Here, we shall introduce the new notion of $C^{*}$-algebra valued bipolar b-metric spaces as a generalization of usual metric spaces, $C^{*}$-algebra valued metric space, b-metric spaces. In the above-mentioned spaces, we shall define $({\alpha }_{\mathfrak{A}}-{\psi }_{\mathfrak{A}})$ contractions and prove some fixed point theorems for these contractions. Some existing results from the literature are also proved by using our main results. As an application Ulam–Hyers stability and well-posedness of fixed point problems are also discussed. Some examples are also given to illustrate our results.

We present the novel concept of graphical bipolar metric-type space in this article, which combines the notions of graph theory with fixed point theory. We prove that every bipolar metric space is a graphical bipolar metric space but the converse is not true in general. Various concepts like covariant mapping, contravariant mapping, and Cauchy bisequence are also discussed within the context of graphical bipolar metric-type spaces. Furthermore, in this study, we show that fixed point results exist in graphical structures of bipolar metric spaces and a series of examples are provided to support the main results within the realm of the graph structure.

In this article, the new Ishikawa iteration method is presented to find the approximate solution of an ordinary differential equation with an initial condition. Additionally, some numerical examples with initial conditions are given to show the properties of the iteration method. Furthermore, the results of absolute errors are compared with Euler, Runge-Kutta and Picard iteration methods. Finally, the present method, namely the new modified Ishikawa iteration method, is seen to be very effective and efficient in solving different type of the problem. MSC:65K15, 65L07, 65L06, 65L70.

Common coupled fixed point theorems are examined in this paper for comparable mappings ensuring nonlinear contraction in ordered partial metric spaces. Given theorems enlarge and universalize some conclusions of Gnana Bhaskar and Lakshmikantham (2006).

In this paper a class of self-mappings on cone Banach spaces which have at least one fixed point is considered. More precisely, for a closed and convex subset C of a cone Banach space with the norm ∥ x ∥ C = d ( x , 0 ) , if there exist a, b, c, r and T : C → C satisfies the conditions 0 ⩽ Φ q ( r ) < Φ q ( a ) + 2 ( Φ q ( b ) + Φ q ( c ) ) and a Φ p ( d ( T x , T y ) ) + b Φ p ( d ( x , T x ) ) + c Φ p ( d ( y , T y ) ) ⩽ r Φ p ( d ( x , y ) ) for all x , y ∈ C , then T has at least one fixed point.

We initiate the investigation of the topological aspects of bipolar metric spaces. In this context, some concepts that generalize open and closed balls and sets, accumulation points, closure and interior operators for bipolar metric spaces, of which little is known about their topological behaviors, are discussed. In addition, some essential properties regarding these notions were obtained, and counterexamples were provided for some expected but not satisfied properties.