CUBO A Mathematical Journal

Vol.10, N
o
¯ 04, (45–66). December 2008

Fixed Points for Operators on Generalized Metric Spaces

Adrian Petruşel, Ioan A. Rus and Marcel Adrian Şerban

Department of Applied Mathematics,

Babeş-Bolyai University Cluj-Napoca,

Kogălniceanu 1, 400084, Cluj-Napoca, Romania

emails: petrusel{iarus, mserban}@math.ubbcluj.ro

ABSTRACT

The purpose of this paper is to present the fixed point theory for operators (singlevalued

and multivalued) on generalized metric spaces in the sense of Luxemburg.

RESUMEN

El proposito de este art́ıculo es presentar la teoria de punto fijo para operadores (uni-

variados y multivaluados) sobre espacios métricos generalizados en el sentido de Lux-

emburg.

Key words and phrases: Generalized metric in the sense of Luxemburg, Pompeiu-Hausdorff gen-

eralized functional, weakly Picard operator, fixed point, strict fixed point, generalized contraction,

fibre generalized contraction, data dependence, pseudo-contractive multivalued operator.

Math. Subj. Class.: 47H10, 54H25.



46 Adrian Petruşel, Ioan A. Rus and Marcel Adrian Şerban CUBO
10, 4 (2008)

1. Introduction

Let X be a nonempty set. A functional d : X × X → R+ ∪ {+∞} is said to be a generalized

metric in the sense of Luxemburg on X ([9], [13]) if:

i) d(x, y) = 0 ⇔ x = y;

ii) d(x, y) = d(y, x);

iii) x, y, z ∈ X with d(x, z), d(z, y) < +∞ ⇒ d(x, y) ≤ d(x, z) + d(z, y).

The pair (X, d) is called a generalized metric space. In a generalized metric space, the concepts

of open and closed ball, Cauchy sequence, convergent sequence, etc. are defined in a similar way

to the case of a metric space.

There are some contributions to fixed point theory for singlevalued operators (W.A.J. Lux-

emburg [13], J.B. Diaz and B. Margolis [7], C.F.G. Jung [9], S. Kasahara [10], G. Dezso [6],...) and

multivalued operators (H. Covitz and S.B. Nadler [5], P.Q. Khanh [11],...) on a generalized metric

space in the sense of Luxemburg.

The aim of this paper is to establish some new fixed point theorems for operators on a

generalized metric space and, in this framework, to study the basic problems of the metrical fixed

point theory.

2. Generalized metric spaces in the sense of Luxemburg

We start our considerations by presenting some examples of generalized metric spaces.

Example 2.1 Let X be a nonempty set and d : X × X → R+ ∪ {+∞}, given by

d(x, y) =

{

0, if x = y,

+∞, otherwise.

Example 2.2 Let X := C(R) and d : X × X → R+ ∪ {+∞} given by d(x, y) := sup
t∈R

|x(t) − y(t)|.

Example 2.3 Let X := C(R) (the space of all continuous functions on R) and d : X × X →

R+ ∪ {+∞} given by d(x, y) := sup
t∈R

(|x(t) − y(t)| · e−τ |t|), where τ > 0.

Example 2.4 (Generic example) Let (Xi, di), i ∈ I be a family of metric spaces such that each

two elements of the family are disjoint. Denote X :=
⋃

i∈I

Xi. If we define

d(x, y) :=

{

di(x, y), if x, y ∈ Xi

+∞, if x ∈ Xi, y ∈ Xj , i 6= j
,

then the pair (X, d) is a generalized metric space.



CUBO
10, 4 (2008)

Fixed Points for Operators ... 47

The following characterization theorem of a generalized metric space was given by Jung.

Theorem 2.5 (Jung [9]) Let (X, d) be a generalized metric space. Then there exists a partition

X :=
⋃

i∈I

Xi of X such that di := d|X
i
×X

i

is a metric, for each i ∈ I. Moreover, (X, d) is complete

if and only if (Xi, di) is complete, for each i ∈ I.

Notice that the above partition is induced by the following equivalence relation: x ∼ y ⇔

d(x, y) < +∞.

Let (X, d) be a generalized metric space. Then, the partition X :=
⋃

i∈I

Xi given by Jung’s

theorem is called the canonical decomposition of X into metric spaces. Moreover, if x ∈ X, then

there exists i(x) ∈ I such that x ∈ Xi(x).

We will denote Bd(x0; r) := {x ∈ X|d(x0, x) < r} and ˜Bd(x0; r) := {x ∈ X|d(x0, x) ≤ r}.

If x ∈ Xi, then ˜Bd(x0; r) = ˜Bdi (x0; r) and Bd(x0; r) = Bdi (x0; r).

If (X, d) is a generalized metric space, then the metric topology induced on X is given by:

τd := {Y ⊆ X|y ∈ Y ⇒ ∃r > 0 : Bd(y, r) ⊂ Y }.

By this definition, it follows that:

(xn)n∈N ⊂ X, x
∗ ∈ X, xn

τ
d

→ x∗ ⇔ d(xn, x
∗
) → 0.

A subset Y of X is said to be d-closed (closed with respect to the topology induced by d) if

and only if (yn)n∈N ⊂ Y with d(yn, y) → 0, as n → +∞ implies y ∈ Y . Also, Y is d-open if for

each y ∈ Y there exists a ball B(x0, r) := {x ∈ Y |d(x0, x) < r} ⊂ Y .

Let us remark that if X :=
⋃

i∈I

Xi is the canonical decomposition of X, then Xi is d-closed and

d-open, for each i ∈ I.

Definition 2.6 Two generalized metrics d1 and d2 on X are said to be:

(a) topological equivalent if τd1 = τd2 ;

(b) metric equivalent if there exist c1, c2 > 0 such that:

i) d1(x, y) < +∞ implies d2(x, y) ≤ c1d1(x, y);

ii) d2(x, y) < +∞ implies d1(x, y) ≤ c2d2(x, y).

Remark 2.7 If d1 is a generalized metric on X, then there exists a bounded metric d2 on X,

topological equivalent to d1 (for example take d2(x, y) := min{d1(x, y), 1}).

3. Functionals on generalized metric spaces

Throughout this section (X, d) will be a generalized metric space in the sense of Luxemburg.



48 Adrian Petruşel, Ioan A. Rus and Marcel Adrian Şerban CUBO
10, 4 (2008)

Let us consider now the following families of subsets of the space (X, d):

P (X) := {Y ⊆ X| Y 6= ∅} ; Pb(X) := {Y ∈ P (X)| Y is bounded };

Pcl(X) := {Y ∈ P (X)| Y is closed }; Pb,cl(X) := {Y ∈ P (X)| Y is bounded and closed }.

Consider now some functionals on P (X) × P (X) (see also [3], [16]).

(i) the gap functional Dd defined by:

Dd : P (X) × P (X) → R+ ∪ {+∞}

Dd(A, B) := inf{d(a, b)| a ∈ A, b ∈ B}.

(ii) the excess generalized functional ρd defined by:

ρd : P (X) × P (X) → R+ ∪ {+∞},

ρd(A, B) := sup{D(a, B)| a ∈ A}.

(iii) the Pompeiu-Hausdorff generalized functional Hd defined by:

Hd : P (X) × P (X) → R+ ∪ {+∞},

Hd(A, B) := max{ρ(A, B), ρ(B, A)}.

(iv) the delta functional δd defined by:

δd : P (X) × P (X) → R+ ∪ {+∞}

δd(A, B) := sup{d(a, b)| a ∈ A, b ∈ B}.

Let A, B ∈ P (X). For the rest of the paper, we denote

Ai := A ∩ Xi and Bi := B ∩ Xi,

where Xi are the sets from the characterization Theorem 2.1.

From (i), Theorem 2.5 and Example 2.4 we have:

Lemma 3.1 Let (X, d) be a generalized metric space and A, B ∈ P (X). Then:

(i) D(A, B) = inf
i∈I

D(Ai, Bi);

(ii) D(A, B) < +∞ if and only if there exists i ∈ I such that Ai 6= ∅ and Bi 6= ∅.

A useful result is:

Lemma 3.2 Let (X, d) be a generalized metric space x ∈ X and A ∈ P (X). Then D(x, A) = 0

if and only if Xi(x) ∩ A 6= ∅ and x ∈ A (where Xi(x) denotes the unique element of the canonical

decomposition of X where x belongs).



CUBO
10, 4 (2008)

Fixed Points for Operators ... 49

From (iv) and Theorem 2.5 we obtain:

Lemma 3.3 Let (X, d) be a generalized metric space and A, B ∈ P (X). Then δ(A, B) < +∞ if

and only if there exists i ∈ I such that A, B ∈ Pb(Xi). In particular, A ∈ Pb(X) if and only if

there exists i ∈ I such that A ∈ Pb(Xi).

From (ii), Theorem 2.5 and Example 2.4 we have:

Lemma 3.4 Let (X, d) be a generalized metric space and Y, Z ∈ P (X). Then ρ(Y, Z) < +∞ if

and only if there exists η > 0 such that for each y ∈ Y there is z ∈ Z such that d(y, z) < η.

Proof. If ρ(Y, Z) < +∞, then there is η > 0 such that ρ(Y, Z) < η. Thus D(y, Z) < η for each

y ∈ Y . Hence there exists z ∈ Z such that d(y, z) < η.

Suppose now there is η > 0 such that for each y ∈ Y there exists z ∈ Z with d(y, z) < η.

Then, y, z ∈ Xi, where Xi is an element of the partition of the generalized metric space X. Hence

D(y, Z) ≤ η, for each y ∈ Y . Thus, ρ(Y, Z) ≤ η. 2

Let (X, d) be a generalized metric space, Y ∈ P (X) and ε > 0. An open neighborhood of

radius ε for the set Y is the set denoted Vε(Y ) and defined by:

Vε(Y ) := {x ∈ X| D(x, Y ) < ε}.

Let us remark that Vε(Y ) =
⋃

i∈I,Yi 6=∅

Vε(Yi).

In the usual case of a metric space (X, d) the following equivalent definitions of the Pompeiu-

Hausdorff functional are well-known.

(iii)
′
Hd(A, B) := inf{ε > 0|A ⊂ Vε(B), B ⊂ Vε(A)},

and

(iii)′′ Hd(A, B) := sup
x∈X

|D(x, A) − D(x, B)|.

We have:

Lemma 3.5 Let (X, d) be a generalized metric space. Then, the definitions (iii), (iii)′ and (iii)′′

are equivalent.

We can also prove the following result.

Lemma 3.6 Let (X, d) be a generalized metric space and A, B ∈ P (X). Then the following

assertions are equivalent:

(a) H(A, B) < +∞;

(c) there exists η > 0 such that [for each a ∈ A there exists b ∈ B such that d(a, b) < η] and

[for each b ∈ B there exists a ∈ A such that d(a, b) < η].

Lemma 3.7 Let (X, d) be a generalized metric space. Then the following assertions hold:



50 Adrian Petruşel, Ioan A. Rus and Marcel Adrian Şerban CUBO
10, 4 (2008)

i) Let ε > 0 and Y, Z ∈ P (X) such that H(Y, Z) < +∞. Then for each y ∈ Y there exists

z ∈ Z such that d(y, z) ≤ H(Y, Z) + ε.

ii) Let q > 1 and Y, Z ∈ P (X) such that H(Y, Z) < +∞. Then, for each y ∈ Y there exists

z ∈ Z such that d(y, z) ≤ qH(Y, Z).

Proof. i) Let Y, Z ∈ P (X) and ε > 0. Suppose that H(Y, Z) < +∞. Then, supposing, by

contradiction, there is y ∈ Y such that for every z ∈ Z we have d(y, z) > H(Y, Z) + ε. If

d(y, z) < +∞ then since H(Y, Z) ≥ D(y, Z) ≥ H(Y, Z) + ε we get a contradiction. If d(y, z) = +∞

then, we get a contradiction to the supposition H(Y, Z) < +∞, since, by Lemma 3.6, there is η > 0

such that for each y ∈ Y there is z ∈ Z with d(y, z) < η. 2

Lemma 3.8 Let (X, d) be a generalized metric space and A, B ∈ P (X).

Then:

a) H(A, B) = sup
i∈I

H(A ∩ Xi, B ∩ Xi);

b) A ∈ Pcp(X) ⇔ card{i ∈ I|A ∩ Xi 6= ∅} < +∞ and Ai ∈ Pcp(Xi).

Remark 3.9 Let (X, d) be a generalized metric space. Then Pcp(X) * Pb(X). Consider, for

example, x, y ∈ X with d(x, y) = +∞, then {x, y} is compact but it is not bounded.

4. Singlevalued operators on generalized metric spaces

4.1 General considerations

Let X be a nonempty set, s(X) := {(xn)n∈N|xn ∈ X, n ∈ N}, c(X) ⊂ s(X) and Lim : c(X) → X

an operator. By definition the triple (X, c(X), Lim) is called an L-space if the following conditions

are satisfied:

(i) If xn = x, for all n ∈ N, then (xn)n∈N ∈ c(X) and Lim(xn)n∈N = x.

(ii) If (xn)n∈N ∈ c(X) and Lim(xn)n∈N = x, then for all subsequences, (xni )i∈N, of (xn)n∈N
we have that (xni )i∈N ∈ c(X) and Lim(xni )i∈N = x.

By definition an element of c(X) is convergent sequence and x := Lim(xn)n∈N is the limit of

this sequence and we write xn → x as n → ∞.

In what follows we will denote an L-space by (X, →).

Actually, an L-space is any set endowed with a structure implying a notion of convergence

for sequences. For example, Hausdorff topological spaces, metric spaces, generalized metric spaces

in Perov’ sense (i.e., d(x, y) ∈ Rm+ ), generalized metric spaces in Luxemburg’ sense (i.e., d(x, y) ∈

R+ ∪ {+∞}), K-metric spaces (i.e., d(x, y) ∈ K, where K is a cone in an ordered Banach space),

gauge spaces, 2-metric spaces, D-R-spaces, probabilistic metric spaces, syntopogenous spaces, are

such L-spaces. For more details see Fréchet [8], Blumenthal [4] and I.A. Rus [22].



CUBO
10, 4 (2008)

Fixed Points for Operators ... 51

Let (X, d) and (Y, ρ) be two generalized metric spaces and f : X → Y .

Definition 4.1 The operator f : (X, d) → (Y, ρ) is said to be:

a) continuous, if xn → x
∗

implies f (xn) → f (x
∗
);

b) closed, if xn → x
∗

and f (xn) → y
∗

imply f (x∗) = y∗;

c) α-Lipschitz if α > 0 and

d (x, y) < +∞ =⇒ ρ (f (x) , f (y)) ≤ α · d (x, y) .

d) α-contraction if f is α-Lipschitz with α < 1.

4.2 Weakly Picard operators on L-spaces

Let (X, →) be an L-space and f : X → X. We denote by f 0 := 1X , f
1

:= f , f n+1 := f ◦ f n,

n ∈ N the iterate operators of f . Also:

Ff := {x ∈ X | f (x) = x},

I (f ) := {Y ∈ P (X) | f (Y ) ⊆ Y } .

Definition 4.2 (I.A. Rus [22]) Let (X, →) be an L-space. Then f : X → X is said to be

1) a Picard operator if:

i) Ff = {x
∗};

ii) (f n (x))
n∈N

→ x∗ as n → +∞, for all x ∈ X.

2) a weakly Picard (briefly WP) operator if the sequence (f n (x))
n∈N

converges for all x ∈ X

and the limit (which may depend on x) is a fixed point of f .

If f : X → X is a weakly Picard operator, then we define the operator f ∞ : X → X by:

f ∞(x) := lim
n→∞

f n(x).

Notice that f ∞(X) = Ff . Moreover, if f is a Picard operator and we denote by x
∗

its unique fixed

point, then f ∞(x) = x∗, for each x ∈ X.

Definition 4.3 Let (X, →) be an L-space, c > 0 and d : X × X → R+. By definition, the operator

f is called c-weakly Picard with respect to d, if f is a weakly Picard operator and

d (x, f ∞ (x)) ≤ c · d (x, f (x)) , for all x ∈ X.

If f is Picard operator and the above condition holds, then f is said to be c-Picard.

Theorem 4.4 (Characterization Theorem) (I.A. Rus [25], [22]) Let (X, →) be an L-space and

f : X → X be an operator. Then, f is a weakly Picard operator if and only if there exists a

partition of X, X =
⋃

λ∈Λ

Xλ, such that:



52 Adrian Petruşel, Ioan A. Rus and Marcel Adrian Şerban CUBO
10, 4 (2008)

a) Xλ ∈ I (f ), for all λ ∈ Λ;

b) f |X
λ

: Xλ → Xλ is a Picard operator, for all λ ∈ Λ.

4.3 Contractions on generalized metric spaces

We present first some important auxiliary results.

Lemma 4.5 Let (X, d) be a complete generalized metric space and f : X → X be an α-contraction.

The following statements are equivalent:

i) Ff 6= ∅;

ii) there exists x ∈ X such that d (x, f (x)) < +∞;

iii) there exist x ∈ X and n (x) ∈ N such that d
(

f n(x) (x) , f n(x)+1 (x)
)

< +∞;

iv) there exists i ∈ I such that Xi ∈ I (f ).

Proof. i) =⇒ ii) Let x∗ ∈ Ff . We have

d (x∗, f (x∗)) = d (x∗, x∗) = 0 < +∞.

ii) =⇒ iii) We choose n (x) = 0;

iii) =⇒ i) Since f is an α-contraction we have that (f n (x)) is a Cauchy sequence. This

implies f n (x) → x∗, as n → +∞. From the continuity of f it follows that x∗ ∈ Ff .

ii) =⇒ iv) Since d (x, f (x)) < +∞, there exists i ∈ I such that x ∈ Xi. Let y ∈ Xi then

d (x, y) < +∞. We have:

d (x, f (y)) ≤ d (x, f (x)) + d (f (x) , f (y)) ≤ d (x, f (x)) + α · d (x, y) < +∞

which implies f (y) ∈ Xi.

iv) =⇒ ii) Let x∈Xi. Since Xi ∈I (f ), we get that f (x)∈Xi. Therefore d (x, f (x)) < +∞.2

Lemma 4.6 Let (X, d) be a complete generalized metric space and f : X → X be an α-contraction.

We suppose that:

i) there exists x ∈ X such that d (x, f (x)) < +∞;

ii) if u, v ∈ Ff then d (u, v) < +∞;

Then:

a) Ff = {x
∗};

b) f
∣

∣

X
i(x)

: Xi(x) → Xi(x) is a Picard operator.

Proof. From i) and Lemma 4.5 we have that there exists i ∈ I such that Xi ∈ I (f ), f
n

(x) ∈ Xi
for every n ∈ N, Ff 6= ∅, f

n
(x) → x∗ ∈ Ff ∩ Xi. Let u, v ∈ Ff . Then d (u, v) < +∞ and

d (u, v) = d (f (u) , f (v)) ≤ α · d (u, v) .



CUBO
10, 4 (2008)

Fixed Points for Operators ... 53

Therefore d (u, v) = 0, which implies u = v. Hence Ff = {x
∗}.

Since Xi ∈ I (f ) then d (y, f (y)) < +∞ for every y ∈ Xi and applying again Lemma 4.5 we

get that f
∣

∣

X
i(x)

: Xi(x) → Xi(x) is a Picard operator. 2

Theorem 4.7 Let (X, d) be a complete generalized metric space and f : X → X. We suppose that:

i) f is an α-contraction;

ii) for every x ∈ X there exists n (x) ∈ N such that d
(

f n(x) (x) , f n(x)+1 (x)
)

< +∞.

Then:

a) f is a weakly Picard operator. If in addition, for every x ∈ X we have d (x, f (x)) < +∞,

then f is 1
1−α

-weakly Picard;

b) If, in addition:

b1) for every x ∈ X we have d (x, f (x)) < +∞;

b2) u, v ∈ Ff implies d (u, v) < +∞,

then f is 1
1−α

-Picard.

Proof. a) The first part follows from Lemma 4.5 and Lemma 4.6. For the second conclusion,

notice that for every x ∈ X such that d (x, f (x)) < +∞ and each n ∈ N we have:

d (f n (x) , f ∞ (x)) ≤
αn

1 − α
· d (x, f (x))

which implies

d (x, f
∞

(x)) ≤
1

1 − α
· d (x, f (x)) .

b) From b2) we obtain Ff = {x
∗} and from a) we obtain that f is 1

1−α
-Picard operator. 2

Theorem 4.8 Let (X, d) be a complete generalized metric space and f, g : X → X two operators.

We suppose that:

i) f and g are α-contractions;

ii) d (x, f (x)) < +∞ and d (x, g (x)) < +∞, for every x ∈ X;

iii) there exists η > 0 such that

d (f (x) , g (x)) ≤ η, for all x ∈ X.

Then:

H (Ff , Fg) ≤
η

1 − α
.

Proof. Let x ∈ Ff and y ∈ Fg. From ii) and Theorem 4.7 we have:

d (x, g∞ (x)) ≤
1

1 − α
· d (x, g (x)) =

1

1 − α
· d (f (x) , g (x)) ≤

η

1 − α
.



54 Adrian Petruşel, Ioan A. Rus and Marcel Adrian Şerban CUBO
10, 4 (2008)

Since g∞ (x) ∈ Fg then

D (x, Fg ) ≤ d (x, g
∞

(x)) ≤
η

1 − α
.

By taking the supremum over x ∈ Ff we get

ρ (Ff , Fg) ≤
η

1 − α
.

Using the same technique we have:

ρ (Fg, Ff ) ≤
η

1 − α

which implies the conclusion. 2

Theorem 4.9 (Fibre contraction principle) Let (X0, →) be an L-space and (Xk, dk), k ∈ {0, 1, · · · , p}

(where p ≥ 1) be complete generalized metric spaces. We consider the operators:

fk : X0 × ... × Xk → Xk, k ∈ {0, 1, · · · , p}.

We suppose that:

i) f0 : X0 → X0 is a weakly Picard operator;

ii) fk (x0, ..., xk−1, ·) is an αk-contraction, k ∈ {1, 2, · · · , p};

iii) fk is continuous, k ∈ {1, 2, · · · , p};

iv) for every (x0, x1, ..., xk) ∈ X0 × ... × Xk we have

dk (xk, fk (x0, x1, ..., xk)) < +∞, k ∈ {1, 2, · · · , p}.

Then the operator

gp : X0 × ... × Xp → X0 × ... × Xp

gp (x0, x1, ..., xp) = (f0 (x0) , f1 (x0, x1) , ..., fp (x0, x1, ..., xp))

is weakly Picard.

Proof. We will prove by induction. For p = 1 the conclusion follows by Theorem 3.1 in M.A.

Şerban [31]. We suppose that conclusion holds for k ≤ p and we prove the conclusion for k + 1.

We know that gk+1 = (gk, fk+1), gk are weakly Picard and from ii) fk+1 (x0, ..., xk, ·) is an αk+1-

contraction, so we apply again Theorem 3.1 from M.A. Şerban [31] and we get that gk+1 is weakly

Picard. 2

Theorem 4.10 Let X be a nonempty set, α ∈]0; 1[ and f : X → X an operator. The following

statements are equivalent:

i) Ff = Ff n 6= ∅ for every n ∈ N;

ii) there exists a complete generalized metric d on X such that:

a) f : (X, d) → (X, d) is an α-contraction;



CUBO
10, 4 (2008)

Fixed Points for Operators ... 55

b) d (x, f (x)) < +∞ for every x ∈ X.

Proof. i) =⇒ ii) Ff = Ff n 6= ∅ for every n ∈ N implies that there exists a partition of X,

X =
⋃

i∈I

Xi such that Xi ∈ I (f ), card (Ff ∩ Xi) = 1 and f |Xi is a Bessaga operator (see I.A. Rus

[24]). From Bessaga’s theorem [2] there exists a complete metric di on Xi such that f |Xi : Xi → Xi
is an α-contraction for all i ∈ I. So, d : X × X → R+ ∪ {+∞}

d (x, y) =

{

di (x, y) if x, y ∈ Xi

+∞ if x ∈ Xi, y ∈ Xi, i 6= j

is the complete generalized metric on X that we are looking for.

ii) =⇒ i) is Theorem 4.7. 2

4.4 Graphic contractions

Let (X, d) be a generalized metric space and f : X → X.

Definition 4.11 f : X → X is a graphic contraction if there exists α ∈ [0; 1[ such that:

d
(

f 2 (x) , f (x)
)

≤ α · d (x, f (x)) for all x ∈ X with d (x, f (x)) < +∞.

Theorem 4.12 Let (X, d) be a complete generalized metric space and f : X → X. We suppose

that:

i) f is a closed graphic contraction;

ii) for every x ∈ X there exists n (x) ∈ N such that d
(

f n(x) (x) , f n(x)+1 (x)
)

< +∞.

Then:

a) f is a weakly Picard operator. If, in addition, for every x ∈ X we have that d (x, f (x)) <

+∞, then f is 1
1−α

-weakly Picard;

b) If, in addition:

b1) for every x ∈ X we have d (x, f (x)) < +∞;

b2) if u, v ∈ Ff implies d (u, v) < +∞,

then f is 1
1−α

-Picard.

Proof. a) From i) and ii) we have that for each x ∈ X, the sequence (f n (x)) is Cauchy. Therefore

there exists x∗ ∈ X such that f n (x) → x∗, as n → +∞ and

d (f n (x) , x∗) ≤
αn−n(x)

1 − α
· d

(

f n(x) (x) , f n(x)+1 (x)
)

, n ≥ n (x) .

Since f is closed we get that x∗ ∈ Ff and f
∞

(x) = x∗. This means that f is a weakly Picard

operator.



56 Adrian Petruşel, Ioan A. Rus and Marcel Adrian Şerban CUBO
10, 4 (2008)

If for every x ∈ X we have d (x, f (x)) < +∞, then n (x) = 0 and letting n = 0 in the above

relation, we conclude that f is 1
1−α

-weakly Picard operator.

b) If for u, v ∈ Ff we have d (u, v) < +∞ then Ff = {x
∗}, which means that f is a 1

1−α
-Picard

operator. 2

4.5 Meir-Keeler operators

Let us consider now the case of Meir-Keeler operators on generalized metric spaces.

Definition 4.13 Let (X, d) be a generalized metric space. Then, f : X → X is called a Meir-

Keeler type operator if for each ǫ > 0 there exists η = η(ǫ) > 0 such that for x, y ∈ X with

ǫ ≤ d(x, y) < ǫ + η we have d(f (x), f (y)) < ǫ.

By using an argument similar to the one in the Meir-Keeler fixed point theorem [14] we have:

Theorem 4.14 Let (X, d) be a generalized complete metric space and f : X → X be a Meir-Keeler

type operator. Suppose there exists x0 ∈ X such that d(x0, f (x0)) < +∞.

Then Ff 6= ∅. Moreover, if additionally x, y ∈ Ff implies d(x, y) < +∞, then Ff = {x
∗}.

Proof. Denote xn := f
n
(x0), n ∈ N.

The proof of the theorem can be organized in five steps.

Step 1. We prove that

d(f (x), f (y)) < d(x, y), for each x, y ∈ X with x 6= y and d(x, y) < +∞.

Let x, y ∈ X be such that x 6= y and d(x, y) < +∞. Then by letting ǫ := d(x, y) in the definition

of Meir-Keeler operators we get d(f (x), f (y)) < d(x, y).

Step 2. We can prove, by induction, that d(xn, xn+1) < +∞, for all n ∈ N.

Step 3. We prove that the sequence an := d(xn, xn+1) ց 0 as n → +∞.

If there is n0 ∈ N such that an0 = 0 then xn0 ∈ Ff .

If an 6= 0, for each n ∈ N, then an = d(f (xn−1), f (xn)) < d(xn−1, xn) = an−1. Hence the

sequence (an)n∈N converges to a certain a ≥ 0. Suppose that a > 0. Then, for each ǫ > 0 there

exists nǫ ∈ N such that ǫ ≤ an < ǫ + η, for all n ≥ nǫ. Then, by the Meir-Keeler condition we

obtain an+1 < ǫ, which is a contradiction with the above relation.

Step 4. We will prove that the sequence (xn) is Cauchy.

Suppose, by contradiction, that (xn) is not a Cauchy sequence. Then, there exists ǫ > 0 such

that lim sup d(xm, xn) > 2ǫ. For this ǫ there exists η := η(ǫ) > 0 such that for x, y ∈ X with

ǫ ≤ d(x, y) < ǫ + η we have d(f (x), f (y)) < ǫ. Choose δ := min{ǫ, η}. Since an ց 0 as n → +∞

it follows that there is p ∈ N such that ap <
δ
3
. Let m, n ∈ N∗ with n > m > p such that

d(xn, xm) > 2ǫ. For j ∈ [m, n] we have |d(xm, xj ) − d(xm, xj+1| ≤ aj <
δ
3
. Also, d(xm, xm+1 < ǫ

and d(xm, xn) > ǫ+δ we obtain that there exists k ∈ [m, n] such that ǫ < ǫ+
2δ
3

< d(xm, xk) < ǫ+δ.

On the other hand, for any m, l ∈ N we have: d(xm, xl) ≤ d(xm, xm+1) + d(xm+1, xl+1) +



CUBO
10, 4 (2008)

Fixed Points for Operators ... 57

d(xl+1, xl) = am + d(f (xm), f (xl)) + al <
δ
3

+ ǫ + δ
3
. The contradiction proves that (xn) is

Cauchy.

Step 5. We prove that x∗ := lim
n→+∞

xn is a fixed point of f .

Since f is continuous and xn+1 = f (xn), we get by passing to the limit that x
∗

= f (x∗).

If x∗, y ∈ Ff are two distinct fixed points of f then, by the contractive condition, we get the

following contradiction: d(x∗, y) = d(f (x∗), f (y)) < d(x∗, y). This completes the proof. 2

4.6 Caristi operators

Let (X, d) be a generalized metric space.

Definition 4.15 A space X is said to be sequentially complete in Weierstrass’ sense (see [33]) if

each sequence (xn)n∈N in X such that

+∞
∑

n=0

d(xn, xn+1) < +∞ is convergent in X.

Definition 4.16 Let (X, d) be a generalized metric space. Then, f : X → X is called a Caristi

operator if there exists a functional ϕ : X → R+ such that

d (x, f (x)) ≤ ϕ (x) − ϕ (f (x)) , for every x ∈ X .

Theorem 4.17 Let (X, d) be a sequentially complete (in Weierstrass’ sense) generalized metric

space and f : X → X be a closed Caristi operator. Then f is a weakly Picard operator.

Proof. We remark that if f is a Caristi operator, then d (x, f (x)) < +∞ for every x ∈ X. Denote

by xn := f
n
(x), for n ∈ N. Then:

+∞
∑

n=0

d(xn, xn+1) =

+∞
∑

n=0

d(f
n
(x), f

n+1
(x)).

We will prove that the series

+∞
∑

n=0

d(f n(x), f n+1(x)) is convergent. For this purpose we need to

show that the sequence of its partial sums is convergent in R+. Denote by sn :=

n
∑

k=0

d(f k(x), f k+1(x)).

Then sn+1−sn = d(f
n+1

(x), f n+2(x)) ≥ 0, for each n ∈ N. Moreover sn =
n

∑

k=0

d(f k(x), f k+1(x)) ≤

ϕ(x). Hence (sn)n∈N is upper bounded and increasing in R+. Then the sequence (sn)n∈N is con-

vergent.

It follows that the sequence (xn)n∈N is Cauchy and, from the sequentially completeness of the

space, convergent to a certain element x∗ ∈ X. The conclusion follows from the fact that f is

closed. 2



58 Adrian Petruşel, Ioan A. Rus and Marcel Adrian Şerban CUBO
10, 4 (2008)

4.7 Fixed point theorems in a set with two generalized metrics

Let X be a nonempty set and d, ρ : X × X → R+ ∪{+∞} be two generalized metrics on X. In this

subsection we will present Maia’s fixed point theorem for the case of a set with two generalized

metrics.

Theorem 4.18 Let X be a nonempty set, d, ρ : X × X → R+ ∪ {+∞} two generalized metrics on

X and f : X → X. We suppose that:

i) (X, d) is a complete generalized metric space;

ii) there exists c > 0 such that d (x, y) ≤ c · ρ (x, y) for all x, y ∈ X with ρ (x, y) < +∞;

iii) for every x ∈ X there exists n (x) ∈ N such that ρ
(

f n(x) (x) , f n(x)+1 (x)
)

< +∞;

iv) f : (X, ρ) → (X, ρ) is an α-contraction.

Then f is weakly Picard.

Proof. For each x ∈ X there exists n (x) ∈ N such that ρ
(

f n(x) (x) , f n(x)+1 (x)
)

< +∞. Also,

there exists i ∈ I such that Xi ∈ I (f ) and f
n

(x) ∈ Xi for all n ≥ n (x). Since f : (X, ρ) → (X, ρ)

is an α-contraction, the sequence (f n (x))
n∈N

is Cauchy in (X, ρ). Using conditions ii), iii) and

iv) we get

d
(

f n (x) , f n+p (x)
)

≤ c · ρ
(

f n (x) , f n+p (x)
)

≤ c ·
αn−n(x)

1 − α
ρ

(

f n(x) (x) , f n(x)+1 (x)
)

, n ≥ n (x) ,

so d (f n (x) , f n+p (x)) → 0 as n → +∞. Thus (f n (x))
n∈N

is Cauchy sequence in (X, d), which

implies that f n (x)→x∗ ∈Xi. By condition iv) we have that x
∗ ∈Ff . Hence f is weakly Picard.2

An improved version of Maia’s theorem can be obtained by replacing the assumption ii) with

a more useful condition (from an application point of view), see I.A. Rus [20].

Theorem 4.19 Let X be a nonempty set, d, ρ : X × X → R+ ∪ {+∞} two generalized metrics on

X and f : X → X. We suppose that:

i) (X, d) is a complete generalized metric space;

ii) there exists c > 0 such that d (f (x) , f (y)) ≤ c · ρ (x, y), for all x, y ∈ X with ρ (x, y) <

+∞;

iii) for every x ∈ X there exists n (x) ∈ N such that ρ
(

f n(x) (x) , f n(x)+1 (x)
)

< +∞;

iv) f : (X, ρ) → (X, ρ) is an α-contraction.

Then f is a weakly Picard operator.

Proof. The proof follows the method in Theorem 4.18. 2



CUBO
10, 4 (2008)

Fixed Points for Operators ... 59

5. Multivalued operators in generalized metric spaces

5.1 General considerations

Let (X, d) be a generalized metric space. Let Y, Z be two nonempty subsets of X and T : Y →

P (Z) be a multivalued operator. By definition, t : Y → Z is a selection of T if t(x) ∈ T (x), for

each x ∈ Y . If T : X → P (X) is a multivalued operator, then x∗ ∈ X is a fixed point for T if and

only if x∗ ∈ T (x∗). Denote by FT the set of all fixed points for T . Also, x
∗ ∈ X is called a strict

fixed point for T if and only if {x∗} = T (x∗). We will denote by (SF )T the set of all strict fixed

points of T . By Graph(T ) := {(x, y) ∈ X × X|y ∈ T (x)} we denote the graph of the multivalued

operator T and by T (Y ) :=
⋃

x∈Y

T (x) the image through T of the set Y ∈ P (X).

Recall that if Y ⊆ X, then T (Y ) :=
⋃

x∈Y

T (x). We also denote by T n := T ◦ T · · · ◦ T (the n

times composition).

Recall that, if (X, d) is a metric space, then T : X → Pcl(X) is said to be a multivalued

a-contraction if

a ∈ [0, 1[ and Hd(T (x), T (y)) ≤ ad(x, y), for each x, y ∈ X.

The following result is known as Covitz-Nadler fixed point principle.

Theorem 5.1 (Covitz-Nadler [5]) Let (X, d) be a complete metric space and T : X → Pcl(X) be

a multivalued a-contraction. Then, for each x0 ∈ X there exists a sequence (xn)n∈N in X with

xn+1 ∈ T (xn) for all n ∈ N, which converges to a fixed point of T .

Remark 5.2 From the proof of the above result it follows that for each x ∈ X and each y ∈ T (x)

there exists in X a sequence (xn)n∈N with the properties:

a) x0 = x, x1 = y;

b) xn+1 ∈ T (xn) for all n ∈ N
∗;

c) (xn)n∈N converges to a fixed point of T .

This principle gave rise to the following concept.

Definition 5.3 (Rus-Petruşel-Ŝıntămărian [28], [29]) Let (X, →) be an L-space. Then T : X →

P (X) is a multivalued weakly Picard operator (briefly MWP operator) if for each x ∈ X and each

y ∈ T (x) there exists a sequence (xn)n∈N in X such that:

i) x0 = x, x1 = y

ii) xn+1 ∈ T (xn), for all n ∈ N

iii) the sequence (xn)n∈N is convergent and its limit is a fixed point of T .

A sequence (xn)n∈N in X satisfying the conditions (i) and (ii) in Definition 5.3 is called a

sequence of successive approximations for T starting from (x, y).



60 Adrian Petruşel, Ioan A. Rus and Marcel Adrian Şerban CUBO
10, 4 (2008)

The aim of this section is to establish some fixed point results for multivalued operators of

contractive type on generalized metric space.

5.2 Multivalued contractions on generalized metric spaces

Let us recall first some contractive-type conditions for multivalued operators.

Definition 5.4 Let (X, d) be a generalized metric space. Then T : X → Pcl(X) is called a

multivalued a-contraction if a ∈ [0, 1[ and

Hd(T (x), T (y)) ≤ ad(x, y), for each x, y ∈ X, with d(x, y) < +∞.

Let (X, d) be a generalized metric space. We denote by P(X) the set of all subsets of a

nonempty set X.

Definition 5.5 Let (X, d) be a generalized metric space. If T : X → P (X) is a multivalued

operator, then we consider the following multivalued operators generated by T :

̂T : X → P(X), ̂T (x) := T (x) ∩ Xi(x)

(where Xi(x) denotes the unique element of the canonical decomposition of X where x belongs),

T̃ i : X → P(X), T̃ i(x) := T (x) ∩ Xi

(where Xi denotes an arbitrary element of the canonical decomposition of X).

Then we have:

Lemma 5.6 FT = F ̂T .

Lemma 5.7 FT 6= ∅ ⇔ if there exists i ∈ I such that FT̃ i 6= ∅.

The following result is a straightforward version of Covitz and Nadler alternative theorem in

[5].

Theorem 5.8 Let (X, d) be a generalized complete metric space and T : X → Pcl(X) be a mul-

tivalued a-contraction. Suppose that for each x ∈ X there is y ∈ T (x) such that d(x, y) < +∞.

Then there exists a sequence of successive approximations of T starting from any arbitrary x ∈ X

which converges to a fixed point of T .

The previous result gives rise to the following open question.

Open question. Let T : X → Pcl(X) be a multivalued a-contraction as in the above Covitz-

Nadler fixed point result. Is T a MWP operator ?

Theorem 5.9 Let (X, d) be a generalized complete metric space and T : X → Pcl(X) be a mul-

tivalued a-contraction. Suppose there exists x0 ∈ X and x1 ∈ T (x0) such that d(x0, x1) < +∞.



CUBO
10, 4 (2008)

Fixed Points for Operators ... 61

Then there exists a sequence (xn)n∈N of successive approximations for T starting from x0 which

converges to a fixed point of T .

Proof. Let X :=
⋃

i∈I

Xi be the canonical decomposition of X into metric spaces. Recall that X is

complete if and only if Xi is complete for each i ∈ I. Let j ∈ I such that x0 ∈ Xj .

For x ∈ X we successively have:

D(x, T (x)) < +∞ ⇔ there exists y ∈ T (x) such that d(x, y) < +∞ ⇔ y ∈ T (x) ∩ Xi(x).

Hence

D(x, T (x)) < +∞ ⇔ T (x) ∩ Xi(x) 6= ∅.

Consider now the multivalued operator

T̃ j : X → P(X), T̃ j(x) := T (x) ∩ Xj .

We will prove that T̃
j

|X
j

: Xj → Pcl(Xj ). For this purpose, it is enough to show that

D(x, T (x)) < +∞, for each x ∈ Xj .

For x ∈ Xj we have:

D(x, T (x)) ≤ D(x, T (x0)) + H(T (x0), T (x)) ≤ d(x, x0) + D(x0, T (x0)) + ad(x0, x) < +∞.

Hence T̃
j

|X
j

: Xj → Pcl(Xj ) is a multivalued a-contraction on the complete metric space

(Xj , d|X
j
×X

j

). The conclusion follows from Lemma 5.7 and Theorem 5.1. 2

An answer to the above problem is the following result.

Theorem 5.10 Let (X, d) be a generalized complete metric space and T : X → Pcl(X) be a

multivalued a-contraction. Suppose that for each x ∈ X and y ∈ T (x) we have d(x, y) < +∞ (or

equivalently, for each x ∈ X we have T (x) ⊂ Xi(x)). Then T is a MWP operator.

Proof. From the hypothesis we have that D(x, T (x)) < +∞, for each x ∈ X. Hence, for each

x ∈ X we have that T : Xi(x) → Pcl(Xi(x)). Since (Xi(x), d|X
i(x)

×X
i(x)

) is a complete metric space,

by Theorem 5.1 and Remark 5.2, we conclude that T is a MWP operator. 2

We introduce now the following concepts.

Definition 5.11 (Rus-Petruşel-Ŝıntămărian [29]) Let (X, →) be an L-space and T : X → P (X)

be a MWP operator. Define the multivalued operator T ∞ : Graph(T ) → P (FT ) by the formula

T ∞(x, y) = { z ∈ FT | there exists a sequence of successive approximations of T starting from

(x, y) that converges to z }.

Definition 5.12 (see also Rus-Petruşel-Ŝıntămărian [29]) Let (X, d) be a generalized metric space

and T : X → P (X) be a MWP operator such that for each x ∈ X and y ∈ T (x) we have that

d(x, y) < +∞. Then, T is called a c-multivalued weakly Picard operator (briefly c-MWP operator)

if there exists a selection t∞ of T ∞ such that d(x, t∞(x, y)) ≤ c d(x, y), for all (x, y) ∈ Graph(T ).



62 Adrian Petruşel, Ioan A. Rus and Marcel Adrian Şerban CUBO
10, 4 (2008)

As an example, we have:

Theorem 5.13 Let (X, d) be a generalized complete metric space and T : X → Pcl(X) be a

multivalued a-contraction, such that for each x ∈ X and y ∈ T (x) we have d(x, y) < +∞.

Then T is a 1
1−a

-MWP operator.

We present now an abstract data dependence theorem for the fixed point set of c-MWP

operators on generalized metric spaces.

Theorem 5.14 Let (X, d) be a generalized metric space and T1, T2 : X → P (X) be two multivalued

operators. We suppose that:

i) Ti is a ci-MWP operator, for i ∈ {1, 2}

ii) there exists η > 0 such that H(T1(x), T2(x)) ≤ η, for all x ∈ X.

Then H(FT1 , FT2 ) ≤ η max { c1, c2 }.

Proof. The proof follows in a similar way to Rus-Petruşel-Ŝıntămărian [29]. For the sake of

completeness we present it here.

Let ti : X → X be a selection of Ti for i ∈ {1, 2}. Let us remark that

H(FT1 , FT2 ) ≤ max

{

sup
x∈FT2

d(x, t∞1 (x, t1(x))), sup
x∈FT1

d(x, t∞2 (x, t2(x)))

}

.

Let q > 1. Then we can choose ti (i ∈ {1, 2}) such that

d(x, t∞
1

(x, t1(x))) ≤ c1qH(T2(x), T1(x)), for all x ∈ FT2

and

d(x, t∞
2

(x, t2(x)) ≤ c2qH(T1(x), T2(x)), for all x ∈ FT1 .

Thus we have H(FT1 , FT2 ) ≤ qη max{c1, c2}. Letting q ց 1, the proof is complete. 2

Notice that the above conclusions means that the data dependence phenomenon of the fixed

point set for c-MWP operators holds.

We also have:

Theorem 5.15 Let (X, d) be a generalized complete metric space and T : X → Pcl(X) be a

multivalued a-contraction. Suppose:

(i) (SF )T 6= ∅;

(ii) If x, y ∈ FT then d(x, y) < +∞.

Then FT = (SF )T = {x
∗}.

Proof. We will prove first that (SF )T = {x
∗}. Indeed, if z ∈ (SF )T with z 6= x

∗
, then d(z, x∗) <

+∞ and d(z, x∗) = H(T (z), T (x∗)) ≤ ad(z, x∗), a contradiction. Next we will prove that FT ⊆



CUBO
10, 4 (2008)

Fixed Points for Operators ... 63

(SF )T . Let y ∈ FT . Then d(y, x
∗
) < +∞. Thus d(y, x∗) = D(y, T (x∗)) ≤ H(T (y), T (x∗)) ≤

ad(y, x∗), which implies y = x∗. This completes the proof. 2

5.3 Pseudo-contractive multivalued operators on generalized metric

spaces

In D. Azé and J.-P. Penot [1] the following concept is introduced.

Definition 5.16 (Azé-Penot [1]) Let (X, d) be a metric space. A multivalued operator T : X →

P (X) is said to be pseudo-a-Lipschitzian with respect to the subset U ⊂ X whenever, for all

x, y ∈ U , we have

ρd(T (x) ∩ U, T (y)) ≤ ad(x, y).

Also, the multivalued opeator T is called pseudo-a-contractive with respect to U if it is pseudo-a-

Lipschitzian with respect to U for some a ∈ [0, 1[.

In Azé-Penot [1], the fixed point theory for multivalued pseudo-a-contractive operators with

respect to the open ball Bd(x0, r) of a complete metric space (X, d) is studied. The aim of this

section is to give some fixed point results for multivalued pseudo-a-contractive operators in the

setting of a generalized metric space.

Theorem 5.17 Let (X, d) be a generalized complete metric space and T : X → Pcl(X) be a

multivalued operator. Let X :=
⋃

i∈I

Xi be the canonical decomposition of X. Suppose that there

exists x0 ∈ X such that D(x0, T (x0)) < +∞ and T is pseudo a-contractive with respect to Xi(x0).

Then FT 6= ∅.

Proof. Since D(x0, T (x0)) < +∞ there exists b > 0 and x1 ∈ T (x0) such that d(x0, x1) < b < +∞.

Then x1 ∈ Xi(x0) and thus x1 ∈ T (x0) ∩ Xi(x0). Hence we have D(x1, T (x1)) ≤ ρ(T (x0) ∩

Xi(x0), T (x1)) ≤ ad(x0, x1) < ab. Thus there exists x2 ∈ T (x1) such that d(x1, x2) < ab < +∞.

Thus x2 ∈ T (x1) ∩ Xi(x0). In a similar way, we have D(x2, T (x2)) ≤ ρ(T (x1) ∩ Xi(x0), T (x2)) ≤

ad(x1, x2) < a
2b < +∞.

By induction, we obtain a sequence (xn)n∈N with the following properties:

(a) xn+1 ∈ T (xn) ∩ Xi(x0), for all n ∈ N;

(b) d(xn, xn+1) < a
nb, for all n ∈ N.

From (b) we get that (xn)n∈N is Cauchy and hence convergent in Xi(x0). Thus there exists

x∗ ∈ Xi(x0) (since Xi(x0) is d-closed), such that xn → x
∗

as n → +∞. Let us show now that

x∗ ∈ FT . We have D(x
∗, T (x∗)) ≤ d(x∗, xn+1) + D(xn+1, T (x

∗
)) ≤ d(x∗, xn+1) + ρ(T (xn) ∩

Xi(x0), T (x
∗
)) ≤ d(x∗, xn+1) + ad(x

∗, xn)) → 0 as n → +∞. Hence x
∗ ∈ T (x∗). 2

A second answer to the open problem mentioned in Section 3 is the following:

Theorem 5.18 Let (X, d) be a generalized complete metric space and T : X → Pcl(X) be a



64 Adrian Petruşel, Ioan A. Rus and Marcel Adrian Şerban CUBO
10, 4 (2008)

multivalued operator such that for each x ∈ X and y ∈ T (x) we have d(x, y) < +∞. Let X :=
⋃

i∈I

Xi

be the canonical decomposition of X. Suppose that T is pseudo a-contractive with respect to Xi(x),

for each x ∈ X. Then T is a MWP operator.

Proof. Let x0 ∈ X and x1 ∈ T (x) such that d(x0, x1) < b < +∞, for some b > 0. Thus

x1 ∈ T (x0) ∩ Xi(x0). Hence we have D(x1, T (x1)) ≤ ρ(T (x0) ∩ Xi(x0), T (x1)) ≤ ad(x0, x1) < ab.

We obtain that there exists x2 ∈ T (x1) such that d(x1, x2) < ab < +∞. Thus x2 ∈ T (x1) ∩ Xi(x0).

In a similar way, we have D(x2, T (x2)) ≤ ρ(T (x1) ∩ Xi(x0), T (x2)) ≤ ad(x1, x2) < a
2b < +∞.

By induction, we obtain a sequence (xn)n∈N with the following properties:

(a) xn+1 ∈ T (xn) ∩ Xi(x0), for all n ∈ N;

(b) d(xn, xn+1) < a
nb, for all n ∈ N.

From (b) we get that (xn)n∈N is Cauchy and hence convergent in Xi(x0) to a certain x
∗
. As

before, we obtain x∗ ∈ T (x∗). Since x0 ∈ X and x1 ∈ T (x0) were arbitrarily chosen, we get that

T is a MWP operator. 2

Received: February 2008. Revised: February 2008.

References

[1] D. Azé and J.-P. Penot, On the dependence of fixed points sets of pseudo-contractive

multifunctions. Application to differential inclusions, Nonlinear Dyn. Syst. Theory, 6 (2006),

31–47.

[2] C. Bessaga, On the converse of the Banach fixed point principle, Colloq. Math., 7 (1959),

41–43.

[3] G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer Acad. Publ., Dordrecht,

1994.

[4] L.M. Blumenthal, Theory and Applications of Distance Geometry, Oxford University Press,

1953.

[5] H. Covitz and S.B. Nadler, Multi-valued contraction mapping in generalized metric

spaces, Israel J. Math., 8 (1970), 5–11.

[6] G. Dezso, Fixed point theorems in generalized metric spaces, Pure Math. Appl., 11 (2000),

183–186.

[7] J.B. Diaz and B. Margolis, A fixed point theorem for the alternative for contractions on

a generalized complete metric space, Bull. Amer. Math. Soc., 74 (1968), 305–309.

[8] M. Fréchet, Les espaces abstraits, Gauthier-Villars, Paris, 1928.



CUBO
10, 4 (2008)

Fixed Points for Operators ... 65

[9] C.F.K. Jung, On generalized complete metric spaces, Bull. A.M.S., 75 (1969), 113–116.

[10] S. Kasahara, On some generalizations of the Banach contraction theorems, Mathematics

Seinar Notes, 3 (1975), 161–169.

[11] P.Q. Khanh, Remarks on fixed point theorems based on iterative approximations, Polish

Acad. Sciences, Inst. of Mathematics, Preprint 361, 1986.

[12] R. Kopperman, All topologies come from generalized metrics, Amer. Math. Monthly, 95

(1988), 89–97.

[13] W.A.J. Luxemburg, On the convergences of successive approximations in the theory of

ordinary differential equations, Indag. Math., 20 (1958), 540–546.

[14] A. Meir and E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl., 28

(1969) 326–329.

[15] S.B. Nadler jr., Multivalued contraction mappings, Pacific J. Math., 30 (1969), 475–488.

[16] A. Petruşel, Multivalued weakly Picard operators and applications, Scientiae Mathematicae

Japonicae, 59 (2004), 167–202.

[17] A. Petruşel and I.A. Rus, Multivalued Picard and weakly Picard operators, Fixed Point

Theory and Applications (E. Llorens Fuster, J. Garcia Falset, B. Sims-Eds.), Yokohama Pub-

lishers, 2004, 207–226.

[18] S. Reich, Some remarks concerning contraction mappings, Canad. Math. Bull., 14 (1971),

121–124.

[19] S. Reich, Fixed point of contractive functions, Boll. U.M.I., 5 (1972), 26–42.

[20] I.A. Rus, Metrical Fixed Point Theorems, Cluj-Napoca, 1979.

[21] I.A. Rus, Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca,

2001.

[22] I.A. Rus, Picard operators and applications, Scientiae Mathematicae Japonicae, 58 (2003),

191–219.

[23] I.A. Rus, Metric sapces with fixed point property with respect to contractions, Studia Univ.

Babeş-Bolyai Math., 51 (2006), 115–121.

[24] I.A. Rus, Weakly Picard mappings, Comment. Math. Univ. Carolinae, 34 (1993), 769–773.

[25] I.A. Rus, Weakly Picard operators and applications, Seminar on Fixed Point Theory, Cluj-

Napoca, 2 (2001), 41–58.

[26] I.A. Rus, The theory of a metrical fixed point theorem: theoretical and applicative relevances,

Fixed Point Theory, 9 (2008), to appear.



66 Adrian Petruşel, Ioan A. Rus and Marcel Adrian Şerban CUBO
10, 4 (2008)

[27] I.A. Rus, A. Petruşel and G. Petruşel, Fixed Point Theory 1950–2000: Romanian

Contributions, House of the Book of Science, Cluj-Napoca, 2002.

[28] I.A. Rus, A. Petruşel and A. Ŝıntămărian, Data dependence of the fixed point set of

multivalued weakly Picard operators, Studia Univ. Babeş-Bolyai Mathematica, 46 (2001),

111–121.

[29] I.A. Rus, A. Petruşel and A. Ŝıntămărian, Data dependence of the fixed point set of

some multivalued weakly Picard operators, Nonlinear Analysis, 52 (2003), 1947–1959.

[30] I.A. Rus, A. Petruşel and M.A. Şerban, Weakly Picard operators: equivalent definitions,

applications and open problems, Fixed Point Theory, 7 (2006), 3–22.

[31] M.A. Şerban, Fibre contraction theorem in generalized metric spaces, Automation Comput-

ers Applied Mathematics, 16 (2007), No. 1–2, 9–14.

[32] S.-W. Xiang, Equivalence of completeness and contraction property, Proc. Amer. Math.

Soc., 135 (2007), 1051–1058.

[33] P.P. Zabreiko, K-metric and K-normed linear spaces: survey, Collect. Math., 48 (1997),

825–859.


	N4-Petrusel-Rus-Serban