Estimates of the spectrum for a class of mixed type operators

Lu = −k(y)uxx − uyy + a(y)ux + c(y)u of mixed type originally defined on the set C∞ 0,π(Ω) of infinitely differentiable functions satisfying the conditions u(−π, y) = u(π, y) and ux(−π, y) = ux(π, y) and compactly supported with respect to the variable y, where k(y) is a piecewise continuous function on the interval [−1, 1], yk(y) > 0 for y = 0, and k(0) = 0. Note that the operator L admits closure, which will be again denoted by L. Of papers on close topics, we note [1–5]. We define the fractional derivative D xu of order α ≥ 0 with respect to x of a function u(x, y) as the expression [6]