Nonlocal problems and functional-differential equations: theoretical aspects and applications to mathematical modelling

Abstract

This paper presents a review of results on nonlocal problems, functional-differential equations, and their applications, obtained during several last years. The following research areas are covered: the Kato square root problem for functional-differential operators, Vlasov equations and their applications to the modelling of high-temperature plasma, specific properties of differential-difference equations with incommensurable translations, degenerate functional-differential equations and their applications, functional-differential equations with contractions and extensions of independent variables, and operational methods for parabolic and elliptic functional-differential equations.

Inverse problems for Dirac operators with constant delay: uniqueness, characterization, uniform stability

“…Classical methods for solving inverse problems for differential operators [15][16][17][18][19][20][21][22][23][24][25] are not applicable for operators with delay as well as for other classes of nonlocal operators. However, they are often more adequate for modelling various physical processes frequently possessing a nonlocal nature [26][27][28][29][30][31][32].…”

“…are fulfilled. To these w1 (x) and w2 (x), via formulae (31) and (32) there correspond another pair of functions p(x) and q(x) such that (p, q) = (p, q). Then, analogously to the above, relations (28), (35) and (36) will mean that {λ n k,1 ,1 } k∈Z and {λ n k,2 ,2 } k∈Z are subspectra also of the problems B 1,1 ( Q) and B 1,2 ( Q), respectively, with Q = Q.…”

“…Thus, for each j = 1, 2, there exists a unique function w j (x) ∈ L 2 (a − π, π − a) obeying (34). Let q(x) and p(x) be constructed by formulae (31) and (32) with these w 1 (x) and w 2 (x), which, in turn, obviously appear in the representations ( 20) and ( 21) for the characteristic functions of the corresponding problems B 1,1 (Q) and B 1,2 (Q). Then relations ( 28), ( 34) and (35) mean that the given sequences {µ k,1 } k∈Z and {µ k,2 } k∈Z are their subspectra, respectively, which finishes the proof.…”

“…where {e * n,1 (x)} n∈Z and {e * n,2 (x)} n∈Z are the Riesz bases that are biorthogonal to the bases {e n,1 (x)} n∈Z and {e n,2 (x)} n∈Z , respectively; (iii) Construct the functions q(x) and p(x) by formulae (31) and (32).…”

“…representations (20) and ( 21) hold. Construct the functions p(x) and q(x) by formulae (31) and (32) and consider the corresponding problems B 1,1 (Q) and B 1,2 (Q). Then, as in Section 2, one can show that ∆ 1 (λ) and ∆ 2 (λ) are their characteristic functions, respectively.…”