CONTRACTIONS IN GAUGE SPACES
ADELA CHIS¸ AND RADU PRECUPReceived 9 March 2004 and in revised form 30 April 2004
A continuation principle of Leray-Schauder type is presented for contractions with re-spect to a gauge structure depending on the homotopy parameter. The result involves the most general notion of a contractive map on a gauge space and in particular yields homotopy invariance results for several types of generalized contractions.
1. Introduction
One of the most useful results in nonlinear functional analysis, the Banach contraction principle, states that every contraction on a complete metric space into itself has a unique fixed point which can be obtained by successive approximations starting from any ele-ment of the space.
Further extensions have tried to relax the metrical structure of the space, its complete-ness, or the contraction condition itself. Thus, there are known versions of the Banach fixed point theorem for contractions defined on subsets of locally convex spaces: Mari-nescu [18, page 181], in gauge spaces (spaces endowed with a family of pseudometrics): Colojoar˘a [5] and Gheorghiu [11], in uniform spaces: Knill [16], and in syntopogenous spaces: Precup [21].
As concerns the completeness of the space, there are known results for a space endowed with two metrics (or, more generally, with two families of pseudometrics). The space is assumed to be complete with respect to one of them, while the contraction condition is expressed in terms of the second one. The first result in this direction is due to Maia [17]. The extensions of Maia’s result to gauge spaces with two families of pseudometrics and to spaces with two syntopogenous structures were given by Gheorghiu [12] and Precup [22], respectively.
As regards the contraction condition, several results have been established for vari-ous types of generalized contractions on metric spaces. We only refer to the earlier pa-pers of Kannan [15], Reich [27], Rus [29], and ´Ciri´c [4], and to the survey article of Rhoades [28].
We may say that almost every fixed point theorem for self-maps can be accompanied by a continuation result of Leray-Schauder type (or a homotopy invariance result). An elementary proof of the continuation principle for contractions on closed subsets of a Banach space (another proof is based on the degree theory) is due to Gatica and Kirk [10]. The homotopy invariance principle for contractions on complete metric spaces was established by Granas [14] (see also Frigon and Granas [8] and Andres and G ´orniewicz [2]), extended to spaces endowed with two metrics or two vector-valued metrics, and completed by an iterative procedure of discrete continuation along the fixed points curve by Precup [23,24] (see also O’Regan and Precup [19] and Precup [26]). Continuation results for contractions on complete gauge spaces were given by Frigon [7] and for gener-alized contractions in the sense of Kannan-Reich-Rus and ´Ciri´c, by Agarwal and O’Regan [1] and the first author [3].
However, until now, a unitary continuation theory for the most general notion of a contraction in gauge spaces has not been developed. The goal of this paper is to fill this gap solving this way a problem stated in Precup [25]. We are also motivated by a num-ber of papers which have been published in the last decade, such as those of Frigon and Granas [9] and O’Regan and Precup [20], and also by the applications to integral and differential equations in locally convex spaces, see Gheorghiu and Turinici [13].
2. Preliminaries
2.1. Gauge spaces. LetX be any set. A mapp:X×X→R+is called apseudometric(or agauge) on X if p(x,x)=0, p(x,y)=p(y,x), and p(x,y)≤p(x,z) +p(z,y) for every
x,y,z∈X. A familyᏼ={pα}α∈Aof pseudometrics onX(or agauge structureonX) is said to beseparatingif for each pair of pointsx,y∈Xwithx=y, there is apα∈ᏼsuch thatpα(x,y)=0. A pair (X,ᏼ) of a nonempty setXand a separating gauge structureᏼ onXis called agauge space.
It is well known (see Dugundji [6, pages 198–204]) that any familyᏼof pseudometrics on a setX induces onX a structure ᐁof uniform space and conversely, any uniform structure onXis induced by a family of pseudometrics onX. In addition,ᐁis separating (or Hausdorff) if and only ifᏼis separating. Hence we may identify the gauge spaces and the Hausdorffuniform spaces.
For the rest of this section we consider a gauge space (X,ᏼ) with the gauge structure
ᏼ= {pα}α∈A. A sequence (xn) of elements inXis said to beCauchyif for everyε >0 and
α∈A, there is anN withpα(xn,xn+k)≤εfor alln≥Nandk∈N. The sequence (xn) is calledconvergentif there exists anx0∈Xsuch that for everyε >0 andα∈A, there is an
N with pα(x0,xn)≤εfor alln≥N. A gauge space is calledsequentially completeif any Cauchy sequence is convergent. A subset ofXis said to besequentially closedif it contains the limit of any convergent sequence of its elements.
2.2. General contractions on gauge spaces. We now recall the notion of contraction on a gauge space introduced by Gheorghiu [11]. Let (X,ᏼ) be a gauge space withᏼ= {pα}α∈A. A mapF:D⊂X→X is acontractionif there exists a functionϕ:A→Aand
a∈RA
pα
F(x),F(y)≤aαpϕ(α)(x,y) ∀α∈A,x,y∈D, (2.1)
∞
n=1
aαaϕ(α)aϕ2(α)···aϕn−1(α)pϕn(α)(x,y)<∞ (2.2)
for everyα∈Aandx,y∈D. Here,ϕnis thenth iteration ofϕ. Notice that a sufficient condition for (2.2) is that
∞
n=1
aαaϕ(α)aϕ2(α)···aϕn−1(α)<∞, (2.3) suppϕn(α)(x,y) :n=0, 1,...<∞ ∀α∈A,x,y∈D. (2.4)
The above definition contains as particular cases the notion of contraction on a sub-set of a locally convex space introduced by Marinescu [18], for whichϕ2=ϕ, and the most worked notion of contraction on a gauge space as defined in Tarafdar [30], which corresponds toϕ(α)=αandaα<1 for allα∈A.
Given a spaceX endowed with two gauge structuresᏼ= {pα}α∈A andᏽ= {qβ}β∈B, in order to precise the gauge structure with respect to which a topological-type notion is considered, we will indicate the corresponding gauge structure in light of that notion. So, we will speak aboutᏼ-Cauchy,ᏽ-Cauchy,ᏼ-convergent, andᏽ-convergent sequences;
ᏼ-sequentially closed andᏽ-sequentially closed sets;ᏼ-contractions andᏽ-contractions, and so forth. Also we say that a mapF:X→X is (ᏼ,ᏽ)-sequentially continuous if for everyᏼ-convergent sequence (xn) with the limitx, the sequence (F(xn)) isᏽ-convergent toF(x).
We now state Gheorghiu’s fixed point theorem of Maia type for self-maps of gauge spaces [12].
Theorem2.1 (Gheorghiu). LetX be a nonempty set endowed with two separating gauge
structuresᏼ= {pα}α∈A andᏽ= {qβ}β∈B and letF:X→X be a map. Assume that the following conditions are satisfied:
(i)there is a functionψ:A→Bandc∈(0,∞)A,c= {c
α}α∈Asuch that
pα(x,y)≤cαqψ(α)(x,y) ∀α∈A,x,y∈X; (2.5) (ii) (X,ᏼ)is a sequentially complete gauge space;
(iii)Fis(ᏼ,ᏽ)-sequentially continuous; (iv)Fis aᏽ-contraction.
ThenF has a unique fixed point which can be obtained by successive approximations starting from any element ofX.
The following slight extension of Gheorghiu’s theorem will be used in the sequel.
Theorem2.2. LetXbe a set endowed with two separating gauge structuresᏼ= {pα}α∈A andᏽ= {qβ}β∈B, letD0 andD be two nonempty subsets ofX withD0⊂D, and let F:
(i)there is a functionψ:A→Bandc∈(0,∞)A,c= {c
α}α∈Asuch that
pα(x,y)≤cαqψ(α)(x,y) ∀α∈A,x,y∈X; (2.6) (ii) (X,ᏼ)is a sequentially complete gauge space;
(iii)ifx0∈D0,xn=F(xn−1)forn=1, 2,...,andᏼ-limn→∞xn=xfor somex∈D, then
F(x)=x;
(iv)Fis aᏽ-contraction onD.
ThenF has a unique fixed point which can be obtained by successive approximations starting from any element ofD0.
Proof. Take anyx0∈D0 and consider the sequence (xn) of successive approximations,
xn=F(xn−1),n=1, 2,....SinceF(D0)⊂D0, one hasxn∈D0for alln∈N. By (iv), (xn) isᏽ-Cauchy. Next, (i) implies that (xn) is alsoᏼ-Cauchy, hence it isᏼ-convergent to somex∈D, in virtue of (ii). Now, (iii) guarantees that F(x)=x. The uniqueness is a
consequence of (iv).
2.3. Generalized contractions on metric spaces. It is worth noting that a number of fixed point results for generalized contractions on complete metric spaces appear as direct consequences ofTheorem 2.2. Here are two examples.
Let (X,p) be a complete metric space andF:X→Xa map. (1) Assume thatFsatisfies
pF(x),F(y)≤apx,F(x)+py,F(y)+bp(x,y) (2.7) for allx,y∈X, wherea,b∈R+,a >0, and 2a+b <1.
We associate toFa family of pseudometricsqk,k∈N, given by
qk(x,y)=
a rk−bk rk(r−b)
px,F(x)+py,F(y)+
b r
k
p(x,y) forx=y,
0 forx=y.
(2.8)
Here,r=(a+b)/(1−a) andb < r <1. By induction, we can see that
qk
F(x),F(y)≤rqk+1(x,y) ∀k∈N,x,y∈X. (2.9) It is clear thatᏽ= {qk}k∈Nis a separating gauge structure onXand from (2.9) we have thatFis aᏽ-contraction onX. In this case,ϕ:N→Nis given byϕ(k)=k+ 1 andak=r for allk∈N. Also, for anyk∈N, (2.2) means∞n=1rnqk+n(x,y)<∞, which according to (2.8) is true since 0≤b <1 andb < r <1.
Corollary2.3 (Reich-Rus). If(X,p)is a complete metric space andF:X→Xsatisfies (2.7), thenFhas a unique fixed point.
ᏼ-limn→∞xn=x, that is,p(x,xn)→0 asn→ ∞. From (2.7), we have
pxn,F(x)
=pFxn−1
,F(x)
≤apxn−1,xn
+px,F(x)+bpxn−1,x. (2.10)
Passing to the limit, we obtain p(x,F(x))≤ap(x,F(x)), whence p(x,F(x))=0, that is,
F(x)=x. Now the conclusion follows fromTheorem 2.2.
(2) Assume thatFsatisfies
pF(x),F(y)≤amaxp(x,y),px,F(x),py,F(y),px,F(y),py,F(x) (2.11) for allx,y∈Xand somea∈[0, 1). ThenFis aᏽ-contraction, whereᏽ= {qk}k∈Nand
qk(x,y)=
maxpFi(x),Fj(x),pFi(y),Fj(y),
pFi(x),Fj(y):i,j=0, 1,...,k forx=y,
0 forx=y.
(2.12)
We haveq0=pand from (2.11) we obtain
pFi(x),Fj(x)≤aq k(x,y),
pFi(y),Fj(y)≤aqk(x,y),
pFi(x),Fj(y)≤aq k(x,y),
(2.13)
for alli,j∈ {0, 1,...,k}andx,y∈X. It follows that
qkF(x),F(y)≤aqk+1(x,y) (2.14) and also
qk(x,y)=maxpx,Fi(x),py,Fi(y),px,Fi(y),py,Fi(x), (2.15) where the maximum is taken overi∈ {0, 1,...,k}. If, for example,qk(x,y)=p(x,Fi(x)) for somei∈ {1, 2,...,k}, then
qk(x,y)≤px,F(x)+pF(x),Fi(x)≤px,F(x)+aqk(x,y). (2.16) Hence
qk(x,y)≤1−1
ap
x,F(x)≤ 1
1−aq1(x,y). (2.17)
Generally, we can prove similarly that
Corollary2.4 ( ´Ciri´c). If(X,p)is a complete metric space andF:X→Xsatisfies (2.11), thenFhas a unique fixed point.
Proof. Here againᏼ= {p},ᏽ= {qk}k∈N, andq0=p. To check (iii), assumex0∈X,xn=
F(xn−1) forn=1, 2,..., andᏼ-limn→∞xn=x, that is,p(x,xn)→0 asn→ ∞. From (2.11), we obtain
pxn,F(x)
=pFxn−1
,F(x)
≤amaxpxn−1,x,pxn−1,Fxn−1
,px,F(x),
pxn−1,F(x),px,Fxn−1
.
(2.19)
Passing to the limit, we obtain p(x,F(x))≤ap(x,F(x)), whence p(x,F(x))=0, that is,
F(x)=x. Thus we may applyTheorem 2.2.
3. Continuation theorems in gauge spaces
For a mapH:D×[0, 1]→X, whereD⊂X, we will use the following notations:
Σ=(x,λ)∈D×[0, 1] :H(x,λ)=x,
=x∈D:H(x,λ)=xfor someλ∈[0, 1],
Λ=λ∈[0, 1] :H(x,λ)=xfor somex∈D.
(3.1)
Now we state and prove the main result of this paper: a continuation principle for con-tractions on spaces with a gauge structure depending on the homotopy parameter.
Theorem3.1. LetXbe a set endowed with the separating gauge structuresᏼ= {pα}α∈A andᏽλ= {qλ
β}β∈Bforλ∈[0, 1]. LetD⊂Xbeᏼ-sequentially closed,H:D×[0, 1]→Xa map, and assume that the following conditions are satisfied:
(i)for eachλ∈[0, 1], there exists a functionϕλ:B→Bandaλ∈[0, 1)B,aλ= {aλβ}β∈B such that
qλβH(x,λ),H(y,λ)≤aλβqλϕλ(β)(x,y),
∞
n=1
aλβaλϕλ(β)aλϕ2 λ(β)···a
λ ϕn−1
λ (β)q
λ ϕn
λ(β)(x,y)<∞
(3.2)
for everyβ∈Bandx,y∈D;
(ii)there existsρ >0such that for each(x,λ)∈Σ, there is aβ∈Bwith
infqλ
β(x,y) :y∈X\D
> ρ; (3.3)
(iii)for eachλ∈[0, 1], there is a functionψ:A→Bandc∈(0,∞)A,c= {c
α}α∈Asuch that
(iv) (X,ᏼ)is a sequentially complete gauge space;
(v)ifλ∈[0, 1],x0∈D,xn=H(xn−1,λ)forn=1, 2,...,andᏼ-limn→∞xn=x, then
H(x,λ)=x;
(vi)for everyε >0, there existsδ=δ(ε)>0with
qλϕn λ(β)
x,H(x,λ)≤1−aλϕn λ(β)
ε (3.5)
for(x,µ)∈Σ,|λ−µ| ≤δ, allβ∈B, andn∈N.
In addition, assume thatH0:=H(·, 0)has a fixed point. Then, for eachλ∈[0, 1], the mapHλ:=H(·,λ)has a unique fixed point.
Proof. We prove that there exists a numberh >0 such that ifµ∈Λ, thenλ∈Λfor every
λsatisfying|λ−µ| ≤h. This, together with 0∈Λ, clearly impliesΛ=[0, 1].
First we note that from (ii) it follows that for each (x,λ)∈Σ, there existsβ∈Bsuch that the set
B(x,λ,β)=:y∈X:qλ ϕn
λ(β)(x,y)≤ρ∀n∈N
(3.6)
is included inD.
Letµ∈Λand letH(x,µ)=x. From (vi), there ish=h(ρ)>0, independent ofµandx, such that
qλ ϕn
λ(β)
x,H(x,λ)=qλ ϕn
λ(β)
H(x,µ),H(x,λ)≤1−aλ ϕn
λ(β)
ρ (3.7)
for|λ−µ| ≤hand alln∈N. Consequently, if|λ−µ| ≤handy∈B(x,λ,β), then
qϕλn λ(β)
x,H(y,λ)≤qϕλn λ(β)
x,H(x,λ)+qϕλn λ(β)
H(x,λ),H(y,λ)
≤1−aλ ϕn
λ(β)
ρ+aλ ϕn
λ(β)q
λ ϕn+1
λ (β)(x,y) ≤1−aλ
ϕn λ(β)
ρ+aλ ϕn
λ(β)ρ=ρ.
(3.8)
Hence, for|λ−µ| ≤h,Hλis a self-map ofD0:=B(x,λ,β). NowTheorem 2.2guarantees
thatλ∈Λfor|λ−µ| ≤h.
Assuming a continuity property ofH, we derive fromTheorem 3.1the following result.
Theorem3.2. LetXbe a set endowed with the separating gauge structuresᏼ= {pα}α∈A andᏽλ= {qλ
(a)for eachλ∈[0, 1], there exists a functionϕλ:B→Bandaλ∈[0, 1)B,aλ= {aλβ}β∈B such that
qλ β
H(x,λ),H(y,λ)≤aλ
βqϕλλ(β)(x,y), (3.9)
supqλϕn
λ(β)(x,y) :n∈N
<∞, (3.10)
sup
∞
n=1
aλ βaλϕλ(β)a
λ ϕ2
λ(β)···a
λ ϕn−1
λ (β):λ∈[0, 1]
<∞, (3.11)
for allβ∈Bandx,y∈D;
(b)there exists a setU⊂Dsuch thatH(x,λ)=xfor allx∈D\U andλ∈[0, 1]; and for each(x,µ)∈Σ, there isβ∈B,δ >0, andγ >0such that for everyλ∈[0, 1]with |λ−µ| ≤γ,
y∈X:qλβ(x,y)< δ⊂U; (3.12)
(c)for eachλ∈[0, 1], there is a functionψ:A→Bandc∈(0,∞)A,c= {c
α}α∈A such that
pα(x,y)≤cαqλψ(α)(x,y) ∀α∈A,x,y∈X; (3.13) (d) (X,ᏼ)is a sequentially complete gauge space;
(e)His(ᏼ,ᏼ)-sequentially continuous; (f)for everyε >0, there existsδ=δ(ε)>0with
qλϕn λ(β)
x,H(x,λ)≤1−aλϕn λ(β)
ε (3.14)
for(x,µ)∈Σ,|λ−µ| ≤δ, and allβ∈Bandn∈N.
In addition, assume thatH0:=H(·, 0)has a fixed point. Then, for eachλ∈[0, 1], the mapHλ:=H(·,λ)has a unique fixed point.
Proof. Conditions (i), (iii), (iv), and (vi) inTheorem 3.1are obviously satisfied. Assume (ii) is false. Then, for eachn∈N\ {0}, there is (xn,λn)∈Σandynβ∈X\Dwith
qλn
β
xn,ynβ≤1
n for everyβ∈B. (3.15)
Clearly we may assume thatλn→λ.
Fix an arbitraryβ∈B. From (f) we see that for a givenε >0, there is a numberN= N(ε)>0 such that
qλϕi λ(β)
xn,Hxn,λ≤4ε
C (3.16)
for alln≥Nandi∈N, whereCis any positive number with
1 +
∞
i=1
aλβaλϕλ(β)···aλϕi−1
Now, forn,m≥N, using (a), we obtain
qλβxn,xm=qβλ
Hxn,λn,Hxm,λm≤qβλ
Hxn,λn,Hxn,λ +qλβHxm,λm
,Hxm,λ
+qλβHxn,λ
,Hxm,λ
≤qλ β
Hxn,λn
,Hxn,λ
+qλ β
Hxm,λm
,Hxm,λ
+aλβqλϕλ(β)
xn,xm
≤ ε
2C+aλβqλϕλ(β)
xn,xm
.
(3.18)
Similarly,
qλϕλ(β)xn,xm≤2ε
C+a
λ ϕλ(β)q
λ ϕ2
λ(β)
xn,xm (3.19)
and, in general,
qλϕi λ(β)
xn,xm
≤ ε
2C+aλϕi λ(β)q
λ ϕi+1
λ (β)
xn,xm
(3.20) for alli∈N. It follows that for alln,m≥Nand everyl∈N, we have
qλβxn,xm ≤ ε 2C 1 + l
i=1
aλβaλϕλ(β)···aλϕi−1 λ (β)
+aλβaλϕλ(β)···aλϕl λ(β)q
λ ϕl+1
π (β)
xn,xm
≤ε
2+a λ
βaλϕλ(β)···a
λ ϕl
λ(β)M
λ,β,xn,xm
.
(3.21)
Here, M(λ,β,x,y) :=sup{qλ ϕn
λ(β)(x,y) :n∈N}. According to (3.11), for each couple
[n,m] withn,m≥N, we may find anlsuch that
aλ
βaλϕλ(β)···a
λ ϕl
λ(β)M
λ,β,xn,xm
≤ ε
2. (3.22)
Hence
qβλxn,xm≤ε ∀n,m≥N. (3.23) Thus the sequence (xn) isᏽλ-Cauchy. Now (c) guarantees that (xn) isᏼ-Cauchy. Further-more, (d) implies that (xn) isᏼ-convergent. Letx=ᏼ-limn→∞xn. Clearlyx∈D. Then, from (e),ᏼ-limn→∞H(xn,λn)=H(x,λ). HenceH(x,λ)=x.
Now we claim that
qλn
β
x,xn−→0 asn−→ ∞. (3.24)
Indeed, since (x,λ)∈Σandλn→λ, from (f) it follows that for a givenε >0, there is a numberN0=N0(ε)>0 such that
qλn
ϕi λn(β)
x,Hx,λn≤2ε
for alln≥N0andi∈N. Then, forn≥N0, we have
qλn
β
x,xn
=qλn
β
x,Hxn,λn
≤qλn
β
x,Hx,λn
+qλn
β
Hx,λn
,Hxn,λn
≤ ε
2C+a
λn
β qλϕnλn(β)
x,xn
.
(3.26)
Furthermore, as above, we deduce that
qλn
β
x,xn
≤ε ∀n≥N0. (3.27)
This proves our claim. Also (b) guarantees
qλn
β
x,ynβ
≥δ (3.28)
for a sufficiently largenand someβ∈B. Now, from
0< δ≤qλn
β
x,ynβ≤qλβn
x,xn+qλβn
xn,ynβ, (3.29) we derive a contradiction. This contradiction shows that (ii) holds. Also (v) immediately
follows from (e). ThusTheorem 3.1applies.
Remark 3.3. In particular, if the gauge structures reduce to metric structures, that is,
ᏼ= {p}andᏽλ=ᏽ= {q},pandqbeing two metrics onX,Theorem 3.2becomes the first part ofTheorem 2.2of Precup [23] (with the additional assumption that there is a constantc >0 withp(x,y)≤cq(x,y) for allx,y∈X).
4. Homotopy results for generalized contractions on metric spaces
In this section, we test Theorem 3.1 on generalized contractions on complete metric spaces. We begin with a continuation result for generalized contractions of Reich-Rus type.
Theorem4.1. Let(X,p)be a complete metric space,Da closed subset ofX, andH:D×
[0, 1]→Xa map. Assume that the following conditions are satisfied: (A)there exista,b∈R+witha >0and2a+b <1such that
pHλ(x),Hλ(y)
≤apx,Hλ(x)
+pdy,Hλ(y)
+bp(x,y) (4.1) for allx,y∈Dandλ∈[0, 1];
(B) inf{p(x,y) :x∈,y∈X\D}>0;
(C)for eachε >0, there existsδ=δ(ε)>0such that
pH(x,λ),H(x,µ)≤ε for|λ−µ| ≤δ,allx∈D. (4.2) In addition, assume thatH0:=H(·, 0)has a fixed point. Then, for eachλ∈[0, 1], the map
Proof. We will applyTheorem 3.1. Letᏼ= {p}andᏽλ= {qλ
k}k∈N, whereqλkare defined as (2.8) shows, withFreplaced byHλ. In this caseA= {1}andB=N.
Condition (i) inTheorem 3.1is satisfied as the reasoning inSection 2.3shows. Next (B) guarantees (ii) withβ=0, sinceqλ0=p. Now takeψ(1)=0 to see that (iii) holds trivially. Since the space (X,p) is complete, we have (iv), while (v) can be checked as in the proof ofCorollary 2.3. Thus it remains to check (vi), that is, for eachε >0, there existsδ >0 such that
qλ m
x,H(x,λ)≤(1−r)ε (4.3)
for every (x,µ)∈Σ,|λ−µ| ≤δ, andm∈N. This can be proved by using (C) if we observe thatqλ
m(x,H(x,λ)) depends only onp(x,Hλ(x))=p(H(x,µ),H(x,λ)), and
arm−bm rm(r−b)≤
a r−b,
b r
m
≤1. (4.4)
Similarly,Theorem 3.1yields a continuation result for generalized contractions in the sense of ´Ciri´c.
Theorem4.2. Let(X,p)be a complete metric space,Da closed subset ofX, andH:D×
[0, 1]→Xa map. Assume that the following conditions are satisfied: (A)there existsa <1such that
pHλ(x),Hλ(y)
≤amaxp(x,y),px,Hλ(x)
,py,Hλ(y)
,px,Hλ(y)
,py,Hλ(x)
(4.5)
for allx,y∈Dandλ∈[0, 1]; (B) inf{p(x,y) :x∈,y∈X\D}>0;
(C)for eachε >0, there existsδ=δ(ε)>0such thatp(H(x,λ),H(x,µ))≤εfor|λ−µ| ≤ δand allx∈D.
In addition, assume thatH0:=H(·, 0)has a fixed point. Then, for eachλ∈[0, 1], the map
Hλ:=H(·,λ)has a unique fixed point. Acknowledgment
We would like to thank the referees for their valuable comments and suggestions, espe-cially for pointing out the necessity of taking supremum overλin (3.11).
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Adela Chis¸: Department of Mathematics, Technical University of Cluj, 400020 Cluj-Napoca, Romania
E-mail address:[email protected]
Radu Precup: Department of Applied Mathematics, Babes¸-Bolyai University of Cluj, 400084 Cluj-Napoca, Romania
