In this paper we give definition of m-fold completeness and prove a theorem on completeness of elementary generalized solution of corresponding boundary value problems at which the equation describes the process of corrosive fracture of metals in aggressive media and the principal part of the equation has multiple characteristics. | Turk J Math 33 (2009) , 383 – 396. ¨ ITAK ˙ c TUB doi: On completeness of elementary generalized solutions of a class of operator-differential equations of a higher order Rovshan Z. Gumbataliev Abstract In this paper we give definition of m -fold completeness and prove a theorem on completeness of elementary generalized solution of corresponding boundary value problems at which the equation describes the process of corrosive fracture of metals in aggressive media and the principal part of the equation has multiple characteristics. Key Words: Hilbert space, existence of generalized solution, operator-differential equation. 1. Introduction Many problems of mechanics and mathematical physics are connected in part to eigen and adjoint vectors of operator pencils. As an example, we can show the following papers. Study of trace problems for solving some elliptic equations in a semi-cylinder precedes the completeness problems. Necessary and sufficient conditions are formulated for boundary values providing the belongness of the solution to energetic space. As is known, stress-strain state of a plate may be separated into internal and external layers [1,4]. Construction of a boundary layer is related with sequential solution of plane problems of elasticity theory in a semi-strip. In Papkovich’s paper [5] and in others a boundary value problem of elasticity theory in a semi-strip x > 0, |y| ≤ 1 is reduced to the definition of Airy biharmonic functions in the form u= Im Ck ϕk (y)eiσk x , σk >0 where ϕk are Papkovich functions [5,6], σk are corresponding values of a self-adjoint boundary value problem, and Ck are unknown coefficients. In this connection, in [6] there is a problem on representation of a pair of functions f1 and f2 in the form ∞ k=1 Ck Pk ϕk = f1 , ∞ Ck Qk ϕk = f2 , (1) k=1 2000 AMS Mathematics Subject Classification: 46C05, 36D05, 39B42. 383 GUMBATALIEV where Pk , Qk are differential operators defined by boundary conditions .
