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Continuation principles based onessential maps and topological degreeDonal O’Regana

a School of Mathematics, Statistics and Applied Mathematics,National University of Ireland, Galway, Ireland.Published online: 02 Jun 2014.

To cite this article: Donal O’Regan (2014): Continuation principles based on essentialmaps and topological degree, Applicable Analysis: An International Journal, DOI:10.1080/00036811.2014.915522

To link to this article: http://dx.doi.org/10.1080/00036811.2014.915522

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Applicable Analysis, 2014http://dx.doi.org/10.1080/00036811.2014.915522

Continuation principles based on essential mapsand topological degree

Donal O’Regan∗

School of Mathematics, Statistics and Applied Mathematics, National University of Ireland,Galway, Ireland

Communicated by R. Gilbert

(Received 31 March 2014; accepted 13 April 2014)

In this paper, we present a definition based on the notion of the degree of a mapof d–essential and d–L–essential maps in topological spaces and we establish ahomotopy property for both d–essential and d–L–essential maps.

Keywords: continuation methods; d–essential; fixed points

AMS Subject Classifications: 47H10; 47H04

1. Introduction

In this paper, motivated by the notion of an essential map (see [1–6]) and the notion ofthe degree of a map, we present the definition of d–essential and d–L–essential maps incompletely regular topological spaces. Our definition of d–essential relies on the assumptionthat if F, G ∈ A∂U (U , E) with F |∂U = G|∂U and F ∼= G in A∂U (U , E) thend

((F�)−1 (B)

)= d

((G�)−1 (B)

); here, E is a completely regular topological space,

U is an open subset of E , A∂U (U , E) is a class of maps, G� = I × G, F� = I × F , d isany mapping with values in a nonempty set and B = {(x, x) : x ∈ U }.

Many problems which arise naturally in differential and integral equations can beformulated in the form

x ∈ F x, x ∈ U . (1.1)

A simple example is

y′′(t) = − ey(t), t ∈ [0, 1] with y(0) = y(1) = 0 (1.2)

which models the steady-state temperature in a rod with temperature-dependent internalheating. Note (1.2) can be rewritten in the form

y(t) =∫ 1

0G(t, s) ey(s) ds ≡ F y(t) (1.3)

∗Email: donal.oregan@nuigalway.ieDedicated to Robert P. Gilbert with much admiration.

© 2014 Taylor & Francis

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2 D. O’Regan

where

G(t, s) ={

t (1 − s), 0 ≤ t ≤ s ≤ 1s (1 − t), 0 ≤ s ≤ t ≤ 1.

Now (1.3) is of the form (1.1) i.e. x = F x with E = C[0, 1] and one could take U = {u ∈C[0, 1] : |u|0 = supt∈[0,1] |u(t)| < 1} (note maxt∈[0,1]

∫ 10 |G(t, s)| ds = 1

8 ). Topologicaldegree is a useful tool which allows us to establish whether (1.1) has a solution x ∈ U . Inparticular, it allows us to associate a value (usually a number) d which guarantees that ifd

((F�)−1 (B)

)�= d(∅) then (1.1) has a solution. In applications usually for a complicated

F , the value d((F�)−1 (B)

)can be calculated from d

((G�)−1 (B)

)for a simpler G if

F is homotopic (in an appropriate way) to G. The location of solutions (see Section 3) isalso of interest. For example, one can relate (1.2) with the simpler problem y′′(t) = 0,t ∈ [0, 1] with y(0) = y(1) = 0 via the family y′′(t) = − λ ey(t), t ∈ [0, 1], 0 ≤ λ ≤ 1,with y(0) = y(1) = 0.

In Section 2, we present the topological transversality theorem in a very general andapplicable setting. In Section 3, we present Krasnosielski type (see [7]) results.

Let X and Y be Hausdorff topological spaces. Given a class X of maps, X(X, Y )

denotes the set of maps F : X → 2Y (nonempty subsets of Y ) belonging to X.

2. d–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 situations, the following condition will be

assumed: ⎧⎨⎩

for Hausdorff topological spaces X1, X2 and X3,

if F ∈ A(X1, X3) and f ∈ C(X2, X1),

then F ◦ f ∈ A(X2, X3).

(2.1)

Definition 2.1 We say F ∈ A(U , E) if F ∈ A(U , E) and F : U → K (E) is an uppersemicontinuous map; here U denotes the closure of U in E and K (E) denotes the familyof nonempty compact subsets of E .

Definition 2.2 We say F ∈ A∂U (U , E) if F ∈ A(U , E) with x /∈ F(x) for x ∈ ∂U ;here ∂U denotes the boundary of U in E .

For any map F ∈ A(U , E) let F� = I × F : U → K (U × E), with I : U → Ugiven by I (x) = x , and let

d :{(

F�)−1

(B)}

∪ {∅} → � (2.2)

be any map with values in the nonempty set �; here B = {(x, x) : x ∈ U

}.

Definition 2.3 Let F, G ∈ A∂U (U , E). We say F ∼= G in A∂U (U , E) if there exists amap � : U × [0, 1] → K (E) with � ∈ A(U × [0, 1], E), x �∈ �t (x) for any x ∈ ∂U andt ∈ [0, 1], �1 = F , �0 = G (here �t (x) = �(x, t)) and

{x ∈ U : (x, x) ∈ ��(x, t)

for some t ∈ [0, 1]} is relatively compact (here �� : U × [0, 1] → K (U × E) is givenby ��(x, t) = (x , �(x, t))).

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Applicable Analysis 3

Remark 2.1 We note (see the proof in Theorem2.1) if � : U ×[0, 1] → K (E) is a uppersemicontinuous map then M = {

x ∈ U : (x, x) ∈ (x,�(x, t)) for some t ∈ [0, 1]} isclosed so if M is relatively compact then M is compact. If � : U ×[0, 1] → K (E) is an up-per semicontinuous compact map then

{x ∈ U : (x, x) ∈ (x,�(x, t)) for some t ∈ [0, 1]}

is compact.

Remark 2.2 The results below (with (2.1) removed) also hold true if we use the followingdefinition of ∼=. Let F, G ∈ A∂U (U , E). We say F ∼= G in A∂U (U , E) if there existsan upper semicontinuous map � : U × [0, 1] → K (E) with �( . , η( . )) ∈ A(U , E) forany continuous function η : U → [0, 1] with η(∂U ) = 0, x �∈ �t (x) for any x ∈ ∂Uand t ∈ [0, 1], �1 = F , �0 = G and

{x ∈ U : (x, x) ∈ ��(x, t) for some t ∈ [0, 1]}

is relatively compact (here �� : U × [0, 1] → K (U × E) is given by ��(x, t) =(x , �(x, t))).

The following condition will be assumed:

∼= is an equivalence relation in A∂U (U , E). (2.3)

In addition, we assume that{if F, G ∈ A∂U (U , E) with F |∂U = G|∂U and F ∼= G

in A∂U (U , E) then d((F�)−1 (B)

)= d

((G�)−1 (B)

).

(2.4)

Definition 2.4 Let F ∈ A∂U (U , E) with F� = I × F . We say F� : U → K (U × E) isd–essential if d

((F�)−1 (B)

)�= d(∅). We say F� is d–inessential if d

((F�)−1 (B)

)=

d(∅).

Remark 2.3 If F� is d–essential then

∅ �= (F�

)−1(B) = {x ∈ U : (x, F(x)) ∩ (x, x) �= ∅},

and this together with x /∈ F(x) for x ∈ ∂U implies that there exists x ∈ U with(x, x) ∈ F�(x) (i.e. x ∈ F(x)).

Theorem 2.1 Let E be a completely regular topological space, U an open subset ofE, B = {(x, x) : x ∈ U }, d a map defined in (2.2) and assume (2.1), (2.3) and (2.4) hold.Suppose F ∈ A∂U (U , E). Then the following are equivalent:

(i) F� = I × F : U → K (U × E) is d–inessential;(ii) there exists a map G ∈ A∂U (U , E) with G� = I × G and G ∼= F in A∂U (U , E)

such that d((G�)−1 (B)

)= d(∅).

Proof (i) implies (ii) is immediate. Next we prove (ii) implies (i). Suppose there existsa map G ∈ A∂U (U , E) with G� = I × G and G ∼= F in A∂U (U , E) such thatd

((G�)−1 (B)

)= d(∅). Let H : U × [0, 1] → K (E) be a map with H ∈ A(U ×

[0, 1], E), x �∈ Ht (x) for any x ∈ ∂U and t ∈ [0, 1], H0 = F , H1 = G (here Ht (x) =H(x, t)) and

{x ∈ U : (x, x) ∈ H �(x, t) for some t ∈ [0, 1]} is relatively compact (here

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H � : U × [0, 1] → K (U × E) is given by H �(x, t) = (x , H(x, t))). Consider

D = {x ∈ U : (x, x) ∈ H �(x, t) for some t ∈ [0, 1]} .

If D = ∅ , then in particular (H �(x, 0))−1 (B) = ∅ i.e. (F�)−1 (B) = ∅ sod

((F�)−1 (B)

)= d(∅), so F� is d–inessential. Next, suppose D �= ∅. Note D is closed in

E . To see this let (xα) be a net in D with xα → x ∈ U . Now there exists tα ∈ [0, 1] with xα ∈H(xα, tα). Without loss of generality assume tα → t ∈ [0, 1]. Thus, (xα, tα) → (x, t) andthe fact that H : U ×[0, 1] → K (E) is upper semicontinuous guarantees that x ∈ H(x, t),so D is closed. Note then that D is a compact subset of E . Also, since x /∈ Ht (x) forx ∈ ∂U and t ∈ [0, 1] then D ∩ ∂U = ∅. Thus, (note E is a completely regular topologicalspace) there exists a continuous map μ : U → [0, 1] with μ(∂U ) = 0 and μ(D) = 1.Define a map Rμ : U → K (E) by Rμ(x) = H(x, μ(x)) = Hμ(x)(x) = H ◦ τ(x)

and let R�μ = I × Rμ; here τ : U → U × [0, 1] is given by τ(x) = (x, μ(x)). Notice

Rμ ∈ A(U , E) (note (2.1) and H ∈ A(U×[0, 1], E)) and notice Rμ|∂U = H0|∂U = F |∂U

since μ(∂U ) = 0. Thus, Rμ ∈ A∂U (U , E) (note x �∈ Ht (x) for any x ∈ ∂U and t ∈ [0, 1])with Rμ|∂U = F |∂U .

Note also since μ(D) = 1 that(R�

μ

)−1(B) = {

x ∈ U : (x, x) ∩ (x, H(x, μ(x)) �= ∅}= {

x ∈ U : (x, x) ∩ (x, H(x, 1) �= ∅}= (

G�)−1

(B)

so d((

R�μ

)−1(B)

)= d

((G�)−1 (B)

). Thus d

((R�

μ

)−1(B)

)= d(∅).

We now claim

Rμ∼= F in A∂U (U , E). (2.5)

Let Q : U × [0, 1] → K (E) be given by Q(x, t) = H(x, t μ(x)) = H ◦ g(x, t) whereg : U ×[0, 1] → U ×[0, 1] is given by g(x, t) = (x, t μ(x)). Note Q ∈ A(U ×[0, 1], E)

(note (2.1) and H ∈ A(U × [0, 1], E)), Q0 = F and Q1 = Rμ. Also x /∈ Qt (x) forx ∈ ∂U and t ∈ [0, 1] since if there exists t ∈ [0, 1] and x ∈ ∂U with x ∈ Qt (x) thenx ∈ H(x, t μ(x)) so x ∈ D and as a result μ(x) = 1 i.e. x ∈ H(x, t), a contradiction.Finally, note{

x ∈ U : (x, x) ∈ (x, Q(x, t)) = (x, H(x, t μ(x))) for some t ∈ [0, 1]}is closed (note H is upper semicontinuous) and compact. Thus (2.5) holds.

Consequently F� is d–inessential since d((F�)−1 (B)

)= d

((R�

μ

)−1(B)

)= d(∅). �

Remark 2.4 From the proof above (with a minor modification in two places), we seethat the result in Theorem 2.1 (with (2.1) removed) holds if the definition of ∼= is as inRemark 2.2.

Remark 2.5 If E is a normal topological space then the assumption that{x ∈ U : (x, x) ∈ ��(x, t) for some t ∈ [0, 1]}

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Applicable Analysis 5

is relatively compact can be removed in Definition 2.3 (and Remark 2.2) and we still obtainTheorem 2.1.

Remark 2.6 If E is a normal topological space then the result in Theorem 2.1 holds ifF : U → K (E) is an upper semicontinuous map in the definition of A(U , E) is replacedby any map F : U → 2E (and � : U ×[0, 1] → K (E) is replaced by � : U ×[0, 1] → 2E

in Definition 2.3 or � : U × [0, 1] → K (E) is upper semicontinuous is replaced by� : U × [0, 1] → 2E in Remark 2.2) provided that D in Theorem 2.1 is closed. A similarcomment applies if E is a completely regular topological space i.e. in this case we considermaps so that D in Theorem 2.1 is compact.

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 ofE, B = {(x, x) : x ∈ U }, d a map defined in (2.2) and assume (2.1), (2.3) and (2.4) hold.Suppose � and � are two maps in A∂U (U , E) with �� = I × � and �� = I × � andwith � ∼= � in A∂U (U , E). Then �� is d-inessential if and only if �� is d-inessential.

Proof Assume �� is d-inessential. Then (see Theorem 2.1) there exists a map Q ∈A∂U (U , E) with Q� = I × Q and Q ∼= � in A∂U (U , E) such that d

((Q�)−1 (B)

)=

d(∅). Note (since ∼= is an equivalence relation in A∂U (U , E)) also that Q ∼= � inA∂U (U , E). Then Theorem 2.1 (with F = � and G = Q) guarantees that �� isd-inessential. Similarly, if �� is d-inessential then �� is d-inessential. �

Remark 2.7 The result in Theorem 2.2 (with (2.1) removed) holds if the definition of ∼=is as in Remark 2.2.

Remark 2.8 If E is a normal topological space then the assumption that{x ∈ U : (x, x) ∈ ��(x, t) for some t ∈ [0, 1]}

is relatively compact can be removed in Definition 2.3 (and Remark 2.2) and we still obtainTheorem 2.2.

Remark 2.9 If we discuss the existence of fixed points, it is sufficient to consider thefunction d = d1 given by

d1(Q) ={

1 if ∅ �= Q ⊆ U0 if Q = ∅

Whereas, if we discuss degree theory, the values of d are usually integers which can beobtained by means of degree. Note for example Theorem 2.2 establishes the homotopyinvariance of topological degree (i.e. d) as a result (not taken as an axiom of d). From anapplication point of view, if one sets up a homotopy between an easy problem (i.e. a wellknown map � where d

((��)−1 (B)

)�= d(∅), so in particular (��)−1 (B) �= ∅) and a

difficult problem (i.e. a map �), then Theorem 2.2 guarantees that d((��)−1 (B)

)�= d(∅),

so in particular (��)−1 (B) �= ∅. In particular (if we take d1), the existence of a fixed point

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of � can be guaranteed from the existence of a fixed point of � provided an appropriatehomotopy exists (see [5] for examples).

We now show that the ideas in this section can be applied to other natural situations.Let E be a Hausdorff topological vector space, Y a topological vector space, and U anopen 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 bea linear, continuous single valued map with L + T : dom L → Y an isomorphism (i.e. alinear homeomorphism); for convenience, we say T ∈ HL(E, Y ). In our next two results,(2.1) is not assumed (A is a class of maps).

Definition 2.5 Let F : U → 2Y . We say F ∈ A(U , Y ; L , T ) if (L + T )−1 (F + T ) ∈A(U , E).

Definition 2.6 We say F ∈ A∂U (U , Y ; L , T ) if F ∈ A(U , Y ; L , T ) with L x /∈ F(x)

for x ∈ ∂U ∩ dom L .For any map F ∈ A(U , Y ; L , T ) let F� = I × (L + T )−1 (F + T ) : U → K (U × E),

with I : U → U given by I (x) = x , and let

d :{(

F�)−1

(B)}

∪ {∅} → � (2.6)

be any map with values in the nonempty set �; here B = {(x, x) : x ∈ U

}.

Definition 2.7 Let F, G ∈ A∂U (U , Y ; L , T ). We say F ∼= G in A∂U (U , Y ; L , T ) ifthere exists a map � : U × [0, 1] → 2Y with (L + T )−1 (� + T ) : U × [0, 1] → K (E)

upper semicontinuous and with (L+T )−1 (�( . , η( . ))+T ) ∈ A(U , E) for any continuousfunction η : U → [0, 1] with η(∂U ) = 0, L x �∈ �t (x) for any x ∈ ∂U ∩ dom L andt ∈ [0, 1], �1 = F , �0 = G (here �t (x) = �(x, t)) and{

x ∈ U ∩ dom L : (x, L x) ∈ (x, �(x, t)) for some t ∈ [0, 1]}is relatively compact.

Remark 2.10 Suppose we have the following definitions of A(U , Y ; L , T ) and ∼=. Wesay F ∈ A(U , Y ; L , T ) if (L + T )−1 (F + T ) : U → K (E) is upper semicontinuous and(L + T )−1 F ∈ A(U , E). Let F, G ∈ A∂U (U , Y ; L , T ). We say F ∼= G in A∂U (U , Y ;L , T ) if there exists a map � : U ×[0, 1] → 2Y with (L + T )−1 (� + T ) : U ×[0, 1] →K (E) upper semicontinuous and with � ∈ A(U × [0, 1], Y ; L , T ), L x �∈ �t (x) for anyx ∈ ∂U ∩ dom L and t ∈ [0, 1], �1 = F , �0 = G (here �t (x) = �(x, t)) and{

x ∈ U ∩ dom L : (x, L x) ∈ (x, �(x, t)) for some t ∈ [0, 1]}is relatively compact. With these definitions, the next two results hold if (2.1) is assumed.

Finally, notice if we use the definition of A(U , Y ; L , T ) in Definition 2.5 and thedefinition of ∼= just given then the next two results hold if we assume the analogue of(2.1) with A(X1, X3) and A(X2, X3) replaced by A(X1, X3; L , T ) and A(X2, X3; L , T )

where X1 = U × [0, 1], X3 = Y and X2 = U or X2 = U × [0, 1].The following condition will be assumed:

∼= is an equivalence relation in A∂U (U , Y ; L , T ). (2.7)

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Applicable Analysis 7

In addition, we assume that{if F, G ∈ A∂U (U , Y ; L , T ) with F |∂U = G|∂U and F ∼= G

in A∂U (U , Y ; L , T ) then d((F�)−1 (B)

)= d

((G�)−1 (B)

).

(2.8)

Definition 2.8 Let F ∈ A∂U (U , Y ; L , T ) with F� = I × (L + T )−1 (F + T ). We sayF� : U → K (U × E) is d–L–essential if d

((F�)−1 (B)

)�= d(∅). We say F� is d–L–

inessential if d((F�)−1 (B)

)= d(∅).

Remark 2.11 If F� is d–L–essential then

∅ �= (F�

)−1(B) = {x ∈ U : (x, (L + T )−1 (F + T )(x)) ∩ (x, x) �= ∅},

and this together with L x /∈ F(x) for x ∈ ∂U ∩ dom L implies that there exists x ∈U ∩ dom L with (x, x) ∈ F�(x) (i.e. L x ∈ F(x)).

Theorem 2.3 Let E be a Hausdorff topological vector space, Y a topological vectorspace, U an open subset of E, L : dom L ⊆ E → Y a linear single valued map,T ∈ HL(E, Y ), d a map defined in (2.6) and assume (2.7) and (2.8) holds. SupposeF ∈ A∂U (U , Y ; L , T ). Then the following are equivalent:

(i) F� = I × (L + T )−1 (F + T ) : U → K (U × E) is d–L–inessential;(ii) there exists a map G ∈ A∂U (U , Y ; L , T ) with G� = I × (L + T )−1 (G + T ) and

G ∼= F in A∂U (U , Y ; L , T ) such that d((G�)−1 (B)

)= d(∅).

Proof (i) implies (ii) is immediate. Next we prove (ii) implies (i). Suppose there existsa map G ∈ A∂U (U , Y ; L , T ) with G� = I × (L + T )−1 (G + T ) and G ∼= F inA∂U (U , Y ; L , T ) such that d

((G�)−1 (B)

)= d(∅). Let H : U × [0, 1] → 2Y be a

map with (L + T )−1 (H + T ) : U × [0, 1] → K (E) upper semicontinuous and with(L + T )−1 (H( . , η( . ))+ T ) ∈ A(U , E) for any continuous function η : U → [0, 1] withη(∂U ) = 0, L x �∈ Ht (x) for any x ∈ ∂U ∩ dom L and t ∈ [0, 1], H1 = F , H0 = G(here Ht (x) = H(x, t)) and{

x ∈ U ∩ dom L : (x, L x) ∈ (x, H(x, t)) for some t ∈ [0, 1]}is relatively compact. Let H � : U × [0, 1] → K (U × E) be given by

H �(x, λ) = (x , (L + T )−1 (H + T )(x, λ)).

ConsiderD = {

x ∈ U : (x, x) ∈ H �(x, t) for some t ∈ [0, 1]} .

Notice that it is immediate that

D = {x ∈ U ∩ dom L : (x, L x) ∈ (x, H(x, t)) for some t ∈ [0, 1]} .

If D = ∅ , then in particular (H �(x, 0))−1 (B) = ∅ i.e. (F�)−1 (B) = ∅ sod

((F�)−1 (B)

)= d(∅), so F� is d–L–inessential. Next suppose D �= ∅. Note D is

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8 D. O’Regan

a compact subset of E . Also since L x /∈ Ht (x) for x ∈ ∂U ∩ dom L and t ∈ [0, 1] thenD ∩ ∂U = ∅. Thus, there exists a continuous map μ : U → [0, 1] with μ(∂U ) = 0and μ(D) = 1. Define a map Rμ : U → 2Y by Rμ(x) = H(x, μ(x)) = Hμ(x)(x)

and let R�μ = I × (L + T )−1 (Rμ + T ). Notice Rμ ∈ A(U , Y ; L , T ) and notice

Rμ|∂U = H0|∂U = F |∂U since μ(∂U ) = 0. Thus, Rμ ∈ A∂U (U , Y ; L , T ) (noteL x �∈ Ht (x) for any x ∈ ∂U ∩ dom L and t ∈ [0, 1]) with Rμ|∂U = F |∂U .

Note also since μ(D) = 1 that(R�

μ

)−1(B) =

{x ∈ U : (x, x) ∩ (x, (L + T )−1 (H + T )(x, μ(x)) �= ∅

}=

{x ∈ U : (x, x) ∩ (x, (L + T )−1 (H + T )(x, 1) �= ∅

}= (

G�)−1

(B)

so d((

R�μ

)−1(B)

)= d

((G�)−1 (B)

). Thus d

((R�

μ

)−1(B)

)= d(∅).

We now claim

Rμ∼= F in A∂U (U , Y ; L , T ). (2.9)

Let Q : U × [0, 1] → 2Y be given by Q(x, t) = H(x, t μ(x)). Note Q ∈ A(U ×[0, 1], Y ; L , T ), Q0 = F and Q1 = Rμ. Also L x /∈ Qt (x) for x ∈ ∂U ∩ dom L andt ∈ [0, 1] since if there exists t ∈ [0, 1] and x ∈ ∂U ∩ dom L with L x ∈ Qt (x) thenL x ∈ H(x, t μ(x)) so x ∈ D and as a result μ(x) = 1 i.e. L x ∈ H(x, t), a contradiction.Finally, note{

x ∈ U ∩ dom L : (x, L x) ∈ (x, Q(x, t)) = (x, H(x, t μ(x))) for some t ∈ [0, 1]}is closed and compact. Thus (2.9) holds.

Consequently F� is d–L–inessential since d((F�)−1 (B)

)= d

((R�

μ

)−1(B)

)=

d(∅). �

Remark 2.12 From the proof above, we see that the result in Theorem 2.3 holds if we usethe definitions in Remark 2.10. One could also state the analogue of Remark 2.6 in thissituation.

Essentially the same reasoning as in Theorem 2.2 establishes the following result.

Theorem 2.4 Let E be a Hausdorff topological vector space, Y a topological vectorspace, U an open subset of E, L : dom L ⊆ E → Y a linear single valued map,T ∈ HL(E, Y ), d a map defined in (2.6) and assume (2.7) and (2.8) holds. Suppose� and � are two maps in A∂U (U , Y ; L , T ) with �� = I × (L + T )−1 (� + T ) and�� = I × (L + T )−1 (� + T ) and with � ∼= � in A∂U (U , Y ; L , T ). Then �� is d–L–inessential if and only if �� is d–L–inessential.

Remark 2.13 If E is a normal topological vector space then the assumption that{x ∈ U ∩ dom L : (x, L x) ∈ (x, �(x, t)) for some t ∈ [0, 1]}

is relatively compact can be removed in Definition 2.7 (and Remark 2.10) and we still obtainTheorems 2.3 and 2.4.

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Applicable Analysis 9

3. Applications

Let E be a Hausdorff topological space and U an open subset of E .Let F ∈ A∂U (U , E) with F� = I × F . As in Section 2, we say F� : U → K (U × E)

is d–essential (in A(U )) if d((F�)−1 (B)

)�= d(∅); here d is given in (2.2). We say F� is

d–inessential (in A(U )) if d((J �)−1 (B)

)= d(∅).

Let E be a Hausdorff topological space, U1 and U2 are open subsets of E , U1 ⊂ U2and F : U2 → 2E . Let B = {(x, x) : x ∈ U1}, B2 = {(x, x) : x ∈ U2} and B3 = {(x, x) :x ∈ U2 \ U1}. Assume

x /∈ F(x) for x ∈ ∂U1 ∪ ∂U2. (3.1)

Now(F�

)−1(B2)

= {x ∈ U2 : (x, x) ∩ (x, F(x)) �= ∅}= {x ∈ U1 : (x, x) ∩ (x, F(x)) �= ∅} ∪ {x ∈ U2 \ U1 : (x, x) ∩ (x, F(x)) �= ∅}= {x ∈ U1 : (x, x) ∩ (x, F(x)) �= ∅} ∪ {x ∈ U2 \ U1 : (x, x) ∩ (x, F(x)) �= ∅}= (

F�)−1

(B1) ∪ (F�

)−1(B3).

If we assume

d((

F�)−1

(B2) \ (F�

)−1(B1)

)�= d(∅), (3.2)

then d((F�)−1 (B3)

)�= d(∅) (which is useful in applications).

Remark 3.1 For example, if F ∈ A(U1, E), with F� = I × F : U1 → 2U1×E , isd–inessential in A(U1) and F ∈ A(U2, E) is d–essential in A(U2) then d

((F�)−1 (B1)

)=

d(∅) and d((F�)−1 (B2)

)�= d(∅), and a condition has to be put on d to guarantee (3.2).

A similar comment applies if F ∈ A(U1, E) is d–essential in A(U1) and F ∈ A(U2, E)

is d–inessential in A(U2).

Remark 3.2 From an application point of view, (3.2) could be satisfied if one has an axiomof additivity for d . For example, if we consider d1 in Remark 2.9, then the result above(d

((F�)−1 (B3)

)�= d(∅)) guarantees that F has a fixed point in U2 \ U1. As a result (using

d1 and more generally d) the theory above enables us to consider multiplicity of fixed pointsof a map in a very general setting (see [5] for examples).

Let E be a Hausdorff topological vector space, Y a topological vector space and Uan open subset of E . Also, let L and T be as in Section 2. Let F ∈ A∂U (U , Y ; L , T )

with F� = I × (L + T )−1 (F + T ). As in Section 2, we say F� : U → K (U × E) isd–L–essential (in A(U )) if d

((F�)−1 (B)

)�= d(∅); here d is given in (2.6). We say F�

is d–L–inessential (in A(U )) if d((F�)−1 (B)

)= d(∅).

Let E be a Hausdorff topological vector space, Y a Hausdorff topological vectorspace, U1 and U2 are open subsets of E , U1 ⊂ U2, L : dom L ⊆ E → Y a linear

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10 D. O’Regan

single valued map, T ∈ HL(E, Y ), and F : U2 → 2Y . Let B = {(x, x) : x ∈ U1},B2 = {(x, x) : x ∈ U2} and B3 = {(x, x) : x ∈ U2 \ U1}. Assume

L x /∈ F(x) for x ∈ (∂U1 ∩ dom L) ∪ (∂U2 ∩ dom L). (3.3)

If we assumed

((F�

)−1(B2) \ (

F�)−1

(B1))

�= d(∅), (3.4)

then d((F�)−1 (B3)

)�= d(∅); here F� = I × (L + T )−1 (F + T ).

Remark 3.3 There is an analogue of Remark 3.1 in this situation.

References

[1] Gorniewicz L. Topological fixed point theory of multivalued mappings. Dordrecht: KluwerAcademic Publishers; 1999.

[2] Granas A, Dugundji J. Fixed point theory. New York (NY): Springer-Verlag; 2003.[3] O’Regan D. Homotopy principles for d–essential acyclic maps. J. Nonlinear Convex Anal.

2013;14:415–422.[4] O’Regan D. A unified theory for homotopy principles for multimaps. Appl. Anal. 2013;92:

1944–1958.[5] O’Regan D, Precup R. Theorems of Leray-Schauder type and applications. London: Taylor and

Francis Publishers; 2002.[6] Precup R. On the topological transversality principle. Nonlinear Anal. 1993;20:1–9.[7] O’Regan D. Multiple fixed points via essential and inessential multimaps. Indian J. Math.

2013;55:149–161.

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