Fixed point theory plays a fundamental role in solving functional equations [1] arising in several areas of mathematics and other related disciplines as well. The Banach contraction principle is a key principle that made a remarkable progress towards the development of metric fixed point theory. Markin [2] and Nadler [3] proved a multivalued version of the Banach contraction principle employing the notion of a Hausdorff metric. Afterwards, a number of generalizations (see [4–9]) were obtained using different contractive conditions. The study of hybrid type contractive conditions involving singlevalued and multivalued mappings is a valuable addition to the metric fixed point theory and its applications (for details, see [10–14]). Among several generalizations of the Banach contraction principle, Suzuki’s work [[15], Theorem 2.1] led to a number of results (for details, see [13, 16–21]).
On the other hand, Matthews [22] introduced the concept of a partial metric space as a part of the study of denotational semantics of dataflow networks. He obtained a modified version of the Banach contraction principle, more suitable in this context (see also [23, 24]). Since then, results obtained in the framework of partial metric spaces have been used to constitute a suitable framework to model the problems related to the theory of computation (see [22, 25–28]). Recently, Aydi et al.[29] initiated the concept of a partial Hausdorff metric and obtained an analogue of Nadler’s fixed point theorem [3] in partial metric spaces.
The aim of this paper is to obtain some coincidence point theorems for a hybrid pair of singlevalued and multivalued mappings on an arbitrary nonempty set with values in a partial metric space. Our results extend, unify and generalize several known results in the existing literature (see [13, 15, 21, 30]). As an application, we obtain the existence and uniqueness of a common and bounded solution for SuzukiZamfirescu class of functional equations under contractive conditions weaker than those given in [1, 31–34].
Throughout this work, a mapping
is defined by
In the sequel, the letters ℝ,
and ℕ will denote the set of all real numbers, the set of all nonnegative real numbers and the set of all positive integers, respectively. Consistent with [22, 29, 35, 36], the following definitions and results will be needed in the sequel.
Definition 1.1[22]
Let
X be any nonempty set. A mapping
is said to be a partial metric if and only if for all
the following conditions are satisfied:

(P1)
if and only if
;

(P2)
;

(P3)
;

(P4)
.
The pair
is called a partial metric space. If
, then (P1) and (P2) imply that
. But the converse does not hold in general. A classical example of a partial metric space is the pair
, where
is defined as
(see also [37]).
Example 1.2[22]
If
, then
defines a partial metric p on X.
For more interesting examples, we refer to [
23,
27,
28,
35,
38,
39]. Each partial metric
p on
X generates a
topology
on
X which has as a base the family of open balls (
pballs)
, where
for all
and
. A sequence
in a partial metric space
is called convergent to a point
with respect to
if and only if
(for details, see [
22]). If
p is a partial metric on
X, then the mapping
given by
defines a metric on
X. Furthermore, a sequence
converges in a metric space
to a point
if and only if
Definition 1.3[22]
Let
be a partial metric space, then
Lemma A[22, 35]
Let
be a partial metric space,
then
Consistent with [
29], let
be the family of all nonempty, closed and bounded subsets of the partial metric space
, induced by the partial metric
p. Note that closedness is taken from
(
is the topology induced by
p) and boundedness is given as follows:
A is a bounded subset in
if there exists an
and
such that for all
, we have
, that is,
. For
and
, define
and
It can be verified that
implies
, where
.
Lemma B[35]
Let
be a partial metric space and
A
be a nonempty subset of
X, then
if and only if
.
Proposition 1.4[29]
Let
be a partial metric space.
For any
,
we have the following:
 (i)
;
 (ii)
;
 (iii)
implies
;
 (iv)
.
Proposition 1.5[29]
Let
be a partial metric space.
For any
,
we have the following:

(h1)
;

(h2)
;

(h3)
;

(h4)
implies that
.
The mapping
is called a partial Hausdorff metric induced by a partial metric p. Every Hausdorff metric is a partial Hausdorff metric, but the converse is not true (see Example 2.6 in [29]).
Lemma C[29]
Let
be a partial metric space and
and
, then for any
, there exists a
such that
.
Theorem 1.6[29]
Let
be a partial metric space. If
is a multivalued mapping such that for all
, we have
, where
. Then
T
has a fixed point.
Definition 1.7 Let
be a partial metric space and
and
. A point
is said to be (i) a fixed point of
f if
, (ii) a fixed point of
T if
, (iii) a coincidence point of a pair
if
, (iv) a common fixed point of the pair
if
.
We denote the set of all fixed points of f, the set of all coincidence points of the pair
and the set of all common fixed points of the pair
by
,
and
, respectively. Motivated by the work of [4, 13], we give the following definitions in partial metric spaces.
Definition 1.8 Let
be a partial metric space and
and
. The pair
is called (i) commuting if
for all
, (ii) weakly compatible if the pair
commutes at their coincidence points, that is,
whenever
, (iii) ITcommuting [11] at
if
.
Definition 1.9 Let
be a partial metric space and
Y be any nonempty set. Let
and
be singlevalued and multivalued mappings, respectively. Suppose that
, then the set
is called an orbit for the pair
at
. A partial metric space X is called
orbitally complete if and only if every Cauchy sequence in the orbit for
at
converges with respect to
to a point
such that
.
Singh and Mishra [13] introduced SuzukiZamfirescu type hybrid contractive condition in complete metric spaces. In the context of partial metric spaces, the condition is given as follows.
Definition 1.10 Let
be a partial metric space,
and
be singlevalued and multivalued mappings, respectively. The hybrid pair
is said to satisfy
SuzukiZamfirescu hybrid contraction condition if there exists
such that
implies that
for all
and
Lemma D
Let
be a partial metric space,
and
be singlevalued and multivalued mappings, respectively. Then the partial metric space
is
orbitally complete if and only if
is
orbitally complete.
Proof Suppose that
is
orbitally complete and
is an arbitrary element of
X. If
is a Cauchy sequence in
in
, then it is also Cauchy in
. Therefore, by (1.2) we deduce that there exists
y in
X such that
and
converges to
y in
. Conversely, let
be
orbitally complete. If
is a Cauchy sequence in
in
, then it is also a Cauchy sequence in
. Therefore,
For given
, there exists
such that
for all
. Consequently, we have
whenever
. The result follows. □