Remarks on the bifurcation of solutions of a nonlinear eigenvalue problem |
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Authors: | N Bazley B Zwahlen |
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Institution: | (1) Battelle Institute for Advanced Studies, Geneva |
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Abstract: | Conclusions We have investigated solutions of equation (3) when
2 is an eigenvalue of the linearized operator (13) and when it is not. In Section 4 we have shown that for 0 and
2 =
i
2
we have exactly two nontrivial solutions which bifurcate to the right of
i
2
; these solutions are shown to exist in an interval (
i
2
,
i
2
+
0). The method of Section 3 may then be used to extend these two solutions to the right of
i
2
+
0 providing that
2=
i
2
+
0 is not an eigenvalue of the linear operator (13) evaluated at = ±
1. Either a solution can be uniquely extended, or there exists a value of
2where the bifurcation method must be applied again3.While the method used here gives the exact number of solutions bifurcating from
i
2
, other problems remain open; for example, it is still not proven that the two bifurcating branches have i zeros, as is the case for Hammerstein operators with oscillation kernels 4]. The conjecture of Odeh and Tadjbakhsh that there are exactly 2(i+1) nontrivial solutions in the interval
i
2
<
i
+1/2
remains un-answered, although it would be proven if one could show that there is no secondary bifurcation as in the cases of Kolodner 7] and Coffman 8]. |
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Keywords: | |
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