Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Admissible perturbations of a multivalued Picard operator: Ćirić contraction condition; fixed point and stability results

Cristina Gheorghe

TL;DR

The paper addresses strict fixed-point existence and stability for multivalued Picard operators that do not satisfy a Ćirić contraction, by identifying admissible perturbations $T_G$ (via $T_G(x)=G(x,T(x))$) that restore contraction properties under Takahashi-type conditions. It develops data-dependence, Ulam-Hyers stability, well-posedness, Ostrowski stability, and quasi-contraction results for the perturbed setting, all within a unified framework and with explicit constants. The results are reformulated for Takahashi admissible perturbations using convex-metric spaces $(X,d,W)$ and the perturbation $T_W(x)=\{W(x,y,\lambda):y\in T(x)\}$, including a concrete example with the Takahashi convexity operator. Collectively, these findings provide rigorous criteria under which strict fixed points persist, iterations converge, and stability properties hold for a broad class of multivalued operators.

Abstract

This paper studies strict fixed point and stability results for multivalued operators which does not satisfy a Ćirić 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.

Admissible perturbations of a multivalued Picard operator: Ćirić contraction condition; fixed point and stability results

TL;DR

The paper addresses strict fixed-point existence and stability for multivalued Picard operators that do not satisfy a Ćirić contraction, by identifying admissible perturbations (via ) that restore contraction properties under Takahashi-type conditions. It develops data-dependence, Ulam-Hyers stability, well-posedness, Ostrowski stability, and quasi-contraction results for the perturbed setting, all within a unified framework and with explicit constants. The results are reformulated for Takahashi admissible perturbations using convex-metric spaces and the perturbation , including a concrete example with the Takahashi convexity operator. Collectively, these findings provide rigorous criteria under which strict fixed points persist, iterations converge, and stability properties hold for a broad class of multivalued operators.

Abstract

This paper studies strict fixed point and stability results for multivalued operators which does not satisfy a Ćirić type contraction condition, but their admissible perturbation does. We focus on the conditions imposed on the admissible perturbation of a Picard operator 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.

Paper Structure

This paper contains 10 sections, 20 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Admissible perturbation of a multivalued Ćirić type operator
  3. Data dependence of the strict fixed point
  4. Ulam-Hyers stability property of the strict fixed point problem
  5. Well-posedness of the strict fixed point problem
  6. Ostrowski stability property of strict fixed point problem
  7. Quasi-contraction condition for the multivalued operator T
  8. Application: fixed point and stability results for an admissible perturbation in the sense of Takahashi
  9. Main results for Takahashi admissible perturbations
  10. Example: the convexity operator in the sense of Takahashi

Key Result

Lemma 1.4

(see Berinde2016) Let X be a non-empty set and $T:X\rightarrow P(X)$ a multivalued operator. For an admissible perturbation $T_G$ corresponding to an operator $G:X\times X\rightarrow X$, the following affirmations hold true:

Theorems & Definitions (55)

  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Lemma 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Example 1.8
  • Corollary 1.9
  • Definition 2.1
  • ...and 45 more