Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-19T00:43:17.329Z Has data issue: false hasContentIssue false
  • English
  • Français

Weakly continuous accretive operators in general Banach spaces

Published online by Cambridge University Press:  17 April 2009

W.E. Fitzgibbon
W.E. Fitzgibbon
Affiliation:
Department of MathematicsUniversity of HoustonHouston, Texas 77004United States of America
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Global wellposedness theorems are established for a class of abstract Cauchy initial value problems and a class of abstract Volterra equations which have a linear semigroup as a convolution kernel. These existence theorems are used to show that a class of nonlinear operators and a class of perturbed linear operators are m-accretive. The m-accretiveness results are used in turn to represent solutions to the differential and integral equations.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 41 , Issue 2 , April 1990 , pp. 185 - 199
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Ball, J.M., ‘Strongly continuous semigroups, weak solutions and the variation of constants formula’, Proc. Amer. Math. Soc. 63 (1977), 370–373.Google Scholar
[2]Banas, J. and Goebel, K., Measures of Noncompactness in Banach Spaces (Marcel Decker, New York, 1980).Google Scholar
[3]Chow, S.N. and Shuur, J., ‘An existence theorem for ordinary differential equations in Banach Spaces’, Bull. Amer. Math. Soc. 77 (1971), 1018–1020.CrossRefGoogle Scholar
[4]Cramer, E., Lakshmikantham, V. and Mitchell, A., ‘On the existence of weak solutions of differential equations in nonreflexive spaces’, Nonlinear Anal. 2 (1976), 169–177.CrossRefGoogle Scholar
[5]Crandall, M. and Liggett, T., ‘Generation of semigroups of nonlinear transformations on general Banach Spaces’, Amer. J. Math. 93 (1971), p. 265.CrossRefGoogle Scholar
[6]DeBlasi, F., ‘On the property of the unit sphere in a Banach Space’, R.S. Roumaine, Bul. Math. Soc. Sci. 21 (1977), 259–262.Google Scholar
[7]Faulkner, G.D., ‘On the nonexistence of weak solutions to abstract differential equations in nonreflexive spaces’, Nonlinear Anal. 2 (1978), 505–508.CrossRefGoogle Scholar
[8]Fitzgibbon, W.E., ‘Nonlinear perturbation of m-accretive operator’, Proc. Amer. Math. Soc. 44 (1974), 359–364.Google Scholar
[9]Fitzgibbon, W.E., ‘Weakly continuous nonlinear accretive operators in reflexive Banach spaces’, Proc. Amer. Math. Soc. 41 (1973), 229–235.CrossRefGoogle Scholar
[10]Hille, E. and Phillips, R.S., Functional Analysis and Semigroups (Amer. Math. Soc., Providence R.I., 1957).Google Scholar
[11]Kato, T., ‘Accretive operators and nonlinear evolution equations’, Proc. Symp. Pure Math. Vol 18, Part I (1970).CrossRefGoogle Scholar
[12]Kato, T., ‘Nonlinear semigroups and evolution equation’, J. Math. Soc. Japan 19 (1967), 508–520.CrossRefGoogle Scholar
[13]Kato, T., Perturbation theory for Linear Operators (Springer-Verlag, New York, Berlin, Heidelberg, 1966).Google Scholar
[14]Knight, W., ‘Solutions of differential equations in Banach Spaces’, Duke Math. J. 41 (1974), 437–442.CrossRefGoogle Scholar
[15]Martin, R.H., ‘A global existence theorem for autonomous differential equations in a Banach Space’, Proc. Amer. Math. Soc. 26 (1970), 307–314.CrossRefGoogle Scholar
[16]Oharu, S., ‘Note on the representation of semigroups of nonlinear operators’, Proc. Japan Acad. 42 (1966), 1149–1154.Google Scholar
[17]Papageorgiou, N., ‘Kneser's Theorem for differential equations in Banach Spaces’, Bull. Austral. Math. Soc. 33 (1986), 419–434.CrossRefGoogle Scholar
[18]Papageorgiou, N., ‘Weak solutions of differential equations in Banach Space’, Bull. Austral. Math. Soc. 33 (1986), 407–418.CrossRefGoogle Scholar
[19]Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (Springer-Verlag, New York, Berlin Heidelberg, 1983).CrossRefGoogle Scholar
[20]Szep, A., ‘Existence theorem for weak solutions of ordinary differential equations in reflexive Banach Spaces’, Studia Sci. Math. Hungar. 6 (1971), 197–203.Google Scholar
[21]Szufla, S., ‘On the existence of solutions of differential equations in Banach Spaces’, Bull. Acad. Polan. Sci. Ser. Sci. Math. 30 (1982), 507–512.Google Scholar
[22]Webb, G.F., ‘Continuous nonlinear perturbations of linear accretive operators in Banach Space’, J. Junct. Anal. 10 (1972), 191–203.CrossRefGoogle Scholar