Second-order differentiability with respect to parameters for differential equations with adaptive delays
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In this paper, we study the second-order differentiability of solutions with respect to parameters in a class of delay differential equations, where the evolution of the delay is governed explicitly by a differential equation involving the state variable and the parameters. We introduce the notion of locally complete triple-normed linear space and obtain an extension of the well-known uniform contraction principle in such spaces. We then apply this extended principle and obtain the second-order differentiability of solutions with respect to parameters in the W 1,p -norm (1 ⩽ p < ∞).
KeywordsDelay differential equation adaptive delay differentiability of solution state-dependent delay uniform contraction principle locally complete triple-normed linear space
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- 6.Arino O, Sánchez E. Delays included in population dynamics. Mathematical Modeling of Population Dynamics, Banach Center Publications, 2004, 63: 9–146Google Scholar
- 7.Arino O, Sánchez E, Fathallah A. State-dependent delay differential equations in population dynamics: modeling and analysis. Fields Inst Commun, 2001, 29: 19–36Google Scholar
- 9.Bélair J. Population models with state-dependent delays. Lecture Notes in Pure and Appl Math, 1991, 131: 165–176Google Scholar
- 16.Hartung F. On Differentiability of SolutionsWith State-dependent Delay Equations. Ph D dissertation, University of Texas at Dallas, 1995Google Scholar
- 18.Hartung F, Krisztin T, Walther H -O, Wu J. Functional Differential Equations with State-dependent Delays: Theory and Applications. In: Canada A, Drabek P, Fonda A, eds. Handbook of Differential Equations: Ordinary Differential Equations, Vol 3. Amsterdam: Elsevier, North Holland, 2006Google Scholar
- 20.Hartung F, Turi J. Linearized stability in functional differential equations with statedependent delays. Discrete Contin Dynam Systems, 2001, Added Volume: 416–425Google Scholar
- 31.Smith H L. Some results on the existence of periodic solutions of state-dependent delay differential equations. Pitman Res Notes Math Ser, 1992, 272: 218–222Google Scholar