Our paper is devoted to indicating a way of generalizing Mann’s iteration algorithm and a series of fixed point results in the framework of b-metric spaces. First, the concept of a convex b-metric space by means of a convex structure is introduced and Mann’s iteration algorithm is extended to this space. Next, by the help of Mann’s iteration scheme, strong convergence theorems for two types of contraction mappings in convex b-metric spaces are obtained. Some examples supporting our main results are also presented. Moreover, the problem of the T-stability of Mann’s iteration procedure for the above mappings in complete convex b-metric spaces is considered. As an application, we apply our main result to approximating the solution of the Fredholm linear integral equation.

In this paper, we present a series of fixed point results for Mann’s iteration scheme in the framework of $G_b$-metric spaces. First, we introduce the concept of convex $G_b$-metric space by means of a convex structure and Mann’s iteration algorithm is extended to this space. Furthermore, using Mann’s iteration scheme, we prove some fixed point results for several mappings satisfying various suitable conditions on complete convex $G_b$-metric spaces. Some examples supporting our main results are also presented. We also discuss the well-posedness of the fixed point problems and the P property for given mappings. Moreover, as an application, we apply our main result to prove the existence of the solutions to integral equations.

In this paper, we first introduce the concept of rectangular $G_b$-metric space which generalizes the notion of rectangular metric space and $G_b$-metric space. Then, some fixed point results connected with certain contractions are obtained in the setting of rectangular $G_b$-metric spaces. Additionally, we also introduce the concept of convex rectangular $G_b$-metric space by means of the convex structure and study the fixed points of enriched type contractions in this space.

In this paper, we investigated several new fixed points theorems for multivalued mappings in the framework $ b $-metric spaces. We first generalized $ S $-iterative schemes for multivalued mappings to above spaces by means of a convex structure and then we developed the existence of fixed points and approximate endpoints of the multivalued contraction mappings using iteration techniques. Furthermore, we introduced the modified $ S $-iteration process for approximating a common endpoint of a multivalued $ \alpha_{s} $-nonexpansive mapping and a multivalued mapping satisfying conditon $ (E^{'}) $. We also showed that this new iteration process converges faster than the $ S $-iteration process in the sense of Berinde. Some convergence results for this iterative procedure to a common endpoint under some certain additional hypotheses were proved. As an application, we applied the $ S $-iteration process in finding the solution to a class of nonlinear quadratic integral equations.

The aim of this paper is to establish some fixed point results for enriched Kannan-type mappings in convex metric spaces. We first give an affirmative answer to a recent Berinde and Păcurar's question (Remark 2.3) [J. Comput. Appl. Math., 386 (2021), 113217]. Furthermore, we establish the existence and uniqueness of fixed points for Suzuki-enriched Kannan-type mappings in the setting of convex metric spaces. Finally, we present an application to approximate the solution of the Volterra integral equations to support our results.

In this paper, we redefine the concept of a graphical convex metric space involving sets with a graphical structure, and a new class of non-self mappings, called weak proximal enriched $ G- $contractions, is introduced in the aforementioned space. Moreover, we demonstrate the existence of best approximation points for two types of weak proximal enriched $ G- $contractions in graphical convex metric spaces, under suitable control conditions. Additionally, we provide examples to confirm our main results.