Existence and stability results for a $ (k, \eta) $-coupled system of nonlinear implicit differential equations with nonlocal integral boundary conditions

Madeaha Alghanmi; Bashir Ahmad; Madeaha Alghanmi; Bashir Ahmad

doi:10.3934/math.20261322

AIMS Mathematics

2026, Volume 11, Issue 10: 33835-33859. doi: 10.3934/math.20261322

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Research article

Existence and stability results for a $ (k, \eta) $-coupled system of nonlinear implicit differential equations with nonlocal integral boundary conditions

Madeaha Alghanmi ^{1
,
,},
Bashir Ahmad ²

1.
Department of Mathematics, Faculty of Sciences & Arts, King Abdulaziz University, Rabigh 21911, Saudi Arabia
2.
Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received: 03 August 2026 Revised: 29 September 2026 Accepted: 09 October 2026 Published: 19 October 2026
MSC : 26A33, 34K05

This article investigated a new class of coupled systems of nonlinear implicit $ (k, \eta) $-fractional differential equations complemented with nonlocal integral boundary conditions. Implicit fractional differential systems feature the dependence of the nonlinearities involved in the system on the highest order fractional derivatives and allow encoding instantaneous elastic response and rate-dependent memory response. Examples include nonlinear viscoelastic damping of beams/plates, soft biological tissue models, and fractional resistor–inductor–capacitor (RLC) networks with memristive elements. We first transformed the given coupled system into an equivalent fixed point problem and then established sufficient criteria for existence and uniqueness of its solutions via the fixed point theorems due to Schaefer and Banach. The Hyers-Ulam and generalized Hyers-Ulam stability were also discussed for the proposed system. Examples were given to demonstrate the applicability of the main results.
- $ (k, \eta) $-fractional derivative,
- implicit differential equations,
- coupled system,
- existence criteria,
- stability
Citation: Madeaha Alghanmi, Bashir Ahmad. Existence and stability results for a $ (k, \eta) $-coupled system of nonlinear implicit differential equations with nonlocal integral boundary conditions[J]. AIMS Mathematics, 2026, 11(10): 33835-33859. doi: 10.3934/math.20261322

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Abstract

This article investigated a new class of coupled systems of nonlinear implicit $ (k, \eta) $-fractional differential equations complemented with nonlocal integral boundary conditions. Implicit fractional differential systems feature the dependence of the nonlinearities involved in the system on the highest order fractional derivatives and allow encoding instantaneous elastic response and rate-dependent memory response. Examples include nonlinear viscoelastic damping of beams/plates, soft biological tissue models, and fractional resistor–inductor–capacitor (RLC) networks with memristive elements. We first transformed the given coupled system into an equivalent fixed point problem and then established sufficient criteria for existence and uniqueness of its solutions via the fixed point theorems due to Schaefer and Banach. The Hyers-Ulam and generalized Hyers-Ulam stability were also discussed for the proposed system. Examples were given to demonstrate the applicability of the main results.

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