Essential Maps and Coincidence Principles for General Classes of Maps

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1 Filoma 31:11 (2017), hps://doi.org/ /fil o Published by Faculy of Sciences Mahemaics, Universiy of Niš, Serbia Available a: hp:// Essenial Maps Coincidence Principles for General Classes of Maps Donal O Regan a a School of Mahemaics, Saisics Applied Mahemaics Naional Universiy of Irel, Galway Irel Absrac. We inroduce he noion of an essenial map for a general class of maps. In addiion we presen homoopy normalizaion ype properies for hese maps. To he memory of Professor Lj. Ćirić ( ) 1. Inroducion In his paper we inroduce he noion of a Φ essenial map for a variey of new classes of maps. Firs we presen a homoopy ype propery i.e. if wo maps G F are in a class G is Φ essenial F is homoopic o G (in a cerain sense) hen F is Φ essenial. Also we presen a normalizaion ype propery i.e. we presen condiions which guaranee ha a map is Φ essenial. The noion of an essenial map was inroduced by Granas [4] in 1976 he noion was exended in a variey of seings by many auhors (see [3, 6, 7] he references herein). In paricular he heory here is parly moivaed by coninuaion hoeroms for DKT PK maps in [1, 2]. 2. Main Resuls Le E be a compleely regular opological space U an open subse of E. We will consider classes A, B D of maps. Definiion 2.1. We say F D(U, E) (respecively F B(U, E)) if F : U 2 E F D(U, E) (respecively F B(U, E)); here 2 E denoes he family of nonempy subses of E. Definiion 2.2. We say F A(U, E) if F : U 2 E F A(U, E) here exiss a selecion Ψ D(U, E) of F. In his secion we fix a Φ B(U, E) Mahemaics Subjec Classificaion. Primary 54H25, 47H10 Keywords. essenial maps, coincidence poins, opological principles Received: 15 Sepember 2016; Acceped: 18 April 2017 Communicaed by Vladimir Rakočević address: donal.oregan@nuigalway.ie (Donal O Regan)

2 D. O Regan / Filoma 31:11 (2017), Definiion 2.3. We say F A U (U, E) (respecively F D U (U, E)) if F A(U, E) (respecively F D(U, E)) wih F(x) Φ(x) = for x U; here U denoes he boundary of U in E. Definiion 2.4. Le F A U (U, E). We say F : U 2 E is Φ essenial in A U (U, E) if for any selecion Ψ D(U, E) of F any map J D U (U, E) wih J U = Ψ U here exiss x U wih J (x) Φ (x). Remark 2.5. (i). Noe in Definiion 2.1 ha Ψ is a selecion of F if Ψ(x) F(x) for x U. (ii). Noe if F A U (U, E) is Φ essenial in A U (U, E) if Ψ D(U, E) is any selecion of F hen here exiss an x U wih Ψ (x) Φ (x). To see his ake J = Ψ in Definiion 2.4; noe for x U ha Ψ (x) Φ (x) F (x) Φ (x) =. Finally we noe if Ψ (x) Φ (x) for x U hen Ψ (x) Φ (x) F (x) Φ (x). (iii). If F A(U, E) if F D(U, E) hen an example of a selecion Ψ of F in Definiion 2.2 is F iself. In applicaions we see by appropriaely choosing U, E, D hen auomaically here exiss a selecion Ψ of F in Definiion 2.2 (see for example Remark 2.8). Theorem 2.6. Le E be a compleely regular (respecively normal) opological space, U an open subse of E, F A U (U, E) le G A U (U, E) be Φ essenial in A U (U, E). For any selecion Ψ D(U, E) (respecively Λ D(U, E)) of F (respecively G) for any map J D U (U, E) wih J U = Ψ U assume here exiss a map H Ψ,Λ,J defined on U [0, 1] wih values in E wih H Ψ,Λ,J (., η(. )) D(U, E) for any coninuous funcion η : U [0, 1] wih η( U) = 0, Φ(x) H Ψ,Λ,J (x) = for any x U (0, 1), x U : Φ(x) H Ψ,Λ,J (x, ) for some [0, 1] } is compac (respecively closed) H Ψ,Λ,J = Λ, H Ψ,Λ,J = J; here H Ψ,Λ,J 0 1 (x) = H Ψ,Λ,J (x, ). Then F is Φ essenial in A U (U, E). Proof. Le Ψ D(U, E) (respecively Λ D(U, E)) be any selecion of F (respecively G). Now consider any map J D U (U, E) wih J U = Ψ U. We mus show here exiss x U wih J (x) Φ (x). Choose he map H Ψ,Λ,J as in he saemen of Theorem 2.6. Consider Ω = x U : Φ(x) H Ψ,Λ,J (x, ) for some [0, 1] }. Noe Ω since H Ψ,Λ,J 0 = Λ G is Φ essenial in A U (U, E); o see his noe Λ(x) Φ(x) G(x) Φ(x) = for x U, from Remark 2.5 (ii) here exiss y U wih Λ(y) Φ(y). Also Ω is compac (respecively closed) if E is a compleely regular (respecively normal) opological space. Nex noe Ω U =. Thus here exiss a coninuous map µ : U [0, 1] wih µ( U) = 0 µ(ω) = 1. Define a map R µ by R µ (x) = H Ψ,Λ,J (x, µ(x)) = H Ψ,Λ,J (x). µ(x) Noe R µ D U (U, E) R µ U = H Ψ,Λ,J 0 U = Λ U. Now since G is Φ essenial in A U (U, E) hen here exiss x U wih R µ (x) Φ(x) (i.e. H Ψ,Λ,J (x) Φ(x) ) so x Ω. As a resul µ(x) = 1 so µ(x) H Ψ,Λ,J (x) Φ(x) = J(x) Φ(x), we are finished. 1 Noe Theorem 2.6 is a homoopy ype resul. Nex we presen a number of normalizaion ype resuls i.e. resuls which guaranee ha a map is Φ essenial in A U (U, E). In our firs resul E is a opological vecor space Φ B(E, E) is fixed (we say Φ B(E, E) if Φ : E 2 E Φ B(E, E)). Theorem 2.7. Le E be a opological vecor space, U an open subse of E Φ B(E, E). Assume he following condiions hold: (2.1) 0 A U (U, E) 0 D(U, E) where 0 denoes he zero map (2.2) for any map J D U (U, E) wih J U = 0} J(x), x U R J (x) = 0}, x E \ U, here exiss y E wih Φ(y) R J (y)

3 D. O Regan / Filoma 31:11 (2017), (2.3) here is no z E \ U wih Φ(z) 0}. Then he zero map is Φ essenial in A U (U, E). Proof. Le F(x) = 0 for x U (i.e. F is he zero map) le Ψ D(U, E) be any selecion of F. Noe Ψ is he zero map (noe Ψ : U 2 E, Ψ(x) F(x) for x U 0 D(U, E)). Consider any map J D U (U, E) wih J U = 0}. To show ha he zero map is Φ essenial in A U (U, E) we mus show here exiss x U wih J(x) Φ(x). Le J(x), x U R J (x) = 0}, x E \ U. Now (2.2) guaranees ha here exiss y E wih Φ(y) R J (y). There are wo cases o consider, namely y U y E \U. If y U hen Φ(y) J(y), we are finished. If y E \U hen since R J (y) = 0} (noe J U = 0}) we have Φ(y) 0}, his conradics (2.3). Remark 2.8. We firs recall he PK maps from he lieraure. Le Z W be subses of Hausdorff opological vecor spaces Y 1 Y 2 F a mulifuncion. We say F PK(Z, W) if W is convex here exiss a map S : Z W wih Z = in S 1 (w) : w W}, co (S(x)) F(x) for x Z S(x) for each x Z; here S 1 (w) = z : w S(z)}. Le E be a locally convex Hausdorff opological vecor space, U an open subse of E, 0 U, U paracompac Φ = I (he ideniy map). In his case we le D = D A = A. We say Q D(U, E) if Q : U E is a coninuous compac map. We say F A(U, E) if F PK(U, E) F is a compac map (he exisence of a coninuous selecion Ψ of F is guaraneed from [5, Theorem 1.3] noe Ψ is compac since Ψ is a selecion of F F is compac, so Ψ D(U, E)). Noice (2.1), (2.2) (see [2, 5] or noe ha i is immediae from Schauder s fixed poin heorem) (2.3) (noe 0 U Φ = I) hold. We noe ha a compac map above could be replaced by a more general compacness ype map; see [2]. Remark 2.9. We noe here ha an assumpion was inadverenly lef ou in [1, 2]. In [1] he coninuous selecion Ψ of F should be required o saisfy Propery (A) (his was inadverenly lef ou) i.e. if F saisfies Propery (A) hen i should be assumed ha any coninuous selecion Ψ of F saisfies Propery (A) (of course his assumpion is auomaically saisfied for he ype of map considered in he lieraure i.e. Propery (A) usually means ha he map is compac or condensing). Theorem Le E be a Hausdorff opological space, U an open subse of E, Φ B(E, E) F A U (U, E). Assume he following condiions hold: (2.4) (2.5) (2.6) here exiss a reracion r : E U wih r(w) U if w E \ U for any selecion Ψ D(U, E) of F any map J D U (U, E) wih J U = Ψ U here exiss y E wih J r(y) Φ(y) for any selecion Ψ D(U, E) of F here is no y E \ U z U wih z = r(y) Ψ(z) Φ(y). Then F is Φ essenial in A U (U, E).

4 D. O Regan / Filoma 31:11 (2017), Proof. Le Ψ D(U, E) be any selecion of F consider any map J D U (U, E) wih J U = Ψ U. Now (2.5) guaranees ha here exiss a y E wih J r(y) Φ(y). Le z = r(y) noe J(z) Φ(y). There are wo cases o consider, namely y U y E \ U. If y U hen z = r(y) = y so J(y) Φ(y), we are finished. If y E \ U hen z U noe his conradics (2.6). J(z) Φ(y) = Ψ(z) Φ(y) since J U = Ψ U, Remark Noe here is a dual version of Theorem 2.10 if we consider r J insead of J r (above). Le Φ B(U, E), F A U (U, E) suppose (2.4) holds. In addiion assume he following condiions hold: for any selecion Ψ D(U, E) of F any map J D U (U, E) wih J U = Ψ U here exiss w U wih r J(w) Φ(w) for any selecion Ψ D(U, E) of F any map J D U (U, E) wih J U = Ψ U here is no z E \ U y U wih z J(y) r(z) Φ(y). Then F is Φ essenial in A U (U, E). Theorem Le E be a opological vecor space (so auomaically compleely regular), U an open subse of E, Φ B(E, E) F A U (U, E). Assume he following condiions hold: (2.7) here exiss x U wih Φ(x) 0} (2.8) for any selecion Ψ D(U, E) of F any map J D U (U, E) wih J U = Ψ U hen Φ(x) λ Ψ(x) = for x U λ [0, 1] Ω = x U : Φ(x) λ J(x) for some λ [0, 1] } is compac (2.9) here exiss a reracion r : E U (2.10) for any coninuous map µ : E [0, 1] wih µ(e \ U) = 0 any selecion Ψ D(U, E) of F any map J D U (U, E) wih J U = Ψ U here exiss x E wih Φ(x) µ(x) J(r(x)) (2.11) here is no z E \ U wih Φ(z) 0}. Then F is Φ essenial in A U (U, E). Proof. Le Ψ D(U, E) be any selecion of F consider any map J D U (U, E) wih J U = Ψ U. Le Ω = x U : Φ(x) λ J(x) for some λ [0, 1] }.

5 D. O Regan / Filoma 31:11 (2017), Now (2.7) (2.8) guaranee ha Ω is compac Ω U. We claim Ω U. To see his le x Ω x U. Then since J U = Ψ U we have Φ(x) λ Ψ(x), his conradics (2.8). Now here exiss a coninuous map µ : E [0, 1] wih µ(e \ U) = 0 µ(ω) = 1. Le r be as in (2.9) (2.10) guaranees ha here exiss x E wih Φ(x) µ(x) J(r(x)). If x E \ U hen µ(x) = 0 so Φ(x) 0}, his conradics (2.11). Thus x U so Φ(x) µ(x) J(x). Hence x Ω so µ(x) = 1 consequenly Φ(x) J(x). Remark If in Theorem 2.12 he space E is normal hen he assumpion ha Ω is compac in (2.8) can be replaced by Ω is closed. Remark We say F MA(U, E) if F : U 2 E F A(U, E) we say F MA U (U, E) if F MA(U, E) wih F(x) Φ(x) = for x U. Now we say F MA U (U, E) is Φ essenial in MA U (U, E) if for every map J MA U (U, E) wih J U = F U here exiss x U wih J (x) Φ (x). There are obvious analogues of Theorem s 2.6, 2.7, 2.10, 2.12 for MA maps (hese saemens are lef o he reader). For example he analogue of Theorem 2.10 is: Suppose Φ B(E, E), F MA U (U, E) wih (2.4) he following condiions holding: for any map J MA U (U, E) wih J U = F U here exiss y E wih J r(y) Φ(y) here is no y E \ U z U wih z = r(y) F(z) Φ(y). Then F is Φ essenial in MA U (U, E). We now show ha he ideas in his secion can be applied o oher naural siuaions. Le E be a Hausdorff opological vecor space, Y a opological vecor space, U an open subse of E. Also le L : dom L E Y be a linear (no necessarily coninuous) single valued map; here dom L is a vecor subspace of E. Finally T : E Y will be a linear single valued map wih L + T : dom L Y a bijecion; for convenience we say T H L (E, Y). Definiion We say F D(U, Y; L, T) (respecively F B(U, Y; L, T)) if F : U 2 Y (L + T) 1 (F + T) D(U, E) (respecively (L + T) 1 (F + T) B(U, E)). Definiion We say F A(U, Y; L, T) if F : U 2 Y (L + T) 1 (F + T) A(U, E) here exiss a selecion Ψ D(U, Y; L, T) of F. In his secion we fix a Φ B(U, Y; L, T). Definiion We say F A U (U, Y; L, T) (respecively F D U (U, Y; L, T)) if F A(U, Y; L, T)) (respecively F D(U, Y; L, T))) wih (L + T) 1 (F + T)(x) (L + T) 1 (Φ + T)(x) = for x U. Definiion Le F A U (U, Y; L, T). We say F : U 2 Y is L Φ essenial in A U (U, Y; L, T) if for any selecion Ψ D(U, Y; L, T) of F any map J D U (U, Y; L, T) wih J U = Ψ U here exiss x U wih (L + T) 1 (J + T) (x) (L + T) 1 (Φ + T) (x). Theorem Le E be a opological vecor space (so auomaically compleely regular), Y a opological vecor space, U an open subse of E, L : dom L E Y a linear single valued map T H L (E, Y). Le F A U (U, Y; L, T) le G A U (U, Y; L, T) be L Φ essenial in A U (U, Y; L, T). For any selecion Ψ D(U, Y; L, T) (respecively Λ D(U, Y; L, T)) of F (respecively G) for any map J D U (U, Y; L, T) wih J U = Ψ U assume here exiss a map H Ψ,Λ,J defined on U [0, 1] wih values in Y wih (L + T) 1 (H Ψ,Λ,J (., η(. )) + T(. )) D(U, E) for any

6 D. O Regan / Filoma 31:11 (2017), coninuous funcion η : U [0, 1] wih η( U) = 0, (L + T) 1 (H Ψ,Λ,J + T)(x) (L + T) 1 (Φ + T)(x) = for any x U (0, 1), H Ψ,Λ,J = Λ, H Ψ,Λ,J = J (here H Ψ,Λ,J 0 1 (x) = H Ψ,Λ,J (x, )) x U : (L + T) 1 (Φ + T)(x) (L + T) 1 (H Ψ,Λ,J + T)(x) for some [0, 1] } is compac. Then F is L Φ essenial in A U (U, Y; L, T). Proof. Le Ψ D(U, Y; L, T) (respecively Λ D(U, Y; L, T)) be any selecion of F (respecively G) consider he map J D U (U, Y; L, T) wih J U = Ψ U. Choose he map H Ψ,Λ,J as in he saemen of Theorem Consider Ω = x U : (L + T) 1 (Φ + T)(x) (L + T) 1 (H Ψ,Λ,J + T)(x) for some [0, 1] }. Noe Ω is compac Ω U =. Thus here exiss a coninuous map µ : U [0, 1] wih µ( U) = 0 µ(ω) = 1. Define a map R µ by R µ (x) = H Ψ,Λ,J (x, µ(x)) = H Ψ,Λ,J (x). Noe R µ(x) µ D U (U, Y; L, T) R µ U = H Ψ,Λ,J 0 U = Λ U. Now since G is L Φ essenial in A U (U, Y; L, T) here exiss x U wih (L + T) 1 (R µ + T)(x) (L + T) 1 (Φ + T)(x), so x Ω. As a resul µ(x) = 1 so (L + T) 1 (H Ψ,Λ,J 1 + T)(x) (L + T) 1 (Φ + T)(x) = (L + T) 1 (J + T)(x) (L + T) 1 (Φ + T)(x), we are finished. Remark If in Theorem 2.19 he space E is addiionally normal hen he assumpion ha Ω = x U : (L + T) 1 (Φ + T)(x) (L + T) 1 (H Ψ,Λ,J + T)(x) for some [0, 1] } is compac in he saemen of Theorem 2.19 can be replaced by Ω is closed. Remark We say F MA(U, Y; L, T) if F : U 2 Y (L + T) 1 (F + T) A(U, E) we say F MA U (U, Y; L, T) if F MA(U, Y; L, T)) wih (L + T) 1 (F + T)(x) (L + T) 1 (Φ + T)(x) = for x U. Now we say F MA U (U, Y; L, T) is L Φ essenial in MA U (U, Y; L, T) if for every map J MA U (U, Y; L, T) wih J U = F U here exiss x U wih (L + T) 1 (J + T) (x) (L + T) 1 (Φ + T) (x). There is an obvious analogue of Theorem 2.19 for MA maps. References [1] R.P. Agarwal, D. O Regan, Coninuaion mehods for closed, weakly closed, DKT WDKT maps, Compuers Mahemaics wih Applicaions 38 (1999) [2] R.P. Agarwal D. O Regan, An essenial map heory for U κ c PK maps, Topological Mehods in Nonlinear Analysis 21 (2003) [3] G. Gabor, L. Gorniewicz M. Slosarski, Generalized opological essenialiy coincidence poins of mulivalued maps, Se-Valued Analysis 17 (2009) [4] A. Granas, Sur la méhode de coninuié de Poincaré, C.R. Acad. Sci. Paris 282 (1976) [5] D. O Regan, Fixed poin heorems for he B κ admissible maps of Park, Applicable Analysis 79 (2001) [6] D. O Regan, Coincidence poins for mulivalued maps based on Φ-epi Φ-essenial maps, Dynamic Sysems Applicaions 24 (2015) [7] R. Precup, On he opological ransversaliy principle, Nonlinear Analysis 20 (1993) 1 9.