Unique Fixed-Point Results in Fuzzy Metric Spaces with an Application to Fredholm Integral Equations

Department of Mathematics, Gomal University, Dera Ismail Khan 29050, Pakistan Université de Sousse, Institut Supérieur d’Informatique et Des Techniques de Communication, H. Sousse 4000, Tunisia Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa China Medical University Hospital, China Medical University, Taichung 40402, Taiwan Department of Mathematics, COMSATS University Islamabad, Wah Cantt. 47040, Pakistan Department of Mathematics, Taiz University, Taiz 6803, Yemen


Introduction
In 1922, Banach [1] proved a "Banach contraction principle (BCP)," which is stated as "a self-mapping in a complete metric space satisfying a contraction condition has a unique fixed point". This theorem plays a very important role in the theory of fixed points. Many researchers gave generalization and improved the BCP in many directions for single-valued and multivalued mappings in the context of metric spaces by ensuring the existence of fixed point, common fixed point, and coincidence point results with different types of applications, such as differential-type applications, integral-type applications, functional-type applications. In 2004, Ran and Reurings [2] proved a fixed-point theorem in a metric space by using partially ordered sets and they present some applications to matrix equations. While in [3], Nieto and Rodrguez-López extended and improved the result of Ran and Reurings [2] by using increasing mappings and applied the result to get a unique solution for the first-order ordinary differential equation with periodic boundary equations. In 2017, Priskillal and Thangavelu [4] established some fixed-point theorems in complete metric spaces by using ψ-contractive fuzzy mappings with an application to fuzzy differential equations. Some more fixed-point results in the context of metric spaces can be found in [5][6][7][8][9][10][11][12][13][14][15][16].
In 1965, the theory of fuzzy sets was introduced by Zadeh [17]. Lately, this theory is improved, investigated, and applied in many directions. Among them, we state the theory of fuzzy logic, which is based on the notion of relative graded memberships, as inspired by the processing of human perceptions and cognitions. Fuzzy logic can deal with information arising from computational perceptions and cognitions, that is, uncertain, obscure, imprecise, partly true, or without sharp limits. A fuzzy logic permits the inclusion of vague human assessments in computing problems. The fuzzy logic is extremely useful for many people associated with innovative work including engineering (electrical, chemical, civil, environmental, mechanical, industrial, geological, etc.), mathematics, computer software, earth science, and physics. Some of their findings can be found in [18][19][20][21][22][23][24][25].
The other direction of fuzzy sets is used in topology and analysis by many mathematicians. Subsequently, several authors have applied various forms of general topologies and developed the concept of fuzzy spaces. Kramosil and Michalek [26] developed the concept of a fuzzy metric space (FMspace). Later on, Grabeic [27] extended the BCP and proved a fixed-point result in FM-spaces in the sense of Kramosil and Michalek. George and Veeramani [28] modified the concept of FM-spaces with the help of continuous t-norms and proved some basic properties in this direction. In 2002, Gregori and Sapena [29] proved some contractive-type fixed-point theorems in complete FM-spaces in the sense of Kramosil and Michalek [26] and in the sense of George and Veeramani [28]. Rana et al. [30] established some fixed-point theorems in FM-spaces by using implicit relations. Many authors have introduced the number of fixed-point theorems in FMspaces by using the concept of compatible maps, implicit relations, weakly compatible maps, and R-weakly compatible maps (see [31][32][33][34][35][36][37][38] and the references therein). Furthermore, Beg and Abbas [39], Popa [40], and Imad et al. [41,42] obtained some fixed-point and invariant approximation results in FM-spaces. Recently, Li et al. [43] proved some strong coupled fixed-point theorems in FM-spaces with an integral-type application. Later on, Rehman et al. [44] proved some rational fuzzy-contraction theorems in FM-spaces with nonlinear integral-type application.
The purpose of this paper is at obtaining some extended unique fixed-point theorems in FM-spaces without the "assumption that all the sequences are Cauchy" by using the concept of Li et al. [43] and Rehman et al. [44]. We present some illustrative examples and an integral-type application to support our work. By using this concept, one can prove more generalized contractive-type fixed-point and common fixed-point results in FM-spaces with different types of integral equations. Our paper is organized as follows: Section 2 consists of preliminary concepts. In Section 3, we prove some generalized fixed-point results without continuity in FMspaces and we presented some examples in the support of our obtained results. In Section 4, we consider some generalized Ćirić fuzzy contraction results in complete FM-spaces. In Section 5, we present an application of a particular case of the Fredholm integral equation of the second kind by ensuring the existence of a solution.

Preliminaries
The concept of a continuous t-norm is given by Schweizer and Sklar [45].
Lemma 6 (see [46]). Let ðW, M F , * Þ be a FM-space. Let w ∈ W and fw i g be a sequence in W. Then, Definition 7 (see [29]). Let ðW, M F , * Þ be a FM-space and G : W ⟶ W. Then, G is known as a fuzzy contraction, if there is 0 < h < 1 so that for all w, x ∈ W and t > 0.

Generalized Fixed-Point Results in FM-Spaces
In this section, we consider some generalized contraction theorems on FM-spaces for fixed points (by using the "triangular property of the fuzzy metric").
Proof. Fix w 0 ∈ W. Take an iterative sequence fw i g such that w i+1 = Gw i for all i ≥ 0: Now, by view of (8), we have Then, we have for t > 0,
The uniqueness is as follows: let κ * ∈ W be such that G κ * = κ * . Then, in view of (8), we have for t > 0

Journal of Function Spaces
Hence, we get that M F ðκ, κ * , tÞ = 1, so κ = κ * . Thus, G has a unique fixed point in W.

Corollary 9.
Let ðW, M F , * Þ be a complete FM-space so that the fuzzy metric M F is triangular. Let G : W ⟶ W verify that for all w, x ∈ W, t > 0, a ∈ ð0, 1Þ, and b ∈ ½0, 1/4Þ with ða + 2bÞ < 1. Then, G has a unique fixed point.
Example 10. Let W = ½0, 1 be equipped with a continuous t -norm. Let M F : W × W × ð0,∞Þ ⟶ ½0, 1 be a fuzzy metric defined by for all w, x ∈ W and t > 0. Then, ðW, M F , * Þ is a complete FM-space. Now, we define the mapping G : W ⟶ W by , for all w ∈ 0, 1 ½ and t > 0: Then, Hence, all the conditions of Corollary 9 are satisfied with a = 2/3 and b = 1/15. Hence, the self-mapping G has a unique fixed point, that is, Gð4/5Þ = 4/5 ∈ ½0, 1: Corollary 11. Let ðW, M F , * Þ be a complete FM-space so that the fuzzy metric M F is triangular. Let G : W ⟶ W verify that for all w, x ∈ W, t > 0, a ∈ ð0, 1Þ, and c ∈ ½0, 1Þ with ða + 2cÞ < 1. Then, G has a unique fixed point.
for all w, x ∈ W and t > 0. Then, ðW, M F , * Þ is a complete FM-space. Now, we define a mapping G : W ⟶ W by We have

Journal of Function Spaces
Hence, all the conditions of Corollary 11 are satisfied with a = 3/4 and b = 1/9. Then, the self-mapping G has a unique fixed point, that is, Gð7Þ = 7:

Ćirić-Type Fuzzy Contraction Results in FM-Spaces
In this section, we define Ćirić-type fuzzy contraction mappings and we present a unique related fixed-point theorem on a complete FM-space.
Definition 13. Let ðW, M F , * Þ be a complete FM-space. A self-mapping G : W ⟶ W is said to be a Ćirić contraction if there is α ∈ ð0, 1Þ such that for all w, x ∈ W and t > 0. Here, α is called the contractive constant of T.

Theorem 14.
Let a self-mapping G : W ⟶ W be a Ćirić contraction in a complete FM-space ðW, M F , * Þ so that M F is triangular and (33) satisfies with 2α < 1. Then, G has a unique fixed point.
Proof. Fix w 0 ∈ W. Take an iterative sequence fw i g such that w i+1 = Gw i for all i ≥ 0: Now, by using (33), we have After simplification, for t > 0, we get Now, there are three possibilities: for t > 0, which is not possible Let δ ≔ max fα, a/ð1 − αÞg < 1. Using (36) and (38), we have

Journal of Function Spaces
Similarly, Now, from (39) and (40) and by induction, for t > 0 This yields that Since M F is triangular and k > i, we have Hence, fw i g is a Cauchy sequence. Since ðW, M F , * Þ is complete, there is κ ∈ W so that Now, we have to show that Gκ = κ. Since M F is triangular, one writes In view of (33), (42), and (44), we have for t > 0 Hence, This together with (45) and (44), we have Since α ∈ ð0, 1Þ, one gets M F ðκ, Gκ, tÞ = 1. This implies that Gκ = κ.
The uniqueness is as follows: let κ * ∈ W be such that G κ = κ * . Using (33), we have We get that M F ðκ, κ * , tÞ = 1 for t > 0. This implies that κ = κ * . Thus, G has a unique fixed point in W.
Example 15. Let W = ½0,∞Þ be endowed with a continuous t -norm. Let a fuzzy metric M F : W × W × ð0,∞Þ ⟶ ½0, 1 be defined by for all w, x ∈ W and t > 0. Then, ðW, M F , * Þ is a complete FM-space. Now, we define a mapping G : W ⟶ W as We have for t > 0. Hence, all the conditions of Theorem 14 are satisfied with α = 1/3 and G has a unique fixed point, that is, G ð16/3Þ = 16/3:

Application
In this section, we present an integral-type equation. Let W = Cð½0, ξ, ℝÞ be the space of all real-valued continuous functions on the interval ½0, ξ, where 0 < ξ ∈ ℝ. Now, we present a particular case of a Fredholm integral equation (FIE) of the second kind given as follows: where τ ∈ ½0, ξ and K : The binary operation * , being a continuous t-norm, is defined by α * β = αβ for all α, β ∈ ½0, ξ. The standard fuzzy metric M F : W × W × ð0,∞Þ ⟶ ½0, 1 can be expressed as Then easily, we can show that M F is triangular and ðW , M F , * Þ is a complete FM − space.

Theorem 16.
Assume that there is η ∈ ð0, 1Þ so that where Then, the FIE (53) has a unique solution.
Proof. Give G : W ⟶ W as Notice that G is well defined and (53) has a unique solution if and only if G has a unique fixed point in W. Now, we have to show that Theorem 8 is applied to the integral operator G. Then, for all w, x ∈ W, we have the following six cases: (1) If kw − xk is the maximum term in (57), then NðG , w, xÞ = kw − xk. Therefore, in view of (55) and (56), we have

Journal of Function Spaces
This implies that for all w, x ∈ W such that Gw ≠ Gx. The inequality (60) holds if Gw = Gx. Thus, the integral operator G satisfies all the conditions of Theorem 8 with η = a and b = c = 0 in (8). Then, the integral operator G has a unique fixed point, i.e., (53) has a solution in W (2) If kGw − wk is the maximum term in (57), then, N ðG, w, xÞ = kGw − wk. Therefore, using (55) and (56), we have It yields that for all w, x ∈ W such that Gw ≠ Gx (3) If kGx − xk is the maximum term in (57), then Nð G, w, xÞ = kGx − xk. Therefore, by (55) and (56), we have That is, for all w, x ∈ W such that Gw ≠ Gx (4) If kGw − xk is the maximum term in (57), then, NðG, w, xÞ = kGw − xk. Therefore, due to (55) and (56), we have Hence, for all w, x ∈ W such that Gw ≠ Gx (5) If ∥Gx − w∥ is the maximum term in (57), then, Nð G, w, xÞ = kGx − wk. Using (55) and (56), we have It implies that for all w, x ∈ W such that Gw ≠ Gx The inequalities (62), (64), (66), and (68) hold if Gw = Gx. Thus, the integral operator G satisfies all the conditions of Theorem 8 with η = c and a = b = 0 in (8). The integral operator G has a unique fixed point, i.e., (53) has a solution in W.
Therefore, from (55) and (56), we have Journal of Function Spaces That is, for all w, x ∈ W such that Gw ≠ Gx. The inequality (70) holds if Gw = Gx. Thus, the integral operator G satisfies all the conditions of Theorem 8 with η = b and a = c = 0 in (8). The integral operator G has a unique fixed point, i.e., (53) has a solution in W. Now, we present a special type of example for a particular case of an FIE of a second kind.

Conclusion
In this paper, we proved variant unique fixed-point results for some generalized contraction-type self-mappings in complete FM-spaces, without continuity and by using the "triangular property of the fuzzy metric" as a basic tool. We presented illustrative examples. Moreover, we provided an application about a particular case of Fredholm integral equation of second kind. In this direction, researchers can prove more fixed-point results in complete FM-spaces without using continuity via different types of applications.

Data Availability
Data sharing is not applicable to this article as no dataset were generated or analysed during the current study.