Resumen de Green’s function of differential equation with fourth order and normal operator coefficient in half axis

Mehmet Bayramoglu, Kevser Ozden Koklu

• Let H be an abstract seperable Hilbert space. Denoted by H1 = L2 (0, ∞; H), the all functions defined in [0, ∞) and their values belongs to space H, which R ∞ 0 kf(x)k 2 H dx < ∞. We define inner product in H1 by the formula(f, g)H1 = R ∞ 0 (f, g)Hdx f(x), g(x) ∈ H1,H1 forms a seperable Hilbert space[3] where k.kH and (., .)H are norm and scalar product, respectively in H.  In this study, in space H1, it is investigated that Green’s function (resolvent) of operator formed by the diferential expressiony IV + Q(x)y, 0 ≤ x < ∞,and boundary conditionsy 0 (0) − h1y(0) = 0,y 000(0) − h2y 00(0) = 0,where Q(x) is a normal operator mapping in H and invers of it is a compact operator for every x ∈ [0, ∞). Assume that domain of Q(x) is independent from x and resolvent set of Q(x) belongs to |arg λ − π| < ε (0 < ε < π) of complex plane λ, h1 and h2 are complex numbers. In addition assume that the operator function Q(x) satisfies the Titchmarsh-Levitan conditions.