×

A note on the example of J. Andres concerning the application of the Nielsen fixed-point theory to differential systems. (English) Zbl 1040.34022

Summary: The goal of this note is two-fold: to give a slight improvement of the example due to J. Andres in [Proc. AMS 128, 2921–2931 (2000; Zbl 0964.34030)] concerning the application of the Nielsen number to differential equations, and a reprovement of the method, allowing us to avoid one condition (the connectedness of ANR-spaces).

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
47H10 Fixed-point theorems

Citations:

Zbl 0964.34030
PDFBibTeX XMLCite
Full Text: EuDML

References:

[1] Andres J.: A nontrivial example of application of the Nielsen fixed-point theory to differential systems: problem of Jean Leray. Proceed. Amer. Math. Soc. 128, 10 (2000), 2921-2931. · Zbl 0964.34030 · doi:10.1090/S0002-9939-00-05324-7
[2] Andres J.: Multiple bounded solutions of differential inclusions: the Nielsen theory approach. J. Diff. Eqs. 155 (1999), 285-320. · Zbl 0940.34008 · doi:10.1006/jdeq.1998.3582
[3] Andres J., Górniewicz L.: From the Schauder fixed-point theorem to the applied multivalued Nielsen Theory. Topol. Meth. Nonlin. Anal. 14, 2 (1999), 228-238. · Zbl 0958.34015
[4] Andres J., Górniewicz L., Jezierski J.: A generalized Nielsen number and multiplicity results for differential inclusion. Topol. Appl. 100 (2000), 143-209. · Zbl 0940.55007 · doi:10.1016/S0166-8641(98)00092-3
[5] Borsuk K.: Theory of Retracts. PWN, Warsaw, 1967. · Zbl 0153.52905
[6] Brown R. F.: On the Nielsen fixed point theorem for compact maps. Duke. Math. J., 1968, 699-708. · Zbl 0186.57002 · doi:10.1215/S0012-7094-69-03684-9
[7] Brown R. F.: Topological identification of multiple solutions to parametrized nonlinear equations. Pacific J. Math. 131 (1988), 51-69. · Zbl 0615.47042 · doi:10.2140/pjm.1988.131.51
[8] Brown R. F.: Nielsen fixed point theory and parametrized differential equations. Contemp. Math. 72, AMS, Providence, RI, 1989, 33-46.
[9] Cecchi M., Furi M., Marini M.: About the solvability of ordinary differential equations with assymptotic boundary conditions. Boll. U. M. I., Ser. IV, 4-C, 1 (1985), 329-345. · Zbl 0587.34013
[10] Fečkan M.: Multiple solution of nonlinear equations via Nielsen fixed-point theory: a survey. Nonlinear Anal. in Geometry and Topology (Th. M. Rassias, Hadronic Press, Inc., Fl., (2000), 77-97. · Zbl 0978.34016
[11] Granas A.: The Leray-Schauder index and the fixed point theory for arbitrary ANRs. Bull. Soc. Math. France 100 (1972), 209-228. · Zbl 0236.55004
[12] Krasnosel’skij M. A.: The Operator of Translation along Trajectories of Differential Equations. Nauka, Moscow, 1966
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.