J. Korean Math. Soc. 38 (2001), No. 3, pp. 669–681
A GENERALIZATION OF THE
KRASNOSELSKII–PETRYSHYN COMPRESSION
AND EXPANSION THEOREM: AN
ESSENTIAL MAP APPROACH
Ravi P. Agarwal and Donal O’Regan
Abstract. This paper introduces the notions of essential and inessential maps for countably k–set contractive maps. These ideas enable
us to establish a Krasnoselskii–Petryshyn compression and expansion theorem in a cone for countably k–set contractive maps.
1. Introduction
This paper presents new fixed point results for countably condensing
maps. We begin with a fixed point result for self maps which follows
from a multivalued version of Mönch’s theorem [5] established recently
by O’Regan and Precup [7]. Next we introduce the notions of essential
and inessential maps for countably condensing maps, and some properties of these maps will be established. In particular we show if a map
G is essential and G ∼
= F then F is essential. In Section 3 we present
a generalization of the Krasnoselskii–Petryshyn compression and expansion theorem in a cone [3].
For the remainder of this section we present some preliminary results
which will be needed in Section 2 and Section 3. Let E be a Banach
space and PB (E) the bounded subsets of E. The Kuratowskii measure
of noncompactness is the map α : PB (E) → [0, ∞) defined by
α(X) = inf { > 0 : X ⊆ ∪ni=1 Xi and diam (Xi ) ≤ } ;
here X ∈ PB (E). Let Z be a nonempty subset of E and F : Z →
2E (the nonempty subsets of E). F is called (i). countably k–set
Received November 10, 2000.
2000 Mathematics Subject Classification: 47H10.
Key words and phrases: essential and inessential maps, countably k-set contractive maps, fixed point results.
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Ravi P. Agarwal and Donal O’Regan
contractive (k ≥ 0) if F (Z) is bounded and α(F (Y )) ≤ k α(Y ) for
all countably bounded sets Y of Z; (ii). countably condensing if F
is countably 1–set contractive and α(F (Y )) < α(Y ) for all countably
bounded sets Y of Z with α(Y ) 6= 0.
In [7] O’Regan and Precup established the following generalization of
Mönch’s fixed point theorem.
Theorem 1.1. Let D be a closed, convex subset of a Banach space
E and F : D → CK(D) a upper semicontinuous map; here CK(D)
denotes the family of nonempty, convex, compact subsets of D. Assume
there is an x0 ∈ D with the following property holding:
M ⊆ D, M = co ({x0 } ∪ F (M )) and M = C with
(1.1)
C ⊆ M countable, implies M is compact.
Then there exists x ∈ D with x ∈ F (x).
Remark 1.1. In Theorem 1.1 (and also in many of the results in
Sections 2 and 3) it is possible to replace the condition F : D → CK(D)
is upper semicontinuous with
F : D → C(D) is a closed map and F maps
compact sets into relatively compact sets;
here C(D) denotes the family of nonempty, convex subsets of D.
Theorem 1.1 immediately guarantees a new result for countably condensing maps (this is a multivalued version of Daher’s theorem [2]).
Theorem 1.2. Let D be a nonempty, closed, convex subset of
a Banach space E and F : D → CK(D) a upper semicontinuous,
countably condensing map. Then there exists x ∈ D with x ∈ F (x).
Proof. Let x0 ∈ D, M ⊆ D, M = co ({x0 } ∪ F (M )) and M = C
with C ⊆ M countable. If we show M is compact, then (1.1) will hold,
so the result follows from Theorem 1.1.
Now since
C ⊆ co ({x0 } ∪ F (M )),
each x ∈ C can be written as a finite convex combination of points from
{x0 } ∪ F (M ). Thus there exists a countable set MC ⊆ M with
(1.2)
C ⊆ co ({x0 } ∪ F (MC )).
On Krasnoselskii–Petryshyn compression and expansion theorem
671
If α(MC ) 6= 0 then
α(C) ≤ α(co ({x0 } ∪ F (MC )))
=α(F (MC )) < α(MC ) ≤ α(M )
=α(M ) = α(C) = α(C),
a contradiction. Thus α(MC ) = 0 so from (1.2) we obtain,
α(C) ≤ α(F (MC )) ≤ α(MC ) = 0.
Thus C is compact.
Corollary 1.3. Let D be a nonempty, closed, convex subset of
a Banach space E and F : D → CK(D) a upper semicontinuous,
countably k–set contractive (0 ≤ k < 1) map. Then there exists x ∈ D
with x ∈ F (x).
2.
Essential and inessential maps
Throughout this section, we will assume E is a Banach space, C is
a closed, convex subset of E, and U is an open subset of C. We note
that some of the ideas in this section were motivated by a recent paper
of the authors [1].
Definition 2.1. D(U , C) denotes the set of all upper semicontinuous, countably k–set contractive (0 ≤ k < 1) maps F : U → CK(C);
here U denotes the closure of U in C.
Remark 2.1. It is also possible to consider the set of all upper
semicontinuous, countably condensing maps in this section. We leave
the minor adjustments necessary to the reader.
Definition 2.2. We let D∂U (U , C) denote the set of all maps F ∈
D(U , C) with x ∈
/ F (x) for x ∈ ∂U ; here ∂U denotes the boundary of
U in C.
Definition 2.3. A map F ∈ D∂U (U , C) is essential in D∂U (U , C)
if for every map G ∈ D∂U (U , C) with G|∂U = F |∂U we have that
there exists x ∈ U with x ∈ G(x). Otherwise F is inessential in
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Ravi P. Agarwal and Donal O’Regan
D∂U (U , C) i.e. there exists a map G ∈ D∂U (U , C) with G|∂U = F |∂U
and x ∈
/ G(x) for x ∈ U .
We begin with a simple example of an essential map, which is all we
need from an application viewpoint.
Theorem 2.1. Let E be a Banach space, C a closed, convex subset
of E, U an open subset of C and 0 ∈ U . Then the zero map is essential
in D∂U (U , C).
Proof. Let θ ∈ D∂U (U , C) with θ|∂U = {0}. We must show there
exists x ∈ U with x ∈ θ(x). Let D = co (θ(U )) and consider the map
J : D → CK(D) given by
θ(x), x ∈ U
J(x) =
{0}, otherwise.
Notice J : D → CK(D) is an upper semicontinuous, countably k–
set contractive map. To see that J is countably k–set contractive let
Ω ⊆ D be countable. Then since
J(Ω) ⊆ co (θ(Ω ∩ U ) ∪ {0}),
we have
α(J(Ω)) ≤ α(θ(Ω ∩ U )) ≤ k α(Ω ∩ U ) ≤ k α(Ω),
since Ω ∩ U is countable.
Now Corollary 1.3 guarantees that there exists x ∈ D with x ∈ J(x).
Notice x ∈ U since 0 ∈ U , so x ∈ θ(x).
Next we establish a homotopy property for essential maps.
Theorem 2.2. Let E be a Banach space, C a closed, convex subset
of E and U an open subset of C. Suppose F, G ∈ D(U , C) and assume
the following hold:
(2.1)
(2.2)
G is essential in D∂U (U , C)
x∈
/ λ F (x) + (1 − λ) G(x) for x ∈ ∂U and λ ∈ (0, 1].
Then F is essential in D∂U (U , C).
On Krasnoselskii–Petryshyn compression and expansion theorem
673
Proof. Let H ∈ D∂U (U , C) with H|∂U = F |∂U . We must show H
has a fixed point in U . Consider
B = x ∈ U : x ∈ t H(x) + (1 − t) G(x) for some t ∈ [0, 1] .
Notice (2.1) guarantees that B 6= ∅. It is also immediate that B is
closed (in C). In addition (2.2) together with H|∂U = F |∂U guarantees
that B ∩ ∂U = ∅. Thus there exists a continuous µ : U → [0, 1] with
µ(∂U ) = 0 and µ(B) = 1. Define a map Rµ : U → CK(C) by
Rµ (x) = µ(x) H(x) + (1 − µ(x)) G(x).
Notice Rµ ∈ D(U , C) since if Ω ⊆ U is countable then since
Rµ (Ω) ⊆ co (H(Ω) ∪ G(Ω))
we have
α(Rµ (Ω)) ≤ α(H(Ω) ∪ G(Ω)) = max {α(H(Ω)) , α(G(Ω))} ≤ k α(Ω).
Also for x ∈ ∂U we have Rµ (x) = G(x) so x ∈
/ Rµ (x) for x ∈ ∂U .
Consequently Rµ ∈ D∂U (U , C) with Rµ |∂U = G|∂U . Now (2.1) guarantees that there exists x ∈ U with x ∈ Rµ (x) (i.e. x ∈ µ(x) H(x) +
(1 − µ(x)) G(x)). Thus x ∈ B and so µ(x) = 1. As a result we have
x ∈ U with x ∈ H(x).
Next we will establish some “inessential” map type results. These
results will be needed in Section 3 to establish the generalization of the
Krasnoselskii–Petryshyn compression and expansion theorem.
Theorem 2.3. Let E be a Banach space, C a closed, convex subset
of E with α u + β v ∈ C for all α ≥ 0, β ≥ 0 and u, v ∈ C, and U an
open subset of C. Suppose F ∈ D(U , C) with the following holding:
(2.3)
there exists v ∈ C with x ∈
/ F (x) + δ v for all δ ≥ 0 and x ∈ ∂U
and
(2.4)
there exists δ0 > 0 with x ∈
/ F (x) + δ0 v for x ∈ U .
Then F is inessential in D∂U (U , C).
Proof. Let the map J be defined by J(x) = F (x) + δ0 v. Clearly
J ∈ D∂U (U , C) (note if Ω ⊆ U is countable then J(Ω) ⊆ F (Ω)+{δ0 v}).
To show F is inessential in D∂U (U , C) we must show that there exists
a fixed point free θ ∈ D∂U (U , C) with θ|∂U = F |∂U . Consider
B = {x ∈ U : x ∈ F (x) + (1 − t) δ0 v for some t ∈ [0, 1]}.
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If B = ∅ then in particular F has no fixed points in U and thus F is
inessential in D∂U (U , C). It remains to consider the case when B 6= ∅.
Notice B is closed. Also (2.3) guarantees that B ∩ ∂U = ∅. Thus there
exists a continuous µ : U → [0, 1] with µ(∂U ) = 1 and µ(B) = 0.
Define a map Nµ : U → CK(C) by
Nµ (x) = F (x) + (1 − µ(x)) δ0 v.
Notice Nµ ∈ D(U , C) (note if Ω ⊆ U is countable then Nµ (Ω) ⊆
F (Ω) + co ({δ0 v} ∪ {0})). Also if x ∈ ∂U then µ(x) = 1 so Nµ |∂U =
F |∂U . Consequently Nµ ∈ D∂U (U , C) with Nµ |∂U = F |∂U . We claim
Nµ is fixed point free on U . If this is true then F is inessential in
D∂U (U , C). It remains to prove the claim. If there exists x ∈ U with
x ∈ Nµ (x) = F (x) + (1 − µ(x)) δ0 v then x ∈ B and so µ(x) = 0 i.e.
x ∈ F (x) + δ0 v (this contradicts (2.4)).
Our next result generalizes Theorem 2.3.
Theorem 2.4. Let E be a Banach space, C a closed, convex subset
of E with α u + β v ∈ C for all α ≥ 0, β ≥ 0 and u, v ∈ C, and U
an open subset of C. Suppose F ∈ D(U , C) satisfies (2.3), and assume
the following condition holds:
(2.5)
there exists δ0 > 0 with J, defined by J(x) = F (x) + δ0 v,
inessential in D∂U (U , C).
Then F is inessential in D∂U (U , C).
Proof. Now (2.5) implies that there exists τ ∈ D∂U (U , C) with
τ |∂U = (F + δ0 v)|∂U and x ∈
/ τ (x) for x ∈ U . Let γ : U → CK(C) be
given by γ(x) = τ (x) − δ0 v. Notice γ|∂U = F |∂U . Consider
B = {x ∈ U : x ∈ γ(x) + (1 − t) δ0 v for some t ∈ [0, 1]}.
If B = ∅ then γ has no fixed points in U , and since γ|∂U = F |∂U
we have that F is inessential in D∂U (U , C). It remains to consider the
case when B 6= ∅. Notice B ∩ ∂U = ∅ so there exists a continuous
µ : U → [0, 1] with µ(∂U ) = 1 and µ(B) = 0. Let Tµ : U → CK(C)
be defined by
Tµ (x) = γ(x) + (1 − µ(x)) δ0 v.
Clearly Tµ ∈ D∂U (U , C) with Tµ |∂U = γ|∂U = F |∂U . Also if x ∈ U and
x ∈ Tµ (x) then x ∈ γ(x) + (1 − µ(x)) δ0 v, so x ∈ B and µ(x) = 0 i.e.
On Krasnoselskii–Petryshyn compression and expansion theorem
675
x ∈ γ(x) + δ0 v = τ (x) (a contradiction). Thus Tµ ∈ D∂U (U , C) is a
fixed point free map with Tµ |∂U = F |∂U . Consequently F is inessential
in D∂U (U , C).
Theorem 2.5. Let E be a Banach space, C a closed, convex subset
of E with α u + β v ∈ C for all α ≥ 0, β ≥ 0 and u, v ∈ C, U an
open subset of C and 0 ∈ U . Suppose F ∈ D(U , C) and assume the
following conditions are satisfied:
(2.6)
there exists δ0 > 0 and v ∈ C \ {0} with δ0 v ∈ C \ U
and
(2.7)
x∈
/ λ F (x) + δ0 v for all λ ∈ (0, 1] and x ∈ ∂U.
Then φ (= F + δ0 v) is inessential in D∂U (U , C).
Proof. Let ψ be given by ψ(x) = {δ0 v}. Now since δ0 v ∈ C \ U we
have ψ ∈ D∂U (U , C). Consider
B = {x ∈ U : x ∈ t F (x) + δ0 v for some t ∈ [0, 1]}.
If B = ∅ then φ has no fixed points in U so φ is inessential in
D∂U (U , C). It remains to consider the case when B 6= ∅. Now B
is closed and B ∩ ∂U = ∅ from (2.7). Thus there exists a continuous
µ : U → [0, 1] with µ(∂U ) = 1 and µ(B) = 0. Let Lµ : U → CK(C)
be given by
Lµ (x) = µ(x) F (x) + δ0 v.
Notice Lµ ∈ D(U , C) and Lµ |∂U = φ|∂U , so Lµ ∈ D∂U (U , C). If there
exists x ∈ U with x ∈ Lµ (x) then x ∈ B so µ(x) = 0 i.e. x ∈ {δ0 v}
(a contradiction since x ∈ U and δ0 v ∈ C \ U ). Thus Lµ is fixed point
free on U and as a result φ is inessential in D∂U (U , C).
3.
The Krasnoselskii–Petryshyn compression and expansion theorem for countably k–set contractive maps
In this section E = (E, k . k) will be a Banach space and C will be
a closed, convex subset of E with α u + β v ∈ C for all α ≥ 0, β ≥ 0
and u, v ∈ C. Let ρ > 0 with
Bρ = {x : x ∈ C and kxk < ρ},
and of course Bρ = Bρ ∪ Sρ .
Sρ = {x : x ∈ C and kxk = ρ}
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Ravi P. Agarwal and Donal O’Regan
Theorem 3.1. Let E and C be as above and let r, R be constants
with 0 < r < R. Suppose F ∈ D(BR , C) and assume the following
conditions hold:
x∈
/ F (x) for x ∈ SR ∪ Sr
(3.1)
(3.2)
(3.3)
F : Br → CK(C) is inessential in DSr (Br , C)
(i.e. F |Br is inessential in DSr (Br , C))
F : BR → CK(C) is essential in DSR (BR , C).
Then F has a fixed point in Ω = {x ∈ C : r < kxk < R}.
Proof. Suppose F has no fixed points in Ω. Now (3.2) implies that
there exists θ ∈ D(Br , C) with θ|Sr = F |Sr and x ∈
/ θ(x) for x ∈ Br .
Let Φ : BR → CK(C) be defined by
F (x), r < kxk ≤ R
Φ(x) =
θ(x), kxk ≤ r.
Notice Φ ∈ D(BR , C) and Φ has no fixed points in BR (since θ has no
fixed points in Br and F has no fixed points in Ω). This contradicts
(3.3).
Our next result is a generalization of the Krasnoselskii–Petryshyn
compression theorem [3, 8].
Theorem 3.2. Let E and C be as above and let r, R be constants
with 0 < r < R. Suppose F ∈ D(BR , C) and assume the following
conditions hold:
(3.4)
there exists v ∈ C \ {0} with x ∈
/ F (x)+δ v for all δ > 0 and x ∈ Sr
and
(3.5)
x∈
/ λ F (x) for x ∈ SR and λ ∈ (0, 1).
Then F has a fixed point in {x ∈ C : r ≤ kxk ≤ R}.
Proof. Suppose x ∈
/ F (x) for x ∈ Sr ∪SR (otherwise we are finished).
Then
(3.6)
x∈
/ F (x) + δ v for δ ≥ 0 and x ∈ Sr
On Krasnoselskii–Petryshyn compression and expansion theorem
677
and
(3.7)
x∈
/ λ F (x) for x ∈ SR and λ ∈ [0, 1].
Choose M > 0 such that
kyk ≤ M for all y ∈ F (x) and x ∈ Br .
Now choose δ0 > 0 such that
(3.8)
kδ0 vk > M + r.
Let J : Br → CK(C) be given by J(x) = F (x) + δ0 v. Notice J ∈
DSr (Br , C) and also (3.8) guarantees that
x∈
/ J(x) for x ∈ Br .
This together with Theorem 2.3 implies
(3.9)
F : Br → CK(C) is inessential in DSr (Br , C).
Theorem 2.1 guarantees that the zero map is essential in DSR (BR , C).
This together with Theorem 2.2 and (3.7) implies
(3.10)
F : BR → CK(C) is essential in DSR (BR , C).
Finally (3.9), (3.10) and Theorem 3.1 imply that F has a fixed point in
Ω; here Ω = {x ∈ C : r < kxk < R}.
In our next theorem C ⊆ E will be a cone. Let ρ > 0 with
Ωρ = {x ∈ E : kxk < ρ},
∂E Ωρ = {x ∈ E : kxk = ρ},
Bρ = {x : x ∈ C and kxk < ρ},
and
Sρ = {x : x ∈ C and kxk = ρ}.
Notice
Bρ = Ωρ ∩ C and Sρ = ∂C (Ωρ ∩ C) = ∂E Ωρ ∩ C.
Theorem 3.3. Let E = (E, k . k) be a Banach space, C ⊆ E
a cone and let k . k be increasing with respect to C. Also r, R are
constants with 0 < r < R. Suppose F : ΩR ∩ C → CK(C) is a upper
semicontinuous, countable k–set contractive (here 0 ≤ k < 1) map and
assume the following conditions hold:
(3.11)
kyk ≤ kxk for all y ∈ F (x) and x ∈ ∂E ΩR ∩ C
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Ravi P. Agarwal and Donal O’Regan
and
(3.12)
kyk > kxk for all y ∈ F (x) and x ∈ ∂E Ωr ∩ C.
Then F has a fixed point in C ∩ {x ∈ E : r ≤ kxk ≤ R}.
Proof. First we show (3.11) implies (3.5). Suppose there exists x ∈
SR and λ ∈ (0, 1) with x ∈ λ F (x). Then there exists a y ∈ F (x) with
x = λ y and so
R = kxk = |λ| kyk < kyk ≤ kxk = R,
a contradiction. Next notice (3.12) implies (3.4). Suppose there exists
v ∈ C \ {0} with x ∈ F (x) + δ v for some δ > 0 and x ∈ Sr . Then
there exists a y ∈ F (x) with x = y + δ v. Now since k . k is increasing
with respect to C we have
kxk = ky + δ vk ≥ kyk > kxk,
a contradiction. The result now follows from Theorem 3.2.
For our next two results we again assume C ⊆ E is a closed, convex
nonempty set with α u + β v ∈ C for all α ≥ 0, β ≥ 0 and u, v ∈ C.
Theorem 3.4. Let E and C be as above and let r, R be constants
with 0 < r < R. Suppose F ∈ D(BR , C) and assume the following
conditions hold:
(3.13)
x∈
/ F (x) for x ∈ SR ∪ Sr
(3.14)
F : Br → CK(C) is essential in DSr (Br , C)
and
(3.15)
F : BR → CK(C) is inessential in DSR (BR , C).
Then F has at least two fixed points x0 and x1 with x0 ∈ Br and
x1 ∈ Ω; here Ω = {x ∈ C : r < kxk < R}.
Proof. From (3.14) we know that F has a fixed point in Br . Let
Ψ = F |Ω and suppose Ψ : Ω → CK(C) has no fixed points. Now (3.15)
guarantees that there exists an upper semicontinuous, countable k–set
contractive map θ : BR → CK(C) with
θ|SR = F |SR and x ∈
/ θ(x) for x ∈ BR .
On Krasnoselskii–Petryshyn compression and expansion theorem
Fix ρ ∈ (0, r) and consider the map Φ given by

ρ
R

θ
x
, kxk ≤ ρ


R
ρ




(r−ρ) kxk
(R−ρ) r−(R−r) kxk
Φ(x) =
Ψ
x
,

(R−ρ) r−(R−r) kxk
(r−ρ) kxk






Ψ(x), r ≤ kxk ≤ R.
679
ρ ≤ kxk ≤ r
Notice Φ : BR → CK(C) is well defined since if ρ ≤ kxk ≤ r then
(R − ρ) r − (R − r) kxk r ≤ x
≤ R.
(r − ρ) kxk
Note Φ : BR → CK(C) is a upper semicontinuous, countable k–set
contractive map (the proof of countably k–set contractive is essentially
contained in [6]). In addition
Φ|SR = Ψ|SR = F |SR = θ|SR
and Φ|Ω = Ψ|Ω = F |Ω
and Φ has no fixed point in BR (since θ has no fixed points in BR
and F has no fixed points in Ω).
Lets concentrate on Φ : Br → CK(C) (i.e. Φ|Br ). Now
Φ|Sr = Ψ|Sr = F |Sr
so Φ ∈ D(Br , C) with Φ|Sr = F |Sr and Φ has no fixed points in Br .
This of course contradicts (3.14).
Our next result is a generalization of Krasnoselskii–Petryshyn expansion theorem [3].
Theorem 3.5. Let E and C be as above and let r, R be constants
with 0 < r < R. Suppose F ∈ D(BR , C) and assume the following
conditions hold:
(3.16)
x∈
/ λ F (x) for x ∈ Sr and λ ∈ (0, 1)
and
(3.17) ∃ v ∈ C \ {0} with x ∈
/ F (x) + δ v for all δ > 0 and x ∈ SR .
Then F has a fixed point in {x ∈ C : r ≤ kxk ≤ R}.
Proof. Assume x ∈
/ F (x) for x ∈ Sr ∪SR (otherwise we are finished).
Then
(3.18)
x∈
/ λ F (x) for x ∈ Sr and λ ∈ [0, 1]
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Ravi P. Agarwal and Donal O’Regan
and
x∈
/ F (x) + δ v for δ ≥ 0 and x ∈ SR .
(3.19)
Theorem 2.1 guarantees that the zero map is essential in DSr (Br , C),
and this together with Theorem 2.2 and (3.18) implies
(3.20)
F : Br → CK(C) is essential in DSr (Br , C).
Let δ0 > 0 be such that
(3.21)
kδ0 vk > sup kyk + R for all y ∈ F (x).
x∈SR
Note δ0 v ∈ C \ BR . Let φ : Br → CK(C) be given by φ(x) = F (x) +
δ0 v. Note (3.21) implies for λ ∈ [0, 1] and x ∈ SR that
kδ0 v + λ yk > R = kxk for all y ∈ F (x),
and as a result
(3.22)
x∈
/ λ F (x) + δ0 v for all λ ∈ (0, 1] and x ∈ SR .
Now (3.22) together with Theorem 2.5 guarantees that
(3.23) φ (= F + δ0 v) : BR → CK(C) is inessential in DSR (BR , C).
Also notice (3.19), (3.23) and Theorem 2.4 imply that
(3.24)
F : BR → CK(C) is inessential in DSR (BR , C).
Finally (3.20), (3.24) and Theorem 3.4 guarantees that F has a fixed
point in Ω; here Ω = {x ∈ C : r < kxk < R}.
In our final theorem C ⊆ E will be a cone, and Ωρ , ∂E Ωρ (ρ > 0)
are as discussed before Theorem 3.3.
Theorem 3.6. Let E = (E, k . k) be a Banach space, C ⊆ E
a cone and let k . k be increasing with repect to C. Also r, R are
constants with 0 < r < R. Suppose F : ΩR ∩ C → CK(C) is a upper
semicontinuous, countable k–set contractive (here 0 ≤ k < 1) map and
assume the following conditions hold:
(3.25)
kyk > kxk for all y ∈ F (x) and x ∈ ∂E ΩR ∩ C
and
(3.26)
kyk ≤ kxk for all y ∈ F (x) and x ∈ ∂E Ωr ∩ C.
Then F has a fixed point in C ∩ {x ∈ E : r ≤ kxk ≤ R}.
On Krasnoselskii–Petryshyn compression and expansion theorem
681
Proof. Notice (3.16) and (3.17) are true so the result follows from
Theorem 3.5.
Remark 3.1. It is easy to combine the ideas in Theorem 3.3 with
those in Theorem 3.6 to obtain multiplicity results for F . We leave the
details to the reader.
References
[1] R. P. Agarwal and D. O’Regan, Essential and inessential maps and continuation
theorems, Appl. Math. Lett. 13 (2000), no. 2, 83–90.
[2] J. Daher, On a fixed point principle of Sadovskii, Nonlinear Anal. 2 (1978), 643–
645.
[3] P. M. Fitzpatrick and W. V. Petryshyn, Fixed point theorems for multivalued
noncompact acyclic mappings, Pacific J. Math. 54 (1974), 17–23.
[4] G. B. Gustafson and K. Schmitt, Nonzero solutions of boundary value problems
for second order ordinary and delay differential equations, J. Differential Equations 12 (1972), 129–147.
[5] H. Mönch, Boundary value problems for nonlinear ordinary differential equations
of second order in Banach spaces, Nonlinear Anal. 4 (1980), 985–999.
[6] D. O’Regan, Fixed point and random fixed point theorems in cones of Banach
spaces, Math. Comput. Appl. 32 (2000), 1473–1483.
[7] D. O’Regan and R. Precup, Fixed point theorems for set valued maps and existence principles for integral inclusions, J. Math. Anal. Appl. 245 (2000), 594–612.
[8] W. V. Petryshyn, Existence of fixed points of positive k–set contractive maps as
consequences of suitable boundary conditions, J. London Math. Soc. 38 (1988),
503–512.
[9] S. Park, Fixed points for approximable maps, Proc. Amer. Math. Soc. 124 (1996),
3109–3114.
Ravi P. Agarwal
Department of Mathematics
National University of Singapore
10 Kent Ridge
Crescent 119260, Singapore
Donal O’Regan
Department of Mathematics
National University of Ireland
Galway, Ireland