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 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 $(E^{{}^{\prime }})$. 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.