Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3127 https://internationalpubls.com Fixed Point Theorem for H – F – Contractive Mapping in Complete Fuzzy Metric Spaces J. Ravinder1, A. Bernick Raj2* 1 & 2 Department of Mathematics and Actuarial Science, B.S. Abdur Rahman Crescent Institute of Science and Technology, Chennai, Tamil Nadu, India, 600048. Email: 1 ravinder.maths@gmail.com, 2 bernickraj@crescent.education Article History: Received: 02-01-2025 Revised: 25-02-2025 Accepted: 20-03-2025 Abstract: This study introduces a new class of contractive mappings, namely H –F – contractive mappings, in the setting of complete fuzzy metric spaces. We provide the sufficient conditions under which such mappings admit unique fixed points. Our approach builds on classical results by employing a novel functional inequality that involves a strictly increasing function F ∈ F and auxiliary function H ∈ H. An illustrative example demonstrates the applicability of the main results. This work enriches the framework of the fixed point theory in fuzzy metric spaces and opens pathways for further exploration. Keywords: Fixed point, fuzzy metric space, H –F –contractive mapping, fuzzy analysis, contraction principle. 1. Introduction The theory of fuzzy metric spaces, initiated by Kramosil and Michalek [1] and further developed by George and Veeramani [2], has received considerable attention for addressing the uncertainties in mathematical models. Several classical fixed point results, such as Banach, Kannan, and Chatterjea contractions, have been generalized to fuzzy metric settings [3, 4, 5, 6, 7]. In this context, new con- traction types, such as F -contractions and simulation functions have broadened the scope of fixed point analysis. Motivated by these developments, we propose the concept of H –F –contractive mappings. These mappings incorporate an auxiliary function H and transformation function F, which allows us to derive a fixed point theorem that generalizes several known results. Our main contribution lies in establishing a unique fixed point under the new contractive condition. 2. Preliminaries Here, we recall the key definitions required for our analysis. Throughout this paper, N denotes the set of all positive integers, N0 denotes the set of all non-negative integers, R+ = (0, ∞) and I = [0, 1]. Definition 1. [8] A binary operation ∗ : I × I → I is called a continuous t-norm, if the following conditions holds: (i) ∗ is associative and commutative. (ii) For any a ∈ I, a ∗ 1 = a. (iii) For a, b, c, d ∈ I with a ≤ b and c ≤ d, a ∗ c ≤ b ∗ d. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3128 https://internationalpubls.com (iv) ∗ is continuous. For example, a ∗1 b = min{a, b}, a ∗2 b = ab and a ∗3 b = max{a + b − 1, 0} are commonly used continuous t-norms. Definition 2. [2] The 3-tuple (X, M, ∗) is said to be a fuzzy metric space (referred to as GV - Fuzzy Metric Space), if X is an arbitrary non-empty set, ∗ is a continuous t-norm and M is a fuzzy set defined on X2 × R+ satisfying the following conditions; (GV-1) M (x, y, 0) > 0 for all x, y ∈ X. (GV-2) M (x, y, t) = 1 for all t ∈ R+ iff x = y. (GV-3) M (x, y, t) = M (y, x, t) for any t ∈ R+. (GV-4) M (x, z, t+s) ≥ M (x, y, t)∗M (y, z, s) for all x, y, z ∈ X and t, s ∈ R+. (GV-5) M (x, y, .) : R+ → I is continuous. Remarks: 1. [2] For any r ∈ (0, 1), whenever M (x, y, t) > 1 − r for x, y ∈ X and t ∈ R+, there exists t0 ∈ (0, t) such that M (x, y, t0) > 1 − r. 2. For any r1 > r2, there exists r3 such that r1 ∗ r3 > r2, and for any r4, we can find an r5 such that r5 ∗ r5 > r4, where r1, r2, r3, r4, r5 ∈ (0, 1). 3. [3, 2] It is well known and easy to verify that for every x, y ∈ X, M (x, y, .) is a non- decreasing continuous function of R+. Definition 3. Let (X, M, ∗) be a fuzzy metric space. (i) A sequence {xn} converges to x ∈ X, if every δ ∈ (0, 1) and t > 0, there exists n0 ∈ N such that M (xn, x, t) > 1 − δ for all n ≥ n0. This implies that lim ( , , ) 1.n n M x x t → = (ii) A sequence {xn} is said to be a Cauchy, if for every ϵ > 0 and t ∈ R+, there exists n0 ∈ N such that M (xm, xn, t) > 1 − ϵ for all m > n ≥ n0. Moreover, lim ( , , ) 1,n p n n M x x t+ → = for every p ∈ N and t ∈ R+. (iii) A fuzzy metric space in which every Cauchy sequence is convergent is called the complete fuzzy metric space. 3. Main Result In the following, F denotes the set of all strictly increasing functions F:(0,1)→R satisfying conditions 0 lim ( ) p F p +→ = − and 1 lim ( ) . p F p −→ =  In addition, H denotes the set of mappings H:X×X→I satisfies the conditions, for any sequence {xn} ⊂ X, 1lim ( , ) 0n n n H x x + → = and H(x y) = 0, if x = y. Definition 4. For any F ∈ F, a mapping T : X → X is said to be H – F – contractive if there exists a function H ∈ H such that F (M (Tx, Ty, t)) ≥ H(x, y) + F (M (x, y, t)) (1) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3129 https://internationalpubls.com for all x, y ∈ X with .x y Theorem 1. Let (X, M, ∗) be a complete fuzzy metric space. If T : X → X is an H – F – contractive mapping, then T has a unique fixed point in X. Proof: Let x0 ∈ X and xn+1 = Txn for all n ∈ N0. Assume that T : X → X is an H – F – contractive mapping. For some n ∈ N0, if xn+1 = Txn = xn, then xn is a fixed point. Assume that for any n ∈ N0, Txn = xn+1  xn. From Equation (1), for every n ∈ N0 and t > 0, we have F (M (Txn, Tnn+1, t)) ≥ H(xn, xn+1) + F (M (xn, xn+1, t)). (2) Because H(xn, xn+1) > 0 for all n ∈ N, F (M (Txn, Tnn+1, t)) ≥ H(xn, xn+1)+F (M (xn, xn+1, t)) > F (M (xn, xn+1, t)). (3) We know that F is a strictly increasing function, therefore, M (Txn, Tnn+1, t) = M (xn+1, xn+2, t) > M (xn, xn+1, t). Thus, for every t > 0, {M (xn, xn+1, t)} is an increasing sequence bounded from above in I. Hence, for every t > 0, {M (xn, xn+1, t)} converges in I. That is, for every t > 0, there exists δ(t) ∈ (0, 1) and N ∈ N0, such that M (xn, xn+1, t) > 1 − δ(t) for all n ≥ N. Moreover, for every t > 0, suppose that there exists α(t) ∈ I such that as n → ∞ the sequence {M (xn, xn+1, t)} approaches to its limit α(t) from its left side. 1lim ( , , ) ( ) .n n n M x x t t − + →  = And, this imply ( ) ( )1lim ( , , ) ( ) .n n n F M x x t F t − + → = By taking limit both sides of equation (3), we get ( ) ( ) ( )1( ) lim ( , ) ( ) ( ) .n n n F t H x x F t F t  − − − + →  +  This is possible when 1lim ( , ) 0n n n H x x + → = (4) and ( )( ) 0,F t − = which contradict with ( )( ) 0.F t −  Therefore, we have 1lim ( , , ) 1 .n n n M x x t − + → = (5) Furthermore, we need to prove that {xn} is a Cauchy sequence. For m, n ∈ N0 and equation (2) implies that ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 2 1 2 2 1 1 0 ( , , ) ( , , ) ( , ) ( , , ) ( , ) ( , ) ( , , ) . . . ( , ) ( , , ) n m n m n m n m n m n m n m n m n m n m n m n m n m n m m n k n k n n k F M x x t F M Tx Tx t H x x F M x x t H x x H x x F M x x t H x x F M x x t + + + + − + + − + + − + + − + + − + − + − + − + − + + = =  +  + +  + Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3130 https://internationalpubls.com By taking the limits of both sides and together from equations (3) and (5), we obtain ( ) ( ) ( )1 1 0 lim ( , , ) lim ( , ) 1 1 . m n m n m n k n k n n k F M x x t H x x F F− − + + + + − + → → =  +  From Equation (4), it is clear that for every k ∈ N0, 1lim ( , ) 0.n k n k n H x x+ − + → = Hence ( ) ( )1lim ( , , ) 1 ,n m n m n F M x x t F − + + + → = and hence 1lim ( , , ) 1 .n m n m n M x x t − + + + → = Therefore {xn} is a Cauchy sequence. Since (X, M, ∗) is complete, therefore there exists z ∈ X such that lim .n n x z → = (6) It is easy to verify that z is a fixed point of T. Since T is continuous and equation (7), implies that ( ) 1( ) lim lim lim .n n n n n n T z T x Tx x z+ → → → = = = = (7) Finally, to prove the uniqueness of the fixed point z of T, assume that there exists another fixed point u of T in X. That is T (u) = u, where z  u. Now, by using equation (1) and (3), we get F (M (Tz, Tu, t)) ≥ H(z, u) + F (M (z, u, t)) > F (M (z, u, t)) ⇒ F (M (z, u, t)) ≥ H(z, u) + F (M (z, u, t)) > F (M (z, u, t)). As a consequence, we have F (M (z, u, t)) > F (M (z, u, t)), this imply, M (z, u, t) > M (z, u, t). This is a contradiction, hence z = u. That is T has a unique fixed point in X. Example 1. Let X = R, a ∗ b = min{a, b} for all a, b ∈ I and ( , , ) | | t M x y t t x y = + − for all x, y ∈ X and t > 0. Subsequently, (X, M, ∗) is a complete fuzzy metric space. Now, let F ∈ F be defined as 2 1 ( ) (1 ) p F p p p − = − for all p ∈ (0, 1), and let H ∈ H defined as H(x, y) = |x − y|. Now, define T:X→X, ( ) 4 x T x = for all x ∈ X. Figure 1: H – F – contractive From the graph given in the Figure 1, it is clear that, for any x, y ∈ X and t > 0, T is a H – F – contractive. Thus, all the conditions of Theorem 1 are satisfied, therefore T has a unique fixed point Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3131 https://internationalpubls.com z = 0. To verify the above calculations, the reader is referred to the computational data for t = 1 presented to the table shown in the below Table 2. Table 1: Computation of Fixed Point for t = 1, where LHS = H(xn, xn+1) + F (M (xn, xn+1, t)). However, varying the value of t influences the number of iterations required to approach a fixed point. As illustrated in Figure 2, increasing the value of t reduces the number of iterations required to approximate the fixed point x = 0. Figure 2: Effect of t on the Number of Iterations n. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3132 https://internationalpubls.com Example 2 Let X = C[0, 1] be a fuzzy metric space with a ∗ b = ab for all a, b ∈ I and fuzzy metric || || ( , , ) exp f g M f g t t − −  =     for all f, g ∈ X and t > 0. It is clear that (X, M, ∗) is a fuzzy metric space, to verify its completeness, let us consider the sequence of functions 1 ( ) sin( )nf x x x X n    = +     and a function f (x) = x in X. Figure 3: Convergence of M (fn(x), f (x), t) as n → ∞ for t = 1. From the graph shown in Figure 3, it is clear that (X, M, ∗) is a complete fuzzy metric space. Now, for any g ∈ X define 1 ( , ) || || 2 H f g f g = − and for any q ∈ (0, 1) define ( ) ln . 1 p F p p   =   −  It is easy to see that, for any x ∈ I, H(f , f ) = 0 and a sequence of functions {fn} ⊂ X, 1lim ( , ) 0.n n n H f f + → = Also, we have 0 lim ( ) , p F p +→ = − and 1 lim ( ) . p F p −→ =  Now, define T : X × X → X, 1 0 ( )( ) ( )T f x xyf y dy=  (8) for all f, g ∈ X and x, y ∈ I. Let g(x) = sin x in X, then we have, 1 1 2 0 0 ( )( ) ( ) , 3 x T f x xyf y dy xy dy= = =  1 1 0 0 ( )( ) ( ) sin (sin1 cos1).T g x xyg y dy xy y dy x= = = −  And 1 || || sup | ( )( ) ( )( ) | sin1 cos1 0.0322. 3x I f g T f x T g x  − = − = − +  For t = 1, we have ( ) ( ) ( ) ( ) ( , , ) exp || || 0.8534, ( ( ), ( ), ) exp || ( ) ( ) || 0.9683, 1 ( ), ( ) || || 0.07925, 2 1 ( )( ), ( )( ) || ( ) ( ) || 0.0161. 2 M f g t f g M T f T g t T f T g H f x g x f g H T f x T g x T f T g     = − −  = − −  = − = = − = Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3133 https://internationalpubls.com Now, from contraction (1), we obtain F (M (T (f )(x), T (g)(x), t)) ≈ 3.42 > 1.84 ≈ H(f (x), g(x)) + F (M (f (x), g(x), t)). Hence, it can be concluded that for any f, g ∈ X, T is a H – F – contractive, and hence from Theorem 1, T has a unique fixed point in X. Let us consider h∗ ∈ X is a fixed point of T, that is, T (h∗)(x) = h∗(x) ∀ x ∈ I. (9) to find h∗ take any function h from X. By the definition of T in the equation (8), we construct a sequence of functions as follows; 1 1 1 0 0 1 1 1 1 1 2 0 0 1 1 2 2 2 32 0 0 1 1 3 3 3 3 0 0 ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 3 ( )( ) ( ) ( ) ( ) 3 ( )( ) ( ) ( ) 3 T h x xyh y dy x yh y dy cx h x cx T h x xyh y dy x yh y dy h x cx T h x xyh y dy x yh y dy h x cx T h x xyh y dy x yh y dy   = = = =       = = = =       = = = =       = = =             4 1 1 1 0 0 ( ) . . . ( )( ) ( ) ( ) ( ) 3 n n n nn h x cx T h x xyh y dy x yh y dy h x+ =   = = = =       where 1 0 ( ) .c yh y dy=  This implies that, lim ( ) lim 0, 3 1 n nn n cx h x → → = = − and ( ) 1lim ( )( ) lim ( ) lim ( ) lim (0) 0. 3 n n n n n nn n T h x h x cx T h x T + → → → → = =  = Hence from the equation (9), the zero function h∗(x) = 0 for all x ∈ I, is a unique fixed point of T. Remark: In the above Example 2, consider the evaluation x = 0. we observe that 1 0 ( )(0) 0 ( ) 0,T h yh y dy= = which implies that, any function h ∈ X satisfying h(0) = 0 also fulfills the fixed point condition at the specific point, that is T (h)(0) = h(0). For instance, the non-trivial function h(x) = sin x ∈ X, satisfies this condition since h(0) = 0, and consequently T (h)(0) = 0 = h(0). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 3134 https://internationalpubls.com However, the point-wise existence of the fixed point condition does not ensure that T admits a global fixed point on X. This highlights the limitations of point-wise analysis and motivates the use of an iterative contraction framework, where the sequence {Tn(f )} converges uniformly to a unique fixed point in X, providing a more rigorous foundation for the existence and uniqueness results. Definition 5. Let (X, M, ∗) be a complete fuzzy metric space. For any F ∈ F, a self mapping T:X →X is said to be iterated H – F –contractive if there exists a function H ∈ H and N ∈ N such that F (M (Tnx, Tny, t)) ≥ H(x, y) + F (M (x, y, t)) (10) for all x, y ∈ X and n ≥ N. Corollary 1. Let (X, M, ∗) be a complete fuzzy metric space. If T : X → X is an iterated H – F – contractive mapping, then T has a unique fixed point in X. 4. Conclusion We introduced the concept of H –F –contractive mappings and proved the fixed point theorem in complete fuzzy metric spaces. The result generalizes the existing work and provides a versatile framework for further developments in fuzzy fixed point theory. Future work may involve exploring multivalued mappings, partial fuzzy metrics, and applications of control theory. Refrences [1] Michalek Kramosil I. Fuzzy metric and statistical metric spaces. Kyber- netika, 11:326–334, 1975. [2] Veeramani P. George A. On some results in fuzzy metric spaces. Fuzzy Sets and Systems, 64:395–399, 1994. [3] Grabiec M. 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