Let (*X*, *d*) be a *T*_{0} 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* = 1/2*n*. 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 *T*_{0} 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 T*_{0}*qpm 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 *x*_{0} ∈ *X* and let *x*_{
n
}= *T*^{n} *x*_{0} for all *n* ∈ *ω*.

We show that *q*(*x*_{
n
}, *x*_{n+1}) → 0.

Indeed, if *q*(*x*_{
k
}, *x*_{k+1}) = 0 for some *k* ∈ *ω*, then *φ*(*q*(*x*_{
k
}, *x*_{k+1})) = 0 and thus *q*(*x*_{
n
}, *x*_{n+1}) = 0 for all *n* ≥ *k*. Otherwise, (*q*(*x*_{
n
}, *x*_{n+1}))_{n∈ω}is 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*(*x*_{n+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 *ε*_{0} ∈ (0, 1) such that, for each *k* ∈ ℕ, there exist *n*(*k*), *j*(*k*) ∈ ℕ with *j*(*k*) *> n*(*k*) *> k* and *q*(*x*_{n(k)}, *x*_{j(k)}) ≥ *ε*_{0}.

Since *q*(*x*_{
n
} , *x*_{n+1}) → 0, there exists such that *q*(*x*_{
n
} , *x*_{n+1}) *< ε*_{0} 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*(*x*_{n(k)}, *x*_{n(k)+1}) *< ε*_{0}. Then, for each , we obtain

Since *q*(*x*_{m(k)-1}, *x*_{m(k)}) → 0, it follows from the preceding inequalities that where *r*_{
k
} = *q*(*x*_{n(k)}, *x*_{m(k)}). Hence,

Choose *δ >* 0 with . Let such that *q*(*x*_{n(k)}, *x*_{n(k)+1}) *<* (*ε*_{0} - *δ*)/2, and *q*(*x*_{m(k)+1}, *x*_{m(k)}) *<* (*ε*_{0} - *δ*)/2,

for all *k > k*_{0}.

Then,

for some *k > k*_{0}, which contradicts that *ε*_{0} ≤ *q*(*x*_{n(k)}, *x*_{m(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∈ω}is a Cauchy sequence in the metric space (*X*, *d*^{s} ). 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, *d*^{s} (*x*_{
n
} , *x*_{
m
} ) ≤ *ε*. Consequently, (*x*_{
n
} )_{n∈ω}is a Cauchy sequence in (*X*, *d*^{s} ).

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 *d*^{s} (*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 *T*_{0} qpm space (*X*, *d*) for which Theorem 2.2 applies.

*Example 2.3*. Let *X* = [0, 1) and *d* be the weightable *T*_{0} 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* = *x*^{2} for all *x* ∈ *X*. Then, *T* is not *BW*-contractive because *d*(*Tx*, *Ty*) = *y*^{2} - *x*^{2}*> 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*) = *t*^{2} 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 *T*_{0} 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 *T*_{0} 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 *T*_{0} 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 *T*_{0} 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 T*_{0}*qpm 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 *x*_{0} ∈ *X* and let *x*_{1} ∈ *Tx*_{0}. Then, there exists *x*_{2} ∈ *Tx*_{1} such that *q*(*x*_{1}, *x*_{2}) ≤ *φ*(*q*(*x*_{0}, *x*_{1}). Following this process, we obtain a sequence (*x*_{
n
} )_{n∈ω}with *x*_{
n
} ∈ *Tx*_{n - 1}and *q*(*x*_{
n
} , *x*_{n+1}) ≤ *φ*(*q*(*x*_{n-1}, *x*_{
n
} ) for all *n* ∈ ℕ.

As in Theorem 2.2, *q*(*x*_{
n
} , *x*_{n+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 *ε*_{0} ∈ (0, 1) such that for each *k* ∈ ℕ, there exist *n*(*k*), *j*(*k*) ∈ ℕ with *j*(*k*) *> n*(*k*) *> k* and *q*(*x*_{n(k)}, *x*_{j(k)}) ≥ *ε*_{0}.

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∈ω}is a Cauchy sequence in (*X*, *d*^{s} ) (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 *v*_{n+1}∈ *Tz* with

Since *q*(*x*_{
n
} , *z*) → 0 we have *q*(*x*_{n+1}, *v*_{n+1}) → 0, and so *d*^{s} (*z*, *v*_{n+1}) → 0 from Remark 1.3. Consequently, *z* ∈ *Tz* because *Tz* is closed in (*X*, *d*^{s} ).

It remains to be shown that *q*(*z*, *z*) = 0. Indeed, since *z* ∈ *Tz* we can construct a sequence (*z*_{
n
} )_{n∈ℕ}in *X* such that *z*_{1} ∈ *Tz*, *z*_{n+1}∈ *Tz*_{
n
} , *q*(*z*, *z*_{1}) ≤ *φ*(*q*(*z*, *z*_{
n
} )) and *q*(*z*, *z*_{n+1}) ≤ *φ*(*q*(*z*, *z*_{
n
} )) for all *n* ∈ ℕ. Hence (*q*(*z*, *z*_{
n
} ))_{n∈ℕ}is a nonincreasing sequence in [0, ∞) that converges to 0. From Remark 1.3, the sequence (*z*_{
n
} )_{n∈ℕ}is Cauchy in (*X*, *d*^{s} ). 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, *d*^{s} (*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 T*_{0}*qpm 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 T*_{0}*qpm space and let*. *If there exist a Q-function q on* (*X*, *d*) *and a nondecreasing function φ* : (0, ∞) → (0, ∞) *satisfying φ*^{n} (*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 *x*_{0} ∈ *X* and let *x*_{1} ∈ *Tx*_{0}. Then, there exists *x*_{2} ∈ *Tx*_{1} such that *q*(*x*_{1}, *x*_{2}) ≤ *φ*(*q*(*x*_{0}, *x*_{1}). Following this process, we obtain a sequence (*x*_{
n
} )_{n∈ω}with *x*_{
n
} ∈ *Tx*_{n-1}and *q*(*x*_{
n
} , *x*_{n+1}) ≤ *φ*(*q*(*x*_{n - 1}, *x*_{
n
} ) for all *n* ∈ ℕ. Therefore,

for all *n* ∈ ℕ. Since *φ*^{n} (*q*(*x*_{0}, *x*_{1})) → 0, it follows that *q*(*x*_{
n
} , *x*_{n+1}) → 0.

Now, choose an arbitrary *ε >* 0. Since *φ*^{n} (*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∈ω}is a Cauchy sequence in (*X*, *d*^{s} ). 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]).