Kirchhoff-type problems with the non-local fractional $\mathrm{d}(\mathrm{z},.)$-Laplacian operator

In this work, we examine whether nonnegative solutions exist for a Kirchhoff-type issue that is driven by a fractional $\mathrm{d}\left (\mathrm{z},.\right )-$ Laplacian operator, where d is a continuous positive function. We establish the existence of their nonnegative solutions combined with the theory of fractional Sobolev spaces with variable exponents by using approximation methods.