Abstract
We establish, via auxiliary categories, a version of Newton’s method applicable to nonnormable spaces. We illustrate it with an application to perturbation of functional equations on unbounded domains. Under suitable conditions, the known solution of the unperturbed equation initiates a sequence of Newton iterates that converges to the solution of the perturbed equation. The conditions posed are those required to make Newton’s method work. We also generalize the Frobenius–Dieudonné existence-uniqueness theorem for initial value problems for a vector variable.
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References
Binz, Ernst: Continuous Convergence on C(X), Lecture Notes in Math. 469, Springer, 1975.
Dieudonné, J.: Foundations of Modern Analysis, Academic Press, 1969.
Groetch, C. W.: Elements of Applicable Functional Analysis, Marcel Dekker, 1980.
Kantorovich, L. V. and Akilov, G. P.: Functional Analysis, Pergamon, 1982.
Maurin, K.: Analysis, Part 1: Elements, D. Reidel, 1976.
Nel, L. D.: Categorical differential calculus for infinite dimensional spaces, Cahiers Topologic Géom. Différentielle Catégoriques 29 (1988), 257–286.
Nel, L. D.: Infinite dimensional calculus allowing nonconvex domains with empty interior, Monatsh. Math. 110 (1990), 145–166.
Nel, L. D.: Nonlinear existence theorems in nonnormable analysis, in Category Theory at Work, Heldermann, 1991, pp. 343–365.
Nel, L. D.: Introduction to Categorical Methods (Part One and Two), Carleton-Ottawa Mathematical Lecture Note Series 11, Carleton Univ., 1991.
Nel, L. D.: Introduction to Categorical Methods (Part Three), Carleton-Ottawa Mathematical Lecture Note Series 12, Carleton Univ., 1991.
Nel, L. D.: Effective categories of complete separated spaces, in Recent Developments of General Topology and Its Applications, International Conference in Memory of Felix Hausdorff, Berlin, 1992, Akademie Verlag, Berlin, 1992, pp. 242–251.
Nel, L. D.: Differential calculus founded on an isomorphism, Applied Categorical Structures 1 (1993), 51–57.
Tapia, R.: The Kantorovich theorem for Newton's method, Amer. Math. Monthly 78 (1971), 389–392.
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Min, K.C., Nel, L.D. Newton’s Method and Frobenius–Dieudonné Theorem in Nonnormable Spaces. Applied Categorical Structures 5, 205–216 (1997). https://doi.org/10.1023/A:1008690318367
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DOI: https://doi.org/10.1023/A:1008690318367