What does this research mean for the field?

Hyers-Ulam stability can be characterized for closed linear relations in Hilbert spaces, and specific sufficient conditions guarantee that the sum and product of two such stable relations remain stable. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.ESTABLISHES_NEW_DIRECTION.

What question did this study set out to answer?

The research aims to define Hyers-Ulam stability for linear relations and explore its properties in Hilbert spaces.

April 3, 2026Open Access

Hyers-Ulam stability of closed linear relations in Hilbert spaces

Key Points

The research aims to define Hyers-Ulam stability for linear relations and explore its properties in Hilbert spaces.
Introduced the concept of Hyers-Ulam stability in normed linear spaces.
Characterized stability conditions for closed linear relations in Hilbert spaces.
Established sufficient conditions for the stability of sums and products of stable relations.
Characterization of closed linear relations demonstrating Hyers-Ulam stability.
Identification of sufficient conditions ensuring stability in combinations of relations.

Abstract

This paper introduces the concept of Hyers-Ulam stability for linear relations in normed linear spaces and presents several intriguing results that characterize the Hyers-Ulam stability of closed linear relations in Hilbert spaces. Additionally, sufficient conditions are established under which the sum and product of two Hyers-Ulam stable linear relations remain Hyers-Ulam stable.

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