An Issue About the Existence of Solutions for a Linear Non-autonomous MTFDE
This article is concerned with the existence of solution of a certain non-autonomous linear delayed-advanced differential equation. The main objective is to provide the proof of a theorem introduced in Lima et al. (J. Comput. Appl. Math. 234(9):2732–2744, 2010) about existence of solution of a class of mixed-type functional differential equations (MTFDEs). It is an effort to complete the theoretical basis of some computational methods introduced earlier to solve numerically such equations, which were deduced making use of that theorem.
KeywordsMixed-type functional differential equation Method of steps Boundary value problem Existence of solution
Mathematics Subject Classification (2000):PACS: 02.30 Ks 02.60 Lj 2000 MSC: 34K06 34K28 65Q05 34K10
This work was supported by Portuguese funds through the Center for Computational and Stochastic Mathematics (CEMAT), The Portuguese Foundation for Science and Technology (FCT), University of Lisbon, Portugal, project UID/Multi/04621/2013, and Center of Naval Research (CINAV), Naval Academy, Portuguese Navy, Portugal.
The author also acknowledges Professors P. Lima, N. Ford, and P. Lumb for maintaining their availability and the provided suggestions.
- 2.Collard, F., Licandro, O., Puch, L.A.: The short-run dynamics of optimal growth models with delays. In: Computing in Economics and Finance, vol. 117. Society for Computational Economics, European University Institute, Badia Fiesolana, I-50016 San Domenico (FI), Italy (2004). http://www.iue.it/personal/Licandro/
- 13.Lamb C., Van Vleck, E.S.: Neutral mixed type functional differential equations. J. Dyn. Diff. Equat. 10884, 1–42 (2014)Google Scholar
- 16.Lima, P.M., Teodoro, M.F., Ford, N.J., Lumb, P.M.: Analysis and computational approximation of a forward-backward equation arising in nerve conduction. In: Proc. ICDDEA2011, Azores, Springer Proceedings in Mathematics & Statistics NY, vol. 47, p. 475 (2013)Google Scholar
- 18.Mallet-Paret, J., Verduyn-Lunel, S.M.: Mixed-type functional differential equations, hololmorphic factorization, and applications. In: Proc. Equadiff2003, International Conference on Differential Equations, Hasselt 2003, World Scientific, Singapore, pp. 73–89 (2005). ISBN 9812561692Google Scholar
- 19.Marian, O.I.: Functional-differential equations of mixed type, via weakly Picard Operators. Stud. Univ. Babes-Bolyai, Math. LI (2), 83–95 (1999)Google Scholar
- 21.Pontryagin, L.S., Gamkreledze, R.V., Mischenko, E.F.: The Mathematical Theory of Optimal Process. Interscience, New York (1962)Google Scholar
- 22.Teodoro, M.F., Lima, P.M., Ford, N.J., Lumb, P.M.: Numerical modelling of a functional differential equation with deviating arguments using a collocation method. In: Proc ICNAAM 2008, International Conference On Numerical Analysis And Applied Mathematics, Kos 2008, AIP Proceedings, vol. 1048, pp. 553–557 (2008)zbMATHGoogle Scholar
- 24.Teodoro, M.F.: Solving a Forward-backward Equation from Acoustics.Google Scholar
- 25.Teodoro, M.F.: Solving a Forward-backward Equation from Physiology.Google Scholar