 ½0; 1. Thus ðxa ; t a Þ ! ðx; tÞ and the fact that H : U ½0; 1 ! KðEÞ is upper semicontinuous guarantees that x 2 Hðx; tÞ so D is closed. Note then that D, is a compact subset of E. Also since x R Ht ðxÞ for x 2 @U and t 2 ½0; 1 then D \ @U ¼ ;. Thus (note E is a completely regular topological space) there exists a continuous map l : U ! ½0; 1 with lð@UÞ ¼ 0, and lðDÞ ¼ 1. Define a map Rl : U ! KðEÞ by Rl ðxÞ ¼ Hðx; lðxÞÞ ¼ HlðxÞ ðxÞ ¼ H sðxÞ; here s : U ! U ½0; 1 is given by sðxÞ ¼ ðx; lðxÞÞ. Notice Rl 2 AðU; EÞ (note (2.1) and H 2 AðU ½0; 1; EÞ) and notice Rl j@U ¼ H0 j@U ¼ Fj@U since lð@UÞ ¼ 0. Thus Rl 2 A@U ðU; EÞ (note x R Ht ðxÞ for any x 2 @U and t 2 ½0; 1) with Rl j@U ¼ Fj@U . We now claim
Rl ffi F
in A@U ðU; EÞ:
ð2:3Þ
Let Q : U ½0; 1 ! KðEÞ be given by Q ðx; tÞ ¼ Hðx; t lðxÞÞ ¼ H gðx; tÞ where g : U ½0; 1 ! U ½0; 1 is given by gðx; tÞ ¼ ðx; t lðxÞÞ. Note Q 2 AðU ½0; 1; EÞ (note (2.1) and H 2 AðU ½0; 1; EÞ), Q 0 ¼ F and Q 1 ¼ Rl . Also x R Q t ðxÞ for x 2 @U and t 2 ½0; 1 since if there exists t 2 ½0; 1 and x 2 @U with x 2 Q t ðxÞ then x 2 Hðx; t lðxÞÞ so x 2 D and as a result lðxÞ ¼ 1 i.e. x 2 Hðx; tÞ, a contradiction. Finally note
x 2 U : x 2 Q ðx; tÞ ¼ Hðx; t lðxÞÞ for some t 2 ½0; 1 is closed and compact. Thus (2.3) holds. Consequently F is inessential in A@U ðU; EÞ (take J ¼ Rl in the definition of inessential). h
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D. O’Regan / Applied Mathematics and Computation 219 (2012) 2026–2034
Remark 2.3. From the proof above (with a minor modification in two places) we see that the result in Theorem 2.1 (with (2.1) removed) holds if the definition of ffi is as in Remark 2.2. Now Theorem 2.1 immediately yields the following continuation theorem. Theorem 2.2. Let E be a completely regular topological space, U an open subset of E and assume (2.1) and (2.2) hold. Suppose F and G are two maps in A@U ðU; EÞ with F ffi G in A@U ðU; EÞ. Then F is essential in A@U ðU; EÞ, if and only if G is essential in A@U ðU; EÞ. Proof. F is inessential in A@U ðU; EÞ iff there exists a fixed point free map U 2 A@U ðU; EÞ with F ffi U in A@U ðU; EÞ iff (since (2.2) holds) there exists a fixed point free map U 2 A@U ðU; EÞ with G ffi U in A@U ðU; EÞ iff F is inessential in A@U ðU; EÞ. h Remark 2.4. The result in Theorem 2.2 (with (2.1) removed) holds if the definition of ffi is as in Remark 2.2. Remark 2.5. If E is a normal topological space then the assumption that
x 2 U : x 2 Wðx; tÞ for some t 2 ½0; 1 is relatively compact can be removed in Definition 2.3 (and Remark 2.2) and we still obtain Theorem 2.1 and Theorem 2.2. In many applications fixed point results are needed for homotopies H for which the maps Ht may be defined on different domains. The idea is to reduce the study of this family to that of a new family (of course depending on the old one) defined on the same domain. For notational purposes let Z be a topological space and X a subset of Z ½0; 1. We write Xk ¼ fx 2 Z : ðx; kÞ 2 Xg to denote the section of X at k. Let E be a completely regular topological space and let U be an open subset of E ½0; 1. For our next result we assume (2.1) holds and in addition
ffi is an equivalence relation in A@U ðU; E ½0; 1Þ
ð2:4Þ
8 for Hausdorff topological spaces X 1 and X 2 ; if F 2 AðX 1 ; X 2 Þ > > > < and if Wðy; lÞ ¼ ðFðyÞ; lÞ for ðy; lÞ 2 X ½0; 1; then 1 > Wl 2 AðX 1 ; X 2 ½0; 1Þ for each l 2 ½0; 1 and W 2 AðX 1 ½0; 1; X 2 ½0; 1Þ; > > : here Wl ðxÞ ¼ Wðx; lÞ:
ð2:5Þ
and
Theorem 2.3. Suppose N 2 AðU; EÞ with
x R Nðx; kÞ for ðx; kÞ 2 @U:
ð2:6Þ
Let H : U ½0; 1 ! KðE ½0; 1Þ be given by Hðx; k; lÞ ¼ ðNðx; kÞ; lÞ for ðx; kÞ 2 U and l 2 ½0; 1. In addition assume the following conditions hold:
(
H0 is essential in A@U ðU; E ½0; 1Þ; here;
ð2:7Þ
H0 ðx; kÞ ¼ Hðx; k; 0Þ ¼ ðNðx; kÞ; 0Þ for ðx; kÞ 2 U and
ðx; kÞ 2 U : ðx; kÞ 2 Hðx; k; lÞ for some
l 2 ½0; 1 is relatively compact:
Then H1 is essential in A@U ðU; E ½0; 1Þ H1 ðx; kÞ ¼ Hðx; k; 1Þ ¼ ðNðx; kÞ; 1Þ for ðx; kÞ 2 U.
so
in
particular
there
ð2:8Þ exists
a
x 2 U1
with
x 2 Nðx; 1Þ;
here
Proof. First note
ðx; kÞ R Hðx; k; lÞ for ðx; kÞ 2 @U
and
l 2 ½0; 1:
ð2:9Þ
To see this suppose there exists ðx0 ; k0 Þ 2 @U and l0 2 ½0; 1 with ðx0 ; k0 Þ 2 ðNðx0 ; k0 Þ; l0 Þ. Then l0 ¼ k0 and x0 2 Nðx0 ; k0 Þ which contradicts (2.6). Also note (2.5) guarantees that H0 2 A@U ðU; E ½0; 1Þ and H1 2 A@U ðU; E ½0; 1Þ. We will now apply Theorem 2.2. Now (2.5), (2.8) and (2.9) guarantee that
H0 ffi H1
in A@U ðU; E ½0; 1Þ
ð2:10Þ
(note H 2 AðU ½0; 1; E ½0; 1Þ from (2.5) and N 2 AðU; EÞ). Theorem 2.2 guarantees that H1 is essential in A@U ðU; E ½0; 1Þ. Thus there exists a ðx; kÞ 2 U with ðx; kÞ 2 ðNðx; kÞ; 1Þ i.e. x 2 Nðx; kÞ with k ¼ 1 i.e. x 2 U 1 ¼ fy 2 E : ðy; 1Þ 2 Ug and x 2 Nðx; 1Þ. h Remark 2.6. From the proof above it is clear that Ht is essential in A@U ðU; E ½0; 1Þ for every t 2 ½0; 1; here Ht ðx; kÞ ¼ Hðx; k; tÞ ¼ ðNðx; kÞ; tÞ for ðx; kÞ 2 U. If E is a normal topological space then the assumption (2.8) can be removed in the statement of Theorem 2.3.
D. O’Regan / Applied Mathematics and Computation 219 (2012) 2026–2034
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Remark 2.7. The result in Theorem 2.3 holds (with (2.1) and (2.5) removed) if the definition of ffi is as in Remark 2.2 and if the following condition holds: Hð; ; gð; ÞÞ 2 AðU; E ½0; 1Þ for any continuous function g : U ! ½0; 1 with gð@UÞ ¼ 0. We note that (2.2) or (2.4) could be strong assumptions. However one can obtain applicable results even if (2.2) or (2.4) are not assumed. To establish this we will consider new essential maps (a subset of those essential maps in Definition 2.4). Let E be a completely regular topological space and U an open subset of E. Definition 2.5. Let F 2 A@U ðU; EÞ. We say F is essential in A@U ðU; EÞ if for every map J 2 A@U ðU; EÞ with Jj@U ¼ Fj@U there exists x 2 U with x 2 JðxÞ. Theorem 2.4. Let E be a completely regular topological space, U an open subset of E and assume (2.1) holds. Suppose G 2 A@U ðU; EÞ; H : U ½0; 1 ! KðEÞ with H 2 AðU ½0; 1; EÞ and assume the following hold:
Hðx; 0Þ ¼ GðxÞ for x 2 U;
ð2:11Þ
G is essential in A@U ðU; EÞ ðDefinition 2:5Þ
ð2:12Þ
and
x R Hðx; tÞ for x 2 @U
and t 2 ð0; 1:
ð2:13Þ
In addition suppose x 2 U : x 2 Hðx; tÞ for some t 2 ½0; 1 is relatively compact. Let FðxÞ ¼ Hðx; 1Þ for x 2 U. Then F has a fixed point in U. Remark 2.8. (1) From the proof below we see that we can remove (2.1) and remove H 2 AðU ½0; 1; EÞ in the statement of Theorem 2.4 if we assume H : U ½0; 1 ! KðEÞ is an upper semicontinuous map with Hð; gðÞÞ 2 AðU; EÞ for any continuous function g : U ! ½0; 1 with gð@UÞ ¼ 0. (2) Suppose we have the following definition of AðU; EÞ. We say F 2 AðU; EÞ if F : U ! 2E and F 2 AðU; EÞ. With this definition Theorem 2.4 is again true provided
x 2 U : x 2 Hðx; tÞ for some t 2 ½0; 1 is relatively compact in the statement of Theorem 2.4 is replaced by
x 2 U : x 2 Hðx; tÞ for some t 2 ½0; 1 is compact and we note if E is a normal topological space then we just need to assume
x 2 U : x 2 Hðx; tÞ for some t 2 ½0; 1 is closed in the statement of Theorem 2.4. Proof. Consider
D ¼ x 2 U : x 2 Hðx; tÞ for some t 2 ½0; 1 : Notice D – ; since G is essential in A@U ðU; EÞ (i.e. in particular there exists x 2 U with x 2 GðxÞ ¼ Hðx; 0Þ). Note D is a compact subset of E. Next notice that (2.13), with G 2 A@U ðU; EÞ, guarantees that D \ @U ¼ ;. Thus there exists a continuous map l : U ! ½0; 1 with lð@UÞ ¼ 0 and lðDÞ ¼ 1. Define a map Rl : U ! KðEÞ by Rl ðxÞ ¼ Hðx; lðxÞÞ ¼ HlðxÞ ðxÞ ¼ H sðxÞ; here s : U ! U ½0; 1 is given by sðxÞ ¼ ðx; lðxÞÞ. Notice (as in Theorem 2.1) that Rl 2 AðU; EÞ and Rl j@U ¼ Gj@U since lð@UÞ ¼ 0. Thus Rl 2 A@U ðU; EÞ with Rl j@U ¼ Gj@U and since G is essential in A@U ðU; EÞ there exists x 2 U with x 2 Rl ðxÞ (i.e. x 2 HlðxÞ ðxÞ). Thus x 2 D and so lðxÞ ¼ 1. Consequently x 2 Hðx; 1Þ ¼ FðxÞ. h Remark 2.9. We now note that Theorem 2.4 holds if we use the definition of essential (see (2.12)) in Definition 2.4 instead of in Definition 2.5. Now as in the proof above we have Rl 2 A@U ðU; EÞ with Rl j@U ¼ Gj@U . We now show Rl ffi G in A@U ðU; EÞ. To see this let Q : U ½0; 1 ! KðEÞ be given by Q ðx; tÞ ¼ Hðx; t lðxÞÞ ¼ H gðx; tÞ where g : U ½0; 1 ! U ½0; 1 is given by gðx; tÞ ¼ ðx; t lðxÞÞ. Note Q 2 AðU ½0; 1; EÞ; Q 1 ¼ Rl and Q 0 ¼ G. Also x R Q t ðxÞ for x 2 @U and t 2 ½0; 1 since if there exists t 2 ½0; 1 and x 2 @U with x 2 Q t ðxÞ then x 2 Hðx; t lðxÞÞ so x 2 D and as a result lðxÞ ¼ 1 i.e. x 2 Hðx; tÞ, a contradiction. Finally note
x 2 U : x 2 Q ðx; tÞ ¼ Hðx; t lðxÞÞ for some t 2 ½0; 1 is closed (note H is upper semicontinuous) and compact. Thus Rl ffi G in A@U ðU; EÞ and since G is essential in A@U ðU; EÞ (as in Definition 2.4) we can conclude now as in the proof of Theorem 2.4. The result also holds if the definition of ffi is as in Remark 2.2 (if we use the assumption in Remark 2.8).
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Remark 2.10. If E is a normal topological space then the assumption that
x 2 U : x 2 Hðx; tÞ for some t 2 ½0; 1 is relatively compact can be removed in Theorem 2.4. Let E be a completely regular topological space and let U be an open subset of E ½0; 1. For our next result we assume (2.1) and (2.5) hold. Theorem 2.5. Suppose N 2 AðU; EÞ and (2.6) holds. Let H : U ½0; 1 ! KðE ½0; 1Þ be given by Hðx; k; lÞ ¼ ðNðx; kÞ; lÞ for ðx; kÞ 2 U and l 2 ½0; 1. In addition assume (2.7) (Definition 2.5) and (2.8) hold. Then there exists a x 2 U 1 with x 2 Nðx; 1Þ. Proof. As in Theorem 2.3 note (2.9) holds and H 2 AðU ½0; 1; E ½0; 1Þ from (2.5). Now apply Theorem 2.4. h Remark 2.11. If E is a normal topological space then the assumption (2.8) can be removed in the statement of Theorem 2.5. Remark 2.12. The result in Theorem 2.5 holds (with (2.1) and (2.5) removed) if the following condition holds: Hð; ; gð; ÞÞ 2 AðU; E ½0; 1Þ for any continuous function g : U ! ½0; 1 with gð@UÞ ¼ 0. Recall a compact connected set is called a continuum. For our next result we will use Whyburn’s lemma from topology which we state here for convenience. Theorem 2.6. Let A and B be disjoint closed subsets of a compact Hausdorff topological space K such that no connected component of K intersects both A and B. Then there exists a partition K ¼ K 1 [ K 2 where K 1 and K 2 are disjoint compact sets containing A and B respectively. For our next result E will be a completely regular topological space and U an open subset of E ½0; 1. We will also assume (2.1), (2.5), (2.8) and the following condition:
8 > < for Hausdorff topological spaces X 1 and X 2 ; if F 2 AðX 1 ; X 2 Þ; v 2 CðX 1 ; ½0; 1Þ and if UðyÞ ¼ ðFðyÞ; v ðyÞÞ for y 2 X 1 ; > : then U 2 AðX 1 ; X 2 ½0; 1Þ:
ð2:14Þ
Theorem 2.7. Suppose N 2 AðU; EÞ and (2.6) holds. Let H : U ½0; 1 ! KðE ½0; 1Þ be given by Hðx; k; lÞ ¼ ðNðx; kÞ; lÞ for ðx; kÞ 2 U and l 2 ½0; 1. In addition assume (2.7) (Definition 2.5) and the following condition holds:
X ¼ fðx; kÞ 2 U : x 2 Nðx; kÞg is compact:
ð2:15Þ
Then X contains a continuum intersecting X0 f0g and X1 f1g; here Xt ¼ fx 2 E : ðx; tÞ 2 Xg for each t 2 ½0; 1. Proof. Note A ¼ X0 f0g # X and B ¼ X1 f1g # X are closed (and compact). If there is no continuum intersecting A and B then from Theorem 2.6, X can be represented as X ¼ XH [ XHH where XH and XHH are disjoint compact sets with A # XH and B # XHH . Notice XH and XHH [ @U are closed and disjoint (note XH \ @U ¼ ; since if there exists a ðx; kÞ 2 @U and ðx; kÞ 2 XH then (note ðx; kÞ 2 XH # X) x 2 Nðx; kÞ which contradicts (2.6)). Now there exists a continuous map l : U ! ½0; 1 with lðXHH [ @UÞ ¼ 0 and lðXH Þ ¼ 1. Let
Tðx; kÞ ¼ ðNðx; kÞ; lðx; kÞÞ for ðx; kÞ 2 U: Notice (2.14) guarantees that T 2 AðU; E ½0; 1Þ and in fact T 2 A@U ðU; E ½0; 1Þ since if there exists a ðx; kÞ 2 @U with ðx; kÞ 2 Tðx; kÞ then ðx; kÞ 2 ðNðx; kÞ; lðx; kÞÞ ¼ ðNðx; kÞ; 0Þ so x 2 Nðx; 0Þ which contradicts (2.6). Notice as well (here H0 ðx; kÞ ¼ Hðx; k; 0Þ ¼ ðNðx; kÞ; 0Þ) that
Tj@U ¼ H0 j@U since if ðx; kÞ 2 @U then Tðx; kÞ ¼ ðNðx; kÞ; lðx; kÞÞ ¼ ðNðx; kÞ; 0Þ (note lðXHH [ @UÞ ¼ 0). Now (2.7) guarantees (see Definition 2.5) that there exists a ðx; kÞ 2 U with ðx; kÞ 2 Tðx; kÞ i.e. x 2 Nðx; kÞ and k ¼ lðx; kÞ. Note ðx; kÞ 2 X since ðx; kÞ 2 U and x 2 Nðx; kÞ. Now either ðx; kÞ 2 XH or ðx; kÞ 2 XHH . Case 1. Suppose ðx; kÞ 2 XH . Then lðx; kÞ ¼ 1. Thus k ¼ lðx; kÞ ¼ 1 and x 2 Nðx; kÞ ¼ Nðx; 1Þ i.e. ðx; 1Þ 2 B # XHH which contradicts ðx; 1Þ ¼ ðx; kÞ 2 XH . Case 2. Suppose ðx; kÞ 2 XHH . Then lðx; kÞ ¼ 0. Thus k ¼ lðx; kÞ ¼ 0 and x 2 Nðx; kÞ ¼ Nðx; 0Þ i.e. ðx; 0Þ 2 A # XH which contradicts ðx; 0Þ ¼ ðx; kÞ 2 XHH . h We now show that the ideas in this section can be applied to other natural situations. The coincidence theory presented here is a generalization of results presented in [1,6,9] (in our results we assume the usual Poincaré, Leray–Schauder condition (see (2.21) or (2.28))). First let E be a completely regular topological vector space, Y a topological vector space, and U an open subset of E. Also let L : dom L # E ! Y be a linear (not necessarily continuous) single-valued map; here dom L is a vector subspace of E. Finally T : E ! Y will be a linear, continuous single-valued map with L þ T : dom L ! Y an isomorphism (i.e. a linear homeomorphism); for convenience we say T 2 HL ðE; YÞ.
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Definition 2.6. Let F : U ! 2Y . We say F 2 AðU; Y; L; TÞ if ðL þ TÞ1 ðF þ TÞ 2 AðU; EÞ. Definition 2.7. We say F 2 A@U ðU; Y; L; TÞ if F 2 AðU; Y; L; TÞ with L x R FðxÞ for x 2 @U \ dom L. Definition 2.8. Let F; G 2 A@U ðU; Y; L; TÞ. We say F ffi G in A@U ðU; Y; L; TÞ if there exists a map W : U ½0; 1 ! 2Y with W 2 AðU ½0; 1; Y; L; TÞ; L x R Wt ðxÞ for any x 2 @U \ dom L and t 2 ½0; 1; W1 ¼ F, W0 ¼ G (here Wt ðxÞ ¼ Wðx; tÞ) and
x 2 U \ dom L : L x 2 Wðx; tÞ for some t 2 ½0; 1 is relatively compact. For our next two results we assume the following condition holds:
8 > < if X 2 ¼ U or X 2 ¼ U ½0; 1 and if; F 2 AðU ½0; 1; Y; L; TÞ and f 2 CðX 2 ; U ½0; 1Þ; > : then F f 2 AðX 2 ; Y; L; TÞ:
ð2:16Þ
Remark 2.13. The results below (with (2.16) removed) also hold true if we use the following definition of ffi. Let F; G 2 A@U ðU; Y; L; TÞ. We say F ffi G in A@U ðU; Y; L; TÞ if there exists a map W : U ½0; 1 ! 2Y with ðL þ TÞ1 ðW þ TÞ : U ½0; 1 ! KðEÞ upper semicontinuous and with ðL þ TÞ1 ðWð; gðÞÞ þ TÞ 2 AðU; EÞ for any continuous function g : U ! ½0; 1 with gð@UÞ ¼ 0; L x R Wt ðxÞ for any x 2 @U \ dom L and t 2 ½0; 1; W1 ¼ F, W0 ¼ G and
x 2 U \ dom L : L x 2 Wðx; tÞ for some t 2 ½0; 1 is relatively compact. The following condition will be assumed:
ffi is an equivalence relation in A@U ðU; Y; L; TÞ:
ð2:17Þ
Definition 2.9. Let F 2 A@U ðU; Y; L; TÞ. We say F is L–essential in A@U ðU; Y; L; TÞ if for every map J 2 A@U ðU; Y; L; TÞ with Jj@U ¼ Fj@U and J ffi F in A@U ðU; Y; L; TÞ there exists x 2 U \ dom L with L x 2 JðxÞ. Otherwise F is L–inessential in A@U ðU; Y; L; TÞ i.e. there exists a map J 2 A@U ðU; Y; L; TÞ with Jj@U ¼ Fj@U and J ffi F in A@U ðU; Y; L; TÞ such that L x R JðxÞ for x 2 U \ dom L. Theorem 2.8. Let E be a completely regular topological vector space, Y a topological vector space, U an open subset of E; L : dom L # E ! Y a linear single valued map, T 2 HL ðE; YÞ, and assume (2.16) and (2.17) hold. Suppose F 2 A@U ðU; Y; L; TÞ. Then the following are equivalent: (i) F is L–inessential in A@U ðU; Y; L; TÞ; (ii) there exists a map G 2 A@U ðU; Y; L; TÞ with G ffi F in A@U ðU; Y; L; TÞ such that L x R GðxÞ for x 2 U \ dom L.
Proof. (i) implies (ii) is immediate. Next we prove (ii) implies (i). Suppose there exists a map G 2 A@U ðU; Y; L; TÞ with G ffi F in A@U ðU; Y; L; TÞ such that L x R GðxÞ for x 2 U \ dom L. Let H : U ½0; 1 ! 2Y be a map with H 2 AðU ½0; 1; Y; L; TÞ; L x R Ht ðxÞ for any x 2 @U \ dom L and t 2 ½0; 1; H0 ¼ F; H1 ¼ G (here Ht ðxÞ ¼ Hðx; tÞ) and
x 2 U \ dom L : L x 2 Hðx; tÞ for some t 2 ½0; 1 is relatively compact. Consider
n o D ¼ x 2 U : x 2 ðL þ TÞ1 ðH þ TÞðx; tÞ for some t 2 ½0; 1 : Notice that it is immediate that
D ¼ x 2 U \ dom L : L x 2 Hðx; tÞ for some t 2 ½0; 1 : If D ¼ ; then in particular L x R Hðx; 0Þ for x 2 U \ dom L so F is L–inessential in A@U ðU; Y; L; TÞ. Next suppose D – ;. Note D is a compact subset of E. Also since L x R Ht ðxÞ for x 2 @U \ dom L and t 2 ½0; 1 then D \ @U ¼ ;. Thus there exists a continuous map l : U ! ½0; 1 with lð@UÞ ¼ 0 and lðDÞ ¼ 1. Define a map Rl : U ! 2Y by Rl ðxÞ ¼ Hðx; lðxÞÞ ¼ HlðxÞ ðxÞ ¼ H sðxÞ; here s : U ! U ½0; 1 is given by sðxÞ ¼ ðx; lðxÞÞ. Notice Rl 2 AðU; Y; L; TÞ (note (2.16) and H 2 AðU ½0; 1; Y; L; TÞ) and notice Rl j@U ¼ H0 j@U ¼ Fj@U since lð@UÞ ¼ 0. Thus Rl 2 A@U ðU; Y; L; TÞ (note L x R Ht ðxÞ for any x 2 @U \ dom L and t 2 ½0; 1) with Rl j@U ¼ Fj@U . We now claim
Rl ffi F
in A@U ðU; Y; L; TÞ: Y
ð2:18Þ
Let Q : U ½0; 1 ! 2 be given by Q ðx; tÞ ¼ Hðx; t lðxÞÞ ¼ H gðx; tÞ where g : U ½0; 1 ! U ½0; 1 is given by gðx; tÞ ¼ ðx; t lðxÞÞ. Note Q 2 AðU ½0; 1; Y; L; TÞ (note (2.16) and H 2 AðU ½0; 1; Y; L; TÞ), Q 0 ¼ F and Q 1 ¼ Rl . Also
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L x R Q t ðxÞ for x 2 @U \ dom L and t 2 ½0; 1 since if there exists t 2 ½0; 1 and x 2 @U \ dom L with L x 2 Q t ðxÞ then L x 2 Hðx; t lðxÞÞ so x 2 D and as a result lðxÞ ¼ 1 i.e. L x 2 Hðx; tÞ, a contradiction. Finally note
x 2 U \ dom L : L x 2 Q ðx; tÞ ¼ Hðx; t lðxÞÞ for some t 2 ½0; 1 is compact. Thus (2.18) holds. Consequently F is L–inessential in A@U ðU; Y; L; TÞ. h Remark 2.14. From the proof above we see that the result in Theorem 2.8 (with (2.16) removed) holds if the definition of ffi is as in Remark 2.13. Essentially the same reasoning as in Theorem 2.2 establishes the following result. Theorem 2.9. Let E be a completely regular topological vector space, Y a topological vector space, U an open subset of E; L : dom L # E ! Y a linear single valued map, T 2 HL ðE; YÞ, and assume (2.16) and (2.17) hold. Suppose U and W are two maps in A@U ðU; Y; L; TÞ with U ffi W in A@U ðU; Y; L; TÞ. Then U is L–essential in A@U ðU; Y; L; TÞ if and only if WH is L–essential in A@U ðU; Y; L; TÞ. Remark 2.15. If E is a normal topological vector space then the assumption that
x 2 U \ dom L : L x 2 Wðx; tÞ for some t 2 ½0; 1 is relatively compact can be removed in Definition 2.8 (and Remark 2.13) and we still obtain Theorems 2.8 and 2.9. Let E be a completely regular topological vector space, Y a topological vector space, and U an open subset of E ½0; 1. Also let L : dom L # E ! Y be a linear (not necessarily continuous) single-valued map; here dom L is a vector subspace of E. Now let L : dom L ¼ dom L ½0; 1 ! Y ½0; 1 be given by Lðy; kÞ ¼ ðL y; kÞ. Let T : E ! Y be a linear, continuous single-valued map with L þ T : dom L ! Y an isomorphism (i.e. a linear homeomorphism) and let T : E ½0; 1 ! Y ½0; 1 be given by T ðy; kÞ ¼ ðT y; 0Þ. Notice ðL þ T Þ1 ðy; kÞ ¼ ððL þ TÞ1 y; kÞ for ðy; kÞ 2 Y ½0; 1. For our next result we assume (2.16) (with Y replaced by Y ½0; 1; L replaced by L and T replaced by T ) holds and in addition that
ffi is an equivalence relation in A@U ðU; Y ½0; 1; L; T Þ
ð2:19Þ
8 > < if F 2 AðU; Y; L; TÞ and if Wðy; lÞ ¼ ðFðyÞ; lÞ; ðy; lÞ 2 U ½0; 1; then Wl 2 AðU; Y ½0; 1; L; T Þ for each l 2 ½0; 1 and > : W 2 AðU ½0; 1; Y ½0; 1; L; T Þ; here Wl ðxÞ ¼ Wðx; lÞ:
ð2:20Þ
and
Theorem 2.10. Suppose N 2 AðU; Y; L; TÞ with
L x R Nðx; kÞ for ðx; kÞ 2 @U \ dom L: Y½0;1
Let H : U ½0; 1 ! 2 ditions hold:
(
ð2:21Þ
be given by Hðx; k; lÞ ¼ ðNðx; kÞ; lÞ for ðx; kÞ 2 U and
l 2 ½0; 1. In addition assume the following con-
H0 is essential in A@U ðU; Y ½0; 1; L; T Þ; here
ð2:22Þ
H0 ðx; kÞ ¼ Hðx; k; 0Þ ¼ ðNðx; kÞ; 0Þ for ðx; kÞ 2 U and
ðx; kÞ 2 U \ dom L : Lðx; kÞ 2 Hðx; k; lÞ for some
l 2 ½0; 1 is relatively compact:
ð2:23Þ
Then H1 is essential in A@U ðU; Y ½0; 1; L; T Þ so in particular there exists a x 2 U 1 \ dom L with L x 2 Nðx; 1Þ; here H1 ðx; kÞ ¼ Hðx; k; 1Þ ¼ ðNðx; kÞ; 1Þ for ðx; kÞ 2 U. Proof. First note
Lðx; kÞ R Hðx; k; lÞ for ðx; kÞ 2 @U \ dom L and
l 2 ½0; 1:
ð2:24Þ
To see this suppose there exists ðx0 ; k0 Þ 2 @U \ dom L and l0 2 ½0; 1 with Lðx0 ; k0 Þ 2 ðNðx0 ; k0 Þ; l0 Þ. Then l0 ¼ k0 and L x0 2 Nðx0 ; k0 Þ which contradicts (2.21). Also note (2.20) guarantees that H0 2 A@U ðU; Y ½0; 1; L; T Þ and H1 2 A@U ðU; Y ½0; 1; L; T Þ. We will now apply Theorem 2.9. Now (2.20), (2.23) and (2.24) guarantee that
H0 ffi H1
in A@U ðU; Y ½0; 1; L; T Þ
ð2:25Þ
(note H 2 AðU ½0; 1; Y ½0; 1; L; T Þ from (2.20) and N 2 AðU; Y; L; TÞ). Theorem 2.9 guarantees that H1 is essential in A@U ðU; Y ½0; 1; L; T Þ. Thus there exists a ðx; kÞ 2 U \ dom L with Lðx; kÞ 2 ðNðx; kÞ; 1Þ i.e. L x 2 Nðx; kÞ with k ¼ 1 i.e. x 2 dom L with x 2 U 1 ¼ fy 2 E : ðy; 1Þ 2 Ug and L x 2 Nðx; 1Þ. h
D. O’Regan / Applied Mathematics and Computation 219 (2012) 2026–2034
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Remark 2.16. From the proof above it is clear that Ht is essential in A@U ðU; Y ½0; 1; L; T Þ for every t 2 ½0; 1; here Ht ðx; kÞ ¼ Hðx; k; tÞ ¼ ðNðx; kÞ; tÞ for ðx; kÞ 2 U. If E is a normal topological vector space then the assumption (2.23) can be removed in the statement of Theorem 2.10. Remark 2.17. The result in Theorem 2.10 holds (with (2.16) and (2.20) removed) if the definition of ffi is as in Remark 2.13 and if the following condition holds: ðL þ T Þ1 ðHð; ; gð; ÞÞ þ T Þ 2 AðU; E ½0; 1Þ for any continuous function g : U ! ½0; 1 with gð@UÞ ¼ 0. We note that (2.17) or (2.19) could be strong assumptions. However we can establish some results even if (2.17) or (2.19) are not assumed. To establish this we will consider new essential maps (a subset of those essential maps in Definition 2.9). Let E be a completely regular topological vector space, Y a topological vector space, and U an open subset of E. Also let L and T be as described before Definition 2.6. Definition 2.10. Let F 2 A@U ðU; Y; L; TÞ. We say F is essential in A@U ðU; Y; L; TÞ if for every map J 2 A@U ðU; Y; L; TÞ with Jj@U ¼ Fj@U there exists x 2 U \ dom L with L x 2 JðxÞ. Theorem 2.11. Assume (2.16) holds. Suppose G 2 A@U ðU; Y; L; TÞ; H : U ½0; 1 ! 2Y with H 2 AðU ½0; 1; Y; L; TÞ and assume the following hold:
Hðx; 0Þ ¼ GðxÞ for x 2 U;
ð2:26Þ
G is essential in A@U ðU; Y; L; TÞ ðDefinition 2:10Þ
ð2:27Þ
L x R Hðx; tÞ for x 2 @U \ dom L and t 2 ð0; 1:
ð2:28Þ
and In addition suppose x 2 U \ dom L : L x 2 Hðx; tÞ for some t 2 ½0; 1 is relatively compact. Let FðxÞ ¼ Hðx; 1Þ for x 2 U. Then there exists x 2 U \ domL with L x 2 FðxÞ. Remark 2.18. From the proof below we see that we can remove (2.16) and remove H 2 AðU ½0; 1; Y; L; TÞ in the statement of Theorem 2.11 if we assume ðL þ TÞ1 ðH þ TÞ : U ½0; 1 ! KðEÞ is an upper semicontinuous map with ðL þ TÞ1 ðHð; gðÞÞ þ TÞ 2 AðU; EÞ for any continuous function g : U ! ½0; 1 with gð@UÞ ¼ 0. Proof. Consider
D ¼ x 2 U \ dom L : L x 2 Hðx; tÞ for some t 2 ½0; 1 : Notice (2.27) guarantees that D – ;. Note D is a compact subset of E. Now (2.28) (and (2.27)) guarantees that D \ @U ¼ ;. Thus there exists a continuous map l : U ! ½0; 1 with lð@UÞ ¼ 0 and lðDÞ ¼ 1. Define a map Rl : U ! 2Y by Rl ðxÞ ¼ Hðx; lðxÞÞ ¼ HlðxÞ ðxÞ ¼ H sðxÞ; here s : U ! U ½0; 1 is given by sðxÞ ¼ ðx; lðxÞÞ. Now Rl 2 AðU; Y; L; TÞ and Rl j@U ¼ Gj@U since lð@UÞ ¼ 0. Note also Rl 2 A@U ðU; Y; L; TÞ since L x R Ht ðxÞ for x 2 @U \ dom L and t 2 ½0; 1. Now since G is essential in A@U ðU; Y; L; TÞ there exists x 2 U \ dom L with L x 2 Rl ðxÞ (i.e. L x 2 HlðxÞ ðxÞ). Thus x 2 D and so lðxÞ ¼ 1. Consequently L x 2 Hðx; 1Þ ¼ FðxÞ. h Remark 2.19. We now note that Theorem 2.11 holds if we use the definition of essential (see (2.27)) in Definition 2.9 instead of in Definition 2.10. We follow the proof above so we have Rl 2 A@U ðU; Y; ;L; TÞ with Rl j@U ¼ Gj@U . We now show Rl ffi G in A@U ðU; Y; L; TÞ. To see this let Q : U ½0; 1 ! 2Y be given by Q ðx; tÞ ¼ Hðx; t lðxÞÞ ¼ H gðx; tÞ where g : U ½0; 1 ! U ½0; 1 is given by gðx; tÞ ¼ ðx; t lðxÞÞ. Note Q 2 AðU ½0; 1; Y; L; TÞ; Q 1 ¼ Rl ; Q 0 ¼ G and L x R Q t ðxÞ for x 2 @U \ dom L and t 2 ½0; 1 (if there exists t 2 ½0; 1 and x 2 @U \ dom L with L x 2 Q t ðxÞ then L x 2 Hðx; t lðxÞÞ so x 2 D and as a result lðxÞ ¼ 1 i.e. L x 2 Hðx; tÞ, a contradiction). Finally note
x 2 U \ dom L : L x 2 Q ðx; tÞ ¼ Hðx; t lðxÞÞ for some t 2 ½0; 1 is compact. Thus Rl ffi G in A@U ðU; Y; L; TÞ and since G is essential in A@U ðU; Y; L; TÞ (as in Definition 2.9) we can conclude now as in the proof of Theorem 2.11. The result also holds if the definition of ffi is as in Remark 2.13 (if we use the assumption in Remark 2.18). Remark 2.20. If E is a normal topological space then the assumption that
x 2 U \ dom L : L x 2 Hðx; tÞ for some t 2 ½0; 1 is relatively compact can be removed in Theorem 2.11. Let E be a completely regular topological vector space, Y a topological vector space, and U an open subset of E ½0; 1. Also let L; L; T and T be as described before Theorem 2.10. For our next result we assume (2.16) (with Y replaced by Y ½0; 1, L replaced by L and T replaced by T ) and (2.20) hold.
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D. O’Regan / Applied Mathematics and Computation 219 (2012) 2026–2034
Theorem 2.12. Suppose N 2 AðU; Y; L; TÞ and (2.21) holds. Let H : U ½0; 1 ! 2Y½0;1 be given by Hðx; k; lÞ ¼ ðNðx; kÞ; lÞ for ðx; kÞ 2 U and l 2 ½0; 1. In addition assume (2.22) (Definition 2.10) and (2.23) hold. Then there exists a x 2 U 1 \ dom L with L x 2 Nðx; 1Þ. Proof. As in Theorem 2.10 note (2.24) holds and H 2 AðU ½0; 1; Y ½0; 1; L; T Þ from (2.20). Now apply Theorem 2.11. h Remark 2.21. If E is a normal topological space then the assumption (2.23) can be removed in the statement of Theorem 2.12. Remark 2.22. The result in Theorem 2.12 holds (with (2.16) and (2.20) removed) if the following condition holds: ðL þ T Þ1 ðHð; ; gð; ÞÞ þ T Þ 2 AðU; E ½0; 1Þ for any continuous function g : U ! ½0; 1 with gð@UÞ ¼ 0. In our next result E will be a completely regular topological vector space, Y a topological vector space, and U an open subset of E ½0; 1. Also let L; L; T and T be as described before Theorem 2.10. We will also assume (2.16), (2.20), (2.23) and the following condition:
8 > < if F 2 AðU; Y; L; TÞ; v 2 CðU; ½0; 1Þ and if UðyÞ ¼ ðFðyÞ; v ðyÞÞ for y 2 U; > : then U 2 AðU; Y ½0; 1; L; T Þ:
ð2:29Þ
Theorem 2.13. Suppose N 2 AðU; Y; L; TÞ and (2.21) holds. Let H : U ½0; 1 ! 2Y½0;1 be given by Hðx; k; lÞ ¼ ðNðx; kÞ; lÞ for ðx; kÞ 2 U and l 2 ½0; 1. In addition assume (2.22) (Definition 2.10) and the following condition holds:
X ¼ fðx; kÞ 2 U \ dom L : L x 2 Nðx; kÞg is compact:
ð2:30Þ
Then X contains a continuum intersecting X0 f0g and X1 f1g; here Xt ¼ fx 2 E : ðx; tÞ 2 Xg for each t 2 ½0; 1. Proof. Note A ¼ X0 f0g # X and B ¼ X1 f1g # X are closed (and compact). If there is no continuum intersecting A and B then from Theorem 2.6, X can be represented as X ¼ XH [ XHH where XH and XHH are disjoint compact sets with A # XH and B # XHH . Notice XH and XHH [ @U are closed and disjoint (note XH \ @U ¼ ; since if there exists a ðx; kÞ 2 @U and ðx; kÞ 2 XH then (note ðx; kÞ 2 XH # X) L x 2 Nðx; kÞ which contradicts (2.21)). Now there exists a continuous map l : U ! ½0; 1 with lðXHH [ @UÞ ¼ 0 and lðXH Þ ¼ 1. Let
Tðx; kÞ ¼ ðNðx; kÞ; lðx; kÞÞ for ðx; kÞ 2 U: Notice (2.29) guarantees that T 2 AðU; Y ½0; 1; L; T Þ and in fact T 2 A@U ðU; Y ½0; 1; L; T Þ since if there exists a ðx; kÞ 2 @U with Lðx; kÞ ¼ ðL x; kÞ 2 Tðx; kÞ then ðL x; kÞ 2 ðNðx; kÞ; lðx; kÞÞ ¼ ðNðx; kÞ; 0Þ so L x 2 Nðx; 0Þ which contradicts (2.21). Notice as well (here H0 ðx; kÞ ¼ Hðx; k; 0Þ ¼ ðNðx; kÞ; 0Þ) that
Tj@U ¼ H0 j@U ; since if ðx; kÞ 2 @U then Tðx; kÞ ¼ ðNðx; kÞ; lðx; kÞÞ ¼ ðNðx; kÞ; 0Þ (note lðXHH [ @UÞ ¼ 0). Now (2.22) guarantees (see Definition 2.10) that there exists a ðx; kÞ 2 U \ dom L with Lðx; kÞ 2 Tðx; kÞ i.e. L x 2 Nðx; kÞ and k ¼ lðx; kÞ. Note ðx; kÞ 2 X since ðx; kÞ 2 U \ dom L and L x 2 Nðx; kÞ. Now either ðx; kÞ 2 XH or ðx; kÞ 2 XHH . Case 1. Suppose ðx; kÞ 2 XH . Then lðx; kÞ ¼ 1. Thus k ¼ lðx; kÞ ¼ 1 and L x 2 Nðx; kÞ ¼ Nðx; 1Þ i.e. ðx; 1Þ 2 B # XHH which contradicts ðx; 1Þ ¼ ðx; kÞ 2 XH . Case 2. Suppose ðx; kÞ 2 XHH . Then lðx; kÞ ¼ 0. Thus k ¼ lðx; kÞ ¼ 0 and L x 2 Nðx; kÞ ¼ Nðx; 0Þ i.e. ðx; 0Þ 2 A # XH which contradicts ðx; 0Þ ¼ ðx; kÞ 2 XHH . h References [1] J. Bryszewski, L. Górniewicz, A Poincaré type coincidence theorem for multi-valued maps, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976) 593–598. [2] A. Granas, Sur la méthode de continuité de Poincaré, C.R. Acad. Sci. Paris 282 (1976) 983–985. [3] D. O’Regan, Continuation principles and d-essential maps, Math. Comput. Model. 30 (1999) 1–6. [4] D. O’Regan, Topological principles for extension type maps, Dynam. Syst. Appl. 10 (2011) 541–550. [5] D. O’Regan, J. Peran, Fixed points for better admissible multifunctions on proximity spaces, J. Math. Anal. Appl. 380 (2011) 882–887. [6] D. O’Regan, R. Precup, Theorems of Leray–Schauder Type and Applications, Taylor and Francis Publishers, London, 2002. [7] S. Park, Fixed point theorems for better admissible multimaps on almost convex spaces, J. Math. Anal. Appl. 329 (2007) 690–702. [8] R. Precup, On the topological transversality principle, Nonlinear Anal. 20 (1993) 1–9. [9] S. Reich, A Poincaré type coincidence theorem, Amer. Math. Monthly 81 (1974) 52–53.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Coincidence theory for multimaps Donal O’Regan School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland
a r t i c l e
i n f o
Keywords: Continuation methods Essential maps Continua of fixed points Coincidences
a b s t r a c t Several continuation principles in a variety of settings are presented which guarantee the existence of coincidence points for a general class of multimaps. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Essential maps for single-valued maps were introduced by Granas [2] and extended by Precup [8]. These notions were considered by O’Regan in [3,4] and O’Regan and Precup in [6]. In this paper we present homotopy principles for essential type multimaps. Moreover we present a unified theory for establishing the existence of fixed points and coincidence points for a general class of maps. In addition we present very general results for homotopies H for which the maps Hk may be defined on different domains. Also in this paper we discuss continua of fixed points and coincidence points. Let X and Y be Hausdorff topological spaces. Given a class X of maps, XðX; YÞ denotes the set of maps F : X ! 2Y , (nonempty subsets of Y) belonging to X, and Xc the set of finite compositions of maps in X. We let
FðXÞ ¼ fZ : Fix F – ; for all F 2 XðZ; ZÞg; 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 2 Uc is upper semicontinuous and compact valued; and (iii) Bn 2 FðUc Þ for all n 2 f1; 2; . . .g; here Bn ¼ fx 2 Rn : kxk 6 1g. We say F 2 Ukc ðX; YÞ if for any compact subset K of X there is a G 2 Uc ðK; YÞ with GðxÞ # FðxÞ for each x 2 K. Recall Ukc is closed under compositions. The class Ukc contains almost all the well-known maps in the literature (see [7] and the references therein). It is also possible to consider more general maps (see [5,7]) and in this paper we will consider a class of maps which we will call A. 2. Essential maps Let E be a completely regular topological space and U an open subset of E. We will consider a class A of maps. In some results the following condition will be assumed:
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.08.044
D. O’Regan / Applied Mathematics and Computation 219 (2012) 2026–2034
8 > < for Hausdorff topological spaces X 1 ; X 2 and X 3 ; if F 2 AðX 1 ; X 3 Þ and f 2 CðX 2 ; X 1 Þ; > : then F f 2 AðX 2 ; X 3 Þ:
2027
ð2:1Þ
Definition 2.1. We say F 2 AðU; EÞ if F 2 AðU; EÞ and F : U ! KðEÞ is an upper semicontinuous map; here U denotes the closure of U in E and KðEÞ denotes the family of nonempty compact subsets of E. Definition 2.2. We say F 2 A@U ðU; EÞ if F 2 AðU; EÞ with x R FðxÞ for x 2 @U; here @U denotes the boundary of U in E. Definition 2.3. Let F; G 2 A@U ðU; EÞ. We say F ffi G in A@U ðU; EÞ if there exists a map W : U ½0; 1 ! KðEÞ with W 2 AðU ½0; 1; EÞ, x R Wt ðxÞ for any x 2 @U and t 2 ½0; 1, W1 ¼ F, W0 ¼ G (here Wt ðxÞ ¼ Wðx; tÞ) and x 2 U : x 2 Wðx; tÞ for some t 2 ½0; 1 is relatively compact. Remark 2.1. We note (see the proof in Theorem 2.1) if U : U ½0; 1 ! KðEÞ is an upper semicontinuous map then M ¼ x 2 U : x 2 Uðx; tÞ for some t 2 ½0; 1 is closed so that if M is relatively compact then M is compact. If U : U ½0; 1 ! KðEÞ is an upper semicontinuous compact map then
x 2 U : x 2 Uðx; tÞ for some t 2 ½0; 1 is compact. Remark 2.2. The results below (with (2.1) removed) also hold true if we use the following definition of ffi. Let F; G 2 A@U ðU; EÞ. We say F ffi G in A@U ðU; EÞ if there exists an upper semicontinuous map W : U ½0; 1 ! KðEÞ with Wð; gðÞÞ 2 AðU; EÞ for any continuous function g : U ! ½0; 1 with gð@UÞ ¼ 0; x R Wt ðxÞ for any x 2 @U and t 2 ½0; 1; W1 ¼ F; W0 ¼ G and x 2 U : x 2 Wðx; tÞ for some t 2 ½0; 1 is relatively compact. The following condition will be assumed:
ffi is an equivalence relation in A@U ðU; EÞ:
ð2:2Þ
Definition 2.4. Let F 2 A@U ðU; EÞ. We say F : U ! KðEÞ is essential in A@U ðU; EÞ if for every map J 2 A@U ðU; EÞ with Jj@U ¼ Fj@U , and J ffi F in A@U ðU; EÞ there exists x 2 U with x 2 JðxÞ. Otherwise F is inessential in A@U ðU; EÞ i.e. there exists a fixed point free map J 2 A@U ðU; EÞ with Jj@U ¼ Fj@U and J ffi F in A@U ðU; EÞ. Theorem 2.1. Let E be a completely regular topological space, U an open subset of E and assume (2.1) and (2.2) hold. Suppose F 2 A@U ðU; EÞ. Then the following are equivalent: (i) F is inessential in A@U ðU; EÞ; (ii) there exists a fixed point free map G 2 A@U ðU; EÞ with G ffi F in A@U ðU; EÞ. Proof. (i) implies (ii) is immediate. Next we prove (ii) implies (i). Suppose there exists a fixed point free map G 2 A@U ðU; EÞ with G ffi F in A@U ðU; EÞ. Let H : U ½0; 1 ! KðEÞ be a map with H 2 AðU ½0; 1; EÞ; x R Ht ðxÞ for any x 2 @U and t 2 ½0; 1; H0 ¼ F, H1 ¼ G (here Ht ðxÞ ¼ Hðx; tÞ) and x 2 U : x 2 Hðx; tÞ for some t 2 ½0; 1 is relatively compact. Consider
D ¼ x 2 U : x 2 Hðx; tÞ for some t 2 ½0; 1 : If D ¼ ; then in particular x R Hðx; 0Þ ¼ FðxÞ for x 2 U so F is inessential in A@U ðU; EÞ. Next suppose D – ;. Note D is closed in E. To see this let ðxa Þ be a net in D with xa ! x 2 U. Now there exists ta 2 ½0; 1 with xa 2 Hðxa ; ta Þ. Without loss of generality assume t a ! t 2 ½0; 1. Thus ðxa ; t a Þ ! ðx; tÞ and the fact that H : U ½0; 1 ! KðEÞ is upper semicontinuous guarantees that x 2 Hðx; tÞ so D is closed. Note then that D, is a compact subset of E. Also since x R Ht ðxÞ for x 2 @U and t 2 ½0; 1 then D \ @U ¼ ;. Thus (note E is a completely regular topological space) there exists a continuous map l : U ! ½0; 1 with lð@UÞ ¼ 0, and lðDÞ ¼ 1. Define a map Rl : U ! KðEÞ by Rl ðxÞ ¼ Hðx; lðxÞÞ ¼ HlðxÞ ðxÞ ¼ H sðxÞ; here s : U ! U ½0; 1 is given by sðxÞ ¼ ðx; lðxÞÞ. Notice Rl 2 AðU; EÞ (note (2.1) and H 2 AðU ½0; 1; EÞ) and notice Rl j@U ¼ H0 j@U ¼ Fj@U since lð@UÞ ¼ 0. Thus Rl 2 A@U ðU; EÞ (note x R Ht ðxÞ for any x 2 @U and t 2 ½0; 1) with Rl j@U ¼ Fj@U . We now claim
Rl ffi F
in A@U ðU; EÞ:
ð2:3Þ
Let Q : U ½0; 1 ! KðEÞ be given by Q ðx; tÞ ¼ Hðx; t lðxÞÞ ¼ H gðx; tÞ where g : U ½0; 1 ! U ½0; 1 is given by gðx; tÞ ¼ ðx; t lðxÞÞ. Note Q 2 AðU ½0; 1; EÞ (note (2.1) and H 2 AðU ½0; 1; EÞ), Q 0 ¼ F and Q 1 ¼ Rl . Also x R Q t ðxÞ for x 2 @U and t 2 ½0; 1 since if there exists t 2 ½0; 1 and x 2 @U with x 2 Q t ðxÞ then x 2 Hðx; t lðxÞÞ so x 2 D and as a result lðxÞ ¼ 1 i.e. x 2 Hðx; tÞ, a contradiction. Finally note
x 2 U : x 2 Q ðx; tÞ ¼ Hðx; t lðxÞÞ for some t 2 ½0; 1 is closed and compact. Thus (2.3) holds. Consequently F is inessential in A@U ðU; EÞ (take J ¼ Rl in the definition of inessential). h
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Remark 2.3. From the proof above (with a minor modification in two places) we see that the result in Theorem 2.1 (with (2.1) removed) holds if the definition of ffi is as in Remark 2.2. Now Theorem 2.1 immediately yields the following continuation theorem. Theorem 2.2. Let E be a completely regular topological space, U an open subset of E and assume (2.1) and (2.2) hold. Suppose F and G are two maps in A@U ðU; EÞ with F ffi G in A@U ðU; EÞ. Then F is essential in A@U ðU; EÞ, if and only if G is essential in A@U ðU; EÞ. Proof. F is inessential in A@U ðU; EÞ iff there exists a fixed point free map U 2 A@U ðU; EÞ with F ffi U in A@U ðU; EÞ iff (since (2.2) holds) there exists a fixed point free map U 2 A@U ðU; EÞ with G ffi U in A@U ðU; EÞ iff F is inessential in A@U ðU; EÞ. h Remark 2.4. The result in Theorem 2.2 (with (2.1) removed) holds if the definition of ffi is as in Remark 2.2. Remark 2.5. If E is a normal topological space then the assumption that
x 2 U : x 2 Wðx; tÞ for some t 2 ½0; 1 is relatively compact can be removed in Definition 2.3 (and Remark 2.2) and we still obtain Theorem 2.1 and Theorem 2.2. In many applications fixed point results are needed for homotopies H for which the maps Ht may be defined on different domains. The idea is to reduce the study of this family to that of a new family (of course depending on the old one) defined on the same domain. For notational purposes let Z be a topological space and X a subset of Z ½0; 1. We write Xk ¼ fx 2 Z : ðx; kÞ 2 Xg to denote the section of X at k. Let E be a completely regular topological space and let U be an open subset of E ½0; 1. For our next result we assume (2.1) holds and in addition
ffi is an equivalence relation in A@U ðU; E ½0; 1Þ
ð2:4Þ
8 for Hausdorff topological spaces X 1 and X 2 ; if F 2 AðX 1 ; X 2 Þ > > > < and if Wðy; lÞ ¼ ðFðyÞ; lÞ for ðy; lÞ 2 X ½0; 1; then 1 > Wl 2 AðX 1 ; X 2 ½0; 1Þ for each l 2 ½0; 1 and W 2 AðX 1 ½0; 1; X 2 ½0; 1Þ; > > : here Wl ðxÞ ¼ Wðx; lÞ:
ð2:5Þ
and
Theorem 2.3. Suppose N 2 AðU; EÞ with
x R Nðx; kÞ for ðx; kÞ 2 @U:
ð2:6Þ
Let H : U ½0; 1 ! KðE ½0; 1Þ be given by Hðx; k; lÞ ¼ ðNðx; kÞ; lÞ for ðx; kÞ 2 U and l 2 ½0; 1. In addition assume the following conditions hold:
(
H0 is essential in A@U ðU; E ½0; 1Þ; here;
ð2:7Þ
H0 ðx; kÞ ¼ Hðx; k; 0Þ ¼ ðNðx; kÞ; 0Þ for ðx; kÞ 2 U and
ðx; kÞ 2 U : ðx; kÞ 2 Hðx; k; lÞ for some
l 2 ½0; 1 is relatively compact:
Then H1 is essential in A@U ðU; E ½0; 1Þ H1 ðx; kÞ ¼ Hðx; k; 1Þ ¼ ðNðx; kÞ; 1Þ for ðx; kÞ 2 U.
so
in
particular
there
ð2:8Þ exists
a
x 2 U1
with
x 2 Nðx; 1Þ;
here
Proof. First note
ðx; kÞ R Hðx; k; lÞ for ðx; kÞ 2 @U
and
l 2 ½0; 1:
ð2:9Þ
To see this suppose there exists ðx0 ; k0 Þ 2 @U and l0 2 ½0; 1 with ðx0 ; k0 Þ 2 ðNðx0 ; k0 Þ; l0 Þ. Then l0 ¼ k0 and x0 2 Nðx0 ; k0 Þ which contradicts (2.6). Also note (2.5) guarantees that H0 2 A@U ðU; E ½0; 1Þ and H1 2 A@U ðU; E ½0; 1Þ. We will now apply Theorem 2.2. Now (2.5), (2.8) and (2.9) guarantee that
H0 ffi H1
in A@U ðU; E ½0; 1Þ
ð2:10Þ
(note H 2 AðU ½0; 1; E ½0; 1Þ from (2.5) and N 2 AðU; EÞ). Theorem 2.2 guarantees that H1 is essential in A@U ðU; E ½0; 1Þ. Thus there exists a ðx; kÞ 2 U with ðx; kÞ 2 ðNðx; kÞ; 1Þ i.e. x 2 Nðx; kÞ with k ¼ 1 i.e. x 2 U 1 ¼ fy 2 E : ðy; 1Þ 2 Ug and x 2 Nðx; 1Þ. h Remark 2.6. From the proof above it is clear that Ht is essential in A@U ðU; E ½0; 1Þ for every t 2 ½0; 1; here Ht ðx; kÞ ¼ Hðx; k; tÞ ¼ ðNðx; kÞ; tÞ for ðx; kÞ 2 U. If E is a normal topological space then the assumption (2.8) can be removed in the statement of Theorem 2.3.
D. O’Regan / Applied Mathematics and Computation 219 (2012) 2026–2034
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Remark 2.7. The result in Theorem 2.3 holds (with (2.1) and (2.5) removed) if the definition of ffi is as in Remark 2.2 and if the following condition holds: Hð; ; gð; ÞÞ 2 AðU; E ½0; 1Þ for any continuous function g : U ! ½0; 1 with gð@UÞ ¼ 0. We note that (2.2) or (2.4) could be strong assumptions. However one can obtain applicable results even if (2.2) or (2.4) are not assumed. To establish this we will consider new essential maps (a subset of those essential maps in Definition 2.4). Let E be a completely regular topological space and U an open subset of E. Definition 2.5. Let F 2 A@U ðU; EÞ. We say F is essential in A@U ðU; EÞ if for every map J 2 A@U ðU; EÞ with Jj@U ¼ Fj@U there exists x 2 U with x 2 JðxÞ. Theorem 2.4. Let E be a completely regular topological space, U an open subset of E and assume (2.1) holds. Suppose G 2 A@U ðU; EÞ; H : U ½0; 1 ! KðEÞ with H 2 AðU ½0; 1; EÞ and assume the following hold:
Hðx; 0Þ ¼ GðxÞ for x 2 U;
ð2:11Þ
G is essential in A@U ðU; EÞ ðDefinition 2:5Þ
ð2:12Þ
and
x R Hðx; tÞ for x 2 @U
and t 2 ð0; 1:
ð2:13Þ
In addition suppose x 2 U : x 2 Hðx; tÞ for some t 2 ½0; 1 is relatively compact. Let FðxÞ ¼ Hðx; 1Þ for x 2 U. Then F has a fixed point in U. Remark 2.8. (1) From the proof below we see that we can remove (2.1) and remove H 2 AðU ½0; 1; EÞ in the statement of Theorem 2.4 if we assume H : U ½0; 1 ! KðEÞ is an upper semicontinuous map with Hð; gðÞÞ 2 AðU; EÞ for any continuous function g : U ! ½0; 1 with gð@UÞ ¼ 0. (2) Suppose we have the following definition of AðU; EÞ. We say F 2 AðU; EÞ if F : U ! 2E and F 2 AðU; EÞ. With this definition Theorem 2.4 is again true provided
x 2 U : x 2 Hðx; tÞ for some t 2 ½0; 1 is relatively compact in the statement of Theorem 2.4 is replaced by
x 2 U : x 2 Hðx; tÞ for some t 2 ½0; 1 is compact and we note if E is a normal topological space then we just need to assume
x 2 U : x 2 Hðx; tÞ for some t 2 ½0; 1 is closed in the statement of Theorem 2.4. Proof. Consider
D ¼ x 2 U : x 2 Hðx; tÞ for some t 2 ½0; 1 : Notice D – ; since G is essential in A@U ðU; EÞ (i.e. in particular there exists x 2 U with x 2 GðxÞ ¼ Hðx; 0Þ). Note D is a compact subset of E. Next notice that (2.13), with G 2 A@U ðU; EÞ, guarantees that D \ @U ¼ ;. Thus there exists a continuous map l : U ! ½0; 1 with lð@UÞ ¼ 0 and lðDÞ ¼ 1. Define a map Rl : U ! KðEÞ by Rl ðxÞ ¼ Hðx; lðxÞÞ ¼ HlðxÞ ðxÞ ¼ H sðxÞ; here s : U ! U ½0; 1 is given by sðxÞ ¼ ðx; lðxÞÞ. Notice (as in Theorem 2.1) that Rl 2 AðU; EÞ and Rl j@U ¼ Gj@U since lð@UÞ ¼ 0. Thus Rl 2 A@U ðU; EÞ with Rl j@U ¼ Gj@U and since G is essential in A@U ðU; EÞ there exists x 2 U with x 2 Rl ðxÞ (i.e. x 2 HlðxÞ ðxÞ). Thus x 2 D and so lðxÞ ¼ 1. Consequently x 2 Hðx; 1Þ ¼ FðxÞ. h Remark 2.9. We now note that Theorem 2.4 holds if we use the definition of essential (see (2.12)) in Definition 2.4 instead of in Definition 2.5. Now as in the proof above we have Rl 2 A@U ðU; EÞ with Rl j@U ¼ Gj@U . We now show Rl ffi G in A@U ðU; EÞ. To see this let Q : U ½0; 1 ! KðEÞ be given by Q ðx; tÞ ¼ Hðx; t lðxÞÞ ¼ H gðx; tÞ where g : U ½0; 1 ! U ½0; 1 is given by gðx; tÞ ¼ ðx; t lðxÞÞ. Note Q 2 AðU ½0; 1; EÞ; Q 1 ¼ Rl and Q 0 ¼ G. Also x R Q t ðxÞ for x 2 @U and t 2 ½0; 1 since if there exists t 2 ½0; 1 and x 2 @U with x 2 Q t ðxÞ then x 2 Hðx; t lðxÞÞ so x 2 D and as a result lðxÞ ¼ 1 i.e. x 2 Hðx; tÞ, a contradiction. Finally note
x 2 U : x 2 Q ðx; tÞ ¼ Hðx; t lðxÞÞ for some t 2 ½0; 1 is closed (note H is upper semicontinuous) and compact. Thus Rl ffi G in A@U ðU; EÞ and since G is essential in A@U ðU; EÞ (as in Definition 2.4) we can conclude now as in the proof of Theorem 2.4. The result also holds if the definition of ffi is as in Remark 2.2 (if we use the assumption in Remark 2.8).
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Remark 2.10. If E is a normal topological space then the assumption that
x 2 U : x 2 Hðx; tÞ for some t 2 ½0; 1 is relatively compact can be removed in Theorem 2.4. Let E be a completely regular topological space and let U be an open subset of E ½0; 1. For our next result we assume (2.1) and (2.5) hold. Theorem 2.5. Suppose N 2 AðU; EÞ and (2.6) holds. Let H : U ½0; 1 ! KðE ½0; 1Þ be given by Hðx; k; lÞ ¼ ðNðx; kÞ; lÞ for ðx; kÞ 2 U and l 2 ½0; 1. In addition assume (2.7) (Definition 2.5) and (2.8) hold. Then there exists a x 2 U 1 with x 2 Nðx; 1Þ. Proof. As in Theorem 2.3 note (2.9) holds and H 2 AðU ½0; 1; E ½0; 1Þ from (2.5). Now apply Theorem 2.4. h Remark 2.11. If E is a normal topological space then the assumption (2.8) can be removed in the statement of Theorem 2.5. Remark 2.12. The result in Theorem 2.5 holds (with (2.1) and (2.5) removed) if the following condition holds: Hð; ; gð; ÞÞ 2 AðU; E ½0; 1Þ for any continuous function g : U ! ½0; 1 with gð@UÞ ¼ 0. Recall a compact connected set is called a continuum. For our next result we will use Whyburn’s lemma from topology which we state here for convenience. Theorem 2.6. Let A and B be disjoint closed subsets of a compact Hausdorff topological space K such that no connected component of K intersects both A and B. Then there exists a partition K ¼ K 1 [ K 2 where K 1 and K 2 are disjoint compact sets containing A and B respectively. For our next result E will be a completely regular topological space and U an open subset of E ½0; 1. We will also assume (2.1), (2.5), (2.8) and the following condition:
8 > < for Hausdorff topological spaces X 1 and X 2 ; if F 2 AðX 1 ; X 2 Þ; v 2 CðX 1 ; ½0; 1Þ and if UðyÞ ¼ ðFðyÞ; v ðyÞÞ for y 2 X 1 ; > : then U 2 AðX 1 ; X 2 ½0; 1Þ:
ð2:14Þ
Theorem 2.7. Suppose N 2 AðU; EÞ and (2.6) holds. Let H : U ½0; 1 ! KðE ½0; 1Þ be given by Hðx; k; lÞ ¼ ðNðx; kÞ; lÞ for ðx; kÞ 2 U and l 2 ½0; 1. In addition assume (2.7) (Definition 2.5) and the following condition holds:
X ¼ fðx; kÞ 2 U : x 2 Nðx; kÞg is compact:
ð2:15Þ
Then X contains a continuum intersecting X0 f0g and X1 f1g; here Xt ¼ fx 2 E : ðx; tÞ 2 Xg for each t 2 ½0; 1. Proof. Note A ¼ X0 f0g # X and B ¼ X1 f1g # X are closed (and compact). If there is no continuum intersecting A and B then from Theorem 2.6, X can be represented as X ¼ XH [ XHH where XH and XHH are disjoint compact sets with A # XH and B # XHH . Notice XH and XHH [ @U are closed and disjoint (note XH \ @U ¼ ; since if there exists a ðx; kÞ 2 @U and ðx; kÞ 2 XH then (note ðx; kÞ 2 XH # X) x 2 Nðx; kÞ which contradicts (2.6)). Now there exists a continuous map l : U ! ½0; 1 with lðXHH [ @UÞ ¼ 0 and lðXH Þ ¼ 1. Let
Tðx; kÞ ¼ ðNðx; kÞ; lðx; kÞÞ for ðx; kÞ 2 U: Notice (2.14) guarantees that T 2 AðU; E ½0; 1Þ and in fact T 2 A@U ðU; E ½0; 1Þ since if there exists a ðx; kÞ 2 @U with ðx; kÞ 2 Tðx; kÞ then ðx; kÞ 2 ðNðx; kÞ; lðx; kÞÞ ¼ ðNðx; kÞ; 0Þ so x 2 Nðx; 0Þ which contradicts (2.6). Notice as well (here H0 ðx; kÞ ¼ Hðx; k; 0Þ ¼ ðNðx; kÞ; 0Þ) that
Tj@U ¼ H0 j@U since if ðx; kÞ 2 @U then Tðx; kÞ ¼ ðNðx; kÞ; lðx; kÞÞ ¼ ðNðx; kÞ; 0Þ (note lðXHH [ @UÞ ¼ 0). Now (2.7) guarantees (see Definition 2.5) that there exists a ðx; kÞ 2 U with ðx; kÞ 2 Tðx; kÞ i.e. x 2 Nðx; kÞ and k ¼ lðx; kÞ. Note ðx; kÞ 2 X since ðx; kÞ 2 U and x 2 Nðx; kÞ. Now either ðx; kÞ 2 XH or ðx; kÞ 2 XHH . Case 1. Suppose ðx; kÞ 2 XH . Then lðx; kÞ ¼ 1. Thus k ¼ lðx; kÞ ¼ 1 and x 2 Nðx; kÞ ¼ Nðx; 1Þ i.e. ðx; 1Þ 2 B # XHH which contradicts ðx; 1Þ ¼ ðx; kÞ 2 XH . Case 2. Suppose ðx; kÞ 2 XHH . Then lðx; kÞ ¼ 0. Thus k ¼ lðx; kÞ ¼ 0 and x 2 Nðx; kÞ ¼ Nðx; 0Þ i.e. ðx; 0Þ 2 A # XH which contradicts ðx; 0Þ ¼ ðx; kÞ 2 XHH . h We now show that the ideas in this section can be applied to other natural situations. The coincidence theory presented here is a generalization of results presented in [1,6,9] (in our results we assume the usual Poincaré, Leray–Schauder condition (see (2.21) or (2.28))). First let E be a completely regular topological vector space, Y a topological vector space, and U an open subset of E. Also let L : dom L # E ! Y be a linear (not necessarily continuous) single-valued map; here dom L is a vector subspace of E. Finally T : E ! Y will be a linear, continuous single-valued map with L þ T : dom L ! Y an isomorphism (i.e. a linear homeomorphism); for convenience we say T 2 HL ðE; YÞ.
D. O’Regan / Applied Mathematics and Computation 219 (2012) 2026–2034
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Definition 2.6. Let F : U ! 2Y . We say F 2 AðU; Y; L; TÞ if ðL þ TÞ1 ðF þ TÞ 2 AðU; EÞ. Definition 2.7. We say F 2 A@U ðU; Y; L; TÞ if F 2 AðU; Y; L; TÞ with L x R FðxÞ for x 2 @U \ dom L. Definition 2.8. Let F; G 2 A@U ðU; Y; L; TÞ. We say F ffi G in A@U ðU; Y; L; TÞ if there exists a map W : U ½0; 1 ! 2Y with W 2 AðU ½0; 1; Y; L; TÞ; L x R Wt ðxÞ for any x 2 @U \ dom L and t 2 ½0; 1; W1 ¼ F, W0 ¼ G (here Wt ðxÞ ¼ Wðx; tÞ) and
x 2 U \ dom L : L x 2 Wðx; tÞ for some t 2 ½0; 1 is relatively compact. For our next two results we assume the following condition holds:
8 > < if X 2 ¼ U or X 2 ¼ U ½0; 1 and if; F 2 AðU ½0; 1; Y; L; TÞ and f 2 CðX 2 ; U ½0; 1Þ; > : then F f 2 AðX 2 ; Y; L; TÞ:
ð2:16Þ
Remark 2.13. The results below (with (2.16) removed) also hold true if we use the following definition of ffi. Let F; G 2 A@U ðU; Y; L; TÞ. We say F ffi G in A@U ðU; Y; L; TÞ if there exists a map W : U ½0; 1 ! 2Y with ðL þ TÞ1 ðW þ TÞ : U ½0; 1 ! KðEÞ upper semicontinuous and with ðL þ TÞ1 ðWð; gðÞÞ þ TÞ 2 AðU; EÞ for any continuous function g : U ! ½0; 1 with gð@UÞ ¼ 0; L x R Wt ðxÞ for any x 2 @U \ dom L and t 2 ½0; 1; W1 ¼ F, W0 ¼ G and
x 2 U \ dom L : L x 2 Wðx; tÞ for some t 2 ½0; 1 is relatively compact. The following condition will be assumed:
ffi is an equivalence relation in A@U ðU; Y; L; TÞ:
ð2:17Þ
Definition 2.9. Let F 2 A@U ðU; Y; L; TÞ. We say F is L–essential in A@U ðU; Y; L; TÞ if for every map J 2 A@U ðU; Y; L; TÞ with Jj@U ¼ Fj@U and J ffi F in A@U ðU; Y; L; TÞ there exists x 2 U \ dom L with L x 2 JðxÞ. Otherwise F is L–inessential in A@U ðU; Y; L; TÞ i.e. there exists a map J 2 A@U ðU; Y; L; TÞ with Jj@U ¼ Fj@U and J ffi F in A@U ðU; Y; L; TÞ such that L x R JðxÞ for x 2 U \ dom L. Theorem 2.8. Let E be a completely regular topological vector space, Y a topological vector space, U an open subset of E; L : dom L # E ! Y a linear single valued map, T 2 HL ðE; YÞ, and assume (2.16) and (2.17) hold. Suppose F 2 A@U ðU; Y; L; TÞ. Then the following are equivalent: (i) F is L–inessential in A@U ðU; Y; L; TÞ; (ii) there exists a map G 2 A@U ðU; Y; L; TÞ with G ffi F in A@U ðU; Y; L; TÞ such that L x R GðxÞ for x 2 U \ dom L.
Proof. (i) implies (ii) is immediate. Next we prove (ii) implies (i). Suppose there exists a map G 2 A@U ðU; Y; L; TÞ with G ffi F in A@U ðU; Y; L; TÞ such that L x R GðxÞ for x 2 U \ dom L. Let H : U ½0; 1 ! 2Y be a map with H 2 AðU ½0; 1; Y; L; TÞ; L x R Ht ðxÞ for any x 2 @U \ dom L and t 2 ½0; 1; H0 ¼ F; H1 ¼ G (here Ht ðxÞ ¼ Hðx; tÞ) and
x 2 U \ dom L : L x 2 Hðx; tÞ for some t 2 ½0; 1 is relatively compact. Consider
n o D ¼ x 2 U : x 2 ðL þ TÞ1 ðH þ TÞðx; tÞ for some t 2 ½0; 1 : Notice that it is immediate that
D ¼ x 2 U \ dom L : L x 2 Hðx; tÞ for some t 2 ½0; 1 : If D ¼ ; then in particular L x R Hðx; 0Þ for x 2 U \ dom L so F is L–inessential in A@U ðU; Y; L; TÞ. Next suppose D – ;. Note D is a compact subset of E. Also since L x R Ht ðxÞ for x 2 @U \ dom L and t 2 ½0; 1 then D \ @U ¼ ;. Thus there exists a continuous map l : U ! ½0; 1 with lð@UÞ ¼ 0 and lðDÞ ¼ 1. Define a map Rl : U ! 2Y by Rl ðxÞ ¼ Hðx; lðxÞÞ ¼ HlðxÞ ðxÞ ¼ H sðxÞ; here s : U ! U ½0; 1 is given by sðxÞ ¼ ðx; lðxÞÞ. Notice Rl 2 AðU; Y; L; TÞ (note (2.16) and H 2 AðU ½0; 1; Y; L; TÞ) and notice Rl j@U ¼ H0 j@U ¼ Fj@U since lð@UÞ ¼ 0. Thus Rl 2 A@U ðU; Y; L; TÞ (note L x R Ht ðxÞ for any x 2 @U \ dom L and t 2 ½0; 1) with Rl j@U ¼ Fj@U . We now claim
Rl ffi F
in A@U ðU; Y; L; TÞ: Y
ð2:18Þ
Let Q : U ½0; 1 ! 2 be given by Q ðx; tÞ ¼ Hðx; t lðxÞÞ ¼ H gðx; tÞ where g : U ½0; 1 ! U ½0; 1 is given by gðx; tÞ ¼ ðx; t lðxÞÞ. Note Q 2 AðU ½0; 1; Y; L; TÞ (note (2.16) and H 2 AðU ½0; 1; Y; L; TÞ), Q 0 ¼ F and Q 1 ¼ Rl . Also
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L x R Q t ðxÞ for x 2 @U \ dom L and t 2 ½0; 1 since if there exists t 2 ½0; 1 and x 2 @U \ dom L with L x 2 Q t ðxÞ then L x 2 Hðx; t lðxÞÞ so x 2 D and as a result lðxÞ ¼ 1 i.e. L x 2 Hðx; tÞ, a contradiction. Finally note
x 2 U \ dom L : L x 2 Q ðx; tÞ ¼ Hðx; t lðxÞÞ for some t 2 ½0; 1 is compact. Thus (2.18) holds. Consequently F is L–inessential in A@U ðU; Y; L; TÞ. h Remark 2.14. From the proof above we see that the result in Theorem 2.8 (with (2.16) removed) holds if the definition of ffi is as in Remark 2.13. Essentially the same reasoning as in Theorem 2.2 establishes the following result. Theorem 2.9. Let E be a completely regular topological vector space, Y a topological vector space, U an open subset of E; L : dom L # E ! Y a linear single valued map, T 2 HL ðE; YÞ, and assume (2.16) and (2.17) hold. Suppose U and W are two maps in A@U ðU; Y; L; TÞ with U ffi W in A@U ðU; Y; L; TÞ. Then U is L–essential in A@U ðU; Y; L; TÞ if and only if WH is L–essential in A@U ðU; Y; L; TÞ. Remark 2.15. If E is a normal topological vector space then the assumption that
x 2 U \ dom L : L x 2 Wðx; tÞ for some t 2 ½0; 1 is relatively compact can be removed in Definition 2.8 (and Remark 2.13) and we still obtain Theorems 2.8 and 2.9. Let E be a completely regular topological vector space, Y a topological vector space, and U an open subset of E ½0; 1. Also let L : dom L # E ! Y be a linear (not necessarily continuous) single-valued map; here dom L is a vector subspace of E. Now let L : dom L ¼ dom L ½0; 1 ! Y ½0; 1 be given by Lðy; kÞ ¼ ðL y; kÞ. Let T : E ! Y be a linear, continuous single-valued map with L þ T : dom L ! Y an isomorphism (i.e. a linear homeomorphism) and let T : E ½0; 1 ! Y ½0; 1 be given by T ðy; kÞ ¼ ðT y; 0Þ. Notice ðL þ T Þ1 ðy; kÞ ¼ ððL þ TÞ1 y; kÞ for ðy; kÞ 2 Y ½0; 1. For our next result we assume (2.16) (with Y replaced by Y ½0; 1; L replaced by L and T replaced by T ) holds and in addition that
ffi is an equivalence relation in A@U ðU; Y ½0; 1; L; T Þ
ð2:19Þ
8 > < if F 2 AðU; Y; L; TÞ and if Wðy; lÞ ¼ ðFðyÞ; lÞ; ðy; lÞ 2 U ½0; 1; then Wl 2 AðU; Y ½0; 1; L; T Þ for each l 2 ½0; 1 and > : W 2 AðU ½0; 1; Y ½0; 1; L; T Þ; here Wl ðxÞ ¼ Wðx; lÞ:
ð2:20Þ
and
Theorem 2.10. Suppose N 2 AðU; Y; L; TÞ with
L x R Nðx; kÞ for ðx; kÞ 2 @U \ dom L: Y½0;1
Let H : U ½0; 1 ! 2 ditions hold:
(
ð2:21Þ
be given by Hðx; k; lÞ ¼ ðNðx; kÞ; lÞ for ðx; kÞ 2 U and
l 2 ½0; 1. In addition assume the following con-
H0 is essential in A@U ðU; Y ½0; 1; L; T Þ; here
ð2:22Þ
H0 ðx; kÞ ¼ Hðx; k; 0Þ ¼ ðNðx; kÞ; 0Þ for ðx; kÞ 2 U and
ðx; kÞ 2 U \ dom L : Lðx; kÞ 2 Hðx; k; lÞ for some
l 2 ½0; 1 is relatively compact:
ð2:23Þ
Then H1 is essential in A@U ðU; Y ½0; 1; L; T Þ so in particular there exists a x 2 U 1 \ dom L with L x 2 Nðx; 1Þ; here H1 ðx; kÞ ¼ Hðx; k; 1Þ ¼ ðNðx; kÞ; 1Þ for ðx; kÞ 2 U. Proof. First note
Lðx; kÞ R Hðx; k; lÞ for ðx; kÞ 2 @U \ dom L and
l 2 ½0; 1:
ð2:24Þ
To see this suppose there exists ðx0 ; k0 Þ 2 @U \ dom L and l0 2 ½0; 1 with Lðx0 ; k0 Þ 2 ðNðx0 ; k0 Þ; l0 Þ. Then l0 ¼ k0 and L x0 2 Nðx0 ; k0 Þ which contradicts (2.21). Also note (2.20) guarantees that H0 2 A@U ðU; Y ½0; 1; L; T Þ and H1 2 A@U ðU; Y ½0; 1; L; T Þ. We will now apply Theorem 2.9. Now (2.20), (2.23) and (2.24) guarantee that
H0 ffi H1
in A@U ðU; Y ½0; 1; L; T Þ
ð2:25Þ
(note H 2 AðU ½0; 1; Y ½0; 1; L; T Þ from (2.20) and N 2 AðU; Y; L; TÞ). Theorem 2.9 guarantees that H1 is essential in A@U ðU; Y ½0; 1; L; T Þ. Thus there exists a ðx; kÞ 2 U \ dom L with Lðx; kÞ 2 ðNðx; kÞ; 1Þ i.e. L x 2 Nðx; kÞ with k ¼ 1 i.e. x 2 dom L with x 2 U 1 ¼ fy 2 E : ðy; 1Þ 2 Ug and L x 2 Nðx; 1Þ. h
D. O’Regan / Applied Mathematics and Computation 219 (2012) 2026–2034
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Remark 2.16. From the proof above it is clear that Ht is essential in A@U ðU; Y ½0; 1; L; T Þ for every t 2 ½0; 1; here Ht ðx; kÞ ¼ Hðx; k; tÞ ¼ ðNðx; kÞ; tÞ for ðx; kÞ 2 U. If E is a normal topological vector space then the assumption (2.23) can be removed in the statement of Theorem 2.10. Remark 2.17. The result in Theorem 2.10 holds (with (2.16) and (2.20) removed) if the definition of ffi is as in Remark 2.13 and if the following condition holds: ðL þ T Þ1 ðHð; ; gð; ÞÞ þ T Þ 2 AðU; E ½0; 1Þ for any continuous function g : U ! ½0; 1 with gð@UÞ ¼ 0. We note that (2.17) or (2.19) could be strong assumptions. However we can establish some results even if (2.17) or (2.19) are not assumed. To establish this we will consider new essential maps (a subset of those essential maps in Definition 2.9). Let E be a completely regular topological vector space, Y a topological vector space, and U an open subset of E. Also let L and T be as described before Definition 2.6. Definition 2.10. Let F 2 A@U ðU; Y; L; TÞ. We say F is essential in A@U ðU; Y; L; TÞ if for every map J 2 A@U ðU; Y; L; TÞ with Jj@U ¼ Fj@U there exists x 2 U \ dom L with L x 2 JðxÞ. Theorem 2.11. Assume (2.16) holds. Suppose G 2 A@U ðU; Y; L; TÞ; H : U ½0; 1 ! 2Y with H 2 AðU ½0; 1; Y; L; TÞ and assume the following hold:
Hðx; 0Þ ¼ GðxÞ for x 2 U;
ð2:26Þ
G is essential in A@U ðU; Y; L; TÞ ðDefinition 2:10Þ
ð2:27Þ
L x R Hðx; tÞ for x 2 @U \ dom L and t 2 ð0; 1:
ð2:28Þ
and In addition suppose x 2 U \ dom L : L x 2 Hðx; tÞ for some t 2 ½0; 1 is relatively compact. Let FðxÞ ¼ Hðx; 1Þ for x 2 U. Then there exists x 2 U \ domL with L x 2 FðxÞ. Remark 2.18. From the proof below we see that we can remove (2.16) and remove H 2 AðU ½0; 1; Y; L; TÞ in the statement of Theorem 2.11 if we assume ðL þ TÞ1 ðH þ TÞ : U ½0; 1 ! KðEÞ is an upper semicontinuous map with ðL þ TÞ1 ðHð; gðÞÞ þ TÞ 2 AðU; EÞ for any continuous function g : U ! ½0; 1 with gð@UÞ ¼ 0. Proof. Consider
D ¼ x 2 U \ dom L : L x 2 Hðx; tÞ for some t 2 ½0; 1 : Notice (2.27) guarantees that D – ;. Note D is a compact subset of E. Now (2.28) (and (2.27)) guarantees that D \ @U ¼ ;. Thus there exists a continuous map l : U ! ½0; 1 with lð@UÞ ¼ 0 and lðDÞ ¼ 1. Define a map Rl : U ! 2Y by Rl ðxÞ ¼ Hðx; lðxÞÞ ¼ HlðxÞ ðxÞ ¼ H sðxÞ; here s : U ! U ½0; 1 is given by sðxÞ ¼ ðx; lðxÞÞ. Now Rl 2 AðU; Y; L; TÞ and Rl j@U ¼ Gj@U since lð@UÞ ¼ 0. Note also Rl 2 A@U ðU; Y; L; TÞ since L x R Ht ðxÞ for x 2 @U \ dom L and t 2 ½0; 1. Now since G is essential in A@U ðU; Y; L; TÞ there exists x 2 U \ dom L with L x 2 Rl ðxÞ (i.e. L x 2 HlðxÞ ðxÞ). Thus x 2 D and so lðxÞ ¼ 1. Consequently L x 2 Hðx; 1Þ ¼ FðxÞ. h Remark 2.19. We now note that Theorem 2.11 holds if we use the definition of essential (see (2.27)) in Definition 2.9 instead of in Definition 2.10. We follow the proof above so we have Rl 2 A@U ðU; Y; ;L; TÞ with Rl j@U ¼ Gj@U . We now show Rl ffi G in A@U ðU; Y; L; TÞ. To see this let Q : U ½0; 1 ! 2Y be given by Q ðx; tÞ ¼ Hðx; t lðxÞÞ ¼ H gðx; tÞ where g : U ½0; 1 ! U ½0; 1 is given by gðx; tÞ ¼ ðx; t lðxÞÞ. Note Q 2 AðU ½0; 1; Y; L; TÞ; Q 1 ¼ Rl ; Q 0 ¼ G and L x R Q t ðxÞ for x 2 @U \ dom L and t 2 ½0; 1 (if there exists t 2 ½0; 1 and x 2 @U \ dom L with L x 2 Q t ðxÞ then L x 2 Hðx; t lðxÞÞ so x 2 D and as a result lðxÞ ¼ 1 i.e. L x 2 Hðx; tÞ, a contradiction). Finally note
x 2 U \ dom L : L x 2 Q ðx; tÞ ¼ Hðx; t lðxÞÞ for some t 2 ½0; 1 is compact. Thus Rl ffi G in A@U ðU; Y; L; TÞ and since G is essential in A@U ðU; Y; L; TÞ (as in Definition 2.9) we can conclude now as in the proof of Theorem 2.11. The result also holds if the definition of ffi is as in Remark 2.13 (if we use the assumption in Remark 2.18). Remark 2.20. If E is a normal topological space then the assumption that
x 2 U \ dom L : L x 2 Hðx; tÞ for some t 2 ½0; 1 is relatively compact can be removed in Theorem 2.11. Let E be a completely regular topological vector space, Y a topological vector space, and U an open subset of E ½0; 1. Also let L; L; T and T be as described before Theorem 2.10. For our next result we assume (2.16) (with Y replaced by Y ½0; 1, L replaced by L and T replaced by T ) and (2.20) hold.
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Theorem 2.12. Suppose N 2 AðU; Y; L; TÞ and (2.21) holds. Let H : U ½0; 1 ! 2Y½0;1 be given by Hðx; k; lÞ ¼ ðNðx; kÞ; lÞ for ðx; kÞ 2 U and l 2 ½0; 1. In addition assume (2.22) (Definition 2.10) and (2.23) hold. Then there exists a x 2 U 1 \ dom L with L x 2 Nðx; 1Þ. Proof. As in Theorem 2.10 note (2.24) holds and H 2 AðU ½0; 1; Y ½0; 1; L; T Þ from (2.20). Now apply Theorem 2.11. h Remark 2.21. If E is a normal topological space then the assumption (2.23) can be removed in the statement of Theorem 2.12. Remark 2.22. The result in Theorem 2.12 holds (with (2.16) and (2.20) removed) if the following condition holds: ðL þ T Þ1 ðHð; ; gð; ÞÞ þ T Þ 2 AðU; E ½0; 1Þ for any continuous function g : U ! ½0; 1 with gð@UÞ ¼ 0. In our next result E will be a completely regular topological vector space, Y a topological vector space, and U an open subset of E ½0; 1. Also let L; L; T and T be as described before Theorem 2.10. We will also assume (2.16), (2.20), (2.23) and the following condition:
8 > < if F 2 AðU; Y; L; TÞ; v 2 CðU; ½0; 1Þ and if UðyÞ ¼ ðFðyÞ; v ðyÞÞ for y 2 U; > : then U 2 AðU; Y ½0; 1; L; T Þ:
ð2:29Þ
Theorem 2.13. Suppose N 2 AðU; Y; L; TÞ and (2.21) holds. Let H : U ½0; 1 ! 2Y½0;1 be given by Hðx; k; lÞ ¼ ðNðx; kÞ; lÞ for ðx; kÞ 2 U and l 2 ½0; 1. In addition assume (2.22) (Definition 2.10) and the following condition holds:
X ¼ fðx; kÞ 2 U \ dom L : L x 2 Nðx; kÞg is compact:
ð2:30Þ
Then X contains a continuum intersecting X0 f0g and X1 f1g; here Xt ¼ fx 2 E : ðx; tÞ 2 Xg for each t 2 ½0; 1. Proof. Note A ¼ X0 f0g # X and B ¼ X1 f1g # X are closed (and compact). If there is no continuum intersecting A and B then from Theorem 2.6, X can be represented as X ¼ XH [ XHH where XH and XHH are disjoint compact sets with A # XH and B # XHH . Notice XH and XHH [ @U are closed and disjoint (note XH \ @U ¼ ; since if there exists a ðx; kÞ 2 @U and ðx; kÞ 2 XH then (note ðx; kÞ 2 XH # X) L x 2 Nðx; kÞ which contradicts (2.21)). Now there exists a continuous map l : U ! ½0; 1 with lðXHH [ @UÞ ¼ 0 and lðXH Þ ¼ 1. Let
Tðx; kÞ ¼ ðNðx; kÞ; lðx; kÞÞ for ðx; kÞ 2 U: Notice (2.29) guarantees that T 2 AðU; Y ½0; 1; L; T Þ and in fact T 2 A@U ðU; Y ½0; 1; L; T Þ since if there exists a ðx; kÞ 2 @U with Lðx; kÞ ¼ ðL x; kÞ 2 Tðx; kÞ then ðL x; kÞ 2 ðNðx; kÞ; lðx; kÞÞ ¼ ðNðx; kÞ; 0Þ so L x 2 Nðx; 0Þ which contradicts (2.21). Notice as well (here H0 ðx; kÞ ¼ Hðx; k; 0Þ ¼ ðNðx; kÞ; 0Þ) that
Tj@U ¼ H0 j@U ; since if ðx; kÞ 2 @U then Tðx; kÞ ¼ ðNðx; kÞ; lðx; kÞÞ ¼ ðNðx; kÞ; 0Þ (note lðXHH [ @UÞ ¼ 0). Now (2.22) guarantees (see Definition 2.10) that there exists a ðx; kÞ 2 U \ dom L with Lðx; kÞ 2 Tðx; kÞ i.e. L x 2 Nðx; kÞ and k ¼ lðx; kÞ. Note ðx; kÞ 2 X since ðx; kÞ 2 U \ dom L and L x 2 Nðx; kÞ. Now either ðx; kÞ 2 XH or ðx; kÞ 2 XHH . Case 1. Suppose ðx; kÞ 2 XH . Then lðx; kÞ ¼ 1. Thus k ¼ lðx; kÞ ¼ 1 and L x 2 Nðx; kÞ ¼ Nðx; 1Þ i.e. ðx; 1Þ 2 B # XHH which contradicts ðx; 1Þ ¼ ðx; kÞ 2 XH . Case 2. Suppose ðx; kÞ 2 XHH . Then lðx; kÞ ¼ 0. Thus k ¼ lðx; kÞ ¼ 0 and L x 2 Nðx; kÞ ¼ Nðx; 0Þ i.e. ðx; 0Þ 2 A # XH which contradicts ðx; 0Þ ¼ ðx; kÞ 2 XHH . h References [1] J. Bryszewski, L. Górniewicz, A Poincaré type coincidence theorem for multi-valued maps, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976) 593–598. [2] A. Granas, Sur la méthode de continuité de Poincaré, C.R. Acad. Sci. Paris 282 (1976) 983–985. [3] D. O’Regan, Continuation principles and d-essential maps, Math. Comput. Model. 30 (1999) 1–6. [4] D. O’Regan, Topological principles for extension type maps, Dynam. Syst. Appl. 10 (2011) 541–550. [5] D. O’Regan, J. Peran, Fixed points for better admissible multifunctions on proximity spaces, J. Math. Anal. Appl. 380 (2011) 882–887. [6] D. O’Regan, R. Precup, Theorems of Leray–Schauder Type and Applications, Taylor and Francis Publishers, London, 2002. [7] S. Park, Fixed point theorems for better admissible multimaps on almost convex spaces, J. Math. Anal. Appl. 329 (2007) 690–702. [8] R. Precup, On the topological transversality principle, Nonlinear Anal. 20 (1993) 1–9. [9] S. Reich, A Poincaré type coincidence theorem, Amer. Math. Monthly 81 (1974) 52–53.