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On the boundary value problem $ u''+u=\alpha u\sp{-}+p(t),$ $ u(0)=0=u(\pi )$

Authors: L. Aguinaldo and K. Schmitt
Journal: Proc. Amer. Math. Soc. 68 (1978), 64-68
MSC: Primary 34B15
MathSciNet review: 0466707
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Abstract: In this note we consider the boundary value problem $ u'' + u = a{u^ - } + p(t),\;u(0) = 0 = u(\pi ),\;\alpha > 0$, and show that a necessary and sufficient condition for the problem to be solvable is that $ \smallint_0^\pi {p(s)\sin s\;ds\; \leqslant 0} $. We thus answer in the affirmative a question posed by S. Fučik.

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  • [1] Svatopluk Fučík, Boundary value problems with jumping nonlinearities, Časopis Pěst. Mat. 101 (1976), no. 1, 69–87 (English, with Loose Russian summary). MR 0447688
  • [2] J. Mawhin, Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces, J. Differential Equations 12 (1972), 610–636. MR 0328703

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Keywords: Nonlinear boundary value problems, coincidence degree theorems, continuation theorems
Article copyright: © Copyright 1978 American Mathematical Society