Admissible perturbations of a multivalued Picard operator: \'Ciri\'c contraction condition; fixed point and stability results

TL;DR

This paper investigates fixed point and stability results for multivalued operators that do not satisfy a Cirić contraction, but whose admissible perturbations do, extending fixed point theory under perturbations.

Contribution

It introduces conditions on admissible perturbations of a Picard operator ensuring fixed point and stability results despite the original operator not satisfying a Cirić contraction.

Findings

Fixed point results hold under admissible perturbations.

Stability results are established for perturbed operators.

Results are illustrated with examples.

Abstract

This paper studies strict fixed point and stability results for multivalued operators which does not satisfy a \'Ciri\'c type contraction condition, but their admissible perturbation does. We focus on the conditions imposed on the admissible perturbation $T_G$ of a Picard operator $T:X\rightarrow P(X)$ such that the strict fixed point and stability results still hold for T. The results obtained are reformulated in terms of admissible perturbations in the sense of Takahashi and illustrated with some examples.