Coincidence Continuation Theory for Multivalued Maps with Selections in a Given Class

: This paper considers the topological transversality theorem for general multivalued maps which have selections in a given class of maps.


Introduction
To motivate this study first fix a map Φ (an important case is when Φ is the identity). Many coincidence problems between a map F and Φ (i.e., finding a (coincidence) point x with F(x) ∩ Φ(x) = ∅) arise naturally in applications. For a complicated map F the idea here is to try to relate it to a simpler and solvable coincidence problem between a map G and Φ (i.e., we assume we have a (coincidence) point y with G(y) ∩ Φ(y) = ∅) where the map G is homotopic (in an appropriate way) to F and from this we hope to deduce that there is a coincidence point between F and Φ (i.e., we hope to deduce that there is a (coincidence) point x with F(x) ∩ Φ(x) = ∅). To achieve this we consider general (instead of specific) classes of maps and we present the notion of homotopy for this class of maps which are coincidence free on the boundary of the set considered. In particular, in this paper, we look at multivalued maps F and G with selections in a given class of maps and with F ∼ = G in this setting. The topological transversality theorem in this setting will state that F is Φ-essential if and only if G is Φ-essential (essential maps were introduced in [1] and extended by many authors in [2][3][4][5]). In this paper we discuss the topological transversality theorem in a very general setting using a simple and effective approach. In this paper, we consider a generalization of Φ-essential maps, namely the d-Φ-essential maps.

Topological Transversality Theorems
A multivalued map G from a space X to a space Y is a correspondence which associates to every x ∈ X a subset G(x) ⊆ Y. In this paper let E be a completely regular topological space and U an open subset of E.
We will consider classes A, B and D of maps.

Definition 1.
We say F ∈ D(U, E) (respectively F ∈ B(U, E)) if F : U → 2 E and F ∈ D(U, E) (respectively F ∈ B(U, E)); here 2 E denotes the family of nonempty subsets of E and U denotes the closure of U in E.
In this paper we use bold face only to indicate the properties of our maps and usually D = D etc. Examples of F ∈ D(U, E) might be that F : U → K(E) is an upper semicontinuous compact map and F has convex values or F : U → K(E) is an upper semicontinuous compact map and F has acyclic values; here K(E) denotes the family of nonempty compact subsets of E.

Remark 3.
It is of interest to note that in our results below alternatively we could use the following definition for ∼ = in D ∂U (U, E): 1] is compact (respectively, closed), H 0 = Ψ and H 1 = Λ. Note here if we use this definition then we will also assume for any map Θ ∈ D(U × [0, 1], E) and any map f ∈ C(U, U × [0, 1]) then Θ • f ∈ D(U, E); here C denotes the class of single valued continuous functions. Now we are in a position to define homotopy ( ∼ =) in our class A ∂U (U, E). Definition 6. Let F, G ∈ A ∂U (U, E). We say F is homotopic to G in the class A ∂U (U, E) and we write Next, we present a simple and crucial result that will immediately yield the topological transversality theorem in this setting. (1) Proof. Let Ψ ∈ D ∂U (U, E) be any selection of F and consider any map J ∈ D ∂U (U, E) with J| ∂U = Ψ| ∂U . It remains to show that there exists an Ω is compact (respectively, closed) if E is a completely regular (respectively, normal) topological space. Next note Ω ∩ ∂U = ∅ and now we can deduce that there exists a continuous map (called a Urysohn map) µ : Now we consider a generalization of Φ-essential maps, namely the d-Φ-essential maps (these maps were motivated from the notion of the degree of a map). Let E be a completely regular topological space and U an open subset of E. For any map

Remark 4.
If F is d-Φ-essential then for any selection Ψ ∈ D(U, E) of F (with Ψ = I × Ψ) we have Now we define homotopy in this setting for our class D ∂U (U, E).
Definition 8. Let E be a completely regular (respectively, normal) topological space and let Ψ, Λ ∈ D ∂U (U, E). We say Ψ is homotopic to Λ in the class D ∂U (U, E) and we write H(x, t)).

Remark 5.
There is an analogue Remark 3 in this situation.

Remark 6.
It is also easy to extend the above ideas to other natural situations [3,4]. Let E be a (Hausdorff) topological vector space (so automatically completely regular), Y a topological vector space, and U an open subset of E. 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 + T : dom L → Y an isomorphism (i.e., a linear homeomorphism); for convenience we say T ∈ H L (E, Y). We say F ∈ A(U, Y; L, T) if (L + T) −1 (F + T) ∈ A(U, E) and we could discuss Φ-essential and d-Φ-essential in this situation.
Finally, we consider the above in the weak topology situation. Let X be a Hausdorff locally convex topological vector space and U a weakly open subset of C where C is a closed convex subset of X. We will consider classes A, B and D of maps.
Definition 10. We say F ∈ WD(U w , C) (respectively F ∈ WB(U w , C)) if F : U w → 2 C and F ∈ D(U w , C) (respectively F ∈ B(U w , C)); here U w denotes the weak boundary of U in C.
Definition 11. We say F ∈ WA(U w , C) if F : U w → 2 C and F ∈ A(U w , C) and there exists a selection Ψ ∈ WD(U w , C) of F. Now we fix a Φ ∈ WB(U w , C) and present the notion of coincidence free on the boundary, Φ-essentiality and homotopy in this setting.
Definition 12. We say F ∈ WA ∂U (U w , C) (respectively F ∈ WD ∂U (U w , C)) if F ∈ WA(U w , C) (respectively F ∈ WD(U w , C)) with F(x) ∩ Φ(x) = ∅ for x ∈ ∂U; here ∂U denotes the weak boundary of U in C.
Definition 13. We say F ∈ WA ∂U (U w , C) is Φ-essential in WA ∂U (U w , C) if for any selection Ψ ∈ WD(U w , C) of F and any map J ∈ WD ∂U (U w , C) with J| ∂U = Ψ| ∂U there exists a x ∈ U with J (x) ∩ Φ (x) = ∅.
Theorem 5. Let X be a Hausdorff locally convex topological vector space and U a weakly open subset of C where C is a closed convex subset of X. Suppose F ∈ WA ∂U (U w , C) and G ∈ WA ∂U (U w , C) is Φ-essential in WA ∂U (U w , C) and      for any selection Ψ ∈ WD ∂U (U w , C) (respectively, Λ ∈ WD ∂U (U w , C)) of F (respectively, of G) and any map J ∈ WD ∂U (U w , C) with J| ∂U = Ψ| ∂U we have Λ ∼ = J in WD ∂U (U w , C).

Proof.
A slight modification of the argument in Theorem 1 guarantees the result; we just need to note that X = (X, w), the space X endowed with the weak topology, is completely regular.

Assume
∼ = in WD ∂U (U w , C) is an equivalence relation (10) and      if F ∈ WA ∂U (U w , C) and if Ψ ∈ WD ∂U (U w , C) is any selection of F and J ∈ WD ∂U (U w , C) is any map with Ψ| ∂U = J| ∂U then Ψ ∼ = J in WD ∂U (U w , C).

(11)
A slight modification of the proof of Theorem 2 guarantees the topological transversality theorem in this setting. Theorem 6. Let X be a Hausdorff locally convex topological vector space and U a weakly open subset of C where C is a closed convex subset of X and assume (10) and (11) hold. Suppose F and G are two maps in WA ∂U (U w , C) with F ∼ = G in WA ∂U (U w , C). Now F is Φ-essential in WA ∂U (U w , C) if and only if G is Φ-essential in WA ∂U (U w , C).