A unified constructive approach to the topological degree in \(R^ n\). (English) Zbl 0682.55002
The paper gives an approach to topological degree theory in Euclidean spaces. The main subject studied in this paper is strictly connected with the computation of the topological degree. The basic idea consists of computing the degree of a continuous function with respect to a bounded open subset of the n-dimensional Euclidean space by means of an auxiliary function defined on an n-polyhedron approximating the given mapping and the open set.
Reviewer: L.Gorniewicz
References:
| [1] | , Topologie I, berichtigter Reprint. Springer, Berlin 1974 |
| [2] | , Triangulations by reflections with applications to approximation. In: Numerische Methoden der Approximationstheorie Band 4 (, ; eds.), pp. 10–32. Birkhäuser, Basel 1978 |
| [3] | Allgower, Utilitas Math. 16 pp 123– (1979) |
| [4] | Allgower, SIAM Rev. 22 pp 28– (1980) |
| [5] | , (eds.), Numerical Solution of Nonlinear Equations. Lecture Notes in Math. 878. Springer, Berlin 1981 |
| [6] | Allgower, Computing 8 pp 157– (1971) |
| [7] | Amann, Math. Z. 130 pp 39– (1973) |
| [8] | , Perspectives in Nonlinearity. W. A. Benjamin, New York 1968 |
| [9] | Browder, Proc. Sympos. Pure Math. 39 pp 49– (1983) · doi:10.1090/pspum/039.2/9827 |
| [10] | Cronin, Math. Surveys 11 (1964) |
| [11] | Nichtlineare Gleichungen und Abbildungsgrade. Springer, Berlin 1974 · doi:10.1007/978-3-642-65941-6 |
| [12] | Delves, Math. Comp. 21 pp 543– (1967) |
| [13] | Grundzüge der modernen Analysis. Deutscher Verlag der Wissenschaften, Berlin 1971 |
| [14] | Lectures on Algebraic Topology. Springer, Berlin 1972 · Zbl 0234.55001 · doi:10.1007/978-3-662-00756-3 |
| [15] | Eaves, Math. Programming 29 pp 77– (1984) |
| [16] | Eiger, ACM Trans. Math. Software 10 pp 367– (1984) |
| [17] | Eisenbud, Bull. Amer. Math. Soc. 84 pp 751– (1978) |
| [18] | Epstein, Proc. London Math. Soc. 16 pp 369– (1966) |
| [19] | Erdelsky, Math. Comp. 27 pp 133– (1973) |
| [20] | (eds.), Fixed Point Theory. Lecture Notes in Math. 886. Springer, Berlin 1981 |
| [21] | Fan, J. Combin. Theory 2 pp 588– (1967) |
| [22] | (ed.), Numerical Solution of Highly Nonlinear Problems. North-Holland, Amsterdam 1980 |
| [23] | A simplicial approach to the fixed point index. In: [20], pp. 73–102. |
| [24] | Führer, Math. Nachr. 54 pp 259– (1972) |
| [25] | Garcia, Math. Oper. Res. 4 pp 390– (1979) |
| [26] | , Differential Topology. Prentice Hall, Englewood Cliffs, N. J. 1974 |
| [27] | Harvey, Quart. Appl. Math. 34 pp 351– (1976) |
| [28] | Einführung in die kombinatorische Topologie. Wiss. Buchges., Darmstadt 1978 |
| [29] | Heinz, J. Math. Mech. 8 pp 231– (1959) |
| [30] | Hermeline, RAIRO Anal. Numér. 16 pp 211– (1982) |
| [31] | Differential Topology. Springer, Berlin 1976 · Zbl 0356.57001 · doi:10.1007/978-1-4684-9449-5 |
| [32] | Hoenders, Computing 30 pp 137– (1983) |
| [33] | Kearfott, Numer. Math. 32 pp 109– (1979) |
| [34] | A derivative-free arc continuation method and a bifurcation technique. In: [5], pp. 182–198. |
| [35] | Zur numerischen Bestimmung des Abbildungsgrades im Rn und seiner Anwendung bei der Lösung nichtlinearer Gleichungssysteme (Dissertation). Karl-Marx-Univ., Leipzig 1983 |
| [36] | Kliesch, Z. Anal. Anwendungen 3 pp 337– (1984) |
| [37] | Kojima, Math. Programming 15 pp 223– (1978) |
| [38] | Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon, Oxford 1964 |
| [39] | Degree Theory. Cambridge Univ. Press, Cambridge 1978 · Zbl 0367.47001 |
| [40] | Topology from the Differentiable Viewpoint. University Press of Virginia, Charlottesville, VA 1965 |
| [41] | Nagumo, Amer. J. Math. 73 pp 485– (1951) |
| [42] | Ni, J. Math. Anal. Appl. 89 pp 386– (1982) |
| [43] | O’Neil, SIAM J. Numer. Anal. 12 pp 673– (1975) |
| [44] | Peitgen, Jber. d. Dt. Math.-Verein. 84 pp 107– (1982) |
| [45] | , The Leray-Schauder continuation method is a constructive element in the numerical study of nonlinear eigenvalue and bifurcation problems. In: [47], pp. 326–409. |
| [46] | , An -pertubation of Brouwer’s definition of degree. In: [20], pp. 331–366. |
| [47] | (eds.), Functional Differential Equations and Approximation of Fixed Points. Lecture Notes in Math. 730. Springer, Berlin 1980 |
| [48] | Simpliziale Topologie und globale Verzweigung (Dissertation). Univ. Bonn, Bonn 1978 |
| [49] | , On computational aspects of topological degree in Rn. In: [47], 410–433. |
| [50] | Prüfer, J. Math. Anal. Appl. 84 pp 133– (1981) |
| [51] | Lehrbuch der Topologie. Deutscher Verlag der Wissenschaften, Berlin 1975 |
| [52] | Brouwer degree: History and numerical computation. In: [22], pp. 389–416 |
| [53] | Siegberg, Amer. Math. Monthly 88 pp 125– (1981) |
| [54] | Sikorski, Math. Comput. 33 pp 722– (1979) |
| [55] | Ein kombinatorischer Umschließungssatz nebst Anwendungen. In: Game Theory and Related Topics ( eds.), pp. 207–218. North-Holland, Amsterdam 1979 |
| [56] | Fifty years of further development of a combinatorial lemma. In: [22], pp. 183–217 |
| [57] | Stenger, Numer. Math. 25 pp 23– (1975) |
| [58] | Stynes, Numer. Math. 33 pp 147– (1979) |
| [59] | Stynes, Appl. Anal. 9 pp 63– (1979) |
| [60] | An n-dimensional bisection method for solving systems of n equations in n unknowns. In: [22], pp. 93–111 |
| [61] | Stynes, Numer. Math. 37 pp 453– (1981) |
| [62] | The Computation of Fixed Points and Applications. Lecture Notes in Economics and Mathematical Systems 124. Springer, Berlin 1976 |
| [63] | Hamiltonian triangulation of Rn. In: [47], pp. 470–483 |
| [64] | Todd, SIAM J. Algebraic Discrete Methods 5 pp 244– (1984) |
| [65] | , Labelling rules and orientation. In: [5], pp. 239–257 |
| [66] | von Hohenbalken, Math. Programming 15 pp 1– (1978) |
| [67] | von Hohenbalken, Math. Programming 21 pp 233– (1981) |
| [68] | Existenz, Eindeutigkeit, Eigenschaften und Anwendungen des Abbildungsgrades im Rn. In: Theory of Nonlinear Operators. Akademie-Verlag, Berlin 1974 |
| [69] | Zeidler, Math. Nachr. 72 pp 51– (1976) |
| [70] | Nonlinear Functional Analysis and its Applications I-Fixed Point Theorems. Springer, New York 1985 · doi:10.1007/978-1-4612-5020-3 |
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