Fixed-Point Results for Mappings Satisfying Implicit Relation in Orthogonal Fuzzy Metric Spaces

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Introduction
Te study of nonlinear analysis is greatly benefted by the adoption of fxed-point theory.It is a multidisciplinary topic that has applications in numerous areas of mathematics and in domains such as biology, chemistry, engineering, physics, game theory, mathematical economics, optimization theory, approximation theory, and variational inequality.It became even more fascinating through its applicability to solve differential and integral equations.It is typical practice to use mathematical techniques to examine diverse structural aspects and the functioning of multiple departments.As a consequence, handling ambivalence and imprecise evidence in a diverse range of situations happens naturally.In 1965, Zadeh [1] presented the idea of fuzzy sets, and since then, it has become an important mechanism for resolving cases involving ambiguity and uncertainty.Kramosil and Michálek [2] presented the conception of a fuzzy metric space in 1975, which was updated by George and Veeramani [3] in an attempt to generate a Hausdorf topology for a particular category of fuzzy metric spaces.Furthermore, many researchers explored fuzzy metric spaces, their applications, and their related scopes in [4][5][6][7][8][9][10][11][12][13][14][15][16][17].
In 2017, Gordji et al. [18] developed the conception of orthogonal sets and introduced orthogonal metric spaces as a generalisation of metric spaces.Later, in [19][20][21][22] etc., the authors added some generalisations of orthogonal metric spaces along with several fxed-point results.In 2018, Hezarjaribi [19] presented the notion of orthogonal fuzzy metric spaces, and a little work has been performed in this generalisation of a metric space [23][24][25].Te research gap addressed in our work lies in the establishment of new fxed-point results in orthogonal fuzzy metric spaces, specifcally focusing on the satisfaction of implicit relation by self-mappings.While existing literature provides valuable insights into fxed-point theory and fuzzy metric spaces, there is a need to explore the implications of considering self-mappings satisfying implicit relation in orthogonal fuzzy metric spaces.
By bridging this research gap, our paper contributes to the theoretical development of generalisation of fuzzy metric spaces and expands the understanding of fxed-point theory.Te consideration of implicit relations and self-mappings enriches the existing framework and ofers a versatile foundation that can be applied across various domains.
Furthermore, the results obtained in this study have direct applications in solving integral equations, as well as in addressing fractional diferential equations.Integral equations arise in various felds, including physics, engineering, and economics, and often pose challenges due to their nonlinearity and complex nature.Te fxed-point results established in our research can provide a valuable framework for developing efcient numerical methods to solve integral equations, leading to accurate solutions and improved computational efciency.Moreover, the utilization of these results can be extended to fractional diferential equations, which have gained signifcant attention in recent years due to their ability to model various phenomena in science and engineering.By incorporating the obtained fxed-point theorems, we can tackle the solution of fractional diferential equations, contributing to the advancement of this important area of research.
Te paper is structured into six sections, each focusing on specifc aspects of the study.In Section 1, the background and motivation for the research are introduced.Section 2 provides a comprehensive overview of basic concepts related to orthogonal fuzzy metric spaces, establishing the foundation for subsequent discussions.Section 3 is dedicated to the main theoretical contribution of the paper, where fxed-point results for functions satisfying implicit relation criteria in complete orthogonal fuzzy metric spaces are established.Tese results are supported by relevant examples, highlighting their signifcance and applicability.In Section 4, the consequences of the proven fxed-point results are explored, revealing their broader implications and generalisations.Furthermore, in Section 5, the practical relevance of established theory is demonstrated by applying it to the existence and uniqueness of solutions of nonlinear Volterra integral equations of the second kind.Tis application showcases the versatility and utility of the developed fxed-point results in real-world problem-solving.Finally, Section 6 provides the concluding remarks, summarizing the key fndings and highlighting avenues for further research.

Preliminaries
Te following are some preparatory considerations for the writing of the article.
Defnition 4 (see [18]).For a nonnull set X and a binary relation ⊥ defned over X, the pair (X, ⊥) is purported to be an orthogonal set if there remains an element r 0 ∈ X so that r 0 ⊥r or r⊥r 0 for each r ∈ X.
Defnition 5 (see [19]).For a fuzzy metric space (X, C, ⋆) and a binary relation ⊥ defned over X, an ordered fourtuple (X, C, ⋆, ⊥) is purported to be an orthogonal fuzzy metric space if there remains an element r 0 ∈ X so that r 0 ⊥r for each r ∈ X. Defnition 6 (see [19]).Let (X, C, ⋆, ⊥) be an orthogonal fuzzy metric space.Ten, 2 Advances in Fuzzy Systems (3) (5) An O-sequence r m   m ∈ N in X is said to be a Cauchy O-sequence if for any 0 < ϵ < 1 and t > 0, there exists a natural number m 0 so that C(r p , r q , t) > 1 − ϵ, for all p, q ≥ m 0 .(6) (X, C, ⋆, ⊥) is said to be an orthogonally complete (O-complete) fuzzy metric space if every Cauchy O-sequence in X converges to a point in X.
[F 1 ]: Example 4. We consider F: Ω 3 ×Ω ⟶ [− 1, 1] satisfying the following conditions: Theorem 7. Let (X, C, ⋆, ⊥) be a complete orthogonal fuzzy metric space and T: X ⟶ X be a ⊥ − preserving self-mapping satisfying the condition below: for r, s ∈ X and F ∈ F, F(C(r, s, t), C(Tr, r, t), C(Tr, s, t), C(Tr, Ts, t)) ≤ 0. ( Ten, T admits a unique fxed point L ∈ X.Moreover, T is ⊥ − continuous at point L. Proof.As (X, C, ⋆, ⊥) is an orthogonal fuzzy metric space, there exists r 0 ∈ X satisfying r 0 ⊥ s for all s ∈ X. Tis gives r 0 ⊥ Tr 0 .Let r m+1 � Tr m for all m ∈ N. As the mapping T is ⊥ -preserving, the sequence r m   m ∈ N is an O-sequence.We consider z m (t) � η C (P m , t) where P m � r m , r m+1 ,  r m+2 , . ..}, t > 0. Ten, lim m ⟶ ∞ z m (t) � z(t) for 0 ≤ z(t) ≤ 1.In the case r m+1 � r m for some m ∈ N, T admits a fxed point L ∈ X.So we presume that r m+1 ≠ r m for all m ∈ N.
Now, for a fxed value l ∈ N, take r � r m− 1 and s � r m+p− 1 in condition (5) where m ≥ l and p ∈ N, we obtain By the nondecreasing nature of C, we have By the nondecreasing nature of F and the nonincreasing nature of z m (t), Tis implies that So, for all m ≥ l, Taking l ⟶ ∞ gives z(t) ≥ μ(z(t)).Tis is a contradiction for the case z(t) ≠ 1. Tus, z(t)� 1.So lim m ⟶ ∞ z m (t) � 1 i.e., for any given ϵ > 0, there exists a natural number m 0 such that z m (t) > 1 − ϵ for all m ≥ m 0 .Ten, C(r m , r m+p , t) > 1 − ϵ for p ∈ N and m ≥ m 0 .Terefore, the sequence r m   is an O-Cauchy sequence.By the Ocompleteness property of (X, C, ⋆, ⊥), this ensures that there exists a point L ∈ X such that lim m ⟶ ∞ Tr m � L. Now, by substituting r � r m and s � L in condition (5), we obtain Advances in Fuzzy Systems F C r m , L, t , C Tr m , L, t , C Tr m , r m , t , C Tr m , TL, t   ≤ 0, (10) which implies F C r m , L, t , C r m+1 , L, t , C r m+1 , r m , t , C r m+1 , TL, t   ≤ 0. ( By leading m ⟶ ∞, we obtain Tus, TL � L, or L is a fxed point of T. We consider M as another fxed point of T such that L ≠ M. By considering r � L and s � M in condition (5), we obtain or, Using the nondecreasing property of F on Ω 3 , we have which implies μ(C(L, M, t)) ≤ C(L, M, t).Also, C(L, M, t) < μ(C(L, M, t)).Tis is a contradiction.Hence, L � M. We now show that the self-mapping T is ⊥-continuous at the point L. We consider an O-sequence s m   in X. Ten, lim m ⟶ ∞ s m � L.
Putting r� Lands � s m in condition (5), we get F C L, s m , t , C(TL, L, t), C TL, s m , t , C TL, Ts m , t   ≤ 0. ( Tis implies { } and defne a binary relation ⊥ on X as r⊥s if and only if r ≤ s.Ten, it is easy to verify the following. (X, ⊥) is an orthogonal set.We defne the OFM by C(r, s, t) � e − |r− s|/t with a continuous t-norm ⋆ defned as, α ⋆ β � min α, β  .
(Z, ⊥) is an orthogonal set.We defne the OFM, C (where in ⋆ is defned as, Tus, all the hypotheses of Teorem 7 are satisfed, and hence, T admits a unique fxed point in (Z, C, ⋆, ⊥).Moreover, the unique fxed point of T is 0.

Consequences
Corollary 8. Let (X, C, ⋆, ⊥) be a complete orthogonal fuzzy metric space and T: X ⟶ X be a ⊥− preserving selfmapping with the following conditions: where F ∈ F.
Ten, T admits a unique fxed point L ∈ X, and at the point L, T k is ⊥ − continuous.
Proof.By Teorem 7, it can be easily proved that T admits a unique fxed point L ∈ X and T k is ⊥ − continuous at point L. Also, since TL � TT k L � T k TL, TL also becomes a fxed point of T, TL � L by the uniqueness of the fxed point of T. □ Corollary 9. Let (X, C, ⋆, ⊥) be a complete orthogonal fuzzy metric space and T: X ⟶ X be a ⊥− preserving selfmapping with the following conditions: Ten, T admits a unique fxed point L ∈ X, and T is ⊥ − continuous at point L.
Corollary 10.Let (X, C, ⋆, ⊥) be a complete orthogonal fuzzy metric space and T: X ⟶ X be a ⊥− preserving selfmapping with the following conditions: Advances in Fuzzy Systems Ten, T admits a unique fxed point L ∈ X.Moreover, at point L, T k is ⊥− continuous.
Corollary 11.Let (X, C, ⋆, ⊥) be a complete orthogonal fuzzy metric space and T: X ⟶ X be a ⊥− preserving selfmapping with the following conditions: where each b i ≥ 0,  3 i�1 b i � 1 Ten, T admits a unique fxed point L ∈ X, and T is ⊥− continuous at point L.
Corollary 12. Let (X, C, ⋆, ⊥) be a complete orthogonal fuzzy metric space and T: X ⟶ X be a ⊥− preserving selfmapping with the following conditions: where each b i ≥ 0,  3 i�1 b i � 1.
Ten, T admits a unique fxed point, L ∈ X.Moreover, at point L, T k is ⊥− continuous.

Application
Te fxed-point theorems obtained in this work can be utilised to develop efective methods for solving integral equations, enabling the analysis and computation of solutions in scenarios where traditional methods may be limited.Furthermore, the applicability of these results extends to fractional diferential equations, as many fractional calculus problems can be reduced to integral equations.
In this section, we apply Teorem 7 to demonstrate the existence and uniqueness of a solution of nonlinear integral equations.
We consider X � u ∈ C([0, P], R) | u(η) ≥ 0 and P ∈ R   and an integral equation of the form: where λ > 0, A(η) is a fuzzy function of η where η ∈ [0, P] and I K : [0, P] × R ⟶ R + is an integral kernel.We aim to show the existence and uniqueness of the solution of (25) by applying Teorem 7.
Proof.We defne an operator T: X ⟶ X as We observe that, for any u, v ∈ X, u⊥v ⇒ Tu⊥Tv.Hence, T is ⊥− preserving.

Conclusion
In this work, some fxed-point results are proved in orthogonal fuzzy metric spaces, with a particular focus on the satisfaction of implicit relations through self-mappings.Several illustrations are provided to validate the established results.Some consequences of the established fxed-point results in orthogonal fuzzy metric spaces are also shown.Our fndings contribute to the theoretical understanding of extension of fuzzy metric spaces and expanding the applicability of fxed-point theory in diverse domains.Te fxedpoint results presented in this paper have practical implications, one of which is the demonstration of the existence and uniqueness of solutions for integral equations.By applying the established fxed-point results, we provide evidence for the viability of proven results in solving integral equations.While our fndings have practical implications for solving integral equations and addressing fractional diferential equations, further research is needed to extend the framework, enhance computational aspects, and explore additional applications in various domains.Te established results can be extended to the best proximity point results in the future.Te fndings in this article would enable researchers to enhance the further exploration of orthogonal fuzzy metric spaces in terms of developing a general framework for implementation of the fxed-point results to obtain techniques to handle real-world problems.