133
Fixed-Point Theory for Generalized Metric Spaces
4.13.1 Definition Let (X, d) be a quasimetric space. A multivalued mapping
T : X → P(X) is called a contraction if there is a λ ∈ [0, 1) such that, for
all x, y ∈ X and for all a ∈ T (x), there exists b ∈ T (y) satisfying d(a, b) ≤
λd(x, y). We say that T is non-expanding if, for all x, y ∈ X and for all
a ∈ T (x), there exists b ∈ T (y) satisfying d(a, b) ≤ d(x, y).
Again, these definitions are clearly extensions of the corresponding definitions made for single-valued mappings and indeed collapse to them in the
case where T is single valued. An obvious and natural definition of continuity
of T is the following: for every Cauchy sequence (x n ) in X with limit x and
for every choice of y n ∈ T (x n ), we have that (y n ) is a Cauchy sequence and
lim y n ∈ T (x). In fact, the weaker definition following, which is implied by the
one just given, suffices for our purposes and will be used throughout.
4.13.2 Definition Let T : X → P(X) be a multivalued mapping defined on
a quasimetric space (X, d). We say that T is continuous if we have lim x n ∈
T (lim x n ) for every ω-orbit (x n ) of T which is a Cauchy sequence.
Once more, this definition collapses to a natural one if T is single valued.
In fact, if T is single valued, it simply states the condition that lim T (x n ) =
lim x n+1 = lim x n = T (lim x n ) for every ω-orbit which is a Cauchy sequence, which is a weaker condition than that of CS-continuity as in Definition
4.6.2(1).
Finally, if (X, d) is a quasimetric space, we define the associated partial
order ≤ d on X by x ≤ d y if and only if d(x, y) = 0, see Section 4.6.
The main result of this section is the following theorem, generalizing the
Rutten-Smyth theorem we gave earlier, Theorem 4.6.3.
4.13.3 Theorem (Rutten-Smyth multivalued) Let (X, d) be a CScomplete quasimetric space, and let T : X → P(X) denote a non-empty
and continuous multivalued mapping on X. Then T has a fixed point if either
of the following two conditions holds.
(a) T is a contraction.
(b) T is non-expanding, and there is x 0 ∈ X and x 1 ∈ T (x 0 ) such that
d(x 0 , x 1 ) = 0, that is, x 0 ≤ d x 1 .
Proof: (a) Let x 0 ∈ X. Since T (x 0 ) = ∅, we can choose x 1 ∈ T (x 0 ). Since T is
a contraction, there is x 2 ∈ T (x 1 ) such that d(x 1 , x 2 ) ≤ λd(x 0 , x 1 ). Applying
this argument repeatedly, we obtain a sequence (x n ) such that for all n ≥ 0
we have x n+1 ∈ T (x n ) and d(x n+1 , x n+2 ) ≤ λd(x n , x n+1 ). Thus, (x n ) is an
ω-orbit. Using the triangle inequality, we obtain
m−1
m−1
d(x n , x n+m ) ≤
d(x n+i , x n+i+1 )
i
≤
λ
n+i d(x 0 , x 1 ).
=0
i=0
Fixed-Point Theory for Generalized Metric Spaces
4.13.1 Definition Let (X, d) be a quasimetric space. A multivalued mapping
T : X → P(X) is called a contraction if there is a λ ∈ [0, 1) such that, for
all x, y ∈ X and for all a ∈ T (x), there exists b ∈ T (y) satisfying d(a, b) ≤
λd(x, y). We say that T is non-expanding if, for all x, y ∈ X and for all
a ∈ T (x), there exists b ∈ T (y) satisfying d(a, b) ≤ d(x, y).
Again, these definitions are clearly extensions of the corresponding definitions made for single-valued mappings and indeed collapse to them in the
case where T is single valued. An obvious and natural definition of continuity
of T is the following: for every Cauchy sequence (x n ) in X with limit x and
for every choice of y n ∈ T (x n ), we have that (y n ) is a Cauchy sequence and
lim y n ∈ T (x). In fact, the weaker definition following, which is implied by the
one just given, suffices for our purposes and will be used throughout.
4.13.2 Definition Let T : X → P(X) be a multivalued mapping defined on
a quasimetric space (X, d). We say that T is continuous if we have lim x n ∈
T (lim x n ) for every ω-orbit (x n ) of T which is a Cauchy sequence.
Once more, this definition collapses to a natural one if T is single valued.
In fact, if T is single valued, it simply states the condition that lim T (x n ) =
lim x n+1 = lim x n = T (lim x n ) for every ω-orbit which is a Cauchy sequence, which is a weaker condition than that of CS-continuity as in Definition
4.6.2(1).
Finally, if (X, d) is a quasimetric space, we define the associated partial
order ≤ d on X by x ≤ d y if and only if d(x, y) = 0, see Section 4.6.
The main result of this section is the following theorem, generalizing the
Rutten-Smyth theorem we gave earlier, Theorem 4.6.3.
4.13.3 Theorem (Rutten-Smyth multivalued) Let (X, d) be a CScomplete quasimetric space, and let T : X → P(X) denote a non-empty
and continuous multivalued mapping on X. Then T has a fixed point if either
of the following two conditions holds.
(a) T is a contraction.
(b) T is non-expanding, and there is x 0 ∈ X and x 1 ∈ T (x 0 ) such that
d(x 0 , x 1 ) = 0, that is, x 0 ≤ d x 1 .
Proof: (a) Let x 0 ∈ X. Since T (x 0 ) = ∅, we can choose x 1 ∈ T (x 0 ). Since T is
a contraction, there is x 2 ∈ T (x 1 ) such that d(x 1 , x 2 ) ≤ λd(x 0 , x 1 ). Applying
this argument repeatedly, we obtain a sequence (x n ) such that for all n ≥ 0
we have x n+1 ∈ T (x n ) and d(x n+1 , x n+2 ) ≤ λd(x n , x n+1 ). Thus, (x n ) is an
ω-orbit. Using the triangle inequality, we obtain
m−1
m−1
d(x n , x n+m ) ≤
d(x n+i , x n+i+1 )
i
≤
λ
n+i d(x 0 , x 1 ).
=0
i=0