Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 216146, pot

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 216146, 11 pages doi:10.1155/2011/216146 Research Article Generalized Lefschetz Sets ´ Mirosław Slosarski ´ Department of Electronics, Technical University of Koszalin, Sniadeckich 2, 75-453 Koszalin, Poland ´ Correspondence should be addressed to Mirosław Slosarski, slosmiro@gmail.com Received January 2011; Accepted March 2011 Academic Editor: Marl` ne Frigon e ´ Copyright q 2011 Mirosław Slosarski This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited We generalize and modify Lefschetz sets defined in 1976 by L Gorniewicz, which leads to more ´ general results in fixed point theory Introduction In 1976 L Gorniewicz introduced a notion of a Lefschetz set for multivalued admissible ´ maps The paper attempts at showing that Lefschetz sets can be defined on a broader class of multivalued maps than admissible maps This definition can be presented in many ways, and each time it is the generalization of the definition from 1976 These generalizations essentially broaden the class of admissible maps that have a fixed point Also, they are a homologic tool for examining fixed points for a class of multivalued maps broader than just admissible maps Preliminaries ˘ Throughout this paper all topological spaces are assumed to be metric Let H∗ be the Cech homology functor with compact carriers and coefficients in the field of rational numbers Q from the category of Hausdorff topological spaces and continuous maps to the category of {Hq X } is a graded graded vector spaces and linear maps of degree zero Thus H∗ X ˘ vector space, Hq X being the q-dimensional Cech homology group with compact carriers of X For a continuous map f : X → Y , H∗ f is the induced linear map f∗ {fq }, where fq : Hq X → Hq Y see 1, A space X is acyclic if i X is nonempty, ii Hq X for every q ≥ 1, iii H0 X ≈ Q 2 Fixed Point Theory and Applications A continuous mapping f : X → Y is called proper if for every compact set K ⊂ Y the set f −1 K is nonempty and compact A proper map p : X → Y is called Vietoris provided that for every y ∈ Y the set p−1 y is acyclic Let X and Y be two spaces, and assume that for every x ∈ X a nonempty subset ϕ x of Y is given In such a case we say that ϕ : X Y is a multivalued mapping For a multivalued mapping ϕ : X Y and a subset U ⊂ Y , we let: ϕ−1 U x ∈ X; ϕ x ⊂ U 2.1 If for every open U ⊂ Y the set ϕ−1 U is open, then ϕ is called an upper semicontinuous mapping; we will write that ϕ is u.s.c Proposition 2.1 see 1, Assume that ϕ : X Y and ψ : Y compact values and p : Z → X is a Vietoris mapping Then 2.1.1 for any compact A ⊂ X, the image ϕ A set; 2.1.2 the composition ψ ◦ ϕ : X 2.1.3 the mapping ϕp : X x∈A T, ψ ◦ ϕ x T are u.s.c mappings with ϕ x of the set A under ϕ is a compact y∈ϕ x Z, given by the formula ϕp x ψ y , is an u.s.c mapping; p−1 x , is u.s.c Let ϕ : X Y be a multivalued map A pair p, q of single-valued, continuous maps is called a selected pair of ϕ written p, q ⊂ ϕ if the following two conditions are satisfied: i p is a Vietoris map, ii q p−1 x ⊂ ϕ x for any x ∈ X Definition 2.2 A multivalued mapping ϕ : X exists a selected pair p, q of ϕ Y is called admissible provided that there Proposition 2.3 see Let ϕ : X Y and ψ : Y composition ψ ◦ ϕ : X Z is an admissible map Z be two admissible maps Then the Proposition 2.4 see Let ϕ : X ϕ×ψ :X×Z Y × T is admissible T be admissible maps Then the map Y and ψ : Z Y is an admissible map, Y0 ⊂ Y , and X0 Proposition 2.5 see If ϕ : X Y0 of ϕ to the pair X0 , Y0 is an admissible map contraction ϕ0 : X0 ϕ−1 Y0 , then the Proposition 2.6 see If p : X → Y is a Vietoris map, then an induced mapping p∗ : H∗ X −→ H∗ Y 2.2 is a linear isomorphism Let u : E → E be an endomorphism of an arbitrary vector space Let us put N u for some n}, where un is the nth iterate of u and E E/N u Since {x ∈ E : un x u N u ⊂ N u , we have the induced endomorphism u : E → E defined by u x u x We call u admissible provided that dim E < ∞ Fixed Point Theory and Applications E Let u {uq } : E → E be an endomorphism of degree zero of a graded vector space {Eq } We call u a Leray endomorphism if i all uq are admissible, ii almost all Eq are trivial For such a u, we define the generalized Lefschetz number Λ u of u by putting −1 q tr uq , Λ u 2.3 q where tr uq is the ordinary trace of uq cf The following important property of a Leray endomorphism is a consequence of a well-known formula tr u ◦ v tr v ◦ u for the ordinary trace An endomorphism u : E → E of a graded vector space E is called weakly nilpotent if for every q ≥ and for every x ∈ Eq , there exists an integer n such that Since for a weakly nilpotent endomorphism u : E → E we have N u E, we get un x q the following Proposition 2.7 If u : E → E is a weakly nilpotent endomorphism, then Λ u Proposition 2.8 Assume that in the category of graded vector spaces the following diagram commutes E′ u E′′ v u′ E′ u 2.4 u′′ E′′ If one of u ,u is a Leray endomorphism, then so is the other and Λ u Λ u Let ϕ : X X, be an admissible map Let p, q ⊂ ϕ, where p : Z → X is a Vietoris −1 mapping and q : Z → X a continuous map Assume that q∗ ◦ p∗ : H∗ X → H∗ X is a Leray endomorphism for all pairs p, q ⊂ ϕ For such a ϕ, we define the Lefschetz set Λ ϕ of ϕ by putting Λ ϕ −1 Λ q∗ p∗ ; p, q ⊂ ϕ 2.5 Let X0 ⊂ X and let ϕ : X, X0 X, X0 be an admissible map We define two admissible X given by ϕX x ϕ x for all x ∈ X and ϕX0 : X0 X0 ϕX0 x maps ϕX : X ϕ x for all x ∈ X0 Let p, q ⊂ ϕX , where p : Z → X is a Vietoris mapping and p z, q : Z → X a continuous map We shall denote by p : Z, p−1 X0 → X, X0 p z q z for all z ∈ Z, p : p−1 X0 → X0 p z p z , and q : Z, p−1 X0 → X, X0 q z q : p−1 X0 → X0 q z q z for all z ∈ p−1 X0 We observe that p, q ⊂ ϕ and p, q ⊂ ϕX0 Fixed Point Theory and Applications X, X0 be an admissible map of pairs and p, q ⊂ Proposition 2.9 see Let ϕ : X, X0 −1 −1 ϕX If any two of the endomorphisms q∗ p∗ : H X, X0 → H X, X0 , q∗ p∗ : H X → H X , −1 q∗ p∗ : H X0 → H X0 are Leray endomorphisms, then so is the third and −1 Λ q∗ p∗ −1 Λ q∗ p∗ − Λ q∗ p−1 ∗ 2.6 Proposition 2.10 see If ϕ : X Y and ψ : Y T are admissible, then the composition ψ◦ϕ : X T is admissible, and for every p1 , q1 ⊂ ϕ and p2 , q2 ⊂ ψ there exists a pair −1 −1 −1 p, q ⊂ ψ ◦ ϕ such that q2∗ p2∗ ◦ q1∗ p1∗ q∗ p∗ Definition 2.11 An admissible map ϕ : X X is called a Lefschetz map provided that the Lefschetz set Λ ϕ of ϕ is well defined and Λ ϕ / {0} implies that the set Fix ϕ {x ∈ X : x ∈ ϕ x } is nonempty Definition 2.12 Let E be a topological vector space One shall say that E is a Klee admissible space provided that for any compact subset K ⊂ E and for any open cover α ∈ CovE K there exists a map πα : K −→ E 2.7 such that the following two conditions are satisfied: 2.12.1 for each x ∈ K there exists V ∈ α such that x, πα x ∈ V , 2.12.2 there exists a natural number n dimensional subspace of E nK such that πα K ⊂ En , where En is an n- Definition 2.13 One shall say that E is locally convex provided that for each x ∈ E and for each open set U ⊂ E such that x ∈ U there exists an open and convex set V ⊂ E such that x ∈ V ⊂ U It is clear that if E is a normed space, then E is locally convex Proposition 2.14 see 1, Let E be locally convex Then E is a Klee admissible space Let Y be a metric space, and let IdY : Y → Y be a map given by formula IdY y for each y ∈ Y y Definition 2.15 see A map r : X → Y of a space X onto a space Y is said to be an mr-map if there is an admissible map ϕ : Y X such that r ◦ ϕ IdY Definition 2.16 see 3, A metric space X is called an absolute multiretract notation: X ∈ AMR provided there exists a locally convex space E and an mr-map r : E → X from E onto X Definition 2.17 see 3, A metric space X is called an absolute neighborhood multiretract notation: X ∈ ANMR provided that there exists an open subset U of some locally convex space E and an mr-map r : U → X from U onto X Fixed Point Theory and Applications Proposition 2.18 see 3, A space X is an ANMR if and only if there exists a metric space Z and a Vietoris map p : Z → X which factors through an open subset U of some locally convex E, that is, there are two continuous maps α and β such that the following diagram is commutative p Z X 2.8 β α U Proposition 2.19 see Let X ∈ ANMR, and let V ⊂ X be an open set Then V ∈ ANMR Proposition 2.20 see Assume that X is ANMR Let U be an open subset in X and ϕ : U an admissible and compact map, then ϕ is a Lefschetz map Let ϕX : X U X be a map Then ϕn X ⎧ ⎪IdX , ⎪ ⎪ ⎨ ϕ , ⎪ X ⎪ ⎪ ⎩ ϕX ◦ ϕX ◦ · · · ◦ ϕX n-iterates for n 0, for n 1, 2.9 for n > We denote multivalued maps with ϕXY : X Y , and ψZ : Z Z If a nonempty set B given A ⊂ X, a nonempty set B ⊂ Y and ϕXY A ⊂ B then a multivalued map ϕAB : A ϕXY x for each x ∈ X by ϕAB x Definition 2.21 see A multivalued map ϕXY : X Y is called locally admissible provided for any compact and nonempty set K ⊂ X there exists an open set V ⊂ X such X is admissible that K ⊂ V and ϕV X : V Proposition 2.22 see Let ϕXY : X Y and ψY Z : Y ψY Z ◦ ϕXY : X Z is locally admissible the map ΦXZ Z be locally admissible maps Then Proposition 2.23 see Let A ⊂ X be a nonempty set, and let ϕXY : X Y is locally admissible admissible map Then a map ϕAY : A Y be a locally Definition 2.24 see 2, A multivalued map ϕX : X X is called a compact absorbing contraction written ϕX ∈ CAC X provided there exists an open set U ⊂ X such that 2.24.1 ϕX U ⊂ U and the ϕU : U ϕX U ⊂ U , 2.24.2 for every x ∈ X there exists n Proposition 2.25 see Let ϕX : X CAC X then ϕX is a Lefschetz map U, ϕU x ϕX x for every x ∈ X is compact nx such that ϕn x ⊂ U X X be an admissible map, X ∈ ANMR, and ϕX ∈ Fixed Point Theory and Applications Proposition 2.26 see Let ϕX ∈ CAC X , and let U ⊂ X be an open set from Definition 2.24 2.26.1 Let B be a nonempty set in X and ϕX B ⊂ B Then U ∩ B / ∅ 2.26.2 For any n ∈ N ϕn ∈ CAC X X 2.26.3 Let V ⊂ X be a nonempty and open set Assume that ϕX V ⊂ V Then ϕV ∈ CAC V Main result Let X be a metric space, ϕX : X ΩAD ϕ X a multivalued map, and let V ⊂ X : V is open, ϕV : V V is admissible, ϕV V ⊂ V 3.1 Obviously the above family of sets can be empty Thus we can define the following class of multivalued maps: ADL X, ΩAD ϕ / ∅ ϕX : X 3.2 All the admissible maps ϕX : X X particularly belong to the above class of maps because X is called acyclic X ∈ ΩAD ϕ We shall remind that the multivalued map ϕX : X if for every x ∈ X the set ϕX x is nonempty, acyclic, and compact It is known from the mathematical literature that an acyclic map is admissible and the maps r, s : Γ → X given by r x, y x, s x, y y for every x, y ∈ Γ, 3.3 where Γ { x, y ∈ X × Y ; y ∈ ϕX x }, are a selective pair r, s ⊂ ϕX −1 X, if the homomorphism s∗ r∗ : H∗ X → Moreover, for an acyclic map ϕX : X H∗ X is a Leray endomorphism, then Lefschetz set Λ ϕX consists of only one element and Λ ϕX −1 Λ s∗ r∗ 3.4 For a certain class of multivalued maps ϕX ∈ ADL we define a generalized Lefschetz set ΛG ϕX of a map ϕX in such a way that the conditions of the following definition are satisfied Let ϕV : V → V be an admissible map One shall say that a set Λ ϕV is well defined −1 if for every p, q ⊂ ϕV the map q∗ p∗ : H∗ V → H∗ V is a Leray endomorphism Definition 3.1 Assume that there exists a nonempty family of sets ΥAD ϕ ⊂ ΩAD ϕ such that if for any V ∈ ΥAD ϕ Λ ϕV is well defined, then the following conditions are satisfied: −1 {Λ s∗ r∗ } see 3.3 , 3.1.1 if ϕX : X X is acyclic, then ΛG ϕX 3.1.2 if ϕX : X X is admissible, then X ∈ ΥAD ϕ and Λ ϕX / {0} 3.1.3 ⇒ ΛG ϕX / {0} , ΛG ϕX / {0} ⇒ there exists V ∈ ΥAD ϕ such that Λ ϕV / {0} 3.5 Fixed Point Theory and Applications From the above definition it in particular results that see 3.1.1 valued map, continuous and Λ f is well defined, then ΛG f if f : X → X is a single- Λ f 3.6 We shall present a few examples proving that Lefschetz sets can be defined in many ways while retaining the conditions contained in Definition 3.1 Example 3.2 Let ϕX : X defined, then we define X be an admissible map, and let ΥAD ϕ {X} If Λ ϕX is well Λ ϕX ΛG ϕX 3.7 The above example consists of Lefschetz set definitions common in mathematical literature and introduced by L Gorniewicz ´ Example 3.3 Let ϕX : X X be an admissible map, and let ΥAD ϕ be a family of sets V ∈ ΩAD ϕ such that there exists p, q ⊂ ϕV and there exists p, q ⊂ ϕX such that the following diagram H∗ (V ) q∗ (p∗ )−1 H∗ (V ) u∗ υ∗ u∗ H∗ (X) −1 q∗ p∗ 3.8 H∗ (X) is commutative It is obvious that X ∈ ΥAD ϕ , hence ΥAD ϕ / ∅ Assume that for any V ∈ ΥAD ϕ Λ ϕV is well defined We define ΛG ϕX Λ ϕV 3.9 V ∈ΥAD ϕ Justification Let us notice that if ϕX is acyclic, then from the commutativity of the above diagram it results −1 −1 {Λ s∗ r∗ }, hence ΛG ϕV {Λ s∗ r∗ } The second that for every V ∈ ΥAD ϕ Λ ϕV and third conditions of Definition 3.1 are obvious Let A ⊂ X be a nonempty set, and let Oε A where d is metric in X x ∈ X; there exists y ∈ A such that d x, y < ε , 3.10 Fixed Point Theory and Applications Example 3.4 Let X, d be a metric space, where d is a metric such that, for each x, y ∈ X be a multivalued map and let K ϕX X Let X × X d x, y ≤ 1, let ϕX : X ΥAD ϕ V ∈ ΩAD ϕ : V O2/n K for some n 3.11 Assume that ΥAD ϕ / ∅ and for all V ∈ ΩAD ϕ Λ ϕV is well defined We define ΛG ϕX U O2/k K , Λ ϕU , where n ∈ N; O2/n K ∈ ΥAD ϕ k 3.12 Justification The first condition of Definition 3.1 results from the commutativity of the following diagram: H∗ (U) u∗ H∗ (X) υ∗ q∗ (p∗ )−1 H∗ (U) u∗ −1 q∗ p∗ 3.13 H∗ (X), −1 where u∗ i∗ is a homomorphism determined by the inclusions i : U → X, v∗ q∗ p∗ The maps p, q are the respective contractions of maps p, q, p, q ⊂ ϕX Condition 3.1.2 results from the fact that X O2 K ∈ ΥAD ϕ and ΛG ϕX Λ ϕO2 K Λ ϕX 3.14 Satisfying Condition 3.1.3 is obvious Before the formulation of another example, let us introduce the following definition and provide necessary theorems X be a map One shall say that a nonempty set B ⊂ X has an Definition 3.5 Let ϕX : X absorbing property writes B ∈ AP ϕ if for each x ∈ X there exists a natural number n such that ϕn x ⊂ B X Let ΘAD ϕ ΩAD ϕ ∩ AP ϕ We observe that if ϕX : X ΘAD ϕ / ∅ since X ∈ ΘAD ϕ Theorem 3.6 see Let ϕX : X all p, q ⊂ ϕX the homomorphism X is admissible then X be an admissible map Then for any V ∈ ΘAD ϕ and for −1 q∗ p∗ : H∗ X, V −→ H∗ X, V is weakly nilpotent (see Proposition 2.9), where p, q denote a respective contraction of p, q 3.15 Fixed Point Theory and Applications Theorem 3.7 Let ϕX : X well defined Then X be an admissible map Assume that for each V ∈ ΘAD ϕ Λ ϕV is Λ ϕX Λ ϕV V ∈ΘAD ϕ 3.16 −1 −1 c0 We observe that a map q∗ p∗ : Proof Let V ∈ ΘAD ϕ , p, q ⊂ ϕX , and let Λ q∗ p∗ H∗ X, V p, q ⊂ ϕ, ϕ : X, V X, V is weakly nilpotent so from H∗ X, V −1 Propositions 2.7 and 2.9 Λ q∗ p∗ Λ q∗ p−1 c0 , where p, q ⊂ ϕV and p, q denote a ∗ respective contraction of p, q Hence c0 ∈ Λ ϕV and Λ ϕX ⊂ V ∈ΘAD ϕ Λ ϕV It is clear that X ∈ ΘAD ϕ and the proof is complete Example 3.8 Let ϕX : X X be a multivalued map, and let ΥAD ϕ ΘAD ϕ 3.17 Assume that the following conditions are satisfied: 3.8.1 ΥAD ϕ / ∅, 3.8.2 for all V ∈ ΥAD ϕ Λ ϕV is well defined, 3.8.3 V ∈ΥAD ϕ Λ ϕV / ∅ We define ΛG ϕX Λ ϕV 3.18 V ∈ΥAD ϕ Justification Condition 3.1.1 results from Proposition 2.7 and Theorem 3.6 Let us notice that if a map X is admissible, then X ∈ ΥAD ϕ and from Theorem 3.7 we get ϕX : X ΛG ϕX Λ ϕX , 3.19 and condition 3.1.2 is satisfied Condition 3.1.3 is obvious It is crucial to notice that the definition of Lefschetz set encompassed in this example agrees in the class of admissible maps with the familiar definition of a Lefschetz set introduced by L Gorniewicz It is possible to create an example see of a multivalued ´ X that is not admissible and satisfies the conditions of Example 3.8 map ϕX : X Example 3.9 Let ϕX : X X be a multivalued map, and let ΥAD ϕ ΘAD ϕ Assume that the following conditions are satisfied: 3.9.1 ΥAD ϕ / ∅, 3.9.2 for all V ∈ ΥAD ϕ Λ ϕV is well defined 3.20 10 Fixed Point Theory and Applications We define ΛG ϕX Λ ϕV 3.21 V ∈ΥAD ϕ Justification Condition 3.1.1 results from Proposition 2.7 and Theorem 3.6 If a map ϕX : X X is admissible, then X ∈ ΥAD ϕ and hence condition 3.1.2 is satisfied Condition 3.1.3 is obvious The definition of a Lefschetz set in Example 3.9 is much more general than the definition in Example 3.8, and as consequence it encompasses a broader class of maps This definition ignores the inconvenient assumption 3.8.3 Let us define a Lefschetz map by the application of the new Lefschetz set definition Definition 3.10 One shall say that a map ϕX ∈ ADL is a general Lefschetz map provided that the following conditions are satisfied: 3.10.1 there exists ΥAD ϕ / ∅ such that conditions 3.1.1 – 3.1.3 are satisfied, 3.10.2 for any V ∈ ΥAD ϕ Λ ϕV is well defined We will formulate, and then prove, a very general fixed point theorem Theorem 3.11 Let X ∈ ANMR Assume that the following conditions are satisfied: 3.11.1 ϕX ∈ CAC X (see Definiation 2.24), 3.11.2 there exists ΥAD ϕ / ∅ such that conditions (3.1.1)–(3.1.3) are satisfied Then ϕX is a general Lefschetz map, and if ΛG ϕX / {0} then Fix ϕX / ∅ Proof From the assumption ΥAD ϕ / ∅, thus we show that for all V ∈ ΥAD ϕ Λ ϕV is well defined Let V ∈ ΥAD ϕ , then from 2.26.3 ϕV ∈ CAC V , so from Propositions 2.19 and 2.25 Λ ϕV is well defined Assume that ΛG ϕX / {0}, then from 3.1.3 there exists V ∈ ΥAD ϕ such that Λ ϕV / {0} By the application of 2.26.3 , Propositions 2.19, and 2.25, we get ∅ / Fix ϕV ⊂ Fix ϕX and the proof is complete The following is a conclusion from Theorem 3.11 Corollary 3.12 Let X ∈ ANMR, ϕX : X X be locally admissible (not necessarily admissible), and let ϕX ∈ CAC X Then ϕX is a general Lefschetz map, and if ΛG ϕX / {0} then Fix ϕX / ∅ Proof Let U ⊂ X be an open set from Definition 2.24, and let K ϕU U ⊂ U We define ΘAD ϕ see Examples 3.8 and 3.9 The map ϕX is locally admissible, so there ΥAD ϕ X is admissible We observe that exists an open set V ⊂ X such that K ⊂ V and ϕV X : V U ∩ V is admissible, compact and U ∩ V ∈ AP ϕ , U ∩ V ∈ ΥAD ϕ since ϕU∩V : U ∩ V hence ΥAD ϕ / ∅ If we define a generalized Lefschetz set now as in Example 3.9, then from Theorem 3.11 we get a thesis Fixed Point Theory and Applications 11 Finally we shall provide an example which shows that the new Lefschetz set definition is more general than the definition of Lefschetz set for admissible maps already familiar in mathematical literature Example 3.13 see Let C be a complex number set, and let f : C \ {0} → C \ {0} be single-valued continuous and compact map Assume that Fix f ∅, and choose an open set V such that f C \ {0} ⊂ V ⊂ C\{0} Let g : V → V be a compact g V ⊂ V and continuous C \ {0} given by map such that Λ g / We define a multivalued map ϕC\{0} : C \ {0} formula ϕC\{0} z ⎧ ⎨f z , for z ∈ V, / ⎩ f z ,g z for z ∈ V The map ϕC\{0} is admissible, so ΥAD ϕ 3.22 ΘAD ϕ / ∅ see Examples 3.8 and 3.9 Let ΛG ϕ Λ ϕU 3.23 {0} 3.24 U∈ΥAD ϕ see Example 3.9 We observe that Λ ϕC\{0} since the only selective pair is the pair IdC\{0} , f ⊂ ϕC\{0} , but Fix f ∅ 3.25 It is clear that V ∈ ΥAD ϕ and Λ ϕV / {0}, since from the assumption Λ g / Hence ΛG ϕX / {0}, ∅ / Fix ϕV ⊂ Fix ϕC\{0} 3.26 References J Andres and L Gorniewicz, Topological Principles for Boundary Value Problems, Kluwer Academic ´ Publishers, Dordrecht, The Netherlands, 2003 L Gorniewicz, Topological Methods in Fixed Point Theory of Multi-Valued Mappings, Springer, New York, ´ NY, USA, 2006 ´ R Skiba and M Slosarski, “On a generalization of absolute neighborhood retracts,” Topology and Its Applications, vol 156, no 4, pp 697–709, 2009 ´ M Slosarski, “On a generalization of approximative absolute neighborhood retracts,” Fixed Point Theory, vol 10, no 2, pp 329–346, 2009 ´ M Slosarski, “Locally admissible multi-valued maps,” admitted for printing in Discussiones Mathematicae Differential Inclusions, Control and Optimization ... map, X ∈ ANMR, and ϕX ∈ Fixed Point Theory and Applications Proposition 2.26 see Let ϕX ∈ CAC X , and let U ⊂ X be an open set from Definition 2.24 2.26.1 Let B be a nonempty set in X and ϕX B ⊂... z q z for all z ∈ p−1 X0 We observe that p, q ⊂ ϕ and p, q ⊂ ϕX0 Fixed Point Theory and Applications X, X0 be an admissible map of pairs and p, q ⊂ Proposition 2.9 see Let ϕ : X, X0 −1 −1... multiretract notation: X ∈ ANMR provided that there exists an open subset U of some locally convex space E and an mr-map r : U → X from U onto X Fixed Point Theory and Applications Proposition 2.18