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Singular quadratic functionals of one dependent variable. (English) Zbl 0838.34036
The authors consider the singular quadratic functionals over \(W^{1,2}_{\text{loc}}([a, +\infty[)\), \[ I(u)= \alpha u^2(a)+ \liminf_{b\to +\infty} \Biggl[\gamma u^2(b)+ \int^b_\alpha (r(s)u^{\prime 2}(s)- p(s) u^2(s))ds\Biggr] \] subjected to boundary conditions \[ \lim_{b\to +\infty} D\Biggl({u(a)\atop u(b)}\Biggr)= 0, \] where \(D\) is a \(2\times 2\) matrix; \(r^{- 1}\), \(p\in L_{\text{loc}}([a,+ \infty[)\), \(r\geq 0\), \(\alpha, \gamma\in {\mathcal R}\). In particular they discuss necessary and sufficient conditions for the nonnegativity of \(I\) either in the case of free end points or under boundary conditions of different types. For recent connected results see the first author and P. Zezza [Comment. Math. Univ. Carolinae 33, 411-425 (1992; Zbl 0779.49026)].

MSC:
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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