Fixed point theory is one of the famous and traditional theories in mathematics and has numerous applications. The Banach contraction mapping is one of the pivotal results of the analysis. It is a very famous tool for solving existing problems in many fields of mathematical analysis and its applications [1]. Hence, numerous theories have developed in extending its notion in many diverse ways. Accordingly, if 
is a contraction on a Banach space 
, then has a unique fixed point in . Many researchers studied the Banach fixed point theorem in diverse ways and show its generalizations and applications. Among them, Bakhtin [2] introduced a very important generalization of the idea of a metric space, which is later used by Czerwick [3, 4] to present the findings of their work.
Fixed point theorems have been studied in many contexts, one of which is the fuzzy setting. The concept of fuzzy sets was initially introduced by Zadeh [8] in 1965. To use this concept in topology and analysis, so many authors have extensively developed the theory of fuzzy sets and its applications. One of the most significant and interesting works in fuzzy topology is to appropriately find the definition of fuzzy metric space and its applications. It is well known that a fuzzy metric space is an important generalization of the metric space. Many authors have considered this problem and have introduced it in different ways as there remains considerable literature about fixed point properties for mappings defined on fuzzy metric spaces, which have been examined by many researchers [9, 10, 11, 12].
Zhu and Xiao [7] and Hu [6] gave a coupled fixed point theorem for contractions in fuzzy metric spaces, and Fang [5] proved some common fixed point theorems under 
-contractions for compatible and weakly compatible mappings on Menger probabilistic metric spaces. Moreover, Elagan and Segi Rahmat [13] studied the existence of a fixed point in locally convex topology generated by fuzzy 
-normed spaces. The purpose of this paper is to generalize and extend the work done in [17] on coupled fixed point theorems in 
-algebra valued fuzzy metric spaces to tripled fixed point theorems in -algebra valued fuzzy metric spaces with application.
2 Preliminary Notes
In this section, we will discuss few of the basic concepts of -algebra-valued metric space.
Definition 2.1 [14] Let be an arbitrary nonempty set, 
a continuous 
-norm, and 
a fuzzy set on 
The 3-tuple 
is called a fuzzy metric space (FMS) if the following conditions are satisfied 
and 
,
(i) 
,
(ii) 
if and only if 
,
(iii) 
,
(iv) 
,
(v) 
is continuous,
(vi) 
.
Definition 2.2 [16] Let be a fuzzy metric space. A sequence 
in is said to converge to 
if 
.
Lemma 2.1 [16] Let be a fuzzy metric space and 
, 
are sequences in such that 
, 
then 
for every continuity point of 
.
Definition 2.3 [15] Let be a nonempty set. Suppose that the mapping 
satisfies:
(i) 
,
(ii) 
if and only if ,
(iii) 
,
(iv) 
.
Definition 2.4 [15] Let 
be a C*-algebra-valued metric space. A mapping 
is called a C*-algebra-valued contraction mapping on , if exists 
with 
such that
.
Definition 2.5 [17] Let 
be a -algebra valued metric space (-AVMS). A mapping 
is a -algebra valued contraction mapping on , if with such that
Definition 2.6 [17] Let be a nonempty set and 
be a FMS. A mapping is said to be a -AVCM if there exists with such that
and .
3 Main results
In this section, we will firstly define the concepts of -algebra valued contraction mapping (-AVCM) in metric and fuzzy metric spaces.
Definition 3.1 Let be a -AVMS. Then, a mapping 
is a -algebra valued tripled contraction mapping on , if with such that
and
Definition 3.2 Let be a nonempty set and be a FMS. A mapping is said to be a -AVCM if there exists with such that
and
and .
Theorem 3.1 Let be a Cauchy FMS. A mapping is a -AVCM if 
has a unique fixed point in .
Proof. If 
let 
. Then, 
is a sequence for 
and define 
in 
. By -algebra, if 
and 
then 
. Then,
where 
Now, if 
and using the triangular inequality of fuzzy metric spaces, we have
Then, is a Cauchy sequence in with respect to . Then, 
is Cauchy and 
such that 
i.e., 
Then,
Hence, 
, and similarly, 
and 
, which shows that 
and 
are tripled fixed point of .
Now, we will show that and are unique tripled fixed point. If 
, 
and 
be tripled fixed point of 
then 
, 
and 
. Hence, by the contraction condition (1), (2) and (3), we have,
Similarly,
and
Since 
, it is a contradiction. Therefore, has a unique tripled fixed point. Hence, this completes the proof.
4 Applications
In this section, we apply the main results to the existence of the solution of integral equations.
Let 
and 
be a Hilbert space, where 
represent a set of Lebesque measurable. For 
, let 
, where 
If 
there exists 
be continuous, and 
and 
for 
and for 
we obtain
and
Hence, 
are the unique solution of the integral equation,
Proof. Let 
be a Cauchy -AVFMS with respect to 
. Suppose be a mapping, we have
Hence,
similarly,
and
for 
, 
and 
. Then, 
are the unique solutions of the integral equations.
References
[1] Kamran T, Postolache M, Ghiura A, Batul S and Ali R, The Banach contraction principle in C*-algebra-valued b-metric spaces with application. Fixed Point Theory and Applications. 2016; 10. (Doi: 10.1186/s13663-015-0486-z)
[2] Bakhtin IA, The contraction mapping principle in quasimetric spaces. Functional Analysis. 1989; 30:26-37
[3] Czerwick S, Contraction mappings in b-metric spaces. Acta Math Inform Univ Ostrav. 1993; 1:5-11
[4] Czerwick S, Nonlinear set-valued contraction mappings in b-metric spaces. Atti Semin Mat Fis Univ Modena. 1998; 46:263-276
[5] Fang J, Common fixed point theorems of compatible and weakly compatible maps in Menger spaces. Nonlinear Anal. 2009; 5-6:1833-1843 (Doi: 10.1016/j.na.2009.01.018)
[6] Hu X, Common coupled fixed point theorems for contractive mappings in fuzzy metric spaces. Fixed Point Theory Appl. 2011; Article ID 363716 (Doi: 10.1155/2011/363716)
[7] Zhu X, Xiao J, Note on “Coupled fixed point theorems for contractions in fuzzy metric spaces”. Nonlinear Anal 2011; 74(16):5475-5479 (Doi: 10.1016/j.na.2011.05.034)
[8] Zadeh L, Fuzzy sets. Inf Control. 1965; 8:338-353 (Doi: 10.1016/S0019-9958(65)90241-X)
[9] Kramosil I, Michalek J, Fuzzy metric and statistical metric spaces. Kybernetika. 1975; 11:326-333
[10] Cho Y, Fixed points in fuzzy metric spaces. J Fuzzy Math. 1997; 5(4):949-962
[11] Gregori V, Sapena A, On fixed-point theorems in fuzzy metric spaces. Fuzzy Sets Syst. 2002; 125(2):245-252 (Doi: 10.1016/S0165-0114(00)00088-9)
[12] Beg I, Abbas M, Common fixed points of Banach operator pair on fuzzy normed spaces. Fixed Point Theory. 2011; 12(2):285-292
[13] Elagan SK, Rahmat MS, Some fixed points theorems in locally convex topology generated by fuzzy n-normed spaces. Iran J Fuzzy Syst. 2012; 9(4):43-54
[14] Khaogong C, Khammahawong K, Fixed point theorem in -algebra-valued fuzzy metric metric spaces with application. Thai Journal of Mathematics. 2021; 19(3):964-970
[15] Ma ZH, Jiang LN, Sun HK, C*-algebra-valued metric spaces and related fixed point theorems. Fixed Point Theory and Application. 2014; 206:1-11 (Doi: 10.1186/s13663-015-0471-6)
[16] Mihet D, On fuzzy contractive mappings in fuzzy metric spaces. Fuzzy Sets and Systems. 2007; 158:915-921 (Doi: 10.1016/j.fss.2006.11.012)
[17] Aniki SA, An extension of mixed monotone mapping to tripled fixed point theorem in fuzzy metric. Theory of Approximation and Applications. 2022; 16(1):101-113 (Doi: 20.1001.1.25382217.2022.16.1.12.8)
(Department of Mathematics, Faculty of Science, Confluence University of Science and Technology, Osara, Kogi State, Nigeria)
