Summary.
The singular Neumann boundary value problem \( (g({x}\ifmmode{'}\else$'$\fi){)}\ifmmode{'}\else$'$\fi = f(t,x,{x}\ifmmode{'}\else$'$\fi),{x}\ifmmode{'}\else$'$\fi(0) = {x}\ifmmode{'}\else$'$\fi(T) = 0 \) is considered. Here f(t, x, y) satisfies local Carathéodory conditions on \( [0,T] \times \mathbb{R} \times (0,\infty ) \) and f may be singular at the value 0 of the phase variable y. Conditions guaranteeing the existence of a solution to the above problem with a positive derivative on (0, T) are given. The proofs are based on regularization and sequential techniques and use the topological transversality method.
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Manuscript received: February 24, 2004 and, in final form, July 23, 2004.
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Agarwal, R.P., O’Regan, D. & Staněk, S. Neumann boundary value problems with singularities in a phase variable. Aequ. math. 69, 293–308 (2005). https://doi.org/10.1007/s00010-004-2767-1
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DOI: https://doi.org/10.1007/s00010-004-2767-1