Fixed point and endpoint theorems of multivalued mappings in convex $b$-metric spaces with an application

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.