Abstract
The purpose of this article is to introduce locally α-ζ-multivalued contraction and rational Ćirić type α-ζ-multivalued contraction in the context of F-metric spaces and prove some endpoint results. We provide a nontrivial example to show the authenticity of our main result. Our results generalize some well-known results of literature. We also present some endpoint results in both graphic F-metric spaces and ordered F-metric spaces. As an application of our main result, we investigate the solution of an integral equation.