In this work, we develop some variational settings related to some singular $p$-Kirchhoff problems involving the $\psi$-Hilfer fractional derivative. More precisely, we combine the variational method with the min-max method in order to prove the existence of nontrivial solutions for the given problem. Our main result generalizes previous ones in the literature.

The aim of this research article is to introduce the notion of graphic $\mathrm{\Theta }$-contractions in the setting of $\mathcal{F}$-metric spaces, and establish some novel fixed point results. To demonstrate the validity of our theoretical contributions, we include some illustrative examples. In addition, we extend our analysis by proving fixed point theorems for orbitally $G$ continuous graphic $\mathrm{\Theta }$-contractions. A significant application of our results is the formulation of image denoising as a fixed point problem. Specifically, we define a contractive operator that refines pixel intensities through iterative updates based on the local neighborhood of each pixel. Using the principles of fixed point theory, we prove the existence and uniqueness of a stable denoised image that corresponds to the fixed point of the constructed mapping. Furthermore, we investigate the convergence properties of the iterative scheme under the assumptions of graphic $\mathrm{\Theta }$-contractions, thereby confirming its reliability and effectiveness.