In this paper, we introduce g-approximative multivalued mappings to a partial metric space. Based on this definition, we give some new definitions. Further, common fixed point results for g-approximative multivalued mappings satisfying generalized contractive conditions are obtained in the setup of ordered partial metric spaces.
The study of fixed points for multivalued contraction mappings using the Hausdorff metric was initiated by Nadler . After this, fixed point theory has been developed further and applied to many disciplines to solve functional equations. The Banach contraction principle has been extended in different directions either by using generalized contractions for multivalued mappings and hybrid pairs of single and multivalued mappings, or by using more general spaces. Dhage [2, 3] established hybrid fixed point theorems and obtained some applications of presented results. Hong and Shen  proved common fixed point results for generalized contractive multivalued operators in a complete metric space. Also, the monotone iterative technique is associated with several nonlinear problems . This technique is also employed to prove the existence of fixed points for multivalued monotone operators (see, for example, ). In , the problem of existence and approximation of coupled fixed points for mixed monotone multivalued operators was studied in ordered Banach spaces under the assumption that operators satisfy the condensing condition and upper demicontinuity.
Hong introduced the concepts of approximative values, comparable approximative values, upper and lower comparable approximative values in . These definitions are a very useful tool for proving the existence of a fixed point of a multivalued operator in an ordered metric space. Then, Abbas and Erduran in  extended the concept of these definitions using g self-mappings, so they introduced g-approximative multivalued mappings and proved coincidence and common fixed point results for a hybrid pair of multivalued and single-valued mappings. Also, they introduced the concepts of g-comparable approximative, g-upper comparable approximative and g-lower comparable approximative multivalued mappings.
In this paper, unless otherwise mentioned, let denote an ordered complete partial metric space with a partial order ≤ and distance .
Definition 1 Let X be an ordered partial metric space. A mapping is said to be (i) weakly L-idempotent if for x in X, (ii) weakly R-idempotent if for x in X. For example, a mapping given by is weakly R-idempotent.
Definition 2 An ordered partial metric space is said to have a subsequential limit comparison property if for every nondecreasing sequence (nonincreasing sequence) in X such that , there exists a subsequence of with (), respectively.
Definition 3 An ordered partial metric space is said to have a sequential limit comparison property if for every nondecreasing sequence (nonincreasing sequence) in X such that implies that (), respectively.
Let X be any nonempty set endowed with a partial order ≤ and let be a given mapping. We define the set by
Note that for each , one has .
Example 1 Let be endowed with the usual order ≤ and g be a self-map on X defined as , and . Then the subset of is .
In order to extend the concept of g-approximative, g-CAV, g-UCAV, g-LCAV multivalued mappings on partial metric spaces, we first adapt the notion of g-approximative to the partial metric framework as follows.
Definition 4 Let X be a partial metric space and . A subset Y of X is said to be g-approximative for some x in X if and the set
Definition 5 Let X be a partially ordered set. A mapping (collection of all nonempty subsets of X) is said to be:
g-approximative multivalued mapping (in short g-AV multivalued mapping), if Fx is g-approximative for each , that is, is nonempty for each x in X.
g-CAV multivalued mapping (g-comparable approximative multivalued mapping) if F is g-approximative and for each , there exists such that gy is comparable to gz.
g-UCAV (g-upper comparable approximative multivalued mapping) if F is g-approximative and for each , there exists such that .
g-LCAV (g-lower comparable approximative multivalued mapping) if F is g-approximative and for each , there exists such that .
If F is single-valued, then g-UCAV (g-LCAV) means that () for .
Definition 6 Let and . A point x in X is said to be: (i) a fixed point of g if , (ii) a fixed point of T if , (iii) a coincidence point of a pair if , (iv) a common fixed point of a pair if .
, and denote the set of all fixed points of g, the set of all coincidence points of the pair and the set of all common fixed points of the pair , respectively.
Definition 7 Let , and . The pair is called (1) commuting if for all , (2) weakly compatible  if they commute at their coincidence points, that is, whenever , (3) -commuting at if .
Definition 8 Let . The map is said to be T-weakly commuting at if .
Definition 9 The map is said to be coincidently idempotent with respect to if for x in . The point x is called a point of coincident idempotency.
Now, we present an example of a hybrid pair for which f is T-weakly commuting at some .
Example 2 Let with the usual metric. Define , by
It can be easily verified that f is T-weakly commuting at .
Example 3 Let with the usual metric. Define , by
Here and f is coincidently idempotent with respect to T.
Let . Ϝ denotes the class of mappings which satisfy the following conditions:
and for each ,
f is continuous,
f is nondecreasing on .
A mapping f is said to be sublinear if whenever . We define .
denotes the family of mappings which satisfy the following conditions:
for each ,
ψ is nondecreasing and right upper semi-continuous,
for each , .
By means of the functions f and ψ given in Ϝ and respectively, a generalized contractive condition was defined in . Let Φ denote the class of mappings for which and for each t in .
Definition 10 For two subsets A, B of X, we say if for each , there exists such that and if each , implies that .
A multivalued mapping is said to be g-nondecreasing (g-nonincreasing) if implies that () for all . F is said to be g-monotone if F is g-nondecreasing or g-nonincreasing. Moreover, in what follows, will be a partially ordered set such that there exists a complete partial metric p on X. Let . Set if and if .
Consistent with [11–13], the following definitions and results will be needed in the sequel.
Definition 11 A partial metric on a nonempty set X is a function such that for all ,
A partial metric space is a pair such that X is a nonempty set and p is a partial metric on X.
Each partial metric p on X generates a topology on X which has as a base the family of open p-balls , where for all and .
If p is a partial metric on X, then the function given by
is a metric on X.
A mapping is said to be continuous at if for every , there exists such that .
Definition 12 Let be a partial metric space and be a sequence in X. Then
converges to a point if and only if ,
is called a Cauchy sequence if there exists (and is finite) .
Definition 13 A partial metric space is said to be complete if every Cauchy sequence in X converges, with respect to , to a point such that .
Lemma 1Letbe a partial metric space. Then
is a Cauchy sequence inif and only if it is a Cauchy sequence in the metric space ,
is complete if and only if the metric spaceis complete. Furthermore, if and only if
Lemma 2Letbe a partial metric space and letbe a continuous self-mapping. Assumesuch thatas . Then
Recently Haydi et al. introduced a partial Hausdorff metric on a partial metric space and they extended Nadler’s fixed point theorem on partial metric spaces using the partial Hausdorff metric.
Let be a partial metric space. Let be a 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 exist and such that for all , we have , that is, .
Let be endowed with the partial metric defined by . From (i) of Proposition 2.2, we have
In view of Proposition 2.3 and Corollary 2.4 in , we call the mapping a partial Hausdorff metric induced by p.
Remark 2 It is easy to show that any Hausdorff metric is a partial Hausdorff metric. The converse is not true (Example 2.6 of in ).
The aim of this paper is to adapt the notion of g-approximative to the partial metric and extend the concept of g-UCAV, g-LCAV, g-CAV mappings. Also, we prove some fixed point theorems for multivalued mappings and give an example associated with the following theorem.
Theorem 1Suppose thatgis a nondecreasing self-map onXandisg-UCAV and the following holds:
for any , whereandand
IfXhas the limit comparison property andis closed, thenFandghave a coincidence pointxinX. Moreover, Fandghave a common fixed point if one of the following conditions holds:
PairisIT-commuting at someandfor someandgis continuous atu.
PairisIT-commuting at someand .
gisF-weakly commuting at someandgis coincidently idempotent with respect toT.
gis continuous atxfor somefor some ; .
is a singleton subset of .
Proof Let . If , then the result is proved. If not, then we proceed as follows. As F is g-UCAV, , is nonempty, so there exists with such that for some and . Similarly, there exists with such that for some , and . We continue to construct a sequence for which either or there exists with and such that
for some in X. On the other hand,
Since f is nondecreasing, we have
If , then we have
a contradiction. So, we have . This yields
Repeating this process, we have
For , we obtain
On taking limit as and using , we have . By the definition of , we get
This yields that is a Cauchy sequence in . Since X is complete and is closed, then is complete, hence is complete. So, we have for some . From Lemma 1, we get
Consequently, is a Cauchy sequence in . Now, we prove that . Suppose that this is not true, then . For large enough n, we claim that the following equation holds:
Indeed, since and , it follows that
So, there exists such that for every . Note that
which, on taking limit as , gives
a contradiction. Hence, and so . Suppose now that (i) holds. Then , where . Since g is continuous at u, so we have that u is a fixed point of g. By given assumption, for all and . Now, we prove that . Suppose that this is not true, then . Using (1.1), since f is nondecreasing and sublinear, we obtain
On taking limit as , we have
which further implies
On taking limit as ,
a contradiction, so and hence . Consequently, . Hence, u is a common fixed point of F and g. Suppose now that (ii) holds. As , so . Now, implies that gx is a common fixed point of F and g. Suppose now that (iii) holds. The result is obvious. Suppose that (iv) holds. As and for some , . By the continuity of g at x, we get . Hence, x is a common fixed point of F and g. Finally, suppose that (v) holds. Let . Then . Hence, x is a common fixed point of F and g. □
Similarly, we have following theorem.
Theorem 2Suppose thatgis a nondecreasing self-map onXandisg-LCAV and the following holds:
for any , whereandand
IfXhas the sequential limit comparison property andis closed, thenFandghave a coincidence pointxinX. Moreover, Fandghave a common fixed point if any one of the conditions (i)-(v) holds as in Theorem 1.
Example 5 Let with . Define , by
It is clear that F is g-UCAV, also is closed and X has the property of limit comparison. We can see easily that g is F-weakly commuting at . Besides, g is coincidently idempotent with respect to F at . In this case, these functions satisfy the condition of (iii) in Theorem 1. Also, we can define , , then and . If , we have and ,
If , , we have , and ,
Also, we have
So, we satisfy the contractive condition. Finally, if , we have , and ,
Hence, all the conditions of Theorem 1 are satisfied. It is clear that , that is, is a common fixed point of F and g.
Corollary 1Suppose thatgis a nondecreasing self-map onXandandare self-mappings which satisfy
for any , where , and
ThenF, ghave a unique coincidence point . Moreover, Fandghave a unique common fixed point if any one of the conditions (i)-(v) holds as in Theorem 1.
Proof Theorem 1 ensures the existence of a coincidence point. To prove the uniqueness, let y be another coincidence point of F and g. If , then . Thus,
a contradiction, therefore . The result follows. □
Theorem 3Suppose thatgis a nondecreasing self-map onXandisg-AV and the following holds:
for any , whereandand
Ifis closed and there existssuch that , thenFandghave a coincidence point . Further, an iterative sequencewithconverges togx, where . Moreover, Fandghave a common fixed point if any one of the conditions (i)-(v) holds as in Theorem 1.
Proof If , then the proof is finished. Otherwise, for any , one has . As F has a g-approximative multivalued map, for , there exists with and
Similarly, for , there exists with and
We continue the process of constructing a sequence such that for , one obtains with such that
On the other hand, we have
The rest of this proof is the same as that of Theorem 1. □
Theorem 4Suppose thatgis a nondecreasing self-map onX, isg-CAV and the following holds:
for any , whereandand
IfXhas the subsequential limit comparison property andis closed, thenFandghave a coincidence point. Moreover, Fandghave a common fixed point if any one of the conditions (i)-(v) holds as in Theorem 1.
Proof Following similar arguments to those given in Theorem 1 and assuming F is g-CAV, we obtain a sequence whose consecutive terms are comparable, satisfy (1.2) and (1.4), and the following hold:
Since X has the subsequential limit comparison property, so has a subsequence whose every term is comparable to gx. Now, we prove . Obviously,
for . For , there exists such that
for all . As
Since is comparable to gx for each k, therefore
Note that f is continuous and . We obtain, by letting ,
This implies that , so we have . By similar arguments to those in Theorem 1, we can show the existence of a common fixed point. □
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