Summary
We investigate the possible application of an abstract mathematical method, the topological index theory and Krasnosel’ski theorem (a brief account of this theory is given at the end of the paper) for solving nonlinear physical problems, particularly when bifurcation phenomena—with one or more parameters—are expected. Various mathematical results are presented, especially in finite-dimensional cases, in which simple conditions for the existence of nontrivial branching solutions are given, together with a discussion concerning «perturbative» expansions. The case of the presence of some symmetry property is also briefly consieered.
Riassunto
Si studia la possibilitá di applicare un metodo matematico astratto, cioé la teoria dell’indice topologico e il teorema di Krasnosel’ski (di cui si trova un breve sommario nell’ultima parte del lavoro) per risolvere problemi fisici non lineari, particolarmente quando ci si attendono fenomeni di biforcazione, anche a piú parametri. Si presentano vari risultati matematici, specie in dimensione finita, dove si possono fornire semplici condizioni per l’esistenza di rami di soluzioni non banali, insieme con una discussione su sviluppi di tipo perturbativo. Si considera anche brevemente il caso in cui é presente qualche proprietá di simmetria.
Резюме
Мы исследуем возможное применения абстрактного математического метода, теории топологических индексов и теоремы Красносельского (краткий обзор этой теории приведен в Приложении), для решения нелинейных физических проблем, в частности, когда ожидаются явления бифуркации с одним или более параметрами. Проводятся различные математические результаты, особенно в случаях конечного числа измерений, где приводятся простые условия для суцествования нетривальных решений. Также проводятся обсуждение “пертурбационных“ разложений. Вкратце рассматривается случай наличия некототого свойства симметрии.
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Even if, in practice, one usually deals with «continuous» branches of solution, the general definition (8)S.-N. Chow andJ. K. Hale:Methods of Bifurcation Theory (New York, 1982); covers also the case that some sequence of solutionsx n ≠0 inX and λn inR p exists, with λn→λ0 andx n →0 whenn→∞.
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In this statement, the requirement that ν is odd is directly substituted by the assumption that a nontrivial solution exists. As already noted, the existence of a nonzero solution of eq. (5) ensures, by means of (4′), that also the initial equation, (1) is nontrivially solved. See alsoM. G. Crandall andP. H. Rabinowitz: in ref. (3),Bifurcation Phenomena in Mathematical Physics and Related Topics, edited byC. Bardos andD. Bessis (NATO Adv. Stud. Inst. Ser., Cargèse, 1979) (Dordrecht, 1980). p. 3.
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In fact, in definingi τ similarly as in eq. (A. 2), only a finite number of eigenvalues λj of the compact operatorK x (0) appears, namely those eigenvalues for which λj -1 is contained in the open interval (0, τ).
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Supported by Dipartimento di Fisica, Pisa and I.N.F.N., sezione di Pisa.
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Cicogna, G., Degiovanni, M. Mathematical hints in nonlinear problems. Nuovo Cim B 82, 54–70 (1984). https://doi.org/10.1007/BF02723577
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DOI: https://doi.org/10.1007/BF02723577