Extension of Kirk-Saliga Fixed Point Theorem in a Metric Space with a Reflexive Digraph

We extend the result of Kirk-Saliga and we generalize Alfuraidan and Khamsi theorem for reflexive graphs. As a consequence, we obtain the ordered version of Caristi’s fixed point theorem. Some concrete examples are given to support the obtained results.


Introduction
Fixed point theory is one of the most useful tools in mathematics; it is used to solve many existence problems such as differential equations, control theory, optimization, and several other branches (for the literature see [1]).The most well-known fixed point result is Banach contraction principle [2]; it is famous for its applications, proving the existence of solution of integral equations by converting the problem to fixed point problem (see [3]).Recall that a point  ∈  is called a fixed point for a map  :  →  if  = .Due to its importance, this theorem found a number of generalizations and extensions in many directions; for more details see [4] and the references therein.In 1976, Caristi (see [5]) gave an elegant generalization of Banach contraction principle, where the assumption that " :  →  is continuous" is dropped and replaced by a weak assumption.Since then, various proofs, extensions, and generalizations are given by many authors (see [6][7][8]).It is worth mentioning that Caristi's fixed point theorem is equivalent to the Ekeland variational principle [8].Also, it characterizes the completeness of the metric space as showed by Kirk in [9].Among those generalizations, there is Kirk-Saliga fixed point theorem (see [10]) which states that any map  :  →  has a fixed point provided that  is complete metric space and there exist an integer  ∈ N and a lower semicontinuous function  :  → [0, ∞) such that  (, ) ≤  () −  (  ) (1) and () ≤ () for any  ∈ .For more on the latter result, one can consult [11].
Recently, Ran and Reurings [12] extend the Banach contraction principle in the context of partially ordered set where the contraction is restricted to the comparable elements which allowed them to give a meaningful application to linear and nonlinear matrix equations.Moreover, Nieto and Rodríguez-López in [13] have weakened the continuity assumption using a more suitable condition where the order is combined with the topological properties.For more details, one can consult [14,15].Also, in [16] Alfuraidan and Khamsi gave an analogue version of Caristi's fixed point theorem in the setting of partially ordered metric space where the inequality holds only for comparable elements.However, the new approach in their work is mixing the concept of the reflexive acyclic digraph with fixed point results.In this article, we discuss an extension of Kirk-Saliga result and we generalize Alfuraidan and Khamsi theorem for reflexive graphs.As a corollary, we obtain the ordered version of Caristi's fixed point theorem.Some concrete examples are given to support the obtained results.Throughout this paper we denote by N the set of all integers and by N * the set of all positive integers.

Preliminaries
We start by recalling some basic notions on graphs borrowed from [17].A metric space (, ) endowed with a digraph  such that () =  is denoted by (, , ).The following notion of regularity is borrowed from Alfuraidan and Khamsi in [16] that considered it for posets.Definition 4. Let (, , ⪯) be a partially ordered metric space.We say that  satisfies the condition (OSC) if for any decreasing sequence {  } ⊆  that is convergent to  ∈ ,  = inf{  :  ∈ N}.
In the setting of digraphs, the analogue of the infimum of chain may be stated as follows.
Definition 5. Let (, , ) be metric space endowed with a digraph.We say that  satisfies the condition (OSCL) if for any sequence {  } ⊆  that is convergent to  ∈  and for all  ∈ N,  +1 ∈ [  ]  ,  ∈ [  ]  for all  ∈ N and if there exists  ∈  such that  ∈ [  ]  , for all  ∈ N, then  ∈ []  .Remark 6.Let (, , ⪯) be a partially ordered metric space.Let  ⪯ be the digraph associated with the order ⪯ (see [16]).One can see that Under the above observations, the (OSCL) property is reduced to the (OSC) condition.
Let  be the first transfinite ordinal and let Ω be the first uncountable transfinite ordinal. is the order type of N "the set of integers" and Ω is the order type of R the set of real numbers.Note that, for each  < Ω,  is countable.
Proposition 7 (see [11]).The following is valid: The ordinal Ω cannot be attained via sequential limits of countable ordinals.That is if {  } is an ascending sequence of countable ordinals, then the ordinal is countable too.
(ii) Each second kind countable ordinal is attainable via such sequences.In other words: if  < Ω is of second kind (ordinal limit), then there exists a strictly ascending sequence {  } of countable ordinals with property (3).
The following result is needed throughout this work; for the proof see [18,Proposition A.6,pp. 284].Proposition 8. Suppose that a sequence {  } ∈Ω ⊆ R is bounded and either nonincreasing or nondecreasing.Then there exists  ∈ Ω such that   =   for all Ω >  ≥ .
We conclude this section by the following useful definitions.Definition 9. Let (, , ) be metric space endowed with a digraph,  ∈ N and  :  → [0, +∞[ a lower semicontinuous function.Let  :  →  be a self-mapping.We say the following: (2)  is a -Caristi mapping if for all  ∈ , (3)  is a -Kirk-Saliga mapping if for all  ∈ ,

Main Results
Theorem 10.Let (, , ) be a complete metric space endowed with a reflexive digraph satisfying the (OSCL) condition.Let  :  →  be a -monotone and -Kirk-Saliga mapping.If there exists an element  0 ∈  such that  0 ∈ [ 0 ]  , then  admits a fixed point in .
Proof.If  = 0 then  = , for all  ∈  such that  ∈ []  .Assume that  ≥ 1 and consider the function  defined from  into [0, +∞[ by The idea of the proof is to construct a transfinite orbit Then we get for each  ≥  0 that and for all  ∈ {0, 1, . . .,  − 1} Since  is lower semicontinuous, we get From (), we have   ∈ [ By passing to limit superior in inequality (13), it follows that Hence, () holds.
Case 2. Assume that  =  + 1 is an immediate successor; we have shown above that () holds.Then   =   ∈ [  ]  and by assumption we get and for all  ⪯ , we have The triangle inequality implies that for each  ⪯ , which completes the proof of () in both cases.Thus, the orbit (  ) ∈Ω is well constructed.Since {(  )} is nonincreasing on {  } and Ω is uncountable, there must exist  0 ∈ Ω such that (  ) is constant for all  ⪰  0 .From ( 0 + 1), we get Hence, We support our result by giving an example of a mapping which is -Kirk-Saliga mapping, for some integer  > 1, but not -Caristi.
(iii)  is -kirk-Saliga mapping in  with  = 3.Indeed, for all  ∈   , but  is not -Caristi mapping, since and  admits a fixed point in  which is 0.
If we remove the (OSCL) property, we are not certain that the fixed point will be obtained.Let us illustrate that by this counterexample.
Example 12. Replace in the above example the digraph  by the digraph   represented in Figure 2, where and we consider the mapping  :  →  defined as follows: One can see that   satisfies (OSC) property but does not satisfy the (OSCL), since 1/2  → 0 and for all  ∈ N, 0 ∈ [1/2  ]   and 1 ∈ [1/2  ]   but 1 ∉ [0]   .The mapping  satisfies all others conditions of Theorem 10 but has no fixed point in .
We conclude this work by a discussion about preordered sets.
Let (, ⩽) be a preordered set; that is, the binary relation "⩽" is reflexive and transitive.
An analogue version of Theorem 10 in the setting of the preordered metric spaces may be stated as follows.

2 International
Journal of Mathematics and Mathematical Sciences the function  :  → [0, +∞[ defined by

Figure 1 :
Figure 1: Graph  (the loops and the isolated vertices are not represented).
Definition 1.Let  be an arbitrary set.(i)Adirected graph, or digraph, is a pair  = (, ),where  is a subset of the Cartesian product ×.The elements of  are called vertices or nodes of  and the elements of  are the edges also called oriented edges or arcs of .An edge of the form (V, V) is a loop on V.