Let (X, d) be a T0 qpm space. A selfmap T on X is called BW -contractive if there exists a function φ : [0, ∞) → [0, ∞) satisfying φ(t) < t and
for all t > 0, and such that for each x, y ∈ X,
If φ(t) = rt, with r ∈ [0, 1) being constant, then T is called contractive.
In their celebrated article [28], Boyd and Wong essentially proved the following general fixed point theorem: Let (X,d) be complete metric space. Then every BW-contractive selfmap on X has a unique fixed point.
The following easy example shows that unfortunately Boyd-Wong's theorem cannot be generalized to complete quasi-metric spaces, even for T contractive.
Example 2.1. Let X = {1/n : n ∈ ℕ} and let d be the quasi-metric on X given by d(1,/n, 1/n) = 0, and d(1/n, 1/m) = 1/m for all n, m ∈ ℕ. Clearly, (X, d) is complete (in fact, it is complete in the stronger sense of [1, 22, 25, 27]). Define T : X → X by T 1/n = 1/2n. Then, T is contractive but it has not fixed point.
Next, we show that it is, however, possible to obtain a nice quasi-metric version of Boyd-Wong's theorem using Q-functions.
Let (X, d) be a T0 qpm space. A selfmap T on X is called BW-weakly contractive if there exist a Q-function q on (X, d) and a function φ : [0, ∞) → [0, ∞) satisfying φ(0) = 0, φ(t) < t and
for all t > 0, and such that for each x, y ∈ X,
If φ(t) = rt, with r ∈ [0, 1) being constant, then T is called weakly contractive.
Theorem 2.2. Let (X, d) be a complete T0qpm space. Then, each BW-weakly contractive selfmap on X has a unique fixed point z ∈ X. Moreover, q(z, z) = 0.
Proof. Let T : X → X be BW-weakly contractive. Then, there exist a Q-function q on (X, d) and a function φ : [0, ∞) → [0, ∞) satisfying φ(0) = 0, φ(t) < t and
for all t > 0, such that for each x, y ∈ X,
Fix x0 ∈ X and let x
n
= Tn x0 for all n ∈ ω.
We show that q(x
n
, xn+1) → 0.
Indeed, if q(x
k
, xk+1) = 0 for some k ∈ ω, then φ(q(x
k
, xk+1)) = 0 and thus q(x
n
, xn+1) = 0 for all n ≥ k. Otherwise, (q(x
n
, xn+1))n∈ωis a strictly decreasing sequence in (0, ∞) which converges to 0, as in the classical proof of Boyd-Wong's theorem.
Similarly, we have that q(xn+1, x
n
) → 0.
Now, we show that for each ε ∈ (0, 1) there exists n
ε
∈ ℕ such that q(x
n
, x
m
) < ε whenever m > n > n
ε
.
Assume the contrary. Then, there exists ε0 ∈ (0, 1) such that, for each k ∈ ℕ, there exist n(k), j(k) ∈ ℕ with j(k) > n(k) > k and q(xn(k), xj(k)) ≥ ε0.
Since q(x
n
, xn+1) → 0, there exists
such that q(x
n
, xn+1) < ε0 for
all
.
For each
, we denote by m(k) the least j(k) ∈ ℕ satisfying the following three conditions:
Note that there exists such a m(k) because q(xn(k), xn(k)+1) < ε0. Then, for each
, we obtain
Since q(xm(k)-1, xm(k)) → 0, it follows from the preceding inequalities that
where r
k
= q(xn(k), xm(k)). Hence,
Choose δ > 0 with
. Let
such that q(xn(k), xn(k)+1) < (ε0 - δ)/2, and q(xm(k)+1, xm(k)) < (ε0 - δ)/2,
for all k > k0.
Then,
for some k > k0, which contradicts that ε0 ≤ q(xn(k), xm(k)) for all
. We conclude that for each ε ∈ (0, 1), there exists n
ε
∈ ℕ such that
Next, we show that (x
n
)n∈ωis a Cauchy sequence in the metric space (X, ds ). Indeed, let ε > 0, and let δ = δ (ε) > 0 as given in Definition 1.2 (W3). Then, for n, m > n
δ
we obtain
, and
, and hence from Remark 1.3, ds (x
n
, x
m
) ≤ ε. Consequently, (x
n
)n∈ωis a Cauchy sequence in (X, ds ).
Now, let z ∈ X such that d(x
n
, z) → 0. Then q(x
n
, z) → 0 by (Q2) and condition (*) above. Hence,
. From Remark 1.3, we conclude that ds (z, Tz) = 0, i.e., z = Tz.
Next, we show the uniqueness of the fixed point. Let y = Ty. If q(y, z) > 0, q(Ty, Tz) = q(y, z) ≤ φ(q(y, z)) < q(y, z), a contradiction. Hence, q(y, z) = 0. Interchanging y and z, we also have q(z, y) = 0. Therefore, y = z from Remark 1.3.
Finally, q(z, z) = 0 since otherwise we obtain q(z, z) = q(Tz, Tz) ≤ φ(q(z, z)) < q(z, z), a contradiction. □
The following is an example of a non-BW-contractive selfmap T on a complete T0 qpm space (X, d) for which Theorem 2.2 applies.
Example 2.3. Let X = [0, 1) and d be the weightable T0 qpm on X given by d(x, y) = max{y -x, 0} for all x, y ∈ X. Clearly (X, d) is complete because d(x, 0) = 0 for all x ∈ X, and thus every sequence in X converges to 0 with respect to
.
Now, define T : X → X by Tx = x2 for all x ∈ X. Then, T is not BW-contractive because d(Tx, Ty) = y2 - x2> y - x = d(x, y), whenever 0 < x < y < 1 < x + y. However, T is BW-weakly contractive for the partial metric p
d
induced by d (recall that, from Remark 1.4, pd is a Q-function on (X, d)), and the function φ : [0, ∞) → [0, ∞) defined by φ(t) = t2 for 0 ≤ t < 1 and
for t ≥ 1. Indeed, for each x, y ∈ X we have,
Hence, we can apply Theorem 2.2, so that T has a unique fixed point: in fact, 0 is the only fixed point of T, and p
d
(0, 0) = 0. (Note that in this example, there exists not r ∈ [0, 1) such that p
d
(Tx, Ty) ≤ rp
d
(x, y) for all x, y ∈ X.)
In the light of the applications of w-distances and Q-functions to the fixed point theory for multivalued maps on metric and quasi-metric spaces, it seems interesting to investigate the extension of our version of Boyd-Wong's theorem to the case of multivalued maps. In Theorem 2.6 below, we shall prove a positive result for the case of symmetry Q-functions, which are defined as follows:
Definition 2.4. A symmetric Q-function on a T0 qpm space (X, d) is a Q-function q on (X, d) such that
If q is a w-distance satisfying (SY), we then say that it is a symmetric w-distance on (X, d).
Example 2.5. Of course, if (X, d) is a metric space, then d is a symmetric w-distance on (X, d). Moreover, it follows from Remark 1.4, that for every weightable T0 qpm space (X, d) its induced partial metric p
d
is a symmetric w-distance on (X, d). Note also that the w-distance constructed in Lemma 2 of [29] is also a symmetric w-distance.
Given a T0 qpm space (X, d), we denote by 2 X and by
the collection of all nonempty subsets of X and the collection of all nonempty
-closed subsets of X, respectively.
Generalizing the notions of a q-contractive multivalued map [[25], Definition 6.1] and of a generalized q-contractive multivalued map [27], we say that a multivalued map T from a T0 qpm space (X, d) to 2 X , is BW-weakly contractive if there exists a Q-function q on (X, d) and a function φ : [0, ∞) → [0, ∞) satisfying φ(0) = 0, φ(t) < t and
for all t > 0, and such that, for each x, y ∈ X and each u ∈ Tx there exists v ∈ Ty with q(u, v) ≤ φ(q(x, y)).
Theorem 2.6. Let (X, d) be a complete T0qpm space and
be BW-weakly contractive for a symmetric Q-function q on (X,d). Then, there is z ∈ X such that z ∈ Tz and q(z, z) = 0.
Proof. By hypothesis, there is a function φ : [0, ∞) → [0, ∞) satisfying φ(0) = 0, φ(t) < t and
for all t > 0, and such that for each x, y ∈ X and u ∈ Tx there is v ∈ Ty with
Fix x0 ∈ X and let x1 ∈ Tx0. Then, there exists x2 ∈ Tx1 such that q(x1, x2) ≤ φ(q(x0, x1). Following this process, we obtain a sequence (x
n
)n∈ωwith x
n
∈ Txn - 1and q(x
n
, xn+1) ≤ φ(q(xn-1, x
n
) for all n ∈ ℕ.
As in Theorem 2.2, q(x
n
, xn+1) → 0.
Now, we show that for each ε ∈ (0, 1), there exists n
ε
∈ ℕ such that q(x
n
, x
m
) < ε whenever m > n > n
ε
.
Assume the contrary. Then, there exists ε0 ∈ (0, 1) such that for each k ∈ ℕ, there exist n(k), j(k) ∈ ℕ with j(k) > n(k) > k and q(xn(k), xj(k)) ≥ ε0.
Again, by repeating the proof of Theorem 2.2, and using symmetry of q, we derive that
a contradiction.
From Remark 1.3, it follows that (x
n
)n∈ωis a Cauchy sequence in (X, ds ) (compare the proof of Theorem 2.2), and so there exists z ∈ X such that d(x
n
, z) → 0, and thus q(x
n
, z) → 0.
Therefore, for each n ∈ ω there exists vn+1∈ Tz with
Since q(x
n
, z) → 0 we have q(xn+1, vn+1) → 0, and so ds (z, vn+1) → 0 from Remark 1.3. Consequently, z ∈ Tz because Tz is closed in (X, ds ).
It remains to be shown that q(z, z) = 0. Indeed, since z ∈ Tz we can construct a sequence (z
n
)n∈ℕin X such that z1 ∈ Tz, zn+1∈ Tz
n
, q(z, z1) ≤ φ(q(z, z
n
)) and q(z, zn+1) ≤ φ(q(z, z
n
)) for all n ∈ ℕ. Hence (q(z, z
n
))n∈ℕis a nonincreasing sequence in [0, ∞) that converges to 0. From Remark 1.3, the sequence (z
n
)n∈ℕis Cauchy in (X, ds ). Let u ∈ X such that d(z
n
, u) → 0. It follows from condition (Q2) that q(z, u) = 0. Since q(x
n
, z) → 0, we deduce by condition (Q1) that q(x
n
, u) → 0. Therefore, ds (z, u) ≤ ε for all ε > 0, from Remark 1.3. We conclude that z = u, and thus q(z, z) = 0. □
Although we do not know if symmetric of q can be omitted in Theorem 2.6, it can be applied directly to obtain the following fixed point result for multivalued maps on partial metric spaces, which substantially improves Theorem 5.3 of [5].
Corollary 2.7. Let (X, p) be a partial metric space such that the induced weightable T0qpm d
p
is complete and
be BW-weakly contractive for p. Then, there is z ∈ X such that z ∈ Tz and p(z, z) = 0.
We conclude this article by showing, nevertheless, that it is possible to prove a multivalued version of the celebrated Matkowski's fixed point theorem [30], which provides a nice generalization of Boyd-Wong's theorem when φ is nondecreasing.
Theorem 2.8. Let (X, d) be a complete T0qpm space and let
. If there exist a Q-function q on (X, d) and a nondecreasing function φ : (0, ∞) → (0, ∞) satisfying φn (t) → 0 for all t > 0, such that for each x, y ∈ X and each u ∈ Tx, there exists v ∈ Ty with
then, there exists z ∈ X such that z ∈ Tz and q(z, z) = 0.
Proof. Let φ(0) = 0. Fix x0 ∈ X and let x1 ∈ Tx0. Then, there exists x2 ∈ Tx1 such that q(x1, x2) ≤ φ(q(x0, x1). Following this process, we obtain a sequence (x
n
)n∈ωwith x
n
∈ Txn-1and q(x
n
, xn+1) ≤ φ(q(xn - 1, x
n
) for all n ∈ ℕ. Therefore,
for all n ∈ ℕ. Since φn (q(x0, x1)) → 0, it follows that q(x
n
, xn+1) → 0.
Now, choose an arbitrary ε > 0. Since φn (ε) → 0, then φ(ε) < ε, so there is n
ε
∈ ℕ such that
for all n ≥ n
ε
. Note that then,
for all n ≥ n
ε
, and following this process
for all n ≥ n
ε
and k ∈ ℕ. Applying Remark 1.3, we deduce that (x
n
)n∈ωis a Cauchy sequence in (X, ds ). Then, there is z ∈ X such that d(x
n
, z) → 0 and thus q(x
n
, z) → 0 by condition (Q2). The rest of the proof follows similarly as the proof of Theorem 2.6. We conclude that z ∈ Tz and q(z, z) = 0. □
Remark 2.9. The above theorem improves, among others, Theorem 3.3 of [1] (compare also Theorem 1 of [31]).