Strict fixed point problem, stability results and retraction displacement condition for Picard operators

The aim of this paper is to give strict fixed point principles for multivalued operators $T:X\to P(X)$ 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.