The existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both p(·)-Harmonic and p(·)-biharmonic operators Δp(x)2u-Δp(x)u=λw(x)|u|q(x)-2u   in  Ω,            u∈W2,p(⋅)(Ω)∩W0-1,p(⋅)(Ω), is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces Lp(·)(Ω) and Wm,p(·)(Ω).