Singular points of functional equations
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In this paper, the authors consider the problem of branching of the solutions in the neighborhood of a given solution for a very general type of nonlinear integral equations and obtain results exactly parallel to some for ordinary algebraic functions over the real or complex field.Abstract:
Hildebrandt and Graves [9](2) showed that if the partial differential dxb(0, 0; h), considered as a linear transformation of X, has a continuous everywhere-defined inverse, then there exists a unique continuous singlevalued function q defined on a neighborhood of the origin in 2) with values in X such that 4(0) =0 and b [k(y), y] =0 for all y in this neighborhood. Graves [8; 2, p. 408] showed that if dx4(0, 0; h) maps onto X, then there will be at least one solution corresponding to sufficiently small y. Cronin [3] recently considered a case in which dA need not map onto X and obtained, under suitable restrictions, theorems concerning the existence of solutions in terms of the topological degree theory. While our methods are closely related to hers, we focus our attention on the problem of studying the branching of the solutions that this situation allows. In a particular case we are able to apply Dieudonne's modification [5] of the Newton polygon method to obtain results exactly parallel to some for ordinary algebraic functions over the real or complex field. It is also seen that the work of E. Schmidt [17], L. Lichtenstein [14], and R. Iglisch [12] for a class of nonlinear integral equations hold valid for a general class of functions defined on Banach spaces. Also, in both their work and that of T. Shimizu [18], the assumption of analyticity can be replaced by that of the existence of a few continuous derivatives. Further, because of the simpler form for the equations we derive, it is possible to study particular cases in terms of initially given data. Our final part indicates briefly how these results can be applied to nonlinear differential equations with fixed end point boundary conditions. It is possible to treat questions of existence and uniqueness of solutions in the neighborhood of a given solution for a very general type of equation.read more
Citations
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Journal ArticleDOI
Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces
TL;DR: In this article, Leray-Schauder degree theory for mappings of the form I T with I the identity and T : cl i2 -+ X completely continuous is presented.
Journal ArticleDOI
The methods of lyapunov and schmidt in the theory of non-linear equations and their further development
M M Vainberg,V A Trenogin +1 more
TL;DR: The theory of branching of solutions in the analytic case was introduced by Lyapunov and Schmidt as mentioned in this paper, and its application to non-linear equations is discussed in Section 2.1.
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Die Lösung der Verzweigungsgleichungen für nichtlineare Eigenwertprobleme
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Alternative problems for nonlinear functional equations
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The numerical treatment of non-trivial bifurcation points
TL;DR: In this paper, the stability of bifurcation problems based on the classical Liapunov-Schmidt theory is analyzed and a well-posed formulation of the problem is derived and numerical methods, based on Newton's method, are suggested for both the computation of the bifurlcation point itself and for moving onto the nearby solution-curves.
References
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Book
Functional Analysis And Semi-Groups
Einar Hille,R. S. Phillips +1 more
TL;DR: The theory of semi-groups has been studied extensively in the literature, see as discussed by the authors for a survey of some of the main applications of semi groups in the context of functional analysis.
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Mappings between function spaces
TL;DR: In this paper, a function K on Z to Q which is bounded on Z and continuous in the strong topology of Q induces a linear continuous operator K on I to 2 by the formula(2)
Journal ArticleDOI
Spectral theory. I. Convergence to projections
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