Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-08T08:05:36.692Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-linear functional differential equations and abstract integral equations

Published online by Cambridge University Press:  14 November 2011

F. Kappel  and
W. Schappacher
F. Kappel
Affiliation:
Institut fuür Mathematik, Universität Graz, Austria
W. Schappacher
Affiliation:
Institut fuür Mathematik, Universität Graz, Austria
Get access Rights & Permissions [Opens in a new window]

Synopsis

The equivalence between solutions of functional differential equations and an abstract integral equation is investigated. Using this result we derive a general approximation result in the state space C and consider as an example approximation by first order spline functions. During the last twenty years C1-semigroups of linear transformations have played an important role in the theory of linear autonomous functional differential equations (cf. for instance the discussion in [9, Section 7.7]). Applications of non-linear semigroup theory to functional differential equations are rather recent beginning with a paper by Webb [17]. Since then a considerable number of papers deal with problems in this direction. A common feature of the majority of these papers is that as a first step with the functional differential equation there is associated a non-linear operator A in a suitable Banach-space. Then appropriate conditions are imposed on the problem such that the conditions of the Crandall-Liggett-Theorem [5] hold for the operator A. This gives a non-linear semigroup. Finally the connection of this semigroup tothe solutions of the original differential equation has to be investigated [c.f. 8, 15, 18]. To solve thislast problem in general is the most difficult part of this approach.

In the present paper we consider the given functional differential equation as a perturbation of the simple equationx = 0. The solutions of this equation generate a very simple C1-semigroup. The solutions of the original functional differential equation generate solutions of an integral equation which is the variation of constants formula for the abstract Cauchy problem associated with the equation x = 0. Under very mild conditions we can prove a one-to-one correspondence between solutions of the given functional differential equation and solutions of the integral equation in the Lp-space setting. In the C-space setting the integral equation inthe state space has to be replaced by a ‘pointwise’ integral equation. Using the pointwise integral equation together with a theorem which guarantees continuous dependence of fixed points on parameters we show under rather weak hypotheses that the original functional differential equation can be approximated by a sequence of ordinary differential equations. Using 1st order spline functions we finally get results which are very similar to those obtained in [1 and 11] in the L2-space setting.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 84 , Issue 1-2 , 1979 , pp. 71 - 91
Copyright
Copyright © Royal Society of Edinburgh 1979

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Banks, H. T. and Burns, J. A.. Hereditary control problems: numerical methods based on averaging approximations. SIAM J. Control Optimization 16 (1978), 169–208.CrossRefGoogle Scholar
2Brezis, H.. Operators maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Math. Studies 5 (Amsterdam: North-Holland, 1973).Google Scholar
3Brezis, H. and Pazy, A.. Convergence and approximation of semigroups of nonlinear operators in Banach spaces. J. Functional Analysis 9 (1972), 63–74.CrossRefGoogle Scholar
4Colonius, F. and Hinrichsen, D.. Differential state space description of nonlinear time invariant hereditary differential systems. In Control of Distributed Parameter Systems (ed. Banks, S. and Pritchard, A.), Proc. 2nd IFAC Symp., Coventry, Great Britain, 28 June–1 July 1977, pp. 247–260 (Oxford: Pergamon, 1978).CrossRefGoogle Scholar
5Crandall, M. G. and Liggett, T. M., Generation of semigroups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (1971), 265–298.CrossRefGoogle Scholar
6Dinculeanu, N.. Linear operators on Lp spaces. In Vector and Operator Valued Measures and Applications (ed. Tucker, D. H. and Maynard, H. B.), pp. 109–124 (New York: Academic Press, 1973).CrossRefGoogle Scholar
7Dunford, N. and Schwarz, J. T.. Linear Operators, Pt I (New York: Interscience, 1966).Google Scholar
8Dyson, J. and Villella-Bressan, R.. Nonlinear functional differential equations in L 1 spaces. J. Nonlinear Analysis 1 (1977), 383–396.CrossRefGoogle Scholar
9Hale, J. K., Theory of Functional Differential Equations (Berlin: Springer, 1977).CrossRefGoogle Scholar
10Imaz, C. and Vorel, Z.. Ordinary differential equations in a Banach space and retarded functional differential equations. Bol. Soc. Mat. Mexicana, Ser. II, 16 (1971), 32–37.Google Scholar
11Kappel, F. and Schappacher, W.. Autonomous nonlinear functional differential equations and averaging approximations. J. Nonlinear Analysis 2 (1978), 391–422.CrossRefGoogle Scholar
12Naito, T.. On autonomous linear functional differential equations with infinite retardations. J. Differential Equations 21 (1976), 297–315.CrossRefGoogle Scholar
13Naylor, A. W. and Sell, G. R.. Linear Operator Theory in Engineering and Science (New York: Holt, Rinehart and Winston, 1971).Google Scholar
14Pazy, A.. Semi-groups of linear operators and applications to partial differential equations. Univ. Maryland, Lecture Notes 10 (1974).Google Scholar
15Plant, A. T.. Nonlinear semigroups of translations in Banach space generated by functional differential equations. J. Math. Anal. Appl. 60 (1977), 67–74.CrossRefGoogle Scholar
16Webb, G. F.. Continuous nonlinear perturbations of linear accretive operators in Banach spaces. J. Functional Analysis 10 (1972), 191–203.CrossRefGoogle Scholar
17Webb, G. F.. Autonomous nonlinear functional differential equations and nonlinear semigroups, J. Math. Anal. Appl. 46 (1974), 1–12.CrossRefGoogle Scholar
18Webb, G. F.. Functional differential equations and nonlinear semigroups in L p -spaces. J. Differential Equations 20 (1976), 71–89.CrossRefGoogle Scholar
19Yosida, K.. Functional Analysis, 5th edn (Berlin: Springer, 1978).CrossRefGoogle Scholar