Pair ( X, m) is named an extended hexagonal b-metric space. The
Pair ( X, m) is named an extended hexagonal b-metric space. The rest of this article is laid out as follows: Section 2 presents a new generalization of FMS, namely, fuzzy extended hexagonal b-metric spaces, by very first delivering essential concepts and ideas applied in exploring the outcomes of this study. Following that, within this section, an example is offered using a focus on the fuzzy extended hexagonal b-metric spaces and explores the notion of convergence sequence, Cauchy sequence and completness in FEHb-MS, relying on certain topological PTPN2 Proteins Formulation options with the examined space. In Section three, by adding added situations towards the functions that offer Banach contraction and fuzzy -contraction in FEHb-MS, we established new fixed point final results in this study. At some point, in Section 4, we examine the existence and uniqueness of options for nonlinear fractional differential equations inside the sense of Caputo derivative applying the fixed point benefits reported within the preceding section.2. Key Outcomes This section starts with an introductory of fuzzy extended hexagonal b-metric spaces (or basically FEHb-MS), too as an example of your space defined. Definition 7. Let X = , b : X X [1, ), a continuous t-norm, and m be a fuzzy set on X X [0, ). Then, m is named fuzzy extended hexagonal b-metric, if, for all c, d, e, f , g, k X and c = e, e = f , f = g, g = k, k = d, the following situations are satisfied: [mh 1] mh (c, d, 0) = 0 for t = 0;[mh 2] mh (c, d, t) = 1, t 0 if and only if c = d; [mh 3] mh (c, d, t) = mh (d, c, t); [mh 4] mh (c, d, b(c, d)(t s u v w)) mh (c, e, t) mh (e, f , s) mh ( f , g, u) mh ( g, k, v) mh (k, d, w) for all t, s, u, v, w 0; [mh 5] mh (c, d, .) : (0, ) [0, 1] is left continuous.Then, ( X, mh , ) is generally known as fuzzy extended hexagonal b-metric space. TIMP-2 Proteins web Instance 1. Let X = 1, 2, 3, 4, 5, 6 and functions b : X X [1, ) and h : X X R such that h is symmetric may be defined as: b(c, d) = c d, c, d X, h (c, d) = 0, c = d;Symmetry 2021, 13,four ofh (1, 2) = 700; h (1, c) = h (two, c) = 50, c X \1, 2, 6; h (3, four) = h (three, five) = h (4, five) = 50; h (c, six) = 150, c X \6. Then, it is straight away evident that ( X, h ) is an extended hexagonal b-metric space. Let mh : X X [0, ) [0, 1] be specified inside the following form: mh (c, d, t) =t , th (c,d)if t 0 if t =0,where , are good actual numbers and t-norm is defined by t1 t2 = mint1 , t2 . ( X, mh , ) is hence shown to be a fuzzy extended hexagonal b-metric space. We observe that the criteria [mh 1], [mh 2], [mh 3] and [mh 5] of Definition 7 are provably accurate. To demonstrate the home [mh 4] for all c, d X, look at the following mh (c, d, b(c, d)(t s u v w)) = For c = 1, d = four, mh (1, 4, b(1, 4)(t s u v w)) = b(1, four)(t s u v w) b(1, 4)(t s u v w) h (1, four) b(c, d)(t s u v w) . b(c, d)(t s u v w) h (c, d)=5(t s u v w) five(t s u v w) 50 50 , 5(t s u v w) 50 mh (two, 3, s) = mh (5, 6, v) = s 50 = 1- ; s 50 s = 1-mh (1, 2, t) = mh (three, five, u) = mh (six, 4, w) =700 t = 1- ; t 700 t 700 u 50 = 1- ; u 50 u v 150 = 1- ; v 150 v w 150 = 1- . w 150 w Because of this, for all t, s, w, u, v 0, we observe that mh (1, four, b(1, 4)(t s u v w)) = 1 – 50 5(t s u v w) 50 700 70(t s u v w) 700 700 70t 700 700 = mh (1, 2, t). t = 1- 1- 1-It can additional be demonstrated thatmh (1, 4, b(1, four)(t s u v w)) mh (two, 3, t), mh (1, four, b(1, four)(t s u v w)) mh (3, five, t), mh (1, 4, b(1, 4)(t s u v w)) mh (5, six, t), mh (1, 4, b(1, four)(t s u v w)) mh (six, 4,.