Eigenvalue type problem in s(., .)-fractional Musielak-Sobolev spaces

In this paper, we introduce the s(., .)-fractional Musielak-Sobolev spaces W-s(x,W-y) L-Phi x,L-y (Omega). Then, we show that there exists lambda(*) > 0 such that any lambda is an element of (0, lambda(*)) is an eigenvalue for the following problem, by means of Ekeland's variational principle (P-a){(-Delta)(s(x,.))(a(x,.)) u = lambda vertical bar u vertical bar (q(x)-2) u in Omega, u = 0 in R-N\Omega, where Omega is a bounded open subset of R-N with C-0,C-1-regularity and bounded boundary.