Work

Nonlinear Fractional Boundary Value Problems: Lyapunov-Type Estimates Derived from a Generalized Gronwall Inequality

Year: 2026

Type: article

Abstract: In this paper, we investigate a class of nonlinear fractional boundary value problems involving the Caputo fractional derivative under two-point boundary conditions. By combining the Green function of the associated linear problem with a generalized Gronwall inequality, we derive pointwise estimates for solutions expressed explicitly in terms of the Mittag–Leffler function. In contrast to existing Lyapunov-type inequalities, which are mainly restricted to linear equations and rely on global supremum norm estimates, our approach preserves the nonlinear structure of the problem and captures the local behavior of solutions. These pointwise estimates lead to a Lyapunov-type inequality for nonlinear fractional equations, extending the classical result of Jleli and Samet beyond the linear framework. Moreover, we show that the obtained Lyapunov condition serves not only as a necessary condition for the existence of nontrivial solutions, but also as a sufficient criterion ensuring Hyers–Ulam stability and uniqueness. An illustrative example is provided to demonstrate the applicability of the theoretical results.

Source: Mathematics

DOI: 10.3390/math14040688

Authors: Nadiyah Hussain Alharthi, Mehmet Zeki Sarıkaya, Rubayyi T. Alqahtani

Institutions: Imam Mohammad ibn Saud Islamic University, Düzce Üniversitesi

Language: English

Topic: Nonlinear Differential Equations Analysis

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,137

Funders: Imam Mohammed Ibn Saud Islamic University, Al-Imam Muhammad Ibn Saud Islamic University