Summary
Let X be a Banach space and {A(t)|t ε [0, T]} a family of closed linear, densely defined m-accretive operators in X. This paper is concerned with the additive perturbation of {A(t)|t ε [0, T]} by a continuous family of nonlinear accretive operators {B(t)|t ε [0, T]}. Namely solutions are provided for the integral equation u(t, τ, x) = W(t, τ)x −\(\mathop \smallint \limits_\tau ^t \)W(t, s)B(s) · · u(s, τ, x)ds, u(τ, τ, x) = x where W(t, s) is the linear evolution operator associated with the linear differential equation v'(t, s, x) + A(t)v(t, s, x) = 0, v(s, s, x) = x.
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References
F. E. Browder,Nonlinear equations of evolution, Ann. of Math.,80 (1964), pp. 485–523.
M. G. Crandall -A. Pazy,Nonlinear evolution equations in Banach spaces, Israel J. Math.,11 (1972), pp. 57–94.
W. A. Coppel,Stability and asymptotic behavior of differential equations, P. C. Health and Company, Boston (1965).
W. E. Fitzgibbon,Time dependent perturbations of time dependent linear accretive operators, Notices Amer. Math. Soc.,20 (1973), Abstract 701-47-17.
J. A. Goldstein,Abstract evolution equations, Trans. Amer. Math. Soc.,141 (1969), pp. 159–185.
J. A. Goldstein,Semigroups of operators and abstract Cauchy problems, Lecture Notes Series, Tulane University (1970).
T. Kato,Integration of the equation of evolution in a Banach space J. Math. Soc. Japan,5 (1953), pp. 208–304.
T. Kato,Linear evolution equation of hyperbolic type, J. of Fac. Sci. Univ. of Tokyo,17, pp. 241–258.
T. Kato,Linear evolution equations of hyperbolic type II, J. Math. Soc. Japan,25 (1973).
T. Kato,On linear differential equations in a Banach space, Comm. Pure App. Math.,9 (1956).
T. Kato -H. Tanabe,On equations of evolution in a Banach space, Osaka Math. J.,14 (1962), pp. 107–133.
D. L. Lovelady,A Hammerstein Volterra integral equation with a linear semigroup convolution kernel (to appear Indiana Journal).
D. L. Lovelady,On the generation of linear evolution equations (to appear Duke Math. J.).
R. H. Martin,A global existance theorem for autonomous differential equations in a Banach space, Proc. Amer. Math. Soc.,26 (1970), pp. 307–314.
R. H. Martin,The logarithmic derivative and equations of evolution in a Banach space, J. Math. Soc. Japan,22 (1970), pp. 411–429.
K. Mauro -N. Yamada,A remark on an integral equation in a Banach space, Proc. Japan. Acad.,49 (1973), pp. 13–17.
I. Segal,Nonlinear semigroups, Annals of Math.,78 (1963), pp. 339–364.
G. F. Webb,Continuous nonlinear perturbation of linear accretive operators in Banach spaces, Journ. Func. Anal.,10 (1972), pp. 191–203.
G. F. Webb,Nonlinear perturbation of linear accretive operators in Banach spaces, Israel J. Math.,12 (1972), pp. 237–248.
K. Yosida,Functional Analysis, Springer-Verlag, New York, 2nd Ed. (1968).
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Entrata in Redazione il 30 luglio 1975.
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Fitzgibbon, W.E. Nonlinear perturbation of linear evolution equations in a banach space. Annali di Matematica 110, 279–293 (1976). https://doi.org/10.1007/BF02418009
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DOI: https://doi.org/10.1007/BF02418009