Bounds on nonlinear operators in finite-dimensional Banach spaces. (English) Zbl 0585.47047
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.
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.
MSC:
| 47J10 | Nonlinear spectral theory, nonlinear eigenvalue problems |
| 47H05 | Monotone operators and generalizations |
| 65L07 | Numerical investigation of stability of solutions to ordinary differential equations |
| 65H10 | Numerical computation of solutions to systems of equations |
Keywords:
Lipschitz-continuous nonlinear maps in finite-dimensional Banach; and Hilbert spaces; Boundedness; monotonicity; invertibility; nonlinear operators; numerical range; Gerschgorin domain; von Neumann spectral; Lipschitz-continuous nonlinear maps in finite-dimensional Banach and Hilbert spacesReferences:
| [1] | Bauer, F.L., Stoer, J., Witzgall, C.: Absolute and monotonic norms. Numer. Math.3, 257-264 (1961) · Zbl 0111.01602 · doi:10.1007/BF01386026 |
| [2] | Bauer, F.L.: On the field of values subordinate to a norm. Numer. Math.4, 103-113 (1962) · Zbl 0117.11004 · doi:10.1007/BF01386300 |
| [3] | Browder, F.E.: The solvability of nonlinear functional equations. Duke Math. J.30, 557-566 (1963) · Zbl 0119.32503 · doi:10.1215/S0012-7094-63-03061-8 |
| [4] | Dahlquist, G.: Stability and error bounds in the numerical integration of ordinary differential equations. Trans. Royal Inst. of Tech., No 130, Stockholm 1959 · Zbl 0085.33401 |
| [5] | Dahlquist, G.:G-stability is equivalent toA-stability. BIT18, 384-401 (1978) · Zbl 0413.65057 · doi:10.1007/BF01932018 |
| [6] | 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 · Zbl 0499.65044 |
| [7] | Deimling, K.: Nonlinear functional analysis. Berlin, Heidelberg, New York: Springer 1985 · Zbl 0559.47040 |
| [8] | Halmos, P.: A Hilbert space problem book. Princeton: Van Nostrand 1967 · Zbl 0144.38704 |
| [9] | Minty, G.J.: Monotone (non-linear) operators in Hilbert space. Duke Math. J.29, 341-346 (1962) · Zbl 0111.31202 · doi:10.1215/S0012-7094-62-02933-2 |
| [10] | von Neumann, J.: Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes. Math. Nachr.4, 258-281 (1951) · Zbl 0042.12301 |
| [11] | Nevanlinna, O.: On the logarithmic norms of a matrix. Report-HTKK-MAT-A94, Helsinki Univ. of Tech., 1976 · Zbl 0356.65116 |
| [12] | 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 · Zbl 0573.65036 |
| [13] | 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 |
| [14] | Nirschl, N., Schneider, H.: The Bauer field of values of a matrix. Numer. Math.6, 355-365 (1964) · Zbl 0126.32102 · doi:10.1007/BF01386085 |
| [15] | Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables. New York: Academic Press 1970 · Zbl 0241.65046 |
| [16] | Riesz, F., Sz-Nagy, B.: Functional Analysis. New York: Ungar 1955 · Zbl 0070.10902 |
| [17] | Schmitt, B.: Norm bounds for rational matrix functions. Numer. Math.42, 379-389 (1983) · Zbl 0541.65022 · doi:10.1007/BF01389581 |
| [18] | Stoer, J.: On the characterization of least upper bound norms in matrix space. Numer. Math.6, 302-314 (1964) · Zbl 0126.32101 · doi:10.1007/BF01386078 |
| [19] | Ström, T.: On logarithmic norms. SIAM J. Num. Anal.2, 741-753 (1975) · Zbl 0321.15012 · doi:10.1137/0712055 |
| [20] | Söderlind, G.: On nonlinear difference and differential equations. BIT24, 667-680 (1984) · Zbl 0557.65043 · doi:10.1007/BF01934923 |
| [21] | Söderlind, G.: An error bound for fixed point iterations. ibid., 391-393 · Zbl 0554.65039 · doi:10.1007/BF02136040 |
| [22] | Zarantonello, E.H.: The closure of the numerical range contains the spectrum. Pac. J. Math.22, 575-595 (1967) · Zbl 0152.34602 |
| [23] | Zenger, C.: On convexity properties of the Bauer field of values of a matrix. Numer. Math.12, 96-105 (1968) · Zbl 0182.49103 · doi:10.1007/BF02173403 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
