Existence and multiplicity of solutions for fractional $\kappa (\xi )$-Kirchhoff-type equation

In this paper, we aim to tackle the questions of existence and multiplicity of solutions to a new class of $\kappa (\xi )$-Kirchhoff-type equation utilizing a variational approach. Further, we research the results from the theory of variable exponent Sobolev spaces and from the theory of space $\psi$-fractional ${\mathcal{H}}_{\kappa (\xi )}^{\mu ,\nu ;\psi }(\mathrm{\Lambda })$. In this sense, we present a few special cases and remark on the outcomes explored.