Some Fixed Point Results of Weak-Fuzzy Graphical Contraction Mappings with Application to Integral Equations

: The present paper aims to introduce the concept of weak-fuzzy contraction mappings in the graph structure within the context of fuzzy cone metric spaces. We prove some ﬁxed point results endowed with a graph using weak-fuzzy contractions. By relaxing the continuity condition of mappings involved, our results enrich and generalize some well-known results in ﬁxed point theory. With the help of new lemmas, our proofs are straight forward. We furnish the validity of our ﬁndings with appropriate examples. This approach is completely new and will be beneﬁcial for the future aspects of the related study. We provide an application of integral equations to illustrate the usability of our theory.


Introduction
The theory of fixed points centers on the process of solving the equation of the form T(µ) = µ. We discuss a new concept that overlaps between metric fixed point theory and graph theory. This new area yields interesting generalizations of the Banach contraction principle [1] in metric spaces endowed with a graph. The fixed point techniques have received considerable attention due to their broad applications in many applied sciences to solve diverse problems in engineering, game theory, physics, computer science, image recovery and signal processing, control theory, communications, and geophysics.
In 1965, Zadeh [2] introduced the fuzzy sets. Kramosil and Michálek [3] introduced the notion of fuzzy metric space. George and Veeramani [4] modified the description of fuzzy metric spaces due to Kramosil and Michálek. Gregori and Sapena [5] introduced the concept of fuzzy contractive mappings. On the other hand, the results were applied to metric spaces provided with a partial order by Ran and Reuring [6]. To find a solution to some special matrix equations was also one of the great charms of the fixed point theorists. To this end, the work of EL-Sayed and Andre' [7] was a pioneer one. Later on, Nieto and Rodriguez Lopez [8] extended the work of [6] and applied their results to solve some differential equations.
One natural question is whether contractive conditions may be found that indicate the presence of a fixed point in an entire metric space, but do not imply continuity?
A mapping T : U → U, where (U, d) is a metric space is said to be a contraction map [1], if there exists 0 < α < 1, such that for all µ, ν ∈ U; d(Tµ, Tν) ≤ αd(µ, ν) The following result was defined by Kannan [9], in which the above question was answered affirmatively. If T : U → U is a complete metric space where (U, d) satisfies inequality: where 0 < α < 1 2 , then a unique fixed point will be in T. The mapping T need not to be continuous, the references therein (see [9,10]).
In 2008, Jachymiski [11] initiated a novel idea in fixed point theory, where the author evoked graph structure on metric spaces instead of order structure. According to this concept, Banach's contraction condition will be satisfied only for the edges of the graph. If G is a directed graph and E(Ĝ) is the set of its edges then the contraction condition is: d(Tµ, Tν) βd(µ, ν), β ∈ (0, 1), ∀(µ, ν) ∈ E(Ĝ) Some noteworthy efforts done on this concept can be seen in [12][13][14][15][16][17]. Starting from these results, we aim to make a methodical study of fixed point theorem in fuzzy metric spaces endowed with a graph.
In 2016, Usman [18] generalized a new class of F-contractions in b-metric spaces and to obtain existence theorems for Volterra-type integral inclusion. In 2017, Kamran et al. [19] introduced a new class of comparison functions to present some fixed point theorems with an extended b-metric space. For various applications of fixed points in metric spaces (see [20][21][22][23]).
In this paper, we introduce the weak-fuzzy contractions conditions from fuzzy cone metric spaces and prove some fixed point results for such mappings in the sense of Grabiec [24]. Without taking the continuity of the mapping T into consideration, our results unify and enrich the results of Jachymski [11], Gregori and Sapena [5] in the framework of fuzzy metric spaces. Our proofs modify the findings in existing literature. We call this contraction weak-fuzzy graphical contraction (wfgc) and discuss some slip-ups in this context. Throughout our discussion, we shall write fuzzy cone metric space as FCM-space in short.
The article is organized into five sections. In Section 2, we define the preliminaries and some basic definitions which help readers to understand our results easily. In Section 3, we establish some novel results of complete FCM-space with a weak-contraction has a unique fixed point endowed with a graph. We define the related definitions before the main result. In Section 4, we validate the obtained results via the existence of solution of an integral equation in graphical mapping. Few interesting examples are provided to explain our results. Finally, in Section 5, we discuss the conclusion and future directions of our work.
The classical examples of continuous T-norm are T L , T P and T M defined as; (a) The Minimum operator T M (a, b) = min{a, b}; (b) The product operator T P (a, b) = ab; (c) The Lukasiewicz's norm T L (a, b) = max{a + b − 1, 0}.

Definition 2 ([25])
. Let E be a real Banach space, a subset P of E is called cone if; (i) P = ∅, closed and P = {θ}, where θ is the zero element of E; (ii) If a, b ≥ 0 and µ, ν ∈ P, then aµ + bν ∈ P; (iii) If both µ, −µ ∈ P, then µ = θ. The cone P is called normal if there is a number k > 0 such that for all µ, ν ∈ E Throughout in our discussion we suppose E is a Banach space, P is a cone in E with int(P) = φ and ≤ is a partial ordering with respect to P.
The following definition of fuzzy metric space was introduced by George and Veeramani [4]. We are concerned with this concept of fuzzy metric space.

Example 1.
A function M r be defined as; ,ḿ > 0, Then (M r , ·) is a fuzzy metric on U. As a particular case if we take g 1 (ς) = ς n , n ∈ N, and m = 1, M r (µ, ν, ς) = ς n ς n + d(µ, ν) A well-known standard fuzzy metric is obtained for n = 1. If we are using g 1 as a constant function, g 1 (ς) = k > 0, andḿ = 1, we get and so (M r , ·) is a standard fuzzy metric on U.
is monotone non-decreasing and continuous and ψ(t) = 0 if and only if t = 0.
The following "Fuzzy cone Banach contraction theorem" is obtained in [27]. Recently Choudhry [26] have introduced the following weak-contractive condition in metric spaces.
It is clear ς-uniformly continuous and if mapping T is a fuzzy contractive mapping similar to those in [11,17], following the principles of the graphs.
Let ∆ denote the diagonal of the Cartesian product U × U. Consider the graphĜ so that the collection of its vertices V(Ĝ) coincides with U, and the set of its edges E(Ĝ) contains all its loops, i.e., E(Ĝ) ⊇ ∆. We assume thatĜ has no parallel edges. Therefore, we have The Ĝ character refers to the undirected graph obtained fromĜ ignoring the edge path.
In fact, it would be more convenient for us to consider Ĝ as a graph that is symmetrical to the set of its edges. According to this convention, If µ and ν are vertices in a graphĜ, then a path inĜ from µ to ν of length l is a sequence If there is a path between any two vertices ofĜ, the graphĜ is called connected. A graphĜ is weakly connected if Ĝ is connected. The subgraph consists of all edges and vertices which are contained in some path ofĜ. In this case νRω if there is a path inĜ from ν to ω.

Fixed Point Results of Weak-Fuzzy Graphic Contractions
We now determine that a weak-contraction has a unique fixed point endowed with a graph in a complete FCM-space. Before the main outcome, we define the related definitions. We assume that U is a non-empty set in this section,Ĝ is a graph directed to V(Ĝ) = U, and E(Ĝ) ⊇ ∆. First, in the setting of fuzzy metric spaces, we define the Cauchy equivalent sequence and Weak-fuzzy contraction.
Definition 12. Let (U, M r , * ) be a fuzzy metric space andĜ be a graph. Two sequences (µ n ) n∈N and (ν n ) n∈N in U are said to be Cauchy equivalent if each sequence is Cauchy and lim n→∞ M r (µ n , ν n , ς) = 1, ∀ς > 0.

Remark 2.
If T is weak-fuzzy graphical contraction mapping, then it is a fuzzy contraction of botĥ G −1 -fuzzy and Ĝ -fuzzy. Definition 14. Let (U, M r , * ) be a fuzzy metric space and T : U → U be a mapping. We denote the n th iterate of T on µ ∈ U by T n µ and T n µ = TT n−1 µ, ∀ n ∈ N with T 0 µ = µ. T is called a Picard Operator (PO), if T has a unique fixed point u and T is called Weakly Picard Operator (WPO) if there exists a fixed point u µ ∈ U such that lim n→∞ M r T n µ, u µ , ς = 1, for all ς > 0. Note that every Picard Operator is Weakly Picard Operator. Furthermore, the fixed point of (WPO) need not be unique. We will denote the set of all fixed points of T by Fix- The following lemma will be useful in this sequel. Lemma 3. Let T : U → U be a weak-fuzzy contraction, then given µ ∈ U and ν ∈ [µ] Ĝ , we have lim n→∞ M r (T n µ, T n ν, ς) = 1, ∀ς > 0.
That is by property of ψ it is contradiction unless that is µ 1 = µ 2 . Hence T has a unique fixed point. This completes the proof of uniqueness of the fixed point. Letting n → ∞ in the above inequality we obtain M r (µ * , Tµ * , ς) = 1, ∀ς > 0. Thus, Tµ * = µ * ,, i.e., µ * ∈ [µ] Ĝ is a fixed point of T and so by Theorem 3 T| [µ] Ĝ is a Picard Operator.
To prove (B) : Let U T = ∅ and graphĜ is weak connected then [µ] Ĝ = U, ∀µ ∈ U T and so by (A), mapping T is a Picard Operator (PO).
To prove (D) : If T ⊆ E(Ĝ), then (µ, Tµ) ∈ E(Ĝ) ∀ µ ∈ U, so U = U T . The result follows from (A). Example 2. Let (U, M r , * ) be a complete fuzzy cone metric space and let U = , * be a minimum norm. Let M r be defined by Obviousely, T(µ) and ψ are continuous functions. Then we have From the above inequality we conclude that (5) is satisfied. Thus, mapping T is a weak-fuzzy contraction.
LetĜ be a directed graph with V(Ĝ) = U and E where U o and U e be two subsets of even numbers and odd numbers from the set of . Then it is easy to see that mapping T is weak-fuzzy graphical contraction (wfgc).

Conclusions
We have introduced the concept of weak-fuzzy graphical contractions in the framework of fuzzy cone metric spaces, which is a new expansion to the current writing in the context of fuzzy cone metric spaces. The obtained results revamp and extend some wellknown results in the existing state-of-art, by relaxing continuity of the mapping involved. With the help of some novel lemmas, we provide that our proofs are straight forward. Non-trivial examples are presented to show the novelty of the established results. We conclude that research in fixed point theory with the graphical structure of contraction mappings is a field of active research in seeking the presence and uniqueness of fixed point for mappings that fulfill various contractive conditions. Any interested researchers may use this opportunity to carry out their future research in this field.