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Asymptotic Solutions of Equations in Banach Space

  • C. A. Swanson (a1) and M. Schulzer (a1)

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The equation Px = y in Banach spaces has aroused considerable interest, particularly in view of the various situations in applied analysis which it encompasses, and consequently it has been the topic of numerous investigations (2; 9; 10; 12). Detailed references may be found in (10). The equation is of special interest because of its interpretation as an integral equation; and in turn, many problems related to differential equations can be reformulated as integral equations (5; 7; 13).

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References

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1. Banach, S., Théorie des opérations linéaires, Monografje Matematyczne, I (Warsaw, 1932).
2. Bartle, R. G., Newton's method in Banach spaces, Proc. Amer. Math. Soc., 6 (1955), 827831.
3. Borel, E., Sur quelques points de la théorie des fonctions, Thèse, Annales de l'Ecole Normale (1895).
4. Carleman, T. G. T., Les fonctions quasi-analytiques (Paris, 1926) chapter 5.
5. Coddington, E. A. and Levinson, N., Theory of ordinary differential equations (McGraw- Hill, New York, 1955).
6. de Bruijn, N. G., Asymptotic methods in analysis (P. Noordhoff, Groningen, Netherlands, 1958).
7. Duff, G. F. D., Partial differential equations (University of Toronto Press, Toronto, 1956).
8. Erdélyi, A., Asymptotic expansions (Dover, New York, 1956).
9. Kaazik, Yu. Ya. and Tamme, E. E., On a method of approximate solution of functional equations, Dokl. Akad. Nauk SSSR (N.S.), 101 (1955), 981984 (Russian).
10. Kantorovich, L. V., Functional analysis and applied mathematics, Translated by C. D. Benster, National Bureau of Standards (U.C.L.A., 1952).
11. Kolmogoroff, A. N. and Fomin, S. V., Elements of the theory of functions and functional analysis I (Graylock, Rochester, N.Y., 1957).
12. McFarland, J. E., An iterative solution of the quadratic equation in Banach space, Proc. Amer. Math. Soc, 9 (1958), 824830.
13. Riesz, F. and Nagy, B. Sz., Functional analysis (Frederick Ungar, New York, 1955).
14. van der Corput, J. G., Asymptotic expansions I, Technical report I, Contract AF-18(600)- 958 (University of California, Berkeley, 1954).
15. Zaanen, A. C., Linear analysis (Interscience, New York, 1953).
16. Zagadskii, D. M., An analogue of Newton's method for non-linear integral equations, Dokl. Akad. Nauk SSSR (N.S.), 59 (1948), 10411044 (Russian).
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Asymptotic Solutions of Equations in Banach Space

  • C. A. Swanson (a1) and M. Schulzer (a1)

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