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Common Fixed Points of Intuitionistic Fuzzy Maps for Meir-Keeler Type Contractions.

1. Introduction

The theory of fixed points has been revealed as a very powerful and important tool in the study of nonlinear phenomena. In particular, fixed point techniques have been applied in such diverse fields as biology, chemistry, economics, engineering, game theory, and physics. The Banach fixed point theorem [1] (also known as a contraction mapping principle) is an important tool in nonlinear analysis. It guarantees the existence and uniqueness of fixed points of self-mappings on complete metric spaces and provides a constructive method to find fixed points. Many extensions of this principle have been done up to now. In 1976, Jungck [2] studied coincidence and common fixed points of commuting mappings and improved the Banach contraction principle. In 1986, Jungck [3] introduced the notion of compatible maps for a pair of self-mappings and existence of common fixed points. In 1969, Meir and Keeler [4] obtained a valuable fixed point theorem for single valued mappings [PHI] : X [right arrow] X that satisfies the following condition:

Given [epsilon] > 0, there exists a [delta] > 0 such that

[epsilon] [less than or equal to] d (x, y) < [epsilon] + [delta] implies d ([PHI]x, [PHI]y) < [epsilon]. (1)

In 1981, Park and Bae [5] extended it to a pair of commuting single valued mappings. A variety of extensions, generalizations, and applications of this followed; e.g., see [6, 7]. In 1993, Beg and Shahzad [8] derived and proved random fixed point of two random multivalued operators satisfying the M-K [4] condition in Polish spaces. In 2001, Lim [9] wrote on characterization of M-K-contractive maps. In 2012, Abdeljawad et al.'s paper [10] contains a study of M-K type coupled fixed point on ordered partial metric spaces and Chen et al. [11] established and proved common fixed point theorems for the stronger M-K cone-type function in cone ball-metric spaces. In 2013, Karapinar et al. [12] studied the existence and uniqueness of a fixed point of the multidimensional operators in partially ordered metric space which satisfied M-K type contraction condition and improved the results mentioned above and the recent results on these topics in the literature. In the same year, Abdeljawad [13] established and proved M-K-contractive fixed point and common fixed point theorems. Patel et al. [14] formulated and proved a more generalized version of [13]. In 2014, Singh et al. [15] derived a new common fixed point theorem for Suzuki-M-K contractions. In 2015, Redjel et al. [16] proved fixed point theorems for ([alpha], [psi])-Meir-Keeler-Khan mappings. Abtahi [17,18] established and proved fixed point theorems in 2016 and common fixed point theorems in 2017 for M-Kt ype contractions in metric spaces. Popa and Patriciu [19] derived and proved a general theorem of M-K type for mappings satisfying an implicit relation in partial metric spaces in 2017.

Fuzzy sets were introduced by Zadeh [20] in 1965 to represent/manipulate data and information possessing nonstatistical uncertainties. In 1986, the concept of an intuitionistic fuzzy set (IFS) was put forward by Atanassov [21], which can be viewed as an extension of fuzzy set. Intuitionistic fuzzy sets not only define the degree of membership of an element, but also characterize the degree of nonmembership. IFS has much attention due to its significance to remove the vagueness or uncertainty in decision-making. IFS is a tool in modeling real life problems such as psychological investigation and career determination. Abbasizadeh and Davvaz [22] introduced intuitionistic fuzzy topological polygroups. In 2017, Azam et al. [23] proved coincidence and fixed point theorems of intuitionistic fuzzy mappings with applications. In the same year, Azam and Tabassum [24] established and proved fixed point theorems of intuitionistic fuzzy mappings in quasi-pseudometric spaces. Recently, Kumam et al. [25] and Shoaib et al. [26] derived and proved some fuzzy fixed point results for fuzzy mappings in complete b-metric spaces. Humaira et al. [27] established and proved fuzzy fixed point results for [PHI]-contractive mapping and gave some applications. Ertrk and Karakaya [28] stated and proved n-tuplet coincidence point theorems in intuitionistic fuzzy normed spaces. Xia Li et al. [29] worked on the intuitionistic fuzzy metric spaces and the intuitionistic fuzzy normed spaces (see also [30-32]).

In this paper, the main focus is to establish the existence of fixed point and common fixed point theorems of M-K type contraction for intuitionistic fuzzy set-valued maps in complete metric spaces. Some nontrivial examples have been furnished in the support of the main results.

2. Preliminaries

We start this section by recalling some pertinent concepts.

Definition 1 (see [33]). Let (X, d) be a metric space. The set of all nonempty closed and bounded subsets of X is denoted by CB(X). The function H defined on CB(X) x CB(X) by

[mathematical expression not reproducible] (2)

for all A, B [member of] CB(X) is a metric on CB(X) called the Hausdorff metric of d,

where

[mathematical expression not reproducible]. (3)

Definition 2 (see [20]). Let X be an arbitrary nonempty set. A fuzzy set in X is a function with domain X and values in [0,1]. If A is a fuzzy set and x [member of] X, then the function-value A(x) is called the grade of membership of x in A. [I.sup.X] stands for the collection of all fuzzy sets in X unless and until stated otherwise.

Definition 3 (see [21]). Let X be a nonempty set. An intuitionistic fuzzy set is defined as

A = {<x, [[mu].sub.A] (x), [v.sub.A] (x)>: x [member of] X}, (4)

where [[mu].sub.A] : X [right arrow] [0,1] and [v.sub.A] : X [right arrow] [0,1] denote the degree of membership and degree of nonmembership of each element x to the set A, respectively, such that

0 [less than or equal to] [[mu].sub.A] (x) + [v.sub.A] (x) [less than or equal to] 1, [for all]x, y [member of] X. (5)

The collection of all intuitionistic fuzzy sets in X is denoted by [(IFS).sup.X].

Definition 4 (see [34]). Let A be an intuitionistic fuzzy set and x [member of] X; then [alpha]-level set of an intuitionistic fuzzy set A is denoted by [[A].sub.[alpha]] and is defined as

[[A].sub.[alpha]] = {x [member of] X : [[mu].sub.A] (x) [greater than or equal to] [alpha], [v.sub.A] (x) [less than or equal to] 1 - [alpha]},

if [alpha] [member of] (0,1]. (6)

A generalized version of [alpha]-level set of an intuitionistic fuzzy set A was investigated in [35, 36].

Definitions (see [35, 36]). Let L = {([alpha], [beta]) : [alpha] + [beta] [less than or equal to] 1, ([alpha], [beta]) [member of] (0,1] x [0,1)} and let A be an IFS on X; then ([alpha], [beta])-cut set of A is defined as

[A.sub.([alpha],[beta]) = {x [member of] X : [[mu].sub.A] (x) [greater than or equal to] [alpha], [v.sub.A] (x) [less than or equal to] [beta]}. (7)

Definition 6 (see [23]). Let X be an arbitrary set and let Y be a metric space. A mapping S : X [right arrow] [(IFS).sup.Y] is called an intuitionistic fuzzy mapping.

Definition 7. Mappings [PHI] : X [right arrow] [(IFS).sup.X] and [psi] : X [right arrow] X are said to be ([alpha], [beta]) compatible if whenever there is a sequence {[x.sub.n]} [subset or equal to] X satisfying [lim.sub.n [right arrow] [infinity]] [psi][x.sub.n] [member of] [lim.sub.n [right arrow] [infinity]] [[[PHI][x.sub.n]].sub.([alpha],[beta])] (provided [lim.sub.n [right arrow] [infinity]] [psi][x.sub.n] and [lim.sub.n [right arrow] [infinity]] [[[PHI][x.sub.n]].sub.([alpha],[beta])] exist and [psi] [[[PHI][x.sub.n]].sub.([alpha],[beta])] [member of] CB(X)), then [lim.sub.n [right arrow] [infinity]] H[([psi][[PHI][x.sub.n]].sub.([alpha], [beta]), [[PHI][psi][x.sub.n]].sub.([alpha], [beta]) = 0.

Lemma 8 (see [37]). Let {[Y.sub.n]} be a sequence in CB(X) and H([Y.sub.n], Y) [right arrow] 0 for Y [member of] CB(X). If [x.sub.n] [member of] [Y.sub.n] and [lim.sub.n[right arrow][infinity]] d([x.sub.n], x) = 0, then x [member of] Y.

3. Main Results

Theorem 9. Let X be a complete metric space and let [PHI] : X [right arrow] [(IFS).sup.X], [psi] : X [right arrow] X be ([alpha], [beta]) compatible mappings. Suppose for each x [member of] X there exists ([alpha], [beta]) [member of] (0,1] x [0,1) such that [[PHI]x].sub.([alpha],[beta])] [member of] CB(X) and [[union].sub.x[member of]X] [[PHI]x].sub.([alpha],[beta])] [subset or equal to] [psi]X and the following condition is satisfied:

for [epsilon] > 0 there exists a [delta] > 0 such that (8)

[epsilon] [less than or equal to] d ([[psi]x, [psi]y) < [epsilon] + [delta] implies d (u, v) < [member of], (9)

u [member of] [[[PHI]x].sub.([alpha], [beta])],

v [member of] [[[PHI]x].sub.([alpha], [beta])],

[[[PHI]x].sub.([alpha],[beta])] = [[[PHI]y].sub.([alpha],[beta])] (10)

when [psi]x = [psi]y.

If [psi] is continuous, then [PHI] and [psi] have a common fixed point.

Proof. Let [x.sub.0] [member of] X, and consider the following sequences [x.sub.n] and [y.sub.n] in X and [Y.sub.n] [member of] CB(X), [y.sub.n] = [psi][x.sub.n] [member of] [[[PHI][x.sub.n-1]].sub.([alpha],[beta]), n [greater than or equal to] 0 (which is possible due to the hypothesis [[union].sub.x[member of]X] [[[PHI]x].sub.([alpha],[beta])] [subset or equal to] [psi]X). Then for each [member of] > 0 there exists [delta] > 0 such that [member of] [less than or equal to] ([psi][x.sub.m], [psi][x.sub.n]) < [epsilon] + [delta] implies d([psi][x.sub.m+1], [psi][x.sub.n+1]) < [epsilon]. It follows that d([y.sub.n], [y.sub.n+1]) < d([y.sub.n-1], [y.sub.n]). Thus, the sequence {d([y.sub.n], [y.sub.n+1])} is nonincreasing and converges to the greatest lower bound of its range, which we denote by l.

Now l [greater than or equalto] 0; in fact, l = 0. Otherwise, if l > 0, take N so that n [greater than or equal to] N implies l [less than or equal to] d([y.sub.n], [y.sub.n+1]) < l + [delta]. It implies that d([y.sub.n+1], [y.sub.n+2]) < l which contradicts the fact that l = [inf.sub.n] d([y.sub.n+1], [y.sub.n+2]). Hence d([psi][x.sub.n], [[[PHI][x.sub.n]].sub.([alpha],[beta])] [less than or equal to] d([psi][x.sub.n], [psi][x.sub.n+1]) [right arrow] 0. Now it is to prove that {[y.sub.n]} is a Cauchy sequence. Suppose that d([y.sub.n], [y.sub.n+1]) = 0 for some n > 0. Then d([y.sub.m], [y.sub.m+1]) = 0 for all m > n; otherwise d([y.sub.n], [y.sub.n+1]) = 0 < d([y.sub.n+1], [y.sub.n+2]), a contradiction. Hence, {[y.sub.n]} is a Cauchy sequence.

Now assume that d([y.sub.n], [y.sub.n+1]) = 0 for each n. Define [zeta] = 2[epsilon] and choose (without loss of generality) [delta], 0 < [delta] < [epsilon], such that (9) is satisfied. Since d([y.sub.n], [y.sub.n+1]) [right arrow] 0, there exists an integer N such that ([y.sub.i], [y.sub.i+1]) < [delta]/6 for i [greater than or equal to] N. We now let q > p > N and show that d([y.sub.p], [y.sub.q]) [less than or equal to] [zeta], to prove that {[y.sub.n]} is indeed Cauchy. Suppose that

d([y.sub.p], [y.sub.q]) [greater than or equal to] 2[epsilon] = [zeta]. (11)

First, we show that there exists an integer m > p such that

[epsilon] + [delta]/3 d([y.sub.p], [y.sub.m]) < [epsilon] + [delta], (12)

where p and m are of opposite parity. Let k be the smallest integer greater than p such that

d([y.sub.p], [y.sub.k]) > [epsilon] + [delta]/2 (13)

(which is possible due to (11) as [delta] < [epsilon]). Moreover,

d([y.sub.p], [y.sub.k]) < [epsilon] + 2[delta]/3. (14)

For otherwise,

[epsilon] + 2[delta]/3 [less than or equal to] d([y.sub.p], [y.sub.k-1]) + d ([y.sub.k-1], [y.sub.k]). (15)

Since k - 1 [greater than or equal to] p [greater than or equal to] N, therefore d([y.sub.k-1], [y.sub.k]) < [delta]/6. It implies that

d([y.sub.p], [y.sub.k-1]) > [epsilon] + [delta]/2, (16)

which contradicts the fact that k is the smallest such that (13) is satisfied. Thus,

[epsilon] + [delta]/2 < d([y.sub.p], [y.sub.k]) < [epsilon] + [delta]/3. (17)

If p and k are of opposite parity, we can take k = m in (17) to obtain (12). If p and k are of the same parity, p and k + 1 are of opposite parity. In this case,

d ([y.sub.p], [ys.ub.k+1]) [less than or equal to] d ([y.sub.p], [y.sub.k]) + d ([y.sub.k], [y.sub.k+1]) [less than or equal to] [epsilon] + 2[delta]/3 + [delta]/6

= [epsilon] + 5[delta]/6. (18)

Moreover,

[mathematical expression not reproducible]. (19)

Thus,

[epsilon] + [delta]/3 < d ([y.sub.p], [y.sub.k+1]) < [epsilon] + 5[delta]/6. (20)

Putting m = k + 1, we obtain (12). Hence (12) holds. Now,

[mathematical expression not reproducible], (21)

a contradiction.

Hence {[y.sub.n]} = {[psi][x.sub.n]} is a Cauchy sequence. By completeness of the space, there exists an element z [member of] X such that d([y.sub.n], z) [right arrow] 0; continuity of [psi] implies that d([psi][y.sub.n], [psi]z) [right arrow] 0. Hence, H([[[PHI][y.sub.n]].sub.([alpha],[beta])], [[[PHI]z].sub.([alpha],[beta])]) [less than or equal to] sup{d(u, v) : u [member of] [[[PHI][y.sub.n]].sub.([alpha],[beta])] v [member of] [[[PHI]z].sub.([alpha],[beta])] < d([psi][y.sub.n], [psi]z) [right arrow] 0.

Since {[psi][x.sub.n]} is a Cauchy sequence in X and

[mathematical expression not reproducible], (22)

it follows that [Y.sub.n] is a Cauchy sequence in CB(X). By completeness of CB(X), there exists Y [member of] CB(X) such that H([Y.sub.n], Y) [right arrow] 0. Since [y.sub.n+1] [member of] [Y.sub.n] and d([y.sub.n+1], z) [right arrow] 0, Lemma 8 implies that z [member of] Y, that is, [lim.sub.n[right arrow][infinity]] [psi][x.sub.n] [member of] [lim.sub.n[right arrow][infinity]] [[[PHI][x.sub.n].sub.([alpha],[beta])]. Compatibility of [psi] and [PHI] further implies that

[mathematical expression not reproducible]. (23)

Since d([psi][y.sub.n+1], [[PHI][y.sub.n]].sub.[alpha],[beta]]) [less than or equal to] H([psi][[[PHI][x.sub.n]].sub.([alpha],[beta])], [[[PHI][psi][x.sub.n]].sub.([alpha],[beta])]), therefore [psi]z [member of] [[[PHI]z].sub.[alpha],[beta]], that is, [lim.sub.n[right arrow][infinity]] [psi][y.sub.n] [member of] [lim.sub.n[right arrow][infinity]] [[[PHI][y.sub.n]].sub.([alpha],[beta])] [lim.sub.n[right arrow][infinity]] H([psi][[[PHI][x.sub.n]].sub.([alpha],[beta])], [[[PHI][psi][x.sub.n]].sub.([alpha],[beta])]), = H([psi][[[PHI]z].sub.([alpha],[beta])], [[[PHI][psi]z].sub.([alpha],[beta])]) = 0.

Let b = [psi]z; then, by (9) we have

[mathematical expression not reproducible]. (24)

Thus b = [psi]b.

Now,

[mathematical expression not reproducible]. (25)

Hence, b [member of] [[[PHI]b].sub.([alpha],[beta])].

Definition 10 (see [38]). Let (X, d) be a metric space and let [PHI] : X [right arrow] [(IFS).sup.X] be an intuitionistic fuzzy map. A single valued map [psi] : X [right arrow] X is said to be a selection of [PHI] : X [right arrow] [(IFS).sup.X], if there exists ([alpha], [beta]) [member of] (0,1] x [0,1) such that

[psi]x [member of] [[[PHI]x].sub.([alpha],[beta])], x [member of] X. (26)

Theorem 11. Let Y be a compact subset of a complete metric space X and let [PHI] : Y [right arrow] [(IFX).sup.Y] be a mapping which satisfies the following conditions:

Given [epsilon] > 0, there exists a [delta] > 0

such that for all x, y [member of] Y, (27)

[epsilon] [less than or equal to] max {d(x, [[[PHI]x].sub.([alpha],[beta])], d(y, [[[PHI]y].sub.([alpha],[beta])}

< [epsilon] + [delta] (28)

implies H([[[PHI]x].sub.([alpha],[beta])], [[[PHI]y].sub.([alpha],[beta])]) < [epsilon].

Then, there exists a subset W of Y such that [[[PHI]w].sub.([alpha],[beta])] = W for each w [member of] W. Moreover, for each w [member of] W there exists a selection of [PHI] having w as a unique fixed point.

Proof. Let [x.sub.0] be an arbitrary fixed element of X. Two sequences [[x.sub.n]] and {[r.sub.n]} of elements in X and R, respectively, will be constructed. [[[PHI][x.sub.0]].sub.([alpha],[beta])] is a closed subset of Y and therefore is compact. There exists a point [x.sub.1] [member of] [[[PHI][x.sub.0]].sub.([alpha],[beta])] such that d([x.sub.0], [x.sub.1]) = d([x.sub.0], [[[PHI][x.sub.0]].sub.([alpha],[beta])] = [r.sub.0]. Similarly, there exists [x.sub.2] [member of] [[[PHI][x.sub.1]].sub.([alpha],[beta])] such that d([x.sub.1],[x.sub.2]) = d([x.sub.1], [[[PHI][x.sub.1]].sub.([alpha],[beta])] = [r.sub.1]. By induction, we prove that sequences {[x.sub.n]} and {[r.sub.n]} are such that [x.sub.n] [member of] [[[PHI][x.sub.n-1]].sub.([alpha],[beta])] d([x.sub.n], [x.sub.n+1]) = d([x.sub.n], [[[PHI][x.sub.n]].sub.([alpha],[beta])]) = [r.sub.n], n [greater than or equal to] 0. From inequality (28), we have

d([x.sub.n], [[[PHI][x.sub.n]].sub.([alpha],[beta])]) [less than or equal to] H ([[[PHI][x.sub.n- 1]].sub.([alpha],[beta])], [[[PHI][x.sub.n]].sub.([alpha],[beta])])

< max{d([x.sub.n-1], [[[PHI][x.sub.n-1]].sub.([alpha],[beta])]), d([x.sub.n], [[[PHI][x.sub.n]].sub.([alpha],[beta])])}. (29)

If d([x.sub.n], [[[PHI][x.sub.n]].sub.([alpha],[beta])]) > d([x.sub.n-1], [[[PHI][x.sub.n-1]].sub.([alpha],[beta])]), then (29) implies that d([x.sub.n], [[[PHI][x.sub.n]].sub.([alpha],[beta])]) < d([x.sub.n], [[[PHI][x.sub.n]].sub.([alpha],[beta])]), a contradiction. Hence,

d ([x.sub.n], [[[PHI][x.sub.n]].sub.([alpha],[beta])]) < d([x.sub.n-1], [[[PHI][x.sub.n-1]].sub.([alpha],[beta])]). (30)

Thus, {[r.sub.n]} is a monotone nonincreasing sequence of nonnegative real numbers. Therefore, {[r.sub.n]} converges to inf{[r.sub.n] : n [greater than or equal to] 0}. Suppose inf{[r.sub.n] : n [greater than or equal to] 0} = r > 0. Take N so that n [greater than or equal to] N implies that

r [less than or equal to] [r.sub.n] < r + [delta]. (31)

It follows that

[r.sub.n+1] [less than or equal to] H([[[PHI][x.sub.n]].sub.([alpha],[beta])], [[[PHI][x.sub.n+1]].sub.([alpha],[beta])]) < r, (32)

which is a contradiction to the assumption that inf{[r.sub.n] : n [greater than or equal to] 0} = r > 0. Hence [r.sub.n] [right arrow] 0.

That is,

d([x.sub.n], [[[PHI][x.sub.n]].sub.([alpha],[beta])]) [right arrow] 0. (33)

It follows that H([[[PHI][x.sub.n]].sub.([alpha],[beta])], [[[PHI][x.sub.m]].sub.([alpha],[beta])]) [right arrow] 0. By completeness of (CB(Y), H) (see [39]), there exists a set W [member of] CB(Y) such that H([[[PHI][x.sub.n]].sub.([alpha],[beta])], W) [right arrow] 0. Let w [member of] W; then w [member of] [[[PHI]w].sub.([alpha],[beta])]. If not, let d(w, [[[PHI]w].sub.([alpha],[beta])]) = c > 0; then

[mathematical expression not reproducible]. (34)

In a limiting case when n [right arrow] [infinity], we have c < c, a contradiction. Hence,

w [member of] [[[PHI]w].sub.([alpha],[beta])]. (35)

Now,

[mathematical expression not reproducible]. (36)

Hence, [[[PHI]w].sub.([alpha],[beta])] = W for all w [member of] W.

Next, we will prove that there exists a selection of [PHI] which has a unique fixed point. For each u [member of] Y, [[[PHI]u].sub.([alpha],[beta])] is compact. Therefore, for w [member of] Y there exists [u.sub.w] [member of] [[[PHI]u].sub.([alpha],[beta])]) such that

d(w, [u.sub.w]) = d(w, [[[PHI][x.sub.0]].sub.([alpha],[beta])])). (37)

Let [psi] : Y [right arrow] Y defined as [psi]u = [u.sub.w] be a selection of [PHI] : Y [right arrow] [(IFX).sup.Y]. Then, for each u [member of] Y we have [psi]u = [u.sub.w] [member of] [[[PHI]u].sub.([alpha],[beta])]. Let [psi]w = v(= [w.sub.x]); then d(w, v) = d(w , [[[PHI]w].sub.([alpha],[beta])]) = 0. This implies that

v = w = [psi]w. (38)

Now,

[mathematical expression not reproducible]. (39)

It follows that the fixed point of [psi] is unique.

The following examples show that our results generalize a number of previous theorems.

Example 12. Let X be the set of all nonnegative integers with the Euclidean metric. Let [psi] : X [right arrow] X be defined as [psi]x = 2[x.sup.2] and let [PHI]: X [right arrow] [(IFS).sup.X] be an intuitionistic fuzzy map defined as

[mathematical expression not reproducible], (40)

where

[[OMEGA].sub.x] = {u [member of] [psi]X : u [less than or equal to] x}. (41)

For [alpha] = 3/4 and [beta] = 1/5,

[[[PHI]x].sub.(3/4,1/5)] = {t [member of] [psi]X : t [less than or equal to] x}. (42)

For [epsilon] > 0, there exists [delta](= [epsilon]) such that all the hypotheses of Theorem 9 are valid to obtain common fixed point of [psi] and [PHI]. Previously known results are not applicable to this example (even in the case when [PHI] is single valued, that is, [PHI]x = max{t [member of] [psi]X : t [less than or equal to] x}) since [psi][PHI]x [not equal to] [PHI][psi]x at x [not equal to] 0.

Example 13. Let X = R with the Euclidean metric, Y = [-20, 20], and A = ]10, 20[.

For X [member of] Y, define

[[GAMMA].sub.x] = {t : 2 - 1/x [less than or equal to] t [less than or equal to] 4 - 1/x}. (43)

Define intuitionistic fuzzy map [PHI] : Y [right arrow] [(IFS).sup.Y] as follows:

when X [member of] A,

[mathematical expression not reproducible]. (44)

When x [not member of] A,

[mathematical expression not reproducible]. (45)

For [alpha] = 1/2 and [beta] = 2/3,

[mathematical expression not reproducible]. (46)

For [epsilon] > 0, there exists [delta](= 59[epsilon]), such that [PHI] satisfies all the assumptions of Theorem 11. In this case, W = [2,4] [member of] CB(Y) such that [[[PHI]w].sub.(1/2, 1/3)] [member of] W for all w [member of] W and corresponding to w [member of] W the mapping [psi] : Y [right arrow] Y defined as

[mathematical expression not reproducible], (47)

is a selection of [PHI].

Data Availability

No data were used to support this study.

https://doi.org/10.1155/2018/1989423

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Shazia Kanwal (iD) (1) and Akbar Azam (iD) (2)

(1) Department of Mathematics, GC University, Faisalabad-38000, Pakistan

(2) Department of Mathematics, COMSATS University, Islamabad-44000, Pakistan

Correspondence should be addressed to Shazia Kanwal; shaziakanwal690@yahoo.com

Received 29 May 2018; Accepted 2 October 2018; Published 21 October 2018

Academic Editor: Jose A. Sanz
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Title Annotation:Research Article
Author:Kanwal, Shazia; Azam, Akbar
Publication:Advances in Fuzzy Systems
Date:Jan 1, 2018
Words:5023
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