On generalized solution of a class of higher order operator-differential equations

In this paper the sufficient conditions on the existence and uniqueness of a generalized solution on the axis are obtained for higher order operator-differential equations, the main part of which is multi characteristic. | Turk J Math 32 (2008) , 305 – 314. ¨ ITAK ˙ c TUB On Generalized Solution of a Class of Higher Order Operator-Differential Equations Rovshan Z. Humbataliyev Abstract In this paper the sufficient conditions on the existence and uniqueness of a generalized solution on the axis are obtained for higher order operator-differential equations, the main part of which is multi characteristic. Key Words: Operator-differential equations, Hilbert spaces, existence of generalized solution. 1. Introduction Let H be a separable Hilbert space, and A be a positive-definite self-adjoint operator in H with domain of definition D (A). Denote by Hγ a scale of Hilbert spaces generated by the operator A, . Hγ = D (Aγ ) , (γ ≥ 0) , (x, y)γ = (Aγ x, Aγ y) , x, y ∈ D (Aγ ). We denote by L2 ((a, b) ; Hγ ) (−∞ ≤ a < b ≤ +∞) a Hilbert space of vector-functions f (t) determined in (a, b) almost everywhere with values from H measurable, square integrable in the Bochner’s sense f L2 ((a,b);H) ⎛ b ⎞1/2 2 = ⎝ f dt⎠ . γ a Assume L2 ((−∞, +∞) ; H) ≡ L2 (R; H) . 2000 AMS Mathematics Subject Classification: 39B42, 46C05, 36D05 305 HUMBATALIYEV Further, we define a Hilbert space for natural m ≥ 1 [1]. W2m ((a; b) ; H) = u u(m) ∈ L2 ((a; b) ; H) , Am u ∈ L2 ((a, b) ; Hm) with norm u W m ((a,b);H) = 2 (m) 2 u 1/2 2 L2 ((a,b);H) + Am u L2 ((a,b);H) . Here and in sequel the derivatives are understood in the sense of distributions theory [1]. Here we assume W2m ((−∞, +∞) ; H) ≡ W2m (R; H) . We denote by D (R; H)a set of infinitely-differentiable functions with values in H. In the space H we consider the operator – differential equation P d dt u (t) ≡ − d2 + A2 dt2 m u (t) + 2m Aj u(2m−j) (t) = f (t) , j=0 (1) t ∈ R = (−∞, +∞) , where f (t) and u (t) are vector-valued functions from H, and coefficients A and Aj j = 0, 2m satisfy the following conditions: 1) A is a positive-definite self-adjoint operator in H ; 2) the operators Aj j = 0, 2m are .