Banach's contraction principle has been improved and extensively studied on several generalized metric spaces. Recently, Complex-valued $S$-metric spaces have been introduced and studied for this purpose. In this paper, we investigate some generalized fixed point results on a complete Complex valued $S$-metric space. To do this, we prove some common fixed point (resp. fixed point) theorems using different techniques by means of new generalized contractive conditions and the notion of the closed ball. Our results generalize and improve some known fixed point results. We provide some illustrative examples to show the validity of our definitions and fixedpoint theorems.

In this article, we delve into the concept of convexity within fuzzy metric spaces and delve into their structural characteristics. We present several theorems concerning the existence of coincidence points in fuzzy convex metric spaces. Furthermore, we introduce the notion of star-shaped subsets within fuzzy convex metric spaces. Within these star-shaped subsets, we showcase various fixed point theorems for mappings of the non-expansive type that commute. In the final section, we expand the definition of fuzzy convex metric spaces and provide a significant example of such a space. Additionally, we uncover specific fixed point theorems applicable to multi-valued mappings that are non-expansive.

In this paper, we introduce the concept of a w-compatible mappings and utilize the same to discuss the ideas of coupled coincidence point and coupled point of coincidence for nonlinear contractive mappings in the context of Complex valued metric spaces besides proving existence theorems which are following by corresponding unique coupled common  xed point theorems for such mappings. Some illustrative examples are also given to substantiate our newly proved results.

In [24], Khan et al. established some fixed point theorems in complete and compact metric spaces by using altering distance functions. In [16] Gordji et al. described the notion of orthogonal set and orthogonal metric spaces. In [18] Gungor et al. established fixed point theorems on orthogonal metric spaces via altering distance functions. In [25] Lotfy et al introduced the notion of $\alpha_{*}$-$\psi$-common rational type mappings on generalized metric spaces with application to fractional integral equations. In [28] K. Royy et al. described the notion of Branciari $S_b$-metric space and related fixed point theorems with an application. In this paper, we introduce the notion of the common fixed point ($\alpha_*$-$\psi$-$\beta_{i}$)-contractive set-valued mappings on orthogonal Branciari $S_{b}$-metric space with the application of the existence of a unique solution to an initial value problem.

In 1984, Khan et al. established some fixed point theorems in complete and compact metric spaces by altering distance functions. In 2020, Lotfy et al. introduced the $\alpha_{*}$-$\psi$-common rational type mappings on generalized metric spaces applied to fractional integral equations. In 2022, Roy et al. described the notion of Branciari  $ S_b$-metric space and related fixed point theorems with an application. In this paper, we introduce the notion of fixed point theorems for $\alpha_{*}$-$\psi$-$\beta_{i}$-contractive set-valued mappings on Branciari $S_{b}$-metric space with application to fractional integral equations.

For a non-zero normed linear space A, we consider Cb (K), the Complex-valued, bounded and continuous functions space on K with ∥, ·, ∥, ∞, , where K = B(0) 1 (the closed unit ball of A). Also for a non-zero element φ,∈,A ∗,with ∥, φ, ∥,≤,1, we consider the space Cbφ,(K) as the linear space Cb (K) with the new norm ∥, f∥,φ,= ∥, fφ, ∥, ∞,for all f ∈,Cb (K). Some basic properties such as, proximinality, E-proximinality, strongly proximinality and quasi Chebyshev for certain subsets of Cb (K) are characterized with the norms ∥, ·, ∥,φ,and ∥, ·, ∥, ∞, . Some examples for illustration and for comparison between the norms ∥, ·, ∥,φ,and ∥, ·, ∥, ∞,on Cb (K) are presented.