DOI: 10.1515/dema-2025-0320 ISSN: 2391-4661

Extremal and uniqueness results for a singular p -Laplacian Caputo-Hadamard fractional equation involving parameter dependence

Limin Guo
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Abstract

Owing to the non-ideal properties of the Green’s function associated with the boundary value problem of Caputo-Hadamard fractional differential equations, a complete theoretical framework for studying such solutions has remained elusive. To address this challenge, this paper introduces novel substitutions and conducts a meticulous analysis of the Green’s function’s properties. This approach enables us to successfully derive the maximum and minimum iterative solutions for a singular p -Laplacian Caputo-Hadamard fractional differential equation. Subsequently, by employing the theory of mixed monotone operators, we establish the existence and uniqueness of positive solutions. Furthermore, we demonstrate that the unique solution

u λ ⋆ ${u}_{\lambda }^{\star }$
is parameter-dependent, exhibiting the asymptotic behavior
lim λ → ∞ ‖ u λ ⋆ ‖ = 0 ${\mathrm{lim}}_{\lambda \to \infty }{\Vert}{u}_{\lambda }^{\star }{\Vert}=0$
and
lim λ → 0 + ‖ u λ ⋆ ‖ = ∞ ${\mathrm{lim}}_{\lambda \to {0}^{+}}{\Vert}{u}_{\lambda }^{\star }{\Vert}=\infty $
. The validity of our main theoretical findings is confirmed through a numerical example, with graphical illustrations depicting the solution’s variation with the parameter.

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