Summary
We consider Lipschitz-continuous nonlinear maps in finite-dimensional Banach and Hilbert spaces. Boundedness and monotonicity of the operator are characterized quantitatively in terms of certain functionals. These functionals are used to assess qualitative properties such as invertibility, and also enable a generalization of some well-known matrix results directly to nonlinear operators. Closely related to the numerical range of a matrix, the Gerschgorin domain is introduced for nonlinear operators. This point set in the complex plane is always convex and contains the spectrum of the operator's Jacobian matrices. Finally, we focus on nonlinear operators in Hilbert space and hint at some generalizations of the von Neumann spectral theory.
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References
Bauer, F.L., Stoer, J., Witzgall, C.: Absolute and monotonic norms. Numer. Math.3, 257–264 (1961)
Bauer, F.L.: On the field of values subordinate to a norm. Numer. Math.4, 103–113 (1962)
Browder, F.E.: The solvability of nonlinear functional equations. Duke Math. J.30, 557–566 (1963)
Dahlquist, G.: Stability and error bounds in the numerical integration of ordinary differential equations. Trans. Royal Inst. of Tech., No 130, Stockholm 1959
Dahlquist, G.:G-stability is equivalent toA-stability. BIT18, 384–401 (1978)
Dahlquist, G., Söderlind, G.: Some problems related to stiff nonlinear differential systems. In: Computing Methods in Applied Sciences and Engineering V. (R. Glowinski, J. Lions, eds.). Amsterdam: North-Holland 1982
Deimling, K.: Nonlinear functional analysis. Berlin, Heidelberg, New York: Springer 1985
Halmos, P.: A Hilbert space problem book. Princeton: Van Nostrand 1967
Minty, G.J.: Monotone (non-linear) operators in Hilbert space. Duke Math. J.29, 341–346 (1962)
von Neumann, J.: Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes. Math. Nachr.4, 258–281 (1951)
Nevanlinna, O.: On the logarithmic norms of a matrix. Report-HTKK-MAT-A94, Helsinki Univ. of Tech., 1976
Nevanlinna, O.: Matrix valued versions of a result of von Neumann with an application to time discretization. Report-HTKK-MAT-A224, Helsinki Univ. of Tech., 1984
Nevanlinna, O.: Remarks on time discretization of contraction semigroups. In: Proc. VIth Int'l Conf. on Trends in the Theory and Practice of Nonlinear Analysis (V. Lakshmikantham, ed.). Amsterdam: North Holland (to appear). Also available as: REPORT-HTKK-MAT-A225, Helsinki Univ. of Tech., 1984
Nirschl, N., Schneider, H.: The Bauer field of values of a matrix. Numer. Math.6, 355–365 (1964)
Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables. New York: Academic Press 1970
Riesz, F., Sz-Nagy, B.: Functional Analysis. New York: Ungar 1955
Schmitt, B.: Norm bounds for rational matrix functions. Numer. Math.42, 379–389 (1983)
Stoer, J.: On the characterization of least upper bound norms in matrix space. Numer. Math.6, 302–314 (1964)
Ström, T.: On logarithmic norms. SIAM J. Num. Anal.2, 741–753 (1975)
Söderlind, G.: On nonlinear difference and differential equations. BIT24, 667–680 (1984)
Söderlind, G.: An error bound for fixed point iterations. ibid., 391–393
Zarantonello, E.H.: The closure of the numerical range contains the spectrum. Pac. J. Math.22, 575–595 (1967)
Zenger, C.: On convexity properties of the Bauer field of values of a matrix. Numer. Math.12, 96–105 (1968)
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Söderlind, G. Bounds on nonlinear operators in finite-dimensional banach spaces. Numer. Math. 50, 27–44 (1986). https://doi.org/10.1007/BF01389666
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DOI: https://doi.org/10.1007/BF01389666