Strict fixed point problem, stability results and retraction displacement condition for Picard operators
Cristina Gheorghe, Adrian Petruşel
TL;DR
This work develops strict fixed point principles for multivalued Picard operators under Ćirić-type and Ćirić–Reich–Rus-type contraction conditions. By employing retraction–displacement-type bounds and the Pompeiu–Hausdorff metric, it establishes the existence and uniqueness of a strict fixed point $x^*$ and proves convergence of the multivalued iterates to $\{x^*\}$, with explicit quantitative estimates. The authors also derive various stability notions—Ostrowski stability, Ulam–Hyers stability, and data-dependence (well-posedness in Reich–Zaslavski and Ostrowski sense)—under these contraction frameworks. These results generalize classic multivalued fixed point theorems to stronger Picard-type operators and provide verifiable conditions for uniqueness and convergence in complete metric spaces. Open questions and comparative notes on existence criteria for $SFix(T)$ are discussed, along with several sufficient conditions from the literature.
Abstract
The aim of this paper is to give strict fixed point principles for multivalued operators $T:X\rightarrow P(X)$ satisfying some contraction conditions of Ćiri\' c and of Ćiri\' c-Reich-Rus type. We are interested, under which conditions, the multi-valued operator has a unique strict fixed point and, additionally, when the sequence of its multi-valued iterates $(T^n(x))_{n\in \mathbb{N}}$ converges to this unique strict fixed point. Moreover, some stability properties, such as data dependence on operator perturbation, Ulam-Hyers stability, well-posedness in the sense of Reich and Zaslavski and Ostrowski property of the strict fixed point problem are established.