On the solvability of a boundary-value problem for second-order partial differential operator equations

Abstract

In this paper the solvability conditions of one class of elliptic type second order operator-differential equations. In this class the principal part of the operator whose spectrum is in some sector, is a normal operator. The obtained conditions are expressed by the properties of the coefficients.

Sufficient conditions for regular solvability of an arbitrary order operator-differential equation with initial-boundary conditions

“…Nowadays many papers concerning the study of initial value problems of the operator-differential equations in Banach spaces have been published. In both semiaxis and finite interval, second-order operatordifferential equations with zero weight exponential are studied [3,5,6,23,24]. Gasymov [8][9][10] analyzed both the solvability of operator-differential equations and the multiple completeness of some eigen-and associated vectors of corresponding operator pencils.…”

“…Moreover, the solvability of operatordifferential equations in Hilbert spaces with exponential weight has been extensively studied. Second-, third-, and fourth-order operator-differential equations with multiple characteristics with exponential weight have been studied on the semiaxis and the whole axis [2,22]. Moreover, general higher-order operator-differential equations with multiple characteristic in a Sobolev-type space with exponential weight have not been studied yet.…”

“…(2) for j = 2, 2 2,2 (β) + 10 = 0, 2α 1,2 (β) -2α 2,2 (β)α 4,2 (β) + α 2 3,2 (β) -10 = -β, 2α 3,2 (β)α 2 4,2 (β) + 5 = 0;…”

“…1 Ain Shams University, Cairo, Egypt. 2 Helwan University, Cairo, Egypt. 3 Egyptian Russian University, Cairo, Egypt.…”

“…Let β ∈ [0, b -1 s ). Then the polynomial operator pencils are invertible on the imaginary axis and have the following representations:P j (λ; β; A) = F j (λ; β; A)F j (-λ; β; A), j = 1, n, where F j (λ; β; A) = n+1 s=1 λEω j,s (β)A ≡ n+1 m=0 α m,j (β)λ (n-m) A m ,Re ω j,s (β) < 0, s = 1, n + 1, and the numbers α m,j (β) > 0, m = 0, n + 1; α ν,j (β) > 0, ν = 0, n + 1, satisfy the following systems of equations:P j (λ; β; A) = (iλ) 2 E + A 2 n+1β(iλ) 2j A (2(n+1)-2j) j (β)α ν,j (β)λ (n-m+1) (-λ) (n-ν+1) A m+ν .Examples(i) if m = 1 and n = 2, then the following systems of equations are satisfied:(1) for j = 1, 2 (β) + α 2 1,2 (β) -3 = -β, 2α 1,2 (β)α2 2,2 (β) + 3 = 0.…”