FIXED POINT THEORY AND ITS APPLICATIONS
BANACH CENTER PUBLICATIONS, VOLUME 77
INSTITUTE OF MATHEMATICS
POLISH ACADEMY OF SCIENCES
WARSZAWA 2007
NONLINEAR CONTRACTIVE CONDITIONS:
A COMPARISON AND RELATED PROBLEMS
JACEK JACHYMSKI
Institute of Mathematics, Technical University of Łódź
Wólczańska 215, 93-005 Łódź, Poland
E-mail: jachym@p.lodz.pl
IZABELA JÓŹWIK
Institute of Mathematics and Centre of Mathematics and Physics
Technical University of Łódź
al. Politechniki 11, 90-924 Łódź, Poland
E-mail: ijozwik@p.lodz.pl
Abstract. We establish five theorems giving lists of nonlinear contractive conditions which
turn out to b e mutually equivalent. We derive them from some general lemmas concerning subsets
of the plane which may be applied both in the single- or set-valued case as well as for a family of
mappings. A separation theorem for concave functions is proved as an auxiliary result. Also, we
discuss briefly the fol lowing problems for several classes of contractions: stability of procedure
of successive approximations, existence of approximate fixed points, continuous dependence of
fixed points on parameters, existence of invariant sets for iterated function systems. Moreover,
James Dugundji’s contribution to the metric fixed p o int theory i s presented. Using his notio n
of contracti o ns , we also establish an extension of a domain i nvariance theorem for contractive
fields.
1. Introduction. Let (X, d) be a c omplete metric space and T be a selfmap of X. We
say that x
∗
in X is a contractive fixed point (abbr. CFP) of T if x
∗
= T x
∗
and the Picard
iterates T
n
x converge to x
∗
as n → ∞ for all x ∈ X. There are numerous results in the
literature giving sufficient conditions for the existence of a CFP. It seems, however, that
the Banach Principle is still the most important here for its simplicity and an amazing
efficiency in applications. Nevertheless, in this paper we wish to give a detailed analysis of
2000 Mathematics Subject Classification: Primary: 47H10, 54H25; Secondary: 26A15, 26A18,
26A48, 26A51, 54C10.
Key words and phrases: Nonlinear contractive conditions, ϕ-contra ctive map, contractive
fixed point, approximate fixed point, domain invariance theorem, separation theorem, semi-
continuous function, monotonic function, concave function, subadditive function.
The paper is in final form and no version of it will be published elsewhere.
[123]
124 J. JACHYMSKI AND I. JÓŹWIK
several contractive conditions; in particular, our main purpose is to illuminate connections
between them. Part of this study has already been done in our articles [Ja97P] and [Ja99],
and we shall partially use this material in Sections 3 and 4. In general, the rest of the
results given in Sections 2, 5 and 6 seem t o be new.
As far as we know, the first significant generalization of Banach’s Principle was ob-
tained by Rakotch [Ra62] in 1962 (the problem was suggested by H. Hanani). We say
that T is a Rakotch contraction if there is a decreasing function α : R
+
→ [0, 1] such that
α(t) < 1 for all t > 0, and
d(T x, T y) ≤ α(d(x, y))d(x, y) for all x, y ∈ X. (1)
(Here R
+
denotes the set of all non-negative reals; in the sequel we use ‘decreasing’ for
‘non-increasing’ and ‘increasing’ for ‘non-decreasing’.) Then each Rakotch contraction
has a CFP.
Subsequently, in 1968 Browder [Br68] introduced a more general definition. Given a
function ϕ : R
+
→ R
+
such that
ϕ(t) < t for all t > 0, (2)
we say that T is ϕ-contra ctive if
d(T x, T y) ≤ ϕ(d(x, y)) for all x, y ∈ X. (3)
Then we call T a Browder contraction if ϕ is increasing and right continuous. Such a T
has a CFP. (Actually, that was shown by Browder [Br68] under the additional assumption
that (X, d) be bounded.) T his result, in turn, was extended by Boyd and Wong [BW69]
in 1969 who observed that it was enough to assume only the right upper semi-continuity
of ϕ, i.e.,
lim sup
s→t
+
ϕ(s) ≤ ϕ(t) for all t ∈ R
+
.
(Moreover, the boundedness of (X, d) is superfluous.) Then T is said to be a Boyd–Wong
contra ction.
Another natural generalization of Banach’s Principle was given in [KV72] in 1972. We
say that T is a Krasnosel’ski˘ı contraction if given a, b ∈ R
+
with 0 < a < b, there is an
L(a, b) ∈ [0, 1) such that for all x, y ∈ X,
d(T x, T y) ≤ L(a, b)d(x, y) if a ≤ d(x, y) ≤ b. (4)
However, we have shown in [Ja97P] that this definition is equivalent to the one of Browder.
In particular, this corrects a remark in [GD03, p. 16].
Further two conditions were proposed by Geraghty in 1973 [Ge73] and 1974 [Ge74].
T is said to be a Geraghty (I) contraction if it satisfies (1) with a function α : R
+
→ [0, 1]
having the property that given a sequence (t
n
)
n∈N
,
α(t
n
) → 1 implies t
n
→ 0. (5)
This definition was slightly modified in [Ge74]—the above property of α was replaced
there by the following: Given a sequence (t
n
)
n∈N
,
if (t
n
)
n∈N
is decreasing and α(t
n
) → 1, then t
n
→ 0. (6)
NONLINEAR CONTRACTIVE CONDITIONS 125
Then we call T a Geraghty (II) contraction. It turns out that this class of mappings
coincides with the Boyd–Wong class as shown by Heged¨us and Szil´agyi [HS80]. On the
other hand, in Section 2 we shall show that T is a Geraghty (I) contraction if and only if
it is a Rakotch contraction. Also, we shall consider in Sec tion 3 the following variant of
the above conditions: T is a Geraghty (III) contraction if it satisfies (1) with a function
α : R
+
→ [0, 1) such that given a sequence (t
n
)
n∈N
,
if (t
n
)
n∈N
is bounded and α(t
n
) → 1, then t
n
→ 0. (7)
We shall show this definition is eq uivalent to the one of Browder.
Another variant of Browder’s condition was given by Matkowski [Ma75] in 1975 who
replaced the continuity assumption on ϕ by t he c ondition:
lim
n→∞
ϕ
n
(t) = 0 for all t > 0. (8)
Then T is said to be a Matkowski contraction. Though t his class of mappings is essentially
wider than Browder’s class (see Proposition 2 and Example 2), in some sense both these
conditions are equivalent. More precisely, in [Ja99] we have shown that if T is a Matkowski
contraction, then the second iterate T
2
satisfies Browder’s condition. Hence, since by
Browder’s theorem, T
2
has a CFP, so does T.
In 1976 James Dugundji [Du76] established a very general coincidence theorem f rom
which he derived a fixed point theorem for the following class of mappings. We say T
is a Dugundji contraction if the function (x, y) 7→ d(x, y) − d(T x, T y) is positive definite
mod ∆(X), the diagonal in X ×X, i.e., given an ε > 0, there is a δ > 0 such that for all
x, y ∈ X,
d(x, y) − d(T x, T y) < δ implies d(x, y) < ε. (9)
As shown in [Du76], Dugundji’s theorem yields Browder’s result under the assumption
that (X, d) be bounded. However, in Section 5 we present that without the bounded-
ness assumption, the class of Browder’s contractions is essentially wider than the one of
Dugundji. Nevertheless, we also show that each Dugundji contraction enjoys a nice pro-
perty: It has a fixed point which is both contractive and approx imate . Recall that a
mapping T has an a p p roximate fixed point (abbr. AFP) x
∗
if x
∗
= T x
∗
and given a
sequence (x
n
)
n∈N
,
d(x
n
, T x
n
) → 0 implies x
n
→ x
∗
. (10)
(Let us note that this definition of an AFP is different from that given in [MR03].) On
the other hand, it is not clear if each Browder contraction has such a property. Also, in
Section 5 we give an exte nsion of a domain invariance theorem for Dugundji contractive
fields (see Theorem 10). It seems to be interesting here that the domain of t he mapping
need not be open, and the normed linear space need not be complete whereas these
additional assumptions were used by Dugundji and Granas [DG78] in the proof of their
domain invariance theorem for Browder contractions (also see [GD03, p. 11] for the case
of Banach contractions). We close Section 5 with a theorem on continuous dependence
of fixed points on parameters for a family of Dugundji contractions (see Theorem 11).
126 J. JACHYMSKI AND I. JÓŹWIK
Yet another contractive definition was given by Dugundji and Granas [DG78] in 1978.
A mapping T is said to be a Dugundji–Granas contraction if
d(T x, T y) ≤ d(x, y) − Θ(x, y) f or all x, y ∈ X, (11)
where the function Θ : X × X → R
+
is compactly positive on X, i.e., given a, b ∈ R
+
such that 0 < a < b,
inf{Θ(x, y) : a ≤ d(x, y) ≤ b} > 0.
As observed in [DG78], the above definition is equivalent to that of Krasnosel’ski˘ı [KV72].
A domain invariance theorem for such mappings was also given in [KV72].
In Section 6 we study the class of ϕ-contractive maps with an increasing and conti-
nuous ϕ satisfying the following limit condition
lim
t→∞
(t − ϕ(t)) = ∞ (12)
which was introduced by Matkowski [Ma77] in 1977; independently, it also appeared in
Walter’s [Wa81] paper in 1981. Such contractive maps are said to be Matkowski–Walter
contra ctions. (Incidentally, both authors considered more general c onditions than (3).)
This class of ϕ-contractions is of some import ance in the theory of iterated function
systems (see Theorem 13) and the asymptotic fixed point theory.
We close our shortened history of studies on nonlinear contractive conditions with
recalling the following definition given by Burton [Bu96] in 1996. We say that T is a large
contra ction if given an a > 0, there is an L(a) ∈ [0, 1) such that for all x, y ∈ X,
d(T x, T y) ≤ L(a)d(x, y) if d(x, y) ≥ a. (13)
Somewhat unexpectedly, we are coming back here to the beginning point since Burton’s
condition turns out to be equivalent to the one of Rakotch (see Theorem 1).
Finally, we would like to emphasize that our study is far from being comprehensive
since there is a huge number of pap e rs dealing with contractive type conditions. Many
useful references concerning this topic may be found, e.g., in the books [KS01, Chapter 1]
and [RPP02].
Throughout this paper we use the convention that inf ∅ = ∞ and—since we are
working on the half-line R
+
—sup ∅ = 0.
2. Rakotch contractions. We begin with the following auxiliary result concerning a
subset of the quadrant R
+
× R
+
. (The idea of such an approach goes back to the paper
by Heged¨us and Szil´agyi [HS80].) Given a function ϕ : R
+
→ R
+
, set
E
ϕ
:= {(t, u) ∈ R
+
× R
+
: u ≤ ϕ(t)}.
Lemma 1. Let D be a subset of R
+
× R
+
such that for any u ∈ R
+
,
(0, u) ∈ D implies u = 0. (14)
The following statements are equivalent:
(i) there is a decreas ing function α : R
+
→ [0, 1] such that α(t) < 1 fo r all t > 0, and
D ⊆ E
ϕ
, where ϕ(t) := tα(t) (t ∈ R
+
);
NONLINEAR CONTRACTIVE CONDITIONS 127
(ii) given an a > 0, there is an L ∈ [0, 1) such that (t, u) ∈ D a nd t ≥ a imply u ≤ Lt,
i.e.,
sup
n
u
t
: (t, u) ∈ D and t ≥ a
o
< 1;
(iii) there is a function α : R
+
→ [0, 1) such that (5) holds and D ⊆ E
ϕ
, where ϕ(t) :=
tα(t) (t ∈ R
+
);
(iv) there i s a strictly increasing and concave function ϕ : R
+
→ R
+
satisfying (2) and
such that D ⊆ E
ϕ
;
(v) there is a subadditive function ϕ : R
+
→ R
+
satisfying ( 2) and such that D ⊆ E
ϕ
;
(vi) there is a subadditive function ϕ : R
+
→ R
+
which is continuous at 0 a nd such
that ϕ(t
n
) < t
n
for s o m e sequence (t
n
)
n∈N
convergent to 0, and D ⊆ E
ϕ
;
(vii) there is an increasing and right continuous function ϕ : R
+
→ R
+
satisfying (2)
such that lim sup
t→∞
ϕ(t)/t < 1 and D ⊆ E
ϕ
;
(viii) there is an upper semi-continuous function ϕ : R
+
→ R
+
satisfying (2) such that
lim sup
t→∞
ϕ(t)/t < 1 and D ⊆ E
ϕ
.
The pro of of Lemma 1 will be preceded by the following separation theorem for
concave functions which extends Matkowski’s result [Ma93, Lemma 1]. Moreover, our
proof seems to be simpler.
Lemma 2. Assume that functions ϕ, λ : R
+
→ R
+
are such that ϕ(t) < λ(t) for all t > 0,
and λ is concave and strictly increasing. If given an s > 0,
sup
ϕ(t)
λ(t)
: t ≥ s
< 1,
then there is a concave and strictly increasing function ψ : R
+
→ R
+
such that
ϕ(t) < ψ(t) < λ(t) for all t > 0.
Proof. Set
Φ := {η : R
+
→ R
+
| η is concave, increasing and ϕ(t) ≤ η(t) for t > 0}.
Clearly, Φ 6= ∅ since λ ∈ Φ. Define
π(0) := 0 and π(t) := inf{η(t) : η ∈ Φ} for t > 0.
Then ϕ(t) ≤ π(t) for t > 0. It is e asily seen that since every η ∈ Φ is increasing, so is π.
Moreover, since the infimum of any family of concave functions which are equibounded
from below is concave, we infer π|
(0,∞)
is concave. Hence, also π is concave since π is
non-negative and π(0) = 0. We show π(t) < λ(t) for t > 0. Let s > 0. By hypothesis,
there is an α ∈ (0, 1) such that
ϕ(t) ≤ αλ(t) for t ≥ s
which implies ϕ(t) ≤ αλ(t) + (1 − α)λ(s) for t > 0. Indeed, this holds for t ≥ s since
λ(s) > 0; if 0 < t < s, then by hypothesis,
ϕ(t) < λ(t) = αλ(t) + (1 − α)λ(t) < αλ(t) + (1 − α)λ(s).
Set
µ
s
(t) := αλ(t) + (1 − α)λ(s) for t ∈ R
+
.