On some fixed point theorems for multivalued F-contractions in partial metric spaces


 Altun et al. explored the existence of fixed points for multivalued 
 
 
 
 F
 
 F
 
 -contractions and proved some fixed point theorems in complete metric spaces. This paper extended the results of Altun et al. in partial metric spaces and proved fixed point theorems for multivalued 
 
 
 
 F
 
 F
 
 -contraction mappings. Some illustrative examples are provided to support our results. Moreover, an application for the existence of a solution of an integral equation is also enunciated, showing the materiality of the obtained results.


Introduction and preliminaries
Metric fixed point theory has been the centre of extensive research for several researchers. Banach's theorem has been enriched, unified, extended and generalized by many researchers. Fixed point theory has become an important tool for solving many nonlinear problems related to science and engineering because of its applications. We refer the readers to some noteworthy papers for more details of this topic [1][2][3][4][5][6][7].
In 1969, the study of fixed points for multivalued mappings on complete metric spaces was introduced by Nadler [8]. He combined the ideas of multivalued mappings and contractions by providing the following theorem: , be a complete metric space and let → ( ) T X CB X : be a multivalued mapping satisfying for all ∈ x y X , , where ∈ ( ) k 0, 1 and ( ) CB X denotes the collection of non-empty closed and bounded subsets of X. Then T has a fixed point ∈ u X such that ∈ u Tu.
Recently, Wardowski [2] introduced the notion of F-contraction which is defined as: and ( +∞) → F : 0, a mapping satisfying the following: F1: F is strictly increasing, that is for all ∈ + x y , such that < ⇒ ( ) < ( ) x y F x F y . Remark 1. In future discussion, we will denote ( +∞) 0, by + .
Following Nadler [8] and Wardowski [2], Altun et al. [3] introduced the concept of multivalued F-contraction mappings on complete metric spaces. They proved the following theorems: , be a complete metric space and → ( ) T X K X : be a multivalued F -contraction mapping, then T has a fixed point, where ( ) K X is a compact subset of a metric space (X d , ).
, be a complete metric space and → ( ) T X CB X : be a multivalued F -contraction mapping. Suppose that ∈ F Δ F also satisfies: Then T has a fixed point.
Acar et al. [9] extended the work of Altun et al. [3] and proved a fixed point theorem for generalized multivalued F -contraction mappings on complete metric spaces. On the other hand, the notion of metric space has been generalized in many directions. For example, Matthews [10] introduced the concept of partial metric spaces and proved a new fixed point theorem in partial metric setting.
Several researchers tried to extend the notion of multivalued F-contraction mappings to partial metric spaces in recent years. For instance, Paesano and Vetro [11] generalized the concept of multivalued F-contraction mappings in 0-complete partial metric spaces.
The following are some useful definitions and preliminaries required to establish the main results: Matthews [10] defined partial metric space as follows: , called the partial metric, such that for all ∈ x y z X , , the following axioms hold: Clearly, by (P1)-(P3), if ( ) = p x y , 0, then = x y. But, the converse is in general not true.
Each partial metric p on X generates a T 0 topology τ p on X whose basis is the collection of all open : , , p for all ∈ x X, and ε is a real number.
, be a partial metric space. Let A be any non-empty subset of the set X and x be an element of the set X. It is well known in [17] that ∈ x Ā, where Ā is the closure of A, if and only if ( , . Also, the set A is said to be closed in ( ) X p , if and only if = A Ā.
The following definition is of Matthews [10]: , be a partial metric space. Then, , is said to be complete if every Cauchy sequence { } x n in X converges with respect to the topology τ p to a point ∈ x X such that ( Bukatin et al. [14] proved the following lemma which will be useful in our future discussion: , be a partial metric space. Then the mapping × → [ ∞) p X X : 0 , s given by , defines a metric on X. Thus, associated with any partial metric space ( ) Bukatin et al. [14] also proved the following lemma: , be a partial metric space. Then: , is complete if and only if the metric space ( ) X p , s is complete.
In 1905, Pompeiu [18] defined the concept of distance between two closed sets, in the context of complex analysis, in his PhD thesis. Later in 1914, Hausdorff [19] considered all the basic concepts introduced by Pompeiu in his book Grundzüge der Mengenlehre, but in the general setting of a metric space, and adopted an alternative way to symmetrize the asymmetric distances ( ) , and commonly named Hausdorff metric as follows: , be a metric space and ( ) CB X denotes the collection of all non-empty bounded closed subsets of X.
, is a complete metric space.
Altun et al. [3] defined multivalued F -contraction as follows: be a mapping. Then T is said to be a multivalued F-contraction if ∈ F Δ F and there exists > τ 0 such that

3)
Aydi et al. [20] provided the following definition and lemma: On some fixed point theorems for multivalued F-contractions in partial metric spaces  153 denote the collection of all non-empty bounded and closed subsets of a partial metric space sup , : sup , : p . Then, the mapping H p is called the partial Hausdorff metric, on ( ) CB X p induced by the partial metric p.
p This paper intends to extend the notion of multivalued F-contraction mappings of Altun et al. [3] to a complete partial metric space.

Main results
We present the generalization of Definition 1.10 of Altun et al. [3] to partial metric spaces.
, be a partial metric space and The following lemma is very useful in our results: Lemma 2.2. Let X be a partial metric space and ( ) K X a compact subset of X. Let ⊆ ( ) A K X , and define a function , then the following statements are equivalent:  . Hence, f is continuous. □ Now, we will extend Theorem 1.3 to partial metric spaces: , be a complete partial metric space and let → ( ) T X K X : be a multivalued F -contraction, then T has a fixed point in X.
Since Tx 1 is compact, then there exists ∈ x Tx , , .  . From (2.5) the following holds for all ∈ n :   Next, we will present a fixed point theorem for multivalued F -contraction mapping satisfying (F1)-(F 3) plus an additional condition ( ) F 4 imposed on the function F as defined in Theorem 1.4.
The following theorem extends Theorem 1.4 to partial metric spaces: , be a complete partial metric space and let → ( ) T X CB X : p be a multivalued F -contraction, if F also satisfies: , , . Then, from (2.11), we have y Tx Then, from (2.12), there exists ∈ x Tx 2 1 such that, If ∈ x Tx 2 2 , then the proof is complete, otherwise by the same manner we can choose ∈ x Tx 3 2 such that, n , for all ∈ n and using (2.13) the following is true: . From (2.14) the following holds for all ∈ n ,  By Lemma 1.7 we obtain that for any To see this, we consider the following cases: