Keywords
- Boundary Value Problem
- Functional Differential Equation
- Continuation Method
- Point Index
- Global Bifurcation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References
ALEXANDER, J.C., YORKE, J.A.: The homotopy continuation method, Numerically implementable topolgogical procedures, Trans. AMS, 242 (1978), 271–284
ALEXANDER, J.C.: The topolgocial theory of an embedding method, in: "Continuation Methods", H.J. Wacker, ed., New York: Academic Press, 1978.
ALEXANDER, J.C.: Numerical continuation methods and bifurcation, in "Functional differential equations and approximation of fixed points", H.O. Peitgen and H.O. Walther, eds., Berlin, Heidelberg, New York: Springer Lecture Notes, 1979.
ALEXANDER, J.C., YORKE, J.A.: A numerical continuation method that works generically, University of Maryland, Dept. of Math., MD 77-9-JA, TR 77-9.
ALLGOWER, E.L., GEORG, K.: Simplicial and Continuation Methods for Approximating Fixed Points and Solutions to Systems of Equations, SIAM Review (to appear).
ALLGOWER, E.L., KELLER, C.L., REEVES, T.E.: A program for the numerical approximation of a fixed point of an arbitrary continuous mapping of the n-cube or n-simplex into itself, Aerospace Research Laborations, Report ARL 71-0257, 1971.
AMANN, H.: Lectures on some fixed point theorems, Monografias de Matemática, Instituto de matemática pura e aplicada, Rio de Janeiro.
AMANN, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Review, 18.4 (1976).
AMBROSETTI, A., HESS, P.: Positive solutions of asymptotically linear elliptic eigenvalue problems, to appear.
AMBROSETTI, S., RABINOWITZ, P.H.: Dual variational methods in critical point theory and applications, J. Functional Analysis, 14.4 (1973), 349–381.
ANGELSTORF, N.: Global branching and multiplicity results for functional differential equations, in "Functional differential equations and approximation of fixed points", H.O. Peitgen and H.O. Walther, eds., Berlin, Heidelberg, New York: Springer Lecture Notes, 1979.
BERNSTEIN, S.: Sur la généralisation du problème de Dirichlet, Math. Ann. 69 (1910), 82–136.
BORSUK, K.: Theory of Retracts, Warszawa: PWN, Polish Scientific Publishers, 1967.
CHOW, S.N., MALLET-PARET, J., YORKE, J.A.: Finding zeros of maps: Homotopy methods that are constructive with probability one, Math. Comp. 32 (1978), 887–899.
CRONIN, J.: Fixed Points and Topological Degree in Nonlinear Analysis, Providence: Amer. Math. Soc., 1964.
DEIMLING, K.: Nichtlineare Gleichungen und Abbildungsgrade, Berlin, Heidelberg, New York: Springer, 1974.
DOLD, A.: Fixed point index and fixed point theorems for Euclidean neighborhood retracts, Topology 4 (1965), 1–8.
DOLD, A.: Lectures on Algebraic Topology, Berlin, Heidelberg, New York: Springer Verlag, 1972.
DUGUNDJI, J., GRANAS, A.: KKM maps and Variational Inequalities, Annali della Scuola Norm. Sup. di Pisa, 5.4 (1978), 679–682.
EAVES, B.C.: Homotopies for computation of fixed points, Mathematical Programming 3 (1972), 1–22.
EAVES, B.C.: Properly labeled simplices, Studies in Optimization, 10, MAA, Studies in Mathematics, G.B. Dantzig and B.C. Eaves, (eds.), (1974) 71–93.
EISENACK, G., FENSKE, C.C.: Fixpunkttheorie, Mannheim, Wien, Zürich: Bibliographisches Institut, 1978.
FENSKE, C.C., PEITGEN, H.O.: Repulsive fixed points of multivalued transformations and the fixed point index, Math. Ann. 218 (1975), 9–18.
FILIPPOV, A.F.: Differential equations with many valued discontinuous right hand side, Dok. Akad. Nauk SSSR 151 (1963), 65–68.
GAINES, R.E., MAWHIN, J.L.: Coincidence Degree and Nonlinear Differential Equations, Berlin, Heidelberg, New York: Springer Verlag, 1977.
HADELER, K.P.: Delay equations in biology, in "Functional differential equations and approximation of fixed points", H.O. Peitgen and H.O. Walther, eds., Berlin, Heidelberg, New York: Springer Lecture Notes, 1979.
JEPPSON, M.M.: A search for the fixed points of a continuous mapping, Mathematical Topics in Economics Theory and Computation, R.H. Day and S.M. Robinson, eds., Philadelphia: SIAM, (1972) 122–129.
JÜRGENS, H., SAUPE, D.: Methoden der simplizialen Topologie zur numerischen Behandlung von nichtlinearen Eigenwert-und Verzweigungsproblemen, Diplomarbeit, Bremen, 1979.
KEARFOTT, B.: An efficient degree-computation method for a generalized method of bisection, Numer. Math., to appear.
KLEIN, F.: Neue Beiträge zur Riemannschen Funktionentheorie, Math. Annalen 21, 1882/3
KNASTER, B., KURATOWSKI, C., MAZURKIEWICZ, S.: Ein Beweis des Fixpunktsatzes für n-dimensionale Simplexe, Fund. Math. 14, (1929), 132–137.
KRASNOSEL'SKII, M.A.: Topological Methods in the Theory of Nonlinear Integral Equations, Oxford: Pergamon, 1963.
KRASNOSEL'SKII, M.A.: Positive solutions of operator equations, Groningen: Noordhoff, 1964.
KRONECKER, L.: Über Systeme von Funktionen mehrerer Variablen I (1869), Ges. Werke Bd. I, Teubner, Leibzig (1895), 177–234.
LASOTA A., OPIAL, Z.: Fixed point theorems for multi-valued mappings and optimal control problems, Bull. Acad. Pol. Sci. 16 (1968), 645–649.
LASRY, J.M., ROBERT, R.: Analyse non lineaire multivoque, Cahiers de Matématiques de la Decision, Université Paris IX Dauphine, No 7611.
LERAY, J., SCHAUDER, J.P.: Topologie et équations fonctionelles, Ann. Ecole Norm. Sup. (3) 51 (1934), 45–78.
MA, T.W.: Topological degree for set-valued compact vector fields in locally convex spaces, Dissertations Math. 92 (1972), 1–43.
MARSDEN, J.E.: Qualitative Methods in Bifurcation Theory, Bull. Amer. Math. Soc. 84.6 (1978), 1125–1148.
MILNOR, J.: Topology from the differentiable viewpoint, 2nd printing, Charlottesville: The University of Virginia Press, 1969.
NIRENBERG, L.: Topics in Nonlinear Functional Analysis, New York University Lecture Notes, 1973–74.
NUSSBAUM, R.D.: A global bifurcation theorem with applications to functional differential equations, J. Functional Analysis 19, (1975), 319–338.
NUSSBAUM, R.D.: Periodic Solutions of Nonlinear Autonomous Functional Differential Equations, in "Functional Differential Equations and Approximation of Fixed Points", H.O. Peitgen and H.O. Walther, eds., Berlin, Heidelberg, New York: Springer Lecture Notes, 1979.
ORTEGA, J.M., RHEINBOLDT, W.C.: Iterative Solution of Nonlinear Equations in Several Variables, New York: Academic Press, 1970.
POINCARÉ, H.: Sur les Groupes des Equationes Lineaires. Acta Mathem. 4, (1884).
PRÜFER, M.: Sperner simplices and the topological fixed point index, Universität Bonn, SFB 72, preprint no. 134, 1977.
PRÜFER, M.: Simpliziale Topologie und globale Verzweigung, Dissertation, Bonn 1978.
PRÜFER, M., SIEGBERG, H.W.: On computational aspects of degree in Rn, in "Functional Differential Equations and Approximations of Fixed Points", H.O. Peitgen and H.O. Walther, eds., Heidelberg, Berlin, New York: Springer Lecture Notes, 1979.
PEITGEN, H.O.: Methoden der topologischen Fixpunkttheorie in der nichtlinearen Funktionalanalysis, Habilitationsschrift, Universität Bonn, 1976.
RABINOWITZ, P.H.: Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487–513.
RABINOWITZ, P.H.: Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math. 3 (1973), 162–202.
RABINOWITZ, P.H.: On bifurcation from infinity, J. Differential Equations 14 (1973), 462–475.
RABINOWITZ, P.H.: A survey on bifurcation theory, in "Dynamical Systems: An International Symposium", vol. I, New York: Academic Press, 1976.
SIEGBERG, H.W.: Abbildungsgrade in Analysis und Topologie, Diplomarbeit, Bonn 1977.
SPERNER, E.: Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes, Abh. Math. Sem. Hamburg 6 (1928), 265–272.
STENGER, F.: Computing the topological degree of a mapping in Rn, Numer. Math. 25 (1975), 23–38.
STYNES, M.J.: An algorithm for the numerical calculation of the degree of a mapping, Ph. D. Thesis, Oregon, State University, Corvallis, 1977.
TEMME, M.M. (ed.): Nonlinear Analysis, Vol. I, II, Amsterdam: Mathematisch Zentrum, 1976.
TODD, M.J.: The Computation of Fixed Points and Applications, Berlin, Heidelberg, New York: Springer Lecture Notes in Economics and Mathematical Systems, 1976.
TODD, M.J.: Hamiltonian triangulations of Rn, in "Functional Differential Equations and Approximations of Fixed Points", H.O. Peitgen and H.O. Walther, eds., Berlin, Heidelberg, New York: Springer Lecture Notes, 1979.
WACKER, H.J.: A Summary of the Developments in Imbedding Methods, in "Continuation Methods", H.J. Wacker, ed., New York: Academic Press, 1978.
ZEIDLER, E.: Existenz, Eindeutigkeit, Eigenschaften und Anwendungen des Abbildungsgrades in Rn, in "Theory of Nonlinear Operators; Proceedings of a Summer School", Berlin: Akademie-Verlag, 1974, 259–311.
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Peitgen, HO., Prüfer, M. (1979). The Leray-Schauder continuation method is a constructive element in the numerical study of nonlinear eigenvalue and bifurcation problems. In: Peitgen, HO., Walther, HO. (eds) Functional Differential Equations and Approximation of Fixed Points. Lecture Notes in Mathematics, vol 730. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0064326
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DOI: https://doi.org/10.1007/BFb0064326
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