In the framework of $G_b$-metric spaces, we introduce the concept of a generalized Ćirić-type contraction and obtain several fixed-point theorems for this contraction. First, we present a significant lemma, which is used to ensure that the Picard sequence is a Cauchy sequence. Using this lemma, we establish three fixed-point theorems satisfying different conditions. Second, we construct new examples to illustrate our results. As applications, we deduce the famous Ćirić fixed-point theorem in terms of $b$-metric spaces using our results. In addition, we obtain Reich-type contraction fixed-point theorems in such a space using the aforementioned lemma. Our results improve and complement many recent findings. In particular, we substantially enlarge the range of the contraction constant in our results. Finally, we consider the existence and uniqueness of solutions for integral equation applying our new results.

Our purpose of this paper is to answer several open questions posed by Doan (AIMS Math., 6 (2021), 7895–7908). First, we present two fixed point theorems, which are positive answers to Doan's questions. Second, we establish a new type of Riech's fixed point theorem to improve a result of Doan. Finally, we offer a straightforward example illustrating that a set-valued mapping satisfying the conditions of our fixed point theorem may has more than one fixed point.