Infinitely of solutions for fractional κ(ξ)$$ \kappa \left(\xi \right) $$‐Kirchhoff equation in Hκ(ξ)ϖ,ν;μ(Λ)$$ {\mathcal{H}}_{\kappa \left(\xi \right)}^{\varpi, \nu; \mu}\left(\Lambda \right) $$

This work aims to develop the variational framework for some Kirchhoff problems involving the ‐Hilfer operator. Precisely, we use the symmetric mountain pass theorem to prove the existence of unfairly of nontrivial solutions. Further, we research the results from the theory of variable exponent Sobolev spaces and from the theory of ‐fractional space .