A generalized coincidence point index

@Applied General Topology c Universidad Polit´ecnica de Valencia Volume 6, No. 1, 2005 pp. 87-100 A generalized coincidence point index N. M. Benkafadar and M. C. Benkara-Mostefa1 Abstract. The paper is devoted to build for some pairs of continuous single-valued maps a coincidence point index. The class of pairs (f, g) satisfies the condition that finduces an epimorphism of the ∨ Cech homology groups with compact supports and coefficients in the field of rational numbers Q. Using this concept one defines for a class of multi-valued mappings a fixed point degree. The main theorem states that if the general coincidence point index is different from {0}, then the pair (f, g) admits at least a coincidence point. The results may be considered as a generalization of the above Eilenberg-Montgomery theorems [12], they include also, known fixed-point and coincidence-point theorems for single-valued maps and multi-valued transformations. 2000 AMS Classification: 54C60, 54H25, 55M20. 58C06. Keywords: Fixed point, Concidence point, Index, Degree, Multi-valued mapping. 1. Introduction Let f, g :X−→ Ybe two continuous single valued maps of Hausdorff topological spaces. The coincidence problem, which is a generalization of the fixed point problem, is concerned with conditions which guarantees the existence of a solution for the equation f(x) = g(x).A such point x∈Xis called a coincidence point of the pair of maps (f, g).The study of this problem has been treated first in 1946 by Eilenberg-Montgomery [12]. Note that the Eilenberg- Montgomery theorem is a natural generalization of the Lefschetz fixed point theorem, it implies also, the fixed point theorems of Kakutani [21] and Wallace [30]. Topological invariants for different classes of pairs of maps have been studied by many authors [9], [14], [15], [20], [22], [23], [27] and others. The purpose of this note is to describe a generalized coincidence point index for a new class 1The authors acknowledge the support of A.N.D.R.U., (Contract No 03/06 Code CU 19905) and M.E.R.S., (Project No B*2501/04/04), Laboratory M.M.E.R.E. 88 N. M. Benkafadar and M. C. Benkara-Mostefa of pairs of continuous maps (f, g) which satisfy the condition that finduces ar-homomorphism [3], [4] for homology with compact carries. Moreover, one gives several applications of the general coincidence point index in fixed point theory for multi-valued mappings. One uses the Dold’s fundamental class around a compact of a finite euclidean space En[10], Hdenotes the ∨ Cech homology functor with compact carries and coefficients in the field of rational numbers Q, from the category Top(2)of Hausdorff topological pairs and continuous maps to the category Lgof graded vector spaces over the set of rational numbers Qand linear maps of degree zero [13], [18], [29]. 2. Maps n-decomposing. Let G1and G2be two additive abelian groups, τ:G1−→ G2be a homomorphism. Definition 2.1 ([3]).A homomorphism τis a called a r-homomorphism if τ admits a right-inverse homomorphism. The definition signifies, since τ:G1−→ G2is a r-homomorphism then there exists a homomorphism σ:G2−→ G1such that τ◦σ=IdG2, where IdG2is the automorphism identity on G2. The following properties are satisfied. Proposition 2.2. A homomorphism τ:G1−→ G2is a r-homomorphism if and only if the following conditions are satisfied : (1) τis an epimorphism; (2) G1=Ker τ ⊕ G , where Gis a subgroup of G1. Proposition 2.3. If G1and G2are two modules over a field Kand if τ: G1−→ G2is an epimorphism then τis a r-homomorphism. Proposition 2.4 ([3]).Let τ1:G1−→ G2and τ2:G2−→ G3be two rhomomorphisms then their composition τ=τ2◦τ1:G1−→ G3is also a r-homomorphism. The notion of r-homomorphisms has been introduced by Borsuk and Kosinsk [3], [4]. Let (X, A) and (Y, B) be two objects of the category Top(2)of Hausdorff topological pairs and continuous maps and f: (X, A)−→ (Y, B) be a morphism from the Hausdorff pair (X, A) into an other Hausdorff pair (Y, B). Let Hbe the ∨ Cech homology functor with compact carries and coefficients in the field of rational numbers Q, from the category Top(2)of Hausdorff topological pairs and continuous maps to the category Lgof graded vector spaces over the set of rational numbers Qand linear maps of degree zero [13], [18], [29]. A generalized coincidence point index 89 Definition 2.5. A continuous single-valued map f: (X, A)−→ (Y, B)is said to be n-decomposing in the rank n≥0on the Hausdorff pair (Y, B)if the homomorphism f∗:Hn(X, A)→Hn(Y, B)induced by f, is a r-homomorphism. The set of the right-inverse homomorphisms of f∗on (Y, B) will be denoted by Ω(f∗;Y, B). The following propositions and corollaries, prove that the class of n-decomposing maps is vast. Definition 2.6 ([3]).A continuous single-valued map f: (X, A)−→ (Y, B)is called a r-map if fadmits a continuous right inverse. Proposition 2.7. Let f: (X, A)−→ (Y, B)be a single-valued map which is a r-map, then fis n-decomposing on (Y, B)for every rank n≥0. Corollary 2.8. A retraction rof a pair (X, A)onto (X′, A′)is n-decomposing on the retract (X′, A′)of (X, A). Definition 2.9 ([3]).A continuous single-valued map f: (X, A)−→ (Y, B) is said to be a h-map if there exists a continuous single-valued g: (Y, B)−→ (X, A)such that their composition f◦gand the identity map Id(Y,B): (Y, B)−→ (Y, B)are homotopic. Proposition 2.10. If f: (X, A)−→ (Y, B)is a h-map, then fis n-decomposing on (Y, B)for every n≥0. Corollary 2.11. A lower retraction r: (X, A)−→ (X′, A′)is n-decomposing on each lower retract (X′, A′)of (X, A). Proposition 2.12. Let f: (X, A)−→ (Y, B)be a continuous single-valued map. If there exists a continuous single-valued map g: (Z, C)−→ (X, A) such that their composition f◦gis n-decomposing on (Y, B),then fis also n-decomposing on (Y, B). Corollary 2.13. Let f: (X, A)−→ (Y, B)be a continuous single-valued map and (Z, C)⊆(X, A). If the restriction of fon (Z, C)is n-decomposing on (Y, B),then fis also n-decomposing on (Y, B). Proposition 2.14. Let f: (X, A)−→ (Y, B)be a n-decomposing on (Y, B) and g: (Y, B)−→ (Z, C)be a n-decomposing on (Z, C),then their composition g◦fis n-decomposing on (Z, C). Definition 2.15 ([5]).A space Xis Q-acyclic provided: (i) Xis non-empty, (ii) Hq(X) = 0 for all q⩾1and (iii) H0(X)≈Q. Proposition 2.16. Let f: (X, A)−→ (Y, B)be a continuous single-valued map such that: (1) fis proper and surjective; (2) f−1(B) = A; (3) f−1(y)is Q-acyclic for every y∈Y. Then the map fis n-decomposing on (Y, B)for every n≥0. 90 N. M. Benkafadar and M. C. Benkara-Mostefa Proposition 2.17. Let Ube an open subset of an Euclidean space Enand K be a compact subset of U, then the injection i: (U, U\K)−→ (En, En\K)is n-decomposing on (En, En\K). 3. Generalized coincidence point index Let Ube an open subset of an euclidean vector space Enwhich has a fixed orientation. Let (f, g) be a pair of continuous single-valued maps defining as follows: (3.1) Uf ←− Xg −→ En where Xis an arbitrary Hausdorff topological space. Definition 3.1. An element x∈Xis said to be a coincidence point of the pair (f, g)if f(x) = g(x). Let S(f, g) be the set of all coincidence points of the pair (f, g) and F(f, g) be the subset of Udefined as follows: F(f, g) = {u∈U|u∈g(f−1(u)}. Lemma 3.2. One has the equality f(S(f, g)) = F(f, g). Proof. The proof is obvious.  Let Kbe a compact subset of Uwhich contains F(f, g).Thus, one obtains the following diagram: (3.2) (U, U\K)f ←− (X, X\f−1(K)) f−g −→ (En, En\ {θ}) Definition 3.3. A pair of continuous single-valued maps (f, g)as above defined, is called n-admissible on (U, U\K)if fis n-decomposing on (U, U\K). The set of all n-admissible pairs on (U, U\K)is denoted PD(U, U\K). Let (f, g)∈ PD(U, U \K), then if σ∈Ω(f∗;U, U\K) the diagram (3.2) induces the following diagram: (3.3) Hn(U, U\K)f∗ ←− Hn(X, X\f−1(K)) (f−g)∗ −→ Hn(En, En\ {θ}) σց m Hn(X, X\f−1(K)) Let OK∈Hn(U, U\K) be the image of 1 under the composite map: Z=Hn(Sn)−→ Hn(Sn, Sn\K)≅Hn(U, U\K) and O{θ}∈Hn(En, En\ {θ}) be the image of 1 under the composition map: Z=Hn(Sn)−→ Hn(Sn, Sn\ {θ})≅Hn(En, En\ {θ}) where Sn=En∪ {∞}. A generalized coincidence point index 91 The elements OKand O{θ}are called the fundamental classes around the compacts Kand {θ}respectively [9], [10]. Definition 3.4. Let (f, g)be a n-admissible pair on (U, U\K).The generalized coincidence point index of (f, g)relatively σ∈Ω(f∗;U, U\K)is defined as being the rational number Iσ(f, g)which verifies the equality (f−g)∗◦σ(OK) = Iσ(f, g)·O{θ}. Definition 3.5. Let (f, g)be a n-admissible pair on (U, U\K).The generalized coincidence point index of (f, g)is defined as being the set of rational numbers I(f, g) = {Iσ(f, g)|σ∈Ω(f∗;U, U\K)}. Proposition 3.6. If the single-valued map f: (X, X\f−1(K)) −→ (U, U\K) verifies the conditions of the proposition (2.16), then the pair (f, g)is nadmissible on (U, U\K)and I(f, g) = {I(f∗)−1(f, g)}. Proof. The single-valued map f: (X, X\f−1(K)) −→ (U, U\K) induces an isomorphism f∗:Hn(X, X\f−1(K)) −→ Hn(U, U\K) therefore Ω(f∗;U, U\K) = {(f∗)−1}. Proposition 3.7. If F(f, g) = ∅,then I(f, g) = {0}. Proof. Suppose that F(f, g) = ∅then using lemma (3.2) one deduces that S(f, g) = ∅.This equality means that f(x)6=g(x) for each x∈X. Therefore for every σ∈Ω(f∗;U, U\K) we have the following commutative diagram: Hn(U, U\K)σ −→ Hn(X, X\f−1(K)) (f−g)∗ −→ Hn(En, En\{θ}) (f−g)∗ց m Hn(En\{θ}, En\{θ}) where (f−g) = f−g. One concludes the proof remarking that (f−g)∗is the trivial homomorphism.  Corollary 3.8. If I(f, g)6={0},then the pair (f, g)admits at least a coincidence point. Proof. This is a consequence of lemma (3.2).  Let g:U−→ Enbe a continuous single-valued map defined from an open subset Uof an Euclidean vector space Enand Kbe a compact subset of U which contains Fix(g) = {x∈U|x=g(x)}.The fixed point index of gdefined in [9] is the rational Igwhich verifies the equality: (i−g)n∗(OK) = Ig·O{θ}, where i:U−→ Enis the natural injection. Proposition 3.9. The generalized coincidence point index of the pair (i, g)is defined and equal to the fixed point index of g. 92 N. M. Benkafadar and M. C. Benkara-Mostefa Proof. First note that F(i, g) = F ix(g) = {x∈U|x=g(x)}.Let Kbe a compact subset of Enwhich contains F(i, g) = Fix(g).So, one has the diagram: Hn(En, En\K) i∗ ←− Hn(U, U\K)(i−g)∗ −→ Hn(En, En\ {θ}). Therefore, I(i, g)·O{θ}= (i−g)∗◦i−1 ∗(OK) = (i−g)∗(OK) = Ig·O{θ}.  Corollary 3.10. If I(i, g)6={0}then gadmits at least a fixed point. Let (f, g) and (f1, g1) be two pairs of continuous single-valued maps defining as follows: (3.4) Uf ←− Xg −→ En and (3.5) Vf1 ←− X1 g1 −→ En, where Uand Vare two open subsets of Enand Xand X1are two Hausdorff topological spaces. Let Kand K1be two compact subsets of Enwhich contain F(f, g) and F(f1, g1) respectively and such that K⊂K1⊂V⊂V⊂U. For instance, one obtains the following diagrams: (3.6) (U, U\K)f ←− (X, X\f−1(K)) f−g −→ (En, En\{θ}) and (3.7) (V, V \K1)f1 ←− (X1, X1\f−1 1(K1)) f1−g1 −→ (En, En\{θ}). Proposition 3.11. Under the above hypotheses, assume that h: (X1, X1\f−1 1(K1)) −→ (X, X\f−1(K)) is a continuous single-valued map such that the following diagram is commutative: (V, V \K1)f1 ←− (X1, X1\f−1 1(K1)) f1−g1 −→ (En, En\ {θ}) i↓ ↓ hm (U, U\K)f ←− (X, X\f−1(K)) f−g −→ (En, En\ {θ}) where iis the natural injection. Then if the pair (f1, g1)∈ PD(V, V \K1)one can infer that (f, g)∈ PD(U, U \K)and I(f1, g1)⊂I(f, g). Proof. Of course, iinduces an isomorphism i∗:Hn(V, V \K1)−→ Hn(U, U\K) which takes OK1in OK. Moreover, if σ∈Ω(f1∗, V, V \K1),then h∗◦σ◦i−1 ∗∈ Ω(f∗, U, U\K). Let (f, g) be a pair of continuous single-valued maps such that: Uf ←− Xg −→ En and h:X1−→ Xbe a continuous single-valued map defined between two Hausdorff topological spaces X1and X. A generalized coincidence point index 93 Proposition 3.12. If h: (X1, X1\(f◦h)−1(K)) −→ (X, X\f−1(K)) is ndecomposing on (X, X\f−1(K)) and the pair (f, g)is n-admissible on (U, U\K), then (f◦h, g ◦h)∈ PD(U, U \K)and I(f◦h, g ◦h)⊂I(f, g). Proof. Note that F(f◦h, g ◦h)⊆F(f, g)⊆K, the composition f◦his ndecomposing on (U, U\K) (see proposition 2.14), and one has the following diagram: (U, U\K)f◦h ←− (X1, X1\(f◦h)−1(K)) f◦h−g◦h −→ (En, En\ {θ}) Let k∈I(f◦h, g ◦h),then there exists σ∈Ω((f◦h)∗, U, U\K) such that (f◦h−g◦h)∗◦σ(OK) = k·Oθ,therefore (f−g)∗◦h∗◦σ(OK) = k·Oθ. Because h∗◦σ∈Ω(f∗;U, U\K),one deduces that k∈I(f, g). Definition 3.13. Two pairs of continuous single-valued maps defined as follows: Ufi ←− Xgi −→ En,i= 0,1, are called equivariant on a compact K⊂Enif there exist: (1) a Hausdorff pair (X, X\X′)such that: (U, U\K)fi ←− (X, X\X′)fi−gi −→ (En, En\ {θ}), i = 0,1, (2) a pair of continuous maps (ϕ, ψ)n-admissible on (U, U\K)such that: (U, U\K)ϕ ←− (X, X\ϕ−1(K)) ϕ−ψ −→ (En, En\ {θ}) (3) a single-valued map h: (X, X\ϕ−1(K)) −→ (X, X\X′)n-decomposing on (X, X\X′)such that the following diagram is commutative: (U, U\K)f0 ←− (X, X\X′)f0−g0 −→ (En, En\ {θ}) m ↑ hm (U, U\K)ϕ ←− (X, X\ϕ−1(K)) ϕ−ψ −→ (En, En\ {θ}) m ↓ hm (U, U\K)f1 ←− (X, X\X′)f1−g1 −→ (En, En\ {θ}) Proposition 3.14. If (fi, gi), i = 0,1are two equivariant pairs on a compact K⊂En, then (fi, gi)∈ PD(U, U \K), i = 0,1,and I(f0, g0) = I(f1, g1). Proof. Assume (f0, g0) and (f1, g1) are equivariant, then f0∗◦h∗=ϕ∗= f1∗◦h∗therefore f0∗,f1∗are both n-decomposing on (U, U\K) and f0∗=f1∗. Moreover, (f0−g0)∗◦h∗= (ϕ−ψ)∗= (f1−g1)∗◦h∗so (f0−g0)∗= (f1−g1)∗. Definition 3.15. Two pairs (fi, gi), i = 0,1defined as follows: (U, U\K)fi ←− (X, X\X′)fi−gi −→ (En, En\{θ}), i = 0,1, are called homotopic on a compact K⊂Enif the following conditions are verified: 94 N. M. Benkafadar and M. C. Benkara-Mostefa (1) there exists a pair of single-valued maps (ϕ, ψ)n-admissible on (U, U\K)× [0,1] such that: (U, U\K)×[0,1] ϕ ←− (X, X\ϕ−1(K×[0,1])) ϕ−Ψ −→ (En, En\{θ}), (2) there exists a single valued map h: (X, X\X′)−→ (X, X\ϕ−1(K×[0,1])), n-decomposing on (X, X\ϕ−1(K×[0,1])), (3) the following diagram is commutative: (U, U\K)f0 ←− (X, X\X′)f0−g0 −→ (En, En\{θ}) χ0↓h↓ m (U, U\K)×[0,1] ϕ ←− (X, X\ϕ−1(K×[0,1])) ϕ−Ψ −→ (En, En\{θ}) χ1↑h↑ m (U, U\K)f1 ←− (X, X\X′)f1−g1 −→ (En, En\{θ}) where χi(x) = (x, i),for every x∈Uand i= 0,1. Proposition 3.16. If (f0, g0)and (f1, g1)are homotopic on a compact K⊂En then (fi, gi)∈ PD(U, U \K), i = 0,1and I(f0, g0) = I(f1, g1). Proof. Of course, χ0∗and χ1∗are both isomorphisms and are equal, so f0∗= f1∗. One deduces also that f0∗and f1∗are both n-decomposing on (U, U\K). In an other hand, from the commutativity of the diagram one obtains that (f0−g0)∗◦h∗= (f1−g1)∗◦h∗= (ϕ−ψ)∗therefore (f0−g0)∗= (f0−g0)∗. Let (f, g) and (f′, g′) be two pairs defined by the following way: Uf ←− Xg −→ En and U′f′ ←− Xg′ −→ Em where Uand U′are two open subsets of Enand Emrespectively. Let Kbe a compact subset of Enwhich contains F(f, g) and K′be a compact subset Ewhich contains F(f′, g′). Proposition 3.17. If the pairs (f, g)and (f′, g′)are n-admissible on (U, U\K) and (U′, U′\K′)respectively then the pair (f×f′, g ×g′)is (n+m)-admissible on (U×U′, U ×U′\K×K′)and I(f×f′, g ×g′)⊃I(f, g)·I(f′, g′). Proof. One has the following equalities: F(f×f′, g ×g′) = F(f, g)×F(f′, g′), OK×K′=OK×OK′∈Hn+m[(U, U\K)×(U′, U′\K′)] = Hn+m(U×U′, U ×U′\K×K′) and the inclusion: K×K′⊃F(f, g)×F(f′, g′). Therefore, if (σ, σ′)∈Ω(f∗, U, K)×Ω(f′ ∗, U′, K′) one obtains the equalities: (f×f′−g×g′)∗◦(σ×σ′)(OK×K′) = A generalized coincidence point index 95 [(f−g)∗◦σ×(f′−g′)∗◦σ′] (OK×OK′) = (f−g)∗◦σ(OK)×(f′−g′)∗◦σ′(OK) = [Iσ(f, g)·Iσ′(f′, g′)] O{θ}. 4. Generalized fixed point degree of multi-valued mappings Let Xand Ybe two Hausdorff topological spaces. A multi-valued mapping taking Xto Yis a relation Fwhich associates to each element x∈Xa non empty subset F(x)⊂Y. Let K(Y) be the collection of all non empty compact subsets of Yand F:X−→ K(Y) be a multi-valued mapping. The subset: ΓX(F) = {(x, y)∈X×Y|y∈F(x)}, of X×Yis called the graph of the multi-valued mapping Fon X. In this case one could define two natural projectors: tF: ΓX(F)−→ X and rF: ΓX(F)−→ Y such tF(x, y) = x, rF(x, y) = yfor every (x, y)∈ΓX(F). For each element x∈Xone has the equality F(x) = rF(t−1 F(x)).The quintuple [X, Y, ΓX(F), tF, rF] is called the canonical representation of the multivalued F:X−→ K(Y). Let [X1, X2, X0, f1, f2] be a quintuple constituted of Hausdorff topological spaces Xi, i = 0,1,2 and continuous maps fj:X0−→ Xi, j = 1,2 and such that f1is onto and the inverse image of each element x∈X1is compact, then the equality F(x) = g◦f−1(x) defines a multi-valued mapping F:X1−→ K(X2).In this case the quintuple [X1, X2, X0, f1, f2] is called a representation of F:X1−→ K(X2). Two quintuples [X1, X2, X0, f1, f2], [X1, X2, X0, g1, g2] are called equivalents if g1◦f−1 1(x) = F(x) = g2◦f−1 2(x) for each x∈X1. A multi-valued mapping F:X−→ K(Y) is called upper semi continuous if F−1 +(V) = {x∈X|F(x)⊂V}is an open subset of Yfor every open subset Vof X. A multi-valued G:X−→ K(Y) is said to be a selector of F:X−→ K(Y) if G(x)⊆F(x) for every element x∈X. Let Hbe the ∨ Cech homology functor with compact carries and coefficient in the set of rational numbers Q. A multi-valued mapping F:X−→ K(Y) is called to be Q-acyclic provided the image F(x) is Q-acyclic for every element x∈X,Fis said to be compact provided F(X) is contained in a compact subset of Y. More properties on multi-valued mappings can be found in [24]. Let F:U−→ K(En) be a multi-valued mapping and Kbe a compact subset of U⊆En.In this case F(tF, rF) = {x∈U|x∈rF(t−1 F)(x)}={x∈ U|x∈F(x)}=Fix(F).