Strict fixed point and stability results for multivalued operators via the admissible perturbation technique

Abstract

Abstract We consider a class of multi-valued operators $$T:X\rightarrow P(X)$$ having the property that there exists an admissible perturbation of it, defined as $$T_G:X\rightarrow P(X)$$ and satisfying a Ćirić type contraction condition. This work focuses on the conditions imposed on the admissible perturbation $$T_G$$ , such that strict fixed point results hold true for T. The main result consists of formulating a theorem that connects the properties of the admissible perturbation with the conclusion that T is a Picard operator. Additionally, we study whether the strict fixed point problem $$T(x)=\{x\}$$ is well-posed, as well as, data dependence, Ulam-Hyers stability and Ostrowski property for the strict fixed point problem. The admissible perturbations in the sense of Takahashi serve as a relevant example for applying the theory formulated in this paper.