In this paper, we introduce the structure of extended cone b-metric-like spaces over Banach algebra as a generalization of cone b-metric-like spaces over Banach algebra. In this generalized space we define the notion of generalized Lipschitz mappings in the setup of extended cone b-metric-like spaces over Banach algebra and investigated some fixed point results. We also provide examples to illustrate the results presented herein. Finally, as an application of our main result, we examine the existence and uniqueness of solution for a Fredholm integral equation.

In present paper, we prove unique fixed point theorems for contractive maps in \(N\) -cone metric spaces. Our results extend and generalize some well-known results of (Banach, Fund Math 3:133–181 1992; Chatterjee, Rend Acad Bulgare Sci 25:727–730 1972; Kannan, Bull Calcutta Math Soc 60:71–76 1968; Rezapour and Hamlbarani, J Math Anal Appl 345:719–724 2008) in the setting of \(N\) -cone metric spaces.

The aim of the article is to establish the structure of partial cone b-metric spaces over Banach algebras. Topological and structural properties are investigated of the new spaces. We also define generalized Lipschitz mappings and give their application in fixed point theory. The results presented in this paper substantially extend and strengthen the results of the literature. Few examples are provided to support our conclusions and as an application we establish the existence and uniqueness of a solution to a class of system of nonlinear integral equations.

In the present paper, we introduce the notion of N-fuzzy metric spaces (NFMSs), Pseudo N-fuzzy metric spaces and describe some of their properties. Also we prove a fixed point theorem using implicit relation in N-fuzzy metric spaces.

In this paper, we introduce the M-cone metric space over Banach algebra as a generalization of both M-metric space and cone metric space over Banach algebra and investigate some fixed point results in the new settings. Some examples are presented as illustrations. Finally, the results are supported by an application to examine the existence and uniqueness of solution for a Fredholm integral equation.

In the present paper, we define a pseudo 𝑆𝑏 −Menger space with counterexamples and prove a decomposition theorem for a pseudo 𝑆𝑏 −Menger space. Also, we prove fixed-point theorems for rational-type contraction in a newly defined 𝑆𝑏−Menger space. Our results extend the results of Gupta V et al [8] in 𝑆𝑏 −Menger spaces. Applications are also given to prove the effectiveness of our results.