Strongly dissipative operators and nonlinear equations in a Fréchet space

Martin, R. H.

doi:10.1090/S0002-9939-1972-0295163-8

Strongly dissipative operators and nonlinear equations in a Fréchet space
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Proc. Amer. Math. Soc. 32 (1972), 161-168 Request permission

Abstract:

Suppose that X is a Fréchet space, Y is a Banach subspace of X, and A is a function from Y into X. Sufficient conditions are determined to insure that the equation $Ax = y\;(y \in Y)$ has a unique solution ${x_y}$ which depends continuously on y. The techniques of this paper use the theory of dissipative operators in a Banach space, and the results are associated with the idea of admissibility of the space y. Also, the equation $Ax = Cx + y$ is considered where C is completely continuous.

References

G. L. Cain Jr. and M. Z. Nashed, Fixed points and stability for a sum of two operators in locally convex spaces, Pacific J. Math. 39 (1971), 581–592. MR 322606
C. Corduneanu, Problèmes globaux dans la théorie des équations intégrales de Volterra, Ann. Mat. Pura Appl. (4) 67 (1965), 349–363 (French). MR 182849, DOI 10.1007/BF02410815
C. Corduneanu, Some perturbation problems in the theory of integral equations, Math. Systems Theory 1 (1967), 143–155. MR 213919, DOI 10.1007/BF01705524
J. Dieudonné, Foundations of modern analysis, Pure and Applied Mathematics, Vol. X, Academic Press, New York-London, 1960. MR 0120319
Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
José Luis Massera and Juan Jorge Schäffer, Linear differential equations and function spaces, Pure and Applied Mathematics, Vol. 21, Academic Press, New York-London, 1966. MR 0212324
R. H. Martin Jr., Lyapunov functions and autonomous differential equations in a Banach space, Math. Systems Theory 7 (1973), 66–72. MR 322301, DOI 10.1007/BF01824808
R. K. Miller, Admissibility and nonlinear Volterra integral equations, Proc. Amer. Math. Soc. 25 (1970), 65–71. MR 257674, DOI 10.1090/S0002-9939-1970-0257674-9
R. K. Miller, J. A. Nohel, and J. S. W. Wong, A stability theorem for nonlinear mixed integral equations, J. Math. Anal. Appl. 25 (1969), 446–449. MR 234238, DOI 10.1016/0022-247X(69)90247-9
François Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967. MR 0225131