CC BY
20Probl. Anal. Issues Anal. Vol. 13 (31), No3, 2024, pp. 79-100
DOI: 10.15393/j3.art.2024.15870
79
UDC 517.98
M. Rafiqüe, T. Nazir, H. Kalita
FIXED POINTS OF FUZZY MULTIVALUED F-CONTRACTIVE MAPPINGS WITH A DIRECTED GRAPH IN PARAMETRIC METRIC SPACES
Abstract. In the structure of a parametric metric space accompanied by directed graph, we introduce the idea of fuzzy multivalued F-contractive mappings. Results related to the existence of common fuzzy fixed points are introduced. The proved results are supported by an example. Our results bring together, sum up, and supplement different familiar related results in the literature. We hope that the acclaimed results in our work will encourage new analysis aspects in fixed-point theory and parallel hybrid models in the literature of fuzzy mathematics supplemented with a graph.
Key words: F-contraction, fuzzy set, fuzzy multivalued mapping, fuzzy fixed point
2020 Mathematical Subject Classification: 46S40, 47H10, 54H25
1. Introduction A trouble in mathematical modeling of workable incidents is the indefiniteness persuaded by our inability to classify events with ample precision. The crisp set theory cannot manage imprecisions. As an effort to deal with problems of insufficient data, fuzzy sets [31] were considered, which gave birth to fuzzy set theory. It provides suitable mathematical tools for handling information with no statistical uncertainty. As a result, fuzzy set theory has gained much regard because of its applications in multiple domains like management sciences, engineering, environmental sciences, medical sciences, and other fields. The mean concepts of fuzzy sets have been modified and upgraded in different aspects; for example, see [7], [13], [18]. In 1973, Markin [23] commenced the investigation of fixed points for multivalued contractions and nonexpansive maps utilizing the Pompeiu-Hausdorff metric.
© Petrozavodsk State University, 2024
In 1981, Heilpern [14] commenced the study of fuzzy multivalued maps and obtained a fuzzy correspondent of Nadler's fixed-point theorem [24]. Later, many authors reviewed the existence of fixed points of fuzzy multivalued maps, for instance, Al-Mazrooei et al. [6], Azam et al. [8], [9], Bose and Sahani [12], Mohammed [16], Mohammed and Azam [17], Qiu and Shu [26], and so on. In 2003, Rus [28] presented the idea of ^-multivalued mappings by upgrading the idea of K-multivalued mapping presented by Latif and Beg [22] in 1997. In 2009, Abbas and Rhoades [4] worked on ^-multivalued mappings and introduced the idea of a generalized ^-multivalued mappings to establish some common fixed-point consequences for such mappings.
In 2004, Ran and Reurings [27] first examined the existence of fixed points in partially ordered metric spaces and afterwards by Nieto and Lopez [25]. In 2007, Jachymski [19] established some generalized fixed-point results in metric fixed-point theory by using graph structure on metric spaces instead of order structure.
In 2012, Wardowski [30] generalized the Banach contraction principle results by introducing an advance contraction named as F-contraction. In 2013, Abbas et al. [1] used the concept of F-contraction mapping with respect to a self mapping on a metric space and acquired some common fixed-point results. In 2013, Sgroi and Vetro [29] got some fixed-point results for F-contraction multivalued maps in metric spaces (see also [5]).
In 2013, Abbas and Nazir [3] acquired some fixed-point consequences for power graphic contraction pair on a metric space equipped with a graph. In the structure of parametric metric space enriched with directed graph, this work aims to validate some common fuzzy fixed-point consequences for fuzzy multivalued generalized graphic F-contraction mappings. It is important to note that our consequences have been proved without Hausdorff metric, in addition our consequences extend and bind together different parallel results in the existing literature ( [21], [22], [28], and [29]).
2. Preliminaries. Denote the set of natural numbers, the set of positive real numbers, and the set of real numbers by the symbols N, R+, R, respectively. Let us describe some definitions and results needed for our main result.
Definition 1. [15] Let Y be a non-empty set. A mapping f: Y x Y x (0, 8) —> [0, 8) is called a parametric metric if the following conditions are satisfied:
1) f (k1, k2, 9) = 0, for all 9 > 0 if and only if k1 = x2;
2) f (k1, k2, 9) = f (x2, xi, 9) for all xi, x2 e Y and 9 > 0;
3) f (k1, x3,9) ^ f (k1, k2,9) + f (k2, x3,9) for all xi, k2, k3 e Y and 9 > 0;
then the pair (Y, f) is called a parametric metric space.
Example 1. [15] Let the set of all functions g, h : (0, 8) ^ R be denoted by X. Define f : Y x Y x (0, 8) ^ [0, 8) by f (h,g,9) = \h(9) - g(9)\ for all h,g e X and all 9 > 0. Then f is a parametric metric on Y and the pair (f, Y) is a parametric metric space.
Definition 2. [15] Consider a parametric metric space (Y,f) having a sequence {nn\; the following definitions are needed for our results:
1) if lim f (xn, k,9) = 0, then {xn} is called convergent to x e X, written as lim xra = k, for all 9 > 0;
2) if for all 9 > 0 we have lim f (xn, xm,9) = 0, then {xn} is called Cauchy sequence in Y;
3) a parametric space (Y, f) is called complete if every Cauchy sequence converges in it.
Definition 3. [15] Let (Y, f) be a parametric metric space and T : Y ^ Y be a mapping. T is said to be a continuous mapping at a in Y, if for any sequence {an} in Y such that lim an = a, then lim Tan = Ta.
Definition 4. A graph G consists of two sets V = {v1,v2,v3),... } and E = {e1,e2,e3,... }; the elements of V are called vertices while the elements of E are called edges. In the literature on the graph theory, the set V(G) represents the vertex set of G and E(G) represents the edge set. Let {v1,v2} be an edge of graph G, as {v1,v2} is a 2-objects set, so we may write {v2,v1} instead of {v1,v2}.
Following Jachymski [19], consider a parametric metric space (Y,f) and a directed graph G with vertices V(G) coinciding with Y. E(G) represents the set having all edges and loops also E(G) 3 A, where A represents the diagonal of Y x Y. Furthermore, E*(G) represents a set having only edges of the graph G. Also, it is supposed that there are no multiple edges in the graph G and the pair (V(G), E(G)) can perceive the graph G.
Definition 5. [19] In a parametric space (Y, f), an operator "q : Y — Y is said to be a Banach G-contraction if
1. for every a,b e X with (a,b) e E(G), we have (rq(a),'q(b)) e E(G), that is, rq preserves edges.
2. There exists 7 e (0,1), such that for all a,b e X with (a, b) e E(G) we have f (rq(a), rq(b)) ^ jf (a, b), that is, rq decreases weights of edges of G.
A (directed) path in a graph G of length I e N between the vertices k1 and k2 is a finite sequence {nn} (where n e {0,1, 2,... ,1}) of vertices, such that k0 = k1, k = k2 and (kj-1, Kj) e E(G) for j e {1, 2,..., I}.
Note that a graph G is connected if there is a (directed) path between every pair of vertices, and it is weakly connected if G is connected, where G represents the undirected graph obtained from G by neglecting the direction of edges. The graph obtained from G by reversing the direction of edges is denoted by G~l. Thus,
E (G-1) = {(k1, k2) e Y x Y : (k2, k1) e E (G)} .
It is important to note that G is such a directed graph that the set of its edges is symmetric, hence we can write:
E(G) = E(G)u E(G-1).
If the set of edges of a graph G is symmetric, then, for k e V(G), [k]g represents the class of equivalence of the relation R defined on V(G) by the rule:
k2Rk3 if there is a directed path in G from k2 to k3 .
If 0 : Y — Y is an operator, set:
Y^ := {k e Y: (k,^(k)) e E(G)}.
Jachymski [20] obtained the following property:
A graph G has a property (P) : for every sequence {_xn} in Y, if xn — k, such as n — 8 and (xn, Kn+1) e E(G), then (xn, k) e E(G).
Theorem 1. [20] Let (Y, f) be a complete parametric metric space and let G be a directed graph, such that V(G) = Y. Let E(G) and the triplet (Y, f, G) have property (P) and rq : Y — Y be a G-contraction. Then the following statements hold:
(i) Yv ^ 0 if and only if rq has a fixed point;
(ii) if Yv ^ 0 and G is weakly connected, then r] is a Picard operator;
(iii) for any k e Yv, rq\[K] ~ is a Picard operator;
(iv) if Yv c E(G), then rq is a weakly Picard operator.
See Berinde [10,11], for further details of Picard operators. [30] Denote by r the sets of all mappings F : R+ — R that satisfy the following conditions:
( Fl) For all x^ x2 e R+, such that xl < x implies that F(xl ) < F(x2), implies F to be strictly increasing.
( F2) For any sequence {xn} of positive real numbers, lim xn = 0 and
lim F (xn) = —8 are equivalent.
( F3) There exists h e (0,1), such that lim xhF(x) = 0.
Recall that an ordinary subset B of Y is determined by its characteristic function xb, where xb : B —> {0,1} is defined as
/1, if b eB, xb(b)- {0, if b pB.
The value xb(b) defines whether an element belongs to B or not. This suggestion is employed to establish fuzzy sets by permitting an element x e A to correspond to any value in the interval [0,1]. Thus, a fuzzy set in Y is a function with domain Y and values in [0,1] = F The set of all fuzzy sets in Y is denoted by Ir. If B is a fuzzy set in Y, then the function value B(x) is called the grade of membership of x in B. The a-level set of a fuzzy set B is denoted by [Band is defined as follows:
it
mi J t-K p Y:B(x)> 0}, if« = 0, L Ja Ux p Y : B (x) ^ a}, if« e (0,1],
where X represents the closure of set X.
Example 2. Let Y be the set of all individuals in a certain town, and
A = {x e Y| x is an old person}
Then, it is more appropriate to identify an individual to be an old person by a membership function A on Y, because the term «old» is not well defined.
Example 3. Let Y = {1, 2, 3, 4} be equipped with the usual metric. Let T : Y ^ Ir be a fuzzy multivalued map, that is, for each k e Y, T(k) : Y ^ [0,1] is a fuzzy set. For instance, for some a e (0,1], we may define one of the fuzzy sets T(1) by
if k = 1, if k = 2, if k = 3, if k = 4.
Let (Y,f) be a parametric metric space. The family of all nonempty closed subsets of Y is denoted by Pci(Y). Define
FPcl (Y) = {A e Ir : [A]« e Pcl(Y)} for some a e (0,1].
A point k in Y is a fuzzy fixed point of a fuzzy mapping T: Y ^ Ir iff k e [TK]a. Fuz(T) denotes the set of all fuzzy fixed points of a fuzzy mapping T.
Suppose T1, T2: ^ Fpcl (Y) be fuzzy mappings. Set
xtut2 := {k e Y: (k, vh) e E(G) where vH e [Ti(K)]a x [T2(k)]^},
for some a,P e (0,1], where a and ¡3 need not be equal.
In the further discussion, we will take a,0 e (0,1], where a and ¡3 need not be equal, so in the rest of the paper it will not be minded if a and ¡3 are not mentioned to be in (0,1].
Now we give the following definition in the setup of parametric metric space:
Definition 6. Let T1,T2: Y ^ Fpcl (Y) be two fuzzy multivalued mappings in parametric metric space (Y, f). Assume that for every vertex k in G and for any vK e [Ti(K)]a, for i e {1, 2} we have (k,vk) e E(G). A pair (T1,T2) is said to form:
(i) a fuzzy graphic F1-contraction, if for every k1 , k2 e Y with (Ki, K2) e E(G) and vH1 e [Ti(Ki)]a, there exists vH2 e [TjK)]?
T(!)(«) =
a,
for i,j e {1, 2} with i ^ j, such that (vK1 ,vK2) e E*(G) and
r + F (f (vki ,Vk2 ,0)) < F (Mi(Ki, K2; vHl ,vH2,0)), (1)
holds, where r denotes a positive real number and
Mi(Ki , K2; Vki ,Vk2 ,d)= max j f (Ki, K2,d),f (ki,Vki ,0),f (k2,Vk2 ,9),
f (K1,VK2 ,d) + f (k2 ,vki ,0) j
(ii) a fuzzy graphic F2-contraction, if for k1 and k2 e Y with (k1, k2) e E(G) and uK1 e \Ti(K1)]a there exists uK2 e [Tj(k2) and i,j e {1, 2} with i ^ j, such that (uK1, uK2) e E* (G) and we have
T + F (f (Uki ,Uk2 ,0)) ^ F (M2(Ki, K2; Uki ,Uk2 ,0)), (2) where r is a positive real number and
M2(Ki, K2; uki ,uk2 ,0) = aif (Ki, K2,6) + Pif (ki,uki ,0) +
+ lif (K2,UK2 ,0) + Sif (Ki ,Uk2 ,0) + 82f (K2 ,Uki ,9) , and p1,j1,81,82 ^ 0, ^ ^ 82 with a1 + (31 + j1 + + 82 ^ 1.
It is important to note that for different selections of mappings F, one may obtain different conductivity conditions.
Remember that a fuzzy map T: Y ^ Fpcl (Y) is called an upper semi-continuous, if for Kn e Y and K* e [TKn]a with Kn ^ k0 and K* ^ k*, we have k* e [Tk0]^.
A subset W of V, that is, W c V, is named a clique of a graph G if for every two vertices belonging to W, there exists an edge connecting them. This is similar to the condition that the subgraph induced by W is complete, that is, for every k, k* e W(G), we have (k, k*) e E(G).
3. Main results. We present common fuzzy fixed point results for two fuzzy multivalued mappings on a parametric metric space enriched with a directed graph. We start with the following result:
Theorem 2. Suppose (Y, f) is a complete parametric metric space enriched with a directed graph G, such that V(G) = Y and E(G) ^ A. If fuzzy mappings T1,T2: Y ^ Fpcl (Y) form a fuzzy graphic F1-contraction pair, then the following statements hold:
(i) Fuz (Tl) = Fuz (T2) * 0 if and only if Fuz(T) * 0 for any i e {1, 2}.
(ii) YT1;T2 * 0 given that Fuz (Ti) x Fuz (T2) * 0.
(iii) The graph G is weakly connected and Yt1,t2 * 0; then Fuz (Tl) = Fuz (T2) * 0 given that either (a), or Tl or T2 are upper semicontinuous, or (b) F is continuous, either Tl or T2 are bounded, and G has property (P).
(iv) Fuz (Tl) x Fuz (T2) is a clique of graph G if and only if Fuz (Tl) x Fuz (T2) is a singleton set.
Proof. To validate (i), let x* be any point of Y and suppose that x* e [Tl(x*)]a, such that x * R [T2 (x*)]p. As it is given that (Tl,T2) form a graphic fuzzy ^-contraction pair, this implies that there exists a x e [T2 (x*)]p with (x*, x) e E* (G), such that
t + F (f(x*, x, 9)) ^ F(Ml(x*, x*; x*, x, 9)),
where
Ml(x*, x*; x*, x, 9) = max j f(x*, x*, 9), f(x*, x*, 9), f(x, x*, 9),
Kx*,x,e)+j(x*,x*,0) j ^ x.,ey
Thus we have
T + F (f(x*, x, 9)) ^ F(f(x*, x, 9)),
a contradiction as r> 0. Hence, x* e [T2(x*)~]p and so Fuz(Tl) c Fuz(T2). Similarly, Fuz(T2) c Fuz(Tl) and, therefore, Fuz(Tl) = Fuz(T2). Also, if x* e [T2(x*)]/3, then we have x* e [Tl(x*)]a. The converse statement can be proved easily by simple steps.
To prove (ii), let Fuz (Tl) x Fuz (T2) * 0. Then there exists x e Y, such that x e \Tl(x)]a x [T2(x)]p. As A c E(G), we deduce that Ati,T2 * 0.
To validate (iii), assume that x0 is an arbitrary point of Y. If x0 e [Tl (x0)]a or x0 e [T2 (x0, then the proof is completed. So, assume that x 0 R [Tl (x0)]a and x0 R [T2 (x0. Now, for i, j e {1, 2} with i * j, if xl e \Ti(x0)]a, then there exists x2 e [Tj(xl)]^ with (xl, x2) e E*(G), such that
t + f (f(xl,X2,9)) ^ F(mi(xo,xi; Xl,X2,d)),
where Mi(xo, Ki; Ki, K2, 9) =
iff a\ t( o\ t( o\ f (кQ, K2,d) + f (ki, Ki, = maxjj (kq, Ki,9), f (kq, Ki, 9),f (kx, k2, 9)
= max j f (kq, Ki,6),f (ki, k2,6n ^
s f tf a\ ft a\ f (кQ, Ki,d),f ^ K2
< max j (kq, Ki,0),f (ki,k2,0), —
2
f (kq, k2,q) , <
2
= max{/(kq, Ki,9),f (ki, K2,9)}. If M1(kq, k1; k1) k2,6) = f (k1, k2,6), then
T + F (f (Ki, K2,d)) < F(f (Ki, K2,d)),
gives a contradiction as r > 0. Therefore, Mi(kq, k1; k1, k2,9) = = f (kq, k1, 9) and we have
r + F (f (ki, K2,9)) < F (f (kq, ki,0)) .
Correspondingly, for the point k2 in [Tj (Ki)]a, there exists k3 e [Ti(K2)\@ with (k2,k3) e E* (G) such that
T + F (f (K2, K3,9)) < F(Mi(Ki, k2; k2„ k3,6)),
where Mi(Ki, K2; K2, K3, 9) =
f (Ki, K3,d)+f (K2, K2,
= max j/ (ki , K2,d),f (ki, K2,9), f (k2 , k3
2
= max{/(ki, K2,6),f (k2, k3,9)}.
In case M1(k1, k2; k2, k3,9) = f (k2, k3,9):
t + F (f (K2, K3, 9)) < F(f (K2, K3, 9)),
gives a contradiction as r > 0. Therefore, M1(k1,k2; K2,K3,9)=f (k1,k2, and we have:
T + F (f (K2, K3,9)) < F (f (Ki, K2,9)).
Enduring this way, for X2n e [Tj(x2n—l)]a, there exist X2n+1 e [T (x2n)]p with (x2n, x2n+1) e E* (G), such that
T + F (f(x2 ni x2n+1, 0)) ^ F(Ml(x2 n— l, x2n; x2n, x2n+1, 0)),
that is,
T + F (f(x2n, X2n+1, 0)) ^ F (f(x2n—l, X2n)) ■
In the similar manner, for x2n+1 e[Tj(x2n)]a, there exist x2n+2 e[Ti(x2n+ l)]p, such that for (x2n+1, x2n+2) e E*(G) implies
T + F (f(x2n+1, X2n+ 2, d)) ^ F (f(x2n, X2n+1, 0)) ■
Hence, we got a sequence {xn} in Y, such that for xn e [Tj(xn—l)]a, there exist xn+l e [Ti (xn)]^ with (xn, xn+1) e E*(G) and it satisfies
t + F (f(xn, xn+ l,d)) ^ F (f(xn—l, xn, 0)).
Therefore,
F (f(xn, xra+1, 0)) ^ F (f(xn—l, xn, 0)) — t^F (f(xn—2, xn—l, 0)) — 2t ^
^ ... ^F (f(xo, xl, 9))—nr. (3)
From (3), we have got lim F (f(xn, xn+1, 0)) = —8 that together with
n—>8
( F2) yields lim f(xn, xn+1, 0) = 0.
n—8
Now, by ( F3), there exists h e (0,1), such that
lim [ f(xn, xn+1, 0)]hF (f(xn, xn+1,0)) = 0.
n—8
From (3), we have
[ f(xn, xn+1, 0)]hF (f(xn)xn+1, 0)) — [f(xn, xn+1, 0)]hF (f(xo, xn+1, 0)) ^
^ — nr[f(xn, xn+1, 0)]h ^ 0.
On taking limit as n — 8, we get lim n[f(xn, xn+1, 0)]h = 0.
n—8
So, limn h f(xn, xn+1,0) = 0 and there exists nl e N, such that
n—8
nh f(xn, xn+1, 0) ^ 1 for all n ^ nl. So, we get
1
f(xn, xn+1, 0) ^
n
l/h
for all n ^ n1. Now consider m,n e N, such that m > n ^ n1; we get: f (ku, Km, 9) ^
8 i
^ f(xn, Kn+1, 9) + f (Kn+1, Kn+ 2, 9) + ... + f {xm-i, Km, 9) ^ V:
7
l=n
1/h '
8 i
By the convergence of the series ^ —h, we get f (xn, Km, 9) ^ 0 as
%i/h
n,m ^ 8. Therefore, {xn} is a Cauchy sequence in X. Since X is complete, there exists an element k* e Y, such that k n ^ k* as n ^8.
Now, if Ti is upper semicontinuous, then, as K2n eX, K2n+1 e [Ti (K2n)]a with K2n ^ k* and K2n+1 ^ k* as n ^8 implies that k* e [Ti (k*)]^. Using (i), we get k* e [Ti (K*)]a = [Tj (k*)]$. In the same fashion, the result holds when Tj is upper semicontinuous.
Assume that F is continuous. Since K2n converges to k* as n ^ 8 and (K2n, K2n+1) e E (G), we have (K2n, k*) e E (G). For K2n e [Tj (K2n-i)]a, there exists vn e [Ti (k*)]$, such that (K2n, vn) e E* (G). As {vn} is bounded, limsup vn = v*, and liminf vn = v* both exist. Assume that
n—>8 n—8
v* ^ x*. Since (T1,T2) form a graphic F1-contraction,
T + F (f(K2n, Vn, 0)) ^ F(M1 (K2n-1, K*; K2n, vn, 9)),
where
M1(K2n-1, K*; K2n, Vn, 9) = max j f(K2n-1, K* , 9), f(K2n-1, K2n, 9),
f, * m f(K2n-1, Vn, 9) + f(K*, K2n, 9) ) J(K , Vn, 9),-2-}.
Taking lim sup implies
r + F (f(K*, v*, 9)) ^F (f(K*, v*, 9)),
a contradiction. Hence, v* = k*. In the same way, taking the liminf yields v* = k*. Since vn e [Ti (K*)]a for all n ^ 1 and [Ti (K*)]a is a closed set, it follows that k* e [Ti (K*)]a. Now, from (i), we get k* e [Ti(K*)]/3 and, hence, Fuz(T1) = Fuz(T2).
Finally, to verify (iv), suppose that the set Fuz (T1) x Fuz (T2) is a clique set of G. We need to verify that Fuz (T1)xFuz (T2) is singleton. Assume the contrary: that there exists and u, such that
v,u e Fuz (T]) x Fuz (T2), but u ^ v. As (v,u) e E*(G) and T1 and T2 form a graphic ^-contraction, so for (vx,uy) e E* (G) implies
t + F (f (v,u,d)) < F(Mi(v,u; v,u,6)) =
= F(] max{/(v,u,9),f(v,v,9),f(u,u,9),
f (v,u,d) + f (u,v,d)
= F (f (v,u,6))
a contradiction as r > 0. Hence, v = u. Conversely, if Fuz(Ti) x Fuz(T2) is singleton, then it follows that Fuz (Ti) x Fuz(T2) is a clique set of G. □
V i v 1 )
Example 4. Let Y = {0} y {an = —L : n e N} = V (G),
E (G) = {(k, k*) : k < k* where k, k* e V (G)} and E * (G) = {(k, k*) : k < k* where k, k* e V (G)}.
Let V (G) be endowed with the parametric metric defined as
f (k, k*
f
Bmax{K, k*}, k ^ k*
0,
K K* .
Define Ti : Y ^ Fpcl (Y) as follows:
Ti(K)(t) = Now, for a = 2, we have
f i
2
i
V 3 :
if t = Kj^,
elsewhere.
[Ti(k)\ I = {t : [Ti(K)(t)\a > 1} = {ki}
and T2 : Y ^ Fpcl (Y) as in the following cases: 1) For k = ^
T2(K)(t) "
2, if t = Ki,
i, elsewhere.
2) For x = xn with n > 1
T2(x)(t) =
Now, for /3 = i,
n
2 '
1
V 3 '
if t P {Ml, xn-1},
elsewhere.
PMx)] i = {t : [T(x)(t)], > 1} =
{_Kl}, if K = Kx^
{xx, Kn_x}, if k = Kn with n > 1.
Take F (u) = lnu + u, u > 0, and r = 1. For (uK,uK*) p E*(G), and a = ft = -, we observe the following cases:
(i) If k = Kl) x* = Km, for m > 1, then for vK = xx p [Tx (x)\ i, there exists vx* = Km_x p [T2 (k*)\ i, such that
f (vK, vK*, d)ef (v*' vk* 'd)-Mpk' , vk* , o) ^
^ f (vK,VK* ,0)efpv"'v~*'0)-fpKK*'&) = 0
2
m2 — m
,—mO
<
< d
' ^ ' e-6 = e~6 f (x, x*,0) ^ e~dMx (x, x* ; v„, v>
(ii) If k = Kn, k* = xn+l with n > 1, then for vK = xx p [Tx (k)\ i there exists vK* = xn_x p [T2 (x*)\ i, such that:
f (vx, Vx*, d)ef ,v**,eq-Mx*;v*'v«* 'eq ^
^ f (vx,Vx* ,0)ef^>v~*> ,n2 + 2n + 1
î(k,vh* ,e)+f (k*,vh* ,e)
= e
n — n o-3n-i
-ee -s— <
< a" — ' 0 = e0 2
f (x,Ux* ,0) + f (x*,Ux
^ e d Mx (x, k* ; vK ,vK* ,9).
(iii) When x = xn, x* = xm with m > n > 1, then for vK = xx p \Tl(x)\ 1 there exists vx* = xn_x p [T2 (k*)\ i , such that
2
f (vx, vx*, 9)e f {vk'v** < d)-Mx*;v*'v** <ff) ^
2 2
^ f(vx, Vk* , 9) ef pv*,v>*, ff)-fPx' VK, 0) = Q^^l e < e-0 =
2 2
= e -e f (x, Vk, 9) ^ e-dMi(x, x*; v„, vy
Now we show that for x^ x* e Y, vke[T2(x)\ i ; there exists vK* e \Ti(h*)\ i , such that (vx, vx*) e E* (G) and (1) is satisfied. For this, we consider the following cases:
(I) If x = xn, x* = x\ with n > 1, we have for vx = xn-i e [T2 (x)\i, there exists vx* = xi e [Ti (x*)\ i, such that:
f(vx, vx*, 9)ef{v",vk*,0)-Mpx'k*vk>vk*,e) ^
22
^ f(vx, vx*, 9)ef pvk' vk* , o)-f(x,x*oq = Qn_^e-n < 0n^e- =
= e-d f(x, x*, 0) ^ e-dMi (x, x*; vx, vx*, 9).
(II) In case x = xn, x* = xm with m > n > 1, for vx = xn-i e [T2 (x)\ i, there exists vx* = xi e [T2 (x*)\i, such that
f(ux, Ux*, 9)ef Puk' uK* '0)-Mpx' x*Uk' uK* ,0) ^
^ f(Ux ,Ux* , 9)e f Puk 'UK* '0)-f Px'uK* , S) = Qn—n eepn2-n-m2-m) <
m2 m
< o Tme= ^ J(x*, Vx*, Q) ^ e-dMi (x, x*; Vx,v.
Henceforth, for all x, x* in V (G), condition (1) is satisfied. Hence, all the requirements of Theorem 2 are satisfied. Moreover, {1} is the common fuzzy fixed point of Ti and T2 with Fuz(Ti) = Fuz(T2).
The following result generalizes Theorem 3.4 in [28].
Theorem 3. Let (Y, f) be a complete parametric metric space endowed with a directed graph G, such that V(G) = Y and E(G) 3 A. lfTi,T2 : Y ^ Fpcl (Y) form a fuzzy graphic F2-contraction pair, then the following statements hold:
(i) Fuz (Ti) or Fuz (T2) ^ 0 if and only if Fuz (Ti) = Fuz (T2) ^ 0.
(ii) YTliT2 ^ 0 provided that Fuz (Ti) x Fuz (T2) ^ 0.
*
(iii) If YTl>T2 ^ 0 and G is weakly connected, then Fuz(Ti) = Fuz(T2) ^ 0, provided that either (a), or Tx or T2 is upper semicontinuous, or (b) F is continuous, either Tx or T2 is bounded, and G has property (P).
(iv) Fuz(T\) x Fuz(T2) is a clique set of G if and only if Fuz (Tx) x Fuz (T2) is a singleton set.
Proof. To validate (i), let x* p [Ti(x*)\a. Assume x* R [T2 (x*)\fi; then, since (Tx,T2) form a fuzzy graphic F2-contraction pair, there exists a x p[T2 (x*)\p with (k*, k) p E* (G), such that
t + F (f (k*, k, d)) ^ F(M2(x*, x*; x*, x, 9)),
where
M2(k*, k*; k*, k,9) = axf (k*, x*,6) + /3xf (k*, x*,d) + (k, x*,d) +
+ Sif (k*, k, 0) + S2f (K*, K*,d) = (>yi + 5i)f (k, K*,d).
Thus, we have
r + F (f (k*, k, d)) ^ F((n + 5i)f (k*, k, d)) ^ F(f (k*, k, d)),
a contradiction as r > 0. Hence, k* p [T2 (k*)\p and, so, Fuz(Ti) c Fuz(T2). Similarly, Fuz(T2) c Fuz(Tx) and, therefore, Fuz(Ti) = Fuz(T2). Also, if x* p [T2(k*)\^, then we have k* p [Tx(x*)\a. Converse can be proved by straightforward steps.
To validate (ii), let Fuz (Tx) x Fuz (T2) ^ 0. Then there exists x p Y, such that x p \Tl(x)\a x [T2(x)\p. Since A c E(G), we conclude that Yti,t2 ^ 0.
To validate (iii), suppose that x0 is an arbitrary point of Y. For i, j p {1, 2}, with i ^ j, take xx p [Ti(x0)\a; there exists x2 p [Tj(xx)\@ with (xx, x2) p E* (G), such that
t + F (f (xi, K2,9)) ^ F(M2(xo, Ki; Ki, K2,0)),
where
M2(kQ, Ki; Ki, K2,d) = aif (x0, Ki,d) + ftif (x0, Ki,d) +
+ Jif (Ki, K2, 9) + +Sif (Ko, K2, d) + 62! (Ki, Ki, 0) ^
^ (ai + fti + 5i)f (xo, Ki,d) + (-fi + 5i)f (Ki, K2,d).
If f(x0, xi, 9) ^ ¡(x-]^, x2, 9), then we have
T + F (f(xi, X2, 9)) ^ F ((ai + Pi + 'i + 25i)f(xi, X2, 9)) ^ F (f(xu X2,9)),
gives a contradiction as r > 0. Therefore,
r + F (f(xi, X2, 9)) ^F (f(xo, Xi, 9)).
Continuing this process, for x2n e [Tj(x2n-i)]a, we see that there exists X2n+1 e [TiX2n]^s, such that for (x2n,X2n+1) e E* (G), we have
T + F (f (X2n, X2n+ i, 9)) ^ F (M2(x2n-i, X2n; X2n, X2n+ i, 9)) ,
where
M2(x2n-i, X2n; X2n, X2n+ i, 9) =
= aif (X2n-i, X2n, 9) + Pif (X2n-i, X2n, 9) + 'if (X2n, X2n+ i, 9) + + Sif (X2n-i, X2n+ i, 9) + $2f (X2n, X2n, 0) ^ ^ (ai + Pi + 8i)f(x2n-i, X2n, 9) + ('i + Si)f(x2n, X2n+ i, 9).
If f(x2n-i, X2n, 9) ^ f(x2n, X2n+1, 9), then
T + F (f(x2n, X2n+i, 9)) ^ F ((ai + Pi + yi + 261,9)f(x2n, X2n+ i, 9)) ^
^ F (f(x2n, X2n+ i, 9)) ,
gives a contradiction as > 0. Therefore,
T + F (f(x2n,X2n+ i, 9)) ^ F (f(x2n-i,X2n, 9)) .
In a similar way, for X2n+1 e [Tj (x2n)]a, there exists X2n+2 e[T (x2n+i)]p with (x2n+ i, x2n+2) e E* (G), such that
T + F (f(x2n+ i, X2n+ 2, 9)) ^ F (f(x2n, X2n+ i, 9)) .
Hence, we obtain a sequence {xn} in Y, such that for xn e [Tj(xn-i)]a, there exists xn+i e [T (xn)]p with (xn, xn+^ e E* (G), such that
T + F (f(xn, Xn+i, 9)) ^ F (f(xn-i, Xn, 9)).
Therefore,
F (f(xn, Xn+1,9)) ^ F (f(xn-i, Xn, 6))-T ^ F (f(Xn-2, Xn-i, 6))-2t ^
^ ... ^F (f(xo, xi, d))-nr.
Consequently, lim F (f(xn, xn+i, 9)) = —8 together with (F2) gives
n—>8
lim f(xn, xn+ i, 9) = 0. Following arguments similar to those in proof of
n—8
Theorem 2, {xn} is a Cauchy sequence in Y. Since Y is complete, there exists an element x* e Y, such that xn ^ x* as n ^ 8.
Now, if Ti is upper semicontinuous, then, as x2n e Y, x2n+i e[Ti(x2n)]a with x2n ^ x* and x2n+ i ^ x* as n ^ 8 implies that x* e [Ti (x*)]p. Using (i), we get x* e [Ti (x*)]a = [Tj (x*)] p. In the same manner, the result holds when Tj is upper semicontinuous.
Assume that F is continuous. Since x2n converges to x* as n ^ 8 and (X2n, X2n+ i) e E (G), we have (x2n, x*) e E (G). For X2n e [Tj (x2n-i)]a, there exists vn e [Ti (x*)]p, such that (x2n, vn) e E* (G). As {vn} is bounded, lim sup vn = v*, and liminf vn = v* both exist. Assume that
n—8 n—8
v* ^ x*. Since (Ti,T2) form a fuzzy graphic F2-contraction, T + F (f(x2n, Vn, 9)) ^ F (M2 (X2n-i, X*; X2n, Vn, 9)),
where
M2 (X2n- i , X*; X2n, Vn, 9) = af (X2n-i , X*, 9) + Pp(x2n-i, X2n, 9) +
+ 'f (x*, Vn, 9) + Sif (X2n-i, Vn, 9) + 62f (x*, X2n, 9).
Taking lim sup implies
t + F (f(x*, v*, 9)) ^ F ((' + Si )f(x*, v*, 9)) ^ F (f(x*, v*, 9)),
a contradiction. Hence, v* = x*. Similarly, taking the liminf gives v* = x*. Since vn e Ti (x*) for all n ^ 1 and [Ti (x*)]a is a closed set, it follows that x* e [Ti (x*)]a. Now, from (i) we get x* e [Ti(x*)]a and, hence, Fuz (Ti) = Fuz (T2).
Finally, to validate (iv), assume that the set Fuz (Ti) x Fuz (T2) is a clique of G. We are to prove that Fuz (T1)xFuz (T2) is singleton. Suppose the contrary: that there exist and u, such that
u,v e Fuz (Ti) x Fuz (T2) but u ^ v. As (v,u) e E*(G), and Ti, and T2 form a graphic F2- contraction, so, for (va,u,b) e E* (G), we have:
r + F (f (v,u,d)) ^ F (M2(v,u; v,u,d)) =
= F (aif (v,u,d)+ pif (v,v,d)+ jif (u,u,d) + + Sif (v,u,6) + 82 f (u,v,6)) =
= F ((ai + Si + 82) f (v, u, 9)) ^ F (f (v, u, 9)),
a contradiction as r > 0. Hence, v = u. Conversely, if Fuz(Tx) x Fuz(T2) is singleton, then it follows that Fuz(Tx) x Fuz(T2) is a clique of G. □
Remark 1. Let (Y, f) be a complete parametric metric space enriched with a directed graph G. If we replace (2) by either of the following three conditions:
r + F (f (va, vb, 9)) ^ F (aif (a, b, 9) + ftj (a, vb, 9) + ^f (b, vb, 9)), (4) where ai,fti,yi ^ 0 and ai + + ^ 1, or
T + F (f (va, vb, 9)) ^ F(h[f (a, vb, 9) + f (b, vb, 0)\), (5)
where h p [0, i \, or
r + F (f (va,vb,9)) ^ F (f (a,b,9)). (6)
Then the deductions acquired in Theorem 3 remain true. Remark 2.
1) If E(G) := Y x Y, then, clearly, G is connected and our Theorem 2 improves and generalizes (i) Theorem 2.1 in [2], (ii) Theorem 1.9 in [4], (iii) Theorem 4.1 in [22], (iv) Theorem 3.4 of [28], and (v) Theorem 3.1 of [29].
2) If E(G) := Y x Y, then Theorem 3 improves and extends Theorem 3.4 in [28], and Theorem 3.4 in [29].
3) If E(G) := Y x Y, then our Remark 1 extends and generalizes (i) Theorem 3.4 in [28] and (ii) Theorem 4.1 of [22].
4) If E(G) := Yx,Y then our Remark 1 improves and generalizes Theorem 4.1 in [22].
5) If we take Tx = T2 in graphic Fi-contraction pair and graphic F2-contraction pair, then we obtain the fixed point results for graphic Fi-contraction and graphic F2-contraction of a single map.
4. Problem Statement. The results of this paper expand the common fuzzy fixed point theory of multivalued mappings by incorporating fuzzy graphic F-contractions. The concepts discussed within the context of parametric metric spaces are foundational. Therefore, these ideas can be enhanced when applied to more generalized metric structures, such as 6-metric spaces, controlled metric spaces, semi-metric spaces, and quasi-metric spaces. Additionally, the component of the fuzzy multivalued map can be extended to include L-fuzzy mappings, intuitionistic fuzzy mappings, soft multivalued maps, and others.
Acknowledgment. The authors are thankful to the anonymous referees for their remarks and suggestions, which helped to improve the presentation of the paper.
References
[1] Abbas M., Ali B., Romaguera S. Fixed and periodic points of generalized contractions in metric spaces. Fixed Point Theory Appl., 2013, vol. 243, pp. 1-11. DOI: https://doi.org/10.1186/1687-1812-2013-243
[2] Abbas M., Alfuraidan M. R., Nazir T. Common fixed points of multivalued F-contractions on metric spaces with a directed graph. Carpathian J. Math., 2016, vol. 32, pp. 1-12.
DOI: https://doi .org/10.37193/CJM.2016.01.01
[3] Abbas M., Nazir T. Common fixed point of a power graphic contraction pair in partial metric spaces endowed with a graph. Fixed Point Theory Appl., 2013, vol. 20, pp. 01-08.
DOI: https://doi.org/10.1186/1687-1812-2013-20
[4] Abbas M., Rhoades B. E. Fixed point theorems for two new classes of multi-valued mappings. Appl. Math. Lett., 2009, vol. 22, pp. 1364-1368. DOI: https://doi .org/10.1016/j.aml.2009.04.002
[5] Acar O., Durmaz G., Minak G. Generalized multi-valued F-contractions on complete metric spaces. Bulletin Iran. Math. Soc., 2014, vol. 40, pp. 1469-1478.
[6] Al-Mazrooei A. E., Ahmad J. Fixed point theorems for fuzzy mappings with applications. J. Intell. Fuzzy Syst., 2019, vol. 36, pp. 3903-3909.
DOI: https://doi .org/10.3233/JIFS-181687
[7] Atanassov K. T. Intuitionistic fuzzy sets. In Intuitionistic fuzzy sets, Phys-ica, Heidelberg, 1999, vol. 35, pp. 1-137.
DOI: https://doi .org/10.3233/JIFS-181687
[8] Azam A., Arshad M., Vetro P. On a pair of fuzzy ip-contractive mappings. Math. Comput. Modelling, 2010, vol. 52, pp. 207-214.
DOI: https://doi.org/10.1016/j.mcm.2010.02.010
[9] Azam A., Beg I. Common fixed points of fuzzy maps. Math. Comput. Modelling, 2009, vol. 49, pp. 1331-1336.
DOI: https://doi.org/10.1016/j.mcm.2008.11.011
[10] Berinde V. Approximating fixed points of weak contractions using the Picard iteration. Nonlinear Anal. Forum, 2004, vol. 9, pp. 43-53.
[11] Berinde M., Berinde V. On a general class of multivalued weakly Picard mappings. J. Math. Anal. Appl., 2007, vol. 326, pp. 772-782.
DOI: https://doi.org/10.1016Zj.jmaa.2006.03.016
[12] Bose R. K., Sahani D. Fuzzy mappings and fixed point theorems. Fuzzy Sets and Systems, 1987, vol. 21, pp. 53-58.
DOI: https://doi.org/10.1016/0165-0114(87)90152-7
[13] Goguen J. A. L-fuzzy sets. J. Math. Anal. Appl., 1967, vol. 18, pp. 145 -174. DOI: https://doi.org/10.1016/0022-247X(67)90189-8
[14] Heilpern S. Fuzzy mappings and fixed point theorem. J. Math. Anal. Appl., 1981, vol. 83, pp. 566-569.
DOI: https://doi.org/10.1016/0022-247X(81)90141-4
[15] Hussain N., Salimi P., Parvaneh V. Fixed point results for various contractions in parametric and fuzzy b-metric spaces. J. Nonlinear Sci. Appl., 2015, vol. 8, no. 5, pp. 719-739.
DOI: http://dx.doi.org/10.22436/jnsa. 008.05.24
[16] Shagari M. S. On fuzzy soft set-valued maps with application. J. Nig. Soc. Phys. Sci., 2020, vol. 2, pp. 26-35
DOI: https://doi .org/10.46481/jnsps.2020.48
[17] Shagari M. S., Azam A. Fixed point theorems of fuzzy set-valued maps with applications. Probl. Anal.-Issues Anal., 2020, vol. 9, pp. 68-86.
DOI: https://doi .org/10.15393/j3.art. 2020.6750
[18] Shagari M. S., Azam A. Fixed points of soft-set valued and fuzzy set-valued maps with applications. J. Intell. Fuzzy Syst., 2019, vol. 37, pp. 3865-3877. DOI: https://doi .org/10.3233/jifs-190126
[19] Jachymski J. The contraction principle for mappings on a metric space with a graph. Proc. Amer. Math. Soc., 2008, vol. 136, pp. 1359-1373.
[20] Jachymski J., Jozwik I. Nonlinear contractive conditions: a comparison and related problems. Banach Center Publ., 2007, vol. 77, pp. 123-146. DOI: https://doi .org/10.4064/bc77-0-10
[21] Kannan R. Some results on fixed points. Bull. Calcutta. Math. Soc., 1968, vol. 60, pp. 71-76. DOI: https://doi.org/10.2307/2316437
[22] Latif A., Beg I. Geometric fixed points for single and multivalued mappings. Demonstratio Math., 1997, vol. 30, pp. 791-800.
DOI: https://doi .org/10.1515/dema-1997-0410
[23] Markin J. T. Continuous dependence of fixed point sets. Proc. Amer. Math. Soc., 1973, vol. 38, pp. 545-547.
[24] Nadler S.B. Multi-valued contraction mappings. Pacific J. Math., 1969, vol. 30, pp. 475-488. DOI: https://doi.org/10.2140/pjm.1969.30.475
[25] Nieto J. J., Lopez R. R. Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order, 2005, vol. 22, pp. 223-239. DOI: https://doi.org/10.1007/s11083-005-9018-5
[26] Qiu D., Shu L. Supremum metric on the space of fuzzy sets and common fixed point theorems for fuzzy mappings. Information Sciences, 2008, vol. 178, pp. 3595-3604.
DOI: https://doi .org/10.1016/j.ins.2008.05.018
[27] Ran A. C. M., Reurings M. C. B. A fixed point theorem in partially ordered sets and some application to matrix equations. Proc. Amer. Math. Soc., 2004, vol. 132, pp. 1435-1443.
DOI: https://doi.org/10.1090/S0002-9939-03-07220-4
[28] Rus I. A., Petrusel A., Sintamarian A. Data dependence of fixed point set of some multivalued weakly Picard operators. Nonlinear Anal., 2003, vol. 52, no. 8, pp. 1944-1959.
DOI: https://doi.org/10.1016/S0362-546X(02)00288-2
[29] Sgroi M., Vetro C. Multi-valued F-contractions and the solution of certain functional and integral equations. Filomat, 2013, vol. 27 no. 7, pp. 1259-1268. DOI: https://doi.org/10.2298/FIL1307259S
[30] Wardowski D. Fixed points of new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl., 2012, vol. 94, pp. 01-06. DOI: https://doi.org/10.1186/1687-1812-2012-94
[31] Zadeh L. A. Fuzzy sets. Information and Control., 1965, vol. 8, pp. 338353.
DOI: https://doi.org/10.1016/S0019-9958(65)90241-X
Received March 21, 2024. In revised form, June 03 , 2024. Accepted June 05, 2024. Published online July 12, 2024.
Muhammad Rafique Department of Mathematics, COMSATS University Islamabad, 44000, Pakistan E-mail: [email protected]
Talat Nazir
Department of Mathematical Sciences,
University of South Africa, Florida 0003, South Africa
E-mail: talatn@unisa .ac.za
Hemanta Kalita
Mathematics Division, School of Advanced Sciences and Languages, VIT Bhopal University, Bhopal-Indore Highway, Kothrikalan, Sehore, Madhya Pradesh, India E-mail: [email protected]
