# Implicit linear difference equation in Frechet spaces

 1Gefter, SL, 1Piven, AL1V.N. Karazin Kharkiv National University Dopov. Nac. akad. nauk Ukr. 2017, 6:3-8 https://doi.org/10.15407/dopovidi2017.06.003 Section: Mathematics Language: Russian Abstract:  An criterion of the existence and the uniqueness for a solution of the implicit linear difference equation $Ax_{n +1} + Bx_{n} = g_{n}$, where A and B are continuous operators, which act on Frechet spaces, is proved. Explicit formulas for the solution of this equation are found. For the case of Banach spaces, the results are specified. Keywords: Frechet space, implicit difference equation
References:
1. Baskakov, A.G. (2001). On the invertibility of linear difference operators with constant coefficients. Russ. Math., No. 5, pp. 1-9.
2. Benabdallakh, M., Rutkas, A.G. & Solov'ev, A.A. (1990). Application of asymptotic expansions to the investigation of an infinite system of equations Axn+1 + Bxn = fn in a Banach space. J. Soviet Math., 48, Iss. 2, pp. 124-130. https://doi.org/10.1007/BF01095789
3. Vlasenko, L.A. (2006). Evolutionary models with implicit and degenerate differential equations. Dnepropetrovsk: Sistemnyie Technologii (in Russian).
4. Gorodnii, M.F. (1991). Bounded and periodic solutions of a difference equation and its stochastic analogue in Banach space. Ukr. Math. J., 43, Iss. 1, pp. 32-37. https://doi.org/10.1007/BF01066900
5. Horodnii, M.F. & Vyatchaninov, O.V. (2009). On the boundedness of one recurrent sequence in a banach space. Ukr. Math. J., 61, Iss. 9, pp. 1529-1532. https://doi.org/10.1007/s11253-010-0294-x
6. Slusarchuk, V.E. (2003). Stability of Solutions of Difference Equations in a Banach Space. Rivne: Vyd-vo UDUVHP (in Ukrainian).
7. Bernhard, P. (1982). On singular implicit linear dynamical systems. SIAM J. Control Optim., 20, No. 5, pp. 612-633. https://doi.org/10.1137/0320046
8. Bondarenko, M. & Rutkas, A. (1998). On a class of implicit difference equations. Reports of the National Academy of Sciences of Ukraine, No. 7, pp. 11-15.
9. Campbell, S.L. (1980). Singular systems of differential equations I, Research Notes in Mathematics, Vol. 40. San Francisco, London, Melbourne: Pitman Advanced Publ. Program.
10. Campbell, S.L. (1982). Singular systems of differential equations II, Research Notes in Mathematics, Vol. 61. San Francisco, London, Melbourne: Pitman Advanced Publ. Program.
11. Gohberg, I., Goldberg, S. & Kaashoek, M.A. (1990). Classes of linear operators, Vol. 1, Operator Theory: Advances and Applications, Vol. 49. Basel: Birkhäuser. https://doi.org/10.1007/978-3-0348-7509-7