Set-valued fixed-point theorems for generalized contractive mappings without the Hausdorff metric

In this paper, the concept of a set-valued contractive mapping is considered by using the idea of a generalized distance, such as the τ -distance, in metric spaces without using the concept of the Hausdorff metric. Furthermore, under some mild conditions, we provide the existence theorems for fixed-point problems of the considered mapping. Hence, our results can be viewed as a generalization and improvement of many recent results.