NONLOCAL BOUNDARY-VALUE PROBLEMS WITH A SHIFT A. L. Skubachevskii We consider two types of nonlocal problems: i) elliptic differential equations in a domain Q with nonlocal boundary conditions connecting the values of the unknown function on certain pieces of the boundary with its values on shifts of these pieces in the domain Q, and 2) elliptic differential-difference equations. The interconnection of these two problems allows us to apply theorems concerning the solvability and the spectrum of the second problem (Sec. 4) to a study of the solvability and the spectrum of the first problem (Sec. 5). Analogous results are presented in Sec. 6 for ordinary differential equations with nonlocal conditions on an interval. We remark that problems of the solvability and asymptotics of the characteristic values of nonlocal problems with a shift were studied in [1-4] and elsewhere for ordinary differen- tial equations. Elliptic equations with nonlocal conditions on shiftsof the boundary in a domain were considered for the first time in [5]. i. We consider the equation (i) Y, i<mD~R=~qD~u (X)---- / (x) . (x ~ Q) I, with the boundary conditions (2) O~-*u (~) = 0 (x ~_ @Q; t~ --~ 1, m). Ovlz- Z 9 ., Here QCR n is a bounded domain; moreover, either OQ~C ~, or Q=(0, d) • G(~C | if n > ~); a = (a,..... a,t), fi --- (~i..... ~n); R~gq ----- PQR~glQ; IQ: L 2 (Q)-+ L~ (B~) is the opera- tion of extending a function to be zero outside of Q; P0: L~ (R~)-+ L~ (Q) is the operation of restricting a function on Q; _n~u (x) = Y,,~,~a,~t~,~ (x) u (x + h), (3) a=gh'~_ C = (Rn), f c~_L~(Q) are complex-valued functions; ~EcR" is a finite set of vectors with integer coordinates; v is the unit exterior normal vector to OQ We denote by wk(Q) the Sobolev space of complex-valued functions with the scalar product (u, v)w~-(O ) ---= ~ ' ~ , ~; D~u (x)D=v (x) dx, where D= = D~' , . . D~ ~, Dj = --iO/Ox~. We denote by ~,l; (Q) the closure of the set C+ (Q) in W ~ (Q) , and we denote by W I~,N(Q) the direct product W h'(Q) x . . . x W ~(Q). DefSnition i. We call the Eq. (i) and the differential-difference operator corresponding to it str0ngly elliptic in the domain Q if, for all ~ ~ C= (Q) , (4) lie (~l~l, , ,- D~Ra6~.Dgu' u),~(,+> /.> c, ]l tt I15~,,(+> -- c: t[ ", Ilk<o). If~K Here, and in the sequel, c, ci, and ki will denote positive constants. 2. We consider at first the properties of difference operators (see [6]). LEMMA i. The operators H~f~: L.z(Q)--~L~(Q) are bounded; H~ =- PQII~IQ , where S. Ordzhonikidze Moscow Aviation Institute. Translated from Matematicheskie Zametki, Vol. 38, No. 4, pp. 587-598, October, 1985. Original article submitted February 19, 1985. 0001-4346/85/38Z:5-0833509.50 9 1986 Plenum Publishing Corporation 833