Z-Numbers Bring Reliability Into Fractional Differential Equation Analysis

A new study published in the journal Boundary Value Problems on 2 October 2026 presents a generalized framework for analyzing a class of Ψ-Hilfer fractional Hammerstein integro-differential equations, treating uncertainty and the reliability of available information together through so-called Z-numbers. The paper, authored by Alireza Hatami, Reza Saadati, H. M. Srivastava and Donal O’Regan, is published as open access research.

Z-numbers are a construct designed to describe uncertain quantities while also capturing how trustworthy the underlying information is, rather than representing uncertainty alone. In this work, the authors apply that generalized Z-number setting to a family of fractional integro-differential equations of Hammerstein type, formulated with the Ψ-Hilfer fractional operator. The stated goal is to study three linked questions for these equations: whether solutions exist and are unique, how well approximate solutions can represent exact ones, and whether the equations are stable in the Ulam–Hyers–Rassias sense.

According to the abstract, the analysis rests on a fixed-point approach carried out in a generalized metric space. Working within that setting, the authors establish sufficient conditions under which solutions to the equations exist and are unique. Alongside these existence and uniqueness results, the paper derives best-approximation estimates, which quantify how closely solutions can be approximated, and proves Ulam–Hyers–Rassias stability results for the equations under consideration.

To demonstrate that the theoretical conditions are not merely formal, the authors include an illustrative example. The abstract states that this example is provided to verify the main assumptions of the theory and the corresponding contraction condition, thereby demonstrating the applicability of the established results. The paper is a peer-reviewed, accepted research article shared early with a permanent DOI, and the published version is subject to further edits before the final Version of Record replaces it.

Subject of Research: Solvability, best approximation and Ulam–Hyers–Rassias stability of Ψ-Hilfer fractional integro-differential equations using a generalized Z-number framework

Article Title: A generalized Z-number approach to solvability, best approximation, and Ulam–Hyers–Rassias stability of Ψ-Hilfer fractional differential equations

Article References: Hatami, A., Saadati, R., Srivastava, H. M., & O’Regan, D. (2026). A generalized Z-number approach to solvability, best approximation, and Ulam–Hyers–Rassias stability of Ψ-Hilfer fractional differential equations. Boundary Value Problems. https://doi.org/10.1186/s13661-026-02373-y

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DOI: 10.1186/s13661-026-02373-y

Keywords: fractional differential equations, Z-numbers, Ulam-Hyers-Rassias stability, fixed-point theory, Hammerstein integro-differential equations, best approximation, existence and uniqueness, generalized metric spaces, Ψ-Hilfer operator, Boundary Value Problems, generalized, Z-number