Analysis of a $\psi$-Hilfer fractional Kirchhoff equation in a new fractional Orlicz space

Abstract

This paper presents a novel functional framework by defining and analyzing the $\psi$-Hilfer fractional Orlicz space ${\mathcal{O}}_G^{\alpha ,\gamma ,\psi }(\Lambda ,\mathbb{R})$. This space extends traditional function spaces by integrating fractional calculus with the flexibility of Orlicz spaces, allowing for a broader class of function growth conditions. A key contribution of this work is the qualitative analysis of this newly introduced space. We investigate fundamental structural properties such as reflexivity, completeness, and separability, which play a crucial role in functional analysis and the study of variational problems. Additionally, we establish a continuous embedding of ${\mathcal{O}}_G^{\alpha ,\gamma ,\psi }(\Lambda ,\mathbb{R})$ into a suitable Orlicz spaces. This result provides deeper insight into the relationship between our proposed space and existing functional frameworks, ensuring its applicability in mathematical analysis. As a practical application, we employ Ricceri’s three critical points theorem to demonstrate the existence of three weak solutions for a class of fractional Kirchhoff type equations. This application underscores the effectiveness of our newly developed space in solving variational problems, particularly in the context of critical point theory. Overall, this work bridges fractional calculus, Orlicz spaces, and variational analysis, providing a novel mathematical setting for studying complex differential equations.