ε -Compatible Map and New Approach for Common Fixed Point Theorems in Partial Metric Space Endowed with Graph

In 2016, Muralisankar and Jeyabal introduced the concept of ε − Compatible maps and studied the set of common fixed points. They generalized the Banach contraction, Kannan contraction, Reich contraction and Bianchini type contraction to obtain some common fixed point theorems for ε − Compatible mappings which don’t involve the suitable containment of the ranges for the given mappings in the setting of metric spaces. Motivated by this new concept of mappings, we establish a new approach for some common fixed point theorems via ε -compatible maps in context of complete partial metric space including a directed graph G=(V,E). By the remarkable work of Jachymski in 2008, we extend the results obtained by Muralisankar and Jeyabal in 2016. In 2008, Jachymski obtained some important fixed point results introduced by Ran and Reurings (2004) using the languages of graph theory instead of partial order and gave an interesting approach in this direction. After that, his work is considered as a reference in this domain. Sometimes, there are some mappings which do not satisfy the contractive nature on whole set M(say) but these can be made contractive on some subset of M and this can be done by including graph as shown in our Example 2.6 which is provided to substantiate the validity of our results.


Introduction and Preliminaries
French mathematician Maurice Frechet in 1906 introduced the concept of metric space. By following the work of Frechet, several generalization of metric space came out by several authors and one of the generalization is partial metric space that introduced by Mathews [14] to obtain appropriate mathematical models in the theory of computation.
In the research of modern mathematics, the theory of fixed point is one of the most powerful tools. By the fixed point theorems, we mean that Theorems concerning the existence and properties of fixed point. The classical theorem formulated by Stephen Banach in 1922, in non-linear functional analysis which state that a contraction mapping on complete metric space (M,d) has a unique fixed point and also conveys the method how to compute the fixed point through iteration process. This famous Banach principle also gave birth to many other remarkable fixed point theorem such as Ciric (1974), Rhoadas (1996), Kirk (1983), Park (1998), Nulder (1969) and etc.
Study of common fixed point has been at the centre of high-spirited research activity. In 1976, Jungck [4] generalized the Banach contraction principle for more than one mapping to obtain common fixed point theorems for commuting mappings such that one of them is continuous. His theorem states that, Theorem 1.1 [4]. Let (M, d) be a complete metric space and let ᶂ and ᶃ be commuting self-maps of M satisfying the conditions: i) ᶂ M ⊆ ᶃM ii) d(ᶂ , ᶂƴ) ≤ d(ᶃ , ᶃƴ), for all , ƴ ∈M and some ∈ (0, 1). If ᶃ is continuous then ᶂ and ᶃ have a unique common fixed point.
Onwards, Jungck and Rhoades [3] have introduced the concept of compatibility of maps which is weaker condition than weak commutativity and obtained some common fixed point results for these kind of mappings. Generally, common fixed point theorem involves conditions on commutativity, continuity, completeness and suitable containment of ranges of the involved mappings apart from an propitious contraction condition and investigators in this area are aimed at weakening one or more of these conditions. In this direction, one more step is taken out by other researchers to find common fixed point via -compatible map which doesn't involve suitable containment of ranges of the involved mappings.
In current trend of research, the study of fixed point theory involving graph inhabits/occupies a noteworthy place in many fields. In 2008, Jachymaski [9] was studied first the Banach Contraction Principle on a metric space with a graph. The results obtained byJachymaski generalize the many recent results of other researchers obtained in a partially ordered metric space. According to Jachymaski, using the language of graph theory instead of a partial ordering is more convenient. A lot of work in fixed point theory in various spaces with graph has been done by changing some conditions on mapping and spaces, by redefining some definitions on mappings like G-contraction, G-kannan, G-monotone non-expansive, -contraction, G-graphic contraction, and etc. Now, we recall some basic definitions, notations and relevant concepts to partial metric space and common fixed point theorem.
Let M be partial metric space with partial metric ϸ, let sequence { n} in M and ∈M. The mapping are said to be commutating if ᶂᶃ =ᶃᶂ , ∀ ∈M. ii) The mapping is compatible if d(ᶂᶃ n, ᶃᶂ n) → 0 as n→ ∞ whenever{ n} is a sequence in M such that ᶂ n, ᶃ n → t , for some t M and weakly compatible if they commute at their coincidence point i.e, if ᶂ = ᶃ ⇒ ᶂᶃ = ᶃᶂ , ∈M. iii) The mapping ᶂ and ᶃ are said to be semi-compatible mapping if { n} be a sequence in M such thatᶂ n → , ᶃxn → implies ᶂ ᶃ n → ᶃ and both mappings commute at their coincidence point and semi-weakly compatible if they commute at the fixed point of either ᶂ or ᶃ i.e, If ᶂ ᶚ = ᶚ or ᶃ ᶚ = ᶚ then ᶂᶃ ᶚ = ᶃᶂᶚ.
Taking limit n→ ∞ and by (1), (2), we get lim Next, Applying limit n, m→ ∞ and by Lemma 1.7, (3), it follows that Therefore by Lemma 1.7., it is also Cauchy sequence in partial metric space (M, p).
Throughout this paper, suppose triplet (M, ϸ, G) be a partial metric space, G be a directed graph where the set V(G)=M and the edge set E(G) contains no parallel edges such that ∆⊆ E (G), ∆ denote the diagonals of the Cartesian product M×M. ̃ means the undirected graph which is obtained from G by ignoring the direction of edges and E(̃) = E(G) ∪ E( −1 ).
Inspired by the some recent work on metric spaces with a graph (see [1,2,[7][8][9]15]), we establish some interesting common fixed point results from metric space to partialmetric space through a new approach. In this paper, we obtain some common fixed point theorems in a complete partial metric space endowed with graph using -compatible maps.

Main Results
In this section, we present some new type of common fixed point theorems by using the concept of −compatible in context of partial metric space and also obtain some examples to understand the usability of the definitions and results. For any > 0, whatever we take there always exists = 0, =1 such that |f -g |= 0 and |fg -gf | = 0, for any function where is continuous function at 0. That is, the pair f and g is -compatible. For =3, f = g = 9, but |fg -gf |≠ 0. Hence, the pair f and g are not compatible and also not weakly compatible. Example 2.3. From the Figure 1., we can easily observe that any two parallel graphs on R never are -compatible since they have the fixed distance so that they never come to close at any point therefore, they are not -compatible but they are always compatible since these graph are translation maps.
For an example, if we take f = + 3 and g = + 7 then f and g are compatible, infact commutating maps are not -compatible because |f -g |= 4 for all M and if we take any < 4, we couldn't find an element M such that d(f , g ) < i.e, this is the necessary condition for -compatible mappings.
Then f and g have common fixed point. Furthermore, if , * are fixed points of f and g with ( , *) E(̃) then f and g have unique common fixed point.
Or in another way, we can also prove f and g are -compatible since f(0) = g(0)=0 and fg(0) = gf(0) = 0 i.e, there exist a coincidence point and they commute at coincidence point.
Thus all the conditions of theorem 2.4 are satisfying and therefore, 0 is common fixed point of self-mappings f and g.
Since a+b+c<1 implies that (a+c) < (1-b) Now from (7), we have (1 − )+2b which implies b > 1 that is not possible. Thus, contractive condition doesn't hold for = 0, ƴ=1 and the given problem has no common fixed point by applying Theorem 2.1 with contractive condition on whole set. But this is possible if we restrict the contractive condition on some subset of [0,1] and this is possible by inducing the graph.
Then f and g have common fixed point. Moreover, if u, v are fixed point of f and g with (u,v) E(̃) then f and g have unique common fixed point.

Conclusions
In this paper, we presented some common fixed point theorems with graph for − compatible maps which doesn't involve the containment of the ranges for the given mappings. Generally, we see that common fixed point results for more than one mapping existing in the literature involve the suitable containment of the ranges for the given mappings which is a strong assumption in its own. In addition, we presented some examples to demonstrate the definitions and our results.